Copyright 1921, By FREDERICK P. SPALDING PRESS OF BRAUNWORTH & CO. BOOK MANUFACTURERS BROOKLYN, N. Y. PREFACE THIS book is designed to present, in a brief and systematic man- ner, the fundamental principles involved in the design and construc- tion of masonry structures. The term Masonry has been construed to include concrete, and the field covered by the title is a very wide one. It has therefore been necessary to select for discussion those types which seem most adequately to illustrate the principles, and no attempt has been made to cover fully the details of all classes of masonry structures. The purpose has been to provide an introduction to the subject, which may later be followed by intensive study in more detailed works upon the various branches. This gives a general view of the subject as a whole, and is the natural method of approach. The Author has derived much assistance from a number of books which deal more fully with various portions of the subject. These are mentioned at the ends of articles or chapters to which they specially relate. They should be studied by students desiring a more complete presentation of the subject. Special acknowledgment is also due to Professors A. Lincoln Hyde and Guy D. Newton of the University of Missouri for reading and criticising portions of the manuscript and for assistance in pre- paring the illustrations. F. P. SPALDING. COLUMBIA, MISSOURI, September, 1920. 436264 iii CONTENTS CHAPTER I DEVELOPMENT OF MASONRY CONSTRUCTION PAGE ART 1. INTRODUCTION 1 1. Definition; 2. Uses of Masonry. ART. 2 EARLY HISTORY 3 3. Ancient Masonry; 4. Roman and Medieval. ART. 3. RECENT DEVELOPMENTS 5 5. The Cement Industry; 6. Reinforced Concrete. CHAPTER II CEMENTING MATERIALS ART. 4. LIME 9 7. Classification; 8. Common Lime; 9. Hydraulic Lime; 10. Hydrated Lime; 11. Specifications. ART. 5. HYDRAULIC CEMENT 16 12. Setting and Hardening; 13. Portland; 14. Natural; 15. Puzzolan; 16. Sand Cement; 17. Soundness; 18. Chemistry of Cement. ART. 6. SPECIFICATIONS AND TESTS FOR CEMENT 26 19. Specifications; 20. Purpose of Tests; 21. Compressive Strength; 22. Special Tests. ART. 7. SAND FOR MORTAR 31 23. Quality; 24. Tests; 25. Mechanical Analysis; 26. Voids; 27. Specific Gravity; 28. Density; 29. Strength Test; 30. Washing; 31. Specifications. ART. 8. CEMENT MORTAR 39 32. Proportioning; 33. Mixing; 34. Yield; 35. Mixtures of Lime with Cement; 36. Strength. ART. 9. GYPSUM PLASTERS 48 37. Classification; 38. Properties and Uses. v vi CONTENTS CHAPTER III STONE MASONRY PAGE ART. 10. BUILDING STONE 51 39. Qualities; 40. Classification; 41. Strength; 42. Durability. ART. 11. STONE CUTTING 62 43. Tools; 44. Methods of Finishing; 45. Machinery. ART. 12. WALLS OF STONE MASONRY 70 46. Classification; 47. Parts of Wall; 48. Stonework; 49. Trimmings; 50. Specifications. ART. 13. STRENGTH OF STONE MASONRY 78 51. Compressive; 52. Capstones and Templets; 53. Lintels and Corbels. ART. 14. MEASUREMENT AND COST 82 54. Measurements; 55. Cost. CHAPTER IV BRICK AND BLOCK MASONRY ART. 15. BUILDING BRICKS 85 56. Clay and Shale; 57. Sand Lime; 58. Cement; 59. Tests. ART. 16. BRICK MASONRY 93 60. Joints; 61. Bond; 62. Strength; 63. Efflorescence; 64. Measure- ment and Cost. ART. 17. TERRA COTTA CONSTRUCTION 102 65. Structural Tiling; 66. Block Construction; 67. Architectural Terra Cotta. ART. 18. GYPSUM AND CEMENT CONCRETE BLOCKS 107 68. Gypsum Wall Blocks; 69. Roof and Floor Blocks; 70. Hollow Concrete Blocks. CHAPTER V PLAIN CONCRETE ART. 19. AGGREGATES FOR CONCRETE 110 71. Materials; 72. Tests for Coarse Aggregates. ART. 20. PROPORTIONING CONCRETE 116 73. Arbitrary; 74. Voids; 75. Analysis Curves; 76. Trial; 77. Fine- ness Modulus and Surface Area; 78. Yield. CONTENTS vii PAGE ART. 21. MIXING CONCRETE .. c 125 79. Preparing Materials; 80. Hand Mixing; 81. Machine. ART. 22. PLACING CONCRETE 129 82. Transporting; 83. Depositing; 84. Freezing; 85. Contraction Joints; 86. Finishing Surfaces. ART. 23. WATERTIGHT CONCRETE 136 87. Permeability; 88. Integral Waterproofing; 89. Exterior Coatings. ART. 24. DURABILITY OF CONCRETE 140 90. Destructive Agencies; 91. Sea Water; 92. Alkalies; 93. Fire Resistance. ART. 25. STRENGTH OF PLAIN CONCRETE 144 94. Compression; 95. Tests; 96. Tension and Transverse. ART. 26. COST OF CONCRETE WORK 149 97. Materials; 98. Labor; 99. Total Costs. CHAPTER VI REINFORCED CONCRETE ART. 27. GENERAL PRINCIPLES 153 100. Object of Reinforcement; 101. Bond Strength; 102. Reinforcing Steel; 103. Modulus of Elasticity; 104. Reinforced Concrete in Tension. ART. 28. RECTANGULAR BEAMS WITH TENSION REINFORCEMENT 158 105. Flexure Formulas; 106. Tables; 107. Shear; 108. Diagonal Tension; 109. Bond; 110. Design. ART. 29. T-BEAMS WITH TENSION REINFORCEMENT 178 111. Flexure Formulas; 112. Shear and Bond; 113. Diagrams. ART. 30. BEAMS REINFORCED FOR COMPRESSION 186 114. Flexure Formulas; 115. Tables and Examples. ART. 31. SLAB AND BEAM DESIGN 194 116. Bending Moments and Shears; 117. Loadings; 118. Design. ART. 32. CONCRETE COLUMNS 206 119. Plain Concrete Columns; 120. Longitudinal Reinforcement; 121. Hooped Reinforcement; 122. Eccentric Loads. CHAPTER VII RETAINING WALLS ART. 33. PRESSURE OF EARTH AGAINST A WALL 214 123. Theories; 124. Computations; 125. Graphical Method. viii CONTENTS PAGE ART. 3*.- SOLID MASONRY WALLS 223 126. Stability; 127. Empirical Design; 128. Design Using Formulas. ART. 35. REINFORCED CONCRETE WALLS 229 129. Types; 130. Cantilever Walls; 131. Counterforted Walls. ART. 36. CONSTRUCTION OF RETAINING WALLS 244 132. Foundations; 133. Drainage and Back-filling; 134. Gravity Walls. CHAPTER VIII MASONRY DAMS ART. 37. GRAVITY DAMS 247 135. Stability; 136. Graphical Analysis; 137. Design of Profile; 138. Diagonal Compressions; 139. Horizontal Tension; 140. Uplift. ART. 38. DAMS CURVED IN PLAN 258 141. Curved Gravity Dams; 142. Arch Dams; 143. Multiple Arch Dams. ART. 39. REINFORCED CONCRETE DAMS 266 144. Arch Dams; 145. Flat Slab and Buttress Dams. ART. 40. CONSTRUCTION OF MASONRY DAMS 268 146. Foundations; 147. Masonry; 148. Qxe&ow Dams. CHAPTER IX SLAB AND GIRDER BRIDGES ART. 41. LOADINGS FOR SHORT BRIDGES 271 149. Highway Bridges; 150. Distribution of Concentrated Loads; 151. Railway Bridges. ART. 42. DESIGN OF BEAM BRIDGES 273 152. Slab Bridges; 153. T-Beam Bridges; 154. Girder Bridges. CHAPTER X MASONRY ARCHES ART. 43. VOUSSOIR ARCHES 282 155. Definitions; 156. Stability. ART. 44. LOADS FOR MASONRY ARCHES 286 157. Highway Bridges; 158. Railway Bridges; 159. Dead Loads. CONTENTS ix PAGE ART. 45. DESIGN OF Voussom ARCHES 289 160. Methods; 161. Thickness of Arch; 162. Stability. ART. 46. THE ELASTIC ARCH 295 163. Analysis; 164. Temperature; 165. Direct Thrust. ART. 47. DESIGN OF REINFORCED CONCRETE ARCH . . . . : 300 166. Selection of Dimensions; 167. Division of Arch Ring; 168. Analysis; 169. Stresses. ART. 48. TYPE OF CONCRETE ARCHES 309 170. Arrangement of Spandrels; 171. Methods of Reinforcement; 172. Hinged Arches; 173. Unsymmetrical Arches; 174. Elastic Piers. ART. 49. OTHER METHODS OF ANALYSIS 314 175. Influence Lines. 176. Arbitrary Divisions. CHAPTER XI CULVERTS AND CONDUITS ART. 50. CULVERTS 321 177. Types; 178. Area of Waterway; 179. Pipe Culverts; 180. Box Culverts; 181. Arch Culverts. ART. 51. CONDUITS 332 182. Types; 183. Gravity Conduits; 184. Pressure. CHAPTER XII FOUNDATIONS ART. 52. FOUNDATION MATERIALS 342 185. Examination of Soil; 186. Bearing Capacity; 187. Tests of Bear- ing Capacity. ART. 53. SPREAD FOUNDATIONS 347 188. Distribution of Loads; 189. Masonry Footings; 190. Grillage; 191. Reinforced Concrete. ART. 54. PILE FOUNDATIONS 358 192. Classification; 193. Pile Drivers; 194. Timber Piles; 195. Bear- ing Power; 196. Concrete Piles; 197. Sheet piling. ART. 55. COFFERDAMS 373 198. Types; 199. Sheet Pile; 200. Crib. ART. 56. Box AND OPEN CAISSONS 376 201. Box Caissons; 202. Types of Open Caissons; 203. Single-wall Timber; 204. Cylinder; 205. Dredging through Wells. X CONTENTS PAGE ART. 57. PNEUMATIC CAISSONS. 387 206. Compressed Air; 207. Construction; 208. Sinking; 209. Physio- logical Effect. ART. 58. BRIDGE PIERS AND ABUTMENTS 393 210. Locations and Dimensions; 211. Stability; 212. Construction; 213. Types of Abutments. MASONRY STRUCTURES CHAPTER I DEVELOPMENT OF MASONRY CONSTRUCTION ART. 1. INTRODUCTION 1. Definition. The term masonry in its original significance means " a construction of dressed or fitted stones and mortar." It is thus properly limited to stone masonry. Custom has, however, extended the use of the term to cover any construction composed of pieces of inorganic non-metallic material fitted together into a mono- lithic block. This includes all structural work in stone, brick, and tile, as well as concrete construction. The word brick was formerly used to designate a small block of burned clay. Similar blocks of other materials have recently come into use, and we now have several kinds of bricks; as clay brick, sand-lime brick, cement brick, etc. Glazed and other ornamental and surfacing tiles are commonly employed, while hollow tiles of various kinds are rapidly coming into use. All construction formed of bricks or tiles cemented together may be classed as brick masonry. The term stone masonry is used to designate any work in which stones are fitted and cemented together so as to form a structure. Stone masonry is further subdivided into rubble masonry, squared- stone masonry, and ashlar or cut-stone masonry. Concrete is ordinarily formed by mixing broken stone or gravel with cement mortar to a mobile condition and placing it in forms in the position in which it is to be used. It is then left to harden and forms a monolithic block. Ordinary concrete cannot be economically employed where tensile stresses are developed in the structure on account of the low tensile resistance of the concrete. It is therefore common, when it is desired to use concrete in such situations, to embed steel rods in the con- 2 DEVELOPMENT OF MASONRY CONSTRUCTION crete to take the tensile stresses, leaving the concrete to carry com- pression only. This construction is known as reinforced concrete. 2. Uses of Masonry. Masonry in some form is now used in nearly all kinds of engineering and architectural construction. The selection of the type of masonry to be used in any particular structure is ordinarily largely a matter of cost, the latter factor depending upon the suitability of the construction to the use to which it is to be put, and the availability and costs of the necessary materials and labor. These factors are subject to local variation and need to be considered in each instance. Brick masonry is largely used in the construction of buildings, being usually cheaper than stone, and when of good quality showing both strength and durability. Very pleasing architectural effects are readily obtained by proper selection and arrangement of materials in brickwork. Brick masonry is frequently used in the construction of large sewers and in the arch ring of small arched bridges, and is readily adapted to such uses, but is gradually giving way to concrete. Hollow-tile construction is being quite commonly applied in building operations, and is replacing ordinary brickwork in many instances. It is sometimes faced with brick in exterior walls, and is used for partitions and in solid floor construction on account of its lightness and low cost. Stone masonry is largely used in architectural construction, where the appearance and permanence of the structure are of special importance. It is almost universally employed in monumental construction, being at once the most durable material known to man and the one capable of producing the most imposing and most beau- tiful effect. Many engineering structures such as retaining walls, bridge piers, and abutments and arch bridges are often constructed of stone masonry, or are faced with stone. Concrete is, however, gradually replacing stone masonry for such work on account of lower cost and facility of construction, except where facing of stone is used for appearance or durability. Concrete is almost universally employed in foundations, having replaced stone masonry for this purpose. In the construction of tunnels, subways, and other underground work, it is usually the cheapest and most convenient material. In heavy masonry, such as retaining walls, dams, piers, and abutments, concrete is com- monly used, alone or with a facing of stone masonry. The use of reinforcement makes it possible to apply concrete in many types of construction to which masonry has heretofore been EARLY HISTORY 3 inapplicable. For short-span bridges reinforced concrete is rapidly replacing wood and steel, and, on account of its durability, is a much more economical material for such use. Reinforced concrete is extensively used in fireproof building construction for floors, beams, and columns, and is frequently used in connection with hollow tile for this purpose. It is sometimes used for the walls of buildings but is apt to be more expensive than brick, on account of the forms necessary in such work. ART. 2. EARLY HISTORY 3. Ancient Masonry. The art of masonry construction dates from the earliest records of authentic history. The most fruitful source from which to obtain a knowledge of the history of the more ancient peoples is in a study of the remains of their masonry struc- tures. The earliest important constructions of which we have any remains are probably those of Chaldea and Assyria, with which the great constructions of Egypt may be classed. The dates of few of them are known with accuracy. The earliest of the Chaldean remains are supposed to date from about 2500 B.C. Alongside of these are the remains of the second Babylonian Empire, founded about 600 B.C. Stone and timber were lacking in Chaldea, and hence the natural development of their primitive construction was toward the use of brick. In the earlier and more crude structures, sun- dried brick of rough form were used; later, hard-burned bricks were employed. In some of the early buildings both classes were used, the burned bricks being employed as facing to protect the sun-dried from the weather. The burned bricks of the earliest times are still found to be sound and hard, and many of the sun-dried still keep their shapes. These bricks were of square, flat form, the burned ones varying from 11 to 13 inches square and 2J to 3 inches thick; the sun-dried were somewhat larger. According to Professor Rawlinson, the cementing material in some of the early Chaldean structures was either a coarse clay, sometimes mixed with straw, or a bitumen of good quality which still unites the bricks so firmly that they can with difficulty be separated. In the later Babylonian construction the character of the materials shows improvement, and elaborate ornamentation is introduced. Ornamentation was accomplished by enameling and 4 DEVELOPMENT OF MASONRY CONSTRUCTION carving the bricks and by the use of colors. Ordinary lime mortar was used. Assyria, unlike Chaldea, had plenty of stone. The type of construction used by the Assyrians, however, was probably derived from that of the Chaldeans. Brick was the principal material employed, although frequently stone was used to face the brick walls, and sculptures were freely used. The great halls of their palaces were ornamented with sculptures; the entire walls in some cases to a height of 10 or 12 feet were covered with figures in relief, representing scenes from life, and usually commemorating the great- ness of the monarch for whom they were erected. The arch was used by the Assyrians to a limited extent for nar- row openings, the arches being of brick, which were made narrower at one end than the other, in order to fit in the arch. The art of construction in Egypt was much more advanced than in Assyria and Babylonia and was probably of an earlier date. The ancient Egyptians were very skillful in working stone. Their tem- ples were built of large blocks of stone, well squared, and laid so that the joints are scarcely visible. They quarried granite and trans- ported large blocks for long distances. They also cut and polished granite. The great pyramid has a base of 764 feet square and is approxi- mately 486 feet high, and is built in courses, of great blocks of lime- stone, from 2 to 5 feet thick and as much as 30 feet in length. The early Egyptian masonry is remarkable both on account of the great size of the materials and the exactness with which they are fitted together, no mortar being employed. In Greece and Italy remains are found of Cyclopean masonry built of stones of large size and carefully adjusted joints. The walls of Mycenae were built of irregular blocks of great size, the spaces being filled with smaller stones. Greek Masonry. The masonry of the Greeks was arranged in courses and the joints carefully fitted and equal to the best Egyptian workmanship. The carving of artistic forms was here for the first time developed to a high degree of excellence. The Egyptians had used the system of the column and entab- lature in their temples. The Greeks introduced the pediment, and improved the artistic design of the buildings, bringing the propor- tioning and ornamentation of such structures to a most wonderful perfection. 4. Roman and Medieval Construction. In the system of con- struction developed by the Romans the walls were built of coarse RECENT DEVELOPMENTS 3 concrete or rough cemented rubble, and were usually faced with brick or marble. Sometimes, in less important construction, small blocks of tufa, set irregularly, formed the surface of the walls, which were stuccoed on their interior surfaces. The art of building was greatly developed during the Roman period. The introduction of the arch changed the whole system of construction. In the Romanesque architecture, the circular arch was the principal feature, the structures consisting mainly of heavy walls supporting semicircular arched roofs. Roman arches were constructed of cut stone, brick, or concrete. The introduction of the pointed arch, and later of the use of arched ribs with piers and buttresses to transmit the loads to the foundations, marks another advance in the art of construction. This made possible a disposition of the materials of the structures to better advantage, and led to more economical construction. During medieval times the use of stone masonry was brought to a high state of perfection. Random ashlar or rubble was com- monly used in buildings in preference to coursed ashlar. Beautiful and imposing effects were attained by the use of materials of rather small size, and great skill was developed in the cutting of ornamental forms. The Romans used lime mortar in their ordinary construction. They also discovered that if certain materials of volcanic origin were pulverized and mixed with lime, the resulting mortar possessed the property of hardening under water. The mortar used by the Romans in their aqueducts and other hydraulic works was made from this material, obtained from near the foot of Vesuvius. Similar materials were later found and used in Germany and France. ART. 3. RECENT DEVELOPMENTS 5. The Cement Industry. The discovery by the Romans of the hydraulic properties of volcanic lava, and the location of other materials possessing the same properties, made possible the con- struction of subaqueous masonry work. No considerable progress, however, was made in such work. About the middle of the eighteenth century Smeaton, a noted English engineer, discovered that lime made from certain limestones containing clay possessed hydraulic properties. This discovery opened new possibilities in under-water work, and these hydraulic limes were used to a limited extent during the next half century. In 1796 James Parker, an Englishman, burned limestone con- 6 DEVELOPMENT OF MASONRY CONSTRUCTION taining a larger proportion of clay and ground the product. He thus produced the first natural cement, which he called Roman cement. This process was patented, and the manufacture of natural cement resulted. In 1818 Canvas White, an engineer of the Erie Canal, located rock suitable for making natural cement in Madison County, New York, and the first cement produced in the United States was made in the same year. Five years later the manufacture of natural cement was begun at Rosendale, New York. The production of cement in this region extended, and cement was thus provided for most of the hydraulic construction in this country for a considerable period. Later, as the development of the country proceeded, and demands for cement increased, deposits of cement rock were found at many other places. Natural cement plants were established along the James River in Virginia; in the Lehigh Valley, in Pennsyl- vania; at Louisville, Kentucky; Utica, Illinois; Milwaukee, Wis- consin, and a number of other localities. In 1824 Joseph Aspdin, of Leeds, England, discovered that by burning a mixture of slaked lime and clay at high temperature, hydraulic cement was produced. Aspdin named this material Portland Cement, on account of its resemblance to Portland stone, then largely used in England. In 1845 the manufacture of Port- land cement was begun on a commercial scale by J. B. White & Sons, in Kent. During the period between 1830 and 1850 Vicat, in France, made a number of studies which were of great value in extending knowledge of the new material. Plants were soon established in France and Germany for the manufacture of Portland cement, and the industry became an important one throughout Europe. During the next few years, 1865 to 1880, John Grant made a series of investi- gations of the properties of Portland cement and methods of using it in mortars and concrete. His papers before the Institution of Civil Engineers had a marked influence in shaping the methods of use of cement. From 1880 to 1900 the Portland cement industry developed rapidly in Europe, and numerous studies were made concerning the composition and properties of the material. LeChatelier, Alex- andre, Candlot, and Feret, in France, Tetmajer in Switzerland, Michaelis and Bohme in Germany, Faija in England, and a number of others, investigated all phases of the subject, greatly improving the quality of the cement and showing methods of employing it in construction to secure the best results. RECENT DEVELOPMENTS 7 In 1875 Mr. D. O. Saylor began the manufacture of cement at Coplay, Pennsylvania. From this beginning, the American Port- land cement industry has developed. Great improvements in methods of manufacture and in the control of the character of the product have been made in this country. The studies of Newberry, Richardson, and others have contributed to definite knowledge of the proper composition of the material, while committees of the National Engineering Societies and many independent investigators have perfected methods of testing cement and of using it in con- struction. This industry has now reached immense proportions in the United States, and the use of Portland cement has extended in all directions, modifying largely the types and methods of construction used in all classes of structures. 6. Reinforced Concrete. In the early use of concrete, it was commonly employed as a filler in heavy construction, and was not possessed of great strength. Walls of concrete were usually protected by facings of stone or brick masonry. In recent years, however, the availability of cementing materials of high grade has made possible the use of concrete in many classes of construction for which stone or brick masonry was formerly employed. The facility with which concrete may be applied to many uses makes it highly desirable material, and since the introduction of Portland cement its use has rapidly increased. This use has been further extended in the past few years by the development of reinforced concrete construction. In 1850 Lambot, in France, constructed a boat of reinforced concrete, and in 1855 patented his invention in England. Fra^ois Coignet, in 1861, applied reinforced concrete to the construction of beams, arches, pipes, etc. In 1861 Joseph Monier, a gardener of Paris, constructed tubs and small water tanks of concrete in which a wire frame was imbedded. In 1867 Monier patented his reinforcement, which consisted of a mesh formed of wires or rods placed at right angles to each other. He also exhibited some work at the Paris Exposi- tion in the same year. Nothing came of this invention for a number of years, but in 1887 Wayss and Bauschinger published, hi Germany, the results of an investigation showing the value of the Monier system, and giving formulas for use in design. The next few years saw considerable development of this type of construction in Austria, and Melan, an Austrian engineer, invented a system of reinforcement for arches in which I-beams were bent to the form of the arch and enclosed in concrete. Hennebique, 8 DEVELOPMENT OF MASONRY CONSTRUCTION in France, began making reinforced concrete slabs about 1880, and patented his system of slab reinforcement in 1892. The first use of reinforced concrete in the United States seems to have been by Ernest L. Ransome, in 1874. The next year W. E. Ward constructed a building in New York, in which reinforced concrete walls, roof, and floor beams were used. In 1877 H. P. Jackson used reinforced concrete in building construction in San Francisco. About 1884 Ransome began applying reinforced con- crete to important work in California, and in that year took out a patent for the first deformed bar. In 1894 the Melan system of arch-bridge construction was intro- duced into the United States by Mr. Fr. von Emperger, who built the first important arch bridges. At about the same time Mr. Edwin Thacher began the construction of arch bridges using bar reinforcement. During the period from 1890 to 1900 the use of reinforced con- crete steadily increased, while the applications of plain concrete had been extending rapidly, as the increasing supply of cement provided material for a better grade of construction. Since 1900 the use of reinforced concrete has rapidly increased. The use of massive slab construction for railroad bridges was intro- duced by the C. B. & Q. Railroad at Chicago. Fireproof building construction of concrete has become common, and concrete has become the standard material for short-span highway bridges. Many investigations have been made concerning the properties of the materials and the strengths of various structural forms; the work of Considere, in France, and of Talbot at the University of Illinois, being specially notable. Principles for rational design have been established and recognized standards of practice are rapidly forming. CHAPTER II CEMENTING MATERIALS ART. 4. LIME 7. Classification. The cementing materials employed in the construction of masonry and concrete structures include common lime, hydraulic lime, Portland cement, natural cement, and puzzolan. These materials are formed by the calcination of limestones, or of mixtures of limestones with siliceous or argillaceous materials, and their properties vary with the nature and porportions of the sub- stances combined in them. Common Lime. When limestone composed of nearly pure car- bonate of lime is burned, the resulting clinker, known as quicklime, possesses the property of breaking up, or slaking, upon being treated with a sufficient quantity of water. The slaking of lime is due to its rapid hydration when in contact with water, and the process is accompanied by a considerable increase in the volume of the mass of lime and by a rise in temperature. If the quantity of water be only sufficient to cause the hydration of the lime, the quicklime is reduced to a dry powder; while if the water be in excess it becomes a paste. The slaked lime thus formed possesses the further property, when mixed to a paste with water and allowed to stand in the air, of hardening and adhering to any surface with which it may be in contact. This hardening of common limes will take place only when exposed to the air and allowed to become dry. When lime is nearly pure and its activity very great it is known Sisfat lime. If the lime have mixed or in combination with it considerable impurities of inert character, which act as an adulteration to lessen the activity of the lime, causing a partial loss of the property of slaking and diminishing its power to harden, it is known as meager or poor lime. Hydraulic Lime. When the limestone contains about 10 to 20 per cent of silica or clay mixed with the carbonate of lime, the 9 10 CEMENTING MATERIALS material resulting from the burning is known as hydraulic lime. This clinker will slake when treated with water like common lime, but with reduced activity. The slaked lime thus obtained pos- sesses the further property, when mixed with water to a paste, of hardening under water and without contact with the air. In hydraulic lime the silica and alumina are combined with a portion of the lime, forming compounds which harden under water, while part of the lime is left uncombined. This free lime expands when hydrated by addition of water, causing the material to slake. Hydraulic Cement. When the proportion of siliceous or argil- laceous materials in limestone, or mixed with it, is sufficient to combine with all the lime, leaving no lime in a free state, the prod- uct of burning is known as hydraulic cement. This clinker will not slake, but must be reduced to powder by grinding. The cement powder, when mixed with water, has the property of setting and hardening under water, and of adhering firmly to any surface with which it may be in contact. Portland Cement is the name given to hydraulic cement which is formed by burning and grinding an intimate mixture of powdered limestone and argillaceous matter in accurately determined pro- portions. In making Portland cement, the ingredients are care- fully proportioned to secure the complete combination of the lime with the silica and alumina into active material, and it is necessary to reduce the materials to a very fine state and secure uniform incorporation of the ingredients before burning. Natural Cements are made by burning limestones which con- tain proper proportions of argillaceous materials, and grinding the resulting clinker to powder. Natural cements are less rich in lime than Portland cements, complete combination of the argillaceous materials not being effected. They are burned, like lime, without the pulverization of the raw materials, and require a much lower temperature in burning than Portland cement. The term Puzzolan is commonly applied to a class of materials which, when made into a mortar with fat lime or feebly hydraulic lime, impart to the lime hydraulic properties and cause the mortar to harden under water. It derives its name from Pozzuoli, a city of Italy near the foot of Mount Vesuvius, where its properties were first discovered. It was extensively used by the Romans in their hydraulic constructions, being mixed with slaked lime for the for- mation of hydraulic mortar. Puzzolan is essentially a silicate of alumina in which the silica exists in a condition to be attacked LIME 11 readily by caustic alkalies, and hence easily combines with the lime in the mortar. Puzzolan Cement is formed by mixing slaked lime with puzzolan and grinding the mixture to a fine powder. Certain materials of volcanic origin are frequently used for this purpose in Europe, while considerable quantities of cement of this class have been made by the use of . blast furnace slag, both in Europe and the United States. 8. Common Lime. Common lime is such as does not possess hydraulic properties. It is divided into fat or rich lime and meager lime, according to the quantity of impurities of an inert character it may contain. When made into paste and left in air it slowly hardens. The process of hardening consists in the gradual forma- tion of carbonate of lime through the absorption of carbonic acid from the air, accompanied by the crystallization of the mass of hydrated lime as it gradually dries out. In common lime the final hardening takes place very slowly, working inward from the surface, as it is dependent upon contact of the mortar with the air. When the lime is nearly pure the resulting carbonate is likely to be some- what soluble, and consequently to be injured by exposure. Nearly all limes, however, contain small amounts of silica and alumina, and these ingredients, even when in quantities too small to render the lime hydraulic, impart a certain power to set, causing the harden- ing to take place with greater rapidity and without entire dependence upon contact with air. It also renders the material less soluble and more durable in exposed situations. Nearly pure limes, consisting mainly of calcium oxide, are very caustic and become hydrated very rapidly when brought into con- tact with water. This hydration, or slaking, produces a rise in temperature and increase in volume, which vary in amount accord- ing to the purity of the lime, the volume being doubled or tripled for good fat lime. When the lime is derived from a magnesian limestone, it may contain a considerable proportion of magnesia mixed with the lime. Limes containing more than about 15 per cent of magnesia are usually called magnesian limes. The presence of magnesia has the effect of rendering the lime less active, causing it to expand less upon slaking. The magnesian limes harden more slowly, but usually gain a higher ultimate strength than the high- calcium limes. The common method of slaking lime consists in covering the quicklime with water, using two or three times the volume of the lime. This method is known as drowning. The lime is usually 12 CEMENTING MATERIALS spread out in a layer perhaps 6 or 8 inches thick, in a mixing box, the water poured over it and allowed to stand. Sufficient time must be allowed for all of the lumps to be reduced. When the lime contains much foreign matter, the operation frequently requires several days. Too great quantity of water is to be avoided, the amount being such as will reduce the lime after slaking to a thick pasty condition. All the water should be added at once, as the addition of water after the hydration is in progress causes a lower- ing of temperature and checks the slaking. For the same reason, the lime should be covered after adding water, and not stirred or disturbed until the slaking is completed. The covering is often effected by spreading a layer of sand over the lime, the sand being afterward used to mix with it in making mortar. A second method of slaking is sometimes employed having for its object the reduction of the slaked lime to powder, and known as slaking by immersion. This is accomplished in two ways. By the first method, the lime is suspended in water in baskets for a brief period to permit the absorption of the necessary water, after which it is removed and covered until slaking takes place and the lime falls to powder. By the second method, sprinkling is substituted for immersion, the lime being placed in heaps and sprinkled with the necessary quantity of water, then covered with sand and allowed to stand. Lime is commonly sold as quicklime, and should be in lumps and not air slaked. When it is old and has been exposed to the air it is likely to have absorbed both moisture and carbonic acid, thus becoming less active, the portion combined with carbonic acid being inert. A simple test of the quality of quicklime is to immerse a lump for a minute, then place in a dish and observe whether it swells, cracks, and disintegrates, with a rise of temperature. Slaking some days in advance of use is desirable in order to insure the complete reduction of the lime, and it is quite common to slake lime several weeks before it is to be used. Common lime is ordinarily used in construction as a mortar, mixed with sand. The quantity of lime in the mortar should be just sufficient to fill the voids in the sand, without leaving any part formed entirely of lime. Mortar of rich lime shrinks in hardening, while masses composed entirely of lime on the interior are likely to remain soft, so that an excess of lime may be an element of weak- ness. If too little lime be used the mortar may be porous and weak. The proportions ordinarily required are between one part lime to two parts sand, and one part lime to three parts sand. LIME 13 In mixing lime mortar, sand is spread over the lime paste and worked into it with a shovel or hoe. The proper proportions of sand and lime may be judged by observing how the mortar works. If too much sand be used it will be brittle, or " short "; while too much paste will cause it to stick and cake so that it will not flow from the trowel. Mortar of common lime should not be employed in heavy masonry or in damp situations. Where the mass of masonry is large, the lime mortar will become hardened with great difficulty, and after a long time. The penetration of the final induration due to the absorption of carbonic acid is very slow. The observations of M. Vicat showed that carbonization extended only a few millimeters the first year and afterward more slowly. The induration of the lime along the surfaces of contact with a harder material is usually more rapid than in the interior of the mass of lime, and the strength of adhesion to stone or brick is often greater than that of cohesion between the particles of mortar. 9. Hydraulic Lime. Hydraulic lime is obtained by burning limestone containing silica and alumina in sufficient quantities to impart the ability to harden under water. The hydraulic elements are present in such quantities that they combine with a portion of the lime, forming silicates and aluminates of lime, leaving the remainder as free lime in an uncombined state. The hydraulic activity of a lime or cement, that is, its ability to harden under water, depends primarily upon the relative propor- tions of the hydraulic ingredients and of lime. Silica and alumina are considered to be the effective hydraulic ingredients, and it is common to designate the ratio of the sum of the weights of silica and alumina to that of lime in the material its hydraulic index. The hydraulic index gives, therefore, within certain limits, a measure of the hydraulicity of the various classes of limes. It is to be remem- bered, however, that there are other factors to be considered in judging of the action of lime than this simple proportion. The other ingredients may by their combinations withdraw portions of the active elements so as to modify the effective ratio between them, while the activity of the lime depends largely upon the state of combination in which the active elements exist. This is not shown by analysis, and may be greatly modified by the manipulation given the material during manufacture. Limes with hydraulic index less than 10/100 possess little if any hydraulic properties, and are known as common limes. When the hydraulic index is between 10/100 and 20/100 the lime is feebly 14 CEMENTING MATERIALS hydraulic, and may require from twelve to twenty days to set under water. Hydraulic lime proper includes that of index from about 20/100 to 40/100. These may harden in from two to eight or ten days. The quantity of free lime in the material is dependent upon the degree of burning, as well as upon the amount of lime contained by the stone. If the stone be underburned, the combination of the hydraulic elements with the lime is not complete, and more of the lime remains in a free state. For this reason, a stone of high hydraulic index may, when underburned, yield a lime, but burned at a high temperature becomes unslakable. The best limes are usually those which can be burned at a high temperature to complete the chemical combinations. It is necessary that sufficient free lime be present to cause the lime to slake properly, but it is also desirable that the quantity of uncombined lime be as small as possible, as the setting properties are due to the silicates and alu- minates, while the hydrated lime remains inert during the initial hardening of the mortar. According to Professor LeChatelier, limestone for hydraulic lime should contain but little alumina, as the aluminates are hydrated during the slaking of the lime, while the silicates are not affected, the heat of the slaking preventing their hydration. The following is given as an average analysis of the best French hydraulic lime: Silica 22 Alumina 2 Oxide of iron 1 Lime ... 63 Magnesia 1.5 Sulphuric acid 0.5 Water. . 10 100 It is important that the slaking be very thorough, as the pres- ence of unhydrated free lime in the mortar while hardening is an element of danger to the work. Any lime becoming hydrated after the setting of the mortar may, by its swelling, cause distortion and perhaps disintegration of the mortar. After the lime has been reduced to powder by slaking, it is forced through sieves which permit the passage of all pulverized particles but hold those of appreciable size, including the underburned rock LIME 15 and the overburned parts which refuse to slake. The residue left from the sifting of hydraulic lime is known as grappiers. This material is mainly composed of hard material more rich in silica and alumina than the other portions of the lime. The grappiers are frequently ground and sold as cement, and when properly handled may form cement of fairly good quality. 10. Hydrated Lime. When quicklime is slaked with the quantity of water necessary completely to hydrate it, and the resulting mate- rial is bolted to remove all unslaked particles, the result is a very fine white powder, commercially known as hydrated lime. This lime is sold on the market in barrels or bags, and it is in convenient form for use. Lime in this form may be kept for considerable periods without deterioration, provided it is protected from con- tact with moisture. Hydrated lime ordinarily weighs about 40 pounds per cubic foot, and contains approximately 75 per cent of quicklime. By mixing with about an equal weight of water, it may be reduced to lime paste, or lime putty, as it is commonly called in building oper- ations. Lime paste occupies a slightly greater volume than the hydrated lime from which it is prepared. The use of hydrated lime for mixing with cement mortar in ordinary masonry construction is rapidly increasing. It is also frequently used in small proportions in Portland cement concrete to make the concrete flow more smoothly, and sometimes to decrease the permeability of the mortar. (See Art. 23.) 11. Specifications for Lime. In ordinary building operations lime is commonly employed in the form of quicklime and slaked where used. Usually the quality of the lime has been judged by its activity in slaking and no particular tests are specified. Tests of composition by chemical analysis and of completeness of slaking by washing through sieves are, however, frequently employed. Hydrated lime is now largely used for mixing with cement mortar and for plastering work, and this use is rapidly extending. The tests employed for hydrated lime include chemical analysis, fine- ness, and permanence of volume or soundness. The American Society for Testing Materials has adopted standard specifications giving methods for making these tests. These speci- fications are given in the Book of Standards of the Society or may be obtained in pamphlet form from the Secretary of the Society. As they are now undergoing revision they are subject to change and will not be given here. 16 CEMENTING MATERIALS ART. 6. HYDRAULIC CEMENT 12. Setting and Hardening of Cement. When cement powder is mixed with water to a plastic condition and allowed to stand, it gradually combines into a solid mass, taking the water into con- bination, and soon becomes firm and hard. This process of com- bination among the particles of the cement is known as the setting of the cement. Cements of different character differ very widely in their rate and manner of setting, some occupying but a few minutes in the operation, while others require several hours. Some begin setting immediately and take considerable time to complete the set, while others stand for considerable time with no apparent action and then set very quickly. The points where the set is said to begin and end are necessarily arbitrarily fixed, and are determined by finding when the mortar will sustain a needle carrying a specified weight. The initial set is supposed to be when the stiffening of the mass has become per- ceptible; the final set, when the cohesion extends through the mass sufficiently to offer such resistance to any change of form as to cause rupture before deformation can take place. After the completion of the setting of the cement, the mortar continues to increase in cohesive strength over a considerable period of time, and this subsequent development of strength is called the hardening of the cement. The process of hardening appears to be quite distinct from, and independent of, that of setting. A slow-setting cement is apt, after the first day or two, to gain strength more rapidly than a quick-setting one; but it does not necessarily do so. The ultimate strength of the cement is also quite independent of the rate of setting. A cement imperfectly burned may set more quickly and gain less ultimate strength than the same cement properly burned, but of two cements of different composition the quicker-setting may be the stronger. There is as wide variation in the rate of hardening of different cements as in the rate of setting; some gain strength rapidly and attain their ultimate strengths in a few weeks, while others harden much more slowly at first and continue to gain in strength for several years. The rate of early hardening gives but little indication of the ultimate action of the cement, as the final strength of the mortar may be the same however rapidly the strength is attained. The rate at which cement sets seems to depend upon the pres- HYDRAULIC CEMENT 17 ence of certain aluminates of lime, the rapidity of set increasing with the percentage of alumina in the material. The final harden- ing is attributed mainly to the silicates of lime, which are the impor- tant elements in giving strength and durability to the mortar. The formation of these active elements in the cement depends upon the manipulation of the material in manufacture, as well as upon the composition of the raw materials. In an underburned cement, the relative proportions of aluminates to silicates is large and the set is rapid. Calcium Sulphate. The addition of a small amount of sulphate of lime to cement has the effect of slackening the rate of set. Such addition is frequently made by manufacturers to reduce the activity of fresh cement, by grinding a small amount of gypsum with the cement. Effect of Sand. Cement is ordinarily employed in mortar formed by mixing it with sand, and the action of the mortar is necessarily largely affected by the nature and quantity of sand used. When the cement is finely ground and the sand of good quality, a mortar composed of equal parts of each, as a general thing, finally attains a strength as high as, or higher than, that of neat cement. Cements of different characters, however, vary considerably in their power to " take sand " without loss of strength; some of the weaker ones may not be able to take more than half their weight of standard sand, while others can be mixed with considerably more than their own weight without loss of strength at end of six months or one year after mixing. All have a certain limit within which they may be made stronger by an admixture of good sand than they would be if mixed neat. Clean and sharp sand usually gives higher strength in mortar than that containing admixtures of clay or earth, or that composed of rounded grains, coarse sand usually giving greater strength than that which is very fine. It is often difficult, however, to judge of the quality of sand without experimenting with it. In some cases a small amount of fine clay appears to increase the strength of mor- tar, while a judicious mixture in the sand of grains of various sizes may be of value in reducing the volume of interstices. Mortar composed of sand and cement usually possesses greater ability to adhere to other surfaces when coarse sand is used that when the sand is fine. Effect of Water. The quantity of water used in mixing mortar is one of the most important elements; the less the quantity, pro- vided there be sufficient to thoroughly dampen the mass of cement, 18 CEMENTING MATERIALS the quicker the set. With some Portland sements, changing the quantity of water used in mixing from 20 to 25 per cent of the weight doubles or even triples the time required for the mortar to set. When the quantity of water used in mixing is sufficient to reduce the mortar to a soft condition, the hardening as well as the setting becomes slow, and the strength during the early period is less than when a less quantity of water is used. This difference disappears to a considerable extent with time, and the mortar mixed wet may eventually gain as much strength as though mixed with less water. Cement mortar kept under water hardens more rapidly in the early period than that exposed to the ah*. Nearly any cement mortar will harden more rapidly and gain greater strength if kept moist during the operation of setting and the first period of harden- ing than if it be exposed at that time to dry air. Sudden drying out about the time of completing setting causes a considerable loss of strength in cement mortar, and frequently the mortar so treated is filled with drying cracks. This result is usually more marked when the mortar has been mixed quite wet. Effect of Temperature. The temperature of the water used in mixing and that of the air in which the mortar is placed during setting has an important bearing upon the time required for setting; the higher the temperature, within certain limits, the more rapid the set. Some cements which require several hours to set when mixed with water at temperature of 40 F. will set in a few minutes if the temperature of the water be increased to 80 F. Below a certain inferior limit, ordinarily from 30 to 40 F., the mortar sets, with extreme slowness or not at all, while at a certain upper limit, in some cements between 100 and 140 F., a change suddenly occurs from very rapid to very slow rate of set, which then decreases as the temperature increases until the cement ceases to set. The temperature of the air or water in which the mortar is immersed while hardening has a very important effect upon the gain in strength. Heat accelerates the action, while at temperatures near the freezing-point of water the gain in strength is very slow. 13. Portland Cement. The term Portland cement is used to Designate material formed by burning to incipient fusion a finely ground mixture of definite proportions of limestone and argillaceous materials, and grinding the clinker so formed to fine powder. Several classes of materials are used for this purpose. Hard limestone or chalk, consisting of nearly pure carbonate of lime, is frequently employed, mixed with clay or shale to furnish the hydraulic ingredi- ents. In the Lehigh District in Pennsylvania cement rock, con- HYDRAULIC CEMENT 19 sisting of limestone containing silica and alumina in sufficient quanti- ties to make natural cement when burned alone, is mixed with nearly pure limestone to obtain the proper Portland cement composition. In the Michigan district marl and clay excavated in soft and wet condition are used. In a few instances limestone is mixed with blast-furnace slag for the production of Portland cement. This is quite distinct from the manufacture of slag cement (so called) in which the materials are not burned together. To make good Portland cement it is always necessary that the ingredients be very carefully proportioned and that the mixture be very homogeneous. This requires the pulverization of the mate- rials and their uniform incorporation into the mixture before burning. The burning of Portland cement requires high heat to insure complete combination of the lime with the silica and alumina. In underburned cement, a part of the lime may be left as caustic lime, uncombined with the clay. This is apt to produce unsound cement, which may swell and crack after being used. The action of Portland cement seems to depend upon the for- mation, during burning, of certain silicates and aluminates of lime which constitute the active elements of the cement, the other ingre- dients being considered impurities. The ideal cement would be that in which the proportion of lime is just sufficient to combine with all the silica and alumina in the formation of active material. If there be a surplus of clay beyond this point, it forms inert material. Any surplus of lime remains in the cement as free lime and consti- tutes one of the chief dangers in the use of cement, as, although it may not prevent the proper action of the cement when used, it may cause the mortar to swell afterward and become cracked and distorted as the lime slakes. As perfect homogeneity is not attainable in practice, it is always necessary that the clay be somewhat in excess in order that free lime be not formed. The amount of excess of clay necessary depends upon the thoroughness of the burning and the evenness which may be reached in the mixture of the raw materials. The normal composition of Portland cement is usually within the following limits: Silica 20 to 25 per cent Alumina 5 to 9 per cent Iron oxide 2 to 5 per cent Lime 59 to 65 per cent Magnesia 0.5 to 3 per cent Sulphuric acid . . . 0.25 to 2 per cent 20 CEMENTING MATERIALS After the cement clinker resulting from the burning is sufficiently cooled, it is put through grinders and reduced to a fine powder. The degree of fineness to which the cement is ground is always very important in its effect upon the strength of mortar made from the cement. The valuable part of the cement is that which is ground extremely fine to an impalpable powder. The coarse parts are not altogether inert, but are more or less active, depending upon the size of the grains of which they are composed. Cement when used is commonly mixed with sand and the attain- ment of strength in sand mortar, rather than paste of neat cement, is of importance. The more finely ground the cement, the greater its resistance when mixed with sand, both in the earlier and later stages of hardening, and also the sooner will it reach its ultimate strength. The effect of fine grinding is much greater when the proportion of sand to cement is large, as the power of the cement to " take sand " without diminution of strength is thereby greatly increased. The coarser particles of the cement may be considered as practically inert material, which acts as sand rather than as cement in the mortar. The ability of the cement to harden and develop "strength in sand mortar is thus dependent upon the amount of fine material contained in it. Portland cement made from materials containing very small percentages of iron oxide are very light in color or white. These cements usually contain high percentages of alumina, and are con- sequently quick setting. They are lower in strength than normal Portlands. 14. Natural Cement. The term natural cement is used to desig- nate a large number of widely varying products formed by burning rock without pulverization or the admixture of other materials. These cements contain larger proportions of argillaceous materials, with less lime, than Portland cement, and are burned at a lower temperature. The term Roman Cement is used in Europe to designate a class of quick-setting cements formed by burning, at a comparatively low temperature, limestone containing a high percentage of clay. The proportion of alumina in these materials is large and possibly accounts for the quick set. Materials of this character become inert when the temperature of burning is increased to the point where the chemical reactions would become complete. A class of materials intermediate between the Roman cements and the Portland cements is called in Europe Natural Portland Cement. In composition they are similar to Portland cement, but HYDRAULIC CEMENT 21 contain less lime. They are burned at a higher temperature than Roman cements, and are usually slower setting. Natural cements made in the Lehigh region are of this character. These materials may be made into Portland cement by addition of a limestone con- sisting of more nearly pure carbonate of lime. Magnesian Natural Cements are formed by burning Magnesian limestones. The composition of these cements varies from that of the Roman cements to that in which the proportion of magnesia is as great as that of lime. The action of cements of this class is somewhat similar to that of the Roman cements. They may be either slow or quick setting, and gain strength rather slowly, reach- ing a much less ultimate strength than Portland cement. Mag- nesian cements are but little used in Europe, but in the United States they constitute the larger part of the natural cements in use, and many of them have been found by experience to be very useful and reliable materials. The rock from which natural cements are made differs greatly in character in the same locality, and in different strata in the same quarry. In some of the mills the nature of the product is regulated by mixing, in proper proportions, the clinker obtained by burning rock from different strata. Each portion of the rock must be burned in such degree as is suited to its composition, and hence, as the material is not pulverized before burning, it must be burned sepa- rately and mixed afterward. To produce uniformly good cement, therefore, requires close and careful attention; for this reason there is often considerable difference in the quality of cement made by works in the same locality and from very similar materials. Mixed Cements. In localities where both Portland and natural cements are made by the same works, mixtures of the lower grades of Portland with natural cements are sometimes made. These are usually sold as natural cements under the name Improved Cements. The effect of the mixture is to make the setting slower, and to somewhat increase the strength of the natural cement. 