UNIVERSITY ^F , KQRNIA 
 
 DEPARTMENT OF CIVIL ENGINEERING 
 
 BERKELEY. CALIFORNIA 
 
 engineering 
 
UNIVERSITY OF CALIFORNIA 
 DEPARTMENT OF CIVIL ENGINEERING 
 , BERKELEY. CALIFORNIA 
 
MASONRY DAM DESIGN 
 
 .. 
 INCLUDING 
 
 HIGH MASONRY DAMS 
 
 BY 
 
 CHARLES E. MORRISON, C.E., PH.D. 
 
 Formerly of the Civil Engineering Department, Columbia University 
 
 AND 
 
 ORRIN L. BRODIE, C.E. 
 
 Mem. Am. Soc. C. E., Late Asst. Designing Eng., N. Y. Board of Water Supply 
 
 SECOND EDITION, REVISED AND ENLARGED 
 
 FIRST THOUSAND 
 
 NEW YORK 
 
 JOHN WILEY & SONS, INC. 
 
 LONDON: CHAPMAN & HALL, LIMITED 
 
 1916 
 
> 
 
 Engineering 
 Library 
 
 Copyright, 1910, 1916 
 
 BY 
 CHARLES E. MORRISON AND ORRIN L. BRODIE 
 
 THE SCIENTIFIC PRESS 
 
 ROBERT DRUMMOND AND COMPANY 
 
 BROOKLYN. N. Y. 
 
PREFACE TO SECOND EDITION 
 
 IN this revision the authors have attempted to amplify 
 some of the features which, in the earlier edition, were 
 little more than touched upon. Chapters on the Overfall 
 and Arched types of masonry dams have been added, 
 together with cross-sections of a selected series of masonry 
 dams chronologically arranged. The last are for the pur- 
 pose of comparison and showing the development of the 
 masonry dam from the time of the massive Spanish type 
 to the present. 
 
 The design of low and medium-sized as well as that of 
 high-masonry dams, may be prosecuted according to the 
 theory and methods in this work, as general expressions 
 have been written wherever possible. It was considered 
 best to indicate this in the title while the original page 
 captions remain the same. 
 
 The authors are indebted to Miss Bessie N. MacDonald, 
 A.B., for assistance in verifying the more difficult mathe- 
 matical derivations for the Arched Dam. Acknowledg- 
 ment is also made to Mr. Alfred D. Flinn, Member American 
 Society of Civil Engineers, who had kindly furnished one 
 of the authors with a set of cross-sections of dams, a num- 
 ber of which are shown in the series in Appendix III. 
 
 C. E. M. 
 
 O. L. B. 
 
 NEW YORK CITY, 
 February, 1916. 
 
 iii 
 
 793985 
 
PREFACE TO FIRST EDITION 
 
 IT is the practice at Columbia University to require of 
 the third-year students in the Department of Civil Engineer- 
 ing, the execution of the design of a masonry dam, and 
 to aid them in this problem they have heretofore been 
 furnished with "Notes on the Theory and Design of High 
 Masonry Dams," prepared some years ago by Prof. Burr 
 of the Department, and having for their basis the method 
 as set forth by Mr. Edward Wegmann. 
 
 This procedure with which Wegmann is credited, and 
 which was developed through the investigations undertaken 
 in connection with the Aqueduct Commission of the city 
 of New York, for the purpose of determining a correct 
 cross-section for the Quaker Bridge dam, resulted in 
 the first direct method for calculating the cross-section 
 of such structures and is essentially a development of 
 the Rankine theory. 
 
 The studies appeared first in the report made by Mr. 
 A. Fteley to the chief engineer of the Aqueduct Com- 
 mission of the city of New York, dated July 25, 1887, 
 and later in Mr. Wegmann's treatise on "The Design 
 and Construction of Dams." 
 
 Neither in the report nor in the treatise however, 
 have the effects of uplift, due to water permeating the 
 
VI PREFACE 
 
 mass of masonry, and of ice thrust, acting at the sur- 
 face of the water in the reservoir, been considered, and 
 in consequence of this, objection might be legitimately 
 raised that the series of equations determining the cross- 
 section fail to account for these factors. Some difference 
 of opinion may exist as to the relative importance of 
 these considerations, but when a structure of great 
 responsibility is projected, conservatism in design is 
 essential. 
 
 The following presentation which aims to supply these 
 omissions, has been prepared primarily that there may be 
 had in convenient form a text, containing the general 
 treatment and such consideration of these factors as more 
 recent practice requires, together with a brief statement 
 regarding the late investigations undertaken for the pur- 
 pose of determining more accurately the variation of 
 stress in masonry dams. 
 
 The formulae relating to uplift, ice thrust, etc., were 
 deduced by one of the authors and have been used in part 
 in connection with the design of the large dams for the 
 new water supply for the city of New York. 
 
 The computations for the design of a high masonry 
 dam are appended to facilitate the ready comprehension 
 and application of the formulae. 
 
 It is hoped that the presentation may appeal to the 
 practicing engineer as well as the student, and that there 
 may be found therein enough to compensate him for the 
 
 labor involved in its perusal. 
 
 C. E. M. 
 
 O. L. B. 
 
 COLUMBIA UNIVERSITY, 1910. 
 
TABLE OF CONTENTS 
 
 CHAPTER PAGE 
 
 INTRODUCTION ix 
 
 I. UPWARD PRESSURE AND ICE THRUST i 
 
 PART I. Upward Pressure I 
 
 PART II. Ice Thrust 16 
 
 II. PRELIMINARY CONSIDERATIONS 20 
 
 Pressure on a submerged surface. Center of pressure 20, 22 
 
 Distribution of stress in a masonry joint 23 
 
 III. PART I. DEVELOPMENT OF FORMULA FOR DESIGN 31 
 
 Nomenclature 3 1 
 
 Conditions for stability 35 
 
 Factors of safety 36 
 
 Derivation of formulae 42 
 
 PART II. DESIGN FORMULA A, B, C, D, E AND F FOR REFERENCE 53 
 
 IV. INVESTIGATION FORMULAE FOR REFERENCE 69 
 
 Formulae for position of the resultant on horizontal joint or base 
 . Formulae for maximum intensities of pressure on the hori- 
 zontal joints Variation in assumed value for hydrostatic head 
 causing uplift in any section. 
 
 V. THE DESIGN OF "A MASONRY DAM 74 
 
 VI. WEIR OR OVERFALL TYPE OF DAM 99 
 
 VII. THE ARCH DAM 134 
 
 VIII. RECENT CONSIDERATIONS OF THE CONDITION OF STRESS IN A 
 
 MASONRY DAM 169 
 
 APPENDIX I. DERIVATION OF CANTILEVER EQUATIONS 207 
 
 APPENDIX II. MOVEMENTS AND STRESSES IN AN ARCH SUBJECTED TO 
 A UNIFORM RADIAL LOAD WITH DERIVATION OF Equa- 
 tion 8 FOR ARCH CROWN DEFLECTION 221 
 
 APPENDIX III. CROSS- SECTIONS OF EXISTING MASONRY DAMS 231 
 
 INDEX 269 
 
 vii 
 

 t 
 
 INTRODUCTION 
 
 THE method of analysis by which an economical cross- 
 section of a gravity type high masonry dam may be most 
 directly calculated, and the one which is most generally 
 adopted in engineering practice, was first devised by Mr. 
 Edward Wegmann through studies made for the Aqueduct 
 Commissioners of New York City, in connection with the 
 design of the New Croton Dam, and it is that method 
 which will be employed here, though it will receive some 
 modification in certain particulars and be elaborated in 
 certain others. 
 
 In determining the cross-section by the series of equa- 
 tions developed in that analysis, no account is taken of 
 uplift due to water penetrating the foundation or the mass 
 of masonry above, nor of the ice thrust acting horizontally 
 against the up-stream face of the dam, at the surface of 
 the water in the reservoir, though reference is made to it. 
 Present practice requires, however, that these two factors 
 be recognized where a structure of great responsibility is 
 proposed, and in this respect at least will the analysis 
 be amplified. 
 
 IX 
 
MASONRY DAM DESIGN 
 
 Including High Masonry Dams 
 
 CHAPTER I 
 
 UPWARD PRESSURE AND ICE THRUST 
 PART I UPWARD PRESSURE 
 
 ALTHOUGH it had been appreciated for a number of 
 years that a complete analysis of a high masonry dam 
 required the consideration of uplift and ice thrust, until 
 comparatively recently no structure of this type had been 
 designed which allowed for these two factors in the com- 
 putations. 
 
 In fact, it may be said that prior to the year 1853 
 masonry dams were built without a rational consideration 
 of any of the forces acting in or upon them, for it was not 
 until then that de Sazilly first indicated the principles 
 upon which dam design is based, by providing for a suf- 
 ficient safety factor against sliding and overturning and 
 by assigning a maximum limit of pressure against the 
 crushing of the material. 
 
 Some time later Rankine added to the theory by pre- 
 scribing the well-known requirement that the line of 
 resultant pressure for reservoir, full or empty, should lie 
 within the middle third of the structure, to preclude the 
 possibility of tension in any joint, and suggested that the 
 
2 HIGH MASONRY DAM DESIGN 
 
 'e^t-KlJ^ 
 
 limit of pressure should be made less for the down-stream 
 
 edge than for the up-stream. 
 
 In 1884, when the Aqueduct Commission of New York 
 City came to design the New Croton Dam, a structure 
 between 275 and 300 feet high, then the highest in the 
 world, and exceeding the next highest by about 100 feet, 
 it was found necessary to modify some of the older con- 
 clusions with respect to dam design, in order to make 
 the theory applicable to their particular problem. 
 
 Thus, where heretofore the prescribed limit of crushing 
 strength of masonry had been assumed to be between 
 6 and 10 tons per square foot, they increased it to 16 
 tons, as it had been demonstrated that such pressures 
 actually existed in dams still doing duty, and since, with 
 the lower values, the computations would have given a 
 horizontal face at a joint 300 feet below the top. Both 
 upward pressure and ice thrust were considered, but both 
 in turn were disregarded. The former because it was 
 felt that the condition of the masonry and of the founda- 
 tion was such that the entering of water would be a remote 
 possibility, and the latter because it was believed that 
 the mass of the masonry was sufficiently great to care for 
 any additional forces due to the ice thrust. 
 
 It remained, therefore, for the engineers of the Wa- 
 chusett dam in Massachusetts to be the first in the United 
 States actually to incorporate uplift and ice thrust in the 
 design of a high masonry dam. They may have been led 
 to this precaution by the fact that the structure was 
 located only one -half mile above a town of some 13,000 
 inhabitants, where a failure would result in enormous 
 loss of life, and where it was, in consequence, necessary 
 
HIGH MASONRY DAM DESIGN 3 
 
 to be particularly careful, but at any rate they were the 
 first to adopt these l;wo considerations in the design of a 
 high masonry dam. 
 
 Their assumption for uplift was two-thirds of the 
 static head at the up-stream edge, diminishing as a straight 
 line to zero at the down-stream edge, and for ice thrust, 
 47,000 pounds per linear foot of dam, or equivalent to 
 the crushing strength of ice one foot thick. 
 
 To-day, in the light of experience, no structure of 
 this character would be built without careful consideration 
 of both these elements, and it is doubtful if, under any 
 circumstances, they would be eliminated entirely, though 
 they might not receive the same weight they were given 
 in the computations for the Wachusett Dam. 
 
 That engineers are not fully agreed on the matter 
 of uplift and ice thrust and that a considerable diversity 
 of opinion exists in the profession with respect to them, 
 may perhaps be partially explained by the fact that the 
 former does not lend itself to an exact treatment, while, 
 with regard to the latter, there are no exact data as to 
 the expansive force of ice acting at the surface of a res- 
 ervoir. Furthermore, there are many high masonry dams 
 now standing which were designed with no consideration 
 being given to these two factors, and this would seem 
 to refute the argument that they are necessary consid- 
 erations for safety. 
 
 It is recognized, however, that the influence of upward 
 pressure and ice thrust on the stability of masonry 
 dams, together with the actual internal distribution 
 of stress in very large masses of masonry are probably 
 the most indefinite factors in the design of such structures. 
 
4 HIGH MASONRY DAM DESIGN 
 
 Uplift. It may not be out of place to explain here 
 at some length what " uplift " means and how it may be- 
 come active. 
 
 In a masonry mass, especially a large concrete mass, 
 cracks can be formed by temperature changes, due to the 
 setting of the concrete in the first place, and to sub- 
 sequent daily and seasonal exterior temperature varia- 
 tions. 
 
 Contraction joints, provided to meet the effect of such 
 temperature changes in the body of the masonry, are often 
 built in large dams. 
 
 Discontinuity of the mass of a large masonry structure 
 like a dam, owing to interruption and resumption of con- 
 struction work from day to day, is also evidenced by 
 joints, mostly horizontal, perhaps, but, in spite of the 
 utmost attempts to preserve continuity, often unavoid- 
 able. 
 
 Temperature cracks, contraction and construction joints, 
 then, all tend to affect permeability to a greater or less 
 degree, admitting water to the body of a dam according 
 to the pressure exerted by that water. 
 
 Besides, as it has been observed that water under 
 sufficient head has passed through 30 feet thickness of 
 good concrete and that under enormous pressures water 
 has been made to ooze through cast-steel cylinders, it may 
 be appreciated from the above considerations that water 
 from a reservoir may enter the masonry mass of the 
 dam. In fact, it has been frequently found in high masonry 
 dams that, following construction and upon filling the res- 
 ervoir, small issuing streams or leaks have appeared on the 
 down -stream face. These leaks have been observed at 
 
HIGH MASONRY DAM DESIGN 5 
 
 > 
 
 the base, down-stream, as wfcll as higher up, so it may be 
 safely assumed that water from the reservoir may pene- 
 trate the natural foundations as well as the masonry 
 above, especially if the former consist of a porous or 
 stratified formation. 
 
 Therefore, in a structure of considerable height, which 
 retains a body of water behind it, there may be exerted 
 a powerful vertical force acting upward under the dam 
 or on some joint of the masonry above. This force is 
 commonly termed " uplift," and will, of course, depend, 
 for its amount, upon the hydrostatic head. Furthermore, 
 this force, due to the water pressure, tends to counter- 
 balance the downward, vertical component of all forces 
 acting in or upon the structure. 
 
 Obviously, were this " uplift " to become sufficiently 
 great it might actually float the structure off its foundation 
 or off any joint, whereupon the horizontal water thrust 
 back of the dam would complete the destruction by 
 sliding it down-stream. 
 
 Upward pressure, therefore, should receive considera- 
 tion, both from the standpoint of its effect in the foundation 
 of the dam, and also in any of the joints above. 
 
 Naturally, it is much more difficult to ascertain the 
 condition of this with respect to the foundation, as the 
 latter 's physical characteristics are never revealed until 
 actual work has begun on the structure and the site is 
 uncovered. For this reason, it should be made imperative 
 to examine by exploration, drill holes, etc., as completely 
 as possible, the nature of the foundation, so that its true 
 state may be at least approximately known, and so that, 
 also, proper provision for upward pressure may be made. 
 
6 HIGH MASONRY DAM DESIGN 
 
 An examination of the outcropping rock at a dam site 
 will never be sufficient to determine the nature of the 
 foundation below, as the latter may not conform with the 
 exposed surface. Core borings -should be made at frequent 
 intervals. They should be driven well into the bed rock 
 to develop the character, and where there is limestone with 
 a likelihood of cavities, particular care should be exercised. 
 Upon such cavities being uncovered, they should be filled 
 with grout and concrete, so as to preclude the entrance 
 of water. Very porous sandstone or the existence of 
 seams and strata may give rise to very dangerous con- 
 ditions. Thus, an examination by borings at the 
 Austin, Pa., dam site would have indicated the porosity 
 of the rock, and might have been the means of prevent- 
 ing the disaster which followed. All foundations should 
 be tested for tightness by applying air or water pressure 
 to the drill holes. 
 
 Even in the best foundation, however, it may be said 
 that there is no absolutely water-tight condition. 
 
 All water getting into the dam should be collected in 
 a chamber or tunnel, carried outside and measured for 
 quantity. Thereby a measure of the water-tightness of 
 the dam may be ascertained. 
 
 Treatment of Uplift. There is not the clearest con- 
 ception among engineers as to how to allow for this up- 
 ward pressure, but in some of the more recent discussions, * 
 it has been suggested that perhaps three general con- 
 ditions may be recognized. 
 
 * " Provision for Uplift and Ice Pressure in Designing Masonry Dams." 
 By C. L. Harrison, Trans. Am. Soc. C.E., Vol. LXXV, p. 142. 
 
HIGH MASONRY DAM DESIGN 7 
 
 1. The case where no ^upward pressure could exist 
 because the foundation rock, and the joints of the dam, 
 were so tight that no water could possibly enter. Evi- 
 dently for such a condition provision for upward pressure 
 in the design of the dam is unnecessary. 
 
 2. Second, the case at the other extreme, where the 
 rock is of such a nature that water may freely enter the 
 foundation, and as freely leave it from the lower edge of 
 the dam. Here it is quite evident that the water would 
 enter with the full hydrostatic pressure acting at that 
 point or elevation, and if the water flowed away freely 
 from the down-stream edge, the hydrostatic head would 
 be zero at this latter point. It might be a fair assump- 
 tion to conclude that the pressure varied as a straight line 
 between the up-stream and down-stream edge,* which 
 would give an equivalent pressure over the entire base 
 of one-half the hydrostatic head assumed acting at that 
 point. 
 
 3. The third case might be represented by that which 
 would be intermediate between cases i and 2 ; in other 
 words, where there was easy access to the foundation, 
 but not such easy access from it. Under these conditions 
 the pressure at the heel would be assumed equal to the 
 hydrostatic head, while at the toe it would be equal to 
 that pressure represented by the head of the issuing stream. 
 
 It therefore becomes a question for the engineer to 
 decide, from a knowledge of the condition of the founda- 
 tion, as to what degree of entering water and consequent 
 
 * Cf. Proceedings Am. Soc. C.E. for May, 1915, " Experiments on Uplift." 
 These, however, are upon too small a scale to yield conclusions other than 
 those applying to the experiments themselves. 
 
8 HIGH MASONRY DAM DESIGN 
 
 uplift may exist, and to provide for it accordingly. It 
 is just because the matter is based upon judgment that 
 such a diversity of opinion prevails. 
 
 Generally it is quite likely that none of the above 
 conditions will strictly apply, but rather varied combina- 
 tions of them, so that it becomes difficult to conclude how 
 to dispose of this question. 
 
 Some engineers demand that the structure shall be 
 designed for the full static head acting over the entire 
 base, while others advise that no allowance whatsoever 
 be made, but it is generally conceded that some dams, 
 dependent on the kind of their foundations, need provision 
 for uplift. 
 
 As an example of the former, there may be cited the 
 dam at Marklissa, Prussia, over the Queis, * while the New 
 Croton Dam is an example of a very important structure 
 of this type where such provision was absolutely elim- 
 inated in the design. 
 
 There are several ways in which upward pressure may 
 be cared for: First, by adding a sufficient section to the 
 dam to offset the upward pressure, and second, by providing 
 drainage wells and galleries to intercept all entering water, 
 carrying it away through a discharge gallery, or conduit, 
 to the lower side of the dam, and at the same time by 
 carefully providing for as impervious an up-stream face 
 as possible. 
 
 In the foundation an adequate cut-off, of width and 
 depth determined by examination of conditions disclosed 
 during the progress of foundation excavation, is often 
 
 * Trans. Am. Soc. C.E., Vol. LXXV. 
 
HIGH MASONRY DAM DESIGN 9 
 
 
 
 advisable. The exploratory borings should usually indi- 
 cate beforehand this necessity, so that the final extent to 
 which a cut-off trench is taken remains to be decided during 
 its actual excavation. Borings may be extended from its 
 faces and bottom to reach seams and pockets to be filled 
 by grouting under pressure before the concrete of the 
 cut-off trench is placed. 
 
 The drainage wells, slightly inclined to the vertical, 
 and the cut-off are placed as near as consistent to the 
 up-stream face of the dam, and galleries are built longi- 
 tudinally to the up-stream face. While the wells and gal- 
 leries may be nearly completely effective in intercepting 
 percolation, they cannot be considered absolutely so, and 
 consequently may allow some water to get down-stream. 
 Such seepage would then result in upward pressure down- 
 stream from the wells and galleries, and if the water had 
 connection in any way with the reservoir, pressure on a 
 joint due to the full static head might result. These last 
 remarks apply, but with less force, perhaps, to the 
 foundation cut-off. 
 
 The theory of this intercepting drainage system is 
 that any water having gotten into the dam due to the 
 static pressure acting on faults or cracks in the com- 
 paratively more impervious up-stream face, will be 
 caught and prevented from going any further into the 
 structure. 
 
 It should not be assumed that because of an impervious 
 up-stream face, and because of drains and cut-offs, no 
 water reaches the body of the dam below the latter, for 
 there may be construction joints, and contraction cracks 
 in the face, and in places the cut-offs and the down- 
 
10 HIGH MASONRY DAM DESIGN 
 
 stream portion may be less pervious than the up-stream 
 part, as a consequence of which uplift may exist in the 
 latter. 
 
 Additional means of protection should be provided in 
 the form of drainage channels to lead the water away from 
 the down -stream portion of both the foundation and the 
 body of the dam. 
 
 As an example of intercepting drains, at the Cataract 
 Dam, which furnishes the water supply for Sydney, Australia, 
 " the upper face of masonry was built with special care 
 to a depth of 2 or 3 feet, and this alone is relied upon 
 to prevent seepage. The rest of the dam is built of good, 
 though more pervious, masonry, and throughout the whole 
 were placed 6-inch rectangular conduits filled with broken 
 stone parallel to and about 6 feet back from the up-stream 
 face. These are collected into 6-irich earthenware pipes, 
 laid at right angles to the longitudinal axis of the dam, 
 with exits on the down-stream face." * (Cf. cross-section, 
 Olive Bridge Dam, page 96.) 
 
 These systems of drainage naturally tend to eliminate 
 upward pressure and consequently increase the stability 
 of the dam, and would seem justifiable in the case of all 
 important structures. 
 
 In small dams drainage wells are not so easily 
 provided and the protection is relatively less complete, 
 because there is a certain minimum distance from the 
 upper face within which the drains cannot well be extended. 
 
 With the correction for upward pressure applied in 
 the form of increased section, the water entering the dam 
 is wasted, which is a considerable item of cost, while in 
 * Mr. Allen Hazen, Trans. Am. Soc. C.E., Vol. LXXV, p. 154. 
 
HIGH MASONRY DAM DESIGN 
 
 11 
 
 addition the cost of the Structure is increased by the 
 added masonry. With collecting galleries both of these 
 items may be partially eliminated. The cut-off wall in 
 the foundation will often prove an economy in this respect. 
 The question of cost may become a very important 
 one, as in the case of hydro-electric developments where 
 additional cross -section may mean such an increase in the 
 cost as to cause the abandonment of the project. 
 
 TABLE I 
 WIDTH OF BASE OF DAM GIVEN IN FEET FOR VARYING HEIGHTS 
 
 Height of dam in feet 
 
 5 
 
 IO 
 
 30 
 
 60 
 
 IOO 
 
 250 
 
 Water pressure only (hori- 
 
 
 
 
 
 
 
 zontal) 
 
 1>.2 
 
 6.4 
 
 IQ 
 
 4 
 
 65 
 
 161 
 
 Uplift as described below. . . . 
 
 4.2 
 
 8. 4 
 
 25 
 
 5i 
 
 84 
 
 211 
 
 The effect of uplift in its tendency to increase the mass 
 of masonry is shown in Table I, for six triangular dam 
 cross-sections, where the upward pressure is assumed 
 equal to the static head at the heel, and diminishing as 
 a straight line to zero at the toe. For convenience, also 
 the width at the top is taken equal to zero.* 
 
 Table II and Fig. i represent five cross-sections of 
 dams, four of which were directly designed for comparative 
 purposes, showing the effect, not only of uplift, but of 
 ice pressure, upon the top and bottom dimensions, super- 
 elevation necessary, as well as the comparative volumes 
 resulting. These differ from those previously cited, in that 
 the full hydrostatic heads for dams sustaining different 
 heads are there employed for " uplift " comparison, 
 
 * Mr. W. J. Douglas in Trans. Am. Soc. C.E., Vol. LXXV, p. 207. 
 
12 
 
 HIGH MASONRY DAM DESIGN 
 
 TABLE II 
 DIMENSIONS, CONDITIONS, ETC., FOR FIVE CROSS-SECTIONS OF DAMS. 
 
 
 
 
 Super- 
 
 
 
 Percent- 
 
 
 Conditions of 
 
 
 elevation 
 
 
 A __ 
 
 age of 
 
 Cross- 
 sections. 
 
 loading. 
 (Masonry, 140 
 pounds per 
 cu.ft.) 
 
 Top 
 width, 
 in feet, 
 
 above 
 normal 
 reservoir 
 surface, 
 
 Base, 
 in 
 feet. 
 
 x\rea t 
 in 
 square 
 feet. 
 
 excess of 
 area over 
 the area 
 of D. 
 
 
 
 
 in feet. 
 
 
 
 
 Austin, Pa. 
 
 
 2 5 
 
 2 5 
 
 "}O 
 
 840 
 
 o 
 
 A 
 
 Ice, * uplift, and 
 
 o 
 
 O 
 
 O 
 
 V^W _J_ 
 
 
 
 horizontal water 
 
 
 
 
 
 
 
 thrust. ........ 
 
 10. 
 
 IO.O 
 
 4i-5 
 
 1475 
 
 76 
 
 t 
 
 Uplift and horizon- 
 
 
 
 
 
 
 
 tal water thrust. 
 
 6.0 
 
 5-o 
 
 40.0 
 
 992 
 
 19 
 
 ct 
 
 Uplift and horizon- 
 
 
 
 
 
 
 
 tal water thrust. 
 
 IO.O 
 
 IO.O 
 
 39-5 
 
 999 
 
 2O 
 
 D 
 
 Horizontal water 
 
 
 
 
 
 
 
 thrust 
 
 5-o 
 
 5.0 
 
 33-1 
 
 836 
 
 
 
 * 21,500 pounds per lin. ft. of dam. 
 
 t B and C are subjected to the same conditions of loading. Flood level at +2.5 ft. 
 The resultant on any base or horizontal joint of masonry is at the down-stream " middle 
 third" point. 
 
 while but one depth of reservoir is employed in the latter 
 set of comparisons. 
 
 The uplift intensity is assumed as varying uniformly 
 from a maximum at the heel to zero at the toe and the uplift 
 is considered as acting over only a portion of the area of 
 the joint. This is cared for by assuming only a portion 
 of the full hydrostatic head as acting at the up-stream end 
 of the joint considered. Two-thirds of the full up-stream 
 head is used. 
 
 Extent and Distribution. Upward pressure cannot act 
 over the entire area of a joint or base, or the dam would 
 be floating. Total pressure on the parts of the joint in 
 contact must equal the difference between the weight of 
 dam and uplift, and be less than the crushing strength 
 
HIGH MASONRY DAM DESIGN 
 
 13 
 
 of the material or not morA than 500 pounds per square 
 inch. " The assumption that only one-third of the rock 
 
 15.0 
 
 -47.5 
 
 faces are in direct contact gives pressure as great as prob- 
 ably occur, and leads to the conservative assumption of 
 
14 HIGH MASONRY DAM DESIGN 
 
 upward pressure over two-thirds of the base, and varying 
 according to the resistance losses in passing through the 
 rock or masonry." Hence a uniformly varying intensity 
 is usually employed. " Where upward pressure must be 
 allowed for in the base, there is no economy in failing to 
 allow for it in the joints above the base." 
 
 A distinction should be drawn between the uplift 
 conditions which may be encountered in the foundations 
 and those higher up in the dam, and the foregoing as- 
 sumptions in regard to uplift may be modified for special 
 cases. For example, a trapezoidal (instead of a tri- 
 angular) distribution of intensity due to uplift may be 
 found advisable for a foundation. In cases where inter- 
 cepting drainage wells are provided in the body of the 
 masonry, as for the Olive Bridge and Kensico Dams, it 
 would be reasonable to assume a triangular disposition 
 of intensities, with the maximum at the heel as before, 
 but running out to zero at or a little beyond the line of 
 wells. 
 
 It has been suggested as reasonable to assume that 
 if water is to be properly excluded from a masonry dam, 
 the same general methods should be applied as are em- 
 ployed in waterproofing any foundation, such as the use 
 of several layers of tarred felt, a waterproofing surface 
 coat of some kind, or by pouring wet concrete continuously. 
 Objection to the first method would be that such foreign 
 substance, in layers, would form a plane of cleavage that 
 would defeat its very purpose by providing a weak, hori- 
 zontal joint. 
 
 It is known that " joints between two successive days' 
 work in concrete may become planes of entry for the water." 
 
HIGH MASONRY DAM DESIGN 
 
 15 
 
 Furthermore, when mortar 4s used in ashlar or rubble 
 masonry it must be dry enough to handle, and conse- 
 quently it lacks the wet consistency necessary to make 
 it waterproof, thus permitting water to enter at the 
 joints. 
 
 As a general proposition, then, upward pressure and 
 ice thrust may be said to be more or less dependent upon 
 local conditions, as to the extent to which they should be 
 considered in any given case. 
 
 Three cases of failure, undoubtedly due in part to 
 upward pressure for which no allowance was made in 
 the design, may be instanced, but at the same time it should 
 be pointed out that all of these dams were established on 
 poor foundations. 
 
 
 
 Maximum 
 Height. 
 
 Built. 
 
 Failed. 
 