15. Puzzolan Cement. Puzzolan cement is formed by mixing and grinding together definite proportions of slaked lime and puzzolan. In Germany puzzolan cement is made by the use of a natural puz- zolan called trass, consisting of a volcanic earth. In the United States cement of this character is made by the use of specially pre- pared blast-furnace slag. This cement is sometimes called slag cement. Basic slag, containing lime in excess of the silica and with a high alumina content, is used for this purpose. It is made granular by quenching in cooling. 22 CEMENTING MATERIALS It is very important, in making slag cements, that the slag be ground very fine, and be very intimately mixed with the lime. The lime is slaked and bolted and then ground mechanically with the slag so as to insure thorough incorporation into the mixture. In some of the European plants the slag is finely ground and bolted through fine sieves before being mixed with the lime, but more com- mon practice is to slake and bolt the lime and mix with the granular slag before grinding, or to do the pulverizing of the slag in two stages and make the mixture between the first and second grinding. Puzzolan cement is usually very finely ground, and is slow in setting. It is sometimes treated with soda to quicken the set. When allowed to harden in dry air, it is likely to shrink and crack. When used for under-water work, mortar of puzzolan cement fre- quently gives nearly the same strength as good Portland cement. It is essentially a hydraulic material, and it is specially important that it be kept damp during the early period of hardening, in order that the water necessary to proper hardening may not evaporate. The composition of slag cement usually differs from that of Portland cement in having a less quantity of lime, more silica and alumina and more alumina in proportion to silica. 16. Sand Cement. Sand cement is the name given to material formed by grinding together Portland cement and silica sand to extremely fine powder and a very intimate mixture. It is claimed that a considerable amount of sand may be thus mixed with the cement without materially reducing the strength of mortar made by mixing the resulting cement with the usual proportions of sand. The additional grinding reduces all of the cement to impalpable powder, thus increasing the amount of active material. Sand cement as ordinarily made contains equal proportions of Portland cement and silica sand. Cement of this character has recently been made in California by grinding volcanic rock, or tufa, with Portland cement. The tufa used is a puzzolan, and it is claimed that it reacts with the lime of the cement. The results of tests indicate that mortar made from this cement is equal in strength to that of the original Portland. Cement of this kind is now being made by the U. S. Reclamation Service in some of the Western States to reduce the cost of concrete work where Portland cement is expensive and difficult to get. Similar methods are employed in Germany where a puzzolan called trass is used, and in Italy where volcanic lava is ground with the cement. These cements are used for work in sea water to les- HYDRAULIC CEMENT 23 sen the action of the sea salts upon the lime salts of the Portland cement. Sand cement has frequently been used for the purpose of secur- ing impermeable mortar where waterproof work is needed. It is useful for this purpose on account of its extreme fineness. 17. Soundness of Cement. The permanence of any structure erected by the use of cement is dependent upon the ability of the cement, after the setting and hardening processes are complete, to retain its strength and form unimpaired for an indefinite period. Experiment has shown that mortars made from cement of good quality frequently continue to gain strength and hardness through a period of several years, or at least that there is no material diminu- tion in strength with time; and that changes of temperature, or in the degree of moisture surrounding it, produce no injurious effects upon the material. This durability in use is commonly known as the permanence of volume or soundness of the cement. When mortar which has been immersed in water is transferred to dry air, a slight contraction may take place in volume, together with an increase in strength; while a transfer the other way may produce the opposite result; but" no distortion of form or disinte- gration of the mortar will take place in either case if the cement be of good quality. Sometimes cement when made into mortar sets and hardens properly, and later, when exposed to the action of the atmosphere or water, becomes distorted and cracked or even entirely disinte- grated. If the composition deviates but slightly from the normal, this process of disintegration may not show itself for a considerable time and proceeds very slowly. It thus becomes an element of considerable danger, as it is liable to escape detection in testing the cement. The presence of small quantities of free lime in cement is doubt- less one of the most common causes of disintegration in cement mortar. The lime being distributed through the cement in small particles is hydrated very slowly after the cement has set, causing, through its swelling during slaking, strong expansive forces on the interior of the mortar, and producing an increase of volume, loss of strength, and perhaps final disintegration. Free magnesia in cement is supposed to act very much like free lime. The action of magnesia, however, is much slower than that of lime, and for this reason is a more serious defect. Specifications for Portland cement frequently limit the amount of magnesia that may be present in the cement. 24 CEMENTING MATERIALS Most Portland cements probably contain small amounts of the expansive elements, which when in very small quantity act with extreme slowness and perhaps produce no visible effect for several months after the use of the cement; then occurs a decrease of strength, which disappears with time. Cements which gain strength rapidly are quite apt to act in this manner, a depression in the strength curve occurring at from six months to one year after the mortar is made. Cements for use in sea water should contain very little alumina. Some of the salts in the sea water attack these alumina compounds, causing disintegration of the cement and giving rise to expansive action which cracks and breaks up the work. The presence of expansive elements in Portland cement is prob- ably due to incomplete burning or lack of uniformity in the incor- poration of the ingredients rather than to defective composition. The fineness of the cement modifies the action of the free lime, as finely divided material will slake more quickly than coarse grains, and the lime is more apt to become hydrated before setting; or, if the cement be exposed before use, the lime in a fine state will sooner become air slaked. 18. Chemistry of Cement. Professor LeChatelier was the first to explain the composition of Portland cement. He studied sec- tions of clinker under the microscope, and examined the properties of the various compounds formed by the principal ingredients. He concluded l that the tricalcium silicate, 3CaO, SiO2, is the only silicate that is really hydraulic, and that it is the essential active element in cement. In Portland cement he finds it to be the princi- pal component, occurring in cubical crystals. It is formed by combination of silica and lime in presence of fusible compounds formed by alumina and iron. " The dicalcium silicate, 2CaO, SiC>2, possesses the singular property of spontaneously pulverizing in the furnace upon cooling. This silicate does not possess hydraulic properties and will not harden under water. " There are various aluminates of lime, all of which set rapidly in contact with water. The most important is the tricalcium alu- minate, 3CaO, A1 2 O 3 ." Professor LeChatelier gives two limits within which the quantity of lime in Portland cement should always be found. These are, that the proportion of lime should always be greater than that represented by the formula 1 Annales des Mines, September, 1893. HYDRAULIC CEMENT 25 CaO+MgO i ~~~~ ^~ o% and that it should never exceed that given by the formula, CaO+MgO 3. The symbols in these formulas represent the number of equivalents of the substances present, not the weights. Messrs S. B. and W. B. Newberry from a study of the compounds of silica and alumina with lime reached the following conclusions : 1 (1) Lime may be combined with silica in proportion of three molecules to one and still give a product of practically constant volume and good hardening properties, though hardening very slowly. With 3| molecules of lime to one of silica the product is not sound and cracks in water. (2) Lime may be combined with alumina in the proportion of two molecules to one, giving a product which sets quickly but shows good hardening proper- ties. With 2 1 molecules of lime to one of alumina the product is unsound. Assuming that the tricalcic silicate and the dicalcic aluminate are the most basic compounds which can exist in good cement we arrive at the following formula: , SiO 2 ) + F(2CaO, A1 2 O 3 ), in which X and Y are variable quantities depending upon relative proportions of silica and alumina in materials employed. 3CaO, SiO 2 corresponds to 2.8 parts of lime by weight to 1 of silica, while 2CaO, A1 2 O 3 corresponds to 1.1 parts of lime to one of alumina. Per cent lime = Per cent silicaX2.8+Per cent aluminaXl.l. Mr. G. A. Rankin, in an extended study of the composition of Portland cement 2 finds the essential constituents to be the trical- cium silicate, 3CaO, SiO 2 ; the dicalcium silicate, 2CaO, SiO 2 ; and the tricalcium aluminate, 3CaO, A1 2 C>3. He finds that in burning Portland cement, after the carbon dioxide has been driven off, the lime combines with silica and alumina, forming first a fusible alu- minate, 5CaO, A1 2 03, and the dicalcium silicate. At higher tem- peratures these compounds unite with additional lime, forming the tricalcium aluminate and silicate. When the material is not thoroughly burned, and complete equilibrium is not reached, the clinker will contain free lime, CaO, and the aluminate, 5CaO, A1 2 C>3. Magnesia and iron oxide have little influence on the final main 1 Journal Society of Chemical Industry, Nov. 30, 1897. 2 Journal Industrial and Engineering Chemistry, June, 1915. 26 CEMENTING MATERIALS constituents of the cement, but act as fluxes and lower the temper- ature at which the reactions take place. Too high proportion of lime causes cement to be unsound through the presence of free lime. The same results are caused by under- burning or by irregular incorporation of the raw materials into the mixture. As perfect uniformity in the mixture of the ingredients is not attainable in the manufacture of cement, it is necessary that the amount of lime be somewhat less than the theoretic maximum to avoid unsoundness in the cement. The desirable proportion of lime seems to be that which will change the dicalcium silicate to tricalcium silicate as completely as possible without producing unsoundness. The ratio of silica to the sum of alumina and iron in cement materials is known as the silica ratio. It is desirable that the silica ratio be at least 2.5 or possibly 3 in Portland cement. Very little is definitely known concerning the chemical reactions which take place in the setting and hardening of cement mortars. Studies are in progress which it is hoped may throw light upon the subject and tend to more accurate knowledge of the requirements for such materials. ART. 6. SPECIFICATIONS AND TESTS FOR CEMENT 19. Standard Specifications. The specifications of the American Society for Testing Materials are now commonly recognized as standard and used in the purchase of cement in the United States. These specifications were adopted in 1904 and revised in 1908, 1909, and 1916. In specifications for construction of masonry and concrete it is usual to require that the cement meet the requirements of the American Society for Testing Materials, although in ordinary work it is not common to actually apply all the tests. The tests of chemical analysis and specific gravity are used only when special reasons exist for their application in the character of the work to which the cement is to be applied or doubt as to the material offered. It is frequently necessary, on important work, to modify the specifications to suit the peculiarities of the particular construction, This is particularly the case in work to be subjected to the action of sea water, or unusual conditions of service. The general specifications adopted in 1909 were modified in 1916 as to Portland cement only, those for natural cement being left unchanged. The Committee, however, expressed the intention of proceeding with the modification of the requirements for natural SPECIFICATIONS AND TESTS FOR CEMENT 27 cement as soon as possible, and changes may be expected in these at an early date. The methods of making the tests for Portland are to be also applied to natural cement. The 1916 specifications make some important changes from those previously used. The No. 100 sieve is dropped from the test for fineness and the requirements somewhat increased for the No. 200 sieve. The Gillmore needles are introduced as an alternate method in the test for rate of setting. Tensile tests of cement paste are dropped and sole dependence placed on the 1 to 3 mortar test, requirements for which are somewhat increased. The normal test for soundness which had previously been the final test is dropped and the steam test is made the standard. The specifications have been gradually developed through experi- ence with a number of different methods of testing which have been changed from time to time as knowledge of the material has increased and manufacturers have improved the quality of the material they are able to produce. The reliability of the cement on the market has markedly improved within a few years past and the likelihood of finding poor cement and consequently the necessity for tests under ordinary circumstances has greatly diminished. The applica- tion of tests where feasible and upon all important work is, however, desirable. - ! The specifications for Portland cement adopted in 1916 are the result of several years' work of a Joint Committee of the American Society of Civil Engineers, the U. S. Government Engineers, and the American Society for Testing Materials. They are published in the Book of Standards of the Society for Testing Materials, and are also reprinted for distribution to those interested in cement testing by the Portland Cement Association. 20. Purpose of Standard Tests. The tests imposed by the standard specifications are chemical analysis, specific gravity, fine- ness, normal consistency, time of setting, tensile strength, and soundness. Specifications covering all of these are usually employed for cement to be used in important work. The making of the tests for chemical analysis and specific gravity are often omitted when the cement proves satisfactory upon the other tests. The chemical analyses employed for Portland cement are intended to determine whether the cement has been adulterated with inert material, such as slag or ground limestone, and whether magnesia or sulphuric anhydride are present in too large amounts. The test for specific gravity when used for Portland cement is intended mainly to detect adulteration with materials of lower 28 CEMENTING MATERIALS specific gravity. It may also aid in determining the true character of the material and whether the cement is well burned. The specific gravity of Portland cement is usually between 3.10 and 3.20, that of a natural cement 2.75 to 3.10, and puzzolan cement 2.7 to 2.9. Good Portland cement may be lowered in specific gravity by long exposure to the air without serious injury to the cement. For this reason, the specifications allow a second test upon an ignited sample of cement failing upon a first test. The test for normal consistency is made to determine the proper quantity of water to be used in the paste or mortar for tests of time of setting or strength. In the preparation of paste or mortar for these tests, variations in the quantity of water used, or in the methods of mixing and molding the specimens, may produce considerable differences in results. A standard method is therefore prescribed. The time of setting is tested for the purpose of determining whether the cement is suitable for a given use, rather than as a measure of the quality of the cement. Testing for time of setting consists in arbitrarily fixing two points in the process of solidification called the initial set and the final set. This is accomplished by noting the penetration of a standard needle carrying a given weight into the mass of cement. The test for fineness is to determine whether the cement is properly ground. Only the extremely fine powder is of value as cement. The coarse parts, while having some cementing value, are practically inert when used in sand mortar. The test for tensile strength of cement pastes and mortars is made for the purpose of demonstrating that the cement contains the active elements necessary to cause it to set and harden properly. Cement is not usually subjected to tensile stresses in use, but the tensile test has commonly been employed because it offers the easiest way to determine strength, and seems to give a satisfactory means of judging the desired qualities. The proper conduct of any test for strength is a matter requiring care and experience. There are a number of points connected with the conditions and manipulation of the tests which have important effects upon the results. These are the form of the briquette, the method of mixing and molding, the amount of water used in tempering the mortar, the surroundings in which the mortar is kept during hardening, the rate and manner of applying the stress, the temperatures at which all the operations are performed. In order to secure uniform results, it is essential that the tests be standard- ized in all these particulars. SPECIFICATIONS AND TESTS FOR CEMENT 29 Soundness is the most important quality of a cement, as it means the power of the cement to resist the disintegrating influences of the atmosphere or water in which it may be placed. Unsoundness in cement may vary greatly in degree, and show itself quite dif- ferently in different material. Cement in which unsoundness is very pronounced is apt to become distorted and cracked after a few days, when small cakes are placed in water. Those in which the disintegrating action is slower may not show any change of form, but after weeks or months gradually lose coherence and soften until entirely disintegrated. The object in the tests is to accelerate the actions which tend to destroy the strength and durability of the cement. As the tests must be made in a short time, it is necessary to handle the cement in such manner as to cause these qualities to show quickly. Normal Test. The method which has been commonly employed is to make small cakes, or pats, of cement paste about 3 inches in diameter and \ inch thick at the center, with thin edges, upon a plate of glass about 4 inches square. These pats are kept twenty- four hours in moist air and then allowed to stand for twenty-eight days in water, or in the air. The pat during this period should show no signs of cracking, checking, distortion, or disintegration. This is known as the normal test, and has been relied upon as the final test for soundness. This test is defective in requiring too much time and also, in some instances, fails to discover defective material in which the action is very slow. Accelerated Tests. Numerous tests have been proposed for the purpose of hastening the hardening of the cement and causing unsoundness to show more quickly. In most of these tests, heat is employed to accelerate the changes taking place in the cement, and they are known as accelerated tests. These tests have usually been made by subjecting small pats of the cement to the action of hot water or steam and observing whether cracking or disintegration takes place. Sometimes small bars of cement are used and the increase in length of the bar meas- ured after exposure to the hot water or steam. The expansion of unsound cement should be much greater than that of sound cement. The tensile strengths of briquettes of cement which have been exposed to hot water or steam are sometimes measured and compared with the strengths of similar briquettes kept at normal temperatures. The heat should cause a considerable increase in strength of sound cement. The standard steam test consists in observing the effect of steam 30 CEMENTING MATERIALS at about 100 C. upon small pats of the cement. This test was recommended by a committee of the American Society of Civil Engineers in 1904. It has since been included in the specifications of the American Society for Testing Materials in conjunction with the normal pat test, which was the deciding test. In the modified specifications for Portland cement adopted in 1916, the normal test is discontinued and the steam test becomes the standard. The methods for making the standard tests are described in detail, with the specifications, in the Book of Standards of the American Society for Testing Materials, and in the reprint pub- lished by the Portland Cement Association. 21. Tests of Compressive Strength. Tests of compressive strength are seldom used in specifications for cement, on account of the greater ease of making the tensile test and the lighter machines that may be employed for the purpose. These tests have frequently been made for purposes of comparison or to determine special qualities of the material. The standard test piece has usually been a 2-inch cube, prepared in the same manner as the tension specimens. This was recommended by a committee of the American Society of Civil Engineers in 1909. As cement mortar is usually employed in compression, some engineers prefer to use the compression test in their specifications. A new tentative specification with methods of testing was recom- mended by a committee of the American Society for Testing Materi- als in 1916. This has not been adopted by the society as a standard, and may be further modified before such adoption. It is probable that such a standard will be adopted, to be used in conjunction with or to replace the tension test. This proposed specification with the method of making the test is given in Volume I of the Transactions of the Society for 1909. Reprints may be had from the Secretary of the Society. 22. Special Tests. The tests ordinarily employed in determin- ing the quality of cement are enumerated in the preceding sections. Other tests are frequently made to determine special qualities or for the purpose of investigating properties of cements and mortars. Transverse Strength. Tests of the strength of cement mortar under transverse loading are seldom employed as a measure of the quality of the material, but are frequently made with a view to determining the action of the material in service. Propositions have often been made to substitute the transverse for the tensile test in the reception of material. These suggestions have usually been based upon the simplicity of the test and of the apparatus SAND FOR MORTAR 31 with which it may be carried out. The specimen usually employed for this purpose is 1 inch by 1 inch and 6 inches long. It is tested by placing upon knife edges 5 inches apart and bringing the load upon the middle section. Professor Durand-Claye, from a large number of comparative tests, found the unit fiber stress under transverse load to average about 1.9 times the unit stress for tension. Adhesive Strength. The ability of cement mortar to adhere firmly to a surface with which it may be placed in contact is one of its most valuable properties and quite as important as the develop- ment of cohesive strength. Tests for adhesive strength are not employed as a measure of quality, . because of the uncertain char- acter of the test and the difficulty of so conducting it as to make it a reliable indication of value. The adhesive properties of the cement are to a certain extent called into play in tests of sand mor- tar, and may be inferred from comparison of neat and sand tests. Experiments upon the adhesion of mortars to various substances are sometimes made, both for the purpose of comparing the cements or methods of use, and to study the relative adhesions to various kinds of surfaces. Such experiments are quite desirable with a view to the extension of knowledge of this very important quality. The common method of making this test is to prepare briquettes of which one half the briquette is of cement paste or mortar and the other half a block of stone, glass, or other material to be used. The cement half is made in the ordinary form for tensile specimens. The other half is made to fit the cement mold at the middle and arranged at the end to be held by a clip in the testing machine. ART. 7. SAND FOR MORTAR 23. Quality of Sand. As hydraulic cement is commonly mixed with certain proportions of sand, when used in construction, the nature and quality of sand used, and the method of manipulating the materials in forming the mortar have quite as important an effect upon the final strength of the work as the quality of the cement itself. In testing cement a standard sand is employed. This sand may be obtained quite uniform in quality. In the execution of work, however, local sand must generally be used; this varies widely in character, and should always be carefully considered upon any work where the development of strength and lasting qualities are of importance. Size of Sand Grains. It is usual to class as sand all material 32 CEMENTING MATERIALS less than J-inch diameter; pieces larger than this being classed as gravel. Coarse sand is superior to fine sand for use in cement mortar. Coarse sand presents less surface to be coated with cement and the interstices are more easily filled with cement paste. Fine sand requires more water in mixing to the same consistency, and gives usually weaker and more porous mortar than coarse sand. The use of a mixture of grains of different sizes is usually desir- able, giving less voids to be filled by the cement; and it is frequently found, when the cement is not in considerable excess, that .the strength obtained by such a mixture is much greater than is given by either the large or small grains alone. Sand of mixed sizes, giving a minimum of voids, requires less cement to make a mortar of maximum density and strength than that of more uniform sizes. Shape of Grains. Sand with angular grains usually gives better results in mortar than that with rounded grains, and specifications frequently call for sharp sand. This difference is, however, much less important than that of proper gradation of sizes, and should not be given undue weight in the selection of sand for use in mortar. Stone Screenings. The screenings from crushed stone are fre- quently used in place of natural sand. Ordinarily screenings from stone of good quality give mortar of rather better strength than natural sand. This, however, depends in most instances upon the gradation of sizes in the two materials. The sharpness of grain is favorable to the screenings, and the presence of a certain amount of very fine stone dust in the screenings seems to be of value in the mortar. When the screenings are derived from soft rock, the dust may be present in too large amount and need to be screened out before the screenings can be successfully used. Chemical Composition. Sands as commonly used for mortar are composed mainly of silica. In most cases, sand which has a proper granulometric composition is satisfactory for use. The failure of concrete work has, however, in a number of instances been found to be due to the use of sand low in silica. Sand containing less than 95 per cent silica needs to be carefully tested before being used, although some sands as low as 75 per cent silica have given good results. The composition of sands have not been sufficiently studied to determine the differences of composition which cause failure in one case and success in another. The presence of mica in sand or screenings is supposed to injuri- ously affect the strength of mortar in which the material is used. The results of experiments upon the effect of mica are not conclusive, although they seem to indicate that mica may sometimes be injuri- SAND FOR MORTAR 33 ous. Sand containing mica should be carefully tested before being used. Effect of Impurities. Sand for use in mortar should be clean, and as free from loam, mud, or organic matter as possible. In general the presence of any foreign matter is to be avoided, though a small amount of fine clay distributed through sand has sometimes been found to increase the strength of cement mortar, and also helps to make the mortar work more smoothly, sometimes decreasing its permeability. The effect of the clay depends upon the character of the sand and upon the richness of the mortar. Fine clay may help to fill the voids in an otherwise porous mortar with good effect, but may be deleterious in a rich mortar, or when it is not finely divided and uniformly distributed through the sand. In a particular instance, the effect of such an adulteration can be judged only by testing it. Impurities of an organic nature are always objectionable in sand for use in mortar. When it is necessary to use sand containing such impurities, it should be carefully washed and tested. A very small amount of vegetable matter in sand has sometimes caused the failure of mortar to harden properly. Selection of Sand. Sands differ so greatly in their qualities that it is difficult by mere inspection of the materials to judge of their relative values for use in mortar. In choosing sand for use in impor- tant work, it is desirable not only to determine fully the physical characteristics of the available materials, but also to make actual tests of mortar by their use. 24. Tests for Sand. Tests intended to determine the mortar- making qualities of sand may be made in three ways: 1. Mechanical analysis of the sand, with determination of voids in the sand. 2. Density tests of mortars made from the sand with the cement to be used in the work. 3. Strength tests of mortars made from the sand in question with the cement to be used hi the work. The value of sand depends mainly upon its granulometric com- position. The sand which, mixed with a given proportion of cement, gives the most dense mortar yields the strongest mortar. The sand which requires the least cement to make a mortar of maximum density is the most economical sand, when the mortar is properly proportioned. The purpose in testing the sand should be to determine the pro- portions of cement to sand necessary as well as to choose the best sand. 34 CEMENTING MATERIALS 25. Mechanical Analysis. To determine the relative sizes of grains composing sand, the material is screened through a series of sieves of varying degrees of fineness. The sieves are made of standard size, 8 inches in diameter by 2J inches high, those with openings smaller than ^ inch being made of woven brass wire, while the larger sizes are preferably drilled circular openings in sheet brass. These sieves are designated by numbers corresponding to the number of meshes to the linear inch, the size of opening depending upon the diameter of wire used. The size openings usually employed for sand analysis are approximately as follows : No. of Sieve 4 10 20 30 40 50 80 100 200 Size Opening, in. .. 0.25 .073 .0335 .0195 .015 .011 .0067 .0055 .00265 For ordinary examination of sand, when comparing or selecting sand for use, it is unnecessary to separate into so many sizes, and sieves Nos. 4, 10, 20, 50, and 100 are commonly employed. The sieves are made to fit together in nests with a cover and tight bottom to catch the residue from the finest sieves. The sifting may be done by hand, by shaking and jarring the sieves, or mechanical shakers may be used. These may be obtained to work by hand or with small electric motors attached. In making the tests, a sample weighing 50 g. is dried to constant weight at temperature not more than 110 C. (230 F.) and is then sifted through the sieves, so as to separate the grains into various sizes and determine the percentage of each by weight. The material properly classed as sand is that which passes through the No. 4 sieve and is retained on the No. 100 sieve. Sand retained by the No. 10 or No. 20 sieve may be classed as coarse sand; that caught between the No. 20 and No. 50 sieves is medium sand; that which passes the No. 50 sieve is fine sand. Material passing the No. 100 sieve is called dust. Analysis Curves. Comparisons of the granulometric composi- tions of sands are readily made by plotting the results of the sieve analysis as curves. It is usual to plot the sizes of openings as abscissae and percentages passing each size as ordinates. The reciprocals of the numbers of the sieves may be used for size without impairing the value of the results, and probably represent more nearly the actual sizes of grains passing the sieves than does the computed width of opening. Table II gives the results of analyses of sands in common use for mortar, showing something of the variations which may frequently occur. These results are plotted in Fig. 1. Sand No. 1 is a coarse bank SAND FOR MORTAR 35 sand containing a small amount of clay. No. 2 is a medium river sand of good quality. No. 3 is a fine sand. No. 4 is screenings from broken limestone, containing rather high percentage of dust. TABLE II. ANALYSES OF SANDS PERCENTAGES PASSING SIEVES. Sieve No. Sand. Screenings. 1 2 3 4 4 100 100 100 100 10 57.82 85.18 99.82 95.07 20 34.96 56.82 99.42 74.01 30 10.00 33.93 97.33 59.68 40 7.67 20.02 83.74 49.23 50 5.82 13.07 35.13 41.91 80 3.69 7.44 2.27 30.97 100 3.01 5.18 0.96 28.24 200 1.73 0.38 0.65 17.71 100 .Ol .02. .03 .04, .05 .06 .07 .08 .09 .10 .11 .IE 51ZEOF OPENING IN INCHE5 FIG. 1. Analyses of Sands. 36 CEMENTING MATERIALS 26. Determination of Voids. The method most commonly used for void determination is known as the wet method, which consists in filling a measure with the sand to be tested and pouring in water until the voids are completely filled. The volume of water required to fill the voids divided by the volume of sand and multiplied by 100 is the percentage of voids; or the weight of water poured into the sand divided by the weight of water required to fill the measure and multiplied by 100 is the percentage of voids. It is very dif- ficult to eliminate completely the air from the sand in making this test. The test is therefore liable to considerable error unless great care be used in manipulating it. Dry Method. A more accurate method of determining voids is to compare the weight of a measured volume of the sand with the weight of an equal volume of the solid material of which the sand is composed. In measuring the volume of sand, it is necessary to use care to secure the proper degree of compactness. For ordi- nary comparisons the sand should be well compacted by shaking and jarring the measure. The weight of the solid rock is obtained by multiplying the weight of an equal volume of water by the specific gravity of the sand. The difference between the weight of the rock and that of the* sand divided by the weight of the rock and multiplied by 100 is the percentage of voids. If R is the weight of the solid rock and S, the weight of the sand, percentage of voids is This test supposes the sand to be dry. When it is desired to obtain the voids in moist sand, a weighed sample of the sand should be dried at 212 F. and the loss of weight determined. The weight of moisture in the measure of sand to be used in the test may then be computed. This weight is then to be subtracted from the total weight of the moist sand to find the weight of solid material in the sand. If m is the weight of moisture in the volume of sand under test, percentage of voids is R-(S-m). R 100. 27. Specific Gravity. The specific gravity of siliceous sand is quite uniformly 2.65, or the weight per cubic foot of the solid rock is 165 pounds. To assume these values in determining the voids SAND FOR MORTAR 37 in such sand involves slight error in any case. Sands not strictly siliceous may vary in specific gravity from about 2.6 to 2.7. The determination of specific gravity is made by immersing a sample of the material in water at 68 F. and dividing the weight of the sand by the weight of water displaced. This is most con- veniently done by sifting the sand into the water in a graduated glass tube, and reading the increase of volume of the liquid in the tube. Care must be used to introduce the sand slowly so as to eliminate all air bubbles. 28. Density Test. Comparative tests of sands may be made by determining the volume of mortar produced by definite weights of cement and dry sand. The sand that for a given weight of materi- als, when mixed with the same proportion of cement to the required consistency, produces the smallest volume of mortar gives the most dense mortar. In making this test, molds in which the height is large in comparison with the section are convenient, the relative heights to which the mold is filled giving the proportionate volumes. The volume of mortar after setting is what is required, but the measurement before setting, unless the mortar is quite wet, will give practically the same result. Determination of Density. The term density, as commonly applied to mortar, means the ratio of the volume of solid materials contained in the mortar to the whole volume of mortar. The density is obtained by weighing the ingredients before mixing and calculating their solid volumes from these weights and their specific gravities. The weight and volume of the resulting mortar are then measured. The weight of mortar should equal the sum of the weights of the several ingredients. The density equals the sum of the solid vol- umes of sand and cement divided by the measured volume of the mortar. The density of mortars made from the sands shown in Fig. 1, one part cement to three parts sand by volume, are as follows : WEIGHTS USED, GRAMS. Mortar Sand No. Cement. Sand. Water. c.c. 1 358 1026 178 670 0.75 2 358 1128 163 735 0.73 3 358 972 180 730 0.66 4 358 1122 268 790 0.68 38 CEMENTING MATERIALS The method of computation is as follows: Taking specific gravity of cement as 3.1 and specific gravity of sand as 2.65, Density of No. 1 is 358 1026 3.1 2 . 65 _ 670 29. Strength Tests. Tests of the strength of mortars made from sands are the most conclusive evidence of the mortar-making properties of the sands. These tests to be of real value should extend over a period of at least twenty-eight days. They are made in the same manner as the mortar tests for judging cement, and comparisons are sometimes made with the results of tests with standard sand. Table III gives comparative results of tests of the sands shown in Fig. 1. TABLE III. RESULTS OF SAND TESTS Sand No. PACKED SAND. Density, 1 :3 Mortal . TENSILE STRENGTH. Per Cent Voids. Weight, Cu. Ft. 1 : 2 Mortar. 1 : 3 Mortar. 28 Days. 6 Months. 28 Days. 6 Months. 1 2 3 4 Standard. . 36.1 28.6 38.0 28.7 106.4 117.5 100.8 116.7 .75 .73 .66 .68 523 379 223 396 326 603 493 343 567 477 443 253 153 304 268 495 339 265 503 318 30. Washing Test. When it is necessary to examine sand for organic impurities, the silt may be removed from the sand by washing. This is done by shaking a sample of the sand in a bottle with water, letting it settle for a few seconds, and then pouring off the turbid water. This is done repeatedly until the suspended matter is all removed. The wash water is then evaporated, and the amount of silt determined. The silt is ignited in a platinum crucible and the loss on ignition is the percentage of organic matter present. A very small amount, not more than 1 per cent, of organic matter may be a serious detriment, sand containing such impurities should be carefully tested and may need to be washed in order to give satis- factory results in use. CEMENT MORTAR 39 31. Specifications for Sand. Tests have seldom been used as means of judging sand for use in masonry construction. The require- ments have usually been that the sand be coarse, clean, and sharp; the requirement of sharpness is now commonly omitted. Mechanical analysis and void tests are frequently made for the purpose of judging the qualities of available sands on important work, and to aid in properly proportioning mortar, but such tests are not usual in specifications. The Joint Committee of the Engineering Societies on Concrete and Reinforced Concrete has suggested the following as requirements for sand to be used as fine aggregate in concrete work: (a) Fine Aggregate. This should consist of sand, crushed stone or gravel screenings, graded from fine to coarse, and passing when dry a screen having holes i inch in diameter. It is preferable that it be of siliceous material, and should be clean, coarse, free from dust, soft particles, vegetable loam, or other deleterious matter; and not more than 6 per cent should pass a sieve having 100 mashes per linear inch. Fine aggregates should always be tested. Fine aggregates should be of such quality that mortar composed of one part Portland cement and three parts fine aggr gates by weight, when made into briquettes, will show a tensile strength at least equal to the strength of 1 to 3 mortar of the same consistency made with the same cement and standard Ottawa sand. If the aggregate be of poorer quality, the proportion of cement should be increased to secure the desired strength. If the strength developed by the aggregate in the 1 to 3 mortar is less than 70 per cent of the strength of the Ottawa sand mortar, the material should be rejected. To avoid the removal of any coating on the grains, which may effect the strength, bank sand should not be dried before being made into mortar, but should contain natural moisture. The percentage of moisture may be determined on a separate sample for correcting weight. From 10 to 40 per cent more water may be required in mixing bank or artificial sands than for standard Ottawa sand to produce the same consistency. ART. 8. CEMENT MORTAR 32. Proportioning Mortar. In specifying the proportions of ingredients for cement mortar to be used in construction, it is usual to give the ratio of parts of cement to those of sand by volume The relative proportions of sand and cement to be used in any instance depend upon the nature of the work and the necessity for developing strength or water-tightness in the mortrr. The pro- portions commonly used in ordinary work are: for natural cement, one part cement to one part or two parts sand; for Portland cement, one part cement to two parts or three parts of sand. In common practice these ratios are chosen without reference to the particular materials used and the resulting mortars vary widely in character. 40 CEMENTING MATERIALS Good sand in a i to 3 mortar frequently shows greater strength than a poorer one mixed 1 to 2, and gives equally good results in use. The methods of measuring materials also vary, and the relative quantities of cement and sand in the mortar differ correspondingly. Measuring Cement. Cement should always be measured by weight, on account of the variation in volume of the same quantity of cement with different degrees of compactness. In specifying proportions by volume, therefore, it is always desirable to state the weight of cement to be taken as unit volume. Portland cement is usually packed in wooden barrels or in canvas bags. A barrel of cement contains 376 pounds of cement, while a bag contains 94 pounds, or one-quarter barrel. Natural cement is ordinarily packed in barrels of 282 pounds, or bags of 94 pounds (one-third barrel) each. Portland cement as packed in barrels weighs a little more than 100 pounds per cubic foot. A cubic foot of cement paste requires from 95 to 110 pounds of cement. It is common to consider a cubic foot of Portland cement to weigh 94 pounds in porportioning mortar. A bag of cement is then mixed with 2 cubic feet of sand to form 1 to 2 mortar, or with 3 cubic feet of sand to form 1 to 3 mortar. This assumes the volume of a barrel of cement to be 4 cubic feet. This is the recommendation of the Joint Committee of the Engineer- ing Societies. Some engineers use 3.8 cubic feet as the volume of a barrel, or 100 pounds as the weight of a cubic foot. In the same way, 70 pounds is frequently used as the weight of a cubic foot of natural cement. This makes the volume of a sack of natural cement 1J cubic feet. A barrel of natural cement would then have the same nominal volume as a barrel of Portland, 4 cubic feet. The actual volume-weight of natural cement varies con- siderably for different brands. Measuring Sand. It is usual to measure sand by volume. The method of measuring to be used in any particular instance depends upon the method of mixing and handling the mortar. Very com- monly the measuring is done in the barrow or bucket in which the sand is carried to the mixer or platform. Measuring boxes with- out bottoms are often employed to set on the mixing platform, and after filling are removed, leaving the measure of sand. Whatever method of handling the sand is employed, it is important that care- ful attention be given to securing the correct proportion of sand for the mortar. Effect of Moisture. In proportioning mortar by volume, the CEMENT MORTAR 41 moisture content of the sand may be a matter of importance. Damp sand weighs less per unit volume than dry sand. When sand is moistened with a small quantity of water, the grains of sand are coated with a thin film of water, which separates the grains, causing the sand to occupy more space than when dry. When the amount of water becomes sufficient to coat all the grains of sand (about 4 to 7 per cent with ordinary sands), a maximum effect is reached, and an increase in amount of water beyond that point causes a reduc- tion of volume. At saturation (10 to 20 per cent of water), it becomes slightly less in volume than when dry. The solid content in a given volume of moist sand is less than that of the same volume of dry sand, and a mortar mixed with the moist sand will be richer in cement than that mixed with the same sand when dry. This effect is greater with fine then with coarse sand. A given volume of sand measured dry may contain 10 per cent to 15 per cent more solid material than the same volume of the same sand measured in a moist condition. The extent to which differences in moisture condition may effect the volume of the sand depends upon the position in which the sand is placed and the way it is handled in measuring. If dry sand in a bin, or a pile, be moistened with a small quantity of water, the sand will not appreciably swell in the pile, as the particles are held by the weight of the mass above they are not free to move and the water fails to separate them. If the sand be loosened in moving to a new position, it will be found to have increased in volume and will not return to its former dimensions until it has become dry, or wet to saturation. Proportioning by Weight. In Germany it has been quite common to measure the material for mortar by weight. This has been applied in some instances in the United States, and reduces largely the vari- ations in the proportions due to moisture. On important work it may frequently be possible to arrange for weight measurement without materially increasing the cost of handling the material. The ratio of cement to sand is commonly arbitrarily fixed with reference to the particular use to which the mortar is to be put, without considering the character of the sand to be used. For ordinary masonry, or massive concrete, Portland cement is usually employed in 1 to 3 mixtures. When high strength is needed, as in reinforced concrete work, the mixture is 1 to 2. Under specially trying conditions, or sometimes when cement grout is being used, a 1 to 1 mixture may be employed. With natural cement, the mix- tures are 1 to 2 for ordinary work and 1 to 1 where greater strength 42 CEMENTING MATERIALS is needed. Natural cement is not used for reinforced concrete work. The choice of ratios has usually been well on the side of safety, and good results have been obtained in practice by this method, although equally good work at less cost might in many instances have been obtained by more careful study of the materials in proportioning the ingredients of the mortar. In comparing the mortar-making qualities of various sands, it is found that the amount of cement necessary to make mortar of the same strength from different sands depends mainly upon the fineness and density of the sands. The office of the cement paste in mortar is to coat the grains of sand and fill the voids between them. In fine sand the surface to be coated with cement is greater than in coarse sand. Dense sand, with grains of varying sizes, presents less voids to be filled than more uniform sand. It is desirable that careful study be given to the sands to be used in any important work before finally deciding upon the proportions of the materials, and that final judgment be based upon actual tests of the mortar itself. Frequently a mixture of a fine with a coarse sand, or of crusher dust with sand may be so proportioned as to give economical results in the saving of cement, while at the same time improving the mortar. 33. Mixing Mortar. In mixing mortar by hand a water-tight box or platform is used. The required quantity of sand is spread over the floor of the box and the cement distributed evenly over the sand. The cement and sand are then mixed together with a hoe or shovel until the cement is uniformly distributed through the sand, as shown by the even color of the mixture diy. It is important that the dry materials be very thoroughly mixed before water is added. A uniform mixture will not otherwise be obtained. When the mixing of dry materials is complete, water is added and the mass worked into a stiff paste. The quality of the mortar is materially affected by the vigor with which it is worked in bring- ing it to the proper consistency. After the water has been absorbed by the cement, vigorous working will make the mass more plastic, and working should continue until a permanent condition is reached. Quantity of Water. The quantity of water to be used in mixing mortar can be determined only by experiment in each instance it depending upon the nature of the cement and sand, and the pro- portion of cement to sand. The quantity of water used should be the least consistent with reducing the mortar to the required condi- tion of plasticity by vigorous working. Additional water should not be used to save labor in working. CEMENT MORTAR 43 Mixing should be quickly and energetically done, only such quantity being mixed at once as can be used before initial set takes place. A considerable quantity is sometimes mixed dry and left to stand until needed before adding water. If this is done with damp sand, the cement may be acted upon by the moisture in the sand to the injury of the mortar. Quick-setting cements are par- ticularly liable to injury from this cause. Retempering. Masons frequently mix mortar in considerable quantities, and if the mass becomes stiffened before being used, add more water and work again to plastic condition. After the second tempering the cement is much less active than at first, and remains a longer time in a workable condition. This practice is not approved by engineers and is not permitted in good engineering construction, although there is some dispute as to the extent of the injurious effects. Cement when retempered becomes very slow in action, both in setting and hardening. The quicker-setting cements are usually more affected than the slow setting. The strength during the earlier periods of hardening is lessened, although the final strength may not be impaired. Portland cement may ordinarily be used for two, or sometimes three hours after mixing without appreciably affecting its action. When retempered after a longer period it will usually become slower in action, but may in some cases gain as much strength in thirty to sixty days. Continuous working materially improves the strength of mortar, and when allowed to stand after mixing it should be frequently worked. Grout. Mortar when made thin, so that it can be poured into cracks or small openings, is known as grout. Mixtures of cement and sand used in this manner are difficult to handle without sepa- ration of the materials. They should be used only under excep- tional circumstances and when stiffer mortar cannot be applied. 34. Yield of Mortar. The volume of mortar formed by mixing given quantities of cement and sand depends mainly upon the den- sities of the materials. It is affected by the method of preparing the mortar, the uniformity of the mixture, and the degree of com- pactness. The net volume of materials entering into the com- position of mortar is readily found from their weights and densities, but it represents only approximately the resulting volume. An accurate knowledge of the yield of any particular mixture is to be obtained only by experimenting upon the materials to be employed. The amount of cement paste made by a given weight of cement 44 CEMENTING MATERIALS powder varies with the specific gravity of the cement and the amount of water necessary in gaging. The lighter cements require more water and yield less paste for a given volume of cement than the heavier ones. To form a cubic foot of plastic paste requires usually from 80 to 95 pounds of natural cement, while from 95 to 101 pounds of Portland cement are necessary. Table IV gives approximate quantities of materials ordinarily required for 1 cubic. yard of compact plastic mortar. A barrel of cement is taken as 4 cubic feet, corresponding to a weight of 94 pounds per cubic foot for Portland cement and 70 pounds for natural cement. The sand is dry and measured loose. TABLE IV. MATERIALS FOR 1 CUBIC YARD OF MORTAR PROPORTIONS. QUANTITY OF SAND TO 1 SACK CEMENT. MATERIALS FOR 1 Cu. YD. COMPACT, PLASTIC MORTAR. Cement. Sand. Portland, Cu. Ft. Natural, Cu. Ft. Cement, Barrels. Sand, Cu. Yds. 1 6.75 to 7. 85 1 1 1.0 1.3 4.25 to 4. 75 0.63 to 0.70 1 2 2.0 2.7 2.95 to 3. 15 0.87 to 0.93 1 3 3.0 4.0 2. 20 to 2. 37 0.98 to 1.04 1 4 4.0 5.3 1.75 to 1.85 1.03 to 1.09 The differences in quantities are mainly due to variations in the fineness of the sand, in the amount of moisture contained by the sand, and in the compactness given to the mortar. Less materials are required when using fine than when using coarse sand; more materials are required when the sand is moist than when it is dry. The compactness of the mortar is affected by the quantity of water used in mixing and the method of placing the mortar. 35. Mixtures of Lime and Cement. The addition of slaked or hydrated lime to cement mortar causes the mortar to work more smoothly, and makes it easier and more economical to handle in masonry construction. A lean cement mortar may be improved in density and strength by the addition of a small quantity of lime paste. Lime in larger quantities, or lime added to rich mortar, diminishes the strength of the mortar but may sometimes be economical, through cheapen- ing the mortar and improving its working qualities, when high strength is not of special importance. Lime may be used with cement either by mixing lime paste with CEMENT MORTAR 45 cement mortar, or by mixing dry hydrated lime with cement before mixing the mortar. Lime must always be thoroughly slaked before mixing with cement, as unhydrated lime in cement mortar is always a detriment. It is also essential that the mixture be very uniform, and that the mortar be worked to an even color. For this reason, the use of dry hydrated lime is to be preferred over lime paste. In proportioning lime to cement, the method of measurement is important. Hydrated lime from nearly pure limestone contains about 75 per cent of quicklime and ordinary lime paste contains about 40 per cent of lime by weight. About 25 pounds of quick- lime are required to make a cubic foot of lime paste. Experiments upon mixtures of lime and cement show that 10 to 15 per cent of lime (measured as unslaked lime) may be substi- tuted for an equal weight of cement in a 1 to 3 cement mortar with- out sensibly decreasing the strength of the mortar. In some instances when not more than 10 per cent of lime is used the strength is increased and the mortar made more dense. As the proportion of lime is increased the strength of the mortar is lessened. For mortars leaner than 1 to 3 of Portland cement the use of a small amount of lime is usually an advantage. With some natural cements, lime may be used to replace cement to the extent of 25 to 30 per cent of the weight of the cement with- out appreciable loss of strength in the mortar. Cement so treated becomes slower in action and is longer in gaining strength than when used without lime. Mixtures of this kind with either Portland or natural cement are frequently used in mortar for ordinary building operations. Hydrated lime is sometimes added to cement for the purpose of rendering the mortar less permeable where water-tight work is needed, and is also sometimes added to Portland cement concrete in small quantity to make the concrete flow more readily in filling the forms. 