 Bouzey Darn 
 
 , France 
 
 72 
 
 1878-81 
 
 April 27 180^ 
 
 Austin Darn 
 
 Texas. 
 
 68 
 
 I 80 I Q2 
 
 April 7 1 900 
 
 Austin Dam, 
 
 Pa 
 
 so 
 
 IQOQ-IO 
 
 Seot. 1*0 ion 
 
 
 
 
 
 
 In the Bouzey Dam the -foundation was on fissured 
 red sandstone, and quite permeable, and the excavation 
 was carried down to only a fairly good bottom, and by 
 no means to solid rock. The foundation of the Austin, 
 Texas, dam was located partly over a fault 75 feet wide, 
 filled with adobe with occasional streaks of red clay, nor 
 was the foundation trench excavated deep enough, while 
 the protection on the downstream side was insufficient. 
 The Austin, Pa., dam was founded on sandstone, underlaid 
 by shale having fissures filled with clay, sand, and gravel. 
 
16 HIGH MASONRY DAM DESIGN 
 
 PART II ICE THRUST 
 
 It is realized that, in countries where low tempera- 
 tures prevail in winter, the pressure of expanding ice 
 against the face of a dam at the level of the water sur- 
 face may become, before crushing of the ice takes place, 
 a very tangible force tending to destroy the equilibrium 
 of forces and thus overturn the structure. This expan- 
 sion will occur when the ice is formed under low tempera- 
 tures and when higher temperatures later prevail. 
 
 In addition to this, the ice may deliver a considerable 
 blow or impact when it has formed into floes and the 
 wind carries it down to the face of the dam. In this latter 
 connection, however, it is well to remember that under 
 the force of the wind the jagged points of the ice floes 
 would first come into contact with the dam, and these 
 would be broken off. But under any conditions the 
 expansive force of the ice will, without question, be the 
 more important consideration. 
 
 It is to guard against these forces which produce an 
 additional tendency to destroy the equilibrium of the 
 structure, that ice thrust is considered. 
 
 Where the reservoir has sloping sides it would seem 
 reasonable to assume that the expanding ice would tend 
 to slide up the shores, and, as the face of the dam 
 is only a small part of the shore line, that there would 
 be comparatively little force exerted against it. If the 
 walls of the reservoir were vertical, however, this would 
 not apply. 
 
 In very cold climates it may not be wise nor safe to 
 assume that the ice may be kept clear of the dam by means 
 
HIGH MASONRY DAM DESIGN 17 
 
 
 
 of maintaining an o^eri trench next to the upstream face, 
 for on the contrary the full pressure may be exerted. 
 
 In the Cross River Dam and in the Croton Falls Dam, 
 recently erected on the watershed of the Croton River near 
 the new Croton Reservoir, and with no inhabitants imme- 
 diately below them, besides allowing for upward pressure, 
 ice thrust was figured at 24,000 pounds per linear foot for 
 the former and 30,000 pounds per linear foot for the latter. 
 
 However, in this connection, it should be borne in mind 
 that reservoirs for domestic supply are usually drawn 
 down during the period when ice prevails, and that a point 
 of application of the ice thrust thus results much lower 
 than might be assumed, which the structure is better 
 able to resist. But if a storage reservoir is to be kept at 
 high level during the ice season, full pressure should be 
 considered acting at the top. 
 
 The late C. L. Harrison * concluded that under the 
 following conditions it was not necessary to provide for 
 ice pressure : 
 
 "(i) For the ordinary storage reservoir with sloping 
 banks, in climates where the maximum thickness of ice 
 is 6 inches or less for dams with southern exposure this 
 limit may be placed as high as i foot. 
 
 " (2) For reservoirs which are filled during the flood 
 season and from which all the stored water is drawn off 
 each year during the low- water season.' This would in- 
 clude even the large reservoirs on the head-waters of the 
 Mississippi River, where the ice has a thickness of more 
 than 4 feet, and the atmospheric temperatures reach 50 
 below zero. 
 
 * Trans. Am. Soc. C.E., Vol. LXXV, p. 219. 
 
18 HIGH MASONRY DAM DESIGN 
 
 " (3) For storage reservoirs where the water will be 
 drawn off each year during the winter to a level where 
 the dam is strong enough to resist the ice pressure. 
 
 " (4) For reservoirs where the contour of the ground 
 at the high-water level is such that the expansive force 
 of the ice will not reach the dam." 
 
 Ice thrust has a greater influence on the thickness 
 of a dam than the flood water level, down to about no 
 feet below the flow line in the Olive Bridge Dam, at 
 which point the lo-foot flood level begins to require a 
 wider base; and in the Kensico Dam, with a flood level 
 .of 5 feet, the change is about 210 feet below, indicating 
 that, for dams of moderate height, ice pressure has a 
 very great influence. (Cf. 5 Profiles of Fig. i, page 13.) 
 
 In the Olive Bridge Dam it was assumed, that clear 
 block ice i foot thick might be expected to form at the 
 surface, and expand so as to exert its full crushing strength 
 of about 47,000 pounds per linear foot of dam, and this 
 figure was used in the Wachusett Dam; 42,000 pounds 
 per linear foot was recommended in the Quaker Bridge 
 Dam, and 30,000 pounds per linear foot in the Croton 
 Falls Dam, and 24,000 in Cross River Dam, while in the 
 Design of the New Croton Dam ice thrust was dis- 
 regarded. 
 
 In a discussion regarding ice pressure, before the 
 Canadian Society of Civil Engineers, in December, 1891, 
 agreement seemed to have been reached on two points: 
 That thrust from ice less than 3 inches thick can be dis- 
 regarded, and that the thrust can safely be taken at the 
 crushing strength of ice. The ' ' Engineering News " of Jan- 
 uary 12, 1893, and of April 5, 1894, records the compressive 
 
HIGH MASONRY DAM DESIGN 19 
 
 p - ; 
 
 strength of ice ranging all the^way from 100 to 1000 pounds 
 per square inch; 21,500 pounds per linear foot of dam, 
 with ice at 3 inches thick would be equivalent to about 
 600 pounds resistance per square inch. The value of 47,000 
 pounds above noted is equivalent to about 650 pounds 
 per square inch for ice 6 inches thick. 
 
 The uncertainties of the problem are thus increased 
 by the lack of exact and more extended data bearing upon 
 the subject of ice pressures. It is, therefore, largely a 
 matter of judgment as to what should be used as an ice 
 thrust in any given case, as has been indicated by the 
 foregoing discussion. 
 
CHAPTER II 
 PRELIMINARY CONSIDERATIONS 
 
 THE studies involved in the determination of a gravity 
 cross-section demand an investigation along two general 
 lines : 
 
 First, the direct calculation to fix the most economical 
 cross-section under the imposed conditions, and 
 
 Second, studies in the comparison of cross-sections 
 ranging between this one, which may be called the min- 
 imum, and one of an existing masonry dam, where the 
 conditions and responsibility are practically the same as 
 those under consideration. 
 
 Before undertaking such an analysis, however, it will 
 be advisable to consider the manner in which water pressure 
 is exerted against a submerged surface; its amount; the 
 method of determining the point of s application of the 
 resultant ; the assumed distribution of pressure in a masonry 
 joint ; and finally, the action of the forces in and upon the 
 structure. 
 
 It may be stated as a general proposition that water 
 pressure acts in all directions against a submerged object 
 and that it depends for. its value merely upon the " head," 
 or depth of the center of gravity of the figure below the 
 free surface of the liquid. In consequence of this principle 
 it may be shown that the total normal pressure is repre- 
 sented by 
 
 (i) 
 
 20 
 
HIGH MASONRY DAM DESIGN 21 
 
 
 
 where P=the total normal prdtesure; 
 
 7-= the weight of a unit volume of water; 
 A =the total area ; and 
 
 h=ihe vertical distance of the figure's center of 
 gravity beneath the free surface of water. 
 
 The demonstration * may be made by considering the 
 surface to be divided into an infinite number of parts; the 
 total pressure on each one of these elements, depending 
 only upon the weight of water resting upon it, may be 
 written, 
 
 ....... (2) 
 
 in which p = the total normal pressure on the differential 
 
 area; 
 
 a = the differential area ; 
 
 hi = the head on a (practically constant over the 
 differential area). 
 
 If, therefore, we take the sum of the pressures on all of 
 these small areas, we shall obtain the previous equation, 
 which is perfectly general and applies to any surface. 
 In the case of a vertical, rectangular strip of the back 
 of a dam, the application of the formula will give a total 
 pressure of 
 
 p--J*f 
 
 Jo 
 
 where b is the constant breadth of the strip, usually taken 
 as one unit, oc is the variable, and H is the total height 
 of the rectangle. 
 
 * See Merriman's " Hydraulics." 
 
22 HIGH MASONRY DAM DESIGN 
 
 The point on the submerged surface at which this 
 resultant pressure acts may be determined by assuming 
 for an axis the horizontal line in which the surface of 
 the water cuts the plane of the back of the dam, taking 
 the moment of inertia of the surface about this axis, and 
 dividing the result by the static moment of the surface 
 with reference to the same axis. Applying this to the 
 strip referred to above, there will result, 
 
 bH 3 
 1= , the moment of inertia of rectangular strip, 
 
 and 
 
 12 
 
 which is the moment of inertia with regard to the as- 
 sumed axis. 
 
 H bH 2 
 
 is the static moment of the surface about the same axis, 
 hence, 
 
 /! 2H 
 
 is the distance of the center of pressure from the surface 
 of the water. 
 
 In the investigation of the distribution of pressure in 
 a masonry joint subjected to external forces, the material 
 is assumed to be rigid, though in reality it is to a certain 
 
HIGH MASONRY DAM DESIGN 23 
 
 > 
 
 degree elastic. This elasticity gives the distribution of 
 stress an indeterminate law, so that neither the direction 
 nor the intensity is actually known at any point. It is 
 certain, however, that the intensity must be zero at the 
 edges, although it may increase with great rapidity to 
 higher values very near the limits of the joint. Investi- 
 gations have been made within the past few years to 
 obtain more exact information as to this distribution of 
 stress, but so far the results are not completely satisfac- 
 tory. Reference will be made to this matter in Chapter 
 VIII. 
 
 Inasmuch as the exact law of stress variation is not 
 known, one of uniform variation of normal stress has 
 been assumed in all practical treatments of masonry 
 joints. 
 
 Fig. 2 represents the simplest case, in which the 
 pressure is assumed to be uniformly distributed over the 
 joint a 6, with the constant intensity p\ it might be taken 
 as representing any horizontal joint with a superimposed 
 load acting at its center. 
 
 To express this condition of uniform stress algebra- 
 ically, / may be assumed to be the length of the joint 
 from a to b, while the breadth, perpendicular to the plane 
 of the paper, is taken as unity. The area of the joint 
 will then be /, whence, 
 
 W=pl, ....... (5) 
 
 or 
 
 W 
 
 which is the formula for a condition of uniform intensity of 
 stress over the entire joint. 
 
A W| I 
 
 U it- -->{ 
 
 m///////////////////////////W//////////////////////////tffo> 
 
 24 HIGH MASONRY DAM DESIGN 
 
 It may be observed here that this pressure is uniform 
 only because the total load represented by W, acts at 
 the center of the joint, and that 
 when the point of application is 
 changed to some other position, a 
 there will be an increased stress 
 in that direction toward which 
 the load has been moved, and 
 a corresponding decrease in the 
 opposite direction. 
 
 It will be necessary therefore, to consider this varia- 
 tion of pressure in eccentrically loaded joints and also 
 the manner in which the eccentricity in the case of a dam 
 is produced. 
 
 If a b be any plane, horizontal joint in the dam at the 
 distance H below the surface, OY the water surface, and 
 <j> the angle that the back makes with the vertical, then 
 the total pressure on the back acting at a point one- third 
 the distance up from the joint, will be 
 
 sec 
 
 Combining this force with the weight of masonry W 
 above the joint acting through the center of gravity of the 
 section, the resultant R will intersect it at some point as 
 e, on a b, other than the center of figure, called the center 
 of resistance, and it is evident that with a variation of 
 F' and W it may occupy any position along the joint. 
 
 Fig. 3, showing only the vertical component, exhibits 
 such a case, where compression exists over the entire joint 
 
wi 
 
 L -u--- .>) 
 
 'A! 
 
 HIGH MASONRY DAM DESIGN 25 
 
 as in Fig. 2, but where the -center of pressure is not at 
 the center of figure 
 
 If the intensity of pressure 
 at b may be represented by 
 the vertical line p, and the in- 
 tensity of pressure at a by 
 the line p', then, since by the 
 assumption the pressure varies 
 uniformly over the entire joint, 
 the vertical, p", at any point, included between the 
 horizontal a b and the line joining the extremities of p 
 and p f will indicate the intensity of pressure at that 
 point, while the area of the trapezoid will represent the 
 total pressure on the joint. 
 
 The former may be expressed algebraically thus: 
 
 FIG. 3. 
 
 and the latter by, 
 
 I, 
 
 (7) 
 
 (8) 
 
 The determination of the maximum and minimum 
 pressure p and p f may be made as follows : 
 
 Since the static moment of the rectangle p I about a 
 point J-/ from p' is the same as the static moment of the 
 trapezoid about the same point, because the moment of 
 the triangle p p', I about that point is zero, that being 
 the center of gravity of the triangle, there will result by 
 taking moments 
 
 (9) 
 
26 HIGH MASONRY DAM DESIGN 
 
 whence, 
 
 aW 
 
 which is an expression for the intensity of pressure at the 
 point b, on the joint a b. To solve for the value of p' 9 
 the intensity of the pressure at the point a, in a similar 
 manner we may take moments about a point JZ from b, 
 whence, 
 
 or, 
 
 . , . . (12) 
 
 When p f becomes zero, the trapezoid reduces to a tri- 
 angle as shown in Fig. 4, with its center of gravity at a 
 distance from b equal to f /, and, since the center of pres- 
 sure of W must lie vertically above the center of gravity 
 of the triangle graphically representing the variation of 
 pressure over the joint, we shall have, p' = o> = /, and 
 Eq. (9) reducing to 
 
 Wl = pP. 
 whence, 
 
 2_W/ 
 
 That is to say, the maximum pressure p for this condition 
 is twice the value as obtained from Eq. (6), 
 
HIGH MASONRY DAM DESIGN 
 
 27 
 
 In Fig. 5 is represented & case in which tension exists 
 over a portion of the joint, p' is here negative. 
 
 Wl 
 
 FIG, 4. 
 
 FIG. 5. 
 
 Although both masonry and the best hydraulic cement 
 mortar have considerable tensile strength, running up to 
 several hundred pounds per square inch in tests, the 
 latter, together with the continued adhesion of the mortar 
 to the aggregate in concrete, when used, is of uncertain 
 value in this connection. The tensile strength is therefore 
 always neglected in considering the stability of masonry 
 dams or other similar structures, and is an omission which 
 is the more justifiable since it leads to an error on the 
 side of safety. 
 
 In the case represented by Fig. 5, the triangle, whose 
 base is 3*4, and altitude p, is therefore alone considered, 
 and by taking moments about 6, there will result, 
 
 whence, 
 
 Wu = 3M u ,. 
 
 2W 
 
 (15) 
 
 (16) 
 
 If it is desirable to know what the tension in the joint 
 
28 HIGH MASONRY DAM DESIGN 
 
 is, it may be determined from Eq. (12"*. As ^< i.o, the 
 
 resulting value is negative, thus denoting a tension by 
 that equation. 
 
 The pressures at a and b may also be determined as 
 follows: Decomposing the resultant acting on any joint 
 into its vertical and horizontal components, V will repre- 
 sent the total normal or vertical pressure, equal to W, 
 the weight of masonry above the joint, plus the vertical 
 component of the thrust from the water. The horizontal 
 component of the resultant is disregarded, as its effect 
 upon the joint is more or less indeterminate, and since 
 too, it is assumed to be neutralized by the friction acting 
 in the joint. 
 
 The vertical component V, acting through the point of 
 application of the resultant R in the joint, is therefore 
 the factor producing the difference in pressure between 
 a and b, or the uniformly varying stress. 
 
 Assume that at the center of the joint, which is not 
 necessarily vertically below the center of gravity of the 
 mass above, two forces equal and opposite to each other, 
 and of the same value V, are applied normal to the joint. 
 The effect of each is to neutralize the other, but if we 
 consider, apart from the other forces, the one acting down- 
 ward, since it is applied at the center of figure it will 
 
 V 
 produce a uniform stress p over the joint equal to y. 
 
 The two remaining and equal forces V and V, one 
 acting downward at the point of application of R, and the 
 other upward at the center, form a couple whose lever 
 arm is v, and the moment of which is therefore 
 
HIGH MASONRY DAM DESIGN 29 
 
 This moment produces a ufliformly varying stress over 
 the joint, increasing the intensity at b and decreasing it 
 at a by an equal amount. 
 
 To determine its value we have but to consider the fol- 
 lowing : 
 
 M = Vv. ...... (17) 
 
 the moment caused by the couple and producing the 
 varying stress. Also, 
 
 (18) 
 
 where k is the intensity of stress at the maximum dis- 
 tance from the neutral axis; 7, the moment of inertia 
 of the section about such an axis; and d\ the normal dis- 
 tance from the neutral axis to that point where k exists. 
 Since the neutral axis passes through the center of 
 figure of the joint, the value of di is half the length of the 
 joint, while /, the moment of inertia, equals T V 3 , if we 
 consider a horizontal section in the plane of the joint a b 
 extending back from the plane of the paper one unit's 
 distance. Hence, 
 
 ... kl fc 3 W 
 
 or, 
 
 6 Vv 
 
 Here k represents the stress that must be added to the 
 uniform stress to find the intensity of pressure at the 
 
30 HIGH MASONRY DAM DESIGN. 
 
 y 
 toe b and the amount which must be subtracted from -y 
 
 to arrive at the intensity at the heel a. It is expressed in 
 pounds per square inch, but if the distances are measured 
 in feet and the forces in pounds, k will be designated in 
 pounds per square foot. 
 
 While it is customary to consider only the normal 
 component of the resultant pressure acting in a hori- 
 zontal joint and to assume it to vary uniformly, this is 
 probably correct only for horizontal joints in rectangular 
 walls vertically loaded and not subjected to lateral pres- 
 sures. It will be shown later that the maximum stresses 
 exist at or near the down-stream face, and act in a direction 
 parallel to and on planes normal to that face. The fact 
 also that acute edges do not crack off in the inclined faces 
 of dams is in itself a partial confirmation' of the statement. 
 
 Under these circumstances then, the maximum normal 
 pressure in a horizontal joint must be much less than the 
 actual maximum pressure in the dam, and it has been 
 assumed to bear the ratio to the latter of about 9 to 13. 
 
I 
 I 
 
 CHAPTER III PART I 
 DEVELOPMENT OF FORMULAE FOR DESIGN 
 
 Six series of formulae, designated by the letters A, B, 
 C, D, E, and F, will now be presented, in each of which 
 a given set of conditions with respect to the external 
 forces will be involved; but as the method of procedure 
 is practically the same for all cases, only series A will 
 be developed here. 
 
 The following nomenclature will be employed: 
 
 L = the width of the top of the dam cross-section; 
 / = length of a horizontal joint of masonry, to 
 
 be determined; 
 1 = known length of the joint next above joint of 
 
 length /; 
 h = depth of a course of masonry (vertical distance 
 
 between / and /) ; 
 P=line of pressure, reservoir full; 
 P f = line of pressure, reservoir empty ; 
 u = distance from front edge of the joint / to the 
 
 point of intersection of P with the joint /, 
 
 measured parallel to joint /; 
 y = distance from back edge of the joint / to the 
 
 point of intersection of P' with the joint /. 
 
 measured parallel to joint /; 
 y = distance from back edge of the joint / to 
 
 the point of intersection of P f with the 
 
 joint / , measured parallel to joint 1 Q \ 
 
 31 
 
32 HIGH MASONRY DAM DESIGN 
 
 v = distance between P and P' at the joint /, 
 
 measured parallel to joint /; 
 ? = weight in pounds of a cubic foot of water 
 
 (62.5); 
 f = weight in pounds of a cubic foot of mud 
 
 (75-90); 
 J = ratio of unit weight of masonry to unit weight 
 
 of water (often assumed as J) ; 
 J^- = w T eight in pounds of a cubic foot of masonry; 
 //=head of water on joint / (vertical distance of 
 
 joint / below water surface) ; 
 
 H' = depth of earth back fill over joint / on front; 
 HI =head of water on joint / when ice acts at sur- 
 face of water; 
 H Hi=rise of water level, due to flood, wave, etc., 
 
 above normal level for full reservoir; 
 /*i=head of water above mud level (liquid mud 
 
 of weight f) ; 
 
 /J 2 =head of liquid mud on joint /, on back; 
 a = vertical distance from the top of the dam to 
 
 the surface of water (flood) ; 
 
 ai = vertical distance from the top of the dam to 
 the surface of water when ice is considered 
 (ai generally exceeds a) ; 
 6 = vertical distance from water surface to top 
 
 of dam when dam is overtopped; 
 c = ratio of upward thrust intensity, due to 
 hydrostatic head H (or HI, or hi+h 2 ), as- 
 sumed to act at heel of joint / (usually 
 assumed as 
 
HIGH MASONRY DAM DESIGN 33 
 
 > 
 
 Tf= horizontal ice thrust at water surface in pounds 
 
 (47,000); 
 
 (The value here given, for example, was used 
 in studies for design. Our present lack 
 of exact data in regard to ice pressures 
 prevents more than a speculation from 
 being made as to a definite value to be 
 assigned in any case); 
 
 Df = horizontal dynamic thrust of water in pounds;* 
 
 Ef = thrust of earth back fill in pounds (on front); 
 
 Wvf = vertical pressure on inclined upstream face 
 
 above joint I, in pounds; 
 AQ = total area of cross-section of dam above 
 
 joint / ; 
 A = total area of cross-section of dam above 
 
 joint /. 
 
 / = batter of upstream face for vertical distance h ; 
 5 = distance of line of action of W v f from upstream 
 edge of joint /, measured parallel to joint /; 
 d = angle that Ey makes with horizontal; 
 a. = angle of slope of downstream face of dam 
 
 with horizontal; 
 
 /? = angle R makes with the vertical; 
 p = maximum allowable pressure intensity at toe 
 
 (in pounds per square foot) ; 
 
 q = maximum allowable pressure intensity at heel 
 (in pounds per square foot) (p is assumed 
 less than q) p and q may be used to signify 
 the calculated, existent pressure intensities 
 
 * Determined, as in the case of the overfall dam, by the probable velocity 
 of flow against the dam. 
 
34 HIGH MASONRY DAM DESIGN 
 
 corresponding to P and P' respectively, for 
 
 the joint /. 
 /=the coefficient of friction for masonry on 
 
 masonry (usually 0.6 to 0.75); 
 5 = the shearing resistance of the masonry per 
 
 square unit; 
 
 rH 2 
 
 F = ~ =the horizontal static thrust of the water in 
 2 
 
 pounds ; 
 
 the moment of F about any point in the 
 
 joint /; 
 
 total weight, in pounds, of masonry 
 resting on the joint /; 
 
 total weight, in pounds, of masonry 
 resting on the joint / ; 
 R = the resultant of F and W\ 
 R r = the resultant of the reactions ; 
 
 = up ward thrust of water on base /. 
 
 In the figures, hydrostatic pressures are indicated by 
 triangular and trapezoidal areas included within dotted 
 lines, while ice pressure is shown to contrast HI with H. 
 
 As before, if a unit length of one foot of dam be 
 considered, the letters T, D, E, W 9 , A, A , and H 2 will 
 signify volumes. 
 
 It will be observed that, where possible, the several 
 equations have been cleared of the term Ar, thereby 
 simplifying actual calculations. 
 
 In the above table c, in a manner, may be considered 
 to provide for an assumption of a certain proportion of the 
 
HIGH MASONRY DAM DESIGN 35 
 
 ..; 
 
 joint's area being subjected to upward water pressure ; and 
 the distribution, as evidenced by cHlf/2, varying from 
 a maximum intensity at the heel to zero intensity at the 
 toe, is assumed in view of the fact that the tendency to 
 open the joint would begin at the heel while a zero intensity 
 of upward pressure at the toe would presuppose an opening 
 with consequent flow at that point. As the dam would 
 then be failing in its chief function, i.e., to retain water, 
 this flow is not considered to exceed a slight seepage. 
 
 In general four ways are recognized in which a masonry 
 dam may fail : 
 
 1. By overturning about the edge of any joint, due 
 to the line of action of the resultant passing beyond the 
 limits of stability. 
 
 2. By the crushing of the masonry or foundation 
 because of excessive pressure. 
 
 3. By the shearing or sliding on the foundation or any 
 joint, due to the horizontal thrust exceeding the shearing 
 and frictional stability of the material. 
 
 4. By the rupture of any joint due to tension in it. 
 An unsatisfactory foundation might also be mentioned 
 
 as possibly leading to failure, and in view of this, the 
 footing upon which the dam rests should always be most 
 carefully scrutinized. 
 
 To preclude failure from any of the above mentioned 
 causes, it is the practice to design the cross-section of 
 the dam with the following conditions imposed : 
 
 1. The lines of pressure, both for the reservoir full 
 and empty, must not pass outside the middle third of any 
 horizontal joint. 
 
 2. The maximum normal working pressure on any 
 
36 HIGH MASONRY DAM DESIGN 
 
 horizontal joint must never exceed certain prescribed 
 limits, either in the masonry itself or in the founda- 
 tion. 
 
 3. The coefficient of friction in any plane horizontal 
 joint, or between the dam and its foundation, must not be 
 less than the tangent of the angle which the resultant 
 makes with a vertical. 
 
 As may be seen by referring to the figures showing 
 the distribution of pressure on a joint, when the resultant 
 lies within the middle third, tension can exist in no part 
 of it, nor can the safety factor be less than two, if we 
 neglect to consider the upward pressure of water perco- 
 lating through any of the joints or beneath the dam. 
 
 U y ><- v >r< u 
 
 FIG. 6. 
 
 To illustrate the conditions that exist and to derive 
 the value of the safety factor when the resultant cuts the 
 joint at the extremity of the middle third, we may take 
 the case as shown in Fig. 6. Resolving R into its horizontal 
 and vertical components, and taking moments about the 
 center of resistance e, the following equation is obtained: 
 
 where F is the horizontal component of the thrust from 
 the water behind the dam, acting at a point \H above 
 
HIGH MASONRY DAM DESIGN 37 
 
 i 
 
 the plane of the joint, whij W is the vertical compo- 
 nent of the resultant, and as such, includes not only 
 the weight of the masonry, but the vertical component 
 of the thrust from the water as well, provided the 
 latter is considered as acting normal to the back of the 
 dam. 
 
 For the dam to be on the point of rotating about 6, 
 the downstream edge of the joint, it is obvious that the 
 resultant R must pass through that point. Under these 
 
 TT 
 
 circumstances, since the lever arm of F is still , and the 
 
 o 
 
 lever arm of W has been increased to twice its former 
 value or 2 ( j , for the above equation to still hold, F must 
 
 also be increased to twice its former value. This would 
 indicate that when R acts through the point e, the value 
 of H is only one-half as great as is necessary to produce 
 overturning; or, in other words, that the factor of safety 
 
 is two as indicated by the ratio of . It should be 
 
 observed however, that the material near the edge of the 
 joint will crush some time before the resultant has reached 
 it, and that therefore the factor of safety against overturning 
 with R at the limit of the middle third is something less 
 than two. 
 
 When, however, the upward pressure of water acting 
 over the joint due to percolation is taken into considera- 
 tion, the factor of safety will be somewhat modified, as the 
 following demonstration will make clear. 
 
 By referring to Fig. 7, it will be seen that, for example, 
 
 the horizontal water pressure on the back, - , the uplift, 
 
38 
 
 HIGH MASONRY DAM DESIGN 
 
 , and the weight of the masonry, W, constitute the 
 
 forces, with the reaction (not shown) considered as acting 
 on the dam. 
 
 The forces tending to overturn the dam about the 
 
 toe at b are - and , the resultant of which is denoted 
 2 2 
 
 in magnitude and direction by the line, 0, at the normal 
 distance, r, from b. The force resisting this tendency is 
 W, the line of action of which is normally distant (u+v) 
 from b. If nm denotes the force, 0, in magnitude and 
 direction, and n d y the force, W, then n g will represent, 
 in magnitude and direction, the force, R, or the final re- 
 sultant of all the forces, which is opposed by the reaction 
 
HIGH MASONRY DAM DESIGN 39 
 
 (for equilibrium to be assumed) at the point, e, distant 
 u from b. 
 
 The resisting moment about &, then, is W(u+v)=M Q 
 while the overturning moment about the same point is 
 
 Whence, for the ratio of " resisting moment to overturning 
 moment " there may be written: 
 
 MQ MQ ( . 
 
 ' 
 
 and for the ratio of the " resultant moment of the ver- 
 tical components " to the " resultant moment of the 
 horizontal components " of the forces there follows: 
 
 These two expressions for the " factor " evidently 
 become equal to each other only when Mi = o, or when 
 uplift is ignored; also, when the factor of safety is equal 
 to unity (when M Q M2=Mi), or at the point of over- 
 
 AM I 7t 
 
 turning. Therefore, the usual expression, , will not 
 
 v 
 
 be the value for the " factor of safety " against over- 
 turning, according to Eq. (210). 
 