36. Strength of Cement Mortar. The strength of cement mortar is dependent upon the quality and proportions of cement and sand; the quantity of water used in gaging; the method of mixing and thoroughness of working; the temperature and moisture conditions under which it is kept during hardening; the age of the mortar. The effect upon tensile strength of varying proportions of cement and sand is shown in Fig. 2, which gives the relative strengths for an average Portland cement, or cement paste, and mortars with standard sand, for a period of one year after mixing. Individual cements may vary quite widely from the curves shown. Some gain strength more slowly at first and continue to gain for a longer 46 CEMENTING MATERIALS period. Others have greater early strength and show more loss of strength during the period of retrogression. Nearly all Portland cements after gaining strength rapidly for a time reach a maximum and then lose strength for a period. This loss of strength is usually regained later. It seldom occurs in less than three months or more than one year after the mortar is mixed. Cement which gains strength very rapidly and has high early strength is apt to suffer greater loss of strength later than cements 1000 I a 3 A- 5 'v"S* r -\* Jtelfe-v krf&^ FIG. 30. Methods of Finishing Joints. The best pointing mortar is usually composed of Portland cement and sand, 1 to 1. Coloring matter is added when needed. The mortar is used quite dry, like damp earth. When the face of the stone would be stained by Portland cement, a putty made of lime, plaster of paris, and white lead is sometimes employed. Various non-staining cements are also available. 49. Trimmings. In the erection of masonry structures, certain special parts are ordinarily required to be of cut stone, which must be of definite form and dimension. These trimmings have to do with the ornamentation of the structure, finishing about openings, or joining different types of construction. Water-tables with sloping surfaces are used at the top of founda- tion walls, where they join the narrower upper walls. Copings, cornices, window sills, and sometimes belt-courses pro- ject beyond the surface of the wall; they must have sufficient width to be firmly held in the wall, and to balance on the wall in laying. The projections should also have upper surfaces which slope away from the wall, and a drip (called the wash) underneath to cause water to drop off at the outer edge, the drip being made by cutting a groove on the under side of the stone. When cut-stone trimmings are used for a brick wall, they should be dimensioned so that they will fit into the brickwork without splitting the brick. 76 STONE MASONRY Window sills just the width of the opening and not built into the wall at the ends are called slip sills, while those extending into the walls are called lug sills. The ends of lug sills are rectangular, the sloping surface of the sill being made the width of the opening. Lug sills should be bedded only at the ends to prevent cross-bending stresses due to the weight of the wall. When stone lintels are used to span openings, care must be taken in selecting the stone, and making sure that it has the transverse strength necessary to carry the load. When necessary an angle bar or I-beam may be used to support the lintel, a recess being cut into the back of the stone for this purpose. 50. Specifications for Stone Masonry. The following general requirements for stone masonry and special requirements for bridge and retaining wall masonry are recommended by the American Railway Engineers' Association in their Manual for 1915: GENERAL REQUIREMENTS Stone. 3. Stone shall be of the kinds designated and shall be hard and durable, of approved quality and shape, free from seams or other imperfections. Unseasoned stone shall not be used where liable to injury by frost. Dressing. 4. Dressing shall be the best of the kind specified. 5. Beds and joints or builds shall be square with each other, and dressed true and out of wind. Hollow beds shall not be permitted. 6. Stone shall be dressed for laying on the natural bed. In all cases the bed shall not be less than the rise. 7. Marginal drafts shall be neat and accurate. 8. Pitching shall be done to' true lines and exact batter. Mortar. 9. Mortar shall be mixed in a suitable box, or in a machine mixer, preferably of the batch type, and shall be kept free from foreign matter. The size of the batch and the proportions and the consistency shall be as directed by the engineer. When mixed by hand the sand and cement shall be mixed dry, the requisite amount of water then added and the mixing continued until the cement is uniformly distributed and the mass is uniform in color and homo- geneous. Laying. 10. The arrangement of courses and bond shall be as indicated on the drawings, or as directed by the engineer. Stone shall be laid to exact lines and levels, to give the required bond and thickness of mortar in beds and joints. 11. Stone shall be cleansed and dampened before laying. 12. Stone shall be well bonded, laid on its natural bed and solidly settled, into place in a full bed of mortar. 13. Stone shall not be dropped or slid over the wall, but shall be placed without jarring stone already laid. 14. Heavy hammering shall not be allowed on the wall after a course is laid. 15. Stone becoming loose after the mortar is set shall be relaid with fresh mortar. WALLS OF STONE MASONRY 77 16. Stone shall not be laid in freezing weather, unless directed by the En- gineer. If laid, it shall be freed from ice, snow, or frost by warming. The sand and water used in the mortar shall be heated. 17. With precaution, a brine maybe substituted for the heating of the mor- tar. The brine shall consist of 1 pound of salt to 18 gallons of water, when the temperature is 32 F.; for every degree of temperature below 32 F., 1 ounce of salt shall be added. 18. Before the mortar has set in beds and joints, it shall be removed to a depth of not less than 1 inch. Pointing shall not be done until the wall is com- plete and mortar set; nor when frost is in the stone. 19. Mortar for pointing shall consist of equal parts of sand, sieved to meet the requirements, and Portland cement. In pointing, the joints shall be wet, and filled with mortar, pounded in with a "set-in" or calking tool and finished with a beading tool the width of the joint, used with a straight-edge. BRIDGE AND RETAINING WALL MASONRY, ASHLAR STONE Bridge and Retaining Wall Masonry, Ashlar Stone. 20. The stone shall be large and well proportioned. Courses shall not be less than 14 niches or more than 30 inches thick, thickness of courses to diminish regularly from bottom to top. Dressing. 21. Beds and joints or builds of face stone shall be fine-pointed, so that the mortar layer shall not be more than J inch thick when the stone is laid. 22. Joints in face stone shall be full to the square for a depth equal to at least onte-half the height of the course, but in no case less than 12 inches. Face or Surface. 23. Exposed surfaces of the face stone shall be rock-faced, with edges pitched to the true lines and exact batter. The face shall not pro- ject more than 3 inches beyond the pitch lines. 24. Chisel drafts 1 J inches wide shall be cut at exterior corners. 25. Holes for stone hooks shall not be permitted to show in exposed surfaces. Stone shall be handled with clamps, keys, lewis, or dowels. Stretchers. 26. Stretchers shall not be less than 4 feet long with at least one and a quarter times as much bed as thickness of course. Headers. 27. Headers shall not be less than 4 feet long; shall occupy one- fifth of face of wall; shall not be less than 18 inches wide in face; and where the course is more than 18 inches high, width of face shall not be less than height of course. 28. Headers shall hold in heart of wall the same size shown in face, so arranged that a header in a superior course shall not be laid over a joint, and a joint shall not occur over a header; the same disposition shall occur in back of wall. 29. Headers in face and back of wall shall interlock when thickness of wall will admit. 30. Where the wall is 3 feet thick or less, the face stone shall pass entirely through. Backing shall not be permitted Backing. 31a. Backing shall be large, well-shaped stone, roughly bedded and jointed; bed joints shall not exceed 1 inch. At least one-half of the back- ing stone shall be of the same size and character as the face stone and with parallel ends. The vertical joints in back of wall shall not exceed 2 inches. The interior vertical joints shall not exceed 6 inches. 78 STONE MASONRY Voids shall be throughly filled with concrete, or with spalls, fully bedded in cement mortar. 316. Backing shall be of concrete, or of headers and stretchers, as specified in paragraphs 26 and 27, and heart of wall filled with concrete. Paragraphs 31a and 316 are so arranged that either may be eliminated accord- ing to requirements. 32. Where the wall will not admit of such arrangement, stone not less than 4 feet long shall be placed transversely in heart of wall to bond the opposite sides. 33. Where stone is backed with two courses, neither course shall be less than 8 inches thick. Bond. Bond of stone in face, back, and heart of wall shall not be less than 12 inches. Backing shall be laid to break joints with the face stone and with one another. Coping. 35. Coping stone shall be full size throughout, of dimensions indi- cated on the drawings. 36. Beds, joints and top shall be fine-pointed. 37. Location of joints shall be determined by the position of the bed plates as indicated on the drawings. Locks. 38. Where required, coping stone, stone in the wings of abutments, and stone on piers, shall be secured together with iron cramps or dowels, to the position indicated on the drawings. BRIDGE AND RETAINING WALL MASONRY, RUBBLE STONE 39. The stone shall be roughly squared and laid in irregular courses. Beds shall be parallel, roughly dressed, and the stone laid horizontal to the wall. Face joints shall not be more than 1 inch thick. Bottom stone shall be large, selected flat stone. 40. The wall shall be compactly laid, having at least one-fifth the surface of back and face headers arranged to interlock, having all voids in the heart of the wall thoroughly filled with concrete, or with suitable stones and spalls, fully bedded in cement mortar. ART. 13. STRENGTH OF STONE MASONRY 51. Compressive Strength. Stone masonry varies widely in strength according to the character of the construction. The accuracy with which the joints are dressed, the strength of the mor- tar, the bonding of the masonry and size of blocks of stone are more important than the strength of the stone itself. No experimental data are available which show the actual strength of masonry as used. The mortar has usually much less strength than the stone, and in some experiments on brick piers, the mortar seemed to squeeze out, causing the failure of the brick in tension. The loads to which masonry is ordinarily subjected are much less than its actual strength, but when heavy loads are being carried by piers or arches, it is frequently necessary to proportion the sec- tion to the load. STRENGTH OF STONE MASONRY 79 When the masonry is of cut stone with thin joints and Portland cement mortar, the strength of the masonry may be proportioned to the strength of the stone. For rubble with thick joints, the strength of the stone has no material effect upon the strength of the masonry. The loads used in practice vary quite widely according to the views of the designers. Building laws of the various cities differ considerably in the loads allowed. The following may be considered as conservative values for the limits of safe loading: Cut Stone. Dressed stone, with joints not more than f inch in first class Portland cement mortar: Tons per Square Foot. Granite 50 to 60 Hard limestone or marble 35 to 40 Sandstone 25 to 30 The siliceous sandstones may have larger values, while the soft limestones should be reduced. For ashlar of good quality as commonly laid with J-inch joints in Portland cement: Tons per Square Foot. Granite 40 to 45 Limestone, hard 35 to 40 Sandstone 25 to 30 Rubble. For masonry composed of large blocks of squared stone, 1-inch joints, in Portland cement mortar: Tons per Square Foot. Sandstones or limestones 10 to 20 Granite 20 to 30 Uncoursed rubble: In cement mortar 5 to 8 In lime mortar 3 to 5 For an ashlar pier whose height exceeds ten times, or a rubble pier whose height exceeds five times, its least lateral dimension, these figures should be reduced. Piers of small dimensions carrying heavy loads should always be of ashlar. Rubble should not be used for less thicknesses than 20 to 24 inches when it is necessary to develop the full strength of the masonry. Failures of masonry most frequently occur through defective foundation or workmanship. Masonry, to develop its full strength, 80 STONE MASONRY must always be adequately supported, so that unequal pressures are not produced through settlement. Weight of Masonry. In determining loads^ fit is usually necessary to estimate the weight of masonry. This depends upon the specific gravity of the stone and the closeness of the joints. The following table gives approximate weights for the different classes of stone masonry: Pounds per Cubic Foot. Limestone, ashlar 155 to 165 Limestone, squared rubble 145 to 150 Limestone, rough rubble 135 to 140 Granite, ashlar 165 to 170 Granite, squared rubble 155 to 160 Sandstone, ashlar 135 to 150 Sandstone, rubble 120 to 140 52. Capstones and Templets. When loads are to be transferred from the ends of beams or columns to masonry walls or piers, bearing blocks may be necessary properly to distribute the loads over the surface of the masonry. When used under a column or post, these blocks are called capstones; when used in walls to carry the ends of beams, they are templets. In placing bearing blocks, the loads should always be centered on the top of the block, if possible, so as to produce uniform pressure upon the masonry below; in all cases, the center of pressure must be within the middle third of the base to avoid a tendency to open the joint between the bearing block and the masonry, In Fig. 31, \P ..l t 1 /, _ FIG. 31. let P=the vertical load at center of pressure; 61 = the pressure at edge nearest the center of pressure; 62 = the pressure at edge farthest from center of pressure ; I = the length of stone ; x = distance from middle of block to center of pressure; STRENGTH OF STONE MASONRY 81 Zi = distance from nearest edge to center of pressure; 1 2 = distance from farthest edge to center of pressure ; b 2 = width of block. Then, Pl+6Px bl 2 bl 2 and 4PZ-6P/2 Pl- k 2 = bl 2 bl 2 When s = 0, h = l/2 and ki=k 2 = P/lb. When x = l/6, Zi = Z/3, ki = 2P/lb and &2 = 0. If x becomes greater than 1/6, k^ is negative and a tension will be developed in the joint or it will open. In designing a bearing block, k\ must not be greater than the safe load for the masonry. The load is commonly brought on top of a bearing block through an iron plate, which should have such area that the pressure will not be more than one-tenth to one-twelth of the crushing strength of the stone. The bearing block must have sufficient thickness not to break under the transverse load imposed by the upward pressure of the masonry. In designing a templet which is to be built into a wall, the weight of wall resting on the top of the templet must be included in deter- mining the pressure on its base. 53. Lintels and Corbels. A stone lintel is a beam of stone span- ning an opening in a wall. The strength of a lintel is determined by the ordinary beam formulas. The safe modulus of rupture may be taken at about one-twelfth to one-tenth of the ultimate modulus for the stone. Mean values of the safe modulus of rupture are about as follows: granite, 180 lbs./in. 2 ; Limestone, 150; marble, 130; sandstone, 120 lbs./in. 2 . There are, however, certain tough sandstones, specially adapted to this use, which may be used with modulus of 250 to 300 lbs./in. 2 Beams carrying live loads should not rest upon stone lintels. When the load upon a lintel is a solid masonry wall, it is common to assume that the masonry may arch over the opening, so that the actual weight upon the lintel is only that of a triangle whose height is about three-quarters of the span. This assumes that the lintel will yield somewhat, and be relieved of stress before reach- ing the maximum load. It is quite possible that in well-built masonry, with cement mortar, the lintel might be removed without the wall above yielding at all. If, however, there is no yielding of the lintel, the pressure upon its upper surface may be the same as at any other point in the same horizontal plane of the wall. 82 STONE MASONRY Beam A corbel is a block of stone extending beyond the surface of a wall or pier for the purpose of carrying the end of a beam or an overhanging wall, see Fig. 32. The overhang of the corbel is a cantilever beam, which must have sufficient section at the surface of the wall to resist the bending moment due to the load. The corbel must extend sufficiently into the wall, to give a resultant pressure within the middle third of the base of the corbel (R = P-{-W) ) as in the case of bearing blocks. FIG. 32. Corbels. Double corbels may be used when necessary, each being separately treated in determining strength. When weight of wall above (W) is lack- ing, the corbel must be anchored to the wall below by steel ties. ART. 14. MEASUREMENT AND COST 54. Methods of Measurement. In engineering work it is usual to estimate stone masonry in cubic yards of actual masonry. When parts of the work are of special character, requiring cut-stone finish, special prices per cubic yard may be given, or the additional costs of the cut surfaces are paid for by the square yard. In architectural work masonry is measured by the cubic yard or by the perch. A perch may be 16 J, 22, or 25 cubic feet, accord- ing to the custom of the locality in which the masonry is constructed. In the use of the perch as a unit, it is advisable to state the number of cubic feet to be considered a perch. In building work it is common to take outside measurements of walls, thus including the corner masonry twice; it is also custom- ary to measure small openings as solid wall. Commonly openings less than 70 square feet are not deducted. In some cases allow- ances are made for openings more than 6 feet wide. Customs differ in different parts of the country, and it is necessary to know the local usage, unless the method of measurement is stated. 55. Cost of Stone Masonry. So many items are included in the cost of masonry and these items vary so widely in different localities that it is not feasible to give any definite values to the costs of different kinds of work. The items of cost include the price of the rough stone at the quarry, the transportation to place of use, MEASUREMENT AND COST 83 dressing joints and faces of stone, mortar for joints, setting the stonework, and pointing the joints. Rough stones at the quarry are commonly classified into rubble or small stone and dimension stone. Rubble stone includes the more irregular stones and blocks suitable for small ashlar. Dimen- sion stone includes all stone required to be of particular sizes and blocks of large dimensions and definite thicknesses, as required for coursed ashlar. These classes vary according to the kinds of stone in the quarry and the specifications to be met by the stone. Rubble stone is commonly sold by the ton free on board cars at point of delivery. Prices for rubble stone delivered have varied in various localities from $0.50 to $2 per ton, when wages of quarry- men were about $4.50 per* day and common labor $1.50. The cost is largely a matter of locality. A ton of rubble stone may lay from about 16 to 22 cubic feet of masonry. Dimension stone and ashlar in the rough may cost from $0.50 to $1.25 per cubic foot for limestone or sandstone and $0.75 to $1.50 per cubic foot for granite, according to quality and location. Cost of Stone Cutting. The cost of cutting ashlar depends upon the hardness of the stone and the shape in which the blocks are received. Some stratified stones require almost no dressing on the bed joints, while other stones need every joint dressed from an irregular surface. With wages of stone cutters at $5 per day, the following may be considered average costs per square foot for cutting to J-inch joints; granites, 27 to 35 cents; hard sandstones and limestones, 20 to 30 cents; soft stones, 16 to 22 cents. Costs of peculiar face cuttings and of trimmings are so special to particular stones that they are of little value for general use. Sills, lintels, water-tables, and copings are usually sold by the lineal foot. The cost of sawing and machine dressing is usually much less than that for hand dressing, and varies with the way the stone is handled and the organization of the yard. ' Mortar Required. The amount of mortar needed in rubble masonry may vary from about 15 to 35 per cent of the volume of the masonry. Rubble of squared stones with joints 1 inch thick will ordinarily require 15 to 20 per cent, according to the sizes of the stones. For random rubble, stratified stones with flat beds require less than irregular stones. In the use of irregular rubble stones, the careful use of spalls in the larger joints reduces the amount of mortar materially, with saving in cost. The amount of mortar needed for ashlar work depends upon the sizes of the stones. Ordinary ashlar with J-inch joints in courses 84 STONE MASONRY 12 to 20 inches thick requires 4 to 7 per cent of mortar; random ashlar with smaller stones will require more, while with large blocks and thinner joints less will be required. Cost of Laying Masonry. The cost of setting stone varies with the size of the job, the organization of the work, and the skill of the masons, as well as with the character of the work itself. In ordinary rubble or squared-stone work, such as cellar walls or light retaining walls, a mason should lay a cubic yard of masonry in three or four hours. A helper to two masons or a helper to each mason, accord- ing to convenience of work, being required to supply stone and mortar. With masons at 50 cents an hour and helpers at 20 cents, this would cost from $1.80 to $2.80 per cubic yard. In large work, where stone is handled by derricks, and rubble constructed of large blocks, the cost of placing the stone is frequently reduced to $0.85 to $1.25 per cubic yard. The cost of setting ordinary ashlar varies from about $3 to $5 per cubic yard for limestone and sandstone, and from $6 to $9 for granite. The total cost of masonry in place, made up by so many varying items, necessarily varies within wide limits. Ordinary rubble at prices which have existed within the past few years (previous to the War), averages in cost from $5 to $7 per cubic yard. Rubble in heavy construction, usually granite, where the stone was quarried on the work and handled by machinery, has run from $5 to $11 per cubic yard. Sandstone and limestone bridge masonry, with ashlar facings and rubble backing and filling, usually varies from about $8 to $14 per cubic yard. Gillette's " Handbook of Cost Data " gives a number of detailed statements of costs of stone masonry. Such costs vary in about the same ratio as the pay of labor employed. The unsettled state of prices and labor costs since the War make it impracticable to give costs based upon present prices. CHAPTER IV BRICK AND BLOCK MASONRY ART. 15. BUILDING BRICKS 56. Clay and Shale Bricks. The cheapness, ease of construction, and durable qualities of good brick masonry make it one of the most desirable materials for general structural work. It is not as largely used in engineering work as stone or concrete, but in building con- struction it is very extensively employed. The qualities of clay bricks vary widely according to the character of the clay and methods of manufacture, and care must be taken in selection of material in order to secure good results. Composition of Clay Bricks. Clay consists primarily of silicate of alumina. Common clays also usually contain certain percentages of iron oxide, magnesia, lime, and alkalies. These are known as fluxes, having the effect, when in considerable quantities, of making the clay fusible. Fire clays contain a low percentage of fluxes, and withstand a high degree of heat without fusing. Sandy clays contain high proportions of silica in an uncombined state, a factor which, if not in excess, is of value, tending to give stability to the form of the brick. Sand is commonly added to plastic clays for this purpose. The color of the brick is mainly dependent upon the amount of iron oxide present in the clay. The color varies from white, through buff to red as the percentage of iron oxide increases. The presence of iron oxide is also of value in adding strength and hardness to the brick. Lime, when present in appreciable quantities, must be finely divided and uniformly distributed through the clay. If in lumps, the slaking of the lime, subsequent to burning, may cause the brick to become distorted and cracked. When in excess, lime neutralizes the color effect of the iron oxide, making the bricks lighter in color, buff or yellow colors being sometimes due to this cause. Excess of alumina usually makes the clay very plastic and causes it to shrink and crack in drying. 85 86 BRICK AND BLOCK MASONRY Physical Properties. The physical properties of clay are of more importance than the chemical composition. Plasticity is one of the important properties of clay for brick making, as it permits the clay to be worked into a plastic mass, and to be molded into the desired form. Clay shrinks in drying and also in burning, very plastic clay shrinking more than that less plastic. Sand is frequently mixed with clay to reduce excessive shrinkage. The degree of plasticity is sometimes controlled by mixing clays which differ in this respect. When subjected to high heat, clay gradually becomes soft and fuses together, and as the heat is increased the softening and shrink- age progresses until the material finally melts sufficiently to lose its shape. The temperature required for burning varies widely with different clays, and the degree of burning given to brick depends upon the kind of product desired and the fusibility of the clay. Manufacture. There are three methods in use for forming the brick. They are known as the soft-mud, the stiff-mud, and the dry- press methods. The soft-mud process consists in pulverizing the clay or shale and tempering it with water to the consistency of soft mud. This paste is then pressed into wooden molds, which are usually sanded on the surface to prevent the clay sticking, thus giving the brick five sanded surfaces. The stiff-mud process consists in mixing the pulverized clay or shale with sufficient water to form a stiff paste, capable of retaining its form, which is then forced through a die, resulting in a bar of the section of the brick. The bar is then cut into bricks by wires. These bricks may be either side cut or end cut. Dry-press bricks are made by pressing pulverized clay containing a small amount of moisture into steel molds, a method used to secure bricks with smooth faces and sharp edges for face brick. Repressed bricks are made by putting bricks made by the soft- mud or stiff-mud methods into presses and subjecting them to high pressure. The purpose is to give the brick more perfect form and sometimes to imprint a design upon the surface. Bricks made by the wet method must be dried before being placed in the kiln. In some yards this is accomplished by exposing the molded bricks to the air on floors or racks, while in the larger plants the drying is done more rapidly in dryers using artificial heat. The burning is accomplished either in temporary kilns, built of the brick to be burned, or in permanent kilns arranged usually with fire boxes on the outside and a downdraft and intended to give BUILDING BRICKS 87 uniform heat throughout the kiln. This cannot be fully accomplished and all of the brick will not be perfectly burned. The degree of burning received by brick in temporary kilns depends upon the position in the kiln. They must be sorted after burning into various shades, varying from the light underburned to the dark arch brick. Good bricks may be made by any of the methods of manufacture, provided the material is carefully handled and the burning properly regulated. The differences due to method used are mainly those of the form and appearance of the brick. Dry-press brick are usually somewhat softer and weaker than stiff-mud brick of equally good material. CLASSIFICATION OF BRICK Bricks used in structural work may be classified as follows: Common bricks are those used for ordinary brickwork, where appearance is not of special importance. They are burned at moderate temperatures. The best, well-burned common bricks are known as hard or cherry bricks, or sometimes as stock bricks. Those next the fire and heavily burned are known as clinker or arch bricks. Those from the underburned portion of the kiln are known as salmon, pale or soft bricks. The relative proportions of each kind in a kiln vary with the material and the skill used in burning. Pressed, face or front bricks are those made with greater care, so as to secure uniformity of form and color. They are used for facing walls of common brick and where appearance is important, and are usually dry-pressed or re-pressed brick. Vitrified bricks are made from a more refractory clay and burned at a high heat to the point of vitrification, so that considerable softening and shrinkage occurs, though the brick still hold its shape. These bricks are commonly made in larger sizes than common bricks, called paving blocks, and are used in street pavements. They are also frequently used in building construction, where obtainable at moderate prices. Blocks too lightly burned for use in pavements often make good material for building construction. Fire bricks are made from clay which is lacking in fluxing ingredi- ents. They are usually light in color, on account of the absence of iron oxide, and are used when high temperatures are to be resisted. Enameled bricks are made by coating the surface of pressed or re-pressed bricks before burning with a slip, which will burn to the proper color, and covering with a glaze. The enamel is usually applied to a single surface of a brick. 88 BRICK AND BLOCK MASONRY The following designations are also frequently employed: Sewer bricks are those common bricks which are so hard burned as to be practically non-absorbent of moisture, and are commonly used for lining sewers. Compass bricks are shorter on one edge than the other, for use in circular walls. Feather-edge bricks are made wedge shaped, for use in arches. Furring bricks are 'those having a surface grooved for plastering. Ornamental bricks are those having designs stamped in relief upon their faces, or bricks of special forms intended for use in making an ornamental surface design. PROPERTIES OF CLAY AND SHALE BRICK Good building brick should show a uniform compact structure without laminations. They should have plane, parallel faces and sharp edges, and should not show kiln marks on their edges. The dry-pressed and re-pressed bricks are usually smoother and more accurate in shape than those made by the soft-mud or stiff- mud processes, their density and strength being largely dependent upon the degree of burning and the shrinkage in the kiln. The under- burned, salmon bricks are porous and weak, and are usually employed only where strength is not important and in unexposed positions. The well-burned cherry or hard bricks are the best building brick. The overburned clinker bricks are more dense and absorb less water, but may be brittle, and are frequently distorted in shape. The overburned and distorted bricks ar^ sometimes used by architects for special exterior designs with very good effect. Vitrified bricks, as manufactured for use in paving, are superior in strength and density to common bricks. They frequently show kiln marks on one side, due to softening in the kiln. A clay for making vitrified brick must burn at high temperatures and have considerable range of temperature between the point of incipient fusion and the point of vitrification. It is difficult to maintain the temperature uniformly, so as to burn a large portion of the bricks to the right degree, unless the range of temperature is considerable. 57. Sand-lime Bricks, Bricks made of sand cemented with lime have been used in a small way for many years. These bricks, as formerly made, were molded and allowed to harden by standing in the air, or in an atmosphere rich in carbon dioxide (CO2). Bricks of this kind are virtually composed of ordinary lime mortar, but with less lime, and are called mortar bricks. They depend, like BUILDING BRICKS 89 lime mortar, upon the formation of carbonate of lime for their harden- ing, and are weak and of little value as brick, although some struc- tures of such materials have proven substantial and durable. In 1881 Dr. Michaelis of Berlin patented a process of hardening mixtures of lime and sand by the use of steam at high pressure. He discovered that, in the presence of steam at high temperature, the lime combines with a portion of the silica of the sand, forming a silicate of lime, which acts as a cementing medium. This silicate is formed upon the surfaces of the grains of sand and binds the sand into a single hard block. About fifteen years after Michaelis took out his patent, the manufacture of sand-lime bricks was begun in Germany on a com- mercial scale, and soon developed into a considerable industry. In 1901 the first plant was opened in the United States, and the growth of the industry in this country was also very rapid. Manufacture. In the manufacture of sand-lime bricks, four operations are essential: (1) The lime must be completely slaked. (2) A very uniform mixture of the lime and sand must be obtained. (3) The material must be formed into bricks under high pressure. (4) The bricks must be subjected to the action of steam at high pressure for several hours. The methods employed in different plants for performing these operations vary considerably, depending upon the character and condition of the materials employed. Hydrated-lime Process. In this process the lime is first slaked to a powder, or a putty, and then mixed with the sand and pressed. The lime may be slaked by any of the methods ordinarily employed in the manufacture of hydrated lime, or it may be reduced to a paste by the use of an excess of water. It is easier to obtain a uniform mixture of the lime and sand when dry hydrated lime and dry sand are used and the necessary water added afterward. It may, however, be advantageous sometimes to use wet materials, and good results may be obtained by either method if the mixing be thorough and the lime uniformly incorporated in the sand. Caustic Lime Process. Caustic lime is sometimes pulverized and mixed with the sand before slaking. Enough water is then added to slake the lime and reduce the mixture to proper consistency for pressing. High-calcium lime, which slakes quickly, is necessary when this method is used, as sufficient time must be given for the complete slaking to take place before the mixture goes to the press. In some plants the mixture is placed in a silo and allowed to stand 90 BRICK AND BLOCK MASONRY for a few hours before pressing, in order to insure that no unslaked lime is left in the mixture when the brick is formed. The caustic lime process is sometimes modified by grinding the lime with a portion of the sand to a fine powder, which is mixed with the remainder of the sand, and water added to slake the lime and wet the mixture. This is then placed in a silo for a sufficient period to allow the lime to become completely slaked before pressing. It is claimed that grinding the sand and lime together produces an intimate mixture and insures the complete combination during the steaming into silicate which forms the cementing medium of the brick. Grinding the lime and sand together reduces the lime to very fine condition and minimizes the danger from any unslaked particles of lime left in the mixture, and also fills the voids in the sand more completely, making a more dense brick. Molding. The bricks are formed in molds similar to those used for dry clay bricks, and are subjected to high pressure in molding. Hardening. After molding, the bricks are loaded upon cars and run into the steaming cylinders, where they are subjected to steam pressure of from 100 to 110 pounds per square inch for a period of six to ten hours, resulting in the combination of the lime with the silica into the cementing substance and binding the sand into a solid block. The brick continue to harden and gain in strength for a time after their removal from the steaming cylinder, as they gradually dry out. Materials. High-calcium lime seems preferable for this use, on account of this rapid action and the fine subdivision of its particles. Any good lime may, however, be used for the purpose if care be taken to insure that it be completely slaked. The requirements for sand to be used in making sand-lime bricks are not essentially different from those for sand to be used in cement mortar. The graduation of sizes to give a dense material is desirable. The presence of more fine material seems to be needed, however, in order to secure a smooth and compact mixture, and to lessen the wear upon the molds, which may become an important item of cost. Coarse sands seem to give stronger brick, but fine sand produces brick with smoother surfaces. Properties of Sand-lime Brick. In strength and durability, sand- lime bricks do not differ materially from good average clay bricks. When of good quality they possess sufficient strengths for all the purposes for which building brick are ordinarily employed, and are BUILDING BRICKS 91 usually more dense, and absorb less water than common clay bricks. Sand-lime bricks are usually very uniform in size and shape, and are commonly gray in color, the shade depending upon the sand used in manufacturing them, unless artificially colored. 58. Cement Bricks. Bricks made of cement mortar or concrete are used in a number of localities. They are commonly made of mortar, about one part Portland cement to four parts of sand, or sometimes of a richer mortar, 1 to 2 J or 1 to 3, mixed with about an equal quantity of coarser material, varying from J to ^ inch in diameter. These bricks are made by pressing in hand or power presses, a mixture as wet as is feasible to shape well in the press. About two weeks are required for hardening before the bricks can be used. The materials need to be carefully selected, and require the same prop- erties as for mortar for use in masonry or concrete. The strength may vary considerably with the grading of the aggregate, the com- pression given to the blocks, and the moisture conditions under which the bricks are kept during the period of hardening, the greatest strength will result when they are kept warm and thoroughly dampened. The compressive strength at twenty-eight days should not be less than 1000 lb./in. 2 , and the absorption not more than 15 per cent. Cement bricks are usually employed as face bricks. The appear- ance will depend upon the texture of the aggregates used and the method of finishing, which may be smooth or roughened by the use of brushes or acids. Color may be given to the bricks by the use of various mortar colors. 59. Test for Building Brick. In determining the suitability of a brick for structural work, examination is commonly made of the material as to form and texture with reference to the particular needs of the work in hand. Tests for strength and absorption are sometimes included in specifications for important work, but there is no recognized standard to which such tests conform, and com- paratively little data upon which to base a reasonable require- ment. Form. For neat work, the bricks should be uniform in size with plane faces and sharp edges. Care in sorting is usually necessary with clay brick to secure uniformity of color and dimension in par- ticular work. Texture. Good bricks should be uniform and compact in struc- 92 BRICK AND BLOCK MASONRY ture, should be sound and free from cracks, and the broken surfaces should be free from flaws or lumps. Clay brick should be thoroughly burned, and when struck with a trowel or another brick should give a clear ringing sound. Bricks which meet these requirements are usually suitable for all ordinary work. In ordinary building work little care is usually given to inspec- tion of the materials, and defective work frequently results from the use of poor bricks. Seriously defective bricks are so easily detected by inspection that there is usually no excuse for their inclusion in brickwork of good character. A Committee of the American Society for Testing Materials has been for some time studying the matter of a standard specifica- tion and standard tests for building brick. They have suggested tentative methods for classification of brick and for making tests for absorption, compressive strength, and transverse strength. The committee recommends that the standard sizes for building brick shall be 2J by 3| by 8 inches. They also recommend the following classification of bricks: (a) According to the results of the physical tests, the bricks shall be classified as vitrified, hard, medium, and soft bricks on the basis of the following require- ments: Name of Grade. ABSORPTION LIMITS, PER CENT. COMPRESSIVE STRENGTH, (ON EDGE) LB. PER SQ. IN. MODULUS OF RUPTURE, LB. PER SQ. IN. Mean of 5 Tests. Individual Maxi- mum. Mean of 5 Tests. Individual Mini- mum. Mean of 5 Tests. Individual Mini- mum. Vitrified Brick. 5 or less 6.0 5000 or over 4000 1200 or over 800 Hard Brick 5 to 12 15.0 3500 or over 2500 600 or over 400 Medium Brick . 12 to 20 24.0 2000 or over 1500 450 or over 300 Soft Brick 20 or over No Limit 1000 or over 800 300 or over 200 (6) The standing of any set of bricks shall be determined by that one of the three requirements in which it is lowest. The methods proposed for making these tests are given in the Proceedings of the Society for 1919, Part 1, p. 543. Durability Tests. The durability of bricks under difficult weather conditions is one of their most valuable qualities. Tests are some- times made of the effect upon bricks of freezing while in a saturated BRICK MASONRY 93 condition. These tests have been made in various ways, usually by immersing the brick in water, then freezing and thawing it repeat- edly, commonly twenty repetitions, and determining the loss of weight or of strength. Very soft, porous bricks may be disinte- grated by such treatment; those of low absorption and good strength usually show but slight effect. The Committee of the American Society for Testing Materials, in 1913, suggested a method for making this test. They have not, however, found it of sufficient value to include in their later speci- fications. A test in which the brick is saturated with a solution of sodium sulphate, which is then allowed to crystallize in the pores of the brick, has sometimes been used, the results of this action being similar to those of freezing, but much more rapid and severe. A study of this method has been made for the Committee by Pro- fessor Edward Orton, Jr. 1 and it seems probable that it may become a standard method of testing brick. It has not yet been definitely formulated for use in specifications. ART. 16. BRICK MASONRY 60. Joints in Brickwork. In the construction of brick masonry, it is necessary that the joints between the bricks be filled with mor- tar, the purpose of which is to give a firm and even bearing to the bricks, so that the pressure upon them will be uniformly distributed. The mortar should also adhere to the bricks and bind them into a monolithic mass. While thick joints usually make weaker masonry than those that are thin, it is desirable that the joint be as thin as it can readily be made. When attempts are made at too thin joints, they are apt to be imperfectly filled, and thus weaken the masonry. Joints in wall masonry of common brick, as used in building construction, are usually from J to f inch thick. It is common to specify the thickness of joints by stating the thickness for eight courses of brick. It is frequently required that the thickness of eight courses of brick masonry shall not exceed the thickness of eight courses of dry bricks by more than 2 inches. When pressed bricks are used for the face of a wall, the joints in the face are usually from J to -^ inch thick. 1 Proceedings, American Society for Testing Materials, 1919, Part 1. 94 BRICK AND BLOCK MASONRY Pressed bricks, being smoother, may be laid to thinner joints with good effect. In heavy masonry as sometimes used in engineering work, the joints usually of cement mortar are often \ inch thick. Mortar for Brickwork. Lime mortar is more extensively used for ordinary brickwork in building construction than any other. Mixtures of lime and cement mortars in about equal quantities are coming largely into use. The cement materially increases the strength of the mortar and its adhesion to the brick, while the smoothness of the lime mortar is maintained. In important struc- tures, where considerable strength is needed, it is common to use cement .mortar with addition of 10 to 15 per cent of hydrated lime a mixture which retains the strength of the cement but makes the mortar easier to work, and usually secures better work than would result from the use of cement alone. In engineering work, cement mortar is usually employed, but the mixture of hydrated lime with the cement is rapidly coming into use. Laying the Brick. In the construction of a brick wall the two outer courses are first laid, by spreading a bed of mortar where the brick is to be placed, and against the surface of the last brick laid, then shoving the brick horizontally into place so as to squeeze the mortar into the bottom of the vertical joint between the bricks. A bed of mortar is placed between the outside bricks and the filling bricks are shoved and pressed into place. Mortar is then slushed or thrown with some force into the upper part of the vertical joints to fill them completely. Bricks should be thoroughly wet before being laid, in order to prevent the water being absorbed from the mortar by the brick. Good adhesion cannot be had between mortar and dry, porous bricks. In finishing joints upon the face of the wall, a flush joint may be made by pressing back the mortar with the flat edge of the trowel. This is usually done upon interior walls. A weather joint may be made, as FIG. 33. Weather . ^. 00 , J . . , ' Joint shown in Fig. 33, by using the point of the trowel held obliquely. 61. Bond of Brickwork. Brickwork is always laid in horizontal courses, and lateral bond is secured by several different arrange- ments of the brick in the courses. Common Bond is the bond most commonly used in the United BRICK MASONRY 95 States, for walls of common brick. In this bond, one course of headers is used to four to six courses of stretchers on the face of the wall, as shown in Fig. 34. 1 II II II II II I 1 1 1 I II 1 II 1 II I II II II I II 1 1 II 1 II II 1 FIG. 34. Common Bond. In Flemish Bond (Fig. 35), alternate headers and stretchers are used in each course, each header being placed over the middle of the stretcher in the course below. Small closers are introduced next to the headers at the corners. ii ir II II II II II II II 1 II II II II II II FIG. 35. Flemish Bond. English Bond consists of alternate layers of headers and stretchers (Fig. 36). This construction, like the Flemish bond, makes very strong work. English bond in which the alternate courses of stretchers break joints with each other is called Cross English bond. 96 BRICK AND BLOCK MASONRY Hoop-iron Bond. This consists in placing pieces of hoop iron longitudinally in the joints to strengthen the bond, the ends of the iron being turned down into vertical joints. Pressed Brick Facing. In applying a facing of pressed bricks to a wall of common bricks, it is quite common to lay all of the face bricks as stretchers. When this is done bond may be obtained by metal ties or by diagonal bond. 1 J II I FIG. 36. English Bond. Metal Ties are sometimes used as shown in Fig. 37. When the joints in the face and backing cannot be brought to the same level, the metal tie may be bent, but this is not desirable, and frequent level joints should always be possible. These ties may consist of a thin piece of galvanized iron bent over a wire at the ends, or it may be a piece of galvanized wire bent into a loop at the ends to grasp the mortar. mm/m/////, V//////////A FIG. 37. Metal Ties for Face Brick. Diagonal Bond consists in breaking off the back corners of face bricks and inserting bricks diagonally to bond with the face brick. These bonds are not very strong, and the face bricks are not con- sidered as adding to the strength of the wall or carrying any load. Stronger work is obtained by using occasional courses of headers, BRICK MASONRY 97 or courses of alternate headers and stretchers as in the Flemish bond. This is usually possible by using care in regulating the thickness of joints in the backing, even when the bricks are not of the same sizes. Hollow Brick Walls. For the purpose of providing air space in a wall to prevent the passing of moisture or changes of temper- ature through it, hollow construction is sometimes adopted. This consists in building a double wall with a narrow air space between the outer and inner portions. It is necessary for proper strength that the two portions of the wall be bonded in some way, either by occasional headers which span the opening or by metal ties. The headers constitute a con- nection between the masonry of the two walls, and are sometimes objected to as likely to cause moisture to pass from one wall to the other. The metal ties may be provided with a drip at the middle which insures the complete isolation of the walls from each other. Such walls require more careful work and are more expensive to construct than solid walls. When loads are to be carried, one of the walls must be capable of bearing them. 62. Strength of Brick Masonry. In tests which have been made on the crushing strength of brick piers, failure occurred by the lateral bulging of the piers. When pressure is applied longitudinally upon the pier, a lateral expansion normal to the direction of pressure results. This causes tension upon the brickwork and the pier yields through breaking the bricks in tension and pulling apart of joints. The transverse strength of the bricks may also be called into play when they are not bedded with perfect evenness a fact proven by a series of tests on brick piers at the Watertown arsenal in 1907, in which bricks set on edge gave somewhat higher strengths than when laid flat. Piers in which the joints were broken at every third or sixth course gave slightly better results than those breaking joints at every course, as was also observed in piers tested in 1884. The strength of brickwork depends upon the bond as well as upon the adhesion of the mortar and the strength of the bricks. In masonry to be subjected to heavy loads, careful attention should be given to the bonding of the work and to the complete filling of the vertical joints in laying the masonry. The advantage of using strong mortar in such work is demon- strated by many tests made at Watertown arsenal and reported by the Ordnance Department of the United States Army in " Tests of Metals, etc." That the strength of brick masonry in piers is some- what proportional to the strength of the bricks is also demonstrated by these tests. 98 BRICK AND BLOCK MASONRY A series of tests made by A. N. Talbot and D. A. Abrams at the University of Illinois Experiment Station in 1908 gives very inter- esting results. A summary of these results is given in Table VI. TABLE VI SUMMARY OF TESTS OF BRICK COLUMN Average Values Ratio of Ratio of Crushing Ratio of Ref. Characteristics of Columns. Average Unit Load, Ib. Strength of Column to Strength of Column to Strength of 6-in. Mortar Strength of Column to per sq. in. Strength of Brick Strength of "A" Cubes, Ib. per sq. in. Strength of Cubes Shale Building Brick A Well laid, 1:3 Portland cement mortar, 67 days. 3365 .31 1.00 2870* 1.17 B Well laid, 1:3 Portland cement mortar, 6 months 3950 .37 1.18 .... .... C WeU laid, 1:3 Portland cement mortar, eccen- trically loaded, 68 days. 2800 .26 .83 .... .... D Poorly laid, 1 : 3 Portland cement mortar, 67 days. 2920 .27 .87 2870* 1.05 E Well laid, 1:5 Portland cement mortar, 65 days. 2225 .21 .66 1710 1.30 F Well laid, 1:3 natural cement mortar, 67 days. 1750 .16 .52 305 5.75 G Well laid, 1 : 2 lime mortar, 66 days 1450 .14 .43 Underburned Clay Brick H Well laid, 1:3 Portland cement mortar, 63 days. 1060 .27 .31 2870* .37 * Average value based on 1 : 3 tests of 1 : 3 Portland cement mortar cubes sixty days old. In the testing of brick piers it has been found that the initial yielding of the pier usually occurs at about one-half the breaking load. The safe load should be taken at not more than one-tenth to one-twelfth of the breaking load, on account of the many elements of uncertainty concerning the actual strength, chances for defective work, etc. BRICK MASONRY 99 A committee of engineers and architects recommended to the City of Chicago in 1908, the following values to be used as safe working pressures for brick masonry in building construction: Common Brick of Crushing Strength Equal to 1800 lb./in. 2 Lb. per Sq. In. Tons per Sq. Ft. In Iiin6 mortar 100 7 2 In lime and cement mortar 125 9.0 In natural-cement mortar 150 10 8 In Portland-cement mortar 175 12.6 Select, Hard, Common Brick, of Crushing Strength Equal to 2500 Lb per Sq. In. In 1 part Portland cement, 1 lime paste and 3 sand In 1 * 3 Portland cement mortar 175 200 12.6 14 4 Pressed and Sewer-brick, of Crushing Strength Equal to 5000 Lb. per Sq. In. In 1 : 3 Portland cement mortar Paving brick, in 1 : 3 Portland cement mortar 250 350 18.0 25.2 The building code of the City of St. Louis, in 1917, gives the following allowable compression on brick masonry: Vitrified paving brick, one part Portland cement, three parts sand . . Strictly hard pressed brick, one part Portland cement, three parts sand Ordinary hard and red brick, one part Portland cement, three parts sand Ordinary hard and red brick, one part Portland cement, one lime, three sand Merchantable brick, good lime mortar Per sq. in. 300 250 200 175 100 Vitrified paving brick and strictly hard brick shall not crush at less than five thousand (5000) pounds pressure per square inch. Ordinary hard and red brick shall not crush at less than two thousand and three hundred (2300) pounds pressure per square inch. Merchantable brick shall not crush at less than one thousand and eight hundred (1800) pounds pressure per square inch. 63. Efflorescence. The appearance of brick masonry is sometimes marred by a white coating which exudes from the masonry and is deposited upon its surface. This is called efflorescence, and is caused by soluble salts in the brick or the mortar, usually the latter, which 100 BRICK AND BLOCK MASONRY are dissolved by water when the wall is wet and deposited on the surface as the water evaporates. Such deposits usually consist of salts of soda, potash, or magnesia contained in the lime or cement, or of sulphate of lime or magnesia from the brick. Efflorescence may be prevented by keeping the wall dry. The use of impervious materials, and making the masonry itself imperme- able, render the appearance of efflorescence improbable. When a wall is in a damp situation, a damp-proof course at the base of the wall to prevent moisture rising in the masonry is desirable. If the masonry is permeable and is dampened by rain, some waterproof coating may be applied to the surface of the wall. There are various patented preparations for this purpose, and the Sylvester process is sometimes successfully used. This consists in applying first a wash of aluminum sulphate (1 pound to 1 gallon of water), and then a soap solution (2.2 pounds of hard soap per gallon of water). These applications are made twenty-four hours apart. The soap solution is applied at boiling temperature. The walls must be dry and clean, and the air temperature should not be below about 50 F. when the application is made. Efflorescence may usually be removed by scrubbing with a weak solution of hydrochloric acid. 64. Measurement and Cost. Measurement of brickwork is usually made by estimating the number of thousand bricks. It is assumed that an 8- or 9-inch wall contains 15 bricks per square foot of surface; a 13-inch wall, 22 \ bricks; a 17- or 18-inch wall, 30 bricks, etc. These numbers are employed without regard to the actual size of the bricks, adjustments in price per thousand being made for various sizes. The methods of estimating are sometimes rather complicated and are subject to rules established by custom. The plain wall is the standard of measurement, openings less than 80 square feet are usually not deducted; larger openings are measured 2 feet less in width than they actually are. Hollow walls and chimneys are measured solid. A pier is sometimes measured as a wall whose length is the cir- cumference and whose thickness is the width of the pier. Some- times one-half the circumference is taken as the length. Stone trimmings are not deducted from the brickwork measure- ments. Various rather complicated rules are used in estimating footings, pilasters, detached chimneys, etc. Having estimated the work in thousands of brick by these rules, a price per thousand, suited to the plain wall, is used for the entire BRICK MvlASGNBtfi ^ i ; /, 101 job. When pressed brick facing is used, the area of such facing is separately estimated. If an ashlar facing be used, its thickness is not included in that of the brick wall. In engineering work, brickwork is usually measured, like stone masonry, by the cubic yard of actual masonry. Number of Bricks Required. The actual number of bricks needed for the construction of masonry varies with the size of the bricks and the thickness of joints. For ordinary brickwork, with common bricks of the usual (8iX4X2J inches) size, and joints J to f inch thick, 1000 bricks will lay about 2 cubic yards of masonry. If the joints be J to f inch thick, 1000 bricks will lay about 2| cubic yards. With common bricks of ordinary size in masonry walls, six bricks will usually be required per square foot of wall surface for each width of brick in the thickness of the wall. For ordinary pressed-brick fronts, 6 to 6^ bricks are required per square foot of actual wall sur- face. In average building construction, deductions for openings will reduce the number by about one-third of those required for solid wall. Mortar Required. For ordinary building construction with J to f-inch joints, 0.5 to 0.6 cubic yard of mortar is required per 1000 bricks. This needs for 1 to 3 portland cement mortar, about 1.5 barrels of cement and 0.6 cubic yard of sand; for lime mortar about 200 pounds (2J bushels) of lime and 0.6 cubic yard of sand. In heavy masonry with joints \ to f inch, about 0.35 to 0.40 cubic yard of mortar per cubic yard of masonry, or approximately one barrel of cement and 0.4 cubic yard of sand for 1 to 3 Portland cement mortar. Labor of Laying Bricks. A bricklayer on ordinary work may lay from about 125 to 175 common bricks per hour, according to the skill of the workman and the organization of the work. He should place somewhat less than half as many face bricks. The number of bricks laid may be somewhat less with cement mortar than with lime mortar. On thin walls, with careful work, one helper may be needed for two bricklayers. On common brickwork, in building construction, one helper may be needed for each mason. In recent work with masons at 60 to 70 cents per hour, helpers at 30 to 35 cents per hour, lime at 40 to 50 cents per bushel, the cost of laying common bricks in the walls of buildings has run from $5 to $8 per 1000 bricks. Costs for scaffolding, for machinery and labor in erection of brickwork necessarily vary materially with the conditions under which the work must be done. 102 BRICK; ANfe BLG.CK .MASONRY At prices which have existed since the World War, these figures would be largely increased. Costs have varied widely in different localities and are now very unstable. ART. 17. TERRA COTTA CONSTRUCTION 65. Structural Tiling. Hollow tiling for use in building con- struction is made in many different forms. It is employed either as the main structural material or as fireproof covering for other materials. The materials of which the tiles are made are similar to those used in making bricks, but requiring usually higher grade and more refractory materials. Shales or semi-fire clays, similar to those used for paving bricks, are frequently employed for this purpose, or some- times fire clays are mixed with plastic clays to prevent fluxing at moderate temperatures. Tiling for use in construction may be made either dense or porous according to the qualities desired. Dense Tiling is made from materials which vitrify at high tem- peratures (above 2000 F.) and is burned to the point of vitrifica- tion like paving bricks. This material when of good quality pos- sesses high strength and is practically non-absorbent. It is used in outer walls of buildings, or for floor and wall construction when strength is needed. Hollow blocks as made for ordinary wall construction are not usually vitrified, but are burned to a less degree than the best dense tiling. They must be hard burned to be of value. In the rapid growth of the tile industry, attempts have been made to produce hollow tiling from inferior materials, and soft tiles lacking in strength and durability have sometimes been offered. Care must be exercised in selecting tiling to make sure of its quality. Porous Tiling, or Terra Cotta Lumber, is made from refractory plastic clays by mixing sawdust with the clay in forming the blocks, and burning at high temperature. The sawdust burning out leaves the material light in weight and porous. These blocks may be cut with, a saw, and nails or screws may be driven into them without difficulty. This tiling does not possess the strength of good dense tiling, but is tough and less brittle, and is largely used in fireproofing and for interior walls and partitions. Tiling of less porosity but possessing somewhat the character of th6 terra cotta lumber is sometimes made by mixing ground coal with the clay before burning. It is claimed that this makes a better TERRA COTTA CONSTRUCTION 103 fireproofing than the dense tiling. These blocks are sometimes known as semi-porous tiling. The forms and sizes of hollow blocks depend upon the uses to be made of them. For walls or partitions, the blocks are usually in 12-inch lengths, and of rectangular or interlocking sections. Rectangular blocks are made in various sizes 12-inch widths may be had from 2 inches to 8 inches thick. Widths of 6 and 8 inches are made in thicknesses from 2 to 5 inches. They are divided by webs into cells, as shown in Fig. 38. In the heavier tiling, intended Fro. 38. Hollow Rectangular Blocks. for use where loads are to be carried, and in outside walls, the shells are at least 1 inch and the webs at least f-inch in thickness, and the cells not more than 3J or 4 inches in width. In lighter tiling, used as filler in concrete work or for light partitions, the webs are f to J inch, and cell openings may be 5 or 6 inches. Interlocking blocks are made in various shapes, with the object of improving the bond of the wall, and eliminating joints extending through the wall. These blocks are often used in outside walls to prevent moisture passing through the wall and provide air spaces in all parts of the wall. Fig. 39 shows one of the common forms of interlocking tile. Hollow blocks for use in fire protection are made in many shapes to fit around structural members of other materials. They are also made to fit together in round or flat arches to support floors between steel beams. 104 BRICK AND BLOCK MASONRY Good tiling must be well burned, true in form and free from checks or cracks, and should give a ringing sound when struck with metal. The following requirements for hollow tile are given in the Build- ing Code of the city of St. Louis for 1917: All hollow tile used in the construction of walls or partitions shall be hollow shale or terra cotta, well manufactured and free from checks and cracks, each piece or block to be molded square and true and to be hard burned so as to give a good clear ring when struck, and not to absorb more than twelve (12) per cent of its own weight in moisture. Each of said blocks shall develop an ultimate crushing strength of not less than three thousand (3000) pounds per square inch of available section of web area, and shall not be loaded when in the wall more than eighty (80) pounds per square inch of effective bearing area. Tiles shall FIG. 39. Interlocking Tile. have outer shells or walls not less than three-quarters (f) of an inch thick and shall be additionally reinforced by continuous interior walls or webs which shall not be less than one-half (^) inch thick, and so arranged that no void shall exceed four (4) inches in cross-section at any point. It is further provided that the building commissioner may require a test to be made of such blocks before allow- ing the same to be placed in the wall, if, in his judgment, there be any doubt as to whether such blocks, proposed to be used, meet the requirements above specified. 66. Block Construction. In the construction of walls of ordi- nary hollow rectangular blocks, the blocks are usually laid so as to break joints and extend through the walls. They should be so placed that the vertical webs in each course are directly above those in the course below. Such construction is shown in Fig. 38. In using tile with horizontal cells, jamb blocks and corner blocks are made with the cells vertical. When very light walls are used, longitudinal reinforcement, consisting of thin band iron or of special forms of wire mesh, is placed in the joints. This is necessary for 2-inch partitions or for 3-inch partitions more than 10 feet high. TERRA GOTTA CONSTRUCTION 105 Tiles with vertical cell openings are made by some makers. Fig. 40 shows construction with standard tiling of this type. FIG. 40. Walls of Natco Hollow Blocks. Portland cement mortar, or mortar of lime and cement, is used in laying hollow blocks. In walls which are to carry considerable loads, Portland cement with 10 to 15 per cent of hydrated lime by volume (4 to 6 per cent by weight) should be used in 1 to 3 mortar with well-graded sand. For walls which are not to carry loads, a larger amount (equal volumes) of lime may be used. The surfaces of tiles are often grooved to aid the adhesion of the mortar in the joints. When the finish of the wall is to be plaster or stucco, the surface of the tile is grooved to hold the plaster. If brick veneer is to be applied, or if the surface of the tile is to be used for exterior finish, a smooth finish may be desirable. Floor Construction. The method of using hollow blocks in flat arch floor construction is shown in Fig. 41. These arches vary from FIG. 41. Flat Arch Floor Construction. about 3 to 6 feet in span and from 6 to 12 inches in depth. The blocks required consist of the skewback, the fillers and the key-block. 106 BRICK AND BLOCK MASONRY The skewbacks are usually made of such form as to enclose the bottom of the I-beam for fire protection. Such arches are now commonly made by the end-construction method in which the cell openings run lengthwise of the arch. The blocks do not break joints, but form a series of independent arches side by side. A number of different shapes are offered for these arches by different makers, lighter weight being obtained than with side-construction arches for the same strength. Hollow blocks are frequently used as fillers in reinforced con- crete floors, the blocks filling spaces between the webs of the T-beams of concrete, as shown in Fig. 42. Blocks 12 inches wide are usually FIG. 42. Hollow Block Fillers in Concrete Floors. employed for this purpose, the depth depending upon the span and loading of the floor. Strength of Block Masonry. Comparatively few data are avail- able upon the strength of constructions of terra-cotta blocks. A very carefully constructed wall of natco tile (see Fig. 40) was tested by R. W. Hunt & Company. The wall was 36f inches long, 8 inches thick, and 12 feet 2\ inches high, and was twenty-eight days old when tested. It failed under a load of 436,000 pounds, giving a compression of 3110 lb./in. 2 on the net section of the web, or about 1500 lb./in. 2 of gross area. Tests of a wall of Denison tile (see Fig. 39) faced with brick, forty-two days old, was made at the labo- ratory of the Bureau of Standards. This wall was 5 feet, 1 inch in length, 12J inches thick, and 31 feet high. It carried a load of 686,000 pounds, or about 900 lb./in. 2 of gross area. Good dense tiling should have a crushing strength of 3000 to 6000 lb./in. 2 of net section. When laid in masonry the allowable load is usually not more than one-fifteenth of the ultimate strength of the block. Carefully laid masonry of good quality hollow blocks may be allowed to carry a load of 200 lb./in. 2 of net section of block, or in general about 5 tons per square foot of gross area. 67. Architectural Terra-cotta. Terra-cotta for exterior finish or ornamental work is usually made from a mixture of clays, carefully selected to secure the desired qualities. The clay is ground, mixed, tempered, and worked to a proper condition of plasticity. It is GYPSUM AND CEMENT CONCRETE BLOCKS 107 then formed into the desired shapes in plaster molds or by hand, modeled as may be necessary, and dried. After drying, it is given a surface treatment, by spraying with a liquid upon the surface, which determines the kind of finish to be given in burning and its color. The blocks of terra-cotta may have a length up to 30 inches, and depth of 6 to 10 inches, with height according to the requirements of the work. They are constructed as hollow shells with webs about 1J inches thick, and cells 6 inches or less in width. These blocks are built into the body of the wall by bonding the masonry into and filling the cells. Several kinds of surface finish are used for terra-cotta. Standard terra-cotta is that in which no special finish is applied, leaving the block somewhat porous. Vitreous terra-cotta has a spray applied to the surface which causes the surface material to vitrify during burn- ing, making the material non-absorbent. Glazed terra-cotta has an impervious coating of glaze upon the surface. When the glaze is deadened, it is called mat-glazed. A variety of colors are available for use with this material, and make its use possible in a wide range of artistic designs. Terra-cotta of good quality is one of the most durable materials for use in the trimming and ornamentation of masonry structures. Being practically non-absorbent, it is not affected by frost, or by the gases in the atmosphere. The facility with which it may be worked into desired forms makes it a desirable material for artistic design. ART. 18. GYPSUM AND CEMENT BLOCK CONCRETE 68. Gypsum Wall Blocks. Blocks made by mixing gypsum plaster (see Section 37) with wood fiber or similar materials are used for partition walls in fireproof building construction. They are made 30 inches long, 12 inches high, and from 3 to 8 inches thick, with tapering openings through the block. They are laid in the wall to break joints and cemented with mortar composed of gypsum cement plaster and sand, usually 1 to 3. They are not used for walls bearing loads, but form very light partitions, and have good soundproof and fireproof qualities. The 3-inch blocks are used to a height of wall of about 12 feet, the 4-inch to 17 feet, and the 6-inch to 24 feet. The material may be cut with a saw, and plaster is applied directly to their surfaces. 108 BRICK AND BLOCK MASONRY The weights of walls of hollow gypsum blocks are approximately as follows : Thickness of block, inches 3 4 5 6 8 Weight of wall,lb per sq. ft 10 13 16 20 26 Three pounds per square foot is added for plaster upon each side of the wall. 69. Roofing and Floor Blocks. Blocks of gypsum, similar in composition to the partition blocks, and reinforced with wire mesh, are made both in solid and hollow form for use in roof construction. They are usually 3 or 4 feet in length and are used to span the open- ings between purlins and form a solid deck upon which the roof covering may be placed. They are made with beveled edges, and are set with their lower edges in contact and the triangular openings between them filled with a grout of cement plaster. Blocks with heavier reinforcement for openings up to 10 feet in span are also now offered. Floor blocks, to be used as fillers in reinforced-concrete floor con- struction, are now available. These are designed to act as forms for the concrete, and require support at the ends of the blocks, which are 2 feet long. A spacer is placed between two adjoining blocks to hold the concrete for the web of the beam, forming a smooth surface on the under side upon which plaster may be placed. A section of floor constructed with these blocks is shown in Fig. 43. FIG. 43. Pyrobar Gypsum Floor Tile. 70. Concrete Blocks. Hollow building blocks of Portland cement concrete are frequently employed in building construction in the same manner as in solid concrete construction, given in Chapter V, and the concrete is proportioned and mixed in the same manner. The blocks are usually made to set in the wall with the webs in a vertical position. Several patented forms are on the market which make blocks to bond in the wall in different ways and giving air spaces more or less effective as insulation against moisture and heat. Such blocks, when well made and properly set, make a sub- GYPSUM AND CEMENT CONCRETE BLOCKS 109 stantial and durable building, and may be used in such manner as to give a pleasing appearance. The color of the blocks may be regulated by choice of the aggregate used upon their exposed faces. The use of coloring matter in the concrete has not usually been very successful, although there are mineral colors available which may be used without material injury to the concrete. Metal molds are commonly employed, and concrete of rather dry consistency is compressed into them by tamping or by hydraulic pressure. This yields concrete of greatest strength and also makes a block which may be quickly removed from the mold. For orna- mental work, sand molds are frequently employed, a wooden pattern being used in forming the mold, and the concrete poured in a wet mixture. The curing of the blocks is important in its effect upon the strength and durability of the concrete, which must not dry out during the period of hardening. After the blocks are removed from the molds, they are allowed to stand in the air until the cement has set, when they may be transferred to a steam chamber, where they are subjected to an atmosphere charged with steam at a temperature about 110 to 130 F. After two or three days in the steam, they may be removed to the open air, but should be sprinkled often enough to keep them continually damp for ten or twelve days. When a steam chamber is not employed, the blocks are cured in the open air, but should be kept wet for a longer period to give time for com- plete hardening. The temperature to which they are subjected during hardening should never go lower than about 50 F. CHAPTER V PLAIN CONCRETE ART. 19. AGGREGATES FOR CONCRETE 71. Materials Used for Aggregates. Concrete as used in con- struction is essentially a mixture of cement mortar with broken stone, gravel, or other coarse material. The mortar serves to fill the voids in the stone and the whole is bound into a solid monolith by the setting and hardening of the cement. The materials mixed with the cement in forming concretes are known as aggregates. The sand or stone chips in the mortar is called the fine aggregate and the coarser gravel or broken stone is the coarse aggregate. In the manufacture of good concrete it is essential that each of the materials be of proper quality, and that they be properly proportioned and incorporated into the mixture. Fine Aggregate. Material which will pass a J-inch screen is usually included under the term fine aggregate, or sand. The requirements for sand and its use in mortar have been discussed in Chapter II. Ordinarily, the sand which makes the strongest and most dense mortar will also give the best results in concrete, though this may not always be the case. The grading of the sand should be such as to reach maximum density when combined in proper proportions with the coarse aggregate to be used in the concrete. Coarse Aggregate. This may consist of any hard mineral sub- stance broken to proper size usually broken stone or gravel, although sometimes broken slag, cinders, or broken brick is used. The value of stone as an aggregate depends upon much the same qualities as are needed for building stone. For high-class concrete work, it is important that the stone should possess strength, and absorb but little water. Stones breaking to cubical shapes give better results than those of shaly or slaty character, while rounded pieces pack closer and show less voids than those with sharp corners. Trap and granite are usually the best of concrete materials. When the concrete is to be subjected to abrasive wear, trap is a superior material. For resistance to direct compression, good granite 110 AGGREGATES FOR CONCRETE 111 is to be preferred. Limestones and sandstones vary greatly in their values as concrete materials, hard limestones and some of the more compact sandstones being desirable materials, while the softer vari- eties are not generally suitable for first-class concrete work. Gravel, when of flint or other hard material, may make excellent concrete. Sizes for Broken Stone. The sizes to which concrete stone should be broken depends upon the use to which the concrete is to be put. In heavy walls or massive work, the upper limit of size may be 2 or 3 inches in diameter. It is desirable to have the stones as large as can be easily incorporated into the mixture. In reinforced work, where the concrete must pass between' and under the reinforcing rods^ it may not be feasible to use stone of more than 1 inch diameter. In stone or gravel for coarse aggregate, as in sand for mortar, the grading of sizes should be such as to give maximum density. For a given stone, the strongest concrete will ordinarily be made by that arrangement of sizes which requires the least mortar to completely fill the voids in the stone, as a surplus of mortar beyond that required for completely filling the voids is an element of weak- ness in the concrete, as well as a waste of the more expensive materials. Stone as ordinarily used in concrete contains all sizes, from the largest allowed to the size of the largest sand. All material retained on a J- or f-inch screen is commonly regarded as coarse aggregate, and stone is used as it comes from the crusher with all the sizes included, only the chips being screened out. Gravel containing sand is sometimes used without screening by mixing with cement. This is not desirable practice, as the sand is seldom in proper quantity or uniformly distributed through the gravel, it should be screened out and proportioned properly to the cement and gravel. In concrete work it is usually necessary to use the materials available in the locality of the work, but where important work is to be done, careful attention should be given to the character of these materials and of the concrete made from them. The design of concrete structures should be based upon full information concern- ing the properties of the concrete to be used, and this is largely a question of aggregates. Poor concrete work has much more fre- quently resulted from the use of poor aggregates than from the use of inferior cement. In many cases it may be feasible and desirable to use materials of low grade in concrete work. Cinder concrete is preferred for some uses on account of its lightness, although it is low in strength. Local materials may be of poor quality, but usable by taking proper pre- 112 PLAIN CONCRETE cautions and designing the work in accordance with the character of the concrete. Failures have sometimes resulted from the use of low-grade materials without investigation of their qualities. Many users of concrete have failed to recognize the importance of the quality of the aggregates and seem to have regarded any stone broken to proper size as good enough for concrete. 72. Tests for Coarse Aggregates. There are at present no standard methods of making tests for concrete aggregates, or stand- ard specifications for such materials. The methods usually employed in testing sand have been discussed in Art. 7. A com- mittee of the American Society for Testing Materials is making a study of concrete aggregates and of the methods of testing them, and it is hoped that this may result in a standard practice in making such tests, and in throwing light upon methods of proportioning and forming the concrete. Mechanical Analysis. To determine the relative quantities of various sizes of stone in aggregate, it is common to make a mechan- ical analysis of the material. This consists in separating the various sizes by screening, and recording the amount retained upon each screen. The following has been adopted by the American Society for Testing Materials, upon recommendation of its Committee on Road Materials, as a standard method for making a mechanical analysis of broken stone or broken slag, except for aggregates used in cement concrete: The method shall consist of (1) drying at not over 110 C. (230 F.) to a con- stant weight a sample weighing in pounds six times the diameter in inches of the largest holes required; (2) passing the sample through such of the following size screens having circular openings as are required or called for by the specifi- cations, screens to be used in the order named: 8.89 cm. (3 in.), 7.62 cm. (3 in.), 6.35 cm. (2J in.), 5.08 cm. (2 in.), 3.81 cm. (H in.), 3.18 cm. (1 in.), 2.54 cm. (1 in.), 1.90 cm. (f in.), 1.27 cm. ( in.), and 0.64 cm. (\ in.); (3) determining the percentage by weight retained by each screen; and (4) recording the mechan- ical analysis in the following manner: Passing . 64 cm. ( J in.) screen Passing 1.27 cm. ( in.) screen and retained on a 0.64 cm. (\ in.) screen Passing 1.90 cm. (f in.) screen and retained on a 1.27 cm. (^ in.) screen Passing 2.54 cm. (1 in.) screen and retained on a 1.90 cm. (f in.) screen.. 100.00 AGGREGATES FOR CONCRETE 113 For materials in which sand is combined with the broken stone or broken slag, the same method is employed together with the fine sieves used for sand (see Art. 7) and the results are recorded in the same manner, beginning with the 200-mesh sieve. Apparent Specific Gravity. The weight of a given volume of the solid material of which the aggregate is composed is often of impor- tance in the determination of voids, or in proportioning concrete, a result obtained by determining the apparent specific gravity. The term apparent specific gravity as here used refers to the material as it exists, and includes the voids in the block of material tested; it may be somewhat less than the true specific gravity. For this purpose, the water to which it is referred need not be distilled, and determinations at ordinary air temperatures are sufficiently accurate. The following method of determining apparent specific gravity of coarse aggregates has been adopted as standard by the American Society for Testing Materials. The apparent specific gravity shall be determined in the follow- ing manner: 1. The sample, weighing 1000 g. and composed of pieces approximately cubical or spherical in shape and retained on a screen having 1.27 cm. ( in.) circular openings, shall be dried to constant weight at a temperature between 100 and 110 C. (212 and 230 F.), cooled, and weighed to the nearest 0.5 g. Record this weight as weight A. In the case of homogeneous material, the smallest particles in the sample may be retained on a screen having 1| in. cir- cular openings. 2. Immerse the sample in water for twenty-four hours, surface-dry individual pieces with the aid of a towel or blotting paper, and weigh. Record this weight as weight B. 3. Place the sample in a wire basket of approximately | in. mesh, and about 12.7 cm. (5 in.) square and 10.3 cm. (4 in.) deep, suspend in water 1 from center of scale pan, and weigh. Record the difference between this weight and the weight of the empty basket suspended in water as weight C. (Weight of saturated sample immersed in water.) 4. The apparent specific gravity shall be calculated by dividing the weight of the dry sample (A) by the difference between the weights of the saturated sample in air (B) and in water (C), as follows: Apparent Specific Gravity = B-C 5. Attention is called to the distinction between apparent specific gravity and true specific gravity. Apparent specific gravity includes the voids in the specimen and is therefore always less than or equal to, but never greater than the true specific gravity of the material. i The basket may be conveniently suspended by means of a fine wire hung from a hook shaped in the form of a question mark with the top end resting on the center of the scale pan. 114 PLAIN CONCRETE The specific gravities and weights per cubic foot of materials commonly used for aggregates are approximately as follows: Specific Gravity. Weight per Cubic Foot. Gravel 2 65 165 Trap . . 2.85-3.00 178-187 Granite 2.65-2 80 165-175 Limestone 2 50-2 75 155-170 Compact sandstone 2 45-2 70 153-168 Porous sandstone 2 . 10-2 . 40 130-150 Cinders .... .... . . 1.40-1 60 90-100 Determination of Voids. The voids in coarse material, such as gravel or broken stone not containing sand or other fine material, may be obtained by filling a measure of known volume with the material, and pouring in water until the measure is full. The volume of water Then, the percentage of voids = X 100. The total volume When the specific gravity of the material is known, the voids may be obtained by weighing a measured volume of the broken stone, subtracting this weight from the weight of an equal volume of the solid material, and dividing by the solid weight. If the aggregate contains fine material, the methods used for sand as given in Art. 7 must be used. It is evident that the percentage of voids in a mass of broken material is not a fixed quantity, but varies with the arrangement of the pieces. If the material were composed of equal cubes, it would be possible to place them side by side so as to leave no voids which could be filled by smaller material. Poured loosely into a measure, such cubes would probably show at least 45 per cent of voids, which would be somewhat modified by shaking down and compacting the mass. When the aggregate contains small pieces which may lie in the voids of the larger ones, the tendency to variation in results according to arrangement is greatly reduced, but the method of filling the measure, and amount of shaking that is given, will somewhat affect the results. Commonly, the material is shoveled into the measure and lightly shaken to get what may be a fair estimate of the voids in the material as it is to be used. When fine material is introduced into a coarse aggregate to fill the voids, particles of the fine material get between the larger pieces AGGREGATES FOR CONCRETE 115 and hold them apart so that the voids to be filled in the larger material are increased, and cannot be completely filled. This is shown by the fact that the volume of the mixture is greater than that of the coarse aggregate even though the volume of fine aggregate used is much less than the volume of voids in the larger material. Selection of Aggregates. The Joint Committee of the Engineer- ing Societies on Concrete and Reinforced Concrete makes the follow- ing recommendations concerning the selection of aggregates in its 1917 report. AGGREGATES Extreme care should be used in selecting the aggregates for mortar and con- crete, and careful tests made of the materials for the purpose of determining the quality and grading necessary to secure maximum density or a minimum percentage of voids. Bank gravel should be separated by screening into fine and coarse aggregates and then used in the proportions to be determined by density tests. (a) Fine aggregate should consist of sand, or the screenings of gravel or crushed stone, graded from fine to coarse, and passing when dry a screen having J in. diameter holes; it preferably should be of siliceous material, and not more than 30 per cent by weight, should pass a sieve having 50 meshes per 1m ear inch; it should be clean, and free from soft particles, lumps of clay, vegetable loam, or other organic matter. Fine aggregate should always be tested for strength. It should be of such quality that mortar composed of 1 part Portland cement and 3 parts fine aggre- gate by weight when made into briquettes, prisms or cylinders will show a tensile or compressive strength, at an age of not less than seven days, at least equal to the strength of 1 : 3 mortar of the same consistency made with the same cement and standard Ottawa sand. If the aggregate be of poorer quality, the propor- tion of cement should be increased to secure the desired strength. If the strength developed by the aggregate in the 1 : 3 mortar is less than 70 per cent of the strength of the Ottawa sand mortar, the material should be rejected. In testing aggregates care should be exercised to avoid the removal of any coaling on the grains which may affect the strength; bank sands should not be dried before being made into mortar, but should contain natural moisture. The percentage of moisture may be determined upon a separate sample for correcting weight. From 10 to 40 per cent may be required in mixing bank or artificial sands than for standard Ottawa sand^to produce the same consistency. Coarse aggregate should consist of gravel or crushed stone which is retained on a screen having | in. diameter holes, and should be graded from the smallest to the largest particles; it should be clean, hard, durable, and free from all dele- terious matter. Aggregates containing dust and soft, flat, or elongated particles should be excluded. The Committee does not feel waranted in recommending the use of blast-furnace slag as an aggregate, in the absence of adequate data as to its value, especially in reinforced concrete construction. No satisfactory specifications or methods of inspection have been developed that will control its uniformity and ensure the durability of the concrete in which it is used. The aggregate must be small enough to produce with the mortar a homo- 116 PLAIN CONCRETE geneous concrete of sluggish consistency which will readily pass between and easily surround the reinforcement and fill all parts of the forms. The maximum size of particles is variously determined for different types of construction from that which will pass a -in. ring to that which will pass a 1^-in. ring. For concrete in large masses the size of the coarse aggregate may be increased, as a larger aggregate produces a stronger concrete than a fine one; however, it should be noted that the danger of separation from the mortar becomes greater as the size of the coarse aggregate increases. Cinder concrete should not be used for reinforced concrete structures except in floor slabs not exceeding 8-foot span. It also may be used for fire protection purposes when not required to carry loads. The cinders should be composed of hard, clean, vitreous clinker, free from sulphides, unburned coal or ashes. ART. 20. PROPORTIONING CONCRETE 73. Arbitrary Proportions. The common method of propor- tioning concrete is by assuming ratios between the volumes of cement, sand, and coarse aggregate. These proportions are varied accord- ing to the character of the work, and sometimes are adjusted to the qualities of the materials. A formula of definite proportions does not always lead to the same result unless the method of measuring the materials is the same, as cement measured loose may vary con- siderably in weight for the same volume. A barrel of cement may measure from 3.5 to 5 feet, according to its degree of compactness. It is desirable to follow the recommendation of the Joint Committee on Concrete and take one sack (94 pounds) of cement as a cubic foot, or a barrel as 4 cubic feet in measuring the materials. Specific fixed proportions have to a certain extent become stand- ard in ordinary practice for various kinds of work. For reinforced concrete in building construction and where it is necessary to develop considerable strength, the porportions of 1 part cement, 2 parts sand, ancl 4 parts broken stone are commonly employed. For positions where strength is of special importance, as in column con- struction, or work in light superstructures of buildings, the propor- tions 1 : 1J : 3, or sometimes 1:1:2, are used. In more massive work and where only compressions are to be carried with ample sections, the proportions 1:3:6 and sometimes 1 : 2J : 5 are employed. The common proportions are based upon the requirement that the volume of fine aggregates shall be one-half that of the coarse aggregate. For materials commonly used, this gives a quantity of mortar sufficient to fill compactly the interstices in the coarse aggregate. The quality of the mortar is varied by changing the ratio of cement to fine aggregate, and the strength of the concrete PROPORTIONING CONCRETE 117 varies accordingly. The ratios between fine and coarse aggregates are often varied when the coarse aggregates contain more or less voids than is usual, and 1:2:3, 1:3:5, 1:2:5 or 1:3:7 con- crete is frequently used. Good results have been obtained in practice by this method of proportioning, when proper attention has been given to the quality of the aggregates. More careful methods of adjusting proportions would often be more economical, and equally good results might sometimes be obtained with less cost for materials. Many users of concrete employ ordinary proportions for all concrete irrespective of the character of the materials, and a wide variation in the quality of the concrete is frequently the result. 74. Proportioning by Voids. A method of proportioning some- times followed is to determine the voids in the aggregates, and use enough cement to fill the voids in the fine aggregate and enough mortar to fill the voids in the coarse aggregate. A small excess of fine materials is used in each case on account of inequalities of mix- ing. If the fine materials would all lie in the voids of the larger materials, this method would always give the desired result, and produce the concrete of maximum density and greatest strength. In practice, however, the voids cannot be completely filled, the volumes of the larger materials are increased by the smaller par- ticles lying between them, and the distribution of fine material through the mass is not uniform. Usually a volume of mortar 5 to 10 per cent in excess of the voids most nearly fills the voids without leaving appreciable excess of mortar. More mortar than this swells the volume of the concrete without increasing density, and has the effect of weakening the con- crete. If, for instance, sand containing 50 per cent voids is used with stone containing 40 per cent voids, and just fills the voids in the stone without increasing the volume, the resulting mixture will have 20 per cent voids. If an excess of sand be used, this excess will give an increase in volume having 50 per cent voids. This method of proportioning is an improvement over that of arbitrary selection of ratios, and usually gives approximately the most desirable proportions. Variations in the relative sizes of the materials, however, may change considerably the proportions neces- sary to give the most dense concrete. A certain sand may easily work into the voids of a given broken stone without materially increasing its volume, while with another stone containing the same percentage of voids but of different sizes, the same sand may produce quite different results, and to secure greatest density would need 118 PLAIN CONCRETE to be differently proportioned. The object should be to get the greatest density in the final mixture of fine and coarse aggregates. The inaccuracies involved in proportioning cement to sand by determining the voids in the sand is explained in Art. 7. When determining the ratio of fine to coarse aggregates by the method of voids, it is usual to proportion cement to sand by adopting an arbitrary ratio between the two, although some users of concrete have used the void method for this purpose also. 75. Proportioning by Mechanical Analysis Curves. Mr. William B. Fuller l has devised a method of proportioning concrete by plot- ting the curves of mechanical analysis of the aggregates to be used, then combining them in such proportions as to give a curve which corresponds as nearly as possible with a certain ideal curve. This ideal curve is supposed to represent the combination of sizes which will give maximum density for the given materials. Mechanical Analysis Curve. The method of plotting the curves of mechanical analysis is shown in Fig. 44. The analyses are made 100 .50 .75 LOO |.5 DIAMETER OF PARTICLES IN INCHES FIG. 44. Curves of Mechanical Analyses. 1.5 by the method outlined in Section 72. In the curves, the ordinates represent percentages of the samples (by weight) which pass through 1 An explanation of this method of proportioning is given by Mr. Fuller in Taylor and Thompson's "Concrete, Plain and Reinforced," Third edition, Chapter X. PROPORTIONING CONCRETE 119 openings whose sizes are shown by their distances from the origin. Fig. 44 shows a sample of stone and one of sand which are to be used in forming concrete. 'From these curves, others may be drawn showing the grading of sizes in various combinations of cement, sand, and stone. Thus for the 1:3:6 concrete, we will have percentages passing openings as follows: Sizes of Openings, inches. PERCENTAGES PASSING. Cement. Sand. Stone. Total. 1.50 10 + 30 + 60 = 100 1.25 10 + 30 + .80X60 = 88 1.00 10 + 30 + .53X60 = 71.8 .75 10 + 30 + .37X60 = 62.2 .50 10 + 30 + .15X60 = 49 .25 10 + .97X30 + .03X60 = 41 .10 10 + .88X30 + 00 = 36.4 .05 10 + .70X30 + 00 = 31 .02 10 + .40X30 + 00 = 22 This curve, corresponding to 10 per cent cement, 30 per cent sand, and 60 per cent stone, is shown on the diagram, as is the curve for 1 : 2J : 6J concrete. The ideal curve is found by sifting the stone and sand into a number of sizes, and then recombining these sizes in varying pro- portions and comparing the results, until the condition of maximum density is obtained. In an extended series of experiments, Messrs. William B. Fuller and Sanford E. Thompson l found that the curve of most desirable grading of materials was a smooth curve, consist- ing of an ellipse at the fine end with a straight line tangent to the ellipse and passing through the point where 100 per cent is reached. The materials tested in these experiments consisted of broken stone, gravel, and sand used in the construction of the Jerome Park Reser- voir, at New York. The equation for the ellipse as determined from these experiments is x and y being the horizontal and vertical coordinates of points on the ellipse measured from the origin of the diagram. Transactions, American Society of Civil Engineers, Vol. LIX, p. 67. 120 PLAIN CONCRETE The values of a and b vary for the different materials and are as follows: Materials. a b Jerome Park stone and screenings 0. 035-0. 14D 29 4-2 2D Cow Bay gravel and sand 0.04 -0.16D 26 4-1 3D Jerome Park stone and Cow Bay sand . . . 04 -0 16D 28 5-1 3D D in the above formulas is the maximum diameter of the coarse aggregate. To use this method of proportioning it is first necessary to determine the ideal curve. Sufficient data are not available to indicate whether the formulas given above are generally represent- ative of broken stone and gravel respectively. To determine the curve in a particular case, the sand and stone should each be sifted into about three sizes. A trial curve may then be assumed and the materials mixed in proportions to agree with the curve and the density of the mixture tested. Curves above and below the first one can be tried until an approximate density is located. 76. Proportioning by Trial. The simplest and usually the most accurate way of determining the ratios of quantities of materials for concrete is that of mixing batches in different proportions and comparing the densities of the resulting concrete. The object should be to secure the mixture of aggregates which will give the greatest density when mixed with the cement and water. For making these tests, it is convenient to use a cylindrical measure 8 or 10 inches in diameter and 12 or 15 inches high. A batch of concrete is mixed in assumed proportions to the consistency to be used in the work, and the height to which it fills the cylindrical measure is noted. Other batches are then prepared with the same total weight of materials, but differing in proportions of aggregates, and measured in the same manner. The greatest density is that which occupies the least volume for the same weight. It is necessary to use a uniform method of filling the cylinders, and is usually desir- able to compact the concrete by light ramming in rather thin layers to prevent voids being left where the concrete is in contact with the surface of the cylinder. Amount of Cement. In this method of porportioning, as in the preceding methods, the object is to determine the proper propor- tions of aggregates to give the most dense concrete. In each case, the amount of cement to be used is assumed as a definite ratio to PROPORTIONING CONCRETE 121 the total weight of aggregates. This ratio depends upon the char- acter of the work and the need for strength in the concrete, and is determined as mentioned in Section 82. In many instances, on important work, it is desirable to test the strength of the concrete as well as the density and modify the proportion of cement to suit the requirements. With different aggregates the strength may be quite different when the same proportion of cement is used, and economy in the use of cement may result from determination of the actual strength of concrete with varying proportions of cement to aggregate. (See Section 102.) More cement is usually required to produce the same strength when the sizes of the coarse aggregates are small than when larger aggregates are used. Stone broken to pass a f-inch screen may require 20 to 25 per cent more cement for the same strength than the same stone broken to pass a 1.5-inch screen. 77. Fineness Modulus and Surface Area. Several studies of methods of proportioning concrete have recently been made, involving extensive experimental investigations and resulting in suggestions of new methods. The tests of Mr. D. A. Abrams in the Structural Materials Laboratory at the Lewis Institute at Chicago led to the conclusion that, for a given ratio of cement to aggregate, the pro- portions requiring the least water to produce the required consistency would give the greatest strength. This would depend primarily upon the grading of the aggregate in size, and Mr. Abrams evolved a method by the use of what he calls the " fineness modulus," based upon the mechanical analysis of the aggregate. The Tyler series of sieves is used, Nos. 100, 48, 28, 14, 8, 4, etc., each of which has openings twice the diameter of those of the preceding ones. Mr. Abrams method is given in Bulletin No. 1 of the Structural Materials Research Laboratory. Mr. L. N. Edwards has proposed 1 a method of proportioning concrete by means of the surface areas of the particles of aggregate. A theoretical study of this method of proportioning has been made by Mr. R. B. Young, 2 in which he claims that the quantity of water necessary to bring a concrete mixture to a given consistency is de- pendent upon the surface area of the aggregates. These and other investigations in progress are throwing much light upon the subject of proportioning concrete and upon its qualities. The concrete is affected by a number of elements, ea^h of which must be considered in determining the best proportions. The ratio 1 Proceedings, American Society for Testing Materials, 1918, Part II. 2 Proceedings, American Society for Testing Materials, 1919, Part II. 122 PLAIN CONCRETE of cement to aggregate, the voids in the aggregates, the surface areas of the aggregates, the quantity of water used in mixing are all important, and are all directly concerned with the grading of sizes of aggregates. Some method based upon mechanical analysis may finally be standardized for general use, when the relative impor- tance of the various factors are more fully understood. Any of the methods proposed may be employed as a guide in selecting propor- tions, but actual trial of the materials in concrete is necessary to give certainty in results. 78. Yield of Concrete. The quantities of materials needed for a cubic yard of concrete vary with the amount of voids in the aggre- gates and the proportions in which they are combined The sizes of the aggregates and the quantity of water used in mixing also influence the yield of concrete. Concrete is made up of a mixture of cement, fine aggregate, and coarse aggregate, or it is a mixture of cement mortar with coarse aggregate. The volume of the concrete is the sum of the volumes of the mortar, the solid material in the coarse aggregate and the unfilled voids in the coarse aggregate. Let C = Volume of cement in cubic feet (bags of 94 pounds each) ; S = Volume of fine aggregate in cubic feet ; R = Volume of coarse aggregate in cubic feet ; F= Volume of voids in coarse aggregate in cubic feet; s= Ratio of sand to cement = S/C] r= Ratio of coarse aggregate to cement = R/C; v = Ratio of voids to total volume of coarse aggregate, V/R. The quantities of ingredients necessary to produce given volumes of cement mortars, and the variations for different materials, are discussed in Section 34, and while these quantities vary considerably with different materials, the volume of mortar produced by the mixture of different proportions of cement and sand is fairly well expressed by the expression : Volume of mortar = aC+6$, in which a and b are coefficients depending upon the character of the sand. The volume of concrete from given quantities of cement, sand and stone may then be expressed by the formula : Q = aC+bS+c(R-V), in which c is a coefficient depending upon the amount of unfilled voids in the stone. For ordinary fairly coarse sands commonly PROPORTIONING CONCRETE 123 used for concrete, a may be taken .67 and b .77. For well compacted, plastic concrete of ordinary materials, c is about 1.10. With these values of the coefficients, the formula becomes: or Q = C[.67+.77H-l.lr(l-t;)]. The volume of cement required to make a cubic yard of concrete is: The number of barrels of cement = C/4. Cubic yards of sand = Cs/27, cubic yard of stone = Cr/27. Table VI gives approximate quantities of materials required for 1 cubic yard of plastic concrete, using stone with differing percent- ages of voids. Average crusher run stone, with chips removed, has about 40 to 45 per cent voids; good natural gravel, screened, may have 35 to 40 per cent voids; mixed stone and gravel often runs from 30 to 35 per cent voids, while carefully graded materials may have voids reduced to from 20 to 30 per cent. Variations in the characters of the materials used, and in the methods of handling and placing the concrete may vary considerably the quantities of materials required. Dry concrete, if thoroughly compacted by ramming, is more dense and occupies less space than plastic or wet concrete, but as ordinarily placed is more porous and occupies more space. Fine sand swells more when mixed with cement and water, and fills more space in plastic concrete, than coarse sand. Coarse broken stone compacts in concrete so as to leave less unfilled voids than smaller stone with the same per cent of voids. Poor work, such as irregular mixing or imperfect com- pacting, results in more porous concrete and requires less materials. Tests of the yield of concrete may easily be made by mixing a batch in the proportions to be used and measuring the resulting concrete. In cases where accurate estimates of quantities are impor- tant and data concerning the particular materials are not at hand, such tests should be made. 124 PLAIN CONCRETE 00 Q O > sS 3d G&U - O dodo O CO ^O i i t^ ir^ co t^ d d d d CO 00 CO t^. t^ CO 1^ 1> d d d d O rH TjH O oo i> i> oo dodo O B .TJ |iH -5 i-i CO GO i-H co eo (M TJH d o o d iO O i i CO CO CO -^ CO o o d o rH O i-l 05 O i I CO 00 O5 00 00 Tf -^ 00 O5 l> t^ CO iO l ra o o o o 0000 OB Q O gS Is (N 00 O - l> O5 t^ d d d d OO O5 !> 00 o o d d (M O r^ (N OS 00 00 O5 d d d d O5 00 OO OS o o d d > 5 M -c g* *5 CO O5 T^ 00 CO CO CO -^ o o o o i i CD 00 O3 ^ <^> ^ ^ d d d d 83^ dodo co oo ^ oo CO ^ ^ CO dodo ft 5 Is to 00 co r OID8. ^ . s* 00 CO O 00 i> oo o t"- d d I-H d r-t 1 1 T^ T^ OS O 00 Oi d T-! d o (M r^ CO H B & os co r^ (N co TJH co o 10 O sqs O 00 t^ O i-H rH O CJ SSS8?3 6 GLUMES. a g o QQ (M CO "^ CO Tj< IO Tj< O co o co t^ 00 l> 00 O ' rH > f P CD M o J i rH|M H|N ,_, ,_, ^H (M ^-1 ~-l (N CO iO Tfl i 1 Tt< 00 O C^l rH IO rH OOO OO CO OO CM CO CO CO TJH rH <* oo o oo (M rH 1C TH 00 TJH CO 00 O .CO 00 00 8 S3 If /,= 16,000 lb./in. 2 and it = 80 lb./in. 2 , Z & = 50z, or for safety, the length between the point where the stress of 16,000 lb./in. 2 exists and the end of the bar must be 50 diameters. Anchoring Bar by Bending. When it is not feasible to secure the length of bar necessary for bond, the end of the bar may be anchored in the concrete by bending to a semi-circle. Experiments indicate that, in general, the full strength of a bar in tension may be developed by bending the end to a semicircle, the diameter of which is four times the diameter of the bar. Short right-angled bends are found to be much less effective than curves through 180. In the case of restrained beams, or cantilevers, when maximum tension occurs near the support, careful attention must be given to the anchorage of the bars. Bars used for diagonal tension reinforce- ment, either vertical stirrups or inclined bars, have maximum tension at the neutral axis, and must have a sufficient embedment on the com- pression side of the neutral axis to resist the maximum tension in the steel. Lateral Spacing of Steel. The horizontal tension rods in a rein- forced concrete beam must be so spaced as to leave a sufficient area of concrete between them to carry the shear communicated to the con- crete by the portion of the bars below the minimum section of con- crete. This would require that for circular bars the horizontal sec- tion between rods be capable of carrying a shearing stress equal to the 176 REINFORCED CONCRETE bond stress on the lower half of the bars. If s c be the clear spacing between the bars and i the diameter of the bar, for the round bar iriu iriu s v = ~2 or S <=W For the values of unit stress recommended by the Joint Com- mittee (v = 6 per cent and u=4 per cent of the ultimate compressive ftiM strength), v = %u, and for round bars, s c = ^-=l.05i. For square bars with sides vertical, s c v = 3iu, or s c = 2i, and for square bars with diagonals vertical, s c = *. For deformed bars these values would be increased in the ratio of 5 to 4. The Joint Committee recommends 1 that: The lateral spacing of parallel bars should not be less than three diameters from center to center, nor should the distance from the side of the beam to the center of the nearest bar be less than two diameters. The clear spacing between two layers of bars should be not less than 1 inch. The use of more than two layers is not recommended, unless the layers are tied together by adequate metal con- nections, particularly at and near points where bars are bent up or bent down. Where more than one layer is used at least all bars above the lower layer should be bent up and anchored beyond the edge of the support. 110. Design of Beams. The methods of applying formulas and tables in the design of rectangular beams is illustrated in the following examples : (7) Design a rectangular beam to have a span of 25 feet and carry a uniform load of 600 pounds per linear foot, in addition to its own weight, using working stresses recommended by the Joint Com- mittee for concrete of 2000 lb./in. 2 compressive strength. Solution. From Table VII, for ft =15, /, = 16,000 and f c = 650, we find fl = 108, p = .0078, j=.874. Assume weight of beam = 300 pounds per linear foot. Then jf-- ^rc in .. lb . o o 843750/108 = 7812, and for 6=12, d = 25.5, for 6 = 14, d = 23.6 Taking 6 = 14 and total depth, ft = 25.5, weight of beam = 14X25. 5X150/144 = 372 pounds per linear foot. The assumed load is too small. Assume weight of beam =400 pounds per linear foot. o lUo 1 Proceedings, American Society of Civil Engineers, December, 1916. RECTANGULAR BEAMS WITH TENSION REINFORCEMENTS 177 For 6=14, d = -J-^j-=24.9. Using 6 = 14 and d = 25, make ft = 27. 27X14X150 Then weight of beam = ^ -- = 394 pounds per linear foot, which agrees with the assumption. Horizontal steel, A =pdb = . 0078X14X25 = 2.73 in. 2 From Table X, seven f-inch square bars give A =2.73 in. 2 five f-inch square bars give A =2.81 in. 2 six f-inch round bars give A =2.65 in. 2 Seven f-inch square bars, spaced 1| inch c. to c. or six f-inch round bars, spaced 2J inches c. to c. might be placed in the width of 14 inches, meeting the requirement of spacing 3 diameters c. to c. We will use five f-inch square bars, spaced 2J inches c. to c. and 2 inches from side of beam. ,, . , T7 co 25X1000 10 1U Maximum Shear, V = ^ = -- ~ - = 12,500 Ib. Z 2i V 12500 The section is sufficient for shear, and no diagonal tension reinforce- ment is necessary. Bond Stress, Table X, for five f-inch bars, 2o = 5X3.00 = 15 in. 2 and (19) bv 14X40.6 Q . ftl , ,. o U = 2o = IS" = 37 ' 91b -/m. 2 , which is less than the allowable stress. 8. A simple beam of 10-foot span to center of bearings, is to carry a load of 400 pounds per linear foot. Design the beam, assuming n = 15. f s = 15,000 lb./in. 2 , / c = 750 lb./in 2 , safe value of unit shear =120 lb./in. 2 , and for diagonal tension on concrete =40 lb./in. 2 Solution. Assuming the weight of beam as 65 pounds per linear foot, the total load is (400+65)10 = 4650 pounds. ,, 1 , 4650X10X12 ftfV7Kn . M = %ul = -- ^ = 69750 in.