 To consider the foregoing discussion for the purpose of 
 developing a graphic treatment for determining the value 
 in either case, it must be remembered that, for the dam to 
 
40 HIGH MASONRY DAM DESIGN 
 
 be on the point of overturning, the ratio of the two moments, 
 M Q and M' of Eq. (2ia), must equal unity or M = M\ -\-Mi, 
 and the line of action, R, in Fig. 7, must pass through b. 
 Inasmuch as W is constant, one or all of the other forces 
 may at this stage be considered variable, in order to bring 
 about the above supposititious condition. According to 
 Eq. (2ia), the distance, r, is constant, and the water pressure 
 on the back and the uplift, therefore, are supposed to be 
 proportionately increased to fulfill this condition, of bring- 
 ing the line of action, R, through b. This seems reasonable 
 from the fact that the horizontal water thrust cannot be 
 considered to increase without a corresponding increase 
 in the uplift. Therefore, the condition necessary to bring, 
 the resultant, R, through b instead of through e, where 
 it actually falls, is that 0=dg be increased to di, in Fig. 7. 
 If be be drawn through b, parallel to (and, therefore, 
 to dg), the ratio sought follows from the similarity of the 
 triangles, nid and nbc, or, the factor of safety, with respect 
 to resisting moment and overturning moment, is equal to 
 
 ,.' di cb 
 
 the ratio, = -p. 
 
 dg cj 
 
 The foregoing conception, Eq. (216), of the " factor of 
 safety " tacitly assumes that only the horizontal thrust 
 of the water is instrumental in moving the center of pres- 
 sure from e to b, and that the uplift merely lessens the 
 resisting moment. 
 
 As the overturning force to be increased is therefore 
 horizontal, and as the length of the line parallel to the 
 overturning resultant and comprehended between the 
 point, b, and the line of action of the resisting force is 
 divided by its segment (comprehended between the actual 
 
HIGH MASONRY DAM DESIGN 41 
 
 resultant and the same line 'of action of the resisting force) 
 
 < 
 to get the ratio, or factor against overturning, it at once 
 
 follows that the division according to Eq. (216) would be 
 . Eq. (210) seems preferable, or the " factor of safety" 
 
 = c , as in Fig. 7. The "factor of safety," however, is 
 
 C J 
 of doubtful value, due to the certain impossibility of the 
 
 structure's rotation about the point 6; but it may be a 
 useful quantity for comparison at times. The expression 
 (210) may give values less than (216) by as much as one- 
 third, in some cases. 
 
 As was stated previously, the frictional and shearing 
 resistance of a joint is assumed to withstand the tendency 
 of the horizontal thrust to slide the upper portion over 
 the lower, so that it is quite customary, even though it 
 should be investigated, to neglect it. 
 
 For equilibrium in this regard, 
 
 (22) 
 
 where F is the horizontal component of the water's thrust, / 
 the coefficient of friction, usually taken between 0.6 and 
 0.75 for masonry, and 5 is the shearing resistance per unit 
 of area. 
 
 In spite of the fact that 5 has an appreciable value, 
 and particularly so for monolithic masses of " cyclopean 
 masonry," the value is practically indeterminate, and 
 consequently usually ignored. Numerous attempts have 
 been made however, to write expressions for it, the most 
 rational of which depends upon the trapezoidal law of 
 the distribution of normal stress ; but this too is unsatis- 
 
42 HIGH MASONRY DAM DESIGN 
 
 factory from a practical standpoint.* We shall neglect 
 5, therefore, in the previous equation, whence, 
 
 . . , . ...- (23) 
 which gives at the limit, 
 
 p 
 
 (24) 
 
 In every design the imposed conditions for equilibrium 
 result in a cross-section in which the back has very much 
 less of a batter than the front. It may be shown also,* 
 that, as the shear along either face is zero, the greatest 
 intensity of stress will act in a direction parallel to the 
 face at, and near, the edge. Since the horizontal compo- 
 nent of the pressure is ignored, this implies that the 
 greatest vertical, or normal working intensity of pres- 
 sure must be less at the downstream face where the 
 inclination is greater than at the heel, in order that the 
 components parallel to the respective faces shall be ap- 
 proximately equal. This is accomplished by using a smaller 
 vertical normal working stress at the toe than at the heel. 
 
 As the up-stream face of a masonry dam is vertical for 
 a considerable distance from the top, and then becomes 
 only slightly inclined to it, it is customary to consider the 
 thrust from the water as acting horizontally. This is the 
 more justifiable since the vertical component of the water 
 resting upon the up-stream face of the dam causes an 
 overturning moment about the center of resistance, op- 
 posite in direction to that induced by the horizontal 
 thrust, and hence is an error on the side of safety. 
 * See Chapter VIII. 
 
HIGH MASONRY DAM DESIGN 43 
 
 > 
 
 It must be evident fron* the equation of pressure, 
 p = -pah, that where this alone governs the resulting theo- 
 retical cross-section, it will be triangular in form with the 
 apex at the surface of the water; but where it is intended 
 there shall be no flow over the crest of the dam, it is 
 customary to carry the masonry some distance above 
 the elevation of the water in the reservoir, not only to 
 allow for fluctuations, but because of economic condi- 
 tions or to provide for a foot or carriage way. The super- 
 elevation and the width of top are therefore arbitrarily 
 assumed and should be taken at about T V the height 
 of the dam, with a minimum width of 5 feet and a maxi- 
 mum superelevation of 20 feet.* 
 
 As no equation can be written simultaneously ex- 
 pressing the three conditions of stability, i.e., that the 
 resultant lie within the middle third, that the maximum 
 pressures shall not exceed certain limits, and that the 
 horizontal components shall not cause sliding, it be- 
 comes necessary to determine the length of joints (usually 
 taken vertically 10 feet apart for a depth of about 100 
 feet and increasing to 20 or 30 feet below), by the aid of 
 that equation involving the limiting conditions which are 
 known to apply, in order that the cross-section be a min- 
 imum, and then to test the joint, if necessary, by the 
 other two. Generally speaking the third condition will 
 be found to hold if the joint has been designed in accord- 
 ance with the other two. 
 
 *Mr. William P. Creager, in Proc. Am. Soc. of C.E., for Nov., 1915, 
 " The Economical Top Width of Non-overflow Dams," shows this width to 
 lie between 10% and 17% of the height, according to design assumptions and 
 concludes that exceptionally wide tops may be used, there being but slight 
 economy in adopting narrow tops. 
 
44 HIGH MASONRY DAM DESIGN 
 
 Considering Fig. 6, in which / is the length of joint, 
 it is seen to be divided . into three parts, u, v, and y, 
 and from this what may be called the fundamental equa- 
 tion of the entire design can be written. 
 
 (25) 
 
 If M represents the overturning moment about e, then 
 we have that at the limit of the middle third, 
 
 M = F- =Wv, . . , . . (26) 
 
 or, 
 
 M . 
 
 v= w . . . ~; . - . . (27) 
 
 As the analysis will result in a cross-section polygonal 
 in outline, composed of trapezoids with bases / and 1 Q 
 and altitudes h, we may write a general equation, 
 
 . . . (28) 
 or, 
 
 whence, 
 \. / ' A-A +(^)k, ..... (29) 
 
 and since; 
 
 W f l + l \j 
 
 T r = A +() h > 
 then, 
 
 M_ 
 
 f x 
 . , . . . (30) 
 
HIGH MASONRY DAM DESIGN 45 
 
 which value of v, if substitute^ in Eq. (25) gives, 
 
 M_ 
 Ar 
 
 The above Eq. (31) is a modification of Eq. (25) and, 
 when proper values have been assigned to u and y, depend- 
 ing upon the existing conditions, is used throughout the 
 entire design in the determination of the length of joints. 
 
 In the upper rectangular portion of the dam, where 
 there is an excess of material above that required by the 
 static pressure of the water, it will be found unnecessary 
 to consider failure from crushing, as the maximum normal 
 pressures are well below the allowed working pressure, 
 and consequently the depth at which the section ceases 
 to be rectangular will be fixed by the fact that the re- 
 sultant may not pass outside the middle third. The 
 algebraic expression for this condition is, 
 
 u ^ J / for reservoir full, . . . . (32) 
 y ^ J / for reservoir empty. . . . (33) 
 
 Below this rectangular portion, trapezoidal sections 
 will be found. At the base of the rectangle, l = l(\=L, 
 
 u = , and, since the center of gravity of the figure is 
 o 
 
 vertically above the center of the joint, y = . 
 
 If we wish to determine the depth to which the rectan- 
 gular portion extends, we may do so by the use of Eq. 
 
46 HIGH MASONRY DAM DESIGN 
 
 (31), which, as shown, must involve the condition that 
 the resultant shall just touch the limit of the middle 
 
 third, i.e., u= . Substituting in Eq. (31), u = , y = , 
 and remembering that A = o 
 
 _L 64 L 
 - 
 
 whence, by dividing by L, 
 
 or, solving for H, _ 
 
 ..... -(34) 
 
 If H=h then a=0, and Eq. (34) reduces to, 
 
 . ..... (35) 
 
 At this depth the rectangle ceases, the sections become 
 trapezoidal, the back face is 
 still vertical but the front 
 face h f , is inclined in order 
 to increase the length of the 
 successive joints and thus 
 maintain the resultant for 
 the reservoir full at the down- 
 stream limit of the middle 
 third. For a considerable distance below the rectangular 
 
 section therefore, Eq. (31) will be used with u = -- to de- 
 
 6 
 
HIGH MASONRY DAM DESIGN 47 
 
 t ermine the length of Joint, afod the back face will remain 
 vertical, but for each new joint the resultant for the reser- 
 voir empty will approach nearer and nearer to the limits 
 
 of the middle third, until finally y=. 
 
 O 
 
 It is therefore expedient to compute the value of y 
 under these conditions to learn exactly at what vertical 
 
 depth or joint this value of y first equals -. 
 
 O 
 
 To do this, moments are taken about the vertical 
 face, for both A and the trapezoid, the latter being found 
 by dividing the trapezoid into a rectangle and a triangle; 
 its value is, 
 
 and hence, 
 
 Substituting the value of A from Eq. (29) in the above, 
 and solving for y there results, 
 
 (36) 
 
 .4, 
 
 This gives a value of y to be substituted in Eq. (31) 
 while the value of u = J/, which has been maintained 
 
48 HIGH MASONRY DAM DESIGN 
 
 since leaving the bottom of the rectangular section, is 
 substituted also. There then results by reduction, 
 
 , (37) 
 
 which is the equation used in the determination of the 
 length of joint from the foot of the rectangular section 
 down to that joint where Eq. (36) first gives a value of 
 
 y=-. At this point the back face must be made to slope, 
 
 o 
 
 while w=y = \l is substituted in Eq. (31) to obtain the 
 following: 
 
 which will determine the length of the joints. 
 
 The second condition will be a factor from here on, 
 for below this section at some point, the intensities of the 
 pressures at the toe will gradually approach and finally 
 equal the allowable limit ^ and the length of the joint 
 will depend primarily upon this. It is therefore necessary, 
 after each application of Eq. (38) to see if the limiting 
 pressure p at the toe, which is smaller than q, at the heel, 
 has been reached. Its value is derived from the equation 
 
 2W zAAr . 
 
 p = j- = 7 -, and when the limiting value of p has been 
 
 realized the value of u thereafter must be derived from, 
 
 . . 
 
 (39) 
 
HIGH MASONRY DAM DESIGN 49 
 
 j& 
 
 in which u, is seen to be dependent upon the normal working 
 pressure p at the toe. (Eq. (39) follows directly from 
 Eq. (9)). 
 
 There is some distance below this joint, however, 
 where y still remains equal to J/, while the value of u 
 is being determined from the above Eq. (39). Under 
 these circumstances, / will be found from the following 
 after substituting the values of y = \l, it from Eq. (39) 
 and A from Eq. (29), all in Eq. (31). 
 
 (40) 
 
 This equation will be used until a joint has been reached 
 where the application of q = j shows its value to be 
 
 equal to or greater than that prescribed for q. Here y 
 will be determined by 
 
 2l l 2 
 
 in which it is seen to depend on q. 
 
 When this point has been reached, u will take its 
 
 value from u= -- /. , and y from the equation 
 
 ql 2 
 
 ~' w ^ c ^ must De substituted in Eq. (31) to 
 
 determine /. This will give after reduction, 
 
50 
 
 HIGH MASONRY DAM DESIGN 
 
 All joints below this point will be found by this last 
 equation. 
 
 Summing up, we may say that Eqs. (34), (37), (38), (40), 
 and (42), are the five equations to be used in determining 
 the length of joints from the top down. Strictly speaking 
 Eq. (34) gives the depth at which the rectangular portion 
 ceases, while Eq. (37) gives the length of joints from the 
 base of the rectangle down to where y = \l\ Eq. (38) the 
 length of joints from the point where y = \l to where p 
 reaches its limiting value; Eq. (40) the length of joints 
 from the point where p equals its limiting value to where 
 q equals its limiting value and Eq. (42) gives the length 
 of all joints below. 
 
 Eqs. (34) and (37) involve the value of y, which is 
 obtained with respect to the vertical back, but when 
 that face begins to slope it is necessary to determine it 
 with regard to the back edge of the joint in question. 
 
 |W a 
 
 FIG. 9. 
 
 In Fig. 9, mn represents the back face of the dam and 
 t is the batter to be determined by taking static moments of 
 A and A Q about the back edge, m, of the joint. 
 
 The trapezoid of the figure is composed of the triangles 
 ht/2 and (l-l -t)h/2 and the rectangle hl . 
 
HIGH MASONRY DAM DESIGN 51 
 
 t 
 
 By taking moments about the edge w, 
 
 For Eqs. (38) and (40) the value of y must be, as be- 
 fore, taken equal to J/, while A has the usual value of 
 A + (l + lo)h/2. Substituting these in Eq. (43) and re- 
 ducing: 
 
 (44) 
 
 For the joints to which Eq. (42) applies the value of y is 
 to be taken from Eq. (41) as was done before. In this 
 case: 
 
 By substituting this value in the first member of Eq. (43) 
 and reducing: 
 
 A ( 4 l-6y ) 
 
 After the value of / is found by the use of Eqs. (38), 
 (40) or (42), / can at once be determined for the same 
 joint by either Eq. (44) or Eq. (45). 
 
 In this manner an entire theoretical cross-section can 
 be determined. It will be noticed that the location of the 
 center of pressure in the middle third of the joint is the 
 governing condition in the upper part of the dam, while 
 the lower portion is fixed by the limiting pressures p and q. 
 
52 HIGH MASONRY DAM DESIGN 
 
 The difficulties preventing the forming of a simple working 
 equation for the entire cross-section arise from the fact 
 that the governing conditions are not introduced simul- 
 taneously nor in the same joint. 
 
 By taking h of the proper value, a polygonal cross- 
 section may be determined by the preceding formulae. 
 This cross-section can be then modified by drawing what 
 may be called "mean" lines, straight, broken or curved 
 along the theoretical faces so as to adapt the latter to a 
 practical arrangement and treatment of the joints and 
 facing blocks, which may be of cut stone or concrete. 
 
 The conditions which have governed the analysis are 
 essentially those of Rankine, i.e., the center of pressure 
 has in all cases been kept within the middle third of the 
 joint and the greatest intensity of pressure, either at the 
 front face or back, has not been allowed to exceed the 
 limit p or q. 
 
CHAPTER HIPART II 
 
 FORMULA FOR DESIGN 
 Series A, B, C, D, E, and F. 
 
 As noted earlier six separate series of formulae for 
 design have been derived and they will be here set 
 forth in suitable form for easy reference and use. As they 
 have been developed by the method just outlined it is 
 unnecessary to follow out the derivation of each series, 
 although there exist some detailed differences in the treat- 
 ment of each. These details however, would become 
 evident to anyone following the deductions throughout. 
 
 Various conditions of " loading," with approved as- 
 sumptions, such as pressure due to expanding ice at the 
 water surface, upward water pressure on the base, etc., 
 referred to in the table of nomenclature previously given, 
 have been introduced and are specifically stated for each 
 case. 
 
 It will be recalled that Eq. (31) is the fundamental 
 expression ' for finding the length /, of any joint; and, as 
 the several conditions are introduced, that the " M " must 
 in each case signify the total overturning moment and 
 not merely the moment of the static water pressure on 
 the back. 
 
 The development of a cross-section, by any one of the 
 following series, may comprise five stages, each stage 
 
 53 
 
54 
 
 HIGH MASONRY DAM DESIGN 
 
 representing the introduction of a governing condition. 
 Hence, for each stage there obtains a main equation for 
 finding the length of joint I, each main equation being 
 supplemented by secondary equations for y, u, and t\ 
 p and q. 
 
 It may be necessary to employ more than one of the 
 series of equations in determining a cross-section. 
 
 Flood Lei 
 
 -*_/ 
 
 FIG. 10. 
 
 For ready reference, the five stages will be set forth 
 and depicted in order as follows: 
 
 Stage I. This stage, it will be remembered, extends 
 from the top of the dam to the joint where the front 
 face commences to batter. It is the rectangular section. 
 y>\L; u>%L (see Fig. 10). Ice pressure is purposely 
 omitted in Fig. 10 to prevent confusion of letters in 
 small space.) 
 
HIGH MASONRY DAM DESIGN 
 
 55 
 
 Stage II. This stage extends from the lower limit of 
 Stage I to the point where the back face commences to 
 batter. u = \l; y^\l (see Fig. u). 
 
 Stage III. This stage extends from the lower limit of 
 Stage II to the point where the intensity of pressure on 
 the toe has reached 
 the maximum allow- 
 able intensity. In this Flood Level 
 stage u = |/; y = \l (see 
 Fig. 12). 
 
 Stage IV. This 
 stage extends from the 
 lower limit of Stage 
 III to the point where 
 the pressure intensity 
 on the heel has reached 
 
 the maximum allow- > 
 
 / 
 
 able intensity. For / 
 
 l_ _ _J-_j- ^ 
 
 this stage u>\l\y = \l 
 
 (see Fig. 12). 
 
 Stage V. In this 
 stage, the limiting in- 
 tensities of pressure 
 at both toe and heel having been reached, y>\l\ u>^l. 
 
 This stage extends from the lower limit of Stage IV 
 downward. (See. Fig. 12). 
 
 The following secondary formulae, supplementary to 
 the main equations of all series, with substitutions as 
 noted, are arranged in order corresponding with the 
 preceding. 
 
 FIG. 12. 
 
56 
 
 Stage L 
 
 t = o 
 
 HIGH MASONRY DAM DESIGN 
 
 L 
 A r A 
 
 Stage II. 
 
 y 
 
 Stage III. 
 
 t = 
 
 With the condition of hydrostatic up- 
 ward pressure on the base obtaining,sub- 
 stitute the formulas in this column in 
 place of those correspond ing, as indicated. 
 
 I 2 A A rT \ / T~ 
 
 P-n, cHW2 v 
 
 \ ** 
 
 2JA 
 
HIGH MASONRY DAM DESIGN 
 
 57 
 
 Stage IV. 
 = /- 
 
 & 
 
 * = 
 
 P = 
 
 
 Stage V. 
 U--11- 
 
 (limiting 
 / \* I I intensity) 
 
 2 
 
 ql2 
 
 u = u 
 
 = /- 
 
 t = 
 
 2 A f-A / 
 / \ 
 
 ^\ (limiting 
 I ) intensity) 
 
 I 
 
 / 3^\ (limiting 
 \ I ) intensity) 
 
 --)(-? 
 
 If T enter the following formulae, H above becomes 
 HI. (See Figs, n and 12.) 
 
 The first column of formulas just given would apply, 
 with the condition of upward pressure on the base due 
 to hydrostatic head, if a proper value of u corresponding, be 
 
58 HIGH MASONRY DAM DESIGN 
 
 taken, that is if the excursion of the force AAf, resulting 
 from the effect of all other forces on AAf be considered, 
 rather than the effect of all the other forces on the re- 
 sultant vertical force. 
 
 It will be found expeditious to design a section, where 
 ice pressure at the level of full reservoir is to be considered 
 in connection with the water surface at some higher flood 
 level, first by series of formulas containing T (cf . Series B and 
 D) and then to investigate successive bases, or joints, thus 
 obtained (beginning, for a high masonry dam, usually 
 at a base, or joint, about 100 feet from the top of the dam) 
 with series of formulae lacking T, or the ice pressure con- 
 dition (cf. Series A and C). A base will ultimately be 
 obtained by these supplementary " Flood level " calcula- 
 tions greater than the base at its same elevation as pre- 
 viously determined by the " Ice Pressure " design. 
 
 Continuing with the design by means of the " Flood level " 
 formulae to the bottom of maximum height required will 
 determine the minimum cross-section area to meet the 
 conditions both of " Flood "and of " Ice." It should be 
 remarked in this connection that when a reservoir level 
 is rising due to flood conditions prevailing, it is evident 
 that ice formation cannot develop, or, in other words, the 
 two conditions cannot be coexistent, hence the difference 
 in designation of hydrostatic heads corresponding. (See 
 Figs, ii and 12.) 
 
HIGH MASONRY DAM DESIGN 59 
 
 SERIES A. 
 
 Conditions: Overturning moment due to horizontal 
 static water pressure on back of dam only. 
 
 Stage I. 
 
 Siape II. 
 
 12 + 
 
 Stage III. 
 
 Stage IV. 
 
 P 
 
 Stage V. 
 
 SERIES B. 
 
 Conditions: Overturning moment due to: 
 
 (a) Horizontal static water pressure on back and 
 
 (b) Ice pressure applied at distance (a\) from top, 
 
 Stage I. 
 
60 HIGH MASONRY DAM DESIGN 
 
 Stage II. 
 
 Stage HI. 
 
 Stage IV. 
 
 P = (Hi3 + 6rHi). 
 
 Stage V. 
 
 SERIES C. 
 
 Conditions: Overturning moment due to: 
 
 (a) Horizontal static water pressure on back. 
 
 (b) Upward water pressure on base. Pressure intensity 
 decreasing uniformly from cH-y at heel to zero intensity 
 at toe. 
 
 Stage I. 
 H = 
 
 Stage II. 
 
 Stage III. 
 cH 
 
HIGH MASONRY DAM DESIGN 61 
 
 Stage IV. 
 
 cH 
 
 p \ h 
 
 J which reduces to 
 
 P' 
 
 Stage V. 
 
 cH 
 
 ~ + ^o 5 which reduces to 
 
 SERIES D. 
 
 Conditions: Overturning moment due to: 
 
 (a) Horizontal static water pressure on back (head = H\). 
 
 (b) Ice pressure applied at distance (a\) from top. 
 
 (c) Upward water pressure on base. Pressure decreasing 
 uniformly from cH\f at heel to zero intensity at toe. 
 
 Stage I. 
 H l = 
 
 Stage II. 
 
 
62 HIGH MASONRY DAM DESIGN 
 
 Stage III. 
 
 Stage IV. 
 
 1 ){ 2 A \ 
 - I -7 + /o ) , 
 
 which reduces to 
 
 Stage 7. 
 
 2A , ,\ 
 -T + /O I, 
 
 which reduces to 
 
 In the preceding series of equations it will be observed 
 that the final expressions for / in stages IV and V are 
 very similar, and that the quantity c in equations (a) of 
 these stages disappears in equations (b). Equations (b) 
 of course, are to be used for purposes of calculation of 
 cross-sections. 
 
HIGH MASONRY DAM DESIGN 
 
 63 
 
 SERIES E. 
 
 Conditions: (See Fig. 13.) Ice pressure neglected in 
 Fig. 13. Overturning moment due to: 
 
 (a) Horizontal static water pressure on back (head = /*i). 
 
 (b) Ice pressure applied at distance (a\) from top. 
 
 (c) Upward water pressure on base; pressure intensity 
 decreasing uniformly from c(hi+h 2 )r at heel to zero in- 
 tensity at toe. 
 
 (d) Mud (liquid) pressure on back (head h 2 ) , commencing 
 at distance h 2 above joint in question. Weight of mud = f . 
 As before, if T be equated to zero, a\ becomes equal to 
 a, in the formulas. 
 
 FIG. 13. 
 Stage I. (hi, of known value, h 2 to be determined.) 
 
 feY , 
 
 Stage II. 
 
 
64 HIGH MASONRY DAM DESIGN 
 
 For trapezoidal section at top, make AQ = o and yo 
 and IQ=L in Stage II. This applies generally. 
 
 Stage III. 
 
 Stage IV. 
 
 Stage V. 
 
 From a study of the formulae thus far developed it 
 will be observed that by reducing certain conditions to 
 zero, with their corresponding quantities, the main equa- 
 tions of a given series reduce to those of a simpler series. 
 
 For instance- 
 
 In Series B make T (for ice pressure condition of load- 
 ing) equal to zero and H\=H and a\=a and the main 
 equations of that series reduce to Series A equations. 
 
 In Series C, by making c (for upward water pressure 
 condition) equal to zero in main equations of Stages I, II, 
 and III and also in equations (a) of Stages IV and V, the 
 equations of Series C reduce to those of Series A. 
 
 Likewise, by making the proper eliminations and sub- 
 stitutions, Series E will reduce to Series D, C, B or A. 
 
HIGH MASONRY DAM DESIGN 
 
 65 
 
 SERIES F. 
 
 This series consists of general formulae for a number 
 of imposed conditions of loading. For any given case, 
 the terms or factors expressing those conditions not ap- 
 pertaining must be eliminated by equating them to zero. 
 (See Fig. 14.) 
 
 FIG. 14. 
 
 Conditions for General Formula. 
 
 Overturning moment due to: 
 
 (a) Horizontal static water pressure on back (head = /*i). 
 
 (b) Upward water pressure on base; pressure intensity 
 decreasing uniformly from cHf or c(/h-f^2)r> a ^ nee ^ to 
 zero intensity at toe. 
 
 (c) Mud (liquid) pressure on back (head h 2 ) as before. 
 
 (d) Dynamic pressure of water, Df. 
 
 (e) Water flowing over top of dam, weight of water, 
 of depth 6, on top of dam being neglected. 
 
66 HIGH MASONRY DAM DESIGN 
 
 For condition of water not overtopping dam, b = o and 
 >=o.* 
 
 For condition of no dynamic pressure, D = o. 
 
 For condition of no upward water pressure, c = o. 
 
 For condition of no mud (i.e., mud being replaced by 
 water) make h 2 = o, h\=H. 
 
 Stage I. 
 
 Rectangular cross-section at top or rectangular dam, 
 l=k=L. 
 
 This may fall under either of two cases, viz. 
 
 Case (i) 
 
 Condition: hi=H\ h 2 = o. 
 
 Case (2) 
 
 Condition: hi of known value; h 2 to be determined. 
 
 As in the preceding series, the value of H or h 2 , of 
 Stage I may be determined by several successive trial 
 substitutions. 
 
 Stage II. 
 
 (a) Trapezoidal cross-section at top of dam or trape- 
 
 * For water surface below top of dam, Series E, containing the distance 'a" 
 must be used. 
 
HIGH MASONRY DAM DESIGN 67 
 
 > 
 
 zoidal dam (spillway) front face battered. (A = o, 1 =L 
 and >>o = o.) Note: For a triangular dam / = o, also. 
 
 (b) Trapezoidal section continued (front face battered). 
 
 Stage III. Both faces battered. 
 
 O/ 
 
 Stage IV. Limiting intensity of pressure, p, introduced. 
 
 Stage V. Limiting intensities, p and q. 
 
68 
 
 HIGH MASONRY DAM DESIGN 
 
 The increased number of overturning loads then, tend 
 to render the right-hand members of the various equations 
 more involved; though after a little practice one may 
 easily carry through a design with surprising rapidity. 
 The slide rule may be used to great advantage and it is 
 suggested that the results be tabulated as determined, 
 in some such form as the following: 
 
 TABLE OF RESULTS. 
 
 c= , T= , etc. 
 
 Joint 
 No. 
 
 H 
 
 H 
 
 h 
 
 H + a 
 
 A 
 
 A 
 
 h 
 
 /o 2 
 
 / 
 
 yo 
 
 y 
 
 u 
 
 t 
 
 p 
 
 Q 
 
 Aoyo 
 
 Etc. 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Etc. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CHAPTER IV 
 FORMULA FOR INVESTIGATION 
 
 THE effect upon the calculation of a cross-section, of 
 backfill on the down-stream face, could, of course, be cared 
 for by introducing that condition into the preceding series 
 of equations; but as this effect as computed, would be, 
 in any case, largely dependent upon assumptions which 
 may vary widely and as the placing of backfill is generally 
 a later consideration with respect to construction, the 
 propriety of such introduction at that stage of design is 
 questionable. 
 
 In the following formulae for investigation therefore, 
 the general conditions of an earth thrust acting at the 
 down-stream face and of a vertical component of thrust of 
 material on the up-stream, inclined face of the dam, are 
 introduced. Moments of forces are taken about a point 
 in the joint distant y from the up-stream edge of the joint 
 in derivations for u. 
 
 By any of these formulae the position of the line of 
 resistance for any given cross-section and respective con- 
 ditions may be determined with regard to any horizontal 
 joint and its down -stream edge; the value of u being the 
 quantity to be sought. 
 
 Any condition may be disregarded by equating its 
 
 term to zero. 
 
 69 
 
70 HIGH MASONRY DAM DESIGN 
 
 The first expression below contains all of the conditions 
 heretofore considered with the additional ones just stated; 
 and from it follow the succeeding expressions for u. It 
 should be remembered that the term T cannot be coex- 
 istent in any expression for stability with b and therefore 
 with D. Nevertheless all of these terms are written with 
 the understanding that the proper eliminations be always 
 made. Three general group equations will be written. 
 
 Formula for Investigation. 
 