-lb. o From Table VII, f or f s = 15,000, / c =750, and n=15, we have R = 138, j = 0.857, and p = .0107. then &d 2 =M/# = 69750/138 = 505. If we assume 6 = 5, we find d = 10, and A = pbd = .0107 X 5 X 10 = 0.535 in. 2 By Table X, five f-inch round bars = .55 in. 2 four f-inch square bars = .56 in. 2 three j^-inch square bars = .57 in. 2 178 REINFORCED CONCRETE The four f -inch square bars will fit in the width of beam with proper spacing, but we will use three y^-inch bars. If the concrete extend 1J inches below the center of the steel, /i=d-j-li = lli inches, and the weight of beam is 5X11.5X150/144 = 60 lb./ft.; this is a little less than our assumed weight. Maximum Shear, V This is less than 120 lb./in. 2 , and the dimensions of the beam are sufficient. Reinforcement for diagonal tension is needed beyond the point 401 40X10 where v =40 lb./m. 2 , or x = ^ = . = 3.7 feet. Reinforcement is 2v m 2 X o4 required to 53.7 = 1.3 foot = 15.6 inches from the support. Vertical Stirrups. If we assume s = %d = 5 inches for the stirrup next the support, we have (13). vbs 54X5X5 For U-shaped stirrup, the section of rod required will be one-half of this, or 0.045 in. 2 one-quarter in round bars are sufficient, and three stirrups may be used, spaced 3, 8, and 13 inches from the middle of support. Bond Stress. For the horizontal steel, Table X, 2o = 3X1.75 = 5.25 in. 2 , and (19) u = bv/2o = 5X54/5.25 = 51.4 lb./in. 2 , which is less than the allowable bond stress, and no anchoring is necessary. fi 15000 For the vertical stirrups (21) ^=1^=. =11.8 inches, or the stirrups need 11.8 inches above the neutral axis for anchor- age; they must therefore have hooked ends. ART. 29. T-BEAMS WITH TENSION REINFORCEMENT 111. Flexure Formulas. In a rectangular reinforced concrete beam, in which the steel carries all the tension, the area of concrete below the neutral axis does not affect the resisting moment of the beam. The office of this concrete is to hold the steel in place and carry the shear, thus connecting the steel with the compression area of concrete. In a T-beam, the flange carrying the compression is connected with a narrow web which holds the steel, as shown in Fig. 52. When the neutral axis is in the flange, such a beam may be computed by the T-BEAMS WITH TENSION REINFORCEMENTS 179 formulas and tables used for a rectangular beam, using the width of the flange, b, as the width of the beam. -1 Jcd\ ~ i i JfZaZZ ~- ci b' FIG. 52. T-Beam with Tension Reinforcement. When the neutral axis is below the bottom of the flange of the T-beam, the compression area is less than that of the rectangular beam, and special formulas are necessary. Fig. 52 represents a beam of this kind. The amount of compression on the web is usually very small and may be neglected without material error, thus greatly simplifying the formulas. The same notation will be employed as in the rectangular beam, letting b = width of flange; 6' = width of web; t = thickness of flange. The position of the neutral axis in terms of the unit stresses may be found as in the rectangular beam, giving and f c (22) (23) The average unit compression on the flange is the half sum of the compressions at the top and bottom of the flange, or The total compression on the concrete is 2kd-t C =/ c -, , bt. 2kd This is the equal to the total tension on the steel, T=f s A=f s pbd. (24) (25) 180 REINFORCED CONCRETE From the equality of (24) and (25) we find (2k-t/d) t P= 2n(li=W'd> - V * (26) and fc.^+M/f. ..... . (27) pn+t/d The distance of the centroid of compression from tne upper face of the beam is 3k-2t/d i_ 2k-t/d '3' therefore (28) 2kt/d 3 The resisting moment of the beam is .-V . . . (29) or Examples. The use of these formulas in the solution of problems arising in the design or investigation of T-beams are illustrated in the following examples : 9. A T-beam has the following dimensions, 6 = 48 inches, = 4 inches, d = 22 inches, 6' = 10 inches. The steel reinforcement con- sists of six j-inch round rods. If the safe unit stresses of steel and concrete are 15,000 and 600 lb./in. 2 respectively, and'n = 15, what is the safe resisting moment of the beam? Solution. From Table X, A =2.65 in 2 , and p = 00 ^ Q = .0025; formula (27) gives .0025X15+ A Using (28) we find M-22 3X.247-2(^) 4 Jd ~ 22 2X.247-A 3- 20 " 39 ' From (22), /,_ 15(1-. 247) T~ ~^T If / c = 600 lb./in. 2 , /, = 600X45.7 = 27,420 lb./in. 2 This is greater than the safe unit stress on steel, and the safe moment will be that which causes a stress of 15,000 lbs./in. 2 on the steel, or from (29), M = 2.65 X 15000 X 20.39 = 810000 in.-lb. T-BEAMS WITH TENSION REINFORCEMENTS 181 10. The flange of the T-beam is 26 inches wide and 4 inches thick. The beam is to carry a bending moment of 520,000 in.-lb. The safe unit stresses for concrete and steel are 600 and 16,000 lb./in. 2 respectively. What area of steel and depth of beam are needed. Solution.-*? (23) fc= 1600 1 ^ 1 6 5 ; 600 = .360. We must now find d by assuming values and testing their suitability. Try d 18; from (28) we have 4 " ' 2X.360-A 3 (9) gives C = M/jd = 520000/16.3 = 31900. From (24) / c = 07 .-fa = 440 lb./in. 2 This is a safe value, but t ~2kd~ a less depth will answer. Trying 15 inches, we find C = 38,000 pounds, and / c = 580 lb./in. 2 ; 15 inches is, therefore, approximately the minimum value for d. For this value of d } Formula (25) gives, A = T/f s = 38000/15000 = 2.375 in. 2 Width of Flange. T-beams without lateral reinforcement in the flanges should have a width of flange not more than three times the width of web, 6 = 36'. When the flange is reinforced at right angles to the length of beam, as in a slab floor with T-beams support, experi- ence indicates that the flange may overhang the web on each side to a distance equal to five or six times the thickness of flange, and still act satisfactorily as compression area for the beam. If the width of flange be greater than this, the extra width is of little value and should not be considered in estimating the strength of the beam. The Joint Committee has recommended the following rules for determining flange width: In beam and slab construction an effective bond should be provided at the junction of the beam and slab. When the principal slab reinforcement is parallel to the beam, transverse reinforcement should be used extending over the beam and well into the slab. The slab may be considered an integral part of the beam, when adequate bond and shearing resistance between slab and web of beam is provided, but its effective width shall be determined by the following rules: (a) It shall not exceed one-fourth of the span length of the beam. (6) Its overhanging width on either side of the web shall not exceed six times the thickness of the slab. In the design of continuous T-Beams, due consideration should be given the compressive stress at the support. 182 REINFORCED CONCRETE Beams in which the T-form is used only for the purpose of providing additional compression area of concrete should preferably have a width of flange not more than three times the width of the stem and a thickness of flange not less than one-third of the depth of the beam. Both in this form and in the beam and slab form the web stresses and the limitations in placing and spacing the longitudinal reinforcement will probably be controlling factors in design. 112. Shear and Bond Stresses. Stresses due to shear in the concrete and bond stresses between the steel and concrete in T-beams are found by the same methods that are used for rectangular beams. The shearing and diagonal tension stresses must be carried by the web of the beam, the area of flange not being considered in finding unit shear. Using the same notation as for rectangular beams and letting b' represent the width of the web of the T-beam, the formulas as applied to T-beams become: y For shear, v = r^, and b'd = -.. (33) vj For vertical stirrups, __vb's Vs Av ~2f s or For diagonal steel, vVs Vs ~w, or s = 2 ~^ (35) For point where it is allowable to turn up steel, A ~ Ax= W2?' and X = L^A*. , fc . ., -< 1: ,.';" . . . (36) For bond stress, b'v V and 20 = =-^ (37) T-BEAMS WITH TENSION REINFORCEMENTS 183 For length of bar to prevent slipping, ' "'';' fc-. .-! . , , .'V >.., , (38) The Width of the Web (&') must be sufficient to provide proper area for carrying shear, as shown in (33), and also to allow for properly spacing the steel, as explained in Section 109. b' should not usually be taken at less than d/3, except in heavy beams where a thickness of d/4. may be allowable. The value of j, when not known, may be assumed as { without material error, and the value of vj in (33) may be taken as f of the allowable unit shear. 113. T-Beam Diagrams. The labor of T-beam computations may be considerably lessened by tabulation 'of some of the terms which enter into the formulas. Some of these tabulations are here given in the form of diagrams. If we place Q=fcj -- ^, , Q will be constant for any particular values of unit stresses and t/d. Substituting in Formula (30) we obtain M = Qbtd and M/bt = Qd or Q = M/W. .... , . . . (39) In Diagram I, values of f c and p are given in terms of various values of d/t and Q for n = 15 and f s = 16,000. This diagram may be used in design of beams when these units are to be employed, or similar diagrams may easily be prepared for other values of f s and n. Diagram II gives values of p and j in terms of f s /f c and d/t, when n = 15. This diagram may be used in reviewing a beam of known dimensions and reinforcement, or in design when values of f s other than that used in Diagram I are to be employed. Examples. The following examples illustrate the use of these diagrams and formulas in computation. 11. A T-beam has dimensions as follows: b = 45 inches, t 4 inches, d = 20 inches, b f = 9 inches. It is reinforced with ten f-inch round steel bars. If the safe unit stresses of steel and concrete are 16,000 and 650 lb./in. 2 respectively, what is the safe resisting moment for the beam? 4 41S Solution. Table X, A = .4418X10=4.418 in. 2 , and p= ' - 4o X 2\j = .0049. From Diagram II, for p = .0049 and d/t = 5, we find /// = 29, and j = .914. If / c = 650, /. = 650X29 = 18850. This is 184 REINFORCED CONCRETE more than is allowable, and the safe resisting moment is that giving a stress of 16,000 on the steel, or using (29) = 1,292,800 in./lb. 12. The dimensions of aT-beam are, 6 = 36 in., t = 3 in., d= 13 in., The beam is reinforced with six J-inch round steel bars. If this = 16,000 3.0 3.5 4.0 4.5 5.O 5.5 6.O 6.5 7.0 r _ ^ * * -"'-^ i i i i i i i i i i i i _i i .5 3.O 5.5 4.O 4-.S 5.O 5.5 6.0 6.5 7.0 VALUES OF 150 150 DIAGRAM I for T-Beam Design. M/bt=Qd. beam is subjected to a bending moment of 550,000 in.-lb., what are the stresses in the steel and concrete respectively? 2 65 Solution. A = .4418 X 6 = 2.65 in. 2 , and p = ^^3 = 0057. d/t= 13/3 =4.33. From Diagram II, we find f s /f e = 27.5 and j=. 903. CCrVAAf) Formula (29) now gives / = 2 . 65 x 903X13 = 1827 lb ' /in ' 2 ' which f e = 18270/27.5 = 660 lb./in. 2 T-BEAMS WITH TENSION REINFORCEMENTS 185 13. The flange of a T-beam is to be 30 inches wide and 5 inches thick. The beam is to sustain a bending moment of 930,000 in.-lb., and a maximum shear of 14,500 pounds. The safe unit stresses 4-5 10 15 10 a.5 3.0 3.5 4:0 4-.S 5.0 5.5 6.0 6.5 70 VALUES OF *% DIAGRAM II for Review of T-Beam. on steel and concrete are 16,000 and 650 lb./in. 2 , and maximum unit shear 120 lb./in. 2 What dimensions of web and area of steel are required? V 14500 Solution. Assuming j f, ^'=105, and from (33) b'd = = vj luo 186 REINFORCED CONCRETE = 138 in. 2 For 6' = 8, d=18 or for b' = 7, d = 20 inches. Either of these values would give proper form to the web. The deeper beam will require less steel and may be used provided it gives suf- ficient width for placing the steel, and if the stress upon the concrete is satisfactory. Assume d = 2Q inches. Then d/t=4 and (31) 930000 ^ == 3QX5X20 ==31Q ' "^ r tnese va ^ ues Diagram I gives / c = 540 lb./in. 2 and p = .0054. A =pbd = . 0054X20X30 = 3.24 in. 2 and (20) From Table X, for six f-inch square bars, A =3.37 in. 2 , 2o= 18.0 in. four j^-inch square bars, A =3. 52 in. 2 , 2o = 15.0 in. four 1-inch round bars, A =3. 14 in. 2 , 2o=12.56 in. The four ^f-inch bars could be placed in the 7-inch width of web in two rows (see Section 109). The six f-inch bars need a width of at least 7\ inches and could be used in two rows by increasing the width of web by J inch. If d be made 21 inches, the steel needed would be A =3. 09 in. 2 and the four 1-inch round bars could be used in two rows in the 7-inch width. At ordinary prices, the saving in steel would more than pay for the increased amount of concrete, and this would make the cheapest beam. ART. 30. BEAMS REINFORCED FOR COMPRESSION 114. Flexure Formulas. It is frequently necessary to place steel in the compression as well as the tension side of a beam. When the size of a rectangular beam is limited, so that the concrete area is insufficient to carry the stress, steel may be used to take the surplus compression. In this case the concrete and steel act together, and the stress upon the steel must be limited to such an amount as will not overtax the compressive strength of the concrete. In this discussion, the following notation will be used, in addition to that employed for rectangular beams : A' = area of cross-section of compression steel; p' = ratio of compression steel area to effective area of beam, (p'=A'/bd); d'= depth of center of gravity of compression steel below compression face of beam; /' s =unit stress in compression steel; C' = total compression on steel. BEAMS REINFORCED FOR COMPRESSION 187 The same principles apply in this case as in that of the beam reinforced for tension only, and the concrete is supposed to carry compression but no tension. It is easily seen that f.-f.^J^, and that Compression on concrete, (41) and compression on steel, C' = A'f',=f',p'bd (42) Tension on steel, ' = Af s =f s pbd (43) Substituting (41) and (42) in (43) and combining with (39) and (40) we find -n(p-p f ). . . (44) Taking moments about the tension steel, we find the resisting moment of the beam, M = Cjd+C'(d-d r ) (45) Example. The use of these formulas in design will be illustrated by the following example : 14. A beam whose dimensions are 6 = 12 in., d = 22 in., d' = 2 in., is to carry a bending moment of 1,100,000 in.-lbs. The safe unit stresses are 700 and 16,000 lb./in. 2 for concrete and steel respectively, n = 15. Find the areas of steel required. Solution. For the given stresses (Table VII), A; = .397 and .; = .868. Formula (41) gives 700 C = -^ X .397 X 12 X 22 = 36680 pounds. From (45), 1, 100,000- 36680 X .868X22 20 By Formula (40), OQ7 2 /', = 15 X 700 X ' 22 = 8085 lb./in 2 188 REINFORCED CONCRETE and (42) A' = 19980/8085 = 2.47 in. 2 Area of tension steel (43), T 36680+19980 115. Tables. The labor of computation may be materially lessened by the use of tables, which may be made in several ways, of which the following seem most convenient for use. Table XI. Transposing the terms of formula (43) we have r c 1 A - -4- A f T"5T Js Js This may be placed in the form, A= Pl bd+^p, ....... (46) Js in which p\ is the ratio of steel for a beam with the same unit stresses and without compression steel. Formula (45) may be put in the form M = Rbd 2 +f' s A'(d-d'), or solving for A' -f s (d-d'y Values of R, pi and f' s , in terms of various values of f s , f c , and d'/d for n=15, are given in Table XI. This table may be used to find the areas of steel required when a beam of given dimensions must carry a bending moment too great to be resisted by tension reinforce- ment only. Table XII. Combining (41), (42) and (45) we have M = ^f c jkbd 2 -i-f s p f (l = d'/d)bd 2 , from which = f c jk+f' s p'(l-d'/d)=G, .... (48) in which G is constant for definite values of unit stresses and steel ratios. In Table XII, values of p' and p are given directly for various values of / c and G when n = 15 and /,= 16,000 lbs./in. 2 To use this table in design, it is only necessary to find G by dividing the bending moment M by bd 2 for the proposed beam and take the required ratios of steel directly from the table. BEAMS REINFORCED FOR COMPRESSION 189 TABLE XI BEAMS WITH COMPRESSION REINFORCEMENT. Values of fs in terms of f s ,f c and d'/d n = 15 fs fc R Pi VALUES OF d'/d. .06 .08 .10 .12 .14 .16 .18 .20 14,000 500 77 .0062 6210 5780 5350 4920 4490 4060 3630 3200 600 102 .0084 7620 7160 6700 6240 5780 5320 4860 4400 700 128 .0107 9030 8540 8050 7560 7070 6580 6090 5600 800 157 .0133 10440 9920 9400 8880 8360 7840 7320 6800 15,000 500 74 .0055 6150 5700 5250 4800 4350 3900 3450 3000 600 99 .0075 7560 7080 6600 6120 5640 5160 4680 4200 700 125 .0097 8970 8460 7950 7440 6930 6420 5910 5400 800 151 .0118 10380 9840 9300 8760 8220 7680 7140 6600 16,000 500 72 .0050 6090 5620 5150 4680 4210 3740 3270 2800 550 83 .0059 6745 6310 5825 5340 4855 4370 3885 3400 600 95 .0068 7500 7000 6500 6000 5500 5000 4500 4000 650 108 .0078 8205 7690 7175 6660 6145 5630 5115 4600 700 121 .0087 8910 8380 7850 7320 6790 6260 5730 5200 750 134 .0097 9615 9070 8525 7980 7435 6890 6345 5800 800 147 .0107 10320 9760 9200 8640 8080 7520 6960 6400 900 174 .0128 11730 11140 10550 9960 9370 8780 8190 7600 18,000 600 88 .0055 7380 6840 6300 5760 5220 4680 4140 3600 700 113 .0072 8790 8220 7650 7080 6510 5940 5370 4800 800 139 .0089 10200 9600 9000 8400 7800 7200 6600 6000 900 165 .0107 11610 10980 10350 9720 9090 8460 7830 7200 20,000 600 83 .0046 7260 6680 6100 5520 4940 4360 3780 3200 700 106 .0060 8670 8060 7450 6840 6230 5620 5010 4400 800 132 .0075 10080 9440 8800 8160 7520 6880 6240 5600 900 157 .0091 11490 10820 10150 9480 8810 8140 7470 6800 A' = M-Rbd* f' s (d-d')' A= Pl bd+ A'f s 190 REINFORCED CONCRETE TABLE XII. BEAMS WITH COMPRESSION STEEL Values for p'. / s = 16,000. n = 15. G = M/bd. z fc G P VALUES OF d'/d .06 .08 .10 .12 .14 .16 .18 .20 500 72 .0050 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 80 .0056 .0014 .0015 .0017 .0019 .0022 .0025 .0030 .0036 100 .0070 .0049 .0054 .0060 .0068 .0077 .0089 .0104 .0125 120 .0085 .0084 .0093 .0103 .0116 .0132 .0153 .0189 .0214 140 .0100 .0119 .0132 .0146 .0165 .0188 .0216 .0264 .0306 160 .0115 .0154 .0171 .0190 .0214 .0243 .0280 .0338 .0397 180 .0130 .0189 .0210 .0233 .0263 .0298 600 95 .0068 .0000 .0000 .0000. 0000 .0000 .0000 .0000 .0000 100 .0072 .0007 .0008 .0009 .0010 .0011 .0012 .0014 .0016 120 .0086 .0035 .0039 .0043 .0047 .0053 .0060 .0068 .0078 140 .0100 .0064 .0070 .0077 .0085 .0093 .0107 .0122 .0141 160 .0114 .0091 .0101 .0111 .0123 .0136 .0155 .0176 .0203 180 .0127 .0120 .0132 .0145 .0161 .0178 .0202 .0230 .0266 200 .0140 .0148 .0163. .0179 .0199 .0220 .0250 .0284 .0328 220 .0155 .0176 .0194 .0214 .0237 .0262 .0298 .0338 .0391 650 108 .0078 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 120 .0086 .0015 .0017 .0019 .0021 .0023 .0025 .0028 .0033 140 .0099 .0041 .0045 .0050 .0055 .0061 .0067 .0076 .0087 160 .0112 .0067 .0074 .0081 .0089 .0099 .0109 .0124 .0141 180 .0125 .0093 .0102 .0112 .0123 .0137 .0151 .0172 .0196 200 .0139 .0119 .0130 .0143 .0157 .0174 .0194 .0219 .0250 220 .0152 .0145 .0159 .0174 .0191 .0212 .0236 .0267 .0304 240 .0165 .0171 .0187 .0205 .0225 .0250 .0278 .0315 .0358 260 .0179 .0197 .0215 .0236 .0259 .0288 .0320 .0363 .0413 700 121 .0087 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 140 .0100 .0023 .0024 .0026 .0029 .0032 .0035 .0040 .0046 160 .0114 .0046 .0050 .0054 .0060 .0066 .0073 .0082 .0094 180 .0128 .0070 .0076 .0083 .0091 .0100 .0111 .0125 .0142 200 .0143 .0094 .0102 .0111 .0122 .0135 .0149 .0167 .0190 220 .0157 .0118 .0128 .0140 .0153 .0169 .0187 .0210 .0230 240 .0172 .0142 .0154 .0168 .0184 .0203 .0227 .0252 .0286 260 .0186 .0165 .0180 .0197 .0215 .0237 .0265 .0295 .0334 280 .0200 .0189 .0206 .0225 .0246 .0272 .0303 .0337 .0382 300 .0215 .0213 .0232 .0254 .0277 .0306 .0341 .0380 .0430 800 147 .0107 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 160 .0116 .0013 .0014 .0016 .0017 .0019 .0021 ,0023 .0025 180 .0131 .0034 .0036 .0040 .0043 .0048 .0052 .0058 .0064 200 .0144 .0055 .0058 .0064 .0069 .0076 .0084 .0093 .0103 220 .0159 .0075 .0080 .0088 .0095 .0105 .0115 .0128 .0143 240 .0173 .0096 .0103 .0112 .0122 .0134 .0147 .0163 .0182 260 .0188 .0116 .0125 .0136 .0148 .0163 .0179 .0198 .0221 280 .0203 .0137 .0147 .0160 .0174 .0192 .0211 .0233 .0260 300 .0217 .0158 .0169 .0184 .0200 .0220 .0242 .0268 .0299 320 .0232 .0178 .0192 .0208 .0227 .0249 .0274 .0303 .0338 BEAMS REINFORCED FOR COMPRESSION 191 TABLE XIII. BEAMS WITH COMPRESSION STEEL Values of f s /f c inTerms of p and p' d' P' VALUES OF p d .008 .009 .010 .011 .012 .014 .016 .018 .020 .022 .024 .06 .004 28.1 26.0 24.3 22.8 21.3 19.3 17.7 16.3 15.1 14.1 13.1 .006 30.1 27.8 26.0 24.3 22.9 20.5 18.8 17.3 16.1 15.0 14.0 .008 32.1 29.7 27.6 25.9 24.4 21.7 19.8 18.3 17.0 15.8 14.9 .010 34.2 31.6 29.4 27.4 25.7 23.2 21.1 19.5 .17.9 16.7 15.7 .012 36.3 33.4 31.0 29.1 27.3 24.5 22.2 20.4 18.9 17.6 16.5 .014 38.4 35.3 32.6 30.6 28.7 25.8 23.4 21.6 19.8 18.4 17.3 .016 40.5 37.2 34.3 32.2 30.2 27.0 24.6 22.8 20.7 19.3 18.2 .018 42 3 39.0 36.1 33.8 31.7 28.3 25.9 23.8 21.7 20.1 18.9 .020 44.1 40.8 37.9 35.3 33.2 29.6 27.2 24.8 22.7 21.0 19.7 .10 .004 27.5 25.6 23.9 22.4 21.1 19.1 17.5 16.1 15.0 14.0 13.1 .006 29.2 27.2 25.3 23.7 22.4 20.2 18.5 17.0 15.8 14.8 13.8 .008 30.8 28.7 26.7 24.9 23.7 21.3 19.4 18.0 16.7 15.5 14.6 .010 32.6 30.1 28.1 26.3 25.0 22.4 20.4 '18.9 17.6 16.4 15.4 .012 34.3 31.5 29.5 27.7 26.2 23.5 21.5 19.7 18.4 17.2 16.1 .014 36.0 33.1 30.9 29.0 27.4 24.6 22.5 20.6 19.2 18.0 16.9 .016 37.6 34.6 32.3 30.2 28.6 25.8 23.5 21.5 20.0 18.7 17.6 .018 38.1 36.1 33.7 31.6 29.8 26.9 24.4 22.4 20.8 19.5 18.3 .020 40.6' 37.7 35.1 32.9 30.9 27.9 25.4 23.4 21.6 20.2 18.9 .14 .004 26.9 25.1 23.4 22.0 20.8 18.8 17.3 16.0 14.8 13.8 12.9 .006 28.4 26.4 24.6 23.2 21.9 19.8 18.2 16.8 15.5 14.5 13.6 .008 29.8 27.6 25.8 24.3 23.0 20.8 19.0 17.5 16.3 15.2 14.3 .010 31.2 28.9 27.0 25.4 24.1 21.8 19.9 18.3 17.1 16.0 15.0 .012 32.5 30.2 28.2 26.5 25.1 22.7 20.7 19.1 17.8 16.7 15.7 .014 33.7 31.4 29.4 27.6 26.1 23.6 21.6 19.9 18.5 17.4 16.3 .016 34.9 32.5 30.5 28.7 27.1 24.5 22.4 20.6 19.2 18.0 16.9 .018 36.1 33.7 31.6 29.8 28.1 25.4 23.2 21.3 19.9 18.7 17.5 .020 37.1 34.9 32.6 30.8 29.0 26.2 24.0 22.0 20.6 19.3 18.1 .18 .004 26.1 24.6 23.1 21.7 20.5 18.6 17.1 15.7 14.6 13.6 12.8 .006 26.9 25.6 24.1 22.7 21.4 19.4 17.8 16.4 15.3 14.3 13.4 .008 28.6 26.6 25.0 23.6 22.3 20.2 18.5 17.1 15.9 14.9 14.0 .010 29.6 27.6 25.9 24.5 23.2 21.0 18.3 17.8 16.5 15.5 14.6 .012 30.6 28.6 26.8 25.3 24.0 21.8 20.0 18.5 17.2 16.1 15.2 .014 31.6 29.6 27.7 26.1 24.8 22.6 20.7 19.2 17.9 16.7 15.8 .016 32.6 30.5 28.6 27.0 25.6 23.3 21.4 19.8 18.5 17.3 16.3 .018 33.6 31.4 29.4 27.8 26.4 24.1 22.1 20.4 19.1 17.9 16.9 .020 34.5 32.2 30.3 28.6 27.2 24.8 22.8 21.0 19.7 18.4 17.4 192 REINFORCED CONCRETE TABLE XIV. BEAMS WITH COMPRESSION STEEL Values of N in Formula, Nf c =M/bd; n = 15 t d r d VALUES OF p' .002 .004 .006 .008 .010 .012 .014 .016 .018 .020 16 .06 .228 .252 .277 .302 .327 .351 .376 .401 .425 .450 .10 .224 .246 .267 .289 .310 .331 .353 .374 .395 .417 .14 .221 .240 .258 .276 295 .313 .331 .349 .368 .386 .18 .219 .234 .250 .265 .281 .297 .312 .328 .343 .359 18 .06 .217 .241 .266 .290 .315 .339 .364 .388 .413 .437 .10 .213 .234 .255 .276 .297 .318 .339 .360 .381 .402 .14 .210 .228 .246 .264 .281 .299 .317 .335 .353 .370 .18 .207 .222 .237 .252 .266 .281 .296 .311 .326 .340 20 .05 .208 .232 .256 .281 .305 .329 .354 .378 .402 .426 .10 .205 .225 .246 .267 .287 .308 .329 .349 .370 .391 .14 .201 .219 .236 .253 .271 .288 .306 .323 .340 .357 .18 .198 .212 .227 .241 .254 .269 .283 .297 .311 .326 22 .06 .199 .223 '.247 .271 .295 .319 .343 .367 .391 .415 .10 .195 .216 .236 .256 .277 .297 .317 .337 .358 .378. .14 .192 .209 .226 .243 .260 .276 .293 .310 .327 .344 .18 .187 .202 .216 .229 .243 .257 .270 .284 .297 .311 24 .06 .191 .215 .239 .263 .286 .310 .334 .353 .381 .405 .10 .187 .207 .227 .247 .267 .287 .307 .327 .347 .367 .14 .184 .200 .217 .233 .249 .266 .282 .299 .315 .332 .18 .181 .193 .207 .220 .233 .246 .259 .272 .285 .295 26 .06 .183 .207 .230 .254 .277 .301 .324 .348 .375 .395 .10 .179 .199 .218 .238 .257 .277 .296 .316 .335 .355 .14 .176 .192 .207 .223 .239 .255 .271 .287 .303 .319 .18 .172 .185 .197 .209 .222 .234 .246 .259 .271 .284 28 .06 .177 .201 .224 .247 .271 .294 .317 .340 .364 .387 .10 .173 .193 .212 .231 .250 .270 .284 .308 .327 .346 .14 .170 .185 .201 .216 .231 .247 .263 .278 .293 .309 .18 .166 .178 .190 .202 .214 .226 .237 .249 .261 .273 30 .06 .172 .195 .218 .241 .264 .287 .310 .334 .357 .380 .10 .168 .187 .205 .224 .243 .262 .281 .300 .319 .338 .14 .164 .179 .194 .209 .224 .238 .254 .269 .284 .299 .18 .160 .171 .183 .194 .205 .217 .228 .239 .251 .262 32 .06 .166 .188 .211 .233 .256 .278 .301 .323 .346 .368 .10 .161 .180 .199 .217 .236 .254 .273 .292 .310 .329 .14 .157 .172 .186 .201 .215 .230 .244 .259 .273 .289 .18 .154 .165 .175 .186 .197 .208 .219 .229 .240 .251 BEAMS REINFORCED FOR COMPRESSION 193 Tables XIII and XIV. If Formulas (39) and (44) be combined a value for f s /f c in terms of p, p' and d'/d may be found. These values are tabulated in Table XIII. If the values of /', from (40) be substituted in (48), it becomes M and making 4Jfc +np '(*=|V^ (l- d ' /d) =N> we have Combining the above value of N with (39) we find that the value of N depends upon n, f s /f c , p' and d'/d. In Table XIV, values of N are tabulated for various values of f s /f c , p' and d'/d when n= 15. These tables may be used in the investigation of beams of known dimensions and reinforcement, for the purpose of finding the safe resisting moment, or the unit stresses under given bending moment. Examples. The use of these tables will be best illustrated by a few examples. 15. Solve Problem 14 (p. 187) by the use of the tables. Solution. As w = 15 and /, = 16,000 lb./in. 2 , Table XII may be used. d'/e*=.09, G=M/bd?= "L = 190. From Table XII, with / c =700, =190 and d'/d =.09, we find directly that p = .0136 and p' = .0093, from which, A = .0136X12X22 = 3.59 in. 2 , and A' -.0093X12X22 = 2.46 in. 2 16. A rectangular beam has the following dimensions; 6 = 10 inches, d = 18 inches, d' = 1.5 inches, and is to carry a bending moment of 550,000 in.-lb. The safe unit stresses are 600 and 14,000 lb./in. 2 for concrete and steel respectively, n = 15. Find the areas of steel required. Solution. d'/d= 1.5/18 = .083. From Table XI for /, = 14,000, f c = 600, and d'/d = .083, we find R = 102, p f = .0084, f s = 7090 lb./in. 2 substituting these values in (47) there results ,, 550000-102X10X18X18 A== - 7090X16.5- ~ = 1 - 88in - 2 and (46) 70QO A = .0084X10X18+,^X1.88 = 2.46 in 2 IttUUU 194 REINFORCED CONCRETE 17. A rectangular beam in which 6 = 10 inches, d=22 inches and d' = 2 inches, is reinforced with 2.6 in. 2 of steel in tension and the same amount in compression. The beam carries a bending moment of 850,000 in./lb. What are the maximum unit stresses upon the steel and concrete respectively? Solution. p = p' = T7^7. = .0118. d'/d = .09. For these values Table XIII gives /.// =26.5 and Table XIV, N = 280. Then formula (49) /,= 627X26.5 =16620 lb./in. 2 18. A rectangular beam has 6=10 inches, d=16 inches, d' = 2 inches, A ' = 2.4 in. 2 , A =2.25 in. 2 . If the safe unit stresses on steel and concrete are 16000 and 650 lb./in. 2 respectively, what is the safe resisting moment for the beam? Solution. p'= Qx =.015, p= ^ =.014, d'/d = .l25. From Table XIII, for these values /,//, = 23.0 and Table XIV, N = 306. If /. = 16,000, f e = 16,000/23 = 696 lb./in. 2 , which is greater than is allowable. The safe moment will therefore be that which produces a stress of 650 lb./in. 2 in the concrete. Substituting in (49), M = 306X650X10X16X16 = 509,184 in.-lb. ART. 31. SLAB AND BEAM DESIGN 116. Bending Moments and Shears. Structural forms in which slabs of concrete are supported by T-beams are very common in rein- forced concrete structures. In this type of construction, the slab is commonly made continuous over the T-beam and forms the flange of the T-beam (see Fig. 53), being built with the beam and a part of it. In determining the bending moments and shears in such con- struction, the loads may usually be taken as uniform, and the slabs and beams as fully or partly continuous, depending upon the method of support. Fully Continuous Beams. If a slab which passes over one or more cross-beams is firmly held at the ends by being built into and tied by reinforcement to a wall or heavy beam, it may be considered as fully continuous, and when uniformly loaded, the positive moments of the middle of the spans are -j^wl 2 and the negative moments at SLAB AND BEAM DESIGN 195 supports -j^wl 2 . . The shear at each end of span in such a beam is f wl. If the movable load covers some of the spans leaving others unloaded, these moments may be somewhat increased. For slabs of this type, it is conservative practice to use -f^wl 2 for both positive and negative bending moments and \wl for maximum vertical shear. Supported Ends. The ends of continuous beams, resting upon side walls or end columns, cannot be considered as fixed, and are to be taken as simply supported. Such a beam, or a slab the ends of which are not fixed, has greater positive moments in the end spans and greater negative moments at the first supports from the ends than fully continuous beams. These moments are usually taken as ^wl 2 for beams of more than two spans. The shear in the end span next the first support may be greater than one-half the load on the span and should be taken as .6 wl. For beams of two spans, the nega- tive moment at the middle support is taken as \wl 2 , and the positive moment as -^wl 2 . The moments for continuous beams of unequal spans, or with concentrated and uneven loading should be carefully determined for each individual case. The Joint Committee makes the following recommendations in its 1916 report: (a) For floor slabs the bending moments at center and at support should wl 2 be taken at - for both dead and line loads, where w represents the \2i load per linear unit and I the span length. (6) For beams the bending moment at center and at support for interior wl z spans should be taken at -, and for end spans it should be taken wl 2 at for center and interior support, for both dead and live loads. (c) In the case of beams and slabs continuous for two spans only, with their ends restrained, the bending moment both at the central support and near the middle of the span should be taken at -. (d) At the ends of continuous beams the amount of negative moment which will be developed in the beam will depend on the condition of restraint or fixedness, and this will depend on the form of construction wl 2 used. In the ordinary cases a moment of : - may be taken; for lo small beams running into heavy columns this should be increased, wl 2 but not to exceed . For spans of unusual length, or for spans of materially unequal length, more exact calculations should be made. Special consideration is also required in the case of concentrated loads. 196 REINFORCED CONCRETE Even if the center of the span is designed for a greater bending moment than is called for by (a) or (6), the negative moment at the support should not be taken as less than the values there given. 117. Loading of Slabs, Beams and Girders. Slabs are commonly used as continuous beams passing over a number of T-beams, of which the slab forms the flange as shown in Fig. 53. "V ^~ . FIG. 53. Reinforced Slab and T-Beam. They are reinforced for tension in one direction, perpendicular to the T-beams, and in computation are considered as rectangular beams one foot in width. The T-beams supporting such slabs frequently rest upon girders, which are used to widen the interval between columns, and permit the T-beams to be spaced close enough for economical design of slab. The load upon a T-beam in such a system is uniformly distributed and consists of the weight of a half span of the slab and its load, on each side of the beam. The loads upon the girders are con- centrated at the points where the T-beams cross, but may usually be taken as uniformly distributed without material error. Double Reinforced Slabs. Slabs of long span and nearly square in plan may be supported on all four sides and reinforced in both directions. It is not feasible to make an accurate analysis of the dis- tribution of loadings in such a slab. When the length and width of slab are equal, it is assumed that the reinforcement in each direction carries one-half the load as uniformly distributed. The loads carried by the mid sections (aaaa, bbbb, Fig. 54) are, however, greater than those carried by the sections next the supports, and the reinforcement should be spaced closer in the middle than at the sides. It is sug- gested that the mid-half area (aaaa, 6666, Fig. 80) of the slab be con- sidered as carrying If times the average load, and the side sections (acca, 6cc6) two-thirds of the average load per square foot of slab. When the slabs are not square, the reinforcement parallel to its shorter dimension carries the greater part of the load. The Joint Committee makes the following recommendation concerning the division of the loads in such slabs: Floor slabs having the supports extending along the four sides should be designed and reinforced as continuous over the supports. If the length of the SLAB AND BEAM DESIGN 197 slab exceeds 1 .5 times its width the entire load should be carried by transverse reinforcement. For uniformly distributed loads on square slabs', one-half the live and dead load may be used in the calculations of moment to be resisted in each direction. For oblong slabs, the length of which is not greater than one and one-half times their width, the moment to be resisted by the transverse reinforcement may be found by using a proportion of the live and dead load equal to that given by the formula r = =- 0.5, where I = length and 6 = breadth of slab. The longitudinal o reinforcement should then be proportioned to carry the remainder of the load. In placing reinforcement in such slabs account may well be taken of the fact that the bending moment is greater near the center of the slab than near the Jb FIG. 54. Double-reinforced Slabs. For this purpose two-thirds of the previously calculated moments may be assumed as carried by the center half of the slab and one-third by the oustide quarters. An interesting discussion of the distribution of stresses in double reinforced slabs may be found in a paper by Mr. A. C. Janni in Trans- actions of the American Society of Civil Engineers, 1917. 118. Problems in Design. The use of the formulas and tables which have been given, in designing slab and beams, will be illustrated by the solution of a few problems. In these examples, the working stresses recommended by the Joint Committee for 2000 pounds con- crete will be used. Example 19. A concrete slab is to be supported by T-beams 6 feet apart c. to c., and to carry a live load of 250 pounds per square 198 REINFORCED CONCRETE foot. The T-beams have a clear span of 17 J feet and are built into brick walls at the ends. Design the slab and beams. Solution. Assume the weight of slab as 50 pounds per square foot, giving a total load of 300 pounds per linear foot for a section of slab 12 inches wide. Taking the slab as fully continuous. ., wl 2 300X6X6X12 = 10,800 m.-lb. i - i - From Table VII, for / s = 16,000 and /< = 650, # = 108, p = .0078 and j=.874. Formula (9) gives 12d 2 = 10,800/1 08 =100, and d = 2.9 inches, use 3 inches. A = p6d = 3Xl2X.0078 = .277 in. 2 From Table XV (p. 199), we select f-inch round bars spaced 4.5 inches apart, A = .29 in. 2 If concrete extends f inch below steel, the thickness of slab is 3J inches, and the weight of slab is 150X3.75/12 = 47 pounds per square foot, which agrees with the assumed load. FIG. 55. T-Beam Design. Reinforcement for negative moment over the supports should be the same as for positive moment at mid-span, and will be provided by turning up every alternate bar at the quarter point on each side of the support and continuing them over the support to the one-third point. Transverse reinforcement to prevent cracks will be provided by using f-inch bars spaced 12 inches apart. Unit shear at ends of slab, V 300X3 bjd 12 X. 874X3 28.61b./in. 2 No diagonal tension reinforcement is necessary. T-beam. Assuming the weight of the web of the T-beam as 150 pounds per linear foot, the total load on the T-beam is 6(250+47) + 150=1930 pounds per linear foot. Taking the bearing upon the wall as 6 inches the effective length of T-beam between centers of bearings is 17.5+.5=18 feet. SLAB AND BEAM DESIGN 199 00 SIO rH C5 rH (N (N CO ** iO t- 00 rH 00^ S2S?5 sss^ss S25S OOOO OOOOOrH rHrH^ OOOO 00000 OrH rHrH - rH |> * CO rHrH (N CO OOOO CO"* t-rH CO rH * O CO 00 C5 CO OOOOOrH rH 1>00 l> rH CO gr3S^ OOOO CO * O CO t^ 00000 9^ 2 rH rH rH O5 CO 2888 o co oo o c^ co lOCOt^CirH iO 8 COIN lOrH o 10 cq o rH rH (N CO C5 O i 1 * 00 CO O CO t^> 00 cq *o cs '* OOOO OOOOrH rH (N(N CO OOOO OOOOrH rH i-H i-H C^ a iO CO * CO rH C^ CO * co "^ co *o "* b- Oi rH CO OO ^O^ (N 00 CO CO T-HT-KMCO t^O-* CicO -* cot- GOO * 00 Ci ^O T^OOCOCS o OOOO OOOrH rH rH (N COCO OOOO OOOOrH rHrH (NO, OF BARS 3 rH C5 rH ^ COCO S&OCO IO CO CO >O rH C4 CO * O5 IO (M rH (M O CO 00 00 00 CO OS CO CO i OOOO OOrH rH rH (?q COCO 1 * OOOO OOOrH rH rH (M (M CO DISTANC 3 rH CO 00 CO c^ co * co . OOOO 00 O CO CO O5 CO O rH rH rH rH O5 %%% rH (^ CO0 OOOO |> IO IO trH CO 00 O . CO O5 IO CO CO *O OO C^l C^ 88 OrH TtHO 00 C5 CO 00 t- !> O5 Cq * t O"* 00 "* rH O5 OOOO rH rH rH rH (N CO ^lj OOOO OOrH rH rH (N COCO 5 CO * CO Oi odo'd T-H T-H T-H (N (N CO 5 OOOO O5 rH T^ (>. rH O rH rH rH Oq 00 N. oot- (NCO - O O5 CO 00 CO rH rH (N (N CO OOOO 00 O5 * CO iO rH TjH 00 Cl CO rH rH rH . O (N iO rH tTt< CO O5 CO O O * t^ i-H IO OOrH rH C5 CO O O5 * rH (M CO CO * IIII 02 OOOO 000000 rH rH rH OOOO ooo oo OOOrH I 1 1 (N CO 00 rH rH COl> IO gjCO CO"* CO O CO 00 OC COCO OO O CO cO O5 CO foiH co co CO CO t^- rH rH (M CO O CO -* * co O cOOOOiM O OCO.COrH & gdddd OrHrHrHrH CO CO rH (N rH 00 \Tt< 00 rH C5 IO rH CO t 00 Tt< rH i-H fHrHi-H p co co O^^OOrH^ CO CO rH 00 00^ rH rH rH 200 REINFORCED CONCRETE The maximum shear 7=1930x9 = 17370 pounds. The area V 17370 uired for shear, assuming j = f, b'd = = -^^-=165. in. 2 vj luo For 6' =8, d=2l or for 6' =9, d=18.5. Take 6' =9, and (2 = 18.5. Width of flange 6 = 2X6X3f +9 = 54 inches, and by (31) ~ M 923400 d from Diagram I, / c = 430 lb./in. 2 , and p = .0035, then A =pbd = . 0035X54X18.5 = 3.50 in. 2 , and from Table X, six f-inch round bars in two rows, 2 inches c. to c., spaced 2.75 inches apart in the rows and 1.75 inches from side of web. 4=3.61 in. 2 As the ends of the beam are built into the walls, some negative moment may be developed at the supports, which might cause cracks to occur unless reinforced. The upper layer of reinforcement will therefore be turned up, two rods at the quarter point and the other midway between the quarter point and support, and extend to the end of the beam (see Fig. 55). If the concrete extend 2 inches below the steel, the weight of web below the slab is 9(18.5+3- 3.75) X 150/144 =166 pounds per linear foot. This is a little greater than the assumed value, but would add less than 1 per cent to the total load and need not be redesigned. For the three bars in bottom of beam at the support. Table X gives 20 = 3X2.75 = 8.25, and the unit bond stress u= b 'v/2o= 9X120/8.25 = 131 lb./in. 2 This is too great for safety, and the bars should be bent into hooks at the ends. v is 120 lb./in. 2 at the supports, and diagonal tension reinforce- ment is needed where v is more than 40 lb./in. 2 Stirrups will be needed for two-thirds of the distance from the support to the mid- span, or 6 feet. If the stirrups be spaced s = d/2 = 9 inches apart, eight stirrups will be needed at each end of the beam. For the stirrups next the support (34) A vb's 120X9X9 q . 2 A ' = ~2f-= 2X16000 = ' 31in> Two J-inch round bars, bent as shown (Fig. 55) may be used for the first four stirrups, and f-inch bars for the four nearer the middle of the beam. SLAB AND BEAM DESIGN 201 Example 20. A reinforced concrete slab, to carry a live load of 200 pounds per square foot, is to rest upon a series of T-beams 5 feet apart c. to c. The T-beams are to be continuous for three spans over girders 15 feet c. to c. The girders are supported by walls at the ends and have a span of 20 feet. Design the slab and beams. Solution. Assume the weight of slab at 40 pounds per square foot; then M = 24QX5 * 5X12 # = 108, p = .0078, j = .874, 12^ = 6000/108 = 55.5 and d=2.15, Take d = 2.25 in. A = 2.25 X 12 X .0078 = .21 in. 2 If concrete extend f-inch below steel, the total depth of slab is 3 inches, and weight of slab is 150XA = 37.5 pounds per square foot. From Table XV j^-inch round bars spaced 4 inches apart give A = .23. in. 2 Negative moment at supports will be provided for by bending these up at the quarter points. For lateral reinforcement to prevent cracks, ^-inch round bars spaced 12 inches c. to c. will be used. T-beams. Assuming weight of web of T-beam as 125 pounds per linear foot, load upon T-beam is 5(200+40) + 125= 1325 pounds per linear foot and total span load is 1325 X 15= 19,875 pounds. Maximum shear in end span next girder is V =19,875 X.6 = 11,- 925 pounds, and 6^=7/^=11,925/105=113 in. 2 7X16 or 8X14 might be used. Try 7X16, then M=Wl/10= 19,875X15X12/10 = 357,750 in.-lb. Taking overhang of flange as six times its depth, 6 = 2X6X3+7 = 43 inches and Formula (31) From diagram 7, / c = 325 lb./in. 2 and p=.0024. ^1 = 1.65 in. 2 Table X, six f-inch round bars, A = 1.84 in. 2 in two rows, If inches apart and spaced 2 inches c. to c. and 1.5 inches from side of web. If concrete extends 2 inches below steel, the weight of web below slab is 7X16X150/144=117 pounds per linear foot, which is within the assumed load. The negative moment at crossing of girder is equal to the posi- tive moment already found. Turn up the upper row of bars on each side to provide for tension at top of beam and run the lower ones through at bottom to provide compression reinforcement as shown in Fig. 56. We now have a beam with compression reinforcement, in which 6 = 7, d = 16, d' = 3, A=A'=1.84, p=p'= 1.84/112 = .0164, 202 REINFORCED CONCRETE Formula (48) gives G = = = 200, and Table XII, for/ 5 =16000, / c = 650, = 200, and d'/d = .!S, we find that p = .0139 and p' = .0219 are required. The area of steel in compression (p .0164) is not sufficient and we must either increase the area of compression steel in the bottom of beam or increase the area of concrete section over the support. Try making d 17 inches. Then Now from Table XII, we find that p = .0123 and p' = .0156 are required. The reinforcement is now sufficient and we will increase FIG. 56. T-Beam and Girder. the depth to 17 inches at the girder, sloping the haunches as shown in Fig. 56. Diagonal Tension. Assuming J as .85, the maximum unit shear = = 118 lb./in. 2 If stirrups be spaced next the girder is v = 85 x 17 8 inches apart, the area required for the end stirrups is (34) 118X7X8 2X16000 = .21 in. 2 Two f-inch bars will answer, or a f-inch bar bent to U-shape around horizontal reinforcement. Stirrups will be needed to 6 feet from girder and 4 feet from end support in the end spans and 5 feet from girder on each end of the middle span. Girders. The girders are simple rectangular beams carrying three concentrated loads at the middle and quarter points. Each load is 1.1 times a span load of the T-beam, or 1.1X19875 = 21862 pounds; assuming that the girder weighs 800 pounds per linear foot, the reaction or shear at the support is 1.5X21862+800X10=40793 pounds and the maximum bending moment M = 40,793 X (120 29,862) X 60 = 3,103,440 in.-lb. bd 2 = 3, 103,440/108 = 28,735. For SLAB AND BEAM DESIGN 203 6 = 20, d=38; for 6 = 18, d=40. Try 6 = 18, d = 40; then A = .0078X 18X40 = 5.61 in. 2 . Ten f-inch square bars (4 = 5.62) placed in two rows If inches c. to c., six bars in lower and four in upper layer (Fig. 56). If concrete extend 2} inches below center of lower layer of steel, the beam is 43 inches deep and weighs 18 X 43 X 150/144 = 806 pounds per linear foot, which agrees with the assumed weight. 40793 The maximum unit shear v = Q7/1x//m =65 lb./in. 2 Diag- onal tension reinforcement will be needed from support to first load (60 inches) . This may be supplied by bending up horizontal steel. The bending moment at first load is 40,793 X 60 -4000X30 = 2,327,580 in.-lb. This is about three-fourths of the moment at the middle and two bars may be bent up at this point. For two bars 4^ = 1.12 in. 2 , and Formula (15) A d f s V2 1.12X16000X1.4 s = -^-r = - grvxio - =21.5 inches. vb 65 X 18 Turn up pairs of bars at 20, 40 and 60 inches from support. The bond stress on four horizontal bars at end of beam is bv 18X65 _ ,. This is rather large unless deformed bars are used, and bars should be bent into hooks at ends. Example 21. A reinforced concrete slab, divided into panels 12 ft.X!4 feet, by T-beam supports is to carry a live load of 150 pounds per square foot. The T-beams are supported by columns at the corners of the panels; their ends resting upon side walls. Design the slab and beams. Solution. Assume the weight of slab at 70 pounds per square foot. The proportion of load carried by the 12-foot span is 14/12 0.5 = .67 (see Section 117). The load on the slab in the 12-foot length is (150+70) X. 67 =147 Ib. per square foot and in the 14-foot length 220 X. 34 = 75 pounds per square foot. If 4/3 of the average load per square foot be borne by the mid-section, the load to be carried by a 12-inch width will be 147X4/3 = 196 pounds per linear foot. 204 REINFORCED CONCRETE 12d 2 = 28224/108 = 261 and d = 4.75 inches. If the concrete extend f inch below steel, the total depth of slab will be 5.5 inches and the weight of slab 150X5.5/12 = 69 pounds per square foot, as assumed. A = .0078X12X4.75 = .44 in. 2 From Table XV, f-inch square bars spaced 3.5 inch c. to c. may be used. Alternate bars in each span will be turned up at the quarter points for negative shear at the supports. The side-sections of the shorter span will carry one-half the moment of the mid-sections, and will need about one-half the rein- forcement. We will space the f-inch bars 6 inches apart for the side sections. For the longer span (14 feet) the load upon the mid-section will be 75X4/3 = 100 pounds per linear foot, and the bending moment ,, 100X14X14X12 1ftnAA . 1U T , . , M - -= = 19,600 in.-lb. If we place the reinf orce- ment in the 14-foot direction on top of that in the shorter span, the effective depth will be about J-inch less, or d = 4.75 0.5 =4.25 inches, and R== ^2 = ^x^25X4: 25 = 86. From Table VII, we find that if / s =16,000 and # = 86, p = .006 and / c = 565 lb./in. 2 A = .006X12 X4.25 = .306 in. 2 and Table XV gives f-inch round bars spaced 4 inches apart, A = .33 in. 2 Use these for mid-half of slab and bars of the same size spaced 7 inches apart for the side-sections. T -Beams. Assuming the longer T-beam to weigh 250 pounds per linear foot, the total load will be 196X12X14+250X14=36,428 pounds. The maximum shear will be V = 36,428 X .6 = 21,857 pounds and section needed for shear &'d = 21,857/105 = 208 in. 2 Try V = 10, d = 2l inches. The load at the middle of the beam is greater than that at the ends; this somewhat increases the moment, but the error will not be more than about 2 per cent if the load be taken as uniformly distributed. ul 35428X14X12 . M = = = 611,990 in.-lb. Taking the width of flange as one-fourth the length of beam 6=45 in. M 611990 d/*- ~ From Diagram I, we find that the neutral axis is in the flange and the beam should be designed as a rectangular section. and SLAB AND BEAM DESIGN 205 # = 29, we see that f e will be less than 350 lb./in. 2 and the steel needed is p = .0021 (p = 7#/100QOO approximately) or A = .0021X45 X 20 =1.89 in. 2 Four fj-inch square bars may be used. Two of these bars to be turned up at the quarter point on each side of the support to provide for tension due to negative moment. We now have at the support a double reinforced beam in which 6 = 10 inches, d = 21 inches, A = A ' = 1.89 in. 2 , p = p' = 1.89/210 = .0090, d r = 2 inches, d'/d = .095 and G = in !^o??? 01 = 136 - From Table XII > lUX^l X^l for /,= 16,000, f e = 650, =136 and d'/d =.095 we find that p = 0096 and p' = .0042 are required. The reinforcement for tension is a little small, but as the beam will be strengthened by the slab rein- forcement parallel to it, the j^-inch bars will probably be ample. Diagonal tension reinforcement will be needed for 5 feet from the supports. If stirrups be spaced 9 inches apart, seven stirrups will be needed. The first stirrup will require A ___ QA . 2 A '~2/ s ~ 2X16000 = ^-inch round bars bent to U-shape will answer for the first three stirrups, the four next the middle of the beam may be f-inch. The loads upon the shorter beams, assuming the beam to weigh 150 pounds per foot, are 100X14X12+150X12 = 18600 pounds. The maximum shear is 18600 X.6 = 11160 pounds. 6^=11160/105 = 106. A section 7 inches X 16 inches might be used, but assuming that the depth must be the same as for the longer beams, we may use 7 inches X 21 inches. Then M== 18600 XJ2X 12 = 26784Q in _ lb ^ and 6 = //4 = 36 inches 267840 As before, the neutral axis is in the flange, # and Table VII, f e will be small and p = .0012. A = .0012X36X21 = 0.90 in. 2 Three f-inch round bars will be used. Part of these bars will be turned up, two on one side and one on the other of each support to provide for tension due to negative moment. Then and from Table XII we find that no compression steel is necessary. Maximum unit shear, v= Q^K = ^ lb./in. 2 Diagonal i X^l X . 01 O 206 REINFORCED CONCRETE tension reinforcement is needed 72X47/87=40 inches from support. Spacing stirrups 10 inches apart, for end stirrups 87X10X10 Ac 2X16000 Use f-inch square bars bent to U-shape. .27 in. 2 ART. 32. CONCRETE COLUMNS 119. Plain Concrete Columns. The strength of plain concrete in compression has been discussed in Section 94. The failure of a short block under compression occurs through lateral expansion and the shearing of the material on surfaces making angles of about 30 with the line of pressure as shown in Fig. 57 (a). As the height of block becomes greater in proportion to its diameter, the resistance of the concrete becomes less certain and plain columns in which the length is more than four times the height frequently fail by shearing diagonally across the column as shown in Fig. 57 (6). This usually T (a) (5) FIG. 57. Crushing of Concrete Columns. occurs where the concrete is of good quality and shows high crushing strength. Weaker concrete usually fails by local crushing. Columns in which the lengths are more than six or eight times the diameters are usually reinforced. The Joint Committee recommends that all columns more than four diameters be reinforced, and that the stress on plain columns be limited to 22.5 per cent of the ultimate crushing strength of the concrete. The use of concrete rich in cement is nearly always advisable in the construction of columns, on account of the greater reliability of such concrete, as well as because of the economy of reduced section CONCRETE COLUMNS 207 allowable with rich concrete. In reinforced columns, concrete of high compressive strength also admits of more economical use of steel, through employing higher unit stresses than are admissible with less rich concrete. Concrete less rich than 1 to 6 (2000 pounds) mixtures (see Section 94) is undesirable in column work and richer mixtures are commonly preferable. 120. Longitudinal Reinforcement. Longitudinal bars in the corners of square columns, or near the exterior surfaces of round columns, diminish the uncertainty of action of the columns through preventing the material yielding at points of local weakness. Such reinforcement should always be stayed by light band reinforcement at frequent intervals as shown in Fig. 58 (a). This will prevent the longitudinal bars breaking away from the column through bending when loaded. When a column containing longitudinal steel is loaded, the con- crete and steel are shortened by the compression to the same extent and the stress carried by each material is proportional to its modulus of elasticity. Let A = cross-section of column ; AS = cross-section of steel; p = steel ratio = As/ A ; n = ratio of moduli of elasticity = E S /E C ', P = total load on columns ; f c = unit compression on concrete ; / s = unit compression on steel = nf c . The total area of concrete is A(l p), and P=f e A(l-p)+f.A.=f e (A-pA)+f c npA, or . .. . ;- . . . ' : . . (50) The Joint Committee recommends the following working stresses : (a) Columns with longitudinal reinforcement to the extent of not less than 1 per cent and not more than 4 per cent, and with lateral ties of not less than j inch in diameter 12 inches apart, nor more than 16 diameters of the longitudinal bar: the unit stress recommended for axial compression, on concrete piers having a length not more than four diameters. The Committee also recommends that the ratio of unsupported length of column to its least width be limited to 15, and that the hoops or bands are not to be counted on directly as adding to the strength of the column. 208 REINFORCED CONCRETE In Formula (50), if we let Z=l + (n l)p = r -r, and tabulate h A values of Z (see Table XVI) in terms of n and p, the computation of columns of this type becomes very simple. TABLE XVI. COLUMNS WITH LONGITUDINAL REINFORCEMENT P Values of Z = , in Terms of n and p fcA P n = 10 n = 12 n = 15 P n = 10 n = 12 n = 15 0.006 1.054 .066 1.084 0.021 .189 .231 1.294 0.007 1.063 .077 1.098 0.022 .198 .242 1.308 0.008 1.072 .088 1.112 0.023 .207 .253 1.322 0.009 1.081 .099 1.126 0.024 .216 .264 1.336 0.010 1.090 .110 I.f40 0.025 .225 .275 1.350 0.011 1.099 .121 1.154 0.026 .234 .286 .364 0.012 1.108 .132 1.168 0.027 .243 .297 .378 0.013 1.117 .143 1.182 0.028 .252 .308 .392 0.014 1.126 .154 1.196 0.029 .261 .319 .406 0.015 1.135 .165 1.210 0.030 .270 .330 .420 0.016 1.144 .176 1.224 0.032 .288 .352 .448 0.017 1.153 .187 1.238 0.034 .306 .374 .476 0.018 1.162 .198 1.252 0.036 1.324 .396 .504 0.019 1.171 .209 1.266 0.038 1.342 .418 .532 0.020 1.180 .220 1.280 0.040 1.360 .440 .560 Example 22. A square column is to carry a load of 95,000 pounds, and to be reinforced with 2 per cent of longitudinal steel. If f c =450 Ib./in. and n=15, find dimensions for column and steel. Solution. From Table XVI, for ft =15 and p = .020, we find Z= 1.280. Then A = ^-~=165, and side of column =13 inches. A,= . 020X165 = 3.30 in. 2 From Table X, four ^f-inch square bars may be used, A s = 3.52 inches. If 1 to 3 concrete of 3000 pounds compressive strength (see Section 94) were used in the above problem, we would have f c = 675, n=10, 7 = 1.18, A = 119 in. 2 and A s = 2.38 in. 2 The quantities of materials required would be reduced about 25 per cent, while the proportion of cement in the concrete would be about doubled. Example 23. A column 14 in.Xl4 in. section is to carry a load of 130000 pounds. If / c = 450 and n=15 find area of steel required. CONCRETE COLUMNS 209 Solution. Z 130000 1 .. IT-TV;- = 1.474 and from Table XVI, 450X14X14 p = .034. Then A s =. 034X14X14 = 6.66 in. 2 This might be four l|-inch round bars at the corners (A s = 7.07), or eight ff-inch square bars at corners and middle of sides (A s = 7.03), or four IJ-inch round bars at corners and four f-inch round bars at middle of sides. (A s = 6.68in. 2 ). The Joint Committee recommends a minimum of 1 per cent of longitudinal steel for columns of more than four diameters in length. This gives rigidity to the column, and security against local yielding in the concrete. High percentages of longitudinal steel are not (a) ft) FIG. 58. Reinforced Concrete Columns. usually economical, because of the greater cost of steel as compared with concrete for resisting compression, particularly when the stresses in the steel are limited by those in the concrete. When the concrete is used for fireproofing, the steel should be covered by at least 2 inches of concrete, and about 1 \ inches of con- crete on the exterior of the column should not be considered in deter- mining the strength of the column. 121. Columns with Hooped Reinforcement. As shown in Section 95, the failure of concrete under compression commonly occurs through shearing due to lateral expansion. If the concrete in the column be held by band or spiral steel (see Fig. 58 (6) ) from yielding to 210 REINFORCED CONCRETE lateral expansion, the resistance to crushing will be materially in- creased. Such reinforcement is either formed of steel bars bent to form a spiral or bands of steel spaced at a uniform distance apart, but in either case, the bands should not be spaced more than about one-sixth of the diameter of the column apart, and must be held in place by longitudinal spacing bars. Experiments upon columns with hooped reinforcement indicate that the deflections under working loads are not decreased by the reinforcement, but the ultimate strength is considerably increased, as compared with columns without such reinforcement. When hcoped reinforcement is used, it is usual to allow a larger unit stress than for plain columns, or those with longitudinal reinforcement only. The effective area of the column is that inside the reinforcement. The concrete outside the hooping is stripped off when a stress is reached at which plain concrete would fail. Hooped reinforcement prevents crushing of the concrete until a load is reached which stresses the steel to its yield point, but does not stiffen the column longitudinally, and columns so reinforced fre- quently fail by bending. This reinforcement is commonly combined with longitudinal steel as shown in Fig. 58 (c). The longitudinal steel serves to stiffen the column against bending, and makes the hooping more effective. In general, it is not advisable to use hooped reinforcement without longitudinal steel, as the same amount of steel would be more effective in strengthening the column if used as longitudinal steel. Experiments indicate that about 1 per cent of steel in closely spaced hooping is sufficient to resist lateral expansion and give in- creased strength in compression. Larger amounts of steel do not materially increase the resistance. The Joint Committee makes the following recommendations : (6) Columns reinforced with not less than 1 per cent and not more than 4 per cent of longitudinal bars and with circular hoops or spirals not less than 1 per cent of the volume of the concrete and as hereinafter specified: a unit stress 55 per cent higher than given for (a), provided the ratio of unsupported length of column to diameter of the hooped core is not more than 10. The foregoing recommendations are based on the following conditions: It is recommended that the minimum size of columns to which the working stresses may be applied be 12 inches out to out. In all cases longitudinal reinforcement is assumed to carry its proportion of stress. The hoops or bands are not to be counted on directly as adding to the strength of the column. Longitudinal reinforcement bars should be maintained straight, and should CONCRETE COLUMNS 211 have sufficient lateral support to be securely held in place until the concrete has set. Where hooping is used, the total amount of such reinforcement shall be not less than 1 per cent of the volume of the column, enclosed. The clear spacing of such hooping shall be not greater than one-sixth the diameter of the enclosed column and preferably not greater than one-tenth, and in no case more than 2 inches. Hooping is to be circular and the ends of bands must be united in such a way as to develop their full strength. Adequate means must be provided to hold bands or hoops in place so as to form a column, the core of which shall be straight and well centered. The strength of hooped columns depends very much upon the ratio of length to diameter of hooped core, and the strength due to hooping decreases rapidly as this ratio increases beyond five. The work- ing stresses recommended are for hooped columns with a length of not more than ten diameters of the hooped core. The Committee has no recommendation to make for a formula for working stresses for columns longer than ten diameters. Let d = effective diameter of column in inches; a = area of the steel bar to be used, in square inches; s = longitudinal spacing of the bands or spirals, in inches; p = ratio of steel to concrete in column. Then a = pds/4: or s=4a/pd ......... (51) From this we may obtain the size of bars necessary for steel of re- quired spacing, or the spacing required for bars of given size. Example 24. A concrete column is to carry a load of 225,000 pounds and be reinforced with 1 per cent of spiral steel and 2 per cent of longitudinal steel. Using the stresses recommended by the Joint Committee for concrete of 2000 pounds compressive strength, find dimensions for concrete and steel. Solution. Without hooped reinforcement, the value of f c would be limited to 22.5 per cent of the compressive strength, or 450 lb./ in. This may be increased 55 per cent when spiral reinforcement is used or / c = 700 lb./in. 2 Using Table XVI, for p = .02 and rc = 15, Z=1.28, from which P 225000 =251 in. 2 , and diameter of column is 18 inches. Longitudinal steel, A s =. 02X251 = 5.02 in. 2 From Table X we see that five 1-inch square bars, spaced about 11 inches apart about the circumference of the column, or nine f-inch square bars spaced about 6 inches apart may be used. For the spiral steel, we find from (51) that if the spacing be made 2| inches, a = .OlXl8X2J/4 = .112 in. 2 , and f-inch round bars may be used. 212 REINFORCED CONCRETE 122. Eccentrically Loaded Columns. When the center of gravity of the load upon a column does not coincide with the gravity axis of the column, bending stresses are produced which must be taken into account in designing the column. In some cases, lateral forces may be acting upon a column, producing bending moments, as in wall columns carrying the ends of beams which are firmly attached to the columns. When these conditions exist, the maximum unit com- pression due to both direct thrust and bending moment at any sec- tion must not exceed the safe values for the concrete, and any tensions which may occur must be taken by proper reinforcement. Let Fig. 59 represent the section of a column under eccentric load. A = area of section of column; A s = area of longitudinal steel hi section; P= longitudinal load on column; e = eccentricity of load ; I c moment of inertia of section about its gravity axis; I s = moment of inertia of steel area about same axis; u distance gravity axis to most remote edge of section ; M bending moment on section, Pe; f c = maximum unit compression on concrete; f' c = minimum unit compression on concrete. / c = is made up of two parts that due to direct thrust and that due to bending moment, and is f _ P , MU /KIN 7 ' A+(n-l)A^/,+(n-l)// and Mu When the stress due to moment is greater than that due to direct thrust, f c becomes negative, showing the stress to be tension. Tensions in columns, if occurring at all, should be very small and need not be specially provided for. The stresses in steel are always less than nf c , and therefore within safe limits. If the section is symmetrical about its gravity axis, u = d/2 t bd 3 and for rectangular sections, I c = -^r and I s = A s d s 2 /4, in which d s \2i is distance between centers of steel on the two sides of column. For circular sections, I c = .049d 4 and I s = .125A s d s 2 , where d is the diameter of the column and d s is diameter of the circle containing the centers oi the steel bars. CONCRETE COLUMNS 213 Example 25. A wall column, 12X16 inches in section, carries the end of a beam which brings a longitudinal load of 60,000 pounds and a bending moment of 180,000 in.-lb. upon the column. The column is reinforced with four 1-inch square steel bars at the corners, the centers of steel being 2 inches from surf apes of concrete. n=15. Find the unit stresses on the concrete. FIG. 59. Column with Eccentric Load. Solution. I c = 12X16X16X16/12=4096. 60000 180000X8 ' 192+14X4 ' 4096+14X144 /' c =242-235 = 71b/in. 2 J S =4X12X12/4 = 144. = 242+235=477 lb./in. 2 , Complete discussions of the principles of reinforced concrete design with applications to structures is given in " Concrete, Plain and Reinforced," by Taylor and Thompson, and in " Principles of Reinforced Concrete Construction/' by Turneaure and Maurer. CHAPTER VII RETAINING WALLS ART. 33. PRESSURE OF EARTH AGAINST A WALL. 123. Theories of Earth Pressure. The lateral pressure of a mass of earth against a retaining wall is affected by so many variable conditions that the determination of its actual value in a particular instance is practically impossible. Several theories, based in each case upon certain ideal conditions, have been proposed, none of which are more than very rough approxi- mations to the conditions existing in such structures. These theo- FIG. 60. Pressure of Earth against a Wall. ries assume that the earth is composed of a mass of particles exerting friction upon each other but without cohesion, or that the pressure against the wall is caused by a wedge of earth which tends to slide upon a plane surface of rupture, as shown in Fig. 60. Formulas for the resultant thrust against the wall have been produced in accord- ance with the various theories by several methods, they differ mainly in the direction given to the thrust upon the wall. Coulomb's Theory. A formula for computing the lateral thrust against a wall was proposed by Coulomb in 1773. Coulomb assumed that the thrust was caused by a prism of earth (BAC, Fig. 60) sliding 214 PRESSURE OF EARTH AGAINST A WALL 215 upon any plane AC which produces the maximum thrust upon the wall. There is a certain slope (AD, Fig. 60) at which the material if loosely placed will stand. This is known as the natural slope, and the angle made by this slope with the horizontal as the angle of fric- tion of the earth. On slopes steeper than the natural slope, there is a tendency for the earth to slide down, and if held by a wall, pressures are produced which depend upon the frictional resistance to sliding. The thrust is assumed by Coulomb to be normal to the wall, and the pressure upon the plane of rupture to be inclined at the angle of friction to the normal to the plane. Let h = height of wall; P = resultant pressure upon a unit length of wall; R = pressure upon the plane of rupture; G = weight of the wedge of earth ; e = weight of earth per cubic foot; = angle of friction of earth; a = angle between the back of wall and plane of rupture. If the back of the wall be vertical and the surface of earth horizon- tal, from Fig. 60, tan a tan (a-\-Y For maximum value of P, a = 45 ^, and the plane of rupture bisects the angle between the back of the wall and the natural slope. Substituting this value, P varies as the square of h, and is therefore applied at a distance h/3 above the base of the wall. This is the same in all of the theories. Poncelet's Theory. In 1840 Poncelet proposed to modify the method of Coulomb by making the thrust upon the wall act at the angle of friction with the normal to the wall. Before the wall can be overturned about its toe (F, Fig. 61) the back of the wall (A B) must be raised and slide upon the earth behind it, thus calling into play the friction of the earth upon the wall as a resistance. As the friction of earth upon a rough masonry wall is greater than that of earth upon earth, a nlm of earth would be carried with the wall and slide upon the earth behind and the angle of friction is usually taken as equal to the natural slope of the earth. 216 RETAINING WALLS Let 6 = the angle made by the back of the wall with the horizontal ; z = the angle made by the earth surface with the horizontal. Following the same procedure as in developing Coulomb's formula, we find the pressure against the wall, P = i^2 Si 2 (-*) sin (4>i), sin 2 FIG. 61. Poncelet's Theory of Pressure. For a vertical wall and horizontal earth surface o), eh 2 cos = 90 and which is the formula proposed by Poncelet. Rankine's Theory. Rankine considered the earth to be made up of a homogeneous mass of particles, possessing frictional resistance to sliding over each other but without cohesion. He deduced for- mulas for the pressure upon ideal plane sections through an unlimited mass of earth with plane upper surface, the earth being subject to no external force except its own weight, and determined the direction of the pressure from these assumptions. Rankine found that the resultant pressure upon any vertical plane section through a bank of earth with plane upper surface is parallel to the slope of the upper surface (see Fig. 62) . Let E = the pressure upon the vertical section; ^ = the angle made by the inclination of the upper surface with the horizontal; Then PRESSURE OF EARTH AGAINST A WALL = the angle of friction of the earth; e = the weight per cubic foot of the earth; $=the height of vertical section through earth. 217 , eS 2 . cos ^ Vcos 2 i cos 2 = cosz-- . , cos i + V cos 2 i cos 2 is Rankine's formula for earth pressure. This pressure acts upon the vertical section at a distance S/3 from its base, and makes an angle i with the horizontal. Rankine's formula may be produced in the same manner as Poncelet's by assuming the pressure parallel to the upper slope. FIG. 62. Pressure of Earth Thus in Fig. 61 if the angle made by P with the normal to the wall be equal to i t we find eS 2 cos 2 sin (0 a) sin(0-j-fr')\ 2 ' cos 2 i ) which may be transformed 1 into Rankine's formula as given above. Weyrauchs theory is practically the same as Rankine's although produced in a different way. Cohesion. In all of the ordinary formulas for earth pressure, the effect of cohesion is neglected. Experiments indicate that this effect is not sufficient to affect very materially the actual pressure upon a wall. It causes the earth to break off and slide upon a concave surface in- stead of a plane surface. At the upper surface of the earth, the cohe- 1 Wm. Cain, Practical Designing of Retaining Walls, 1914, p. 103. 218 RETAINING WALLS sion is sufficient to overcome the lateral thrust and cause the earth to stand in a vertical position, while as the lateral thrust increases with the depth, the cohesion becomes relatively less important and the surface of rupture flattens out. When earth is placed behind a wall after it is constructed cohesion is probably negligible at first, although after the earth has become compacted may develop in some cases so that practically no pressure comes against the wall. It is so uncertain that no reliance should be placed upon it in designing walls. Value of Theories. On account of the variable nature of the material, it is evident that estimates of earth pressures are only rough approximations to the actual pressures. The material assumed as possessing uniform friction and without cohesion does not exist in practice. The general laws developed, however, do give rational methods of reaching reasonable estimates upon which safe designs may be based. Experiments upon sand pressures, and experience with walls in use, indicate that Coulomb's use of horizontal earth pressures, or Rankine's thrust parallel to earth surface, where the surface is near the horizontal, give thrusts much greater than those actually produced upon walls with vertical backs. For such walls, the use of the Poncelet's formulas, taking into account the friction of the earth on the back of the wall, give results which seem to agree fairly well with experiment and experience. For walls leaning forward, so that considerable weights of earth rest upon them, Rankine's formulas may be applied to find the thrust upon the vertical section through the earth at the inner edge of the base of the wall. This thrust, combined with the weight of earth resting upon the wall, gives the thrust against the wall. 124. Computation of Earth Thrusts. When the back of a wall is nearly vertical, the thrust may usually be taken as making the angle of friction with a normal to the surface of the wall, as assumed in the theory of Poncelet. For such walls the thrust may be obtained from the formula already given: eh? sin 2 (0-0) M 2 n //. , ,\7 , -;)- sin 20 \ sin 2 0-sm(0+0)u-f ' / sin (0-;)- s A /- , .. . \sm (Bi) sm eh 2 If we place P=Q, values of Q may be tabulated for various 2i slopes and angles of friction as shown in Table XVII. The values of P obtained by this method are supposed to act against the wall at a PRESSURE OF EARTH AGAINST A WALL 219 distance h/3 above the base, and make the angle of friction with the normal to the wall. TABLE XVIL EARTH PRESSURE AGAINST A WALL Values of Q in Formula (1), P=^r--Q Batter of Back of Wall. SLOPE OF UPPER SURFACE OF EARTH. ANGLE OF FRICTION, . Angle i. Vertical to Hori- zontal. 20 25 30 35 40 45 33 40' l to H .59 .39 .28 29 45' 1 to If .76 .45 .34 .25 Vertical } 26 30' 21 50' Ito2 .54 .46 .39 .35 .32 .30 .23 .22 .61 = 90 18 30' 1 to 3 .72 .52 .40 .33 .28 .21 14 00' 1 to 4 .58 .45 .36 .30 .25 .19 000' Level .43 .37 .30 .26 .21 .18 33 40' to H .72 .50 .37 29 40' to If .90 .56 .44 .35 26 30' v\s J. 4 to 2 .64 .49 .40 .32 1 in 10 = 95 40' 21 50' 18 30' to 3 .70 .60 .56 .48 .44 .40 .36 .34 .30 .28 .82 14 00' to 4 .66 .52 .40 .35 .31 .25 000' Level .48 .40 .34 .30 .26 .22 33 40' 1 to H .91 .62 .49 29 45' Itolf 1.08 .68 .55 .46 26 30' 1 to 2 77 .60 .50 .42 1 in 5 = 101 20' 21 50' 18 30' 1 to 2^ 1 to 3 .93 .80 .68 .66 .57 .54 .48 .45 .42 .38 .36 14 00' 1 to 4 .75 .60 .48 .41 .38 .32 I 000' Level .52 .46 .40 .34 .30 .27 When a mass of earth rests upon a wall, as in a wall with sloping back or a reinforced concrete wall, the formula of Rankine for pres- sure upon a vertical section may be applied. This pressure combined with the weight of the earth resting upon the wall gives the thrust against the wall. The value of the pressure upon the vertical section is given by Rankine's formula: eS 2 . cos i Vcos 2 i cos 2 < _ eS 2 2 cos i + Vcos 2 i cos 2 2 (2) 220 RETAINING WALLS Values of K corresponding to various values of i and $ are tabu- lated in Table XVIII, thus greatly reducing the labor of computing the pressures. E as computed from this formula is supposed to act at a distance S/3 from the bottom of the section and to be parallel to the upper surface of the earth. S in this formula is the height of the earth section and not the height of the wall. TABLE XVIII PRESSURES UPON VERTICAL SECTIONS THROUGH EARTH Values of K in Formula (2) -E=K SLOPE OF UPPER SURFACE OF EARTH. ANGLE OF FRICTION. Angle i. Vertical to Horizontal. 20 25 30 35 40 45 33 40' 29 45' 26 30' 21 50' 18 30' 14 00' 000' ItoU 1 to If 1 to 2 Ito2i Ito3 1 to 4 Level 0.59 0.45 0.39 0.34 0.31 0.29 0.27 0.36 0.32 0.29 0.27 0.24 0.23 0.22 0.26 0.23 0.21 0.20 0.19 0.18 0.18 0.76 0.54 0.45 0.40 0.36 0.33 0.60 0.52 0.45 0.40 0.72 0.59 0.50 Angle of Friction. In order to be able to apply any of the formulas for determining earth pressures, it is necessary to know the weight per unit volume and the angle of friction of the earth. These vary with the kind of material to be filled behind the wall and its condition as to compactness and moisture. The natural slope for the earth is the slope at which the surface of the material will stand when dumped into piles, the frictional resistance keeping the surface layer from sliding or rolling down the slope. The angle of sliding friction of a wedge of earth upon an earth surface may not be the same as the inclination of the natural slope. Values of sliding friction as determined by experiment vary considerably for the same material, and it is possible that much of the variation is due to the methods of testing rathei than to differences in the materials. The natural slope of a particular material may usually be approximately determined without difficulty and its use instead of the angle of sliding friction would ordinarily be safe. Table XIX gives approximate values of the angle of internal friction, the natural slopes and weights of various materials commonly met in construction. PRESSURE OF EARTH AGAINST A WALL 221 TABLE XIX. FRICTION ANGLES AND WEIGHTS OF MATERIALS Kind of Material. Angle of Friction. Natural Slope, Horizontal to Vertical. Weight per Cubic Foot. Clav, dry 35 1 5 to 1 110 Clay, damp 40 1.2 to 1 110 Clay, wet 20 3 to 1 120 Sand dry 35 1 5 to 1 100 Sand, moist 40 1 . 3 to 1 100 Sand, wet 25 2 to 1 115 Gravel and sand 40 1.5 to 1 110 Broken rock .... 45 1 . 2 to 1 110 Surcharged Walls. The formulas for earth pressure already given assume the earth to carry only its own weight and the upper surface to slope from the top of the wall. When the earth behind the wall carries a load upon its surface, as when supporting a railway track or a pile of material of any sort, the pressure against the wall is increased uniformly over its entire depth. If w is the weight of the load per unit area of earth surface, Formula (1) becomes, ....... (3) The point of application of P is at a distance l_ eh+Zw 3'eh+2w ' above the base of the wall. In the same manner for the pressure on a vertical section through the mass of earth, Formula (2) becomes E=l^- (4) and its point of application is at a distance I eS+3w 3 eS+2w S. 125. Graphical Method. When the surface of earth is irregular or broken, the formulas do not apply, although it is usually possible to approximate the plane surfaces with sufficient accuracy. Graph- ical determination of earth pressures may be made when the slope of the surface is not too great. When the surface slope is near the natural slope for the material, these methods cannot be used. 222 RETAINING WALLS A graphical method is shown in Fig. 63. OA is the back of a wall, and A BCD, etc., the upper surface of the earth resting against it. Divide the earth into a number of prisms by the lines OB, OC, etc. On the line oa lay off on some convenient scale ab, be, etc., equal respectively to the weights of the prisms OAB, OBC, OCD, etc. From a draw the lines 061, aci, etc., making the angle of friction () with the normals to OB, OC, OD, etc., respectively. From the points b, c, d, etc., draw the lines 661, cci, ddi, etc., making the angle of friction () with the back of the wall (OA), to intersection with the lines abi, aci, etc., respectively. The lengths bbi,cci, ddi, etc., will then represent, on the scale to which the weights were laid off, the thrusts of the prisms between the back of the wall and the planes OB, OC, etc., respectively. B FIG. 63. Graphical Determination of Earth Pressures. In the figure, ee\ is the maximum thrust, caused by the prism between OA and OE, showing OE to be the plane of rupture. This resultant thrust will act at a distance h/3 from the base of the wall, at the angle of friction with the normal to the wall. Detailed discussions of methods of determining earth pressures are given in "Retaining Walls for Earth" by M. A. Howe, New York, 1896, and in "Practical Designing of Retaining Walls," by Wm. Cain, New York, 1914. An interesting paper by E. P. Goodrich in Trans- SOLID MASONRY WALLS 223 actions, American Society of Civil Engineers, December, 1904, gives results of experiments for determination of internal friction and lateral pressure of earth. ART. 34. SOLID MASONRY WALLS 126. Stability of Walls. A masonry retaining wall may fail in either of three ways: 1. By overturning or rotating about its toe. 2. By crushing the masonry. 3. By sliding on a horizontal joint. Insufficient foundation is probably the most common cause of failure of retaining walls. This is not, however, due to failure of the wall itself, but to lack of sufficient footing or other support when placed upon compressible or soft soils or to lack of proper drainage. This is discussed in Art. 36. FIG. 64. Stresses upon Retaining Walls. In Fig. 64, A BCD is a wall with vertical face supporting a bank of earth as shown. Let P = thrust of earth against the wall; V = Vertical component of P; H = horizontal component of P; W = weight of wall acting through its center of gravity; R = resultant pressure on base A B ; 224 RETAINING WALLS 6 = width of base of wall; a = width of top of wall; d = distance from face of wall to its center of gravity; / c = unit compression on masonry at toe of wall; x = distance from toe of wall to point of application of resultant pressure upon the base; = angle made by R with base of wall. Resisting Moment. The moment of the thrust about the toe of the wall at B is M = HX- VX (26 + a) . o This moment tends to overturn the wall by causing rotation about B, and is resisted by the moment of the weight of wall in the opposite direction. This moment is M w = Wd. When these moments are equal (M W M P = Q), the resultant R obtained by combining P and W passes through B and the wall is on the point of overturning. The ratio M W /M P is the factor of safety against overturning. When M w is greater than M PJ R will cut the base of the wall to the right of B. Placing M T M W M P we have t 6 , 6 from which we find the distance of the point of application of R from B: _3Wd+V(2b+a)-Hh 3(W+V) This point of application of R may also be found graphically as shown in Fig. 63. The resultant R should always cut the base of the wall within its middle third (x>6/3) in order that the pressure may be distrib- uted over the whole section of the base and there may be no tend- ency for the joint to open, or no tensile stress developed at the inner edge (A) of the section. Crushing of Masonry. The unit stress at the toe of the wall (B) must not exceed the safe crushing strength of the masonry. The distribution of stress over the section depends upon the position of the point of application of the resultant (R). When x = b/3, the stress at A will be zero, and the stress at B, f c = -- =- -- -. If x be less than 6/3, the pressure will be distributed over a distance 3x SOLID MASONRY WALLS 225 2(W -4-V) from the toe (B), and the maximum stress, f e = - When oX x is greater than 6/3 the maximum compression, (TT+7)(46-6x) Jc= ^2 ....... (6) Resistance to Sliding depends upon the development of sufficient friction in any joint through the wall to overcome the pressure parallel to the joint. Thus (Fig. 64) in order that no sliding occur at the base of the wall, the frictional resistance in the joint A B must be greater .than the horizontal component of the thrust R. This will be the case when R makes an angle (|3) with the normal to AB that is less than the angle of friction of masonry sliding upon masonry. TT Tan fi = ,) must be less than the coefficient of friction of the masonry. In the construction of heavy walls, resistance to sliding may be increased by breaking joints so that no continuous joint exists through the wall. Joints inclined from the front to the back of the wall are also sometimes used so as to bring the resultant pressure more nearly normal to the joint. 127. Empirical Design. In the practical designing of retaining walls, engineers have commonly used empirical rules given by certain prominent authorities, or have assumed dimensions based upon their own experiences. The uncertain and conflicting nature of the assumptions used in producing the formulas based upon the various theories, and the lack of satisfactory experimental data has caused the use of dimensions shown by experience to be safe and in very many instances probably quite excessive. Trautwine's rules have been extensively used for many years, and are as follows J for vertical walls : When the backing is deposited loosely, as usual, as when dumped from carts, cars, etc., Wall of cut stone, or first-class large ranged rub- ble, in mortar ............................ 35 of its entire vertical height Wall of good common scabbled mortar-rubble, or brick ............ ..................... 4 of its entire vertical height Wall of well-scabbled dry rubble ............... 5 of its entire vertical height With good masonry, however, we may take the height from the ground surface 1 up, instead of the total height as above indicated. When the wall has a sloping or offset back, the thickness above 1 Trautwine's Engineer's Pocket-Book. 226 RETAINING WALLS given may be used as the mean thickness, or thickness at the mid- height. Baker's Rules. Sir Benjamin Baker, from an extended experience in the construction of walls under many differing conditions, and after numerous experiments upon the thrust of earth, gives 1 the following statement of his views upon the design of retaining walls: Experience has shown that a wall one-quarter of the height in thickness, and battering 1 inch or 2 inches per foot on the face, possesses sufficient stability when the backing and foundation are both favorable. The Author, however, would not seek to justify this proportion by assuming the slope of repose to be about 1 to 1, when it is perhaps more nearly 1 to 1, and a factor of safety to be unnecessary, but would rather say that experiment has shown the actual lateral thrust of good filling to be equivalent to that of a fluid weighing about 10 pounds per cubic foot, and allowing for variations in the ground, vibrations, and con- tingencies, a factor of safety of 2, the wall should be able to sustain at least 20 pounds fluid pressure, which will be the case if one-quarter of the height in thickness. It has been similarly proved by experience that under no ordinary conditions of surcharge or heavy backing is it necessary to make a retaining wall on a solid foundation more than double the above, or one-half of the height in thickness. Within these limits the engineer must vary the strength in accordance with the conditions affecting the particular case. The rules of Sir Benjamin Baker give walls considerably lighter than those of Trautwine, and the tendency in recent practice has been to somewhat reduce the thicknesses for walls backed with good materials and built under favorable conditions. Where from lack of drainage or other cause, the backing is liable to get into soft condition, it may be necessary to considerably increase thickness. 128. Using Formulas in Design. The design of a wall to sustain a bank of earth is a comparatively simple matter once the earth pressure has been determined. The difficulties met are those of judging the character of the material and its probable pressure against the wall. It is probable that in most instances the full pressures that theoretically might come upon the wall are not actually developed. The design should be made for the worst conditions which may reasonably be expected to occur, but the construction of heavy walls to provide for bad conditions which are not likely to occur, and which may be met by proper attention to drainage and proper care in placing the backing, is unnecessarily expensive and wasteful. For walls with vertical or nearly vertical backs, . Poncelet's for- mulas, taking into account the friction of the earth on the back of 1 The Actual Lateral Pressure of Earth, Van Nostrand Science Series, and Proceedings, Institution of Civil Engineers, Vol. LXV, p. 183. SOLID MASONRY WALLS 227 the wall, give thicknesses for walls which agree fairly well with the results of experience and not differing greatly from the rules sug- gested by Sir Benjamin Baker. in designing by this method, the pressure of earth is obtained by the use of Formula (1), or from Table XVII, a section of wall is assumed and its sufficiency investigated. Example 1. A masonry wall, 22 feet high, is to support a bank of earth whose surface has an upward slope of 1 to 3 from the top of the wall. The backing is ordinary earth whose friction angle may be taken at 35. Weight of masonry is 150 pounds and of earth 100 pounds per cubic foot. Find proper section for the wall. Solution. Try a rectangular wall with thickness of 7.5 f et. From Table XVII, we find Q = .33. Then D eh 2 n 100X22X22X.33 P = -p-Q = - -7: - = 7986 pounds per foot of length of wall. A - As P makes angle of 35 with normal to back of wall, H = Pcos 35 = 6770 pounds and V = P sin 35 = 4580 pounds. W = 22 X 7 . 5 X 150 = 24,750 pounds. From (5), we find 3X24750X3.75+4580X3X7.5-6770X22 3(24750+4580) = 2.64 and the resultant (R) comes just within the middle third of the base. The crushing stress on the masonry at the toe of the wall is (6) _ (W+ V) (46 - 6or) _ (24750+4580) (4 X 7.5 - 6 X 2.64) fc ~ ~W~ ~ 7.5X7.5 lb ' /tt * which would be quite safe for any ordinary masonry. tan = T^Ty = 24750"- 4580 = <231> r = 13 ' and sliding could not occur. Battered Face. The face of the wall may be battered, so as to diminish the width at top by one- third, using the same width of base without decreasing its stability. Battered Back. A wall with battered back may be used. Assume a top thickness, a = 5.5, and base thickness, 6 = 9.5. The angle made by the back of the wall with the horizontal 6 = 100 20'. From eh 2 Q 100 X 22 X 22 X47 Table XVII, we find Q = .47, then P = e ^ = 228 RETAINING WALLS = 1 1624 pounds. H = P cos (0+ - 90) = 8170 pounds, and V = P sin ((9+0 - 90) = 8270 pounds. W = 5 ' 5 + 9 ' 5 X 22 X 150 = 24750 pounds. SB From (3), _3X24750X3. 83+8270(2X9. 5+5. 5) -8170X22_ Q 3(24750+8270) R cuts the base practically at one-third its width from the toe. 2(W-\-V) The crushing stress at the toe is f c = -- =- -- = 6950 pounds, a little less than for the rectangular wall. tan ft = o^yKfl i 0070 = -247, within safe limits but somewhat more than for the rectangular wall. Example 2. A retaining wall 20 feet high is to support a hori- zontal bank of eaHh carrying a railway track. If the maximum train load is taken at 800 pounds per square foot of surface, and the angle of friction of the earth at 30, find the thickness of wall required by Poncelet's formula, w = 150 pounds and e = 100 pounds per cubic foot. Solution. Assume a thickness of wall of 9 feet. From Table XVII, we have Q = 30. Then (3) 3Q= H = P cos = 10800 X .866 = 93 and H = P cos = 10800 X .866 = 9350 pounds, V = 10800 X .5 = 5400 pounds. JF = 9 X 20 X 150 = 27000 pounds. Using (5), _3X27000X4.5+5400X9X3-9350X20_ Q 3(27000+5400) The resultant thrust cuts the base within the middle third, and a little less width might answer. The crushing stress at the toe of the wall is pounds. ..,.-:" C The minimum thickness allowable for a solid wall is that which causes the resultant thrust (R) to cut the base at a distance x = l/3 from the toe of the wall. For a rectangular wall, the width bears a direct ratio to the height for any particular values for weights of REINFORCED CONCRETE WALLS 229 materials and angles of friction. Table XX gives minimum values of thickness ratio, by Poncelet's formula for walls in which the weight of masonry is taken as 150 lb./ft. 3 and the weight of earth as 100 lb./ft. 3 TABLE XX. MINIMUM THICKNESS OF WALLS BY PONCELET'S FORMULA Values of b/h, when w = 150 lb./ft. 3 and e = 100 lb./ft. 3 Slopes of Earth Sur- face Vertical to Horizontal. ANGLES OF FRICTION. 20 25 30 35 40 45 Ito l\ 0.39 0.31 0.25 ltol| .... .... 0.46 0.35 0.29 0.24 Ito2 0.41 0.34 0.28 0.23 Ito2i .... 0.46 0.39 0.33 0.27 0.23 Ito 3 0.53 0.43 0.37 0.32 0.27 0.23 Ito 4 0.49 0.41 0.35 0.30 0.26 0.22 Level 0.43 0.38 0.33 0.29 0.25 0.22 For walls battered or stepped on the back the minimum thickness given in the table may be used as the average thickness at the middle of the height. This gives a broader base to the wall and gives a larger factor of safety against overturning, but requires the same volume of masonry to keep the resultant thrust within the middle third of the base. Walls computed as rectangular may be battered on the face to an extent which lessens the top thickness by one-third without increas- ing the base thickness. This slightly decreases the resisting moment, but increases the value of x, lessens the pressure at the toe, and does not impair the stability of the wall. ART. 35. REINFORCED CONCRETE WALLS 129. Types of Reinforced Concrete Retaining Walls. There are two types of reinforced concrete retaining walls in common use : 1. The cantilever type and 2, the counterforted type. Both of these depend upon the weight of earth carried by the base of the wall to prevent overturning. They differ in the way in which the face wall is attached to the base. A cantilever wall is shown in Fig. 65, consisting of a vertical stem attached to a base, ACFB. The weight of the mass of earth BFEG, rests upon the base of the wall BF and serves to assist the wall in resisting the overturning moment of the earth thrust. The hori- 230 RETAINING WALLS zontal pressure of earth on EF is carried by the vertical stem CDEF acting as a cantilever beam. The projecting bases FB and AC are also cantilever beams, the one supporting the weight of earth resting upon it, the other resisting the upward thrust of the foundation at the toe of the wall. A counterforted wall is shown in Fig. 66. The face wall CDEF is connected with the base ACFB by narrow counterforts EFB, spaced several feet apart. The counterforts are cantilever beams, each carrying the horizontal earth thrust on the face wall EF for a panel length of wall. The face walls CDEF are slabs holding the earth pressure between counterforts and transferring the pressure A /_ B FIG. 65. Cantilever Wall. FIG. 66. Counterfort Walls. to the counterforts. The base FB is a slab carrying the weight of earth FEGB beween counterforts and holding down the ends of the counterforts. The base AC at the front of the wall is a cantilever carrying the upward thrust of the foundation at the toe of the wall. The cantilever type is commonly used for moderate heights of wall. For walls more than 20 or 25 feet high, the counterforted wall is usually more economical. The quantities of materials required for a counterforted wall are less and the amount of form work more than for a cantilever wall. 130. Design of Cantilever Wall. In designing reinforced con- crete walls, the thrust in the vertical section of earth passing through the inner edge of the base may be computed by Rankine's formula, as given in Section 123: REINFORCED CONCRETE WALI 231 eS 2 . cos i Vcos 2 z cos 2< ^ eS 2 cos i + V (2) Values of K may be taken from Table XVIII. This thrust is parallel to the upper surface of the earth and its horizontal and vertical components are eS 2 eS 2 H = E cos i = K cos i, and V = E sin i K sin i. 2^ The method of design will be illustrated by numerical examples. Example 3. Design a retaining wall to hold a level bank of earth 16 feet high. The base of footing is to be 3 feet below surface of ground and the pressure on the soil is limited to 4000 lb./ft. 2 The backing is ordinary soil with angle of friction of 35. Earth weighs 100 pounds and concrete 150 pounds per cubic foot. Unit stresses will be based upon use of 2000 pounds concrete and plain bars of medium steel. Solution. Assume the base under the wall 12 inches thick. The height of the wall above the base is then 18 feet, and the hori- zontal thrust, taking K from Table XVIII, eS 2 K 100X18X18X27 E= = - - - =4374 pounds per foot of length of wall. z _ The bending moment caused by this thrust upon the section at the top of the base is M= 4374X6X12 = 3 14928 in.-lb. From Table VII, for f e = 650 and f, = 16,000, we find R = 108 and p = .0078. Then Rbd 2 = M becomes 108X12d 2 = 314928, and d=15.5 inches. Assuming the steel to be embedded 1.5 inches in the concrete, the total thickness of the vertical stem at the top of the base will be 17 inches. The steel area required for a length of 12 inches of wall is 'pbd=. 0078X12X15.5 = 1.45 in. 2 From Table XV we find that f-inch bars spaced 5 inches apart will answer the purpose. If we assume 10 inches to be the minimum allowable thickness at the top of the wall and make the faces of the wall plane surfaces, the thickness at all intermediate points will be greater than required for strength. At a point 12 feet below the top, the bending moment 232 RETAINING WALLS is 314928X8/27 = 93312 in.-lb., and the effective depth of beam is 13 inches. Then qqqio and from Table IX, we find p = .0032. The area of steel required is 12 X 13 X. 0032 = 0.5 in. 2 per foot of length, or about one -third of that at the base. Similarly at a section 6 feet below the top, no steel would theoretically be required. If ajl of the bars be carried up 6 feet, every third bar 12 feet and every sixth bar to the top the reinforcement will be amply strong. The lower ends of these bars should be turned up in the base for anchorage. The maximum shear in section at base is 4374 pounds, and V 4374 " = P = 12X. 874X15. 5 = 27 lb ' /m ' No diagonal tension reinforcement is necessary. Overturning Moment. Assume the width of base at about 45 per cent of the total height, or 8.5 feet. Let the inner surface of the vertical stem be vertical, and place the stem at a distance equal to one-third the width of base (6/3) from the toe of the wall. (See Fig. 67.) The moment of the thrust about the toe at A tends to overturn the wall, while the moments of the weights of the wall and earth resting upon it resist overturning. The weight of the vertical stem is W i = (18 X 150) = 3035 pounds. The weight of the base is W 2 = 1.0X8.5X 150= 1275 pounds. The weight of the earth is G= 18X4.25X100 = 7650 pounds. rro. rr eS 2 K 100 X 19 X 19 X. 27 , _ The earth thrust, E = = = 4873 pounds. 2i The moment on the toe at A is Af = 3035X3.65 + 1275X4.25+7650X6.4-4873X6.33 = 34610 ft.-lb. The point of application of the resultant on the foundation soil is equal to the moment about A divided by the vertical component of the resultant, or 34610 _ 20fcc "3035+1275+7650" This brings the resultant within the middle third of the base. REINFORCED CONCRETE WALLS 233 The factor of safety against overturning is 3035X3.65+1275X4.25+7650X6.4 2.12. 4873X6.33 Pressure on Foundation. The total vertical load on the foun- dation is 3035+1275+7650=11,960 pounds. The pressure at the E F If? "a FIG. 67. Design of Cantilever Wall. toe is twice the average pressure, 2X11960/8.5 = 2814 lb./ft. 2 , which is within safe limits. Inner Base Cantilever. The length of the inner cantilever is 4.25 feet. It is subject to the action of three loads: (1) The weight of the earth resting upon it (1800 pounds per linear foot) ; (2) Weight of the cantilever itself (150 pounds per linear foot). 234 RETAINING WALLS (3) Upward pressure of the foundation soil (which is at the end D and 1400 pounds where the cantilever joins the vertical wall at C). The bending moment on the section at C is = 13,396 ft.-lb. or 160,752 in-lb. From Table VII, R = 108 and p = .0078. Then 108 Xl2d 2 = 160752 and rf=11.2 inches. If the steel be placed 1.75 inches below the top surface the thick- ness at C is 13 inches, A =pbd = . 0078X11.2X12= 1.05 in. 2 From Table XV, we find that-f-inch bars spaced 5 inches apart, the same as the vertical reinforcement will answer the purpose. These bars should be anchored by bending or by continuing them through the concrete on the front of the base to a length of at least 50 diameters (37.5 inches). The shear in section at C is 7= 1800X4.25+150X4.25- 1400X4.25/2 = 5312 pounds, V 5312 and * = = This value is rather large for use without diagonal tension reinforce- ment. If we make Iz.b in. Using d=13 inches and embedding the steel 2 inches in the con- crete, the total depth of base at C becomes 15 inches. Outer Base Cantilever. The length of the outer cantilever is 2.83 feet. The forces acting upon it are its own weight acting down- ward, and the thrust of the foundation soil acting upward (2814 lb./ft. 2 at A and 1876 lb./ft. 2 at B). The shear in section at B is F = 2814+187_6 283 _ 150x283 = 62121b _ If the unit shear be limited to 40 lb./in. 2 , V 6212 Making d= 15 inches, the total depth of base at B is 17 inches. REINFORCED CONCRETE WALLS 235 The bending moment at B is or 150,120 in./lb. ** ~ J^72 = 10\/10\/1 C = 55.6, R/f s = 55.6/ 16000 = .0035. From Table IX, p = .0038, and A = .0038X12X15 = .68 in. 2 per foot of length of wall. From Table XV, f-inch bars will answer if spaced like the other reinforcement 5 inches apart. These bars must extend into the base a distance of at least 50 diameters (31 inches) past the section at B. Horizontal bars should be placed longitudinally through the wall near the exposed face to prevent cracking due to contraction; J-inch bars spaced 12 inches apart are sufficient for this purpose. Example 4. A cantilever wall is to be 17 feet high above ground and to support a bank of earth whose surface has an upward slope of 2 horizontal to 1 vertical from the top of the wall. Angle of friction for backing earth = 35. The soil under the base may be safely loaded with 6000 pounds per square foot. Earth filling weighs 100 lb./ft. 3 and concrete 150 lb./ft. 3 Safe values of / c = 500 lb./in. 2 /,= 16,000 lb./in. 2 , and for diagonal tension v = 30 lb./in. 2 n=15. The base of the wall will extend 4 feet below the surface of the ground and the toe of the wall cannot extend beyond its face. Solution. Assume a depth of base of 24 inches and a width of base of 12 feet. (See Fig. 68.) Vertical Wall. The total height of the vertical wall is 19 feet. The thrust on the back of this wall is v eWK 100X19X19X.39 E = ^-= = 7040 pounds. This acts parallel to the surface of the earth and its horizontal com- ponent T = 7040 cos 26 30' = 6300 pounds. The moment of this about the base of the wall is (6300 X 19/3) X 12 = 478,800 in.-lb. From Table VII, R = 72 and p = .005. 12d 2 = 478800/72 = 6648, and d = 24 inches. The total thickness at base is 26 inches. Take top as 12 inches thick, and make face of wall vertical. At base, A = 24X12X.005 = 1.44 in. 2 From Table XV, f-inch square bars 4| inches apart will answer. All bars will extend to 12 feet below top, every third bar to 6 feet below top and every sixth bar to top of wall. 236 RETAINING WALLS Shear at base section is 6300 pounds and 6300 which is within limits without diagonal tension reinforcement. Overturning Moment. The thrust on the vertical section at the inner edge of the base is eS 2 v 100X26. 5X26. 5 W E = -^-K = - jr - X .39 = 13,690 pounds. 2 Z Its horizontal component is H= 13,690 cos 26 30' = 12,250 pounds and its vertical component V = 13,690 sin 26 30' = 6100 pounds. The weight of the base of wall TFi = 12 X 2 X 150 = 3600 pounds. Weight of vertical wall W 2 = ^~Y^ X 19 X 150 = 4560 pounds. Weight of earth on wall and is within the middle third of the base. Pressure on Soil. As the resultant cuts the bottom of the base at one-third the width from the toe, the maximum pressure at the toe is The pressure at the inner edge of the base will be practically nothing. Inner Base Slab. The loading on the horizontal base slab is the difference between the sum of the weights of earth and of the base acting downward, and the soil pressure acting upward. The maximum load will be at the inner edge, where the upward pressure is a minimum. Taking a foot in width along this edge and neglect- ing the upward pressure, the load will be 1000+25X100+2X150 = 3800 pounds per linear foot. The thickness of base slab will probably be determined by require- ments for shear. The maximum shear at edge of counterfort (taking REINFORCED CONCRETE WALLS 241 counterforts as 18 inches thick) is V = 3800(4 -.75) = 12350 pounds, and if no reinforcement be used for diagonal shear, the depth , V 12350 or the full depth must be 32 inches. If the assumed depth of 24 inches be used, 12350 ,. This would require light reinforcement for diagonal shear for 8 inches from the edge of the counterfort and may be met by bending up a part of the tension reinforcement to use for negative moment over the supports. The bending moment in the base slab is and using the 24-inch depth 270480 "12X22X22" Table VII gives p = .0032, and A = .0032X22X12 = .85 in. 2 From Table XV, we find that j-inch round bars spaced 6 inches apart are needed. The negative moments at the counterforts are the same as the positive moments and may be provided for by bending up alternate bars on each side of the support, and extending these across the counterforts to the quarter points in the next panel. Counterforts. The counterforts act as cantilevers to carry the horizontal thrust upon the curtain wall for panel lengths of 8 feet. This thrust is pounds? and its moment about the section at the top of the base is OK M= 110700X^X12 = 11070,000 in.-lb. o Considering the counterfort to act as a T-beam, of which the curtain wall is the flange, and the resultant of the compressive stresses to act at the middle of the base of the curtain wall, we may take this middle point as the center of moments for the tensions in the steel in the back of the counterfort. If the center of gravity 242 RETAINING WALLS of the steel is 3 inches from the surface of the concrete, its lever arm is 8.1 feet, and the total stress in the steel is pounds. 8.1X12 The required steel area is A = 114000 16000=7. 12 in. 2 1 Table X, we find that six 11-inch round bars will answer. These may be placed in two rows, four bars being placed 2 inches and two bars 5 inches from the surface of the concrete. These may be spaced FIG. 69. Design of Counterfeited Wall. 4 inches apart and 3 inches from the sides in a thickness of counter- fort of 18 inches. At a section 16 feet below the top the moment =4718900 in.-lb., and the steel required 4718000 5.4X12X16000 = 4.54 in. 2 At 8 feet below the top M= 1327100 in.-lb. and A =2.46 in. 2 Two bars may be stopped at 16 feet below the top, two at 8 fcvt and the others extend to the top of the counterfort. REINFORCED CONCRETE WALLS 243 The total shear in base section of counterfort is 110,700 pounds, and V 110700 At 16 feet below the top v = 45 lb./in. 2 Reinforcement for diagonal tension is needed from the base to a little above the section 16 feet below the top. This may be provided by the bars to be used for bonding the counterforts to the curtain walls and base slabs. Bonding Bars. The curtain wall and the base slab must be tied to the counterforts by horizontal and vertical bars capable of carry- ing the reactions at the points of support. These will equal the sum of the shears on the two sides of the counterfort. At the bottom of the curtain wall the load per foot of height is 2(4 .75)945 = 6140 pounds and the area of steel required 6140/16000= .38 in. 2 If these bars be placed in pairs and at the same distance apart as the hori- zontal reinforcement in the curtain walls, ^-inch round bars will answer. These should be looped around the steel in the face of the curtain wall, and extend into the counterfort at least 50 diameters for bond strength. For the base slab, the load upon the bonding bars per foot of width 2(4 -.75)3600 = 23,400 pounds, and the area of steel required A = 23400/16000 = 1.46 in. 2 A pair of f-inch square bars spaced 6 inches apart meets this requirement. Base Cantilever. The projection of the base at the toe of the wall is a cantilever, as in the cantilever wall, and carries the upward thrust of the soil. The maximum shear is F= / 5270 + 3830 \ X 3.7-300X3.7 = 15725 pounds. 15725 This cantilever may be made 39 inches at the face of the curtain wall and taper to 12 inches at the toe. The maximum moment is M = 15725 X 2 X 12 = 377400, 377400 12X37X37 R/f s = 23/16000 = 0014, and from Table IX, p = .0015. A = .0015 X 12 X37 = .67 in. 2 , and from Table XV, |-inch bars spaced 5 inches apart may be used. 244 RETAINING WALLS ART. 36. CONSTRUCTION OF RETAINING WALLS 132. Foundations. As stated in Section 126, the most common cause of failure of retaining walls is defective foundations. Careful attention must always be given to the sufficiency of the foundation, footings being arranged so that excessive pressure does not come upon the soil upon which the structure rests. On compressible soils it is important to equalize the pressures so that settlement under the toe of the wall may not cause the wall to tip forward. In constructing gravity walls this is accomplished by using a footing under the main wall which extends sufficiently beyond the base of the wall to cause the pressures to be equalized over the foundation soil, and bring the resultant near the middle of the foundation. Reinforced concrete walls must be given sufficient base to prevent excessive pressures on the foundation soil. The extension of the front base cantilever may often be used as a means of securing good distribution of pressures upon the founda- tion; when this is not feasible, widening the base at the back of the wall may answer the same purpose. When the soil is compressible, there is always some settlement, and this is greatest where the load is greatest. In many instances, there- fore, it may be advisable to extend the footing sufficiently to bring the center of pressure back of the middle of the foundation so as to make the pressure greater at the heel than at the toe of the wall, and produce a tendency to tilt backward. When soft materials are encountered, or when the pressures cannot be safely distributed over the foundation soil, a pile foundation or some other means of securing firm support for the wall must be em- ployed. Methods of constructing such foundations, and the loads which may be borne by soils are discussed in Chapter XII. The depth of foundations should be sufficient to prevent freezing in the soil under the footing of the walls, or of the earth in front of the wall at the depth of the bottom of the footing. This usually requires that the footing extend from 3 to 5 feet below the surface of the ground, depending upon local and climatic conditions. 133. Drainage and Back-Filling. Failures of retaining walls have frequently occurred because of the lack of proper drainage, hence pro- vision should always be made for the ready escape of water from the earth behind the wall. If the water is held in and the back-filling becomes saturated, the weight of the material is increased and ~the angle of friction decreased, thus producing a much heavier pressure CONSTRUCTION OF RETAINING WALLS 245 against the wall. Freezing of wet material behind the wall may also produce dangerous pressures against the back of it. To provide for drainage, weep-holes are commonly left through the base of the wall at intervals of 10 or 15 feet. In concrete walls, these are usually made by the use of drain tile about 3 inches in diameter. In stone masonry walls, the stones are set so as to leave an opening 2 or 3 inches wide through the course of masonry at the base of the wall. When the back-filling is of retentive material through which water will not readily pass, a layer of cinders, gravel, or some other porous material should be placed against the back of the wall to per- FIG. 70. Gravity Wall of Concrete mit the water to reach the drains without difficulty. It is always important that water be not held in the back-filling. The manner of placing the back-filling may sometimes have an important effect upon the pressures against the wall. The layers in which the filling is placed should slope away from the wall. With some materials, there is a tendency for the earth to slide along the surfaces between the layers in compacting and settling into place, which may materially increase the pressure if inclined toward the wall. 134. Gravity Walls. In constructing gravity walls it is common to give the back of the wall a batter by stepping off the surface, thus 246 RETAINING WALLS widening the base and making a smaller projection of footing neces- sary. In walls of stone masonry, the steps are usually the height of one or more courses while in plain concrete walls the steps are usually of uniform height of 2 to 4 feet, to simplify the form work, and for convenience in placing the concrete. Fig. 70 shows a typical section for a wall of this kind, as used for carrying a railway embankment. It is common to batter the back of a masonry wall at the top for 3 or 4 feet (see Fig. 70) to prevent injury if the backing becomes frozen near the surface and is lifted by the expansion. This is known as frost batter, and is commonly 2 or 3 inches to the foot. Concrete is quite largely replacing stone masonry in the construc- tion of retaining walls. For high walls, reinforced concrete is econom- ical and usually employed, while for walls less than 20 or 25 feet high, gravity walls may often be less expensive than reinforced walls. A larger quantity of concrete is required for the gravity wall, but concrete of less rich character may be employed and no steel is needed. For reinforced walls, about 1 to 6 concrete is usually used for the body of the work, while 1 to 9 concrete may commonly be used for gravity walls; footings being made of 1 to 11 or 1 to 12 mixtures. The cost of forms does not vary greatly for the two types of wall. CHAPTER VIII MASONRY DAMS ART. 37. GRAVITY DAMS 135. Stability of Dams. A gravity dam, like a retaining wall, depends upon the weight of the mass of masonry to resist the thrust of the water against it. As the dam carries water pressure instead of earth pressure, the loads to which the dam is subjected are defi- nitely known, and the thrusts are everywhere normal to the surfaces of contact. Let A BCD, Fig. 71, represent a slice, 1 foot thick, of a gravity dam sustaining a head of water as shown. FIG. 71. h = height of water above section AB; H = horizontal pressure of water against the dam; V = vertical pressure of water on back of dam; W = weight of dam above section AB', R = resultant pressure upon section AB', k = horizontal distance from inner edge of base to line of action of V ; 6 = width of base A B', d = distance from outer edge of base to line of action of W; 247 248 MASONRY DAMS x = distance from outer edge of base to point of application of resultant R. The conditions of stability for the dam are the same as for the retaining wall: It must not slide or shear on a horizontal section. It must not overturn about outer edge of section. The masonry must not be crushed by pressure upon the section. Stability against Sliding. Taking the weight of water as 62.5 lb./ft. 3 , the horizontal thrust against the dam above AB is # = 31.25 h 2 . This is the shear upon the section AB. If AB is a joint in the dam, or the base of the dam, H must be resisted by the friction of the masonry upon the masonry below, or upon the foundation under the dam, and the value of H/(W+V) must not exceed the coefficient of friction for the material. If A B is a section in a concrete dam, H is resisted by the shearing strength of the concrete as well as by the friction. Continuous joints are not usually employed in construction of masonry dams, and the interlocking of stones eliminates the tendency to slide without shearing blocks of stone. The possibility of sliding need usually only be considered at the foundation. Stability against Overturning. The overturning moment about the outer edge of the section at A, due to pressure of water, is M = ~X o The resisting moment of the weight of wall is M r = Wd, and the dis- tance from the outer edge A to the point of application of the resultant pressure on the base is W+V W+V If the water face of the dam is vertical, V = and W (2) Assuming that pressures upon AB are distributed with uniform variation from A to B, x should be greater than 6/3 in order that no tension may be developed in the section, as in the gravity retain- ing wall. Stability against Crushing. The total pressure normal to the section AB is TF+F, distributed over the section with center of pres- GRAVITY DAMS 249 sure distant x from A. The maximum normal unit pressure is therefore (see Section 52) /c = p ...... (3) This is approximately the crushing stress in the masonry at the outer edge of the section, or the maximum pressure upon the founda- tion if A B is the base of the dam. When the reservoir is empty and the water pressure is removed, the pressure upon the section AB will be W, with center of pressure distant d from the outer edge. The unit pressure at the outer edge of the section will be /.- and at the inner edge, In dams of unsymmetrical cross-sections it is necessary to con- sider the pressures coming upon the bases of sections when the water pressures are removed, as when the reservoir is empty. In this case, the weight of dam will be the only load, and the centers of pressure due to this weight must always come within the middle third of the base, and the crushing stress be within proper limits, so that removal of the water pressure may produce no harmful effects upon the dam. 136. Graphical Analysis of Profiles of Dams. For low dams carrying small heads of water, trapezoidal cross-sections may be employed, and designs made in the same way as for retaining walls using water pressure instead of earth pressure upon the back face of the dam. As the depth increases such a section becomes increasingly uneconomical and the form of cross-section should be modified so as to make the thickness only that required to carry the load above, and the profile such as to distribute the material to the best advan- tage. Fig. 72 shows a method of graphical analysis applied to the sec- tion of a gravity dam. ookk represents a section of a dam 100 feet high. Take a slice of the dam 1 foot thick and of the section shown and divide this by horizontal planes, a-a, b-b } c-c, etc., into a number of horizontal layers (in this case, each 10 feet thick). The weights of the layers, ooaa, aabb, etc., are now computed and plotted to a convenient scale, in the vertical line 0-K. The distance from to each of the several points, A } B, C, etc., represents the total weight of masonry Wa, Wb, etc., above the corresponding sec- tion a-a, b-b, etc., of the dam. 250 MASONRY DAMS The line of action of the total weight of masonry above each horizontal section must now be found. This may be done by taking moments about a vertical line, or it may be done graphically as follows: FIG. 72. Graphical Analysis of Gravity Dam. The center of gravity of each layer into which the section of the dam has been divided is determined and marked (as shown by the points enclosed by circles). From the point o\, lay off on the line Oi-ki the distances from the centers of gravity of the layers to any vertical line, as o-k f . (The scale used in laying off these distances, oi ai, oi bi, etc., is here made larger than that used for the section GRAVITY DAMS 251 of the dam.) Assume a pole, P, and draw strings to the weight line 0- K, then from a point on the vertical through ki draw the equilib- rium polygon as shown, finding the positions of the resultant lines of action, Wk, Wi, Wh, etc. The distances of these lines from the vertical through o\ are the same as the distances of the respective centers of gravity from the line o-k' on the section of the dam. Plotting these lines of action and drawing them to intersection with the corresponding horizontal sections upon which they act, we find the line of pressure for the dam with no water pressure against it. When water pressure is against the dam to its full height, the horizontal against any portion h feet in depth below the surface is # = 31.25 h 2 . These pressures may be computed for each of the horizontal sections, and each resultant pressure acts on a horizontal line at one-third the height from the section to the surface of the water, as shown,- Ha, Hb, etc. On the horizontal line O-K', make the distances 0-A', 0-5', etc., equal the values of the water pressures Ha, Hb, etc., to the same scale as used for the weights of masonry. The lines A- A', B-B', etc., now represent, in direction and amount, the resultant pressures, Ra, Rb, Re, etc., upon the various horizontal sections aa, bb, cc, etc., of the dam. Lines drawn parallel to these directions through the intersections of the corresponding H and W lines of action give the points of application of these resultants upon the various sections, and locate the lines of pressure with water against the dam to the top. 137. Design of Profile. In designing a profile, for a dam we com- mence at the top with the assumed thickness and find by trial the required base thickness for each horizontal layer, making each thick- ness such that the line of pressure remains everywhere within the middle third of the section. This may be done by the use of the formulas given in Section 135 or by the graphical method of Section 136. The crushing stress upon the masonry must also be kept within safe limits. Let 6 = the width of the section; x = the distance from the outer edge to the point of appli- cation of R', 2/ = the distance from the inner edge to the point of appli- cation of W. The maximum crushing stress at the outer edge of the section is given by the formula, (5) 252 MASONRY DAMS The maximum crushing stress at the inner edge is , _W(4b-6y) The allowable crushing stress depends upon the quality of masonry used, and the conditions under which the dam is to be constructed. In high dams, where the front face of the dam has considerable batter, the pressure allowed at the outer face is often made less than that at the inner face. The maximum pressure at the inner edge of a section occurs when the dam carries no water pressure, and the resultant pressure on the base is vertical. The maximum pressure at the outer edge occurs when full water pressure is on the dam, the batter of the outer face is greater than that of the inner face, and the resultant pressure is inclined, only its vertical component being con- sidered in determining the stress. For these reasons Rankine's recommendation that the allowable unit crushing stress at the outer edge be made less than that at the inner edge has been followed by some designers. Pressures of from 8 to 15 tons per square foot have been allowed in a number of large dams of massive rubble or cyclopean concrete. The profile resulting from this method of design is somewhat irregular and may be modified by fitting it with more uniform batters and smooth curves, thus giving a more pleasing appearance and better profiles for construction purposes, without appreciably affecting its stability. Vertical Water Pressure. As the water face of the dam is nearly vertical, it is usual to disregard the vertical component of the water pressure, which is of small consequence in dams of less than about 180 to 200 feet in height. This component has the effect of diminishing the stress upon the outer edge of the section while somewhat increas- ing the total pressure. Its neglect is therefore a small error on the safe side until a depth is reached at which the slope of the inner face may make it of more importance. The shape of the profile depends upon the top width given to the dam, and the weight of the masonry used. The top width must be sufficient to resist any probable wave action and ice pressure, and should usually be made greater for high dams than for low ones. This is a matter of judgment in each case, about one-tenth of the height of dam being frequently used, with a minimum of about 5 feet and a maximum of 20 feet where no roadway is carried on top of the dam. The dam should always extend to a sufficient height above the GRAVITY DAMS 253 normal water surface to prevent water passing over the dam due to waves of floods for which wasteways might not be quite sufficient. This may require the dam to be raised 5 or 10 feet above the eleva- tion of the expected water surface. In designing the dam, water should be assumed level with the top. The weight of masonry used in dam construction commonly varies from about 135 to 150 pounds per cubic foot. The heavier the masonry is assumed to be, the less the required width of section until a depth is reached at which the width is determined by the necessity of pro- viding sufficient area to carry the weight of masonry above. Below this point, usually about 200 feet below the water surface, the width required is greater for the heavier masonry if the same unit com- pression be allowed. Uplift and Ice Pressure. If water under hydrostatic pressure has access to the interior of the dam, the upward pressure will tend to lift the masonry and diminish its effective weight in the moment which prevents overturning. In this discussion it has been assumed that the dam is constructed water-tight, but as this is not altogether possible, in many instances it may be necessary to allow for upward pressure in designing the profile, or make special provision for drain- age a topic discussed in Section 140. If ice forms on the surface when the reservoir is full, a consider- able pressure may be brought against the top of the dam, which should be considered in its design. This will be a concentrated hori- zontal thrust at the surface of the water equal to the crushing strength of the ice, and has been assumed in a number of important dams at from 2500 to 4500 pounds per linear foot of dam. In storage reser- voirs generally, heavy ice is not likely to occur with full reservoir, and if water be low when freezing occurs no special allowance for ice pressure is necessary. Local conditions must determine the necessity of allowing for ice pressure in each instance. 138. Diagonal Compressions. The common method of analysis, already described, considers only the stresses upon horizontal sec- tions and resolves the diagonal thrusts into normal compressions and parallel shears upon these sections. This method does not give the actual maximum compressions, but by using proper unit stresses has seemed to give satisfactory results in use. Several methods have been proposed for computing more accurately the maximum unit compressions. Diagonal Compression upon Horizontal Section. In 1874 Bou- vier * used the actual diagonal pressure (R, Fig. 72) in computing the 1 Annales des Fonts et Chaussees, 1875. 254 MASONRY DAMS maximum unit compression upon a horizontal section, claiming that the unit compressive stresses produced by R parallel to its line of action are greater than those normal to the section. He con- sidered R to be distributed along A-B, so as to act upon successive small sections normal to its direction as shown in Fig. 73. If /3 is the angle made by R with the normal to section A B, and b is the width of section, the area upon which R acts is AC=b-coS'@, and the maxi- mum intensity of the compressive stresses is = Jcd f c b 2 cos 2 |8 cos 2 0' . . (7) in which f C d is the unit compression at the outer edge of the section parallel to R, and f c is that normal to the section at the same point. B FIG. 73. Professor Unwin 1 has shown that the maximum unit compression at the face of the dam occurs on a section normal to the face, and that the maximum value of this compression at the outer edge of a hor- izontal section through the dam is f cm = ^| , in which f c is the value cos (/ of unit vertical compression and B is the angle made by the batter of the face of the dam with the vertical. A method of finding the maximum diagonal compression and its direction at any point of a horizontal section of the profile of a dam is given by Professor Cain, 2 which agrees practically with Unwin's results for the stress at the edge. Compression upon Inclined Sections. The distribution of pressure upon an inclined section is sometimes investigated and the maximum unit stress at the outer face of the dam found to be greater than that 1 Proceedings, Institution of Civil Engineers, Vol. CLXXII, Part II. 2 Transactions, Am. Soc. C. E., September, 1909. GRAVITY DAMS 255 for a horizontal section. In Fig. 74 using the same profile employed in Fig. 72, the pressure upon the inclined section k-n is that due to the water pressure (H) upon the inner face 0-K of the dam combined with the weight of masonry (W) above the section fc-n. The unit compression at n is obtained in the same manner as for the horizontal section. This stress for this profile is greater than for the same point when obtained by using the horizontal section through n, and about the same as that at the outer edge of the base section k-k. Lateral Distribution of Stress. In the trapezoidal distribution of stress, which considers the stresses to vary uniformly from the inner to the outer edge of the section, it is assumed that the whole width of the dam acts together as a single homogeneous body. It is not probable that this is the case in a wide section. The middle portion of the section carries more and the edges less stress than the assumed distri- bution shows, and for this reason many designers have considered that the ordinary method, with low allowable stress upon the outer edge, as proposed by Ran- FIG. 74. kine, to be sufficiently exact. The ordinary method, taking successive horizontal sections, provides an easy way of determining an approximate profile. Careful study should, however, be given to the possible diagonal stresses in a high dam, and if such stresses exceed the allowable unit compression, the profile should be widened so as sufficiently to reduce them. 139. Horizontal Tension. Experiments have been made by Sir J. W. Ottley and Mr. A. W. Brightmore 1 upon models of dams made of plasticine (a kind of modeling clay), and by Messrs. J. W. Wilson and W. Gore 2 on models made of india rubber. The distribution of stresses through the profile was determined in each case by observing the horizontal and vertical displacement of points in the section. These experiments seemed to confirm, in general, the ordinary theory of the trapezoidal distribution of stresses, and to justify the methods of design in common use. At the base of the dam, where the profile section joins the founda- 1 Proceedings, Institution of Civil Engineers, Vol. CLXXII, p. 89. 2 Proceedings, Institution of Civil Engineers, Vol. CLXXII, p. 107. 256 MASONRY DAMS tion, it was found that a different distribution of stress occurs, ten- sion being developed at the inner edge of the base by the immovability of the foundation. Thus, in Fig. 75 the shear on A-B, due to the horizontal water pressure causes horizontal or diagonal tension (T) in the foundation at the inner edge (A) of the base. In the plasticine models diagonal cracks (A-C) occurred at this point in the foundation. Various methods have been suggested for meeting or reducing this tension by modifying the shape of the profile at the base or reinforc- ing the foundation. This does not seem necessary for dams as usually -b constructed. A high masonry dam is usually on solid rock foundation, and the strength of the rock is such that no break in the foundation is to be anticipated from this cause. In most dams the foundation is FIG 75 in rock at considerable depth below the bed of the stream, .and the lower part of the dam is enclosed on both sides by gravel or other soil which usually may be considered to strengthen the dam, although the full depth of water pressure should be assumed to act upon it. If, however, this filling is soft material, which flows when saturated, it may increase the pressure against the dam and may be considered as a fluid heavier than water. 140. Uplift. If a dam be so constructed that water under pres- sure may penetrate into the ulterior of the dam or under its base, the effect of such pressure must be considered in its design. There is considerable difference of opinion among engineers concerning the necessity of providing for uplift in designing the profiles for dams. Some allow for it in ah 1 cases; while others claim that properly con- structed masonry or concrete will be so nearly water-tight that the effect of uplift may be neglected. Interior Pressure. It is always possible that some water may be forced into imperfect joints in the masonry and, if it be prevented from escaping at the lower side of the dam, have the full hydrostatic pres- sure of the head in the reservoir. For this reason it is important that the water face of the dam be made as nearly impervious as possible, and that the interior of the dam be drained so that any water passing into the masonry may escape without damage. It is evident that uplift of the interior of the masonry can exist only where continuous joints for considerable distances are filled with water under pressure. GRAVITY DAMS 257 If concrete be porous and its voids filled with water under hydrostatic pressure, no uplift occurs until the pressure becomes sufficient to over- come the cohesive strength of the concrete. In properly constructed masonry dams, it is usually unnecessary to consider the effect of uplift on sections above the base of the dam. Upward Pressure on Base. The probability of uplift under the base of a dam depends upon the character of the foundation. Care- ful attention should always be given to the determination of the character of the foundation material to considerable depths below the base of the dam. The kind of material of which the foundation is composed, and the existence of seams in the rock, or of strata of permeable material must be accurately investigated. When the foundation is of solid rock without seams, if care be used in joining the base to the foundation and cut-off wall be used under the inner edge of the base, there is little chance of appreciable uplift under the base. When the foundation is permeable and there is water against both sides, as is frequently the case in dock walls, the full hydrostatic head is usually considered to act under the whole base. This is some- what excessive, as it implies that the dam is floating upon a continuous surface of water. Probably two-thirds of this pressure would repre- sent about the maximum which could reasonably be expected in any case. When the foundation is stratified horizontally, so that water may be expected to pass under the dam and escape below, a uniformly di- minishing upward pressure from the inner to the outer edge of the base may be assumed; the pressure at the inner edge being taken at about two-thirds the hydrostatic head above the dam, and that at the lower edge at zero. The probability of upward pressure on the foundation should always be carefully investigated, and the section, where necessary, increased sufficiently to provide weight of masonry to overcome the overturning moment of this water pressure. This subject is very fully treated in the discussion of a paper by the late C. L. Harrison in the Transactions of the American Society of Civil Engineers for December, 1912. Mr. Harrison's conclusions are: 1. For any stable dam, the uplift in the foundation cannot act over the entire area of any horizontal seam, and in the masonry it cannot act over the entire area of any horizontal joint. 2. The intensity of uplift at the heel of the dam can never be more, and is generally less, than that due to the static head. Also, this uplift decreases in intensity from the heel to the toe of the dam, 258 MASONRY DAMS where it will be zero if the water escapes freely, and will be that due to the static head if the water is trapped. 3. The uplift in the foundation should be minimized by a cut-off wall, under-drainage, and grouting when applicable; and in the dam itself by using good materials and workmanship, and by drainage when advisable. 4. The design should be based on the conditions found to exist at each site after a thorough investigation by borings, test-pits, and otherwise, and modified if found necessary after bed-rock is uncovered. ART. 38. DAMS CURVED IN PLAN 141. Curved Gravity Dams. In constructing dams across narrow valleys, it is often desirable to curve the dam in plan, so as to make it form a horizontal arch, convex upstream. When so arranged, a portion of the water pressure may be transmitted to the sides of the valley by arch action, thus diminishing the overturning moment which would exist in a straight dam of the same section. In certain locations, the shape of the valley and depth of suitable foundations make the use of the curved form economical in saving materials, although the length of dam is increased by the curvature. The curved form for gravity dams has not usually been adopted for the purpose of securing the arch action, although the advantage of the curved form is recognized and the added security obtained by the possibility of the upper part of the dam acting as an arch is worth considering when it does not materially increase the cost. In order to develop free arch action in any horizontal slice of a dam, it would be necessary that the section be free to move horizon- tally when the pressure comes against it. As each section is rigidly connected with these above and below it and the base is attached to a practically immovable foundation, the arch action is very imper- fect. Near the top of a gravity section, deflection of the section may be sufficient to permit a portion of the water pressure to be resisted by the arch, but in the lower half of the dam such resistance is inappreciable. There is no satisfactory way of determining how much of the pressure is borne by the arch in a curved gravity dam. In an analy- sis of the stresses in the Cheeseman dam, Mr. Silas H. Woodward estimated 1 roughly the amount of water pressure carried by the arch action, by determining the deflection at various points in the mid-section of the dam, considering the resistance of horizontal slices 1 Transactions, Am. Soc. C. E., Vol. LIII, p. 108. DAMS CURVED IN PLAN 259 of the darn by arch action, and the resistance of a vertical slice as a cantilever beam, fixed at the bottom to the foundation. He concluded that in the Cheeseman dam, the arch carried about half the water pressure at the top and about 6 per cent at the mid-height of the middle section. Mr. Woodward's analysis seemed to indicate that, while added security might be obtained through arch action at the top of the dam, the lines of pressure of the gravity section were only slightly modified by considering part of the load carried by the arch. His conclusion was that no diminution of the gravity section would be justified because of dependence upon arch action. The use of curved plans for gravity dams may be of advantage in affording a possibility of motion when expansion and contraction take place, without cracking the masonry. The advantages to be gained by using curved plans, however, do not seem sufficient to make them worth while when they involve increase in cost. In constructing gravity dams across narrow valleys where arch action might be developed, the sides of the valley may also offer considerable support to a straight dam, causing horizontal slices of the dam to act as beams supported at the ends. In any such dam the actual stresses are probably considerably less than those obtained by considering the gravity resistance only. 142. Arch Dams. Dams are sometimes constructed which depend for stability mainly upon arch action, and are designed as horizontal arches. A number of dams of this type have been constructed across narrow valleys, with sections much lighter than could be used for gravity dams. In some, the lines of pressure fall quite outside the bases when considered as gravity sections. Let A-B, Fig. 76, represent a horizontal slice, 1 foot thick, through a circular dam. R = radius of water face; t = thickness of section; P = water pressure per foot of length; / c = unit compression on the masonry; h = height of water surface above section; w = weight of water per cubic foot. If the slice be supposed to act freely as an arch and carry the water pressure to the abutments, _PR_whR *-' t > 260 MASONRY DAMS If a limiting value of f c be assumed, the thickness of section required at any depth will be PRwhR For a dam of constant radius (R) the required thickness varies uniformly with h, or the vertical section of the dam is triangular. As the ends of the arch at A and B are built into the sides of the valley and not free to move toward the center when subjected to the water pressure, the lines of thrust of the arch will not be exactly axial as assumed in Formula (8), and bending stresses will develop in the arch, giving a maximum compression somewhat greater than the average value. This effect will usually be small as compared with the stress due to arch action, although French authorities recommend FIG. 76. that the line of thrust be assumed at the outer edge of the middle third at the crown, thus making the maximum compression double the average. The use of vertical expansion joints through the dam, dividing it into voussoirs, has the effect of largely eliminating the bending stresses. In practice the bending stress is commonly neglected, very conservative values for f c being used. When the length of the arch is small as compared with its thick- ness, it becomes a curved wedge which acts as a beam between the abutments supporting its ends, and should be considered as a curved beam a condition frequently occurring near the bottom of a curved dam, where the valley is narrow and the thickness of the dam con- siderable. The thickness obtained by considering such a section as an arch is always sufficient. A masonry structure cannot be considered to act as an arch when the thickness of the arch ring is more than from one-quarter to one- DAMS CURVED IN PLAN 261 third of the radius of its outer surface. The exact limitations within which such action may take place are not definitely known and are seldom of importance in a dam. Resistance of Vertical Cantilever. As a dam is rigidly fastened to the foundation, it is evident that complete arch action cannot take place, and that in the lower part of the dam, the arch can carry very little of the load. A vertical section of the dam may be considered as a cantilever fixed at the bottom as in a gravity dam, and the resist- ance of the cantilever to deflection will limit the extent to which arch action may occur. Attempts have been made by estimating the relative deflections of the horizontal arch and the vertical cantilever at various heights upon the mid-section of the dam, to determine what portion of the load is resisted by each. Such studies have been made by Mr. Silas H. Woodward 1 for the Lake Cheeseman dam, which is a curved dam of gravity section (see Section 141) and by Mr. Edgar T. Wheeler 2 for the Pathfinder dam, which was designed as an arch, and has a section considerably lighter than could have been employed in a gravity dam. The section of the dam has a width of 10 feet at the top, a batter of .25 on the downstream and .15 on the upstream face. These analyses, with accompanying discussions, are interesting as throwing light upon the probable action of such dams when sub- jected to water pressure, but afford no means of determining the actual stresses occurring. The vertical cantilever has the effect of reducing the stresses in the arches, but it is not proposed to consider the combined actions in designing dams, or to attempt to use the actual stresses, as limited by the cantilever resistance in proportioning the arches. In practice, the arches are given sections which would enable them to carry the whole water pressure, and the vertical resistance is considered as a source of additional security. Horizontal Shear. As the dam is fixed at the bottom to the foun- dation and the various horizontal slices are not free to act independ- ently of each other, the thickness at any point should be sufficient to carry the total water pressure above as horizontal shear. If S be the safe unit shear per square foot, the thickness should not be less than 7 rt i= . Such shearing stresses can exist only near the bottom of the 2o dam, where it is rigidly attached to the foundation, and can never reach the assumed value if the water pressures toward the top of the dam are carried by arch action. 1 Transactions, Am. Soc. C. E., Vol. LIII, p. 89. 2 Engineering News, August 10, 1905. 262 MASONRY DAMS Weight of Masonry. Each horizontal slice of an arch dam must carry the weight of the portion of the dam above as a vertical com- pression. This compression is computed as in the gravity section when the dam is empty, and must not exceed a safe unit stress on any part of the section. The weight of masonry above also produces a distortion of the horizontal section. The value of Poisson's ratio for concrete may be taken as approximately one-fifth of the unit horizontal compression produced through the mass of masonry, if prevented from expanding laterally is approximately one-fifth of the unit vertical compression which causes it. The effect of this horizontal compression is to cause an expansion of the horizontal section, increasing the length of the arch ring, and deflecting the crown of the arch upstream. When water pressure is brought against the dam, a portion of the pressure, sufficient to produce com- pression in the arch equal to the unit horizontal pressure due to the vertical load, will be used to bring the arch back to its initial position, and no deflection due to arch action will occur until this pressure has been passed. When the crown of the arch has been deflected upstream by the weight of masonry, stress is brought upon the vertical cantilever by its resistance to bending in that direction. If water pressure be now brought against the dam, the vertical cantilever action will offer no resistance to downstream motion until the pressure upon the arches becomes sufficient to bring the dam back to its original unloaded position. The existence of this initial distortion due to the weight of masonry may depend upon the manner in which the dam is constructed. In order to produce this effect it is necessary that the horizontal layers be completed and hardened in position before the load above is applied. If portions of the work be carried up in vertical sections, or if vertical contraction joints be left, to be afterward grouted, the deflection due to weight of masonry may take place only to a very limited extent. Constant-angle Arches. Arch dams are usually constructed across narrow gorges which can readily be spanned by an arch of moderate radius. The gorges vary in cross-section, being usually much nar- rower at bottom than near the top of the arch. The arch at bottom will therefore be much shorter than at the top and if the same radius be used at top and bottom, or the centers lie in the same vertical line, the central angle included by the dam will be greater at top that at bottom. It has been shown by Mr. Lars R. Jorgensen 1 that a dam with a constant central angle of 133 34' requires, theoretically, 1 Transactions, Am. Soc. C. E., Vol. LXXVIII, p. 685. DAMS CURVED IN PLAN 263 the minimum amount of masonry in its construction, and that angles from 110 to 150 vary but little from the minimum. It has therefore been proposed to vary the radius from the top to the bottom, so as to keep within these ranges of central angles. This makes the radius of the dam vary with the width of the gorge at different elevations. Several dams have been constructed in which this principle has been approximately applied. The topography of the site must be care- fully studied in every instance and the dam fitted to its location, keeping in mind the general principles involved. Temperature Stresses. Comparatively little is known concerning the changes of temperature to be expected in a mass of masonry like a dam, but it is evident that distortions produced by such changes may sometimes be of importance, and careful attention should be given to their probable effect. Temperature above the normal at which the masonry was placed cause deflection upstream through expansion, which may bring bending stresses upon the vertical section when the water is low behind the dam. Contractions due to tem- peratures below the normal, causing tensile stresses which the masonry or concrete is not calculated to bear may cause cracks, or prevent the arch action through shortening the arch near the top. It is desirable that masonry which may be injuriously affected by low temperature, be placed when the temperature is low, thus giving a low normal and probable small range below. Mr. Wisner 1 urges that reinforcement be used on the faces of the upper portion of arched dams to prevent cracks; vertical rods on the downstream face to take up the possible vertical tensions due to expansion, and horizontal rods on the up- stream side to prevent contraction cracks. 143. Multiple-Arch Dams. Dams consisting of a series of con- crete arches supported by buttresses are sometimes used for moderate heights where suitable foundations are available and the cost of gravity dams would be greater. The amount of concrete required is much less than for gravity dams, and where concrete materials are expensive considerable savings in cost may result from their use. The form work required and the thin sections of concrete, make the unit costs much more than for gravity dams, and under favorable conditions for cheap concrete work gravity sections may be cheaper to construct. As the buttresses must carry the thrust of the water pressure, it is essential that they be established upon very substantial and unyielding foundations. Usually this is solid rock, although some dams of this type have been built upon gravel or fissured rock. Where the foundation is stable but of character which may permit 1 Engineering News, August 10, 1905. 264 MASONRY DAMS water to penetrate it, this type of dam has advantages over a gravity dam on account of the less importance of possible uplift. Two types of multiple-arch dams are in use; (1) those in which the axes of the arches are vertical, the water pressures coming hori- zontally against the faces and being transmitted as horizontal thrusts against the buttresses; (2) those with inclined axes, the water pres- sures acting normal to the sloping axes and bringing vertical as well as horizontal thrusts upon the buttresses. Let Fig. 77 represent an inclined arch dam. A slice of the arch ring normal to the axis carries a water pressure which varies from the crown to the springing line, and also carries a portion of its own weight to the buttress. If a slice of the arch ring be divided into voussoirs as shown, the water pressures upon each voussoir (Pi P$) varies with the depth (hi-hs) below the surface of the water. The FIG. 77. Inclined Multiple-Arch Dam. weights of the voussoirs (Gi-Gs) may be considered as divided into components, (N=Gcos6) normal to the section and (W=G sin 6) parallel to the section. The normal components are carried as longi- tudinal thrusts to the foundation, while the parallel components (Wi Ws) are carried by the arch ring to the buttress. Having determined these loads, an approximate line of thrust may be drawn by the method used for voussoir arches (see Section 162), from which stresses may be determined. In designing such an arch, the required thickness at various depths may be approximately determined by finding the thickness for a horizontal arch at the same depth, then using this thickness in the analysis, modifying it as required. Practically an assumed thickness is given the ring at the top and tapered to the required thickness at some point below. DAMS CURVED IN PLAN 265 When the arch axis is vertical, the arch carries only the water pressure, which is uniformly distributed over the face. The weight of the arch, in this case, is normal to the arch section and is carried vertically to the foundation. The thickness required for the arch ring may be found from Formula (8), (Art. 138). The stresses upon the buttresses of a multiple-arch dam may be found by the methods used for gravity dams. In Fig. 78 E-F is a section through the crown of an inclined arch; A BCD being the side projection of the buttress. The form of the buttress must be such that the resultant thrust upon any horizontal section A-B will act approximately at the middle of the section. The loads acting are: H FIG. 78. (1) The horizontal water pressure ( H = ^wh 2 L) due to the depth of water above the plane A-B, upon a length of dam (L) equal to the distance between the middle points of adjacent arches. (2) The vertical water pressure (V= H-tan-O). The center of pressure for the vertical water pressure is at the center of gravity of a horizontal section of the water face of the arch at two-thirds the depth below the surface of the water. (3) The weight (Wi) of the two half arches upon each side of the buttress. The center of gravity for the weight of the arch is at the center of gravity of the center line of a horizontal section of the arch ring which passes through the center of gravity of the vertical section (E-F) of the crown of the arch. If the centers of gravity of the center lines of the arch ring be determined for horizontal sections at the top and bottom of the arch, all intermediate centers will lie upon the line joining these points. 266 MASONRY DAMS (4) The weight of the buttress itself, acting through its center of gravity. The resultant (R) of these loads should cut the base A-B near its middle point, in order to secure uniform distribution of pressure over the section. Buttresses, for dams of this type, are usually made very thin in comparison with their widths, and are therefore stiffened laterally by the use of horizontal struts from buttress to buttress, or by the use of cross walls. The design of these struts is purely a matter of judgment on the part of the designer. In the design of multiple-arch dams, the general lay out is a matter which must depend upon local topography. Each dam is a problem by itself, and must be made to fit its location. It has been found that in some instances, where the conditions are favorable to the con- struction of masonry dams of moderate height, multiple-arch dams may be built at much less cost than gravity structures. Forty to 60 or 70 feet between centers of buttresses are commonly found economical distances. Arches with axes making angles of 30 or 40 with the vertical are apt to show some saving of material as compared with vertical axes, but this is not always the case. The unit cost of construction is usually somewhat greater for inclined arches. In constructing gravity dams, a cheaper grade of masonry may be employed, and the form work costs less than for multiple-arch dams. Careful studies of local conditions, and tentative trial designs are necessary in each case for best results. Temperature Stresses, due to temperatures lower than those at which the arches are constructed, are to be expected in all structures. These produce shortening of the arch and give tensile stresses which may result in cracks when the dam is empty. Horizontal reinforce- ment near the downstream face at the crown and near the upstream face at the springing line is desirable to resist this tendency to crack. ART. 39. REINFORCED CONCRETE DAMS 144. Reinforcement in Arch Dams. Several designers have used steel reinforcement in arch dams, where it has seemed desirable to prevent the possible development of cracks, or to give additional security where the uncertainty concerning stresses made tensions seem possible under certain conditions. Cracks which may result from changes of temperature when dams are empty are frequently guarded against by using reinforcement, as has been mentioned in the previous articles. The stresses in most of these cases are practically indeter- REINFORCED CONCRETE DAMS 267 minate, and the reinforcement is placed according to the judgment of the designer. These structures are sometimes called reinforced concrete dams in published reports, but are not designed as reinforced structures and are not properly so classed. No fully reinforced arch dams have as yet been constructed, and no dams have been designed in which the stresses have been determined by the use of the theory of the elastic arch. 145. Flat Slab and Buttress Dams. For dams of moderate height reinforced flat slab and buttress construction has frequently proven economical. In this type of construction the buttresses are usually placed from 12 to 18 feet apart, and the slabs extending be- tween buttresses are inclined at an angle of 40 to 45 with the ver- tical so that the resultant of the normal water pressures passes near the middle of the base of the buttress. Most of the dams of this type in use have been constructed under the patents of the Ambursen Hydraulic Construction Company. Fig. 79 shows a dam of this type in section through the inclined slab. The loads carried by the slab consist of the normal water pres- FIG. 79. sure and the normal component of its own weight. The slab may be designed by the ordinary method for reinforced concrete beams, but the values used for allowable stresses should be very conservative. The buttress should be made of sufficient width to cause the resultant thrust upon its base to pass approximately through its middle point when fully loaded, and must have sufficient base area 268 MASONRY DAMS to keep the pressure upon the foundation within reasonable limits. Lateral stiffness of the buttresses may be provided by giving them sufficient thickness, and using reinforcement on the sides, or it may be obtained by ties and struts between buttresses. In many dams of this type, cellular construction is adopted, in which the spaces between buttresses are divided into cells by hori- zontal floors, openings through the buttresses providing opportunity to pass under the dam throughout its length. Sometimes vertical walls provide rooms which may be utilized for power house or other purposes. Slab and buttress dams, like any other masonry dams, require firm foundations. For locations where substantial foundations may be obtained for buttresses on porous material, they possess an advan- tage over gravity dams which would be subjected to upward pressure. In these cases it is necessary to provide cut-off walls at the heel of the dam to prevent water passing under and washing out the foundation. ART. 40. CONSTRUCTION OF MASONRY DAMS 146. Foundations. Masonry dams are ordinarily applicable only to situations where foundations of solid rock may be obtained. Careful examinations of the character of the rock should always be made to considerable depths below the foundation in order to make sure that no seams or strata of porous materials exist, which might cause slipping of the foundation when subjected to pressure of water behind the dam. Nearly all of the failures of masonry dams which have been recorded have been due to defective foundations, causing settlements through washing out the foundation materials, or sliding of base of dam, and foundation on seams or soft strata through which water under pressure found its way. Where the depth to solid rock is considerable and the rock or gravel near the surface is of a character to give substantial support to the structure, masonry dams may sometimes be used without carrying the base of the dam into the solid rock. In such cases, curtain walls at the heel of the dam should be carried down to the rock to shut off leakage and possible washing of the foundation. When the rock is seamed or fissured, it may frequently be made tight by grouting, which is done by drilling into it and forcing grout (usually of neat cement and water) under pressure into the fissures until the cracks become sufficiently filled to force grout to the surface through adjacent drill holes. CONSTRUCTION OF MASONRY DAMS 269 When a high dam is to be constructed, the geological structure of the valley should be studied, and core drill borings made over the site of the dam so as to determine fully the stability of the foundation and the probability of leakage around or under the dam. The placing of the foundations of a dam is usually the most diffi- cult part of the work of construction. Commonly it is necessary to divert the water of the stream to be damned, and seepage water must be handled in making the excavations and placing the masonry. The methods used in such work are described in Mr. Chester W. Smith's "Construction of Masonry Dams," New York, 1915. 147. Masonry for Dams. Several types of masonry are some- times used in dams of massive construction. Heavy rubble masonry has commonly been employed, in which the large stones are set in mortar beds, and the vertical joints filled with mortar and small stones carefully placed by masons. The stones are put into place by derricks and must be held and lowered so as to seat evenly upon the mortar bed, being set with careful attention to bond, so that no continuous joints exist in any direction. The complete filling of all joints is important. Cyclopean masonry, in which the large rubble stones are set in beds of concrete and the joints filled with soft concrete, has recently been used to considerable extent. The joints are made thicker than mortar joints, and the labor required in setting the stones and making good joints is much less than in the ordinary rubble. Rubble concrete is masonry in which rubble stones are distributed through a mass of concrete. In some cases, boulders of considerable size are used in such work. This differs from cyclopean masonry mainly in the smaller amount of large stone used and larger quantity of concrete. It uses more cement, but is more rapidly constructed and requires less hand labor. Plain concrete is now frequently used, without large stone, for massive work, as well as for the dams of thin sections. The plant required is less, as derricks are needed for handling the heavy stone, and usually more rapid progress is possible in placing the concrete. The nature and location of materials and character of labor available determine the relative costs of the different methods of construction. The rapidity of construction is usually greater as the quantity of large stone becomes less. 148. Overflow Dams. When water is to flow over the top of a dam or spillway, the section must be modified to provide for the pass- ing of the water with the least disturbance possible, and to take into account the additional head of water above the dam. 270 MASONRY DAMS If the water falls freely over the dam, its crest should be given such form as to eliminate the possibility of causing a vacuum behind the sheet of falling water. The effect of the impact of the falling water must also be taken into account, and provision made for pro- tecting the toe of the dam against erosion, which is frequently done by providing a water cushion into which the stream may fall. In overflow weirs of considerable height, the downstream face of the dam is given approximately the form of the curve that the water would take in falling freely over the weir under maximum head. The water may then follow the surface of the dam, and, by reversing the curve in the lower part of the section, be turned to horizontal direc- tion at the toe of the dam. In designing such a section, the weight of the water on the downstream face is neglected, the pressure on the upstream face being taken as that due to the full head at greatest expected flood. Special attention should also be given to the possi- bility of uplift or of scouring at the toe. CHAPTER IX SLAB AND GIRDER BRIDGES ART. 41. LOADINGS FOR SHORT BRIDGES 149. Highway Bridges. Sidewalks of bridges in towns may be considered as carrying a live load of 100 pounds per square foot of sidewalk area. In the more crowded districts of cities, larger loads are sometimes employed, but in general this is ample for all probable occurrences. Roadways of highway bridges should be able to carry the heaviest motor trucks which may reasonably be expected to come upon them. In the development of truck transportation there is a tendency to in- crease the weights carried by a single truck, and careful attention should be given to this possibility in designing bridges intended to last a long time. A motor truck weighing 20 tons, with 6 tons on one axle and 14 tons on the other, the distance between wheels being 6 feet and between axles 12 feet, may reasonably be assumed as a maximum load for a bridge upon an important country highway or street of a town. This load is a very exceptional one for ordinary highways and probably in most cases a truck weighing 7 to 10 tons is as large as is likely to be met under present conditions, and possibly a road roller may be a more probable maximum load. The use of maximum loads not likely to be exceeded in the near future is always desirable in such work. For country bridges under moderate or light traffic, a truck weigh- ing 8000 pounds on each of two axles, 10 feet apart, may be used as a probable maximum load under present conditions, or a 15-ton road roller, 6 tons on the front wheel, which is 4 feet wide, and 4.5 tons on each of the rear wheels, each 20 inches wide. Street Railway Track. When the bridge is to carry a street railway, the load of a car weighing 50 tons on four axles spaced 5, 14, and 5 feet apart may be assumed as a probable maximum load. This load may be considered as distributed over an area of bridge floor about 35 feet in length and 10 feet in width, giving a maximum uni- form load of about 300 pounds per square foot. 271 272 SLAB AND GIRDER BRIDGES For light traffic roads, a car weighing 35 tons on the same wheel distribution may be used, giving a uniform loading of about 200 pounds per square foot. 150. Distribution of Concentrated Loads. Investigations of the distribution of concentrated loads upon slabs have been made by Mr. Goldbeck for the U. S. Office of Public Roads. These tests l seemed to indicate that for a slab whose width is greater than its span, the effective width of distribution of a concentrated load might be taken at about eight-tenths of the span. From a series of tests at the University of Illinois, Mr. Slater concluded 2 that for a slab whose width is greater, than twice the span, the effective width (e) might be assumed as e = %x+d, where x is the distance from the concentrated load to the nearest support and d is the width over which the load is applied. As the ratio of width to span decreases, the effective width becomes less, the coeffi- cient in the formula becoming about 1.2 when the span equals the width. From tests for the Highway Department of the State of Ohio 3 Professor Morris recommends for a concentrated load applied to the concrete floor of a highway bridge that e = Q.6S+1.7, where e is the effective width in feet for a slab whose width is greater than its span, and S is the clear span in feet. This agrees well with the results of Mr. Slater if the load be placed at the middle of the span (x = S/2). When the load comes upon the floor of the bridge through a pave- ment or fill, it may also be considered as distributed lengthwise over a certain area. For earth fill, the length of distribution may be taken as twice the depth of fill. For gravel or macadam road sur- face, three or four times the depth of surface may be used. In T-beam construction, when a slab is continuous over several girders and a load comes upon the slab immediately over one of the girders, the whole of the load will not be borne by the girder under the load, but a portion of it will be transferred by the slab to adjacent girders. In the Ohio tests mentioned above, this distribution was investigated and the following conclusions reached : (1) The percentage of reinforcement has little or no effect upon the load distribution to the joists, so long as safe loads on the slab are not exceeded. (2) The amount of load distributed by the slab to other joists than 1 Proceedings, American Society for Testing Materials, 1915, p. 858. 2 Proceedings, American Society for Testing Materials, 1913, p. 874. 3 Bulletin No. 28, Ohio State Highway Department, 1915. DESIGN OF BEAM BRIDGES 273 the one immediately under the load, increases with the thickness of the slab. (3) The outside joists should be designed for the same live load as the intermediate joists. (4) The axle load of a truck may be considered as distributed uniformly over 12 feet of roadway. 151. Railway Bridges. For short spans, railway moving loads may be considered as uniformly distributed by the track and ballast. If the heaviest locomotive load per foot of length be distributed over a width of about 10 feet, the result will be well on the safe side. When the bridge is covered by a fill under the tracks, the width of distribution may be increased by twice the depth of fill. The weights for maximum locomotive loads may vary from about 8000 to 10,000 pounds per linear foot of track, or from 800 to 1000 pounds per square foot when distributed over a width of 10 feet. For bridges longer than about 35 feet, it may be preferable to use actual locomotive wheel loads, or to somewhat reduce the load per square foot. ART. 42. DESIGN OF BEAM BRIDGES 152. Slab Bridges. When the span of a bridge is not more than 12 to 15 feet, the simple slab spanning the opening and resting upon the abutments at its ends is usually the most economical form to use. Under heavy loading, the economic limit of length may be only 10 to 12 feet, while for lighter loads, slabs 16 to 20 feet in length may be desirable. The design of a bridge slab will be illustrated by a numerical example. Example 1. Design a highway slab of 11 feet clear span, and width of 18 feet to carry a macadam road with the loading given in Section 149. Solution. Assume weight of road material = 80 pounds per square foot. Weight of slab = 145 pounds per square foot. Total dead load = 225 pounds per square foot. Live load is auto truck with 14,000 pounds on each of two wheels 6 feet apart. From Section 150, effective width, e = .6$-fl.7. As .68 is more than the distance apart of wheels, the loads would over- lap, and we consider both loads distributed over e = .6-fl.7+6 = 14.3 feet. The live load per foot of width is 28000/14.3 = 1950 pounds. This load may be considered as applied over a length of 274 SLAB AND GIRDER BRIDGES 1.7 feet = 20 inches. The effective length of the beam is distance between centers of bearings, or 1 foot more than the clear span. (11 + 1 = 12 feet.) Bending moments, M (live) = 1950/2 (72-5) = 65325 in.-lb. M (impact) =25 per cent of live = 16330 in.-lb. M (dead) = 225 X 12 X 12 X 12/8 = 48600 in.-lb. Total moment, M = 130255 in.-lb. Taking f c = 650, /.= 16,000, n=15, Table VII (p. 163) gives # = 108, p = .0078, j = .874. 12d 2 = 13025/108 =1206, and d=10 inches. Maximum shear occurs when center of live load is 1.7/2 feet from support, in which case, V (live) = 1950X 10.65/12 = 1722 pounds. V (impact) = 25 per cent of live = 430 pounds. V (dead) =225 X 11/2 = 1238 pounds. Total shear V =3390 pounds. Depth required for shear, , V 3390 Make d=10 inches, then allowing concrete to extend 1.5 inches below steel, weight of beam is 12X11.5X150/144=144 pounds per square foot, which is within the assumed weight. Reinforcement. The area of steel required per foot of width, A =pbd=. 0078X12X10 = .936 in. 2 From Table XV (p. 199), we see that f-inch round bars spaced 5.5 inches apart, or f-inch square bars 5 inches apart will answer. For the latter the maximum unit bond stress is V 3390 _ . Fig. 80 shows the slab in longitudinal section. For lateral rein- forcement J-inch round bars, 12 inches apart, are used. To prevent cracking due to negative moment where the slab joins the abutments, ^-inch round bars 12 inches apart are placed in the ends of the slab at the top. Expansion joints, usually tar paper, are often placed on the top of the abutment under the slab, thus preventing the develop- ment of negative moment and allowing for temperature changes. DESIGN OF BEAM BRIDGES 275 153. T-Beam Bridges. When the length of the bridge is too great for a simple slab, it is found economical to use girders to support the slab. If the head room is sufficient and the span not too great, T- beam construction may be used. This consists of a series of T-beams extending from abutment to abutment, girders being placed under the :&#B^^^ ^~%square bar5-5"c-c. ^ Vg round bars- \Z"c-c. m FIG. 80. Slab Bridge. slab to form the stems of the T-beams, and the slab being continuous over the girders for the width of the bridge. Example 2. Design a T-beam highway bridge with clear span of 24 feet, to carry a roadway 18 feet wide, using loadings as in Example 1. Solution. Allowing 12 inches for width of base of guard rail, the full width is 20 feet. Use five girders, spaced 4 feet on centers, the outside girders being 2 feet from end of beam (see Fig. 81). -g square bars, 5* c-c. -4, 1^ d 2. 14 round bars. 3LI [ round Stirrups FIG. 81. T-beam Girder Highway Bridge. Slab. Weight of road material = 80 pounds per square foot. Assume weight of slab = 100 pounds per square foot. Total dead load = 180 pounds per square foot. The live load is a single wheel load of 14,000 pounds distributed over a width .6X4+1.7 = 4.1 feet. The live load per foot of width is 14000/4.1 = 3415 pounds. This may be considered as distributed 276 SLAB AND GIRDER BRIDGES over 2 feet of length. The slab is continuous and taking the moment of the concentrated load as four-fifths of the moment for a simply supported beam, we have M (live) = (3415/2) (24 - 6)f = 24588 in.-lb. M (impact) =25 per cent of 24588 = 6147 in.-lb. M (dead) = 180X4X4X 12/12 = 2880 in.-lb. Maximum moment, M =33615 in.-lb. 12 d 2 = 33615/108 = 311, and d = 5.1 inches. The shear is a maximum when the load is placed next to the support, and assuming width of girder at 12 inches, V. (live) =3415X2.5/4 =2134 pounds. V (impact) =25 per cent of 3415 = 533 pounds. V (dead) = 180X3/2 = 270 pounds. Total shear, V =2937 pounds. and the depth required for shear is V 2967 Using d = 7.0 inches, _ M _ 33615 _ _n 12 fe lo lo Take effective length of girder as 36 feet and we have M (live) = 289 (1 J*~ 1 - 8)2 X 12 = 2528200 in.-lb. ob M (impact) -25 per cent of 2528200 = 632050 in.-lb. M (dead) = 4300X Q 36X36 X 12 = 8359200 in.-lb. Total M =11519450 in.-lb. T 70 J-J-OlcMOU -tnnnnr* bd 2 = r = 106666. DESIGN OF BEAM BRIDGES 281 Assuming 6 = 20 inches, we find d = 73 inches. Table X shows that nine If -inch square bars may be used, or six If -inch square bars will answer. These can be spaced four in the lower and two in upper row. The maximum bond stress for the latter is V 106250 .. Shear. Considering the live loads to be applied over a length of 2 feet, V (live) = 28900(36 - 5. 1) /36 = 24800 pounds. V (impact) =25 per cent of 24800 = 6200 pounds. 7 (dead) =4300X17.5 = 75250 pounds. Total 7 = 106250 pounds. The maximum shear at the middle of the beam occurs when the heavier load is just past the middle point, or 7 = 28900(18-4. 6)/36 = 10760 pounds, and 10760 n .. 2 = 9 lb./in. 2 20 X. 875X73 The maximum shear varies from 83 lb./in. 2 at the support to 9 lb./in. 2 at the middle of the girder. Reinforcement for diagonal tension is needed where v is more than 40 lb./in. 2 If U-shaped stirrups be spaced 12 inches apart, at the abutment, _6t_20X83X12 . 2 ~2f s ~ 2X16000 " By Table X, f-inch round bars are needed. The tops of these bars should be turned into hooks to secure ample bond. Seven stirrups will be used spaced 12 inches apart, three spaced 18 inches and two spaced 30 inches, at each end of the girder. Hangers. To prevent the T-beams breaking loose from the girders, bars passing under the steel in the stem of the T-beam, and extending up into the girder are used to carry the reactions at the ends of the T-beams. These reactions equal the maximum shear upon the T-beams, and the area of steel required is A h = 24245/16000 = 1.52 in. 2 By Table X, we find 1-inch round bars to be needed. These should extend upward a distance sufficient to develop a. bond strength equal to the tensile strength of the bars, or at least 50 diameters. CHAPTER X MASONRY ARCHES ART. 43. VOUSSOIR ARCHES 155. Definitions. A masonry arch is a structure of masonry spanning an opening and carrying its loads as longitudinal thrust, which exert outward as well as vertical thrusts upon the abutments. A voussoir arch is one in which the arch ring is composed of a number of independent blocks of stone or masonry. Parts of an Arch. The principal parts of an arch are as follows : FIG. 83. The under or concave surface of an arch is called the soffit. The outer or convex surface is the back. The crown is the highest part of the arch ring (E-F, Fig. 82). The skewbacks are the joints at the ends of the arch where it rests upon the abutments (C-A, B-D, Fig. 83). The intrados is the intersection of the soffit with a vertical plane perpendicular to the axis of the arch (A-E-B, Fig. 83). The extrados is the intersection of the outer surface with a vertical plane perpendicular to the axis (C-F-D, Fig. 83). The springing lines are the intersections of the skewbacks with the soffit. 282 VOUSSOIR ARCHES 283 The span is the distance between springing lines. The rise is the perpendicular distance from the highest point of the intrados to the plane of the springing lines. The voussoirs are the wedge-shaped stones of which an arch is composed. The keystone is the voussoir at the crown of the arch (E-F) . The springers are the voussoirs next the skewbacks. The haunch is the portion of the arch between the keystone and springers. The arch ring is the whole set of voussoirs from skewback to skew- back. The ring stones are voussoirs showing on the face of the arch. The arch sheeting is the portion of the arch ring not showing at the ends. Backing is masonry above and outside the arch ring. The spandrel is the space between the back of the arch and the roadway above. The walls above the ring stones at the ends of the arch are spandrel walls and the filling between these walls is spandrel filling. Kinds of Arches. A full-centered arch is one whose intrados is a semicircle. A segmental arch is a circular arch whose intrados is less than a semicircle. A pointed arch has an intrados composed of two cir- cular arcs which intersect at the crown. A three-centered arch com- posed of arcs tangent to each other is sometimes called a basket- handled arch. A right arch is one whose ends are perpendicular to its axis. An arch whose ends are oblique to its axis is called a skew arch. Hinged arches are those in which hinged joints are used at crown and skewback. Those without hinges are called solid arches. 156. Theory of Stability. A voussoir arch is supposed to be composed of a number of independent blocks in contact with each other and held in place |W by the pressures between them. In Fig. 84, let A BCD represent a voussoir at any part of an arch ring. If P is the pressure received from the voussoir above and W the external load carried by the voussoir, the resultant, R, of _ these forces will be the pressure transmitted to the voussoir below. If the line of action of this resultant should pass outside of the joint A-D, the arch will fail by the voussoir rotating about the edge of the joint. If the point of application of R is outside the middle third of 284 MASONRY ARCHES A-D, there will be a tendency for the joint to open on the opposite side, and the area of contact between the voussoirs will be reduced. If the line of action of R makes an angle with the normal to the joint A-D greater than the angle of friction for the surfaces upon each other, the voussoirs may slide upon each other, causing failure of the arch. For stability of the arch: (1) The resultant pressures between voussoirs should act within the middle third of the joints. (2) The components of the resultant pressures parallel to the joints (R sin a) should be less than the frictional resistance of the voussoirs to sliding upon each other. (3) The unit pressures at the surfaces of contact should be less than the safe compressive strength of the material of the voussoirs. If 6 represents the width of the joint AD, x the distance of the point of application of R from the nearest edge and a the angle made by R with the normal to the joint, the maximum unit compression will be represented by f e = ws a. (See Section 126.) Usually the angle a is so small that cos a. may be taken as 1 with- out sensible error, or R may be considered as equal to its normal com- ponent. Line of Pressure. If an arch ring be divided into a number of voussoirs, and the points of application of the resultant pressures upon the joints between these voussoirs be determined, the broken or curved line joining these points of application is known as the line of pressure for the arch. In Fig. 85 the line abcdef is called the line of pressure for the half arch, when H is the crown thrust and PI, Pz, etc., are the external loads coming upon the several divisions. The true line of pressure, or of resistance, is a curve circumscribing the poly- gon abcdef. The larger the number of divisions of the arch ring, the more nearly will the polygon approach this curve. In determining the line of pressure, the arch ring is divided into a convenient number of parts, usually six to sixteen on each side of the crown, and the external loads (pi-ps, Fig. 85) coming upon the vari- ous divisions are found. It is now necessary to know certain points through which the line of pressure must pass in order to draw it. If the arch be hinged, the line of pressure must pass through the centers of the hinges and may be drawn without difficulty. In a solid arch, the points of application of the pressures upon the various joints are VOUSSOIR ARCHES 285 not definitely known, and certain assumptions must be made concern- ing them. Any number of different lines may be drawn as these assumptions are varied. Hypotheses for Line of Pressure. If Fig. 85 represent half of a symmetrically loaded arch, the crown pressure H will be horizontal. Assuming its point of application, a, and that its line of resistance passes through a definite point on one of the other joints as /, the amount of H may be found by taking a center of moments at / and writing the moment equation for aU the loads upon the half arch equal to zero. H is then known in amount, direction and point of application and the line of pressure may be drawn, as shown. Several hypotheses have been proposed for the purpose of fixing the position of the line of thrust. Professor Durand-Claye assumed FIG. 85. that the true line of resistance is that which gives the smallest abso- lute pressure upon any joint. This method is outlined in Van Nostrand's Engineering Magazine, Vol. XV, p. 33. Professor Winkler suggested that "for an arch ring of constant cross-section, that line of resistance is approximately the true one which lies nearest to the axis of the arch ring, as determined by the method of least squares." No practicable method of applying this principle to ordi- nary cases of voussoir arches has been devised. Moseley's hypothesis was that the true -line of resistance is that for which the thrust at the crown is the least consistent with stability. This occurs (Fig. 85) when H is at the highest and R at the lowest point it can occupy on the joint. This hypothesis is the basis of Scheffler's method of drawing the line of resistance. Scheffler's theory assumes that H is applied at the upper edge of the middle third of the crown joint, and that the value of H is such as 286 MASONRY ARCHES to cause the line of pressure to touch the lower edge of the middle third at one of the joints (as d, e, or /) nearer the abutment. The joint at which the line of pressure is tangent to the lower edge of the middle third is known as the joint of rupture. The joint of rupture may be found by taking moments about the lower edge of the middle third of each of several joints and solving for H. All loads acting between the joint considered and the crown should be used in obtain- ing the moment, and the one giving the largest value of H is the joint of rupture. The value of H so determined is the least consistent with stability, as a less value causes the line of pressure to pass out- side the middle third at the joint of rupture. Should it be found that the line of pressure passes outside the middle third on the upper side of any of the joints between the joint of rupture and the crown, the point of application of H may be lowered without violating the hypothesis. This leads to the usual statement that "if any line of pressure can be drawn within the middle third of the arch ring the arch will be stable." This is justified by common experience. When the loading upon the arch is not symmetrical, this method of finding the crown thrust cannot be used, and in this case it is usual to select three points through which to pass the line of pressure, one at the crown and one near each abutment. A line of pressure is then passed through these three points, and if the line so found does not remain within the middle third of the arch ring the positions of the points may be changed and new lines constructed. This may be repeated until it is determined whether any line of pressure can be drawn within the middle third. ART. 44. LOADS FOR MASONRY ARCHES 157. Live Loads for Highway Bridges. For the floors of open spandrel arch bridges, live loads should be considered in the same manner as for slab bridges (see Art. 41). In investigations of arch rings, live loads are usually taken as uniformly distributed. The loading which should be used in any design depends upon the location of the bridge, the character of traffic, and the length of span. A heavy (20-ton) motor truck may bring a load of about 140 pounds per square foot upon a bridge of short span (about 40 feet). Bridges 60 to 100 feet span subjected to traffic of motor trucks and heavily loaded wagons may be considered to carry about 100 pounds per square foot. For longer bridges this load may be lessened, bridges over 200 feet being designed for about 75 pounds per square foot. LOADS FOR MASONRY ARCHES 287 For bridges less than 100 feet in length carrying street railways, a load of 1800 pounds per foot of length for each track may be taken. For spans of 200 feet or more, this may be reduced to 1200 pounds per foot of track. These loads are considered as distributed over a width of about 9 feet, giving loads of 200 and 133 pounds per square foot respectively. For spans between 100 and 200 feet, the loads may vary according to the length of span. For light traffic lines on country roads, a load of 1200 pounds per foot of track may be used for arches less than 100 feet in length and 1000 pounds per foot for those 200 feet or more in length. Fre- quently bridges must be built for special service, or where the traffic conditions are unusual and should be designed for any loads that may reasonably be expected to come upon them. Traffic conditions are constantly undergoing important changes, and in determining the loading to be used in any particular instance, it is desirable to con- sider the possible effect upon future traffic of the rapid increase in the use of heavy auto-trucks and traction engines. As masonry arches are structures of permanent character, the probable future development of traffic should be considered and liberal loadings used in design. 158. Live Loads for Railway Arches. Standard locomotive load- ings are used in the design of floor systems for open spandrel arches, as in beam bridges, and are also sometimes employed in investiga- tions of arch rings. Equivalent uniform loadings may, however, commonly be used in arch-ring design. Loadings should correspond with the heaviest locomotive and train loads to be expected. For spans less than about 60 feet, a load of 8000 pounds per foot of track, or 1000 pounds per square foot of road surface is frequently used. When the span is 80 feet or more a load of 5600 pounds per foot of track, or about 700 pounds per square foot, is used, which are approximately the same as Cooper's E 40 loading. Impact is not taken into account in the arch-ring investigation. A concentrated load upon a fill may be considered as distributed downward through the fill at an angle of 45 with the vertical, the top of the distributing slope being taken from the ends of the ties. Wheel loads are taken as distributed over three ties and then trans- mitted to the filling. 159. Dead Loads. In arch bridges, the dead weights of the arch ring and of the filling or structure above constitute the principal loads upon the arch rings. The live loads are much less in amount, and are important mainly as producing unsymmetrical loading when 288 MASONRY ARCHES the load does not extend over the whole arch. In computing the dead load upon an arch ring, the actual weights of the materials to be used should be taken when they are accurately known. It is common to assume the weight of earth filling as 100 pounds per cubic foot, and that of concrete of other masonry as 150 pounds per cubic foot. In open-spandrel arches the dead weights act vertically through the columns or walls supporting the floor of the roadway, and may be readily computed. When the spandrels are filled with earth, each section of the arch ring is assumed to carry the weight of the filling and roadway vertically above it. The earth pressures upon the inclined back of the arch ring are not actually vertical, but may have certain horizontal com- ponents. For arches of small rise, these horizontal pressures are small and may be neglected, but when the rise of the arch is large, the horizontal earth thrusts may be considerable, and should be taken into account, although their omission is usually an error on the safe side. While the amount of horizontal earth pressure cannot be exactly determined, it is usual to use Rankine's minimum value for unit horizontal earth pressure in terms of the unit vertical pressure, which is rr 1 sin 6 in which H is the horizontal and V the vertical unit pressure, and the angle of friction for the earth. For ordinary earth filling, this would make the unit horizontal pressure at any point approximately one-fourth of the unit vertical pressure at the same point, the prob- ability being that a horizontal pressure of at least this amount may always be developed. The methods used for determining pressures upon retaining walls evidently are not applicable to this case. The actual horizontal earth pressure may vary within rather wide limits, and cannot be accurately determined. In retaining-wall design, the maximum earth thrust which may come against the wall is computed, while for the arch we need to know the minimum horizontal pressure which may be relied upon to help sustain the arch. That the actual pres- sure may sometimes be considerably more than the computed mini- mum is quite probable. When an arch carries a continuous masonry wall, as in an opening through the wall of a building, or the spandrel wall at the end of an arch bridge, the wall itself would arch over the opening and be cap- able of self-support if the arch were removed. The load upon the DESIGN OF VOUSSOIR ARCHES 289 arch would therefore be only that due to a triangular piece of wall immediately above the arch as in the case of a stone lintel. (See Section 53.) ART. 45. DESIGN OF VOUSSOIR ARCHES 160. Methods of Design. In designing masonry arches, the form and dimensions of the arch ring are first assumed and the stability of the arch, as assumed, is then investigated. The graphical method of investigation is commonly employed, a line of pressure (see Sec- tion 156) being drawn and the maximum unit compression computed. Stability requires that the line of pressure remain within the middle third of the arch ring and that the unit compression does not exceed a safe value. If the first assumptions are not satisfactory the shape or dimensions of the arch ring may be modified and the new assump- tions tested as before. Arches subjected to the action of moving loads should be tested for conditions of partial loading, which may cause unsymmetrical distortion of arch ring, as well as for full load over the whole arch. For ordinary loadings and spans of moderate length, it is usually suffi- cient to draw the line of pressure for arch fully loaded and with live load extending over half the arch, but in large and important struc- tures, or those with unusual loadings, it may be desirable to test the arch ring with live loads in other positions which seem likely to produce maximum distortions of the line of pressure. 161. Thickness of Arch Masonry. The choice of dimensions for the trial arch ring is necessarily based upon judgment founded upon knowledge of the dimensions of existing arches, which are found to differ widely, and rules have been formulated by several authorities for the purpose of aiding in selecting the dimensions. Crown Thickness. Several different formulas have been proposed for determining the thickness at the crown. Trautwine's formula for the depth of keystone of first-class cut-stone arches, whether circular or elliptical, is . , VRadius+half span Depth of key in feet = j -+.2 foot. For second-class work this depth may be increased about one- eighth part; or for brick or rubble about one-third. Rankine's formula for the depth of keystone for a single arch is Depth in feet = v. 12 radius. 290 MASONRY ARCHES This gives results which agree fairly well with Trautwine's formula. For an arch of a series, Rankine also recommends Depth in feet = V.17 radius. These formulas make the thickness depend upon the span and rise of the arch without regard to the loading. They agree fairly well with many examples of existing arches, but make the thickness rather large for arches of moderate span. Douglas Formulas. In Merriman's American Civil Engineer's Pocket Book, Mr. Walter J. Douglas gives the following rules for thickness at crown: THICKNESS IN FEET AT CROWN FOR HIGHWAY ARCHES Kind of Masonry. SPAN IN FEET=L. Under 20. 20 to 50. 50 to 150. Over 150. First-class ashlar. . . . Second-class ashlar or brick Plain concrete Reinforced concrete. . 0.04(6+L) 0.06(6+L) 0.04(6+L) 0.03(6+L) 0. 020(30 +L) 0. 025(30 +L) 0.020 V 30+L) 0. 015(30 +L) 0. 00012(11000 +L2) 0.00016(1 1000 +L2) 0.00014(11000+7,2) 0.00010(1 1000 +L2) 0.018(75+L) 0. 025(75 +L) 0. 020(75 +L) 0. 016(75 +L) For railroad arches, add 25 per cent for arches 20- to 50-feet span, 20 per cent for 50 to 150 feet, and 15 per cent for those over 150 feet. These formulas give smaller thickness for highway arches of short span than Trautwine's and do not vary the thickness with the rise of the arch. Thickness at Skewback. If the arch ring be made of uniform thick- ness, the unit pressure at the ends will be greater than at the crown. The pressure may often be made fairly uniform by making the thick- ness at any radial joint equal to the crown thickness times the secant of the angle made by the joint with the vertical. In the American Civil Engineer's Pocket Book, Mr. Douglas recommends that the thickness at the springing line of a masonry arch be obtained by adding the following percentages to the crown thickness : (1) Add 50 per cent for circular, parabolic, and catenarian arches having a ratio of rise to span less than one-quarter. (2) Add 100 per cent for circular, parabolic, catenarian, and three- centered arches having a ratio of rise to span greater than one-quarter. (3) Add 150 per cent for elliptical, five-centered and seven- centered arches. Mr. Douglas recommends that the top thickness of abutments be assumed at five times the crown thickness. For a pier between DESIGN OF VOUSSOIR ARCHES 291 arches in a series he suggests a thickness at top of three and one-half times the crown thickness, but places an abutment at every third or fifth span. Trautwine gives a method for design of abutment, approximately as follows (see Fig. 86) : Thickness at springing line in feet , Radius . Rise . n f a-b = - +2feet. Layoff a c = rise, and c 10 span Continue bd downward to bottom of abutment, and upward a distance be = rise/2. From e draw a tangent e f to the extrados. It is also required that the thickness at bottom of abutment gh, shall not be less than two-thirds of the height ag. If the abutment is of rough rubble, 6 inches is added to the thickness to in- sure full thickness in every part. h These rules usually give arches which are amply strong FIG. 86. for heavy railway service and heavier than necessary for highway bridges. For structures of small span, however, when voussoir or plain concrete arches are used, the saving effected by paring them down is small, and rather heavy work is common practice. 162. Investigation of Stability. After assuming dimensions for the arch ring and abutments, the stability of the arch is investigated by the methods outlined in Section 156. The stability of the abut- ment is tested by continuing the line of thrust and determining whether it cuts the base of the abutment within the middle third. The sufficiency of the foundation for the abutment must also be examined and footings provided which will properly distribute the pressure over the soil upon which it is to be placed. The following example will outline the method of procedare. Example. A highway arch is to have a span of 40 feet and a rise of 10 feet. It is to carry a moving load of 200 pounds per square foot. 292 MASONRY ARCHES The depth of fill at crown is 2 feet. The weight of earth fill is 100 and of masonry 150 pounds per cubic foot. We will try a segment al arch. By the Douglas rule, the thickness at crown would be 1.4 feet. By Trautwine's formula, it would be 1.95 feet. Make the crown thickness 18 inches. By the Douglas rule the thickness at springing would be between 1.5 and 2 times the crown thickness. We will try 30 inches. Draw the arch ring as shown in Fig. 87, and divide it into equal parts by radial lines. The line z-t represents the roadway and verticals from the points where the radial divisions cut the extrados divide the earth fill into parts resting upon the sections of the arch ring. These loads, including the weights of the sections of arch ring, are now computed, and their vertical lines of action determined. In finding the loads, it is often convenient to draw the reduced load contour, which is obtained by reducing the height of the sections go that the volume contained by them may be considered to weigh the same per unit as the arch ring. Thus if the earth fill weighs 100 pounds and masonry 150 pounds per cubic foot, the height ax is made two-thirds of az, and the other verticals are reduced propor- tionately, giving the volume a-x-u-g, which has the same weight at 150 pounds as the earth fill at 100 pounds. In the same way x-^y^v-u represents the live load which would come upon half the arch ring reduced to 150 pounds per cubic foot. In the example, the loadings given represent live load extending over the left half of the arch, dead load only upon the right half. The horizontal thrusts against the arch ring are sometimes com- puted by assuming that the unit horizontal thrust bears a definite proportion (usually about one-quarter) to the unit vertical thrust. Thus in Fig. 87, if the vertical load upon the section a-b is 5085 pounds the horizontal component of the load on the section is 5085 ap X-r = 1550 pounds. 4 pb In the example, the horizontal components upon the two lower divi- sions on each side are used, those upon the upper divisions being too small to affect the results appreciably. The horizontal components of the loads are not usually considered in a problem of this kind unless the rise of the arch is large as compared with the span. Having computed the loads, a line of pressure may now be drawn through any three points in the arch ring. Assume that it is to pass through the lower third point of the joint a on the loaded side, the DESIGN OF VOUSSOIR ARCHES 293 294 MASONRY ARCHES middle point at the crown, and the upper third point at the joint n on the unloaded side. The load line is first plotted on a convenient scale by laying off the loads which come upon the various sections in succession, n-m, m-l, etc. ; na is now the resultant of all the loads upon the arch ring. A pole O f is assumed and the strings O'a, O'b, etc., drawn. The equilibrium polygon, shown in broken lines, may now be drawn. Starting from A, the lower third point on joint a, with a line parallel to the string O'a to an intersection with the line of action of the load upon the section a-b. From this intersection, draw a line parallel to O'b to intersection with the line of action of the load on 6-c, and continue it until a parallel to O'n is intersected in N' upon a line through N parallel to the resultant n-a. Connect N' with A, and from 0' draw a line parallel to N'-A to intersection J with the resultant n-a of the loads, thus dividing the resultant into two reactions, n-J and J-a, which would exist at the ends of the span if the horizontal thrust of the arch be neglected. Join the points A and N and from J draw a line parallel to A-N. A pole lying upon this line will give an equilibrium polygon passing through A and N. The distance of the pole from J must now be determined to cause the equilibrium polygon to pass through the middle of the crown joint. The line g a in the force polygon, is the resultant of the loads upon the left half of the arch. From the middle of the crown section, draw G-G' ', parallel to g-a, to intersection with the trial equilibrium polygon. Connect A-G' and A-G. From 0' draw O'k parallel to G'A to intersection with g-a in fc, and from k draw k-O parallel to AG. The point where KO intersects JO is the new pole. From A, the new line of thrust may now be drawn with sides par- allel to the strings, Oa, 06, etc. This passes through the points G and N. By inspection we see that the line of thrust, as thus drawn, is everywhere within the middle of the arch ring. The thrust upon the joint at a is represented by the length of the line 0-a = 27000 pounds, and the maximum unit compression is The unit compression upon any other joint may be found in the same manner. The resultant pressure R upon the base of the abutment is found by combining the weight W of the abutment with the thrust 0a of THE ELASTIC ARCH 295 FIG. 88. the arch against the abutment. The footing under the base of the abutment should be so designed as properly to distribute the load over the foundation soil. ART. 46. THE ELASTIC ARCH 163. Analysis of Fixed Arch. Reinforced concrete arches are commonly constructed as solid curved beams firmly fixed to the abutments. Inan- alyzing them, it is assumed that the abutments are im- movable and the ends of the arch firmly held in their original positions. Let Fig. 88 repre- sent the left half of an arch, fixed in position at the end G-H, and carrying loads which produce thrusts and bending moments throughout the arch ring. The arch may be considered as made up of a, number of small divisions. Suppose A BCD to be one of these divisions, small enough so that its ends are practically parallel and its section area constant. The loads upon the arch bring a bending moment upon the division A BCD, which causes the end CD to take the posi- tion EF. Let M bending moment on the division; s = length of division, AD = BC; e = distance from center of section to outside fiber; ds = elongation of fiber distant e from neutral axis; / nt = Unit stress upon fiber distant e from neutral axis; k = angle of distortion, COE;- E modulus of elasticity of material ; 7 = moment of inertia of section; x and y = horizontal and vertical coordinates of center of section, 0, with reference to center of crown sec- tion, J. The unit fiber elongation in the division A BCD is = . s s Unit stress, f m = also U--B--E. S S 296 MASONRY ARCHES Equating these and solving, If the crown of the arch be free to move, the deflection of A BCD into its new form ABEF will bring the middle point of the arch ring J", to the new position K. Let dx and dy be the horizontal and ver- tical coordinates of K with respect to J. Then from similar tri- angles, JK/OJ = dy/x = k, and 7 Mxs dy = xk=^j- ........ (2) Similarly, Mys dx = yk=- J j r . ..... k . . (3) As the end section GH is fixed in position, the summation of all the angular distortions k, for the left half of the arch gives the dis- tortion at the crown section. The summation similarly of those for the right half must give the same result with opposite sign, or indicating the left and right sides of the arch by the subscripts L and R respectively, and indicating summation by the sign 2, we have 2k L = Sfc, also 2dy L = 2dy R and 2dx L = - 2dx B . Substituting for these distortions, their values as found above, we have for a symmetrical arch : M L s__ M R s *~ ~ 2 ' -* 4 M L XS_ M R XS ~ : '~~' ' * and M L ys M R ys If the length of the divisions of the arch ring be made directly proportional to the corresponding values of the moment of inertia, o O T = constant, the terms -^j in Equations (4), (5), and (6) are con- l JbL stant and may be eliminated, and we have, ....... (7) (8) ..... (9) THE ELASTIC ARCH 297 Fig. 89 represents a symmetrical arch divided into parts the lengths of which are directly proportioned to the moments of inertia of the cross-sections at their middle points. s/I = constant. Sup- pose the arch to be cut at the crown and the separate halves supported by introducing the stresses acting through the crown section as exterior forces. These may be resolved into a horizontal thrust, a vertical shear and a bending moment. H c = horizontal thrust at crown; V c = vertical shear at crown; M c = bending moment at crown. V c is considered to be positive when acting in the direction indicated by the arrows. Moments are taken as positive when they FIG. 89. produce compression on the upper and tension on the lower side of the section. Let M L = bending moment on mid-section of any division in left half of arch; M R = bending moment on mid-section of any division in right half of arch; WIL= moment at middle of any division in left half, caused by external loads between that division and the crown section: WR= moment at mid-section of any division in right half, caused by external loads between that section and the crown section; X and y = coordinates of middle point of any division with respect to middle of crown section. The bending moment at any section of a beam is equal to the moment at any other section plus the moments of the intermediate loads about the center of the section. Therefore, M L = M c +V c x+H c y-m L , (10) M R = M c -V c x+H c y-m R (11) 298 MASONRY ARCHES Substituting these values in Equations (7), (8) and (9), we have, ..... (12) (13) = Q. . . . .(14) Solving for the thrusts and moments at the crown TT _ , . 2n2y 2 -2(2y) 2 _ c ~ In analyzing an arch by this method, the arch is first divided into a number of parts in which s/I is a constant. The loads upon the divisions are then found and mL and mR computed for the several centers of division. The values of H c , V c and M c may then be found from Formulas (15), (16) and (17), after which the line of thrust may be drawn beginning with the known values of H c and V c at the crown. The moment M c is due to the eccentricity of the thrust at the crown, and the point of application for H c may be found by divid- ing M c by H c . This gives the vertical distance of H c from the center of gravity of the crown section. For M c positive, H c is above, and for M c negative, H c is below the center of section. The thrust at any section of the arch may be obtained from the thrust diagram as in the voussoir arch. The bending moment at any section is the moment of the thrust upon the section about the center of gravity of the section. The bending moment at any section may also be obtained by the use of Formula (10) or (11). In analyzing an arch bridge subject to moving loads, it is necessary to assume different conditions of loading and find the thrust and mo- ments resulting from each. For a small arch, it is usually sufficient to make the analysis for arch fully loaded and for moving load over one-half the arch. The maximum stresses will be more accurately determined by dividing the moving load into thirds, and determining the stresses with span fully loaded, one-third loaded, two-thirds loaded, center third loaded, and with two end thirds loaded. If complete analysis be made for the arch under dead load alone, for live load over one end third, and live load over the middle third, the results of these three analyses may be combined to give the five conditions of loading above mentioned. THE ELASTIC ARCH 299 164. Effect of Changes of Temperature. A rise in temperature tends to lengthen and a fall in temperature to shorten the span. If the ends of the arch ring are rigidly held in position, the tendency to change in length is resisted by moments and horizontal thrusts at the supports, which produce moments and thrusts throughout the arch ring. If the arch ring were not restrained, a rise in temperature of t degrees would cause an increase in length = CtL ; L being the length of span and C the coefficient of expansion of the material. The moments throughout the arch ring are therefore those which cor- respond to an actual change in length of span = CL, or from For- mula (3) From this, for a symmetrical arch ring M L ys _ M R ys _ ClL and As there are no exterior loads, m Lj and V c are each equal to zero, and Formula (10) becomes M L = M c -\-H c y. Substituting this in (18) and (19) and solving, we have = EI CtLn nc s *0^-v..2 0/V.A2 \"W The line of thrust consists of a single force H c , and is applied on a horizontal line at a distance, e = 2y/n, below the middle of the crown section. The bending moment at any section due to H c is The direct thrust upon any section of the arch ring is the com- ponent of H c normal to the section. For temperatures below the normal, H c will be negative and may be found from Formula (20) by giving t the negative sign. 165. Effect of Direct Thrust. Axial thrusts on the arch ring produce compressive stresses on the various sections and also tend to shorten the arch ring. As the span length does not change, this 300 MASONRY ARCHES tendency to become shorter causes stresses in the arch ring in the same manner as does lowering temperature. If f e lb./in. 2 be the average unit compression due to axial thrust, the arch ring if unre- strained would be shortened an amount dx=f c I/E, from which, rr I fcLn ii c =-- ^ o s 2n2y 2 and M '=n?" I "/'. (23) As the unit stress f c is not uniform through the arch ring, a value obtained by finding the stresses at several points and averaging them may be used. The stresses due to shortening of the arch ring are comparatively small and are often neglected in the analysis of ordinary arches; in some instances, however, they may be considerable. ART. 47. DESIGN OF REINFORCED CONCRETE ARCH 166. Selection of Dimensions. In designing an arch, it is neces- sary to first assume dimensions for the arch ring, and then investigate for the strength of the arch and the suitability of the assumed dimen- sions to the conditions of service. The methods of investigation usually employed are indicated in Art. 46. The investigation will show whether changes in form or thickness should be made in the arch ring. The shape of the arch should be such as to fit as closely as possible the lines of pressure, and the thickness should be such as to give allowable stresses under all conditions of loading. Example. As an illustration of the method of investigation, we will assume an arch of 60 feet clear span and 12 feet rise, to carry a live load of 100 pounds per square foot of road and a solid spandrel filling, 2 feet deep over the crown, weighing 100 pounds per cubic foot. For ordinary arches with solid spandrel filling, a three-centered intrados, with radii at the sides from three-fifths to three-fourths that at the crown, is apt to give better results than a segmental intrados. We will use an intrados composed of three arcs tangent to each other at the quarter points with radii of 52.5 feet and 31.25 feet respect- ively (see Fig. 91). / L W W Weld's formula l for the crown thickness is t = V ^+TTJ + ^T^+TT^ in which ? l Engineering Record, Nov. 4, 1905. DESIGN OF REINFORCED CONCRETE ARCH 301 t = the crown thickness in inches; L = clear span in feet; TF = live load in pounds per square foot; W = weight of fill at crown per square foot. Applying this formula, we find a crown thickness of 16 inches. The thickness of the arch ring should increase from the crown to the spring- ing line; the thickness at the quarter point may be made a little greater than that at the crown (about 1J to 1J times). We will assume a thickness at the quarter point of 21 inches, and at the spring- ing line of 45 inches. The extrados will now consist of three arcs tangent to each other at the quarter points and giving the desired thicknesses. Fig. 117 shows the arch ring as assumed. FIG. 90. The reinforcement may be assumed at from .4 to .7 per cent of the area of the section at the crown, to be placed at both extrados and intrados. We will use f-inch round bars spaced 7 inches apart. 167. Division of Arch Ring. Having chosen the form and dimen- sions of the arch ring, it is necessary to divide the ring so that the lengths of the divisions shall be directly proportional to the moments of inertia of their mid-sections, s/I = constant. This may be done by trial, assuming a division next the crown, determining the value of s/7 for the assumed division, and finding the corresponding lengths of other sections toward the abutment. Then changing the first assumption as may seem necessary to make the division come out properly at the abutment. More easily, the division may be made graphically as shown in Fig. 90. The line a-k is laid off equal in length to half the arch axis (34.52 feet). The moments of inertia are then computed at several points 302 MASONRY ARCHES along the arch axis, and their amounts laid off normally to the line a-k, and the curves of moment of inertia drawn through the points so located. A trial diagonal is then drawn from A to intersection with the curve in the point B. A vertical from B is drawn to intersection with the upper curve, and a second diagonal parallel to A-B, cutting the lower curve in C. Continue successive diagonals and verticals until the end k is reached. If these do not come out accurately at the end k the inclination of the diagonals may be varied until the division of a-k is made into the correct number of parts. This divides a-k into lengths which are proportional to the average of the moments of inertia at the ends of the divisions. The lengths of the divisions, a 6, b c, etc., are now transferred to the arch axis. The axis of the arch in Fig. 91 is thus laid off into ten divisions on each side of the crown section. The con- stant ratio s/I is found to be 5.1, all measurements being taken in feet. The middle point of the arch axis in each division is now located, and the values of x and y are determined with reference to the middle of the crown section. These values and their squares are tabulated in Table XXI for use in the computations. 168. Analysis. If vertical lines be drawn through the points of division of the arch axis, the weight of the portion of masonry and spandrel filling included between each pair of lines may be considered as the dead load resting upon the included division. The live load is similarly divided for the portion of the arch over which it is considered as acting. In Fig. 91 the live load is taken as extending over the left half of the arch, and the loads are as indicated. The values of m^ and m R are now computed and placed in Table XXI, and include in each instance the moments of all loads between the division con- sidered and the crown section about the center of division. The quantities m L x, m R x, m L y, and m R y may now be computed and placed in the table, and the summations of the various columns obtained. These substituted in Formulas (15), (16) and (17) give, _ 10(2375207+ 1989744) -(512291 +425771) 17.09 Hc ~ 2X10X80.85-2X17.09X17.09 T/ 10408051-8686133 Fc= 2X1769.5 =+487 pounds, DESIGN OF REINFORCED CONCRETE ARCH 303 304 MASONRY ARCHES I Si-t rH i-H 12 S3 Jg rH c5 ~. 00 o ^ to 00 828 iO !> 5 I 1 g g rH O5 (M rH Tf O5 I> 00 to CO rH a to QJ Q O C^ c3 oo 8 | 1 00 O l iH|f I I-H 10 di CO b- ^H O CO co -* (N O I 00 i-H CO CO 00 CO l^ ~ I 1-1 o co 00 -N^T l> (N 1-1 O b- co ^ (N O T 00 1-H CO CO 00 oo i i> i 1-1 O b- CO CO 00 i-( O %> S oco' ,05^0 ' PJ 4-l ^ ^ +H ?>> I ! Ill I S s * 3 -S CULVERTS 329 Concrete box culverts are sometimes constructed with a reinforced slab top resting upon side walls which may or may not be reinforced. The design of short bridges of this type has been discussed in Chap- ter IX. Where many culverts are to be constructed, it is common to adopt specific loadings and work out standard forms and dimensions to be used. Such standards have been adopted by many railways and State highway departments. Table XXXI shows dimensions suit- able for ordinary highway culverts 5 to 8 feet in span, to carry the loadings used in Section 156. The steel is to be placed 1J inches from the lower surface of the slab. - Closed box culverts of reinforced concrete are frequently used for small openings, as they I require less headroom than arched openings and are easily applied when open- ings are too large for convenient use of pipes. The stresses in such a culvert cannot be ac- curately determined on account of the indeter- minate character of the loads. A load applied up- on the top of the culvert produces an equal upward thrust upon the bottom of the culvert, as shown in Fig. 98, which causes a moment tending to bend the top and bottom slabs inward and the sides outward. Let 6 = width of culvert; h = height of sides; w = uniform load per foot; If 1= bending moment in top and bottom slabs; M 2 = bending moment at middle of sides; M 3 = bending moment at corners; 1 1 = moment of inertia of sections of top and bottom; /2 = moment of inertia of sections of sides. If we assume the load to be uniformly distributed over the top, the moments will be as follows : 6/37i +h/I 2 1 1 a r -p*. ZL<*>_ ^'4"squore rod s,5.5 c-c. Alternate rods bent up. S'-o" -^"square rods, if c-c. *i \ /*" ~\ j FIG. = = 8 and 6//1+V/2 330 CULVERTS AND CONDUITS If the sectional area of the sides be made the same as those of the top and bottom, we have ,, wtf b/3+h For a square opening this becomes Mi and For sizes of culverts commonly used wb 2 /12 may be considered the limiting value to which the moment may approximate. The moments in top and bottom slabs are decreased and those in the sides increased as the ratio of height to width is lessened. The pressure of earth against the sides of the culvert produces moments in the top, bottom and sides of the culvert of opposite sign to those produced by the load upon top of the culvert, and therefore tend to reduce the effect of the top load upon the culvert. Such pressures always exist to some extent, but are not accurately known. It is usual to assume that unit horizontal pressure, when considered, is about one-third the unit vertical pressure. The mo- ments caused by the side pressures will always be much less than those due to the vertical loads and not sufficient to overcome those moments. If the side pressures be supposed to exist when the vertical loads are not on the culvert, as may be the case with moving loads, the sides will be subject to positive moments and need reinforcing at the inner surfaces. The existence of side pressures tends to increase the negative moments at the corners, and a box culvert can act as a whole only when the corners are reinforced sufficiently to carry these moments without cracking at the corners. In case the fill upon the culvert is not sufficient to distribute the load over the whole top of the culvert, the moment will be increased. For a concentrated load at the middle of span, the moments will be about double those for the same total load distributed over the span. In highway culverts which are covered only with the thickness of the road surface, the distribution of the load may be considered as in Art. 41. In such culverts, the live load should be increased 25 per cent to allow for impact. When, as is sometimes the case, the corners of the culvert are not reinforced for negative moment, the top becomes a simple beam, resting upon the sides but not rigidly attached to them, and the sides carry only the horizontal earth pressure as simple beams. Such CULVERTS 331 %' round bars, 7"c-c. ^-^ round bars, 7*c-c. 9 FIG. 99. Section for Highway Culvert. construction is shown in Fig. 99, which represents a standard section for a highway culvert de- signed to carry a 20-ton auto truck. The section in Fig. 98 is designed for the same loading. 181. Arch Culverts. For locations where suf- ficient headroom is avail- able, arch culverts are often preferable to those with flat top. Very pleas- ing and artistic effects may frequently be ob- tained by careful design of arches for such use. Under fills of considerable height, arch culverts will commonly be more economical to construct than slab top culverts. Fig. 100 shows a section for a standard highway culvert for use under automobile traffic. The analysis of stresses in arch culverts may be made in the same manner as is given for arch bridges in Chapter X. F IG 100 The horizontal earth pressures on the sides of the arch are usually taken as one-third of the vertical pressures at the same point. These pressures are of greater relative importance than in 10 FIG. 101. Concrete Barrel Culvert. 332 CULVERTS AND CONDUITS bridges of longer span. For short spans, plain concrete is com- monly employed, while for spans greater than about 8 feet, rein- forcement is usually introduced for greater security, although not necessary to carry moments. ART. 51. CONDUITS 182. Types of Conduits. Conduits for carrying water may be de- signed either for gravity flow or for internal pressure. Brick masonry was formerly largely used in the construction of gravity conduits, particularly for larger sewers, but is now being replaced for the most part by the use of concrete. For conduits to carry water under pressure, reinforced concrete or steel pipe is usually employed. A conduit consists essentially of two parts, the invert, which forms the channel for the water, and the top, usually arched, which covers the channel and carries the weight of earth or other loads which may come upon it. The shape of the invert depends upon the require- ments of the service. In sewers, special forms of invert are frequently needed to prevent deposits at times of minimum flow. The designs of sections for various uses may be found in works upon water supply, irrigation, and sewerage. Sewage may sometimes cause disintegration of concrete, and the inverts of conduits intended to carry sewage are therefore commonly lined with vitrified brick a method particularly desirable where the sewage is stale or impregnated with chemicals from manufacturing plants. In conduits carrying water for irrigation, injury to concrete may result from alkalis in the soil unless special precautions are taken. The inverts of carefully constructed concrete conduits usually resist the abrasion of flowing water fully as well as those with brick or stone lining such resistance depending upon the alignment of the conduit and the amount of sediment carried by the water. With clear water and an undisturbed flow, very high velocities may pro- duce no appreciable damage, while the impact caused by changes in the direction of flow cause rapid wear, particularly when sand and gravel are carried by the stream. No conduit is absolutely water-tight, and careful attention should always be given to reducing leakages to a minimum. Usually the most serious leakage occurs at joints where one section joins another, although there will generally be some porous spots through which small -quantities of water may pass. The leakage may commonly be reduced to very small proportions by careful design and construe- CONDUITS 333 tion, reinforcing so as to prevent cracks and using dense and uniform mixtures of concrete. This subject is discussed in Art. 23. Conduits of small size are sometimes made rectangular in section and designed in the same manner as rectangular culverts. Larger conduits are usually of curved form with arched tops. 183. Design of Gravity Conduits. After determining the size and general shape of conduit required for a given service, the design depends upon the character of the soil upon which it is to be placed and the external loads that it must carry. When the invert rests upon a firm foundation, capable of supporting the structure without sen- sible yielding, the invert may be considered as fixed in position and the arch may be designed with ends fixed upon the sides of the invert. The design of such arches may be made by the ordinary method used Horseshoe section. Semi-elliptical section. FIG. 102. Typical Sewer Sections. for arch bridges or culverts. Actual loads, in so far as they can be determined, should be used in such designs. Where the loads are light, such conduits may often be built of plain concrete; usually, however, it is preferable to reinforce arches of more than 4 or 5 feet span. Fig. 102 shows typical forms of standard sewer conduits. The horizontal earth pressure to which the side of a conduit may be exposed cannot be accurately determined. It is customary to use Rankine's minimum value, unit horizontal pressure = w 1 sin l+sin0' in which w is the unit vertical pressure and is the angle of friction of the earth. Taking = 30 for ordinary earth, this makes the unit horizontal pressure at any point equal to one-third of the unit vertical 334 CULVERTS AND CONDUITS pressure at the same point. In some instances it may be necessary to consider the possible effect of variations in horizontal pressures. FIG. 103. As the tendency of such a structure under vertical loading is to deflect outward upon the sides, it is reasonable to assume that at least this minimum horizontal pressure may always be depended upon, or a CONDUITS 335 1 ^2 ^D O^ ^^ CO rH ^^ Ol *C^ O"l O^ OJ C^ 1 C^|OOO5COO5t > -l>-OiO5O5O'> a O o rH~ co" TjT CD" oo" cT o cT o" PH rH rH rH rH 3 i 5 CO Vertical. IIIIIIIIIIHI rr^ _ fl C "' O o 00 3:3 Is ^O5iO(NiOCOOO^OiO 3COC<1OCOI>-I>OO 0 * *O cocot^-t^-ooososo atal on i vision. 'giMTtHCOrHOOl^OOCOfNlNiOCq Q ^-ji ^H CO C^ C^l T^ 'O ^^ ^^ ^^ HQ ^ 1 1 1 1 "o -g ^ ^OOOOOOrHCON-COlM "M ^ S OO OO 00 O5 O^ rH T^ CO 4 'S < 0-COOOO5COcOI> r^ ............ o ^ ^(NrHrHOOOOOrHrHrHrH W . OiOiOOOtO^OOOOOOOOOO II 1 1 1 1 o tiTTVffTT^T? e-i. i>^ i-Aij^-i-^^^T 3 336 CULVERTS AND CONDUITS greater passive pressure if needed. In case of soft, wet earth, the horizontal pressure will be much greater, reaching a maximum when it is practically fluid and exerts normal pressure at all points. Conduits to be supported upon compressible soil are often designed to act as a whole, assuming that all parts of the structure, including the invert, are equally subject to distortion under the loads. Fig. 103 represents a half-section of a conduit of this character. If we assume the middle of the invert, m, to be fixed in position, the mo- ments and thrusts in a slice of the conduit 1 foot thick may be found in the manner used for the elastic arch in Chapter X. The axis of the conduit ring is divided into lengths as shown. The lengths of the divisions, coordinates of the centers of divisions with reference to the crown, and thicknesses of concrete at centers of division, are given in Table XXXIII. The loads given (Table XXXII), are those due to the pressure of 20 feet of earth above the crown of the arch. The weight of the earth is taken at 100 pounds per cubic foot, and the intensity of the horizontal earth pressure at one-third that of the vertical pressure at the same point. In computing the loads, the unit pressures at the middle of the extrados of the division are considered as acting upon areas equal to the horizontal and vertical projections of the extrados of the divisions. The upward pressures upon the base are considered as acting vertically and uniformly distributed horizontally. The computations of loads and their moments about the centers of divi- sion are shown in Table XXXII. The moment and thrust at the crown section, a, may be obtained by the use of the formulas of Section 176. As the loading is sym- metrical about the crown, m L and m R are equal, V c = 0, and Formulas (33) and (35) of Section 176 become c and Table XXXIII. gives the computation of the terms needed in these formulas. As the sections are rectangular, no reinforcement being considered in the computations, the value s/t 3 may be used in the formulas in place of s/I. CONDUITS 337 gt THOCOOOTtHTtHOOrHTHt> rJH C^O5iOoOOI>CiOOT-iCNiq5 rj< Jj OOlt^-COiOOCOi-HOOOOi-H CO 00 CO O ^f CO O^ O^ CO CQ 00 00 Oi 00 t^* O^ ^^ O^ ^O *O "^ ^7 i ICOrHCOT^T-HlOrHCOCO s CO OO 00 ^^ ^^ CD t^* CNI CP O^ T-H t>- i-H COOCDOOOSOSt^-t^CO O jSj C^J^JCOi (CDOUCDt>.OOOCOCD !> CD , 4OOIOOOCOCOO5OOOO5OO CO Bl* rHC s qCDCOOOil>COOO' IT^O TjH T I t^ oo tl 00000 rtn' 888SS2i^ggg ^3> OOrHCOOCDOOO(M(NCOCO ^^ C^l ^^ ^O CO CO CO CO ^O CO d ^^ ? 4 338 CULVERTS AND CONDUITS Substituting in the formulas, we have 4841440X14.48-621536X71.43 617X14.48-(71.43) 2 =6710 pounds. M c 9830 The load diagram is now drawn as shown, and the equilibrium polygon (or line of resistance) constructed, beginning with H c at a distance, e=1.46 feet, above the middle of the crown section. The thrusts acting upon the ends of divisions as found from the load diagram may be resolved into normal thrusts and shears as shown by the broken lines. These are tabulated in Table XXXIV. The moments at the centers of sections at the end of divisions may be obtained by multiplying the normal thrust upon the section by the distance from the center of section to the point at which the equilib- rium polygon cuts the section, or they may be computed by Formula 10 of Section 163, which becomes for symmetrical loading M = M c +H c y-m. Table XXXIV, gives the thrusts and moments with the resulting stresses at the extrados and intrados of the sections. These results show that there are tensions at the intrados of the crown section and in the invert, and at the extrados of sections /, g, and h which must be cared for by reinforcement. This reinforcement should be sufficient to carry the tensions in the section without materially changing the position of its neutral axis. or the compression upon the concrete. To do this, the stress in the steel should be limited to about fifteen times that shown for the rectangular section, or about 6000 lb./in. 2 at sections a and g and 9000 lb./in. 2 at ra. Computing the total tension in these sections, we find that an area of about 2 in. 2 of steel per foot of length is required at a and g and about 4 in. 2 at ra. One- inch square bars spaced 6 inches apart near the intrados at sections a and b, then crossing to the extrados at e and extending along the extrados to section i, with If -inch square bars spaced 6 inches apart near the intrados of the invert would answer the requirement. The maximum shear occurs at section j, the unit shear being about 50 lb./in. 2 , which is not excessive. CONDUITS 339 CQ 3 p 1 . i ^ | I a | 1 | i | g 8 i iCOC^C^IC^iHT I ^ O l> ^ 1 1 lj "a" " a" | s| "o i?7i+ + + + + +i c i'i i -1 ^v, ^i CO ^^ i^ CO CO CO ^^ ^^ ^^ *O CO ^+ + +1 1 1 1 1 I4- + + + a* OOOOCOiOi li^OO^OO^OOOOOi 11 *| OSl^COlMl^C^TtHfrt-COlOrHOiCO + + +i 17777+^-^ 11 g g g g g 8 g coi>-osc-^_ _ V.Y.'"V ':"?:': m N '.'" ".'* . .' s;:-: ;/ :.'.*;. v- ;>' Puddle/.' ;:V-'-V-V '/ m s / FIG. 118. ^ o :