 First, Conditions of retained mud, water, overtopping, 
 etc. (see Fig. 14). 
 
 f 2 h 2 -b)-6W v (y-s) 
 
 y) sin d : 
 
 3 sin a J ) 
 
 U ~ y 
 
 Whence, for conditions of retained mud, water, etc., but 
 no overtopping, by making b = o and D = o, there follows 
 (see Fig. 13): 
 
 From this last expression for u, for conditions of re- 
 tained water, etc., but neither mud nor overtopping, by 
 making hi=Hi\ h 2 = o, there is obtained: 
 
HIGH MASONRY DAM DESIGN 71 
 
 i 
 
 + r (j - y)sin ,_ 
 
 _/ y 
 
 As in the equations for design, when 7 = o, HI =H. (See 
 Figs. 1 1 and 12.) If H' is of such depth that the down-stream 
 batter of the cross-section varies considerably, an approxi- 
 mate solution is possible by assuming some average batter 
 for the lower portion. The expression for earth thrust is 
 general, as is evidenced. After u is determined for each joint, 
 the intensities of maxima pressures can be determined 
 for the given cross-section, the general expression for p, 
 corresponding to above expressions for u, being: 
 
 In connection with the computation for the value of y in 
 an investigation, as indicated above, it is necessary to obtain 
 the position of the centroid of 
 a trapezoid with respect to the ~~ l ~ 
 
 back, or up-stream edge, of jn 
 
 the joint in question. The fol- j\ centroid-f- 
 lowing expression for x, in con- 
 nection with Fig. 15, may prove 
 
 convenient: f~" ~~ l '~ ~~*\ 
 
 FIG. 15. 
 
 j_ - x _____ >, 
 
 It is desirable to consider tension as active in the 
 joint, and if p' t is the intensity, in tons per square 
 
72 HIGH MASONRY DAM DESIGN 
 
 at the down-stream end of the joint, and p" t is the inten- 
 sity at the up-stream edge of the joint, p, above, which, as 
 written, is in pounds per square foot, will take the form : 
 
 and 
 
 In the case where there is liquid mud only on the back, 
 
 A f r' 
 W v becomes equal to ', where A' is the area of the 
 
 r 
 
 superimposed mud. 
 
 In the use of the foregoing formulae, it may be desirable 
 to take account of some such variation of the extent of the 
 uplift intensity on the base or joints, as was indicated 
 in Chapter I, p. 14, in the distinction to be drawn be- 
 tween uplift conditions in the foundations and higher 
 up in the dam. 
 
 For the foundation, a trapezoidal distribution was sug- 
 gested and for the body of the masonry, drained by wells, 
 a triangular distribution of intensities, but of shortened 
 extent down-stream, was proposed. 
 
 In the former case, the total pressure would be in- 
 creased, but its lever arm would tend to be diminished. 
 In the latter case, the pressure would be diminished, but 
 the lever arm increased. The effect on a cross-section 
 design, or on a line of pressure for a given cross-section, 
 would have to be worked out for any particular case. For 
 such a structure as the Kensico Dam, the foregoing changes 
 from the ordinary triangular assumption, while modifying 
 the numerical results as to ''factors" against overturning 
 
HIGH MASONRY DAM DESIGN 73 
 
 f/ / 
 
 and as to resulting pressure Intensities on the various joints 
 and the foundation, did not modify the final cross-section. 
 
 An approximation, however, may be satisfactorily ob- 
 tained by varying the value of c as one works down the 
 cross-section from the top. The amount of the requisite 
 variation may be ascertained by comparing the overturning 
 moment of the uplift, acting as usually assumed, with the 
 value of the overturning moment as desired to be assumed 
 in the given case. This may be done for three different 
 points down the dam, and the ratio found by this com- 
 parison. A curve may then be plotted in terms of these 
 ratios, and the distances from the top, all figured from a 
 cross-section assumed as nearly like the dam under con- 
 sideration as can be anticipated. A curve through the 
 three points will yield the relative values of the variable 
 c's to be used in the formulae, each c for its own elevation. 
 Or else formulae such as these here given may be derived 
 for the given assumption and used directly. 
 
 In the studies for design, referred to before, the analytic 
 work should be checked throughout by the graphic method 
 wherever possible. This should always be done both in 
 designing and investigating cross-sections. 
 
 It should be stated here that, after a cross-section has 
 been fixed upon for a given dam and the faces drawn to 
 chosen batters and curves, the entire cross-section should 
 be investigated as just outlined so as to give the actual 
 values for this final cross-section. 
 
 Again, in comparing different cross-sections, especially 
 of different dams, by superimposing, their water lines 
 should be made to coincide and not their tops for a fair 
 comparison. 
 
CHAPTER V 
 THE DESIGN OF A HIGH MASONRY DAM 
 
 To illustrate the method of applying the preceding 
 formulae to the determination of the theoretical cross- 
 section of a high masonry dam, an actual problem will 
 be presented. For this purpose the Olive Bridge Dam 
 has been selected, not only because it is representative of 
 the type for which the formulae were developed, but be- 
 cause the structure has been recently put into service, 
 and is sufficiently well known to be of more than passing 
 interest. 
 
 It may not be inappropriate, before proceeding to the 
 computations, to refer to certain of the structure's more 
 important features, especially as some were entirely new, 
 and to give a brief description of it. 
 
 The Olive Bridge Dam is the principal structure of a 
 number of dams and dikes which serve to impound the 
 waters of the Ashokan Reservoir. The latter is located 
 about 14 miles west of the Hudson River at Kingston, 
 N. Y., has an available storage capacity of 128 billion 
 gallons, derived from the Esopus watershed, with an area 
 of 255 square miles, and delivers the stored water to the 
 Catskill Aqueduct, whose capacity is 500,000,000 gallons 
 per day, to be conducted to the City of New York, about 
 100 miles away, and on the opposite side of the Hudson 
 
 River. 
 
 74 
 
HIGH MASONRY DAM DESIGN 75 
 
 i 
 
 The reservoir is^formed^by the Olive Bridge Dam, 
 placed across Esopus Creek, and by the West, Middle, 
 East and Hurley dikes, located in the smaller gaps between 
 the hills which create the natural basin. 
 
 The dividing dike with its weir separates the reservoir 
 into two portions, while the waste weir, nearly 1000 feet 
 long, at the eastern end of the large dikes, provides the 
 means of discharging surplus floods safely. All of the 
 weirs are masonry structures. 
 
 The length of the reservoir is 12 miles, with a max- 
 imum width of 3 miles, while the total length of the dam 
 and the dikes is 5^ miles. 
 
 Work was started in the latter part of 1907, and on 
 Sept. 9, 1913, the storage of water in the west basin began. 
 By Oct. 2 in the same year, the first Esopus water could 
 have been delivered by gravity into the Catskill Aqueduct, 
 the water surface of the west basin having reached ele- 
 vation 495, or a depth of about 95 feet behind the dam, 
 the equivalent of 2100 million gallons impounded, but 
 unavailable. 
 
 Three types of dam were considered in the studies. 
 
 (1) An earth dam, to be constructed by sluicing or 
 some other method. 
 
 (2) A composite dam, or one consisting of a masonry 
 core, covered by an earth embankment, the masonry 
 portion being of small section across Esopus gorge and 
 rising to within 50 feet of the water surface for full reser- 
 voir, the earth embankment making up the remainder. 
 
 (3) A masonry dam of the gravity type, extending 
 across the gorge and flanked by earth wings at each side 
 of the valley. 
 
76 HIGH MASONRY DAM DESIGN 
 
 The investigations were so far advanced at the end of 
 1906 that a decision in favor of the third type was 
 reached early in 1907, after 268 study drawings had been 
 made. 
 
 The masonry dam is founded on solid ledge rock. A 
 cut -off trench extends about 40 feet below the stream 
 bed, and grout holes go 20 feet deeper. The main struc- 
 ture is of cyclopean masonry, with concrete face blocks, 
 while the wings, built of acceptable earth found near the 
 site, contain concrete corewalls extending to solid ledge 
 or into very compact impervious earth. 
 
 The masonry portion of the main dam, which rises to 
 an elevation of 210 feet above the stream bed, is 1000 feet 
 long, but the total length of the structure including the 
 earth wings is 4650 feet. 
 
 To prevent temperature cracks in the masonry section, 
 it was decided to divide the dam transversely by means 
 of vertical " expansion joints," the distance between which 
 would be well within those distances at which cracks had 
 heretofore been observed in other structures. These in- 
 tervals varied from 84 to 91 feet. 
 
 The expansion joints (more properly termed contraction 
 joints) are formed by building vertical faces of concrete 
 blocks, shaped as a tongue-and-grooved joint, normal to the 
 axis of the dam, and thus preventing a continuous opening 
 through the structure, should the adjacent sections contract. 
 Vertical inspection wells at each expansion joint afford the 
 opportunity of studying the conditions at these sections. 
 
 There are also two longitudinal inspection galleries 
 built within the dam, one near the top and entered by 
 manholes from the surface, and one near the lower portion 
 
HIGH MASONRY DAM DESIGN 77 
 
 of the dam, both of which ^are connected with the vertical 
 inspection wells. 
 
 The inspection wells afford the opportunity, should it 
 be deemed desirable, of placing a copper strip across 
 each expansion joint, to reduce the quantity of water that 
 may be expected to pass through them. 
 
 The lower longitudinal gallery opens near the center 
 of the dam into a transverse gallery, leading to a measur- 
 ing weir chamber and drain, at the downstream side, where 
 the entire leakage may be gaged and discharged. 
 
 Between the vertical inspection wells, drainage wells 
 1 6 inches in diameter, and about 12 feet apart, slightly 
 inclined downstream from the top, are provided, between 
 the upper and lower longitudinal galleries, to intercept 
 seepage into the masonry and to prevent any water from 
 reaching, and consequently disfiguring, the downstream 
 face of the structure. 
 
 These wells were constructed by laying up large, hollow, 
 porous concrete blocks. 
 
 Small quantities of water which may enter the body 
 of the dam, either through the expansion joints, into the 
 inspection wells, or through the capillary spaces in the 
 masonry, will be conducted by means of the wells, galleries 
 and drains to the gorge below the dam. 
 
 It is proposed later to fill the vertical inspection wells 
 with material that will effectively stop all flow. 
 
 The following calculations indicate in detail the method 
 pursued in the determination of the theoretical cross- 
 section of the Olive Bridge Dam. 
 
 The conditions that governed in the design were as 
 follows : 
 
78 HIGH MASONRY DAM DESIGN 
 
 1. Elevation of top of dam above datum 610 ft. 
 
 2. Elevation of free water surface, reservoir full 590 ft. 
 
 3. Elevation of free water surface, reservoir in max. 
 
 flood 596 ft. 
 
 4. Elevation of free water surface, reservoir in max. 
 
 flood and wind blowing at max. velocity of 40 
 
 miles per hour, with waves piled up 600 ft. 
 
 5. Elevation of bottom of max. section ; rock excavation 
 
 at this point approximately 10 feet deep 390 ft. 
 
 6. Ice pressure, assumed the same as that used in the 
 
 design of Wachusett Dam, applied at elevation 590. 23.5 tons per sq.ft. 
 
 7. Upward pressure due to hydrostatic head assumed 
 
 to be of uniformly varying intensity, and varying 
 from a max. at the heel of the joint to zero at the 
 toe, distributed over f the area of the joint. 
 
 8. At the maximum section the earth is refilled to the 
 
 top of the gorge equivalent to elevation 450 or 500. 
 
 9. Other conditions so liberal that section will be reason- 
 
 ably safe against earthquake and dynamite. 
 
 10. Maximum allowable unit pressure 20 tons per sq.ft. 
 
 11. Top width 23 ft. 
 
 12. Resultant line of pressure, reservoir full and empty, 
 
 shall lie within the middle third at each joint. 
 
 13. 4, or ratio of unit weight of masonry to unit weight 
 
 of water 
 
 14. f y weight of a cubic foot of water 62 . 5 Ibs. 
 
 15. c, ratio of upward thrust intensity due to hydrostatic 
 
 head, assumed to act at heel of joint f 
 
 Flood conditions, Series C, and ice conditions, Series D, 
 will be imposed, and the design prosecuted with respect 
 to each simultaneously. 
 
 From the above it follows that 
 
 a = 10 feet (flood conditions), 
 ai =20 feet (ice conditions), 
 67 = 4512, 
 
 C _2 
 
 A~7' 
 
 JOINT No. i, FLOOD CONDITIONS 
 
 Generally speaking it may be assumed that the top of 
 a masonry dam will be about T V of the height above the 
 
HIGH MASONRY DAM DESIGN 79 
 
 i 
 
 water, but in the case^ under consideration, a superelevation 
 of only 20 feet was employed for full reservoir, and 10 feet 
 for flood conditions. While the choice in this respect is 
 purely arbitrary, the above ratio is the one usually pre- 
 scribed if there are no other governing conditions. 
 
 Since the length of a joint depends upon its depth 
 below the water surface, it is evident that at the surface 
 this dimension should be zero. For various reasons, how- 
 ever, such as the desirability of a foot- walk or a driveway 
 on the crest, a top width is chosen which will satisfy the 
 demands. 
 
 As 23 feet had been decided upon as the top width, 
 for a considerable distance below the water surface the 
 rectangular section will more than satisfy the only con- 
 dition for stability that applies in this portion of the dam, 
 namely, that the resultant of all the external forces for res- 
 ervoir full and reservoir empty shall be within the middle 
 third of the cross-section. It is evident that for reservoir 
 empty, the resultant passes through the center of the 
 joint. It becomes necessary, however, for reservoir full, 
 to determine the depth H at which this resultant first 
 emerges from the middle third of the rectangular section. 
 
 For flood conditions we will use the equation under 
 Stage I, Series C, which is, 
 
 Here a = 10 feet, A = J, L = 23 feet ; and c = f . 
 
 This equation may be solved by successive substi- 
 tions for H, until such a value of H is found that equality 
 results. In the present instance it is found that #=35.1 
 feet satisfies the equation, and hence the rectangular cross- 
 
80 HIGH MASONRY DAM DESIGN 
 
 section of the dam may be carried down to a depth of 
 45.1 feet below the top, since a, the superelevation under 
 flood conditions, is equal to 10 feet. 
 
 JOINT No. ia, ICE CONDITIONS 
 
 In a similar manner for ice conditions, we must use 
 the following formula, under Stage I, Series D, to deter- 
 mine the depth, Hi, at which the resultant first emerges 
 from the middle third of the rectangular section. 
 
 Here a\ = 20 feet, A = J, L = 23 feet, c = f and 6T = 45 1 2. 
 
 By successive substitutions for Hi, it is found that a 
 value of 6.7 feet will satisfy the equation and consequently 
 the rectangular section in this case can be carried down 
 only 6.7 feet below full reservoir, or, since the superelevation 
 a\ is 20 feet, to a depth of only 26.7 feet below the top, 
 before it will be necessary to modify the section. 
 
 A comparison of the two values established, HI -\-a\ = 
 26.7 feet, and H-\-a = 4$.i feet, together with an exam- 
 ination of the profile, shows the very marked effect the 
 assumption of ice pressure has upon increasing the cross- 
 section in the upper levels of the dam. 
 
 The solution for either H or HI may be expedited by 
 the use of the graphic method. Thus, assume at least 
 three values for H, say 30 feet, 40 feet, and 50 feet in 
 the present case, substitute successively in the right-hand 
 member of the above equation and solve. Plot these 
 resulting values as abscissae and the assumed values for 
 H corresponding as ordinates. A smooth curve drawn 
 through the points thus obtained will give a point where 
 
HIGH MASONRY DAM DESIGN 81 
 
 ordinate and abscissae are e%ial, and this will locate the 
 desired value. 
 
 At no point in this portion of the dam does the length 
 of a horizontal joint change, but below a depth of 26.7 
 feet from the top, this dimension will have to be increased 
 in order to comply with the requirement for stability, which 
 prescribes that the resultant of all external forces shall 
 lie within the middle third. This is accomplished by 
 giving a batter to the downstream face of the dam, while 
 the upstream face remains vertical. 
 
 This stage of the design extends from the lower limits 
 of the rectangular section to that elevation where it first 
 becomes necessary to batter the back, and -the formulae 
 to be used are those found under Stage II, Series C, for 
 flood, and Series D for ice conditions. 
 
 JOINT No. 2, FLOOD CONDITIONS 
 
 The investigation for the purpose of determining the 
 length / of joint No. 2 involves the use of an equation in 
 
 which u shall have a value of -, since the resultant of the 
 
 3 
 external forces for the reservoir full reached the limit of 
 
 the middle third at the downstream side at joint No. i, 
 and since, also, it may not pass outside that limit. This 
 is expressed for flood conditions by the following equation 
 from Stage II, Series C : 
 
 The value of 4.9 feet will be given to h, to bring the 
 depth of joint No. 2 to elevation 560, or 40 feet below the 
 
82 HIGH MASONRY DAM DESIGN 
 
 flood level, and 50 feet below the top of the dam, merely 
 because this is an easier figure to work with and because 
 the depth of each successive joint is taken as 10 feet, or 
 some multiple of 10 feet, below the next above. 
 
 [n the above expression the factors take the following 
 values : 
 
 ^0 = 1035 square feet; # = 40 feet; ^=4.9 feet; 
 
 \ and A=$. 
 
 Substituting these values in the above, completing the 
 square, and solving for /, we obtain : 
 
 / = 25 feet. 
 
 As it is necessary to use the factor A at each successive 
 joint, it should be determined at this point 
 
 A = 
 
 All the quantities in the above equation are known, 
 and A is found to equal 1152.5 square feet. 
 
 It will be noted that the equation employed to deter- 
 mine the value of / involves the factor yo, which represents 
 the distance from the up-stream face to the point of appli- 
 cation of the resultant pressure for reservoir empty, on 
 the joint next above the one being investigated. It there- 
 fore becomes necessary after each application of the formula 
 to use the supplementary equation from which the value 
 of y is found, not only for the above purpose of locating 
 the point of application of the resultant, reservoir empty, 
 but because this y becomes in turn y Q for the joint next 
 below. 
 
HIGH MASONRY; DAM DBSIGN 83 
 
 Furthermore, as ir has already been given the value 
 
 of -, the only condition for stability in this Section of the 
 o 
 
 dam involving Stage II is that y shall be equal to or greater 
 
 than-. 
 3 
 
 As all of these factors are already known, it merely 
 requires that they be substituted in the above, which 
 results in 
 
 y*n.6 feet, 
 
 This indicates that the pressure line P', for reservoir 
 iMiipty, lies well within the middle third of Joint No. 2, 
 
 As the limiting depth of the rectangular section under 
 ice conditions was found to be 26.7 feet below the top of 
 the dam, while for flood conditions it was located 45.1 
 feet below the top, it will be necessary to investigate the 
 joint at the latter point and called joint No. i, under the 
 ice conditions, to see what effect the ice will have upon its 
 length. After the J has been determined for this joint, 
 the length of the joint under ice conditions at a depth 
 of 50 feet below the top, called joint No. 2, will be found, 
 so that both joint No, i and joint No. 2 will have values 
 of / undtM- flood i-oMilitions .nul under ice conditions, and 
 thereafter the procedure will be to investigate the length 
 of each successive joint, first , for flood and then for ice. 
 
 The following formula from Stage II, Series D, will 
 be employed: 
 
84 HIGH MASONRY DAM DESIGN 
 
 JOINT No. i, ICE CONDITIONS 
 
 In this expression the following factors are known: 
 c = f, A=f, 7/1=25. i feet, /* = i8.4 feet, A Q =6i$ square 
 feet, 57=4512, y = ii.$ feet, /o = 23 feet; whence, by 
 substitution, completing the square, and solving for /, 
 we obtain the value 
 
 =33.0 feet. 
 To determine A, we have 
 
 A =A +~ h = 1130 square feet. 
 
 The preceding determination of the value of /, shows 
 that at a depth of 45.1 feet below the top of the dam 
 the length of the same joint should be 23 feet under flood 
 conditions, and 33 feet when ice is assumed to act. 
 
 We must now determine y, since its value must be used 
 in the joint next below where it takes the symbol y Q , and 
 the following formula is employed, in which all the quan- 
 tities are known : 
 
 12.7 feet. 
 
 It should be noted here that the denominator in the 
 above is equal to A. 
 Similarly for 
 
HIGH MASONRY --DAM DESIGN 85 
 
 JOINT No. 2, ICE CONDITIONS 
 
 Using the same formulae for I as in joint No. i with 
 new values to some of the factors, thus : 
 
 H! = 30 feet, h = 4. 9 feet, / =33 feet, ^0 = 1130 square 
 feet, ^0 = 12.7 feet. 
 
 We find that 
 
 ^=35-5 feet. 
 
 Computing the value of A 
 
 A =A -\-^-h = 1298 square feet, 
 
 and solving for y with the factors from above and using 
 A =1298 square feet for the denominator, we obtain 
 
 y = i3-3 feet. 
 
 JOINT No. 3, FLOOD CONDITIONS 
 
 The same conditions apply to this joint as for Joint 
 No. 2, so that the same equations will have to be used. 
 In the equation for /, Series C, the factors that change have 
 the following values : 
 
 ^0 = 1153 square feet, H = $o feet, h = io feet, 
 
 lo = 2 5 feet, and y = ii.6 feet, 
 
 and by substituting them in the equation, completing the 
 square, and solving for /, we will obtain 
 
 = 29.5 feet. 
 Using the equation for A we find its value to be 
 
 A = 1426 square feet. 
 
 Similarly, using the equation for y t with the new values 
 for the variables as indicated above we have, 
 
 = i2 feet. 
 
86 HIGH MASONRY DAM DESIGN 
 
 JOINT No. 3, ICE CONDITIONS 
 
 Using the same equation here that was used for Joint 
 No. 2 (Stage II Series D) under ice conditions with the 
 factors taking the following values: ^0 = 1298 square 
 feet, Hi =40 feet, h = io feet, / =35-5 feet, and ;yo = i3-3 
 feet, we obtain the following value for / : 
 
 1=40 feet, 
 
 while the use of the same factors in the determination 
 of A gives 
 
 A = 1676 square feet, 
 
 and y is found to have the value of 
 
 y = 14.5 square feet. 
 
 At this point it would be appropriate to determine 
 whether the maximum allowable pressure per square foot 
 had been reached, and as the ice conditions for Joint 
 No. 3 are more severe than the flood conditions, it will 
 be applied in connection with the former. The formula 
 developed in the theory to be used for this purpose is, since 
 
 u = 
 3 
 
 2 W7 3\ 2W 1676X7 
 Maxf (*--) = = 3Xl6X40 = 6.i tons per sq.ft., 
 
 which shows that at the toe of the dam, 50 feet below 
 the top, the presence of ice causes a pressure well within 
 the prescribed limit of 20 tons per square foot. 
 
HIGH MASONRY- DAM DESIGN 87 
 
 JOINT Nor 4, FL<fbo CONDITIONS 
 
 Using the same equations (for /, Series C, Stage II) to 
 determine /, A, and y, but with the following values 
 for the variable factors, A = i426 square feet, H = 6o feet, 
 h = io feet, /o = 29.5 feet, and y = 12 feet, we find that 
 /=35 feet, A =1749 square feet, and y = 12.7 feet. 
 
 JOINT No. 4, ICE CONDITIONS 
 The same equations (Series D for /) will be employed 
 
 here as in determining the values at Joint No. 3, with 
 
 the following values: A) = 1676 square feet, #1=50 feet, 
 
 h = io feet, /o =40 feet, and yo = 14.5 feet. 
 
 These give I =45 feet, ^=2101 square feet, ^ = 15.9 
 
 feet, and p = 6.S tons per square foot. 
 
 JOINT No. 5, FLOOD CONDITIONS 
 
 With the same equations for /, (Series C) A, y, and p, 
 the values of the variables in which have become ; A = 1 749 
 square feet, H = jo feet, h = io feet, / =35 feet, and y = 12.7 
 feet, we obtain =42.2 feet, A =2135 square feet, ^ = 13.9 
 feet, and p = 7.4 tons per square foot. 
 
 The value of y = 13.9 feet, and of I =42.2 feet, establishes 
 
 the fact that at this point, since = 14.1 feet, the resultant 
 
 \j 
 
 pressure for reservoir empty has passed outside the middle 
 third by 0.2 foot. It would seem proper, therefore, to 
 determine the value of q, the maximum pressure at the 
 heel, to see if the limit of pressure has been exceeded, due 
 to this excursion of the resultant beyond the middle third, 
 and the formula to be employed would be 
 
 2W 
 
88 HIGH MASONRY DAM DESIGN 
 
 which is a modification of 
 
 Using the former equation and substituting the appropriate 
 
 values, we find 
 
 q = 7.46 tons per square foot, 
 
 which is well within the limits prescribed. 
 
 JOINT No. 5, ICE CONDITIONS 
 
 Repeating the use of the equations (Stage II Series D, 
 for /) applied in solving for Joint No. 4, and using the values ; 
 AQ = 2101 square feet, HI =60 feet, h = io feet, /o =45 feet, 
 and ^0 = 15-9 feet, we obtain = 50.6 feet, A =2579 square 
 feet, y = 17.5 feet, and p = 7.4 tons per square foot. 
 
 It should be noted here that while the flood conditions 
 gave a value to y for Joint No. 5, which showed that the 
 resultant, reservoir empty, fell outside the middle third, 
 the ice conditions indicate that the resultant lies within 
 
 the middle third 0.7 feet, since - = 16.8 feet, and ^ = 17.5 
 
 O 
 
 feet. Under these circumstances then, the flood conditions 
 control, and make it necessary to batter the back, while 
 the ice conditions do not. This will make it necessary to 
 use the equations coming under Stage III for the flood 
 
 conditions, where the value of y=- is assigned, while 
 
 \) 
 
 equations under Stage II will be used for ice conditions. 
 
 JOINT No. 6, FLOOD CONDITIONS 
 
 The following equation under Stage III, Series C, 
 must be employed : 
 
HIGH MASONRY DAM DESIGN 89 
 
 t 
 
 In the above expression the factors take the following 
 values. Note that h is here made 20 feet. ^0 = 2135 
 square feet, H = 90 feet, h = 2o feet, 10=42.2 feet. 
 
 Substituting these values in the above, completing the 
 square, and solving for /, we obtain 
 
 / = 66 feet. 
 
 while A is found equal to 3217 square feet, and 
 Max. p = 7.i tons per square foot. 
 It is, of course, unnecessary to determine y here, since 
 
 its value has been established as -. 
 
 3 
 
 It is necessary, however, to determine the value of 
 /, which represents the amount the back face has to be 
 battered in going from Joint No. 5 to Joint No. 6. Using 
 the following equation, in which all the quantities are 
 known, we have 
 
 n0 _ f 
 
 JOINT No. 6, ICE CONDITIONS 
 
 We will again use the equations employed in solving 
 lor the quantities under Joint No. 5, using the values 
 ^0 = 2579 square feet, H\ =80 feet, h = 2o feet, / = 5o.6 
 feet, and ^0 = 17.5 feet. 
 
 There follows from these values 1 = 62.4 feet, A =3709 
 square feet, y = 20.8 feet and p =8.7 tons per square foot. 
 
90 HIGH MASONRY DAM DESIGN 
 
 JOINT No. 7, FLOOD CONDITIONS 
 
 Using the same equations for Joint No. 7 that were 
 employed for Joint No. 6, but increasing the value of h 
 to 30 feet, we obtain from the following values of the 
 factors, A) =32 1 7 square feet, H = i2o feet, h=^o feet, 
 /o = 66 feet, the quantities, = 92.5 feet, ^=5595 square 
 feet, p = 8.8 tons per square foot and t = 1.5 feet. 
 
 JOINT No. 7, ICE CONDITIONS 
 
 It will be noted that at Joint No. 6, the value of 
 y = 20.8 feet is just equal to - = - = 20.8 feet. 
 
 O O 
 
 In consequence of this, it will be necessary to take a 
 value of y for all successive joints equal to -, to prevent 
 
 O 
 
 the resultant, reservoir empty, from passing beyond the 
 middle third. This compels the use of Stage III, under 
 Series D, which will produce a batter in the back face, 
 and the following equation will therefore be employed : 
 
 Here ^0=3709 square feet, HI = IIO feet, ^=30 feet, 
 /o = 62.4 feet, which, if substituted in the above, give 
 
 2 = 85.6 feet, 
 
 while A =5929 square feet, p = io.i tons per square foot, 
 and t = 2.o feet. 
 
 Down to Joint No. 7, two cross-sections have now been 
 developed side by side, one for Flood Conditions and one 
 
HIGH MASONRY DAM DESIGN 91 
 
 for Ice Pressure, but- it will oe noted by reference to the 
 table of results, and to the profile, that to and including 
 Joint No. 5, the length of each joint under Ice Pressure 
 has exceeded the length under Flood Conditions, and that 
 at Joint No. 6, the difference is only slightly in favor of 
 the Flood Condition profile. As a consequence, down to 
 Joint No. 6 the section that must be used is that estab- 
 lished by Ice Pressure formulae. We must, therefore, in 
 examining Joint No. 7, for Flood Conditions, use values 
 for area, weight, etc., which represent the values of the 
 Ice Pressure profile, and not values from the Flood Level 
 profile. This in effect amounts to investigating Joint 
 No. 7 and its successors, whose length has been deter- 
 mined by Ice Pressure formulae, by the equations applying 
 under Flood Level Conditions, and where the latter de- 
 velops a length of joint in excess of that determined by Ice 
 Pressure the greater length will be used. 
 
 We will first complete the cross-section under Ice 
 Pressure by examining Joints No. 8, 9, and 10, each of 
 which is 30 feet below the next above, and in each of which 
 the same formulae apply as in Joint No. 7. The results 
 alone are given. 
 
 At joint No. 8, Ice Conditions 1= 108.6 ft., A = 8,842 sq. ft., =11.9 tons, t= 1.2 ft. 
 
 9, " " 1=131.6 " A =12,445 " " =13.8 " / = o.8 " 
 
 " " " 10, " " 1=155.1 " A = i6,74S " " =I5.7 " 1 = 0.8" 
 
 JOINT No. 7, FLOOD CONDITIONS COMBINED WITH ICE 
 PRESSURE PROFILE ABOVE JOINT No. 6 
 
 Proceeding now to investigate Joint No. 7, under 
 Flood Conditions, with the Ice Pressure profile above Joint 
 No. 6 providing the values of the factors to be used, and 
 
92 HIGH MASONRY DAM DESIGN 
 
 employing the formulae under Stage III, Series C, we 
 find, using A =370Q square feet, H = i2o feet, h=$o feet, 
 and = 62.4 feet, 
 
 / = 82.p feet, A =5889 square feet, 
 p = 10.3 tons per square foot and t = 1.2 ft. 
 
 But a comparison of these results with those obtained 
 for the same joint under Ice Pressure shows that the 
 latter, being larger, controls the length of joint. 
 
 JOINT No. 8, FLOOD CONDITIONS COMBINED WITH ICE 
 PRESSURE PROFILE ABOVE JOINT No. 7 
 
 Similarly, for Joint No. 8, we must employ the factors 
 resulting from the Ice Pressure profile in the solution of 
 the quantities in the Flood , Conditions. These factors 
 become ^0 = 5929 square feet, H = i$o feet, ^=30 feet, 
 and / =85.6 feet. 
 
 From which we derive, by the use of the same formulae 
 employed under Joint No. 7, the following: I = 111.2 feet, 
 .A =888 1 square feet, p = n.6 tons per square foot, and 
 2 = 1.9 feet. 
 
 Here we find the above quantities exceeding in value 
 those determined by the Ice Pressure formulae, so that the 
 former must be employed. In other words, / = 111.2 feet 
 is used in the profile instead of / = 108.6. 
 
 JOINT No. 9, FLOOD CONDITIONS WITH ICE PRESSURE 
 PROFILE COMBINED 
 
 The factors to be used are those just derived, A o =8881 
 square feet, H = iSo feet, h=^o feet, and IQ = 111.2 feet. 
 
HIGH MASONRY DAM DESIGN 
 
 93 
 
 From which we obtain witft the same formulae, = 138 
 feet, A =12, 6 1 9 square feet, = 13.3 tons per square foot, 
 and t = i.6. 
 
 Here again the values of the above quantities are 
 greater than those determined by Ice Pressure formulae, 
 so they must be used. 
 
 JOINT No. 10, FLOOD CONDITIONS WITH ICE PRESSURE 
 PROFILE COMBINED 
 
 The factors are ^0 = 12,619 square feet, H = 22o feet, 
 h=3o feet, and l Q = i$8 feet; whence, / = 161.9 feet, A 
 17,119 square feet, = 15.4 tons per square foot, and 
 t =0.3 foot, which values are again in excess of those obtained 
 from the formulae for Ice Pressure. 
 
 In order to check the above calculations by the graph- 
 ical method, there is submitted in tabular form 
 
 QUANTITES FOR GRAPHIC SOLUTION 
 
 
 
 Water 
 Units. 
 
 Water Units 
 in Tons.* 
 
 w = 
 
 I7.II9X- = 
 
 2I0 2 
 2 
 I62X2IO 2 
 
 4O,OOO 
 22,050 
 H,340 
 752 
 
 1250 
 705 
 354 
 23-5 
 1130 
 
 Water Pressure = 
 
 
 Ice Pressure 
 
 
 
 2 3 
 
 47,000 
 
 Resultant . 
 
 62.5 
 
 
 * Multiply 62.5 into Water Units to get pounds, or by ,03125 for tons. 
 
94 
 
 HIGH MASONRY DAM DESIGN 
 
 "8 
 
 f. -2 
 
 w 5 
 
 itM>ONN>O>-i<N'<t>-iNi-i<N(>ot^Tj-r<:i-ivOO t- roo 
 
 O O Oi/3>ow OO ^O t^ . Tt . 
 
 M -OOIH -Ol- 
 
 00 O w fO >0 2 O M C0i 
 
 . M.M.M-M p MMM, 
 
 
 O O O O O 
 
 lO OJ O* O "S to M o O 
 
 O f5 O >O t* O> O\ lO OOO 
 
 O IO t^ O\ O\ >O O 00 00 
 
 p^MMNMro^rJ-Niorot^iNOOoO <V O -fO -M fl oo t^O ' 
 
 OOOOOOlOiOOOONvOOTt-i 
 W w c^ C^ ro CS fO ! 
 
 1-1 (~0 
 
 ^ O O O O 1010O O O NO O -ttiOO -O-O-O ^ O<NOO 
 
 fC<N (*)<N trOTj-TtioOO aoo -O -co -in g OO M roo 
 
 . ^^o^oooo o.M^a^o.^a.. 1 ^T^T 
 
 jO 1 M" M" M M M M M w N N ro ro 10 10 ; oo ; (N ; o" 6 1000" ri t^ 
 
 . . to if) O rooo 
 . .iHPO.rO>OO\ 
 
 . . >O O W M N 
 
 : ; t ^ : J 00 . V . 
 ;oo ; N JJ ] 1000 N 
 
 - 1^00 00 O O CO fOO \O Oi O\ 
 
 s 
 
 OOOOOOOOOOOOOOOO -5 OOOl 
 
 > IOOO ^ThMMMMMMC^CNrOfOfOf'OfOfOfOCO te rOfOfOi 
 
 TtM g 
 
HIGH MASONRY DAM DESIGN 
 
 95 
 
 The finally adopted cross-sections of the Olive Bridge 
 Dam and the Kensico Dam, in addition to being investi- 
 gated by means of the formulae of Chapter IV, and others 
 
 El. 610 
 
 "Ice Pressure" Condition 
 
 "Flood" 
 
 " with "Ice" " 
 
 Figures refer to combination of Ice 
 Section above joint Xo. 7 with Flood 
 Design below 
 
 Limits of middle thirds marked with, 
 small circles thusio 
 
 00) 
 
 54 54 
 
 CALCULATED CROSS-SECTION 
 
 E1.39C 
 
 NOTE. Earth on down-stream toe would not increase compressive stress beyond 20 T. per 
 square foot, if taken at 100 pounds per cubic foot. 
 
 FIG. 16. 
 
 like them, taking into account variation in distribution 
 of uplift, were subjected to studies based on the method 
 of Ottley and Brightmore, given in Chapter VIII, for 
 determining the shearing stresses to be expected on vertical 
 
96 
 
 HIGH MASONRY DAM DESIGN 
 
 planes. This maximum shear intensity was foundi to occur 
 within 8 or 9 feet of the down-stream edge of the base, 
 in each case and did not exceed 92 pounds per square 
 
 Inspection 
 
 Guile 
 Flow Li 
 
 Concrete drainage > 
 blocks 
 
 El .500 
 Drainage Well 
 
 FIG. 17. Olive Bridge Dam. 
 
 inch. This was with the assumption that the base (at 
 the foot of foundation excavation) made an acute angle 
 with the down-stream face. The shear, of course, was 
 
HIGH MASONRY DAM DESIGN 
 
 97 
 
 zero at the up-stream and down-stream edges of the base. 
 Incidentally, the assumed masonry density was checked 
 with the final design, allowing for all openings within the 
 structure. 
 
 Besides these investigations special studies were under- 
 
 CROSS-SECTION AT TOP 
 2. . 8 68 10ft. 
 
 taken to ascertain the probable effect on the stresses, of 
 the large, stream flow passageway in the lower part of the 
 Olive Bridge Dam left open during construction and sub- 
 sequently closed with concrete; also the effect, in the case 
 of the Kensico Dam, of the lower portion's cracking longi- 
 tudinally, due to temperature changes during construe- 
 
98 HIGH MASONRY DAM DESIGN 
 
 tion. This cracking was assumed to take place so as to 
 divide the lower portion into three parts that would serve 
 as three lines of huge, supporting blocks for the portion 
 above. The stresses that would probably result in each 
 case were found to give no cause for concern. 
 
CHAPTER VI 
 WEIR OR OVERFALL TYPE OF DAM 
 
 IN the following pages the development of the cross- 
 section for the overfall, spillway, or weir type of dam will 
 be considered, the function of which type is to permit the 
 flow of water over the top, or " crest." This class of 
 structure may serve primarily either of two purposes : 
 
 (1) To make accurate measurements of the discharge 
 over the crest, the weir being, in that case, called a 
 1 ' measuring weir. ' ' 
 
 (2) To allow surplus water to escape from the side 
 of a canal or reservoir, the weir being then styled a " waste 
 weir " or " spillway." 
 
 The same structure may, however, serve the double 
 purpose at one and the same time, as, for example, where 
 the flow over a spillway dam is gaged. 
 
 In the first case, the discharge capacity per unit length 
 of weir crest is known in terms of the depth or head of 
 water on the crest, while in the second case, the discharge, 
 usually a maximum, is either known or assumed, and the 
 length of weir necessary for a given head or allowable range 
 in head is thereby determined. 
 
 Among the many proposed, the simplest form 
 of expression for weir discharge is the Francis formula, 
 
 99 
 
100 HIGH MASONRY DAM DESIGN 
 
 which shows the relation among those factors entering 
 into the evaluation of the discharge, as follows : 
 
 Q = CLH* 1 *, 
 in which 
 
 Q volume of discharge per unit of time ; 
 
 C = an empirical coefficient ; 
 
 L= length of weir (corrected for end contractions, if any 
 
 of the issuing sheet) ; 
 jy=head on the crest corrected for the effect of velocity 
 
 of approach. 
 
 It is unnecessary to give here more than passing con- 
 sideration to the subject of the hydraulics of weirs in its 
 various phases; the reader is referred for fuller discussion 
 to almost any work on hydraulics, but especially to weir 
 experimentation. * 
 
 As it is impossible to predetermine exactly the dis- 
 charge that is to pass over a waste weir and as allowance 
 must always be made for unusual storms or floods in 
 providing a length of crest in any given case, a knowledge 
 of the precise discharge capacity of the waste weir will 
 be of less importance than in the case of a measuring weir. 
 
 Nevertheless, in addition to the question of spillway 
 length to be provided, there remains the problem of arriving 
 at a proper form of cross-section for the structure. This 
 latter may be determined from a knowledge of the dis- 
 charge, even though it be inexact, since it leads to approx- 
 
 * Merriman's " Treatise on Hydraulics;" Traut wine's " Engineers' Pock- 
 etbook;" " Water Supply and Irrigation, Paper 200" on " Weir Experi- 
 ments, Coefficients and Formulas," Dept. of Interior, U. S. Geol. Survey; 
 " Hydraulics of Rivers, Weirs and Sluices," by David A. Molitor . 
 
HIGH MASONRY DAM DBSIQlf *->;-', j^ \* iOl 
 
 imate probable velocities attained by the sheet of water 
 in its fall over the crest together with the shape of the 
 sheet. 
 
 One condition to be fulfilled by a spillway dam is that 
 it shall discharge a maximum quantity of water per unit of 
 time for a given length and head on crest, and without 
 endangering the structure in any way. It is evident that 
 for high heads and consequent heavy sheets of water, 
 the best results will obtain with regard to the structure's 
 safety, if the overfall takes place without shock at any point ; 
 that is, if the structure is fitted smoothly to the shape 
 that the sheet would naturally assume in flowing over 
 the crest, and if the cross-section at the bottom, down- 
 stream, is of such form and material as to lead the water 
 away from the structure, without impact or erosion at the 
 toe. 
 
 It has been observed that a discharge over a spillway 
 dam has produced vibrations that may affect the stability 
 of the structure. These may be accounted for as follows : 
 
 If the sheet of falling water leaves the dam's face 
 and then impinges upon it lower down, air will be entrained 
 in the intervening space that will be gradually exhausted 
 by the rapidly moving filaments of the adjacent sheet. 
 As the condition of a vacuum is approached the superior 
 atmospheric pressure deflects the entire sheet of water 
 violently against the face of the dam, causing a shock, 
 provided the mass of falling water is not too great to 
 resist such deflection. The atmospheric pressure, acting 
 also on the masonry of the dam, will tend to force it down- 
 stream, if not of sufficient mass, during maintenance of 
 a vacuum under the sheet. This is the so-called " suction " 
 
^^**j^*S* i o.- 
 
 1-Q2 HIGH MASONRY DAM DESIGN 
 
 down-stream exerted on the masonry. Repeating cycles of 
 first entraining air, then exhaustion, and then readjust- 
 ment, coupled with the consequent changes in the imping- 
 ing sheet lower down the face, produce the vibrations 
 above referred to. 
 
 The danger from these lies in the possibility of action 
 in an up and down-stream direction and of acting sym- 
 pathetically with the structure's rate of vibration, in 
 which case the cumulative effect might lead to the ultimate 
 destruction of the dam. 
 
 The smooth face fitted to the curve of fall would 
 prevent the water suddenly leaving it at any point. Fur- 
 thermore, the insurance against the formation of a vacuum 
 between the face of the dam and the water sheet may 
 be accomplished by so proportioning the face of the dam 
 that the cross-section would extend well into the water 
 sheet throughout its entire extent. 
 
 Fanning * recommended a down-stream face of the 
 weir, " slightly more full than the parabolic curve which 
 the film of water at two-thirds depth on the crest tends 
 to take," in order that the overflowing water might not 
 lose contact with the masonry at any point. The foot 
 of this curve should be joined with the river bed by a 
 vertical curve of approximately 100 feet radius, f (Fanning) 
 tangent to the face curve and to the river bed. He further 
 indicated how the above face curve, thus forming what 
 is known as the ogee cross-section for high weirs, may be 
 
 * " A Treatise on Hydraulic and Water Supply Engineering," Ed. 1899, 
 by J. T. Fanning. 
 
 t For high dams. This radius would be far too high a value for moderately 
 high dams. The slope of the face and valley downstream would regulate 
 the radius at toe to a great extent. 
 
HIGH MASONRY DAM DESIGN 
 
 103 
 
 resolved into steps, for the purpose of breaking the fall of 
 water into a number of smaller falls. The steps " kill " 
 acceleration of the falling sheet. It is obvious that the 
 force of the falling water in such modified section is con- 
 strained to act in a vertical direction. The " steps " 
 
 FIG. 19. 
 
 should be so proportioned that they will project well into 
 the under surface of the sheet. Figures 19 and 20 illus- 
 trate, respectively, these ogee and stepped types and are 
 cross-sections of recent structures built for heads not much 
 over 5 feet. It may be noted that the tops are smooth 
 curves in both types of section, that is, stepping should 
 not occur until the sheet is well over the crest. The 
 stepped type serves where high velocities at the down- 
 
104 
 
 HIGH MASONRY DAM DESIGN 
 
 stream toe are objectionable. In the matter of construction, 
 face stones should be set on edge radially with respect to 
 the curves of the face, especially at the foot. In a con- 
 crete structure, hard stone is often thus embedded at this 
 lower portion to resist erosion. 
 
 A parabola can be determined to fit the shape of the 
 falling sheet and extend just inside its lower surface, or 
 nappe, as the French term it. 
 
 First. General formulae for design Eqs. (2) and (3) and 
 Eqs. (9) and (10), will be derived, as has been done for 
 the gravity dam, in Chapter III, and 
 
 Second. Consideration will be given to the shape of 
 the nappes and the determination of a value for the par- 
 ameter of the parabola, appearing in the general formulae 
 as a factor. 
 
HIGH MASONRY^DAM DESIGN 105 
 
 FORMULA #OR DESIGN. 
 Spillway Dam. 
 
 Derivation. In Series F, Formulae for Design, Stage 
 II, page 67, there appears an expression by which a trap- 
 ezoidal cross-section for a spillway dam may be calcu- 
 lated. From the preceding discussion it is evident that the 
 manner of determining the shape at or near the top re- 
 quires a more detailed treatment ; but the formulae of Series 
 F are sufficient for fixing the cross-section lower down. 
 
 One method would be to round the top of the trape- 
 zoidal section in a practical way to suit the case in hand 
 after comparison with crests of approved existing struc- 
 tures, knowing their discharge capabilities. The structures 
 of Figs. 19 and 20 were designed according to this method. 
 
 In the following, however, a general parabolic section 
 is derived, corresponding to Stage I, or the rectangular 
 section, of the series of formulae for design heretofore 
 given and following the identical principles. 
 
 As the shape of the parabolic section is fixed by the 
 falling sheet of water, there being no possibility of ice 
 thrust during overfall, and as ice thrust near the top, 
 with no overfall, would have to be resisted by proper 
 reinforcement, vertically, near the up-stream face, it would 
 seem reasonable to ignore that feature in the formulae 
 for design. But, as the extent downwards from the crest 
 of the parabolic section would be affected by such thrust 
 near the crest, it is thought desirable to include this factor, 
 for the purpose of investigating its effect, if for no other 
 reason. Formulae, Eqs. (9) anol (10), containing the 
 factor T, with water surface at or below the crest, will 
 
106 
 
 HIGH MASONRY DAM DESIGN 
 
 be given, therefore, immediately after those first derived 
 for overfall conditions. 
 
 ____rx 
 heoretical Crest x 
 
 FIG. 21. 
 
 As before, the middle third limit, for center of result- 
 ant pressure on any horizontal joint, will be retained. 
 
 Likewise, for any given case, the terms or factors 
 expressing other than the imposed conditions of loading 
 must be equated to zero. 
 
HIGH MASONRY DAM DESIGN 107 
 
 '> 
 
 Cases may arise where the highest head of water over- 
 topping the dam does not provide the critical stage for 
 stability, due to a rapidly rising backwater with the in- 
 creased discharge. Such must be especially investigated 
 for determination of the critical head. 
 
 In addition to the nomenclature given in Chapter 
 III, p. 31, et. seq., the following designations, in connection 
 with Fig. 2 1 , will be employed : 
 
 b' = vertical distance from the water surface in the res- 
 ervoir down to the theoretical crest of spillway 
 when the crest is overtopped. 
 
 kb' = vertical distance from the water surface down to 
 the actual crest of spillway when overtopped. 
 (kb'=b, of Chap. III.) 
 
 &=o.888 (Bazin). 
 
 k'b' = horizontal distance between the vertical line through 
 the crest (crest line) and the up-stream vertical 
 face. 
 
 '=0.25 (Bazin). 
 
 X = any abscissa (vertical) of the down-stream face of 
 spillway. 
 
 y = ordinate (horizontal), corresponding to X, origin of 
 
 rectangular co-ordinates being at the actual crest. 
 Y 2 =Kb'X, equation of the parabolic, down-stream face 
 of the spillway cross-section, with respect to the 
 vertical, X axis and the horizontal, Y axis through 
 the actual crest. (Kb ' is the parameter of the 
 parabola.) 
 
 K = the constant (considered later in this chapter) de- 
 termining the parabolic face, above, so that the 
 
108 HIGH MASONRY DAM DESIGN 
 
 curve shall extend within the falling sheet of 
 
 water. 
 
 7^ = 2.25 has been used.* 
 
 y=the ordinate to the centroid of the parabolic seg- 
 ment of the cross-section above any joint at dis- 
 tance X below the origin. 
 y'-fy. 
 
 Area of segment =|XF, down to any level X below 
 the origin. 
 
 Overfall, or Flood Level Design (No Ice). 
 In connection with the following, see Fig. 2 1 . 
 Overturning moment due to : 
 
 (a) Horizontal static water pressure on back (head =h\). 
 
 (b) Upward water pressure on base; pressure intensity 
 decreasing uniformly from cHf 01 c(/&i+fe)r, at heel to 
 zero intensity at toe. 
 
 (c) Mud (liquid) pressure on back (head hz) as before. 
 
 (d) Dynamic pressure of water, DJ-. 
 
 (e) Water flowing over top of dam, weight of water 
 of depth b, on top of dam being neglected. 
 
 For condition of no dynamic pressure, D =o. 
 
 For condition of no upward water pressure, c = o. 
 
 For condition of no mud (i.e., mud being replaced by 
 water) make h% =o, hi = H. 
 
 As in the previous deductions, the fundamental equa- 
 tion for length of joint, /, is 
 
 M 
 
 or l = u+ 
 
 * Mr. Richard Muller in " Engineering Record " of Oct. 24, 1908. 
 
HIGH MASONRY DAM DESIGN 109 
 
 t 
 
 in which ' "-**' 
 
 2A 
 
 L 6A 6A 
 
 In deducing the expression for A, or the total area 
 of the cross-section above the joint /, the top will be con- 
 sidered horizontal up-stream from the crest line. 
 
 A = k r b\hi +h 2 -kb') +f (Jn +h 2 -W) \fKV(h 
 or 
 
 u=- and l = k'b r - 
 3 
 
 (This expression for / may be more conveniently held 
 for the last substitution in derivation.) 
 
 y may be derived by taking static moments about the 
 up-stream edge of the joint /, as follows : 
 
 -kb')+^(h l+ h 2 -W)(l-k f b f )(l(l-k f b f )+k f b f } 
 
 k'b'(hi+h 2 -kb f )+%(hi+h 2 -kb')(l-k'b') 
 whence 
 
 _ 3 
 
 '' 4 + 4(k' 
 
 M 
 
 Substituting the above values for a, y, and A in 
 
 the fundamental equation for /, and reducing, gives 
 
110 HIGH MASONRY DAM DESIGN 
 
 General Formula: 
 
 By substituting in this last expression values of / and 
 I 2 involving K, as suggested above, and reducing, gives 
 
 - 6/ii 3 + (hi +/* 2 ) 2 (7 A - 6c)Kb' 
 
 + (hi +fo,) 1 2k'b'[(8 A - 6c) VKb'(hi +h 2 - W) 
 
 + iSDh 2 +kb'[(6k'b') 2 A -k(b') 2 ( 7 KA -i 2 k)- iSD] . (i) 
 
 From this General Formula, Eq. (i), there may be 
 derived directly Eqs. (2) and (3), formulae for the parabolic 
 cross-section corresponding to those for the rectangular 
 cross-section presented in " Series F, Stage I, Cases (i) 
 and (2) " of Chapter III, p. 66. 
 
 Case (i). Condition hi=H; h 2 =o, or no mud, H, 
 to be determined, underlined in Eq. (2). 
 
 +H\2k'b'[(6c - 8A) VKb'(H - kb') +3 (c - A)fc' 
 
 + i6kk\b') 2 &VKb'(H_-kb')=kb'[iSD-6(k'b') 2 A 
 
 .... . , . (2) 
 
HIGH MASONRY DAM DESIGN 111 
 
 Case (2). Condition hi of known value, fe, to be 
 determined, underlined in Eq. (3). 
 
 'b'[(6c - 8 A) ^/Kb f (hz +hi -kb) 
 +3 (c - A)fc V] - 2k(b') 2 [( 3 c - 7 A) K + 9 k) 
 
 -6h 1 *, . . (3) 
 
 It will be noted that the unknowns, in Eqs. (2) and (3) 
 above, are H and & 2 , respectively, and that the known 
 terms or factors are recurrent, and in any design need be 
 substituted but once. 
 
 A convenient method of using either of the above Eqs. 
 (2) or (3) is as follows: 
 
 (1) Substitute chosen values for the known terms, 
 according to assumptions and conditions, and reduce. The 
 right-hand member of either of the equations reduces to a 
 number, the left-hand member (an expression of the third 
 degree) to terms involving the unknown (H or h*) and 
 numerical factors. 
 
 (2) Substitute at least three successive trial values for 
 the unknown (these values always positive) and compute 
 the corresponding values of the left-hand side of the 
 equation. 
 
 (3) Plot a curve with these trial values for the unknown 
 as ordinates and the values of the left-hand member cor- 
 responding as abscissae. The ordinate corresponding to 
 the predetermined value of the right-hand member will 
 at once yield the correct value of the unknown. 
 
112 HIGH MASONRY DAM DESIGN 
 
 Illustration of Use of Equations. Condition. Hydrostatic 
 water pressure on back, with overtopping. Eliminating all 
 but hydrostatic and overtopping factors in Eq. (2), above, 
 gives, as a result, Eq. (4), following: 
 
 6H* - >/Kb f H 2 A -H\2k'b'A[sVKb'(H-kb' 
 
 ] (4) 
 
 Eq. (4) will serve to illustrate the foregoing remarks, 
 using the numerical coefficients for k, k f and K, respectively, 
 0.89, say; 0.25; and 2.25. Substituting these values in 
 Eq. (4) and reducing, gives an expression in terms of b', 
 A, and H, the last, the unknown to be determined, as 
 follows : 
 
 [%V 2 . 2 5 b'(H-o.8 9 b') 
 +0.06256'] -0') 2 (4.6725A -2.3763)! 
 
 -0.896') 
 - 1.4099). (5) 
 
 Eq. (5), then, contains Bazin's coefficients * and a 
 constant, fixing the parabolic face that will determine a 
 cross-section presumably acceptable as to flow conditions. 
 The conditions of stability down to the depth of water, H, 
 on the base at that depth, is that of the " middle third 
 limit," for the resultant pressures on the joints at and 
 above that base. The loading conditions, for simplicity, 
 comprise only those of horizontal static water pressure 
 on back and "overtopping." 
 
 * See Table III. 
 
HIGH MASONRY DAM DESIGN 
 
 113 
 
 Assuming a head (&') on the theoretical crest of 20 
 feet and A = 2. 2 4 and inserting these values in Eq. (5) 
 gives 
 
 + 53i.6.2 3 V 4 5(#-i 7 .8) =24,984. (6) 
 
 Preliminary to the prosecution of a solution of Eq. 
 (6), it will prove convenient to calculate and plot a curve 
 (or the value, 
 
 17.8), for different values of H, 
 ranging from 18 to 100 in this case. The smooth curve 
 resulting can be used in the subsequent assumptions for 
 ''trial" H 's, to pick off corresponding values of 
 
 17.8) 
 for entry in the left-hand member of Eq. (6). This factor, 
 
 17.8), is useful in indicating at once the lowest 
 positive value that may be assumed for H. In the case 
 here given H cannot have a positive value less than 17.8. 
 
 Without reproducing the curves, the results of the 
 trial values of H, assumed, together with the correspond- 
 ing values of 
 
 17.8) obtained from the prelim- 
 inary curve are contained in the following table, for the 
 case here under consideration : 
 
 Assumed 
 H. 
 
 
 Left-hand Member 
 of Eq. (6). 
 
 V 4S (H_i7.8). 
 
 95 
 
 58.8 
 
 -40,325 
 
 IOO 
 
 60.8 
 
 -11,678 
 
 105 
 
 62.6 
 
 +27425 
 
 From the curve (plotted with the assumed values of 
 H in the first column, and the corresponding numbers of 
 
114 HIGH MASONRY DAM DESIGN 
 
 the third column of the table, as abscissae and ordinates, 
 respectively) the ordinate, of value +24,984, of Eq. (6), 
 gave as its abscissa, a value of H equal to 104.7 ^ eet - That 
 is, this dam may be extended as a " parabolic section " 
 down, until the head of water on its base becomes 104.7 
 feet, with a head of 20 feet on the theoretical crest. The 
 distance below the actual crest would be #&/ = 104. 7 
 -17.8=86.9 feet. 
 
 (It may be said at this point that an investigation for 
 positive roots of Eq. (6), between 18 and 25, for values of 
 H, results, for the left-hand member of that equation, 
 in decreasing values, as H varies from 18 to 25; below 
 1 8, the root would have to be negative. 
 
 The factor V(H 17.8)45 indicates an imaginary quan- 
 tity at this stage.) 
 
 A dam of the above type and subjected to the assumed 
 conditions, would therefore reach the limit of stability 
 contained in the expression " the middle third limit," 
 at nearly 87 feet from its crest. 
 
 This result may be checked by use of the formula for 
 investigation, i.e., calculate the position on the base of 
 the center of pressure for 20 feet head on the theoretical 
 crest and base of parabolic section 84.7 feet below that 
 crest. Making the proper eliminations and substitutions 
 to suit the parabolic section in the first formula for in- 
 vestigation of Chapter IV, p. 70, there results: 
 
 (6a) 
 
 . 
 
 o./i A 
 
 The area A may be expressed in terms of /, as follows : 
 A=$(H-kb')(k'b'+2l). . (66) 
 
HIGH MASONRY DAM DESIGN 115 
 
 t 
 
 Whence, from (6a) and $6) , there is obtained : 
 
 y - 
 
 '' ' 
 
 Substituting in Eq. (6c) the values : 
 (H- kb')=S6. 9 feet; 
 
 l=k'b'+VKb'(H-kb f ) =67.5 feet; 
 H = 104.7 feet; 
 : A = 2. 24; 
 
 y= +-/z 7 /L/-L~n =25 - 66feet; 
 
 4 4(k b + 2L) 
 I y =41.84 feet, 
 
 there results, after reduction, 
 
 u =41.8 19.4 = 22.4 feet. 
 
 But - = 22.5 feet, which is a sufficiently close agree- 
 o 
 
 ment. 
 
 To continue the cross-section below a head of 104.7 
 feet, or the foot of the parabolic section, recourse must 
 be had to Formulae of Design, Series F, Stage II, (b), 
 or, in case an ice pressure cross-section is being calculated, 
 similar formulae of Series E, Stage II, et seq., apply. 
 
 Ice Pressure Design. 
 
 The formulae now to be derived are similar to those 
 of Series E, Stage I, of Chapter III, with the same con- 
 ditions governing for that series, as to loading, that is: 
 
 (a) Horizontal, static water pressure on back, (head hi) ; 
 
 (b) Ice pressure at surface of water, (Tf) ; 
 
116 HIGH MASONRY DAM DESIGN 
 
 (c) Upward water pressure on base ; 
 
 (d) Mud (liquid) pressure on back (head hz), 
 commencing at distance h% above joint in question. 
 
 In the fundamental formula for /, M here differs from 
 the value given in the preceding consideration, in that 
 the overtopping effect and, consequently, dynamic pres- 
 sure, are here supplanted by the effect of the ice pressure, 
 which will be referred to the top of the dam, in location, 
 by the vertical distance a\. This reference will in turn 
 change the expressions for A and /, of the parabolic section. 
 
 The procedure, otherwise, remains the same as for the 
 overfall condition. 
 
 There follow: 
 
 M 
 
 A 7' 
 
 Substituting these in l=u-\ j-+?> an d reducing, gives: 
 
 (7) 
 
 The equation, l = k'b' + VKb'(hi+h 2 +ai), applies here. 
 Substituting this expression for /, together with its square 
 in Eq. (7) and reducing, gives Eq. (8), whence two cases, 
 
HIGH MASONRY DAM DESIGN 117 
 
 previously formulated for the overflow condition, follow 
 directly. 
 
 Eq. (8) is: 
 
 i-\-h2 
 + 7 Kb'(hi +h 2 +ai)] } =-^l ....... (8) 
 
 From Eq. (8) may be obtained : 
 
 Case (i). Condition, hi=Hi\ h 2 =o, or no mud. Hi 
 to be determined. 
 
 (9) 
 
 Case (2). Condition, hi of known value, h 2 to be de- 
 termined : 
 
 ^+(fe+&o 
 
 +6c(k'b' + k'V VKb' 
 
 h2-\-ni 
 
 -6hi*. .... .... (10) 
 
118 HIGH MASONRY DAM .DESIGN 
 
 To recapitulate : 
 
 Eqs. (2) and (3) of this chapter are the working equa- 
 tions for the overfall design; Eqs. (9) and (10) for the ice- 
 pressure design. To continue design for overflow conditions, 
 use formulae of Stage II, (6), et seq., of Series F, and to 
 continue ice-pressure design, use formulae of Stage II, 
 et seq., of Series E, of Chapter III. 
 
 After a section has been finally determined, it may 
 have fitted to it circular curves of suitable radii, or tan- 
 gents, to simplify construction. 
 
 A radius equal to the maximum head, with center on 
 the crest line, often proves suitable for first trial near 
 the crest, down-stream. Further adjustment of the upper 
 part of the parabolic face curve, just down-stream of the 
 crest, is also often advantageous. Here, a little cutting 
 away of the curve by appropriate curves of longer radii, 
 where the general direction of the lower nappe of the water 
 sheet is nearer the horizontal than lower down, will tend 
 to increase the flow. There is less chance of serious results 
 from slight vacuum formation at this point than further 
 down, and, besides, the tendency to vibration from such 
 a cause would be in a vertical direction, more or less, with 
 no serious consequence to the structure. Care, however, 
 should be taken that the adjusted curves should be smoothly 
 continuous and follow the sweep of the parabolic face, 
 entering the sheet without abrupt change at any point, 
 so that the falling sheet of water would take its further 
 course without violence. 
 
 For a low dam, the parabolic curve down the face 
 may be replaced by a tangent to the curve at a point 
 where the water is well on its more vertical course. 
 
HIGH MASONRY DAM DESIGN 119 
 
 The probable patlj of the-^heet of water should always 
 be studied in connection with any design. 
 
 If the lower portion of the face is stepped, the study 
 concerns the flow upon and from each step. The para- 
 bolic face or ogee dam is of advantage in this respect, 
 as a preliminary study of the stepped cross-section, since 
 it shows the minimum sized shape for stability, upon 
 which the steps may be arranged to suit the flow. In this 
 connection, compare Figs. 19 and 20. 
 
 Sufficient masonry should be placed just down-stream 
 of the toe, depending, to great extent, upon local con- 
 ditions. The least thickness of this masonry, other con- 
 siderations being equal, may be approximated by ascertain- 
 ing the probable depth of back-water just below the dam, 
 after the velocity is reduced, beyond the shallower dis- 
 charging sheet.* The difference in head between the upper 
 surface of the sheet and the level of the water of greater 
 depth further down-stream could cause an uplift if the 
 head becomes active beneath the toe protection. The 
 thickness of the protection masonry should be at least 
 sufficient to balance this head by the weight of this 
 masonry. 
 
 The force of the flow from the toe is sometimes broken 
 up by masonry baffles; or the discharge into the still 
 back-water below the dam, the presence of which is pro- 
 vided for the purpose, may be so directed by the curve 
 of the dam face that the same object is attained. 
 
 * In this connection see paper on " The Hydraulic Jump in Open Chan- 
 nel Flow at High Velocity," by Karl R. Kennison, in Proc. Am. Soc. of 
 C. E. for Sept., 1915- 
 
120 HIGH MASONRY DAM DESIGN 
 
 SHAPE OF THE FALLING SHEET OF WATER. 
 Spillway Dam. 
 
 A study of the water sheet, besides outlining a proced- 
 ure, will indicate its possibilities and limitations limita- 
 tions to the procedure because the available data are not 
 so satisfactory or complete as might be wished with 
 respect to their application to the purpose in hand. 
 
 Most investigators have concentrated almost wholly 
 upon determination of the discharge, the question of the 
 shape of the water-sheet receiving comparatively little 
 attention, except incidentally to the effect of its fluctua- 
 tions upon the value of the discharge. An attempt at 
 an approximation, only, can be made in this study, which 
 will be based largely upon the results of M. Bazin's re- 
 searches.* 
 
 M. Boussinesq,f in an account of his investigations, 
 in 1887 emphasized the importance of the relation of the 
 shape of the under side of the sheet to the contraction 
 of the sheet at the crest, he making it the basis of a new 
 theory of flow over weirs. 
 
 So M. Bazin, with great care, determined the profiles 
 both of the upper surface and the lower surface of the 
 sheet for two sharp-crested weirs, one 3.7 feet high and 
 the other 1.15 feet high for heads varying from 6 to nearly 
 1 8 inches. 
 
 By reducing the co-ordinates of the curves of the upper 
 
 * Bazin, H. "Experiences nouvelles sur 1'^coulement en deVersoir, Annales 
 des Fonts et Chausse"es, M6moires et Documents," 1888, 1890. See also 
 translation by Arthur Marichal and J. C. Trautwine, Jr., Proc. Engineers' 
 Club of Philadelphia, Vol. IX, No. 3 and Vol. X, No. 2. 
 
 t " Comptes rendus de 1' Academic des Sciences," July 4, 1887. 
 
HIGH MASONRY -* DAM DESIGN 
 
 121 
 
 0.1 0.^ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 
 
 Nappes for Vertical Weir 
 
 " Inclined Weir _. 
 
 Pig. 22 
 
 FIG. 22 and FIG. 23. 
 
122 
 
 HIGH MASONRY DAM DESIGN 
 
 and lower surfaces to a common scale, expressing them as 
 ratios of the head, in each case, he established that for 
 each value of an abscissa there is a corresponding and 
 sensibly constant value of the ordinate. 
 
 The axes of rectangular co-ordinates pass through the 
 crest, Figs. 22 and 23, at A, horizontal abscissas, x, positive 
 
 TABLE III 
 VALUES FOR SHAPES OF NAPPES 
 
 
 Ordinates' r 
 
 
 
 A V* 
 
 
 X 
 
 b' 
 
 Crest Vertical. 
 
 Crest Inclined Down-stream.* 
 
 
 Upper Nappe. 
 
 Lower Nappe. 
 
 Upper Nappe. 
 
 Lower Nappe. 
 
 -3.00 
 
 0.997 
 
 
 
 
 _ 
 
 I.OO 
 
 0.963 
 
 
 
 
 
 
 o.oo 
 
 0.851 
 
 o.ooo 
 
 (0.730) 
 
 O.OOO 
 
 0.05 
 
 
 0.059 
 
 
 
 O.IO 
 
 0.826 
 
 0.085 
 
 (0.700) 
 
 (o.oii) 
 
 0.15 
 
 
 0. 101 
 
 
 
 O.2O 
 
 0-795 
 
 o. 109 
 
 (0.666) 
 
 (0.005) 
 
 0.25 
 
 (0.778) 
 
 O.II2 
 
 
 
 0.30 
 
 0.762 
 
 0. Ill 
 
 (0.630) 
 
 (-0.014) 
 
 Q..35 
 
 
 0.106 
 
 
 
 0.40 
 
 0.724 
 
 0.097 
 
 (0.585) 
 
 (-0.044) 
 
 0-45 
 
 
 0.085 
 
 
 
 0.50 
 
 0.680 
 
 0.071 
 
 (0-535) 
 
 (-0.083) 
 
 0-55 
 
 
 0.054 
 
 
 
 0.60 
 
 0.627 
 
 0.035 
 
 (o . 480) 
 
 (-0.130) 
 
 0.65 
 
 
 0.013 
 
 
 
 0.70 
 
 0.569 
 
 0.009 
 
 (0.418) 
 
 
 0.80 
 
 (0.507) 
 
 (-0.068) 
 
 (0.350) 
 
 
 0.90 
 
 (0-437) 
 
 (-0.129) 
 
 (0.276) 
 
 
 I .00 
 
 (0.360) 
 
 
 (0.196) 
 
 
 I.IO 
 
 (0.276) 
 
 
 (0.109) 
 
 
 1 .20 
 
 (o.i 86) 
 
 
 (0.009) 
 
 
 1.30 
 
 (0.085) 
 
 
 (-0.098) 
 
 
 Numbers in parentheses have been scaled from Bazin's plotted profiles. 
 
 * Slope i on 2, down-stream. The discharge is increased by nearly 13% over vertical 
 weir's discharge. 
 
HIGH MASONRY DAM DESIGN 123 
 
 jk 
 
 to the right, or down-stream, and negative up-stream; 
 the ordinates, y, positive upward from the crest level and 
 negative downward. 
 
 These co-ordinates refer only to the curves of the 
 nappe, and not to the face curve of the dam. 
 
 Ordinates for the shapes of the nappes as tabulated, 
 likewise are vertical and positive upward; abscissae, hori- 
 zontal and positive to the right, both referred to the 
 sharp crest of the weir, or " theoretical crest " of Fig. 21, 
 as origin of rectangular co-ordinates. 
 
 The ratios in Table III, if multiplied by the head &', 
 in feet, will give the corresponding locations of points on 
 the respective nappes, in feet, for a given case. 
 
 The table gives the values, extended from tabulated 
 values of Bazin from his profiles, and were used in plotting 
 Fig. 22. 
 
 The values for the higher weir were found to be very 
 precise, by comparison of 18 different determinations. The 
 final values are here given and are to be multiplied by the 
 head b f for any given case. 
 
 The values for a weir whose crest is inclined downstream 
 on a slope of i on 2 are also included. 
 
 In addition to the shape determinations, the velocity 
 and pressure heads in the sheet, at the contracted section 
 of the sheet, were carefully found experimentally.* The 
 velocities are plotted in Fig. 24 and curves were drawn 
 through them. Their sensible parallelism is significant, in- 
 dicating simple proportionality for different heads. These 
 
 * M. Bazin, "Annales des Fonts et Chausses, Mmoires et Documents," 
 1890.. 
 
124 HIGH MASONRY DAM DESIGN 
 
 1,02 Upper Nappe 
 
 BAZIN'S EXPERIMENTS 
 VELOCITIES OF FILAMENTS 
 
 IN 
 
 CONTRACTED SECTION OF SHEET 
 (sharp crested weir) 
 
 wer Nappe 
 
 0.093 
 0.074 
 0.055 
 
 0.0 
 
 4 5 G 7 
 
 Feet per Second 
 
 FIG. 24. 
 
HIGH MASONRY DAM DESIGN 125 
 
 > 
 
 velocities were determined for a sharp-crested, vertical 
 weir, 3.7 feet high, with. six different heads, ranging from 
 5.9 to 15.75 inches. As in the preceding experiments, 
 the observed factors were found to vary proportionately 
 with the head. 
 
 The purpose of this study is primarily to establish, 
 for the general formulae of design, preceding, a value for 
 the parameter of the parabolic down-stream face of the 
 weir, encroachment within the water-sheet to be assured. 
 For this purpose the water-sheet's lower' surface must 
 be traced, that is, the probable path of the sheet must 
 be extended. The term " nappe " has been applied to 
 the falling sheet; its application to the upper and lower 
 surfaces of the sheet is also permissible. These may be 
 called, then, the " upper " and " lower " nappes, for con- 
 venience. 
 
 The cases of vertical and inclined weirs depicted in 
 Fig. 22 indicate the effect upon the upper and lower nappes 
 of inclining a vertical, sharp crest down-stream. The weir 
 shown in Figs. 19 and 20 should preferably conform to the 
 lower nappe of the inclined weir, rather than to that of 
 the vertical weir. On the other hand, high overfall dams 
 usually have a vertical up-stream face, so, for this type, 
 the nappe of the vertical weir of Fig. 22 should be em- 
 ployed to shape the crest. As the formulas of design are 
 general, with regard to the distance of the actual crest 
 down-stream of the vertical up-stream face, and to the par- 
 abola of the down-stream face, any condition may be cared 
 for by such proper considerations of the flow sheet in con- 
 nection with determining the parabolic parameter for a 
 given case. 
 
126 HIGH MASONRY DAM DESIGN 
 
 As an aid to judgment, it may be stated that Bazin 
 showed, from his experiments on sharp-crested weirs, both 
 vertical and at various inclinations up- and down-stream : 
 
 I e' \ 
 
 (i) That the thickness [7 of Fig. 21 1 of the nappe 
 
 over the crest diminishes as the weir is inclined down- 
 stream. This diminution, barely perceptible for weirs in- 
 clined up-stream, becomes much greater when we pass 
 to weirs inclined down-stream. 
 
 (2) For a given inclination of weir, ^ increases with 
 
 b 
 
 the head, &', or rather with the ratio of b' to the height of 
 the weir. 
 
 (3) Except for very low weirs, or weirs where the 
 velocity of approach is more perceptible, the ratio of the 
 
 if r 
 
 height of the lower nappe to b' or the value of r- f (see 
 Fig. 21) appears to be independent of the head b f for a given 
 
 7 / 1 
 
 inclination of weir. M. Boussinesq also stated , to 
 
 be a constant. 
 
 (4) The thickness of the sheet measured over the 
 summit of the lower curve (at the crest line in Fig. 21) 
 increases from the greatest observed inclination up-stream 
 to that of i on i, or 45, down-stream, beyond which the 
 thickness diminishes. 
 
 (5) For a weir of constant height, the ratio of the height 
 of the lower nappe (b f b, of Fig. 21) to b' diminishes as 
 the inclination of the weir changes from up-stream to down- 
 stream. The ratio of its chord length to b' (the chord 
 being measured horizontally from the " theoretical " crest 
 
 of Fig. 21) decreases as , . decreases. 
 
 b 
 
HIGH -MASdNRY DAM DESIGN 127 
 
 ' Jfc 
 
 (6) For weirs inclined up-stream, the inclination, al- 
 
 7 / T 
 
 though it modifies considerably the value of , has 
 
 b 
 
 e > 
 but little effect upon that of n which does not vary as 
 
 much as 0.02 as we pass from a vertical weir to an up- 
 stream inclination of 45. For weirs inclined down-stream, 
 
 e > 
 
 however, . diminishes rapidly, and the curve of the sur- 
 o 
 
 face becomes elongated as the inclination increases. 
 
 Further experiments by Bazin on weirs of irregular 
 
 section* indicated that, for different heads, f the ratio f 
 
 did not change very rapidly for the same weir, and that 
 the ratio of the thickness, e ', of the sheet of water at the 
 up-stream edge of the sill, to the total head, &', varies 
 within very extended limits for weirs with sloping faces. 
 This variation is due to three causes, as follows : 
 
 (a) The "width" of the crest, or ratio of the head, 
 
 b f , to the crest width. For squared timbers, for example, 
 e ' 
 
 the value ,, quite large for small heads, diminishes as the 
 b 
 
 head increases, approaching progressively to that which 
 applies to the nappe of a sharp-crested weir ("in thin 
 partition "). This influence ought naturally to be found 
 again in weirs with sloping faces, the crest having a fixed 
 width. 
 
 (b) Inclination of the down-stream face. When slightly 
 inclined to the horizontal it exercises an influence similar 
 
 *"Annales des Fonts et Chausse"es, Me"moires et Documents," 1898, 
 2 me trimestre, pp. 121-264. 
 
 t Heads from o.i to 0.4 meter (3.9 to 15.7 inches). 
 
128 HIGH MASONRY DAM DESIGN 
 
 to enlarging the crest and has the same effect in increas- 
 
 e > 
 
 ing the value of -=-7. 
 b 
 
 (c) Inclination of the up-stream face. The effect of 
 
 this inclination, which modifies the contraction of the sheet 
 
 e ' 
 of water at the passage of the sill, is to dimmish -.-7. 
 
 These results for inclination of the faces * are analogous 
 
 to those established for sharp-crested weirs; but it was 
 
 e ' 
 observed that the values of n instead of growing with the 
 
 head, as for sharp-crested weirs, continue to decrease as 
 for weirs of squared timber, thus fixing the influence of 
 width of crest. From another point of view, it may be 
 
 observed here that the effect of this virtual widening of 
 
 0t 
 
 the crest tends to increase the initial values of over the 
 
 6 
 
 corresponding values for a weir of ~ sharp crest, and these 
 
 e > 
 values of ^7 with increase of head, b f , tend to approach the 
 
 e ' 
 values of r for a sharp-crested weir, hence a comparative 
 
 diminishing of values of jj as the head is increased, may 
 
 result. 
 
 From a further series of experiments, one set of which 
 was upon a weir with a crest 15.7 inches wide and with up- 
 stream face at a constant slope of 2 on i, and a down- 
 stream face inclination varying from i on 2, to i on 4 and 
 
 T on 6, it was developed that the value of r remained 
 * Crests of weirs varied from nearly 4 inches to nearly 8 inches in width. 
 
HIGH MASONRY DAM DESIGN 
 * 
 
 nearly unchanged and its value ranged between 0.87 and 
 0.88. 
 
 Weirs with faces joined by arcs of circles to a hori- 
 zontal crest, and above all weirs with curved profile, con- 
 
 e * 
 duce to small values of 7-7, but it should be remembered, 
 
 in this connection, that the thickness e f , measured properly 
 at the " theoretical crest " or where the sill is entirely 
 replaced by an edge, is not at all comparable with that 
 observed at the most elevated point of the lower nappe. 
 
 On weirs where the crest is joined to the up-stream face 
 
 e ' 
 by a curved surface , (presumably measured at the actual 
 
 crest) reduces to below 0.80; but the crest is not then a 
 
 thin edge. 
 
 e ' 
 
 Free nappes in "thin partition" give for -77 values 
 
 b 
 
 ranging from 0.85 to 0.86. The foregoing, considered in 
 connection with Fig. 22, will, it is believed, indicate the 
 effect upon the shape of the nappes of the inclination 
 of the up-stream face, the breadth of crest, and the down- 
 stream face slope near the crest. 
 
 Although objection may be raised against employing 
 empirical data from observations of comparatively low 
 heads for conditions that might obtain for high heads, 
 it should be remarked that, as the head increased, both for 
 the sharp-crested as well as for the irregularly shaped weirs, 
 the variations of the relations approached better definition 
 than for the low heads. Also, there should be noted the 
 constant ratios that obtained in some of the phases out- 
 lined. In the absence of more extended data, therefore, 
 the several indications suggested above may be employed. 
 
130 HIGH MASONRY DAM DESIGN 
 
 1 
 
 It will suffice to consider, for example, the sharp- 
 crested vertical weir, in connection with Figs. 22, 23, and 24. 
 The problem remains to extend the nappes, as plotted 
 by Bazin's co-ordinates, in terms of the head b f . In Fig. 
 23 verticals through the tenth marks, from i' to 7', have 
 been drawn. 
 
 The filament curve, OPE, of the average velocity is 
 then traced, its position in the vertical (the crest line) 
 through the actual crest being first ascertained. 
 
 An integration of each of the velocity curves of Fig. 24, 
 between the upper and lower nappes and the vertical axis 
 of co-ordinates to which it is referred, yields the discharge 
 for each case (within about 5 per cent of the values obtained 
 by the Francis weir formula, Q =3-33L// 3/2 ). Each dis- 
 charge divided by the vertical distances between the nappes 
 at the section (Fig. 24) will give the value of the average 
 velocity, whence its ordinate, or location above the actual 
 crest, may be read from its velocity curve. This location 
 was found to vary from 41 to 44 per cent of the water- 
 sheet's thickness, above the actual crest. The average 
 for the six heads observed was found to be 43 per cent. * 
 
 The value of the coefficient C in the formula for dis- 
 charge, Q, given in Part I of this chapter, lies between 
 3.40 and 3.50. The average was found to be 3.48 in the 
 cases given. H is taken upon the " theoretical " crest. 
 
 As has been brought out, the ratio, (Fig. 21), 
 is practically constant. The. filaments passing the section 
 
 * Mr. Richard Muller, " Engineering Record," Oct. 24, 1908, uses \\ but 
 this value appears to involve pressure heads rather than velocities in their 
 distribution through the sheet. 
 
HIGH MASQNRY DAM DESIGN 131 
 
 Jk 
 
 XX', Fig. 23, are "sensibly parallel. Bazin demonstrated 
 
 that the slight inclination in their direction did not affect 
 the readings of the instruments obtaining the measure- 
 ments of the velocities. Hence, we may proceed to lay 
 off on each of the verticals i i', 2 2', etc., 0.43 (i i') 
 0.43 (22'), etc., above the curve of the lower nappe. 
 The sheet section lines, of which a a' is one, are next 
 drawn by making each pass through a point on a vertical, 
 such as P, just located, and normal to the straight line 
 joining the two similar points adjacent to P. On these 
 lines, a a, 0.43 of the distance of each, comprehended 
 between the upper and lower nappes, is laid off above 
 the lower nappe. These may be considered points on the 
 path of the filament OPE of average velocity. 
 
 The discharge divided by the thickness of the sheet, 
 such as at a a', will give the average velocity which 
 may be plotted to scale as Vp in Fig. 23, normal to a a', 
 and applied to the point of the filament curve nearest 
 P. The vertical and horizontal components plotted, will, 
 from their resulting values, enable the origin, 0, of the 
 filament curve to be located with sufficient precision. This 
 may be tried by consideration of other points and an 
 approximate location fixed. It will be found that the 
 average velocity of the section XX f applies here, though 
 the horizontal component V* should be used for the sheet 
 extension beyond section a a'. With and V h fixed, 
 the path OPE may be approximately computed with 
 reference to and plotted, by choosing points down the 
 curve and computing the vertical component of the velocity 
 attained, by the laws of falling bodies. Laying off this 
 vertical component for each point, and plotting V h , results 
 
132 HIGH MASONRY DAM DESIGN 
 
 in obtaining the velocities of the sheet, tangential to OPB, 
 for the various points. The. discharge divided by the 
 tangential velocity at each point gives the thickness of 
 the sheet. This, laid off on a line normal to the tan- 
 gential velocity and through the given point, with 0.43 
 below the point, and 0.57 of the thickness of the sheet 
 above the point, determines two points, the one on the 
 lower nappe, and the other on the upper nappe. This 
 procedure may be continued as far down as desirable and 
 smooth curves drawn through the points, so determined, 
 will give the direction and shape of the sheet with suf- 
 ficient accuracy. 
 
 A parabola whose parameter is about 1.86', or K = i.8, 
 will approximate the position of the lower nappe. Its 
 origin is at the highest point of the lower nappe, in the 
 crest line, therefore, for a cross-section to be well within 
 the sheet, its K should be greater than K = i.S. K = 2.2$ 
 will provide a parabola that will meet this requirement 
 adequately and allow the curve of the face between the 
 crest and 7' (Fig. 23) to be somewhat flattened by a curve 
 of longer radius, as suggested earlier. 
 
 A parabolic curve for the nappes of the inclined weir 
 shown in Fig. 22 could be similarly worked out. 
 
 It is reasonable to suppose that the friction of an 
 actual masonry surface upon which the lower nappe is 
 flowing would modify the thickness of the sheet. This 
 would be probable, from a consideration of the similarity 
 in effect of the crest of masonry Jhere and the broad crest 
 effects on the thickness, cited earlier from experiments; 
 however, the curves of the face would tend to have an 
 opposite effect from that caused by the virtual widening 
 
HIGH MASONRY DAM DESIGN 133 
 
 of the crest; but to "what extent cannot be said at present. 
 It is possible that the horizontal component of the velocity 
 near the crest would be cut down in value from that ob- 
 taining with discharge free, and over a sharp crest; but, 
 in basing the determination of a cross-section upon the 
 flow over a sharp crest, there is obtained a larger trajectory 
 for the falling sheet than might otherwise result, and 
 so a preferable cross-section will result, as it is on the safe 
 side, both as to encroachment within the water sheet and 
 stability. 
 
CHAPTER VII 
 THE ARCH DAM 
 
 BEFORE deciding upon the cross-section for a masonry 
 dam, the proposed site should be carefully studied with 
 regard to its topography, to determine the type of struc- 
 ture that can be most advantageously used with existing 
 local conditions, keeping in mind especially the question 
 of economy of material. 
 
 It is evident that under all circumstances the choice 
 must lie between the gravity type of dam heretofore dis- 
 cussed and any one of the arched types about to be briefly 
 touched upon. But where gorges or canyons are en- 
 countered, the selection of the arch most naturally suggests 
 itself, especially since in those of 200 to 500 feet in width 
 moderate spans result. 
 
 The economy of one form over another will depend 
 upon whether, with the greater length but smaller cross- 
 sectional area, the arched type will require more or less 
 material than the straight gravity type with its shorter 
 length but greater cross-sectional area. 
 
 Should the gravity section be discarded in favor of 
 the other form, the further question arises as to which 
 of the various arched types may be used to best advantage, 
 and it is the purpose of the few following paragraphs to 
 refer briefly to these types before proceeding to the dis- 
 cussion of the design of what may be termed a simple 
 arched dam. 
 
 134 
 
HIGH MASONRY DAM DESIGN 135 
 
 Buttress Arch Type. This form of dam consists of 
 a series of plain or reinforced concrete arches, either ver- 
 tical or inclined down-stream, supported at the abutments 
 by buttresses, and may need special attention, particularly 
 if the length of the proposed structure is considerable, 
 to determine whether it may be economically used in a 
 given location. Such an investigation may be made by 
 first considering the arches in connection with the loading 
 producing the stresses in them, and second by analyzing 
 the buttresses with respect to the component arch thrusts 
 transmitted to them and acting in a down-stream direction. 
 
 The former investigation would be undertaken by means 
 of any of the prevailing arch theories applicable to the 
 case in hand, while the latter would be prosecuted by the 
 use of the formulae already established in connection with 
 the gravity type of dam. To apply these formulae it is 
 necessary to consider a unit of thickness of the buttress 
 and to reduce the component down-stream thrust of two 
 adjacent arches acting upon a single buttress to an equiv- 
 alent hydrostatic pressure, so that the conditions may 
 correspond to those in the design of the gravity type dam, 
 where the water pressure is taken as acting over a unit 
 length of the dam. 
 
 This reduction may be accomplished by calculating an 
 equivalent value of f, the weight of a cubic foot of 
 water, whereupon with the thickness of the buttress 
 assumed for each level or stage, and with the masonry 
 density either assumed or known, the formulae for design 
 may be employed directly to solve for the successive lengths 
 of base, /, of the buttress. The sides of the buttresses at 
 each stage may be assumed, for ease, to be vertical planes, 
 
136 HIGH MASONRY DAM DESIGN 
 
 until the calculations are completed, when a proper batter 
 may be given to avoid the successive offsets which would 
 result. It may be advisable also to investigate the but- 
 tresses as " plates " to determine their probable tendency 
 to " buckle." One of the chief advantages of this type 
 of dam, as is quite evident, is its comparative freedom from 
 the effect of uplift. 
 
 Gravity Section. A gravity section may be arched in 
 plan, or a special cross-section for the arch type may be 
 developed, since the arch action may, except for high 
 stresses, be limited in the thicker cross-section. In the 
 gravity sections arched in plan, it may be shown that the 
 arch takes from 5 to 8 per cent of the load, or expressed 
 differently, that the cantilever transmits most of the load 
 to the foundation. 
 
 Arch Section. The section of an arch dam shown in 
 Fig. 25 was investigated by the method described later, 
 and the study, including other dams of like dimension, 
 except for down-stream face batters and bases, led to the 
 conclusion that by thickening the top to at least 5 or 6 
 feet, thus gradually increasing the thickness until it reached 
 the section of Fig. 25 at a depth of about 80 feet, a section 
 could be obtained which would reduce the arch stresses at 
 the top without increasing them below at all appreciably. 
 
 The above-mentioned figure represents in a general 
 way the cross-section of an arched dam, and by the 
 curves indicates the amount of arch action at the various 
 levels. It is evident from the curves that at the founda- 
 tion no arch action exists. The up-stream radius was 
 taken as 350 feet, and the arch span as 600 feet. 
 
 To obtain a tentative cross-section for an arch dam, 
 
HIGH MASONRY DAM DESIGN 
 
 137 
 
 the thin ring formula may be used to calculate the bases 
 at the different levels. 
 
 I = length of base in feet ; 
 P = water pressure in pounds per square foot ; 
 R n = length of the up-stream radius in feet ; 
 
 q = average stress in the ring in pounds per square foot. 
 
 Compression - _ 
 
 140 \ 
 
 Cross-section 
 
 10 
 Tons per Square Foot 
 
 FIG. 25. 
 
138 HIGH MASONRY DAM DESIGN 
 
 The Constant Angle Dam. It may be shown that to 
 obtain an arch dam of minimum volume, when the struc- 
 ture acts as an arch, and with minimum stresses (even 
 near the foundation), any arch slice must be subtended 
 by a horizontal central angle, between abutments, of 
 I 33~34 / -* Practically, this angle maybe reduced to 120. 
 The fact that this type of dam has an ability to act as 
 an arch, to a much greater degree than the ordinary 
 arch dam, follows from the fact that an arch, when 
 loaded, undergoes a deflection proportional to the square 
 of the up-stream radius (see Eq. 8, page 149), and in the 
 " constant angle " type, this radius may be several times 
 shorter at the foundation than at the top. As a con- 
 sequence, the deflection at the base required for the same 
 unit stress would be proportionately less than the deflec- 
 tion required at the top to produce this same unit stress, 
 and as canyons are generally narrower at the bottom than 
 at the top, this condition usually applies. 
 
 The principle underlying the " constant angle " may 
 be developed as follows : 
 
 From the formula just given : 
 
 j_PRn 
 
 ~ 
 
 it is evident that the base / and therefore the cross-sectional 
 area, varies directly as the radius R n . The volume in a 
 given section, however, is equal to the area times the length 
 of the mean arc, which latter may be expressed in terms 
 of the length of the mean radius (designated as R m ) 
 
 * " The Constant Angle Arch Dam," by Mr. Lars R. Jorgensen. Trans. 
 Am. Soc. C.E., Vol. 78, p. 685. 
 
HIGH MASONRY DAM DESIGN 139 
 
 times the subtended angle ^designated as 2<f> n , expressed 
 in terms of TT) . We may therefore write for the volume V : 
 
 But R m , the mean radius, is equal to one-half the span, 
 or width of canyon W, divided by sin #, or, expressed as 
 an equation* 
 
 - 
 
 sin <f> n 
 
 Since the area of the cross-section is proportional to 
 the length of the radius, R n , and the volume to R m , we 
 may write without sensible error : 
 
 sin 2 <f> n sin 2 </> 
 
 This expression, in which C and K are constants, K 
 depending upon the width of the canyon, indicates that the 
 
 volume varies with the term -7? 
 
 sin 2 <f> n 
 
 If we differentiate this expression with respect to $, 
 and equate the differential coefficient to zero for a mini- 
 mum, we have, 
 
 i 
 
 2 COt (j>n 
 
 which equation is satisfied by the value of 66- 47' for <, 
 whence 2< w equals i33-34 / . 
 
 Poisson's ratio for lateral strains is taken into con- 
 sideration in determining the relative arch action in a 
 constant angle type, a value of one-fifth being adopted, 
 and the initial stresses induced axially by the weight of 
 
140 
 
 HIGH MASONRY DAM DESIGN 
 
 the dam on the foundation, together with the water load, 
 
 being utilized to help support the latter. 
 
 Fig. 26 represents a cross-section of a dam, developed 
 
 by the constant-angle principle, 250 feet high, with a 
 
 base of 70 feet and a cross-sec- 
 tional area of 9,668 square feet. 
 An equivalent rectangle of equal 
 base would have to be 138 feet 
 high. 
 
 If the specific gravity of the 
 concrete be assumed at 2.3, the 
 average vertical pressure may be 
 
 expressed as , where H is the 
 a 
 
 height of the dam, and a is the 
 ratio of the total height to that 
 of the equivalent rectangle, whence 
 
 81. 
 
 138 
 
 The mean vertical compression 
 for this section would then be 
 
 2 iH 
 
 =i.27H with the reservoir 
 1.81 
 
 empty, expressed in terms of the head of water, when the 
 latter is at an elevation of the top of the dam. 
 
 With the reservoir full, the radial water pressure is 
 assumed to counteract the strain of the masonry acting 
 in an up-stream and down-stream direction, although there 
 is, of course, no direct opposing force on the down-stream 
 side, acting horizontally up-stream. 
 
 It is reasonable to assume, however, that the reactions 
 
 1 700 000 Lb. 
 
 FIG. 26. 
 
HIGH MASONRY DAM DESIGN 141 
 
 at the abutments at least 'partially provide an indirect 
 force acting from the down-stream side in an up-stream 
 direction. 
 
 As Poisson's ratio is approximate, the full head is used 
 as active in this respect, hence the total resulting initial 
 axial compression at the foundation for the section may 
 be written as 
 
 The height of water, represented by h, that this initial 
 axial compression of o.^$H will resist without causing 
 the arch at the bottom to shorten in length, will result 
 by use of the ring formula 
 
 j_PR n 
 
 * ~ - > 
 
 q 
 
 in which q is represented by 0.4 5 # and P by h, whence 
 
 For the narrow section of Fig. 26, 
 
 hence this section, by application of the above expression, 
 is able to carry as an arch 
 
 /*=o.45#X = 0.42^, 
 
 / 
 
 or 42 per cent of the total head of water, before any 
 shortening in the length of the arch occurs. The re- 
 maining 58 per cent of loading is distributed between the 
 arch and cantilever as explained later on. 
 
14:2 HIGH MASONRY DAM DESIGN 
 
 Arched Dam Investigation. After the plan and cross- 
 section of an arched dam have been settled upon, the 
 structure may be investigated to determine the proportion 
 of the loading resulting from horizontal water pressure 
 which will be cared for by the structure acting as a hori- 
 zontal arch, and that cared for by the structure acting 
 as a vertical cantilever, respectively. From this resulting 
 distribution of loading, the value of the intensities of stress 
 on vertical planes normal to the axis of any arch ring under 
 consideration may be calculated from the resulting thrusts 
 into the sides of the canyon. Stress intensities for hori- 
 zontal joints may be found by combining the stresses 
 due to the horizontal loads assumed by the vertical canti- 
 lever, acting within the elastic limit of the material, with 
 the stresses due to the weight of the dam above the joint 
 in question. 
 
 This investigation for distribution of loading between 
 arch and cantilever may be made independently of the 
 value of the modulus of elasticity of the concrete or material 
 of which the dam is constructed, if considered homogeneous, 
 as will appear. 
 
 Limitations. If, however, the dam be built in sections 
 with transverse, vertical " contraction " joints, dividing 
 it segmentally into portions approaching voussoirs in their 
 nature, it could, under certain conditions, hardly be con- 
 sidered to act as an elastic arch. These joints may be 
 open more or less at times, according to the atmospheric 
 temperature, the season of the year when masonry between 
 them was laid, and the depth of water behind the dam, 
 with its consequent effects of swelling the masonry and 
 affecting its internal temperature. Furthermore, the con- 
 
HIGH MASONRY DAM DESIGN 143 
 
 traction at any time* may be greater at the top than further 
 down, or within the dam and different segments may be 
 simultaneously in different conditions of stress due to 
 contraction. 
 
 In short, the conception of the dam as a horizontal 
 arch, fixed at the ends and along the foundation of the 
 dam and acting to a greater or less degree as a huge vous- 
 soir arch, involves both a consideration of internal tem- 
 perature conditions and a knowledge of the value of the 
 modulus of elasticity of the great mass of the dam, both 
 of which are sufficiently uncertain to render anything but 
 an extended investigation under various assumptions of 
 doubtful value. 
 
 But the limits between which the behavior of the 
 structure may lie, viz., that of an elastic arch, held at the 
 sides and bottom of the gorge, and that of a cantilever, 
 bearing the total load, may therefore be profitably in- 
 vestigated. 
 
 Again, the deflection of such cantilever with an assumed 
 value for the modulus of elasticity of the material may 
 be considered, together with the deflection of the top- 
 most arch slice at the crown, due to the opening of the 
 contraction joints. These may be compared, or they may 
 be reduced to a corresponding temperature range and 
 compared with the maximum possible range at the site 
 of the dam, whence an indication as to the probability 
 of arch action ceasing wholly or in part may be reached. 
 
 Method of Arch and Cantilever Analysis. For the con- 
 sideration of the elastic arch with cantilever action and 
 no contraction joints, or joints tightly closed, the fol- 
 lowing method is elaborated from a method outlined by 
 
144 
 
 HIGH MASONRY DAM DESIGN 
 
 the late R. Shirreffs in a discussion of a paper on the Lake 
 Cheesman Dam and Reservoir by the late Charles L. 
 
 Origin for Joints 1,2,3,4 
 
 FT 
 
 QT 
 
 t if- ^U 
 
 -* T -B r <fc-H 
 
 * Pr 
 
 Joint ( n 
 
 i r =u 7 a T -r- 
 n=_7)__ __J 
 
 Computed 
 position 
 
 J 
 
 EXAMPLE: 
 
 (i) For battered up-stream face, 
 
 -d 2 ) tan 3 + (d4-</ 2 ) tan 8t + (d6-da) tan 5 
 
 -d$ tan Ot + tfidt) tan 
 
 (2) For vertical up-stream face (assume Joint 5 to be base of dam, 6& = 0), then, 
 
 D 2 = A 2 +A3+A4-|-(d3-d2) tan Bi+(dt-dd tan 04. 
 Note: All joints may be referred to any one origin. 
 
 FIG. 27. 
 
HIGH MASONRY DAM DESIGN 145 
 
 Harrison and Mr. -Silas P^Woodard, Members Am. Soc. 
 C.E.* 
 
 The formulae developed in the following pages are ap- 
 plicable to the vertical cantilever, contained between two 
 vertical, radial planes (i foot apart at the extrados), of 
 any arched dam, either of overfall (spillway) or complete 
 retaining type. 
 
 The down-stream face, though it may be curved in 
 vertical profile, should be considered straight between 
 load-points. Thus, in a spillway dam, the load-points 
 at the more curved portions of the vertical poofile may 
 be taken nearer together. The load-points may be ar- 
 bitrarily chosen in positions just so that the portions 
 between successive load-points may be considered as trap- 
 ezoids without essentially altering the cross-section of the 
 dam. The " joints " are taken at the load-points. Gen- 
 erality has been attained by introducing the expression for 
 the moment of inertia of the horizontal cross-section of 
 this vertical cantilever in terms of the variable, x, before 
 integrating. 
 
 The following nomenclature, together with that shown 
 in Fig. 27, applies. 
 
 NOMENCLATURE 
 
 For the Cantilever. The " origin" of a joint is the point 
 of intersection of the down-stream side, or face, of the 
 dam, next below that joint, with the corresponding up- 
 stream face of the dam, both produced, if necessary. 
 
 B = the batter of the down-stream side (or, if the up- 
 stream side is battered also, the combined batter 
 
 * Trans. Am. Soc. C. E., Vol. 53, p. 155. 
 
146 HIGH MASONRY DAM DESIGN 
 
 of the down-stream and corresponding up-stream 
 side) next below the joint, or load-point, in 
 question. 
 
 00= the vertical distance from the origin of any joint 
 
 of the cantilever to any level in the portion 
 
 immediately below the joint (between it and the 
 
 next joint). 
 
 d n = the vertical distance of the joint n below its origin 
 
 (d signifies general expression). 
 
 l=Bx=the length of a horizontal joint of masonry at the 
 depth x below the origin for that joint (same 
 as thickness of arch ring). 
 / B =the length of a horizontal joint of masonry at the 
 
 depth d n . 
 Z nJM =the length of a horizontal joint of masonry at the 
 
 depth d+i, etc. 
 
 m =the total number of load-points, or joints. 
 n=the number of the load-point, or joint, considered, 
 
 beginning with the top load-point. 
 
 Q=the total load of water, in tons, considered as con- 
 centrated at any joint, n, over i foot length 
 of extrados. (See left-hand diagram, Fig. 27.) 
 P n = that part of Q n assumed by cantilever action at 
 
 the center of the dam. 
 
 E = modulus of elasticity of the material of the dam. 
 I = the moment of inertia of the horizontal cross-section 
 of the cantilever, at the level, x. (See Appendix 
 I for derivation.) 
 
 n - 6R n B 4 x 
 
 ' ' ' ' 
 
HIGH MASONRY DAM DESIGN 147 
 
 where R n is the radius of the arch extrados at the 
 portion of the cantilever to which % is taken. 
 M n = the moment of all loads above the joint considered 
 (joint n), (M signifies such moment in general). 
 
 A n = the deflection of each individual portion of the 
 cantilever produced by all of the loads above 
 that portion (see Fig. 27), (A signifies such de- 
 flection, in general). 
 
 O n = the angle of deflection (change of angle) at any 
 load-point, n, having reference only to the 
 portion of the cantilever between that load- 
 point and the one next below (n + i). (See Fig. 
 27). (0 signifies such angle in general.) 
 A,=the total deflection of the cantilever at any load 
 point, n. 
 
 For example, from Fig. 27, 
 
 Dz = A2 -f- A 3 -|- A4 -j- ( ds dz) tan 63 -f- (d d%) tan 
 
 for a dam whose up-stream face is vertical. (See p. 144.) 
 g n = the vertical extent of hydrostatic pressure, the 
 
 resultant of which 
 
 (Q) is concentrated at the joint considered, or load- 
 point n. 
 
 k'n = the portion of g above joint n 
 k n = the portion of g below joint 
 (7'=the head of water on the level at the upper end 
 
 of the distance g n . 
 
 n =the head of water on the level at the lower end 
 of the distance g n . (For example, G\ =G'z.) 
 
 The following two expressions (readily derived) give 
 the relations among g n , k, k' n , G' n , and G n \ 
 
 j-or g n =&+&'. ( 2 ) 
 
148 HIGH MASONRY DAM DESIGN 
 
 ' n + [f (G' n -k'n)} 2 -I (G'n -fe'.), .; < . ( 3 ) 
 
 f(G- n +fe n ), . . (4) 
 
 also, 
 
 ,/ . <?' 
 
 / r / 
 3(6- 
 
 and are convenient for computing the hydrostatic con- 
 centrations for load-points wherever chosen. (See Fig. 27.) 
 
 Eq. (40) may be used to locate the last concentratipn 
 at the lower part of the dam. 
 
 Q n , in tons, may be computed from Eq. (5), following: 
 
 if G' n and g n are expressed in feet. 
 
 For the Arch. (Arch ring depth taken at i foot.) 
 .R n =the radius of extrados at load-point n. (If radius 
 of extrados is constant, throughout, R n =R.) 
 In equations, the extrados radius will be desig- 
 nated by R n . 
 r n =the radius of the axis of the arch at any load-point, n. 
 
 r n =Rn-- ......... (6) 
 
 2 
 
 g n =unit loading on axis of arch (corresponding to 
 extrados unit loading) at load-point n. 
 
 jvo^v ... ... (7) 
 
 =i central angle of arch span at level of load-point, n. 
 = thickness of arch ring (depth i foot) at any level 
 of dam, 
 
HIGH MASONRY DAM DESIGN 
 
 149 
 
 l n =same at level d n = length of joint n. 
 D c = deflection at crown of arch at level of load-point 
 (cantilever " joint.") n. (See Appendix II.) 
 
 2 sin 4> n 
 
 D c = 
 
 -(l -COS <) + COS 2 (j> n - 
 
 1.40 
 
 1.35 
 
 1.30 
 
 1.25 
 
 1.20 
 
 1.15 
 
 g 1.10 
 
 1.05 
 1.00 
 0.95 
 0.90 
 0.85 
 0.80 
 
 3<t>n 
 
 sin fa 
 
 -cos fa) 
 
 + COS fa -4 
 
 2 sin <j> n 
 
 (1 cos <t> n ) +cos 2 4> n - 1 
 
 CC, 
 
 -fl-cos 4>t, 
 
 +cos <f> n - 4 
 
 40 
 
 50 ( 
 
 70 
 
 q n r n 
 El, 
 
 , (8) 
 
 FIG. 28. 
 
 The trigonometric function of fa, in Eq. (8), may be 
 denoted by CC C and can be plotted as a curve, greatly 
 simplifying its application, in the calculations. (See CC C 
 of Fig. 28.) 
 
150 HIGH MASONRY DAM DESIGN 
 
 The Cantilever. The general differential expressions 
 for flexure of a cantilever may be written 
 
 _ 
 
 ........ do) 
 
 in which there may be substituted 
 
 M n =P 1 (x-d l )+P 2 (x-d 2 )+ 
 which may be transformed to 
 
 . . . +P n dn). (n) 
 
 Also, there may be written, for an arch dam with ver- 
 tical up-stream face 
 
 -dn) tan 6 n+2 -\-(d n+B -d n ) tan n+3 + . . . 
 + (d m -d n )tane m ............ (12) 
 
 Where the up-stream face is battered, as shown 
 below joint No. 5 in Fig. 27, let m' be the number of 
 the joint at the base of the vertical portion, or joint 5, 
 as there shown. Eq. (12), with this special significance 
 applied in this case to m', will hold as written, down as 
 far as the base of the portion with vertical up-stream 
 face, if n5 be changed to m'. 
 
 The additional terms, made up of differences of d's 
 times respective tangents of 0's, will be successively of 
 the form, 
 
 tan B m , +2 + . . . 
 
 -d m _ l ) tan B m \ 
 
HIGH MASONRY DAM DESIGN 151 
 
 * 
 
 or, for a dam with fart vertical and part battered up-stream 
 face, as shown in Fig. 27, 
 
 + (d n+2 -d n ) tan 6 n+2 + (d n+3 -d n ) tan n+3 + . . . 
 + (d m ,-d n ) tan O m , + (d m , +l -d m .) tan 8 m , +l 
 
 tan W ' +2 + . . . +(d m -d m _ 1 ) tan ro . (120) 
 
 In case there is no vertical up-stream face (see lower 
 portion of right-hand diagram, Fig. 27) Eq. (i2a) would 
 take the following form : 
 
 -d n+ i) tan n+2 + . . . +(d m -d m _ l ) tan O m . (126) 
 
 Tan 0, in Eqs. (12), (i2a) and (126) above, may be con- 
 sidered equal to 0, in each case. 
 
 Cantilever Deflection Equations. By substituting in 
 Eqs. (9) and (10), above, the expressions for / and M, 
 given in Eqs. (i) and (n), respectively, and integrating 
 with respect to x, the resulting expressions between the 
 limits x=d n+ i and x=d n , and reducing, there may be 
 obtained the respective expressions for A n and n (Eqs. 
 (13) and (14)). (See Appendix I.) 
 
 By means of Eqs. (13) and (14), p. 153, the A and 
 6 for each joint or load-point can be computed, and com- 
 binations, as indicated by any one of Eqs. (12), (120), 
 or (126) that pertains, made for each joint, whence there 
 results for each such joint an equation involving E, and 
 the various P's (the latter being the unknowns) for the 
 deflection, D n . 
 
152 HIGH MASONRY DAM DESIGN 
 
 Arch Deflection Equations. Eq. (8), just preceding, 
 provides for writing the several expressions for the crown 
 deflections of the arch elements of the dam, one expression 
 for each of the load-points assumed. These, too, will 
 be in terms of E and the various P's, as an inspection of 
 Eq. (7) in connection with Eq. (8) will show. 
 
 Resulting Simultaneous Equations. By equating each 
 expression for the arch crown deflection, to the expression 
 for the cantilever deflection at the same level, or load con- 
 centration-point, a series of simultaneous equations may 
 be evolved, E dividing out each time, and PI to P m 
 appearing in each of the m equations as the unknown 
 quantities. 
 
 Solution for Distribution of Loading between Arch and 
 Cantilever Actions. A solution of this set of simultaneous 
 equations will yield the value of the P for each joint or 
 load-point. These loads assumed by the cantilever action 
 may then be subtracted from the respective total hydro- 
 static loadings (Q) whence the amount assumed by each 
 of the horizontal arch laminae results. 
 
 The fact is neglected, however, that the several arch 
 slices throughout the dam actually cannot move freely, 
 in relation to each other. This would tend to stiffen 
 the dam along the arch axis and thereby transmit the 
 arch thrusts in an axial direction from a given arch into 
 the abutment of some lower arch slice. 
 
 The general expressions for Ek n and Ed n follow as 
 Eqs. (13) and (14). These, with equations for I and 
 D C1 (Eqs. (i) and (8)), are derived, as outlined above, in 
 Appendices I and II. 
 
HIGH MASONRY DAM DESIGN 153 
 
 R n l n d n+ i Wn+l 
 
 l n +l f-> -L 
 
 log- 
 
 A.) ( 
 
 n + i 
 
 6(d 2 n + l ~dn 2 ) 2 - 
 
 X2. 30259 log 
 lg-T^ 
 
 . -. . (x 4 ) 
 
154 HIGH MASONRY DAM DESIGN 
 
 These expressions involve only P, d, B, R, I, and E. 
 
 In use of expressions (13) and (14) above, for any 
 joint, n, to which a particular batter, B, naturally applies 
 (with reference to that joint's particular origin), care should 
 be taken that the various d's (cf. P\d\, P^d^ above) are 
 referred to that particular origin of the given joint n. 
 
 It will usually be found that, in Eq. (14) (for E0), 
 two terms, the fourth in the first pair and the third in the 
 last pair of braces are negligible. 
 
 Application of Foregoing Formulae (13), (14) and (12), 
 (i2a), or (i2b), (cantilever equations) .-- The procedure 
 will be conveniently described by referring to forms, in 
 order, for tabulating the factors that enter into the con- 
 struction of the cantilever equations of deflection; that 
 is, the application of the foregoing formulae (13), (14), 
 and (12), (i2a), or (126). Numerical values are given 
 for illustration only, and to facilitate comprehension of the 
 use of the tabulation forms. 
 
 The first step after the cross-section has been fixed, 
 is to choose the load -points, or joints. In an overfall 
 dam these will be more numerous at the upper and lower 
 portions, where the curvature of the down-stream face 
 necessitates shorter tangents to approximate more nearly 
 the curved cross-section by one of rectilinear sides, forming 
 a series of trapezoids, with the joints. The load-points 
 may number from 4 to 10, according to the type of dam 
 and its height and shape. 
 
 Second, calculate the g's by means of Eqs. (3) or (4) 
 and for the last joint, m, locate the concentration for the 
 distance g m , remaining, by means of Eq. (40). (See Fig. 27.) 
 
 Third, from the foregoing, compute by Eq. (5) the 
 
HIGH MASONRY DAM DESIGN 
 
 155 
 
 Q's. The heads on the load-points may be entered into 
 the tabulation as well, together with the lengths of the 
 joints (Ts). 
 
 These may be tabulated conveniently as in Table V. 
 
 TABLE V 
 
 - 
 
 G'n 
 
 *. 
 
 , 
 
 G ,4 
 
 Sn 
 
 32 
 
 Qn 
 
 (tons) 
 
 Head on 
 Qn 
 
 In 
 
 I 
 
 
 
 * 
 
 
 
 
 
 
 2 
 
 
 
 * 
 
 
 
 
 
 
 3 
 
 
 
 * 
 
 
 
 
 
 
 m 
 
 
 * 
 
 
 
 
 
 
 
 * Computed. 
 
 The fourth step is to compute the origins of the joints 
 (see nomenclature for definition of origin). These, once 
 found, should be placed as shown in Table VI, entitled 
 " Distances of Origins above Joints," together with these 
 distances, which are useful in carrying out the provisions 
 of origin reference, noted immediately after and referring 
 to use of Eqs. (13) and (14). 
 
 Suppose the position is desired of, say, Joint No. 2 
 with reference to the origin of Joint No. 3. In the left- 
 hand column of Table VI, No. 3 gives the line and 2 the 
 column, and in line 3 and column 2 is read 44.63. That is, 
 the origin of Joint No. 3 is 44.63 feet above joint No. 2, 
 or No. 2 is 44.63 feet below the origin of Joint No. 3. 
 Where these distances are negative, the distance d in the 
 formulae must be written with the minus sign. 
 
156 
 
 HIGH MASONRY DAM DESIGN 
 
 TABLE VI 
 DISTANCES OF ORIGINS ABOVE JOINTS (IN FEET). 
 
 Joint No. 
 
 i 
 
 2 
 
 3 
 
 
 
 8 
 
 9 
 
 Base. 
 
 I 
 
 12.18 
 
 21.73 
 
 
 
 
 
 
 9-55 
 
 2 
 
 20.26 
 
 29.81 
 
 37.62 
 
 
 
 
 
 17.36 
 
 1 3 
 
 a 
 
 35-08 
 
 44-63 
 
 52.45 
 
 64.26 
 
 
 
 
 29.18 
 
 5 L 
 
 
 
 
 
 
 
 
 
 | 
 
 
 
 
 
 
 
 
 
 8 
 
 -41.42 
 
 -31.87 
 
 -24.05 
 
 
 
 65.34 
 
 75-34 
 
 116.76 
 
 9 
 
 -79.98 
 
 -70.43 
 
 -62.61 
 
 
 
 26.78 
 
 36.78 
 
 21.76 
 
 V 
 
 TABLE VII 
 
 COEFFICIENTS FOR 0, IN EQUATIONS FOR D 
 (Table of Differences) 
 
 
 di 
 
 d z 
 
 d 3 
 
 d* 
 
 
 ds 
 
 dg 
 
 <*, 
 
 9-55 
 
 
 
 
 
 
 
 d* 
 
 o . .... 
 
 17-37 
 
 7.82 
 
 
 
 
 
 
 4J 
 
 W ^ 
 
 .l 
 
 <N 
 J|- 
 
 29.19 
 
 19.64 
 
 11.82 
 
 
 
 
 
 
 
 "^ : 
 
 ^H 
 
 
 
 
 
 
 
 
 + 
 
 ^ i 
 
 
 
 
 
 
 
 
 rfs 
 
 106.76 
 
 97.21 
 
 89-39 
 
 77-57 
 
 
 
 
 ^9 
 
 116.76 
 
 IO7.2I 
 
 99 39 
 
 87-57 
 
 ... 
 
 IO.OO 
 
 
 Base 
 
 121.76 
 
 112. 21 
 
 104.39 
 
 92.57 
 
 
 15.00 
 
 5-0 
 
HIGH MASONRY DAM DESIGN 
 
 157 
 
 At this step in the work, the coefficients for 8, in canti- 
 lever equations for D, might be compiled, as indicated 
 in Table VII. 
 
 For instance, in computing D 2 , it is necessary to use 
 as a factor, among others, d^d^. From Table VII, as 
 given there, for example, the value 19.64 feet corresponds. 
 
 What is really the fifth step is to compute the factors 
 occurring in Eqs. (13) and (14) as tabulated below, in 
 Table VIII. 
 
 Each large factor is here designated by a Roman 
 numeral, for future reference. 
 
 TABLE VIII 
 
 Joint 
 No. 
 
 (D 
 
 (II) 
 
 (Ill) 
 
 (IV) 
 
 2.30.259, d n + l 
 
 
 6(<Wi ~dn) 
 
 "(<Wi-<y 
 
 
 Wn+i-<V> 
 
 K> g ' /v 
 
 R n d n 
 
 l n R n d n+l 
 
 d n+l 
 
 V 2 +i 
 
 I 
 
 0.0013898 
 
 0.0027862 
 
 etc. 
 
 etc. 
 
 2 
 
 O.OOI553I 
 
 0.0018624 
 
 5.032262 
 
 1-0303393 
 
 3 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 
 
 
 
 Joint 
 No. 
 
 (V) 
 
 (VI) 
 
 B 
 
 B. 
 
 B. 
 
 2.30259 , 
 
 Rn 
 
 2.30259, 
 
 VJ>K n 
 
 
 
 
 R n 
 
 D lo a 
 R n 
 
 / n -(3 + V^ w 
 
 / n -(3-v 3 -)K 
 
 I 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 2 
 
 0.00000481 
 
 0.0000181 
 
 0.09285 
 
 o . 00862 
 
 0.000800 
 
 3 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 
 
 
 
 
158 HIGH MASONRY DAM DESIGN 
 
 Also tabulate for reference, the values of the following : 
 
 Rn, Rn 2 , 
 
 / 
 
 2 ~ 
 
 Tabulate for the various joints -, , and , the first 
 
 -D ^) JD 
 
 two of these three quantities for calculating the 0's, and 
 the last two for the A's. These could be added in three 
 
 extra columns to Table VIII. -^ and could be multiplied 
 
 JD n z 
 
 by the various d's and tabulated, to facilitate calculations 
 of EA's below. 
 
 Eqs. (13) and (14) may now be written in terms of (I), 
 (II), (III), etc., of Table VIII, with substitutions for / 
 and R n \ that is, if the portions within the braces of Eqs. 
 (13) and (14), involving (I), (II), (III), (IV), etc., be 
 designated by S n and S' n and S" n and 5 //r n , there may 
 be written : 
 
 EA 3 = 
 
 -D 3 -D 3 
 
 etc. etc. 
 
 The SVs and 5 n 's for all the joints should first be 
 separately computed by aid of Table VIII, and then 
 combined in the last written expressions, above. 
 
HIGH MASONRY DAM DESIGN 159 
 
 Similarly, 
 
 3 +-^(Pi + P 2 +P 3 )S" 2 , 
 .D3 >3 
 
 etc. etc. 
 
 The S'" n 's and S" n 's, should be separately computed 
 as for S' n and 5 n , in advance. 
 
 The resulting expressions for EAi, EA 2 , etc., with 
 0i, E02, etc., will each now consist of n terms; for example, 
 EA 3 or 63 will each have 3 terms, involving Pi, P 2 , and 
 P 3 with numerical coefficients. 
 
 Example : 
 
 The computation for E&2, for instance, may be arranged 
 as follows : 
 
 There will be nine terms involving (I), (II), (III), 
 (IV), (V), and (VI), of Table VIII. Assume / 2 =3.86, 
 B 2 =0.09285, ^2=350- 
 
 Substituting in Eq. (13), 
 
 -( 7 oo-/ n )(II)+(IV)+ 443 - 8o -- 26 V) 
 
 O J 
 
 x[(42oo-6/ B ) (I) -(III) -(2100-1. 268/)(V) 
 -( 2 ioo- 4 .722/ n )(VI)] ..... .-'..... (15) 
 
160 
 
 HIGH MASONRY DAM DESIGN 
 
 S' n and S n comprise the quantities within the two sets 
 of brackets and their determination is illustrated in Table 
 IX. For E&2, '2 and 2 are here shown, for example, the 
 quantities of Joint 2, in Table VIII, being used for (I), 
 (II), etc. 
 
 TABLE IX 
 COMPUTATION OF 5' 2 AND 
 
 For EA 2 
 
 2100-15.44 
 
 
 
 X(i)2 
 
 350 
 
 0.0092561 
 
 
 -(200-3.86)X(II) 2 
 
 1.2964911 
 
 
 + (IV) 2 
 
 + 1.0303393 
 
 For S't 
 
 , 443- 80-0. 268X3- 86 ^ 
 
 
 
 1 350 (X)2 
 
 
 
 16562-3.732X3-86 
 
 
 
 X(V L)z 
 350 
 
 
 
 5', 
 
 -0.2754930 
 
 
 + (4200-6X3- 86) X(I) 2 
 
 +6.4870502 
 
 
 -(IH) 2 
 
 -5.032262 
 
 
 -(2ioo-i.268X3.86)(V) 2 
 
 +0.0101005 
 
 For S 2 
 
 -(2100-4.732X3-86) (VI) 2 = 
 
 +0.0377100 
 
 
 5 2 
 
 + 1.5025987 
 
 
 assuming, for sake of explanation, values for di, 
 25.5384 and 41.5384, respectively, or 
 
 of 
 
HIGH MASONRY DAM DESIGN 161 
 
 ' > 
 
 115. 994478^1X25. 5384 +P 2 X4i-5384)( -0.2754930) 
 
 + 1249. 2i924(Pi+P 2 )( + i. 5025987) 
 A 2 = -816.0966^1 -1327.3873^2 + 1877. o752(Pi+P 2 ) 
 
 = io6o.9786Pi +549.6879P 2 . 
 E&2 should be similarly calculated. 
 
 A separate sheet should be devoted to the computation 
 of each EA n and each E6 n , so that ready reference may be 
 made thereto. 
 
 The A's and 0's, having been thus computed, should 
 next be collected according to formula (12), or (120) or 
 (126), as applicable, and by aid of Table VII. 
 
 To this end, the following form of tabulation (Table X) 
 for each D n will prove convenient, for obtaining and sum- 
 ming coefficients of the unknowns, P, assuming, for 
 example, five load-points. 
 
 Application of Eq. (8). (Arch Equation.) Having writ- 
 ten the expressions for cantilever deflections (ED 5, ED 4, 
 . . . EDi) as indicated in Table X, it remains to con- 
 struct the equations for the deflections of the corre- 
 sponding arches, by the application of Eq. (8). 
 
 The tabulation of Table XI will expedite the formation 
 of the arch deflection equations, shown in the last col- 
 umn of Table XL 
 
 Eq. (8) is based upon the assumption of ends fixed for 
 the arch. Limitations to which this assumption may be 
 subject have been pointed out previously in relation to 
 arch action as a whole; but the usual excavations for the 
 dam into the rock sides of the gorge or canyon, justify to 
 some extent, this assumption. 
 
 For the determination of the angles 4> n of the various 
 
162 
 
 HIGH MASONRY DAM DESIGN 
 
 
 So 
 
 
 
 
 
 Tf 
 
 IO vO t^ 00 
 
 ON 
 
 
 ^2: 
 
 
 J^> 
 
 
 
 
 
 
 
 
 oo_ 
 
 00 10 
 ^t" rx 
 
 ro 
 
 
 
 00 t^ 
 
 to 
 
 
 
 ^ 
 
 to oo 
 
 o 
 
 
 
 vo" o" 
 o 
 
 to 
 
 
 D!? 
 
 ON 
 
 
 t^ 
 
 > 
 
 
 ON ON 
 
 00 
 
 
 
 vd 
 
 VO Ol 
 
 1 
 
 
 
 : l 
 
 i 
 
 
 
 O O^ 
 
 ON C? 
 
 HH -rj- 
 
 ON vO 
 (S CO 
 
 ON 
 
 
 
 s g, 
 
 vO 
 | 
 
 
 
 ON <NJ 
 
 N 2 
 
 CO d 
 
 ! 
 
 0) 
 
 
 rO t^- O 
 
 -I CO Tt- 
 
 I 
 
 
 
 $ 
 
 II 
 
 10 
 
 
 
 
 
 ro 00 VO OO 
 O Tj- O HH 
 
 Q\ vO VO Tt" 
 00 M IN. rj- 
 
 1 
 IO 
 
 ro 
 
 en 
 
 
 
 ON 9 
 
 ON rt- 
 
 ro 
 
 6 
 
 
 CO VO >O HH 
 
 
 Q 
 
 s 
 
 HH , 
 
 00 CN 
 
 HH CN 
 
 g. 
 
 "cu 
 
 
 l^ ^t 1 rO n 
 >O O H t^ 
 
 fc- 
 
 1*3 
 
 
 + + 
 
 
 -4^ 
 
 
 
 
 + 
 
 x ;, 
 
 
 
 
 
 
 
 
 
 9 I 
 
 
 
 ^_ ^ 
 
 ^ 
 
 
 
 O <N t^. <N 
 N CN d IO 00 
 
 g 
 
 TAB 
 
 MPUTATIC 
 
 * 
 
 ? o 
 o 3" 
 
 t^ O 
 
 ^t- ON 
 
 vO OO 
 
 O ON 
 Tj- CO 
 
 ON 
 
 ON 
 
 1 
 
 
 CO ON iO HH ON 
 
 ON ON ro 00 ON 
 OO 00 00 O HH 
 
 IO O< vO ^ ON 
 
 CO HH oo IO 00 
 VO tO VO ON ON 
 
 g 
 
 vO* 
 
 d 
 
 
 + + 
 
 + + 
 
 + 
 
 
 
 + + + + + 
 
 + 
 
 
 
 
 
 
 
 -- 
 
 t^* 00 O tO 04 
 
 M 
 
 
 
 II 
 
 11 
 
 ON 
 
 
 1 
 
 o 
 
 ON ON O HH cO 
 O ^^ ^O O ^O 
 CO ro ro to O 
 10 O HH ON 5 
 
 5 
 * 
 
 
 
 O 
 OO r^* 
 
 10 ro 
 
 I 
 
 
 ; 
 
 N 10 ON ro ro 
 
 OX HH (VJ (SJ HH 
 
 00 
 ON 
 
 
 
 \\ \\ 
 
 II II 
 
 
 
 ii 
 
 II II II U II 
 
 4:1 
 
 
 
 
 
 
 
 . 
 
 
 Jj 
 
 
 
 
 
 
 
 t3 
 
 
 + a 
 
 
 
 
 3 
 
 <S 
 
 
 
 
 40 ^ ^, 
 
 
 
 
 o 
 
 ii 
 
 <f ^ 
 
 
 
 
 Jt ^ ^ *$ ^ 
 
 
 
 
 < 
 
 Ji 
 
 ^ 
 
 
 < 
 
 + 'III 
 
 i 
 
HIGH MASONRY DAM DESIGN 
 
 163 
 
 arch laminae, an average depth of excavation (often about 
 15 feet) may be assumed on the profile of the site. This 
 may be transferred, by means of excavation contours, to 
 the plan of site and points thereon determined at the levels 
 of the assumed load-points, or joints. These points are 
 taken on the foundations, so that they are approximately 
 midway between the upstream and downstream faces of 
 the dam at their respective elevations in rock. A line 
 on the plan through these points is the line of the profile 
 for the axes of the various arches, whence the central 
 angles 4> n are readily obtainable. They are to be entered 
 in the second column of Table XI. 
 
 The trigonometric function of </> n , CC C , may be read 
 directly from Fig. 28, p. 149. 
 
 TABLE XI 
 ARCH EQUATIONS 
 
 Joint 
 No. 
 
 *n 
 
 I 
 
 2 
 
 Rn 
 
 .-?-'. 
 
 Rn 
 r n 
 
 Qn~P n 
 
 (tons) 
 
 s n 
 
 I 
 
 59 
 
 I.I85 
 
 350 
 
 348-815 
 
 1.003397 
 
 0.56 -Pi 
 
 6.0 
 
 2 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 
 
 
 
 
 
 
 Joint 
 No. 
 
 J 
 
 n = - 
 
 * (Qn-r n \ 
 
 Function 
 of $ n 
 
 = cc c 
 
 V 
 
 l n 
 
 ED C (Eq. (8) ) 
 
 r n \ 8 n ) 
 
 Eq. (7) 
 
 I 
 
 o. 09365 -0.16723.?! 
 
 1.0952 
 
 5I338.3562I 
 
 5265.5657- 
 94O2 . 7982^1 
 
 2 
 
 etc. . 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 
 
 
 
164 
 
 HIGH MASONRY DAM DESIGN 
 
 Simultaneous Equations Formed. Equating each arch de- 
 flection to its corresponding cantilever deflection (for in- 
 stance, ED i of Table X with ED e of Table XI, for Joint 
 No. i) and reducing, eliminates the quantity Ei between the 
 two, and the simultaneous equations (involving P with 
 numerical coefficients) as many as there are joints, re- 
 sulting, may be tabulated as indicated in Table XII. 
 These simultaneous equations are to be solved for the 
 values of the quantities P, in tons. 
 
 TABLE XII 
 
 FINAL SIMULTANEOUS EQUATIONS 
 Coefficients of Quantities P. 
 
 Joint No. 
 
 PI 
 
 Pz 
 
 Ps 
 
 I 
 
 + 19249.45065 
 
 +67709.40901 
 
 +3077-35694 
 
 2 
 
 etc. 
 
 etc. 
 
 etc. 
 
 3 
 
 etc. 
 
 etc. 
 
 etc. 
 
 4 
 
 etc. 
 
 etc. 
 
 etc. 
 
 5 
 
 +80.40272 
 
 +70.49011 
 
 + 51-90397 
 
 Joint No. 
 
 PI 
 
 P 6 
 
 = 
 
 I 
 
 +905-599I6 
 
 +80.85725 
 
 +5265.56566 
 
 2 
 
 etc. 
 
 etc. 
 
 etc. 
 
 3 
 
 etc. 
 
 etc. 
 
 etc. 
 
 4 
 
 etc. 
 
 etc. 
 
 etc. 
 
 5 
 
 +29.60060 
 
 +215.65413 
 
 29155.10921 
 
 For nearly all of the above computations, a calculating 
 machine will be found most expeditious. 
 
 The final results may be tabulated in some such form 
 as in Table XIII, in which loads are in tons and intensities 
 of stress, in tons per square foot. 
 
HIGH MASONRY DAM DESIGN 
 
 il 
 
 TABLE XIII 
 RESULTS 
 
 165 
 
 Joint 
 No. 
 
 Elev. 
 
 Total 
 Load 
 Concen- 
 tration 
 ?). 
 
 Load Assumed by 
 
 Percent. 
 Arch 
 Action. 
 
 Maximum Stresses in 
 
 Cantilever 
 
 c*y. 
 
 Arch 
 (Q n -Pn>- 
 
 Horizontal Plane 
 (Cantilever) 
 
 Vertical 
 Radial 
 Planes 
 (Arch)] 
 
 Up- 
 stream. 
 
 Down- 
 stream. 
 
 I 
 
 2 
 
 3 
 
 etc. 
 Base 
 
 96.0 
 80.0 
 SO.o 
 
 14.0 
 
 0.56 
 12.55 
 etc. 
 
 etc. 
 
 -1.87 
 -4.11 
 +4-14 
 
 + 2.43 
 + 16.66 
 
 435 
 133 
 
 etc. 
 
 + 13-9 
 etc. 
 
 ii. 7 
 etc. 
 
 etc. 
 etc. 
 
 Cantilever Stresses. To get the resulting maximum 
 stress intensities in any horizontal joint of the cantilever, 
 combine the average intensity of stress on that joint due 
 to the weight of the superimposed masonry with the 
 stress intensity due to the moment of that masonry plus 
 the moments of the ^cantilever loading, P, all moments 
 taken about the center of gravity of the horizontal section 
 of the cantilever at the given joint. For calculating the posi- 
 tion of the centroid of the horizontal section of the canti- 
 lever at any joint of length /, referred to the down-stream 
 
 edge of the given joint, use the expression (3 n n ). 
 
 3X2^-47 
 
 Or, this resultant moment of the forces about the centroid 
 of the given horizontal section of the cantilever may be 
 thus calculated and entered into the well-known expression, 
 
 kl 
 M=-j-, as written on page 29, Eq. (18), or using here 
 
 C for di and S d (down-stream intensity) and S v (up-stream 
 intensity) for k, to prevent confusion, the last expression 
 may be written in the form 
 
 MC 
 
166 HIGH MASONRY DAM DESIGN 
 
 By substituting l n for Boo in Eq. (i), for 7, reducing, 
 and inserting in Eq. (140), the expression for S& results: 
 
 ( M 
 
 nnn - nn 
 and 
 
 It should be remembered that to the stress intensities 
 found by Eqs. (146) and (14^) should be added the average 
 intensity of stress due to the superimposed masonry, to 
 get the resultant intensity, as previously stated. (In 
 computing the weight of masonry, it will be sufficiently 
 close to use an average horizontal base in each case, of 
 rectangular section in lieu of the trapezoidal bases.) 
 
 Arch Stresses. To get the resulting maximum stress 
 intensity on the vertical radial planes, at any joint-level, 
 n, at the crown, the expression for M c should be employed, 
 viz. : 
 
 12 \ 0o / \30 n +sm 0o cos 0o 4 
 
 and the maximum stress in the arch, S a , is: 
 
 6M C 
 
 (14*) 
 
 The stress resulting from Eq. (14^), for any level, n, 
 must be combined with the axial arch thrust (S/) as found 
 byEq. (i 4 /): 
 
HIGH MASONRY DAM DESIGN 167 
 
 If the stress is desired al* any other point than at the 
 crown, the expression M +, in Eq. (12), of Appendix II, 
 should be employed. 
 
 Arch Dam of Rectangular Cross-section. Special Case. 
 
 The foregoing, it will be remembered, is applicable to 
 a dam of any cross-section, with load points assumed at will. 
 
 In case it is desirable to investigate a dam of rectang- 
 ular vertical cross-section, the load points being chosen 
 at equal intervals a vertical distance, a, apart, with con- 
 stant moment of inertia for the horizontal section of canti- 
 lever, the following expressions, Nos. 15 to 18, apply. 
 
 These are derived in a similar way to those that have 
 preceded. Some slight approximation may have to be 
 made in computing the total load concentrations, if they 
 be considered a constant distance apart, but usually a 
 dam of rectangular cross-section is not very high and this 
 approximation will not be serious. 
 
 Loads, etc., are located with reference to the top of the 
 dam in this case, with d n =na. 
 
 - 2 a) + . . . +P H ( x -na). . (15) 
 
 . (16) 
 
 +[i+a(*-i0JP.J. (17) 
 . . . +A m ) +a[tan n+1 
 + 2 tan 0+ 2 +3 tan n+3 + - +(mn) tan ro ]. (18) 
 As before may be written for tan 0. 
 
168 HIGH MASONRY DAM DESIGN 
 
 Expressions (16), (17) and (18) for the rectangular 
 section of arched dam correspond to expressions (13), (14) 
 and (12), respectively, and fulfill the same offices. Eq. (8) 
 still applies, for the arches to be considered. 
 
 The derivation of Eqs. (16) and (17) are compara- 
 tively simple and are outlined in Appendix I. 
 
 For further references to methods of analysis of arch 
 dams, the reader is referred to Transactions of the Ameri- 
 can Society of Civil Engineers, Vol. 53. 
 
CHAPTER VIII 
 
 RECENT CONSIDERATIONS OF THE CONDITION OF STRESS 
 IN MASONRY DAMS 
 
 CONSIDERABLE discussion has been raised within the 
 past few years, by criticisms being leveled at the present 
 general procedure in the design of high masonry dams. 
 This has properly perhaps, been more pronounced abroad 
 than in this country, since the matter may be said to have 
 been precipitated by the publication of a paper by Mr. 
 L. W. Atcherley of London University, " On Some Dis- 
 regarded Points in the Stability of Masonry Dams."* 
 
 It is the purpose to outline the analysis as presented 
 there, and to call attention to some of the discussion which 
 followed, in order to indicate the status of the theory 
 involved in the design of such structures. 
 
 The paper referred to takes exception to current 
 practice in regard to the matter of design and indicates 
 a need for both re vision . and extension in the analysis, 
 and then, supplementing the generally accepted ideas as 
 to the distribution of normal stress on horizontal planes, 
 by an assumption as to the shear on these planes, proceeds 
 to show that peculiar and unexpected conditions arise. 
 
 * Dept. of Applied Mathematics, University College, University of 
 London. Drapers' Company Research Memoirs. Technical Series II. 
 
 169 
 
170 HIGH MASONRY DAM DESIGN 
 
 It is a fact that, owing ist, to the manner in which 
 masonry structures are built, i.e., of a mixture of stone 
 and cement, and 2d, to the nature of the sections at the 
 springings or areas of support, it is practically impossible 
 to apply to them the general theory of elastic bodies. 
 Consequently, the treatment as it is employed to-day 
 has been developed, but only by the use of certain assump- 
 tions which it may be shown are not precisely exact. 
 
 The basis of the present investigation rests upon the 
 four following formulae, in which the usual distribution 
 of normal unit stress on horizontal planes is accepted, 
 but to which is added an assumed condition as to the 
 distribution of horizontal shear. 
 
 *-lT M 
 
 Al'n 
 
 T = *L[__ T 
 
 * max \ \ v L ]> 
 
 2 A^ ' ' (4) 
 
 c = distance along the horizontal joint from the cen- 
 
 troid to the point of application of the resultant. 
 
 d = distance along the horizontal joint from the cen- 
 
 troid to the point locating the neutral axis. 
 = length of the horizontal joint. 
 C max = maximum compressive stress on the joint, 
 ^max = maximum tensile stress on the joint. 
 
HIGH MASONRY DAM DESIGN 171 
 
 iJp 
 
 Q = vertical component of the resultant force acting 
 
 on the joint. 
 A = area of the joint. 
 5 = shear at any point y in the joint. 
 P = total shear on the joint. 
 y = the distance from the centroid to any point on 
 
 the joint. 
 
 With regard to Eq. (4) it may be stated that it has 
 not heretofore been customary to consider the distribution 
 of shearing stress on horizontal joints. But, if the dis- 
 tribution of normal stresses may be assumed to be repre- 
 sented by Eqs. (i), (2), and (3), with equal validity for 
 the usual types of dam, may the shear at any point be 
 assumed to be represented by Eq. (4). It is believed by 
 Mr. Atcherley that these equations more nearly express the 
 conditions of equilibrium in a dam than the usual ones do, 
 even though the latter tacitly assume the first three by 
 imposing the condition of the middle third, and use a fric- 
 tion condition, instead of one for shear as expressed by 
 Eq. (4). 
 
 In reference to this friction factor there may be some 
 question of doubt, since M. Levy* prescribes an angle 
 of 30 for masonry on masonry, while Rankine gives 36; 
 on the other hand, examination of dams actually built 
 frequently shows the angle to lie somewhere between the 
 above values. 
 
 But whatever its exact value, the friction condition 
 leaves some doubt as to the actual distribution of shear 
 
 *" La Statique graphique." IV e Partie, ' Ouvrages en Majonne- 
 rie," page 92. 
 
172 HIGH MASONRY DAM DESIGN 
 
 over a horizontal joint, the variation of which must be 
 known, in order to determine the tensile and compressive 
 stresses on the vertical sections of the tail (i.e., downstream 
 portion) of the dam. In consequence of this the parabolic 
 law as expressed by Eq. (4) has been assumed and will 
 later be shown to be more nearly correct than any other 
 hypothesis. 
 
 According to the author there is no reason whatever 
 why dams should be tested solely by taking horizontal 
 cross-sections, and asserting that the line of resistance 
 must lie in the middle third, while the stresses across 
 the vertical sections of the tail are absolutely neglected. 
 If the former condition is valid, then no dam ought to 
 be passed unless it can be shown also that there is no ten- 
 sion of any serious value across vertical cross-sections of 
 the tail, parallel to the length of the structure. It is 
 believed that a great number of dams as now designed 
 will be found to have very substantial tension in these 
 sections and this, in the opinion of the author, is a source 
 of weakness in dam construction which has not been 
 properly considered and allowed for. 
 
 If the problem is to be solved on the assumption that 
 a dam is an " isotropic and homogeneous " structure, the 
 general equations for the stresses can be determined only 
 by the following considerations: 
 
 (a) The normal and shearing stresses on the horizontal 
 top and curved flank, i.e., downstream face, are both 
 zero. 
 
 (b) The normal stress on the battered front or up- 
 stream face is equal to the water pressure, and the shear is 
 zero, and 
 
HIGH MASONRY DAM DESIGN 173 
 
 J* 
 
 (c) Either the "stresses or the shifts must be supposed 
 given over the base. 
 
 It follows at once from this that Eqs. (i), (2), and (3) 
 are not absolutely true, but that the shear is fairly closely 
 represented by Eq. (4). 
 
 As far as the present investigation is concerned, however, 
 the enquiry is not as to the validity of the usual treatment; 
 it is obviously faulty. But it is the purpose to try to 
 indicate that, supposing it to be correct, its present partial 
 application, i.e., to horizontal joints only, involves the 
 serious, and, it is believed, often dangerous, neglect of 
 large tension across the vertical sections. 
 
 To justify the above statement, two model dams of 
 wood were employed for experimental purposes, the cross- 
 sections being identical, and agreeing with that of a dam 
 actually constructed. One of these models was sub- 
 divided into horizontal strata to study the effect on such 
 planes, and the other into vertical longitudinal strata, 
 for a similar purpose. The application of the loading 
 was such that it approximated as closely as possible the 
 conditions obtaining in an actual dam. The general con- 
 clusions from these experiments were that: 
 
 (a) The current idea that the critical sections of a dam 
 are the horizontal ones is entirely erroneous. A dam 
 collapses first by the tension on the vertical sections of 
 the tail. 
 
 (b) The shearing of the vertical sections over each 
 other follows immediately on this opening up by tension. 
 
 (c) It is probable that the shear on the horizontal 
 sections is also a far more important matter than is usually 
 supposed. 
 
174 HIGH MASONRY DAM DESIGN 
 
 It follows consequently, that keeping the line of resist- 
 ance within the middle third of the horizontal sections 
 is by no means the hardest part of dam design. It would 
 be surprising if, with all the labor spent on this point, 
 the bulk of existing dam constructions are not, for masonry, 
 under very considerable tension, i.e., a tension across the 
 vertical sections which has been hitherto disregarded. 
 
 It is proposed therefore to lay it down as a rule for the 
 construction of future dams that the stability of the dam 
 from the standpoint of the vertical sections must be con- 
 sidered in the first place. If this be satisfactory, it is 
 believed that the horizontal sections will be found to be 
 stable, but of course the latter must be independently 
 investigated. 
 
 The above conclusions were apparently verified by a 
 combined analytical and graphical treatment in which 
 the algebraical analysis will here be considered first. 
 
 Denoting the total vertical force acting on a horizontal 
 joint by Q , and the total horizontal force acting over the 
 same by PO, under the assumption that the reservoir is 
 full, the variation of the normal pressure on the joint 
 may be represented by the straight line of Eq. (i). 
 
 If the resultant pressure on the joint be assumed to 
 cut it at the extremity of the middle third, then according 
 
 b 2 
 
 to the previous notation, d will have a value of J , pro- 
 
 c 
 
 vided 2b is the length of the joint. This indicates that 
 the line representing the variation of normal pressure 
 over the joint intersects it at the upstream edge, and any 
 vertical between it and the joint itself will represent the 
 normal pressure at that point where the vertical is erected. 
 
HIGH MASONRY DAM DESIGN 175 
 
 ' jfc 
 
 Denoting this by yl\i may be termed " the vertical height- 
 giving pressure," and may also be expressed in terms of 
 height of masonry, if the factors upon which it depends 
 are expressed in cubic feet of masonry. 
 
 Again, we may write an equation of the downstream 
 face, with respect to the same joint so long as that face 
 is a straight line, by making y' =moc. 
 
 Evidently then if this latter line, and the one indicating 
 the variation of pressure over the base, be referred to 
 the same origin, the tip of the tail, the difference in areas 
 included between each and the base will represent the 
 total upward force, in cubic feet of masonry, acting over 
 any assumed portion, " x" of the joint, measured from 
 the tail. 
 
 Representing this upward force by FI its point of 
 application may be easily determined, while the shear 
 may be written as F 2 , being regulated by Eq. (4). 
 
 As FI and F% thus give all the external forces, con- 
 sidering a wedge-shaped piece of dam bounded by the 
 downstream face, a vertical and a horizontal plane, the 
 total shear on the vertical plane must equal FI and the 
 total thrust F 2 , since these internal stresses are held in 
 equilibrium by external forces. Thus FI equals the total 
 shear on the vertical section, at a distance x from the 
 tip of the tail, while F 2 equals the total horizontal thrust 
 over the same. 
 
 If y be expressed in terms of x, and locate the point on 
 the successive vertical planes through which the resultant 
 acts, then the equation will represent the line of resistance 
 on these vertical planes. It is found to be an hyperbola. 
 
 Considering the stresses on the vertical sections, it is 
 
176 HIGH MASONRY DAM DESIGN 
 
 found: First, that the maximum shear may be properly 
 represented by f the mean value, and may be so arranged 
 as to be expressed in terms of FI and mx. Such an equa- 
 tion, representing a straight line, immediately shows the 
 necessity of thickening the tip of the tail which, as a 
 matter of fact, is the usual procedure in actual design. 
 Second, the line representing the maximum tensile stress 
 may be shown to vary as a parabola whose axis is vertical. 
 When the downstream face ceases to be linear, it 
 becomes necessary to apply a graphical solution for the 
 determination of the stresses. This it is unnecessary to 
 reproduce here, but the curves may be said to indicate 
 the following results: 
 
 (1) That the line of resistance for the vertical sections 
 lies outside the middle third for rather more than half 
 the vertical sections. In other words, these sections are 
 subjected to tension. 
 
 (2) That the tensile stresses in the tail are, for masonry, 
 very serious, amounting to nearly 10 tons per square foot 
 at the extreme tip, and to 6 tons per square foot after we 
 have passed the vertical section, where the strengthening 
 of the tail has ceased. 
 
 (3) That the maximum shearing stresses amount to 6 
 tons per square foot at the tip of the tail and 5 tons per 
 square foot after we have passed the vertical section, where 
 the strengthening of the tail has ceased. No undue import- 
 ance should be laid on the actual values of these " maxi- 
 mum ' ' shears on the vertical sections however, as they are 
 obtained from the mean shears by using the round multi- 
 plier 1.5. This round number is assumed because the 
 maximum is certainly greater than the mean shear. The 
 
HIGH MASONRY DAM DESIGN 177 
 
 > 
 
 actual distribution 4 ' of shear on the vertical sections has 
 not been discussed. It could, of course, be found from 
 that on the horizontal sections, if the latter were really 
 known with sufficient accuracy, by the equality of the 
 shears on two planes at right angles. It is sufficient to 
 show that the mean shears on the vertical sections appear 
 to be higher than those on the horizontal section, and 
 thus indicate that the parabolic distribution applied to 
 sections some way above the base, probably under-esti- 
 mates the max'mum shearing in the dam. " 
 
 In other words: Whether the test is made by the line 
 of resistance lying outside the middle third, or by the ex- 
 istence of serious tensile stresses, or by the magnitude of 
 the mean shearing stresses, the vertical sections are critical 
 for the stability in a far higher degree than the horizontal 
 sections. 
 
 In a well-designed dam, all the conditions for stability 
 of the horizontal sections may have been satisfied, yet 
 if the very same conditions be applied to the vertical 
 sections not one of them will be found to be satisfied. 
 It seems accordingly very unsatisfactory that the current 
 tests for stability should, if they are legitimate, be applied 
 to the horizontal instead of to the far more critical vertical 
 sections. In the case of the latter they fail completely; 
 and if higher tension and shear are to be allowed in the 
 vertical sections, then it is absurd to exclude them in the 
 case of the horizontal sections.. It is maintained by the 
 author that the current treatment of dams is fallacious, 
 for it screens entirely the real source of weakness, namely, 
 in the first place the tension, and in the second place the 
 substantial shear, in the vertical sections, and this at dis- 
 
178 HIGH MASONRY DAM DESIGN 
 
 tances from the tail far beyond the usual tail-strengthen- 
 ing range. 
 
 Nor do these theoretical results stand unverified by 
 experiment; they are absolutely in accord with the ex- 
 periments on the model dams. These collapsed precisely 
 as might have been expected from the above investiga- 
 tion, i.e., the dam with vertical sections gave long before 
 the dam with horizontal sections. The former collapsed 
 by opening up of the joints by tension towards the tail, 
 followed almost immediately by a shear of the whole 
 structure. In the case of the horizontally stratified dam, 
 the collapse, which occurred much later, was by shear of 
 the base, followed almost simultaneously by a shear of 
 one or more of the horizontal sections. 
 
 The question then arises as to how far the previously 
 assumed distribution of shear affects the main features 
 of the results, and so the other extreme was taken, i.e., 
 uniform shear, and the effect determined. 
 
 This distribution must be further from the actual 
 than the first hypothesis, yet it is still found: 
 
 (1) That the line of resistance falls well outside the 
 middle third for about half the dam. 
 
 (2) That there exist considerable tensions, 3 to 4 tons 
 per square foot, in the masonry. 
 
 (3) That the average shearing stresses on the vertical 
 sections are greater than on the horizontal sections. As a 
 result of this extreme case, it is believed that the real 
 distribution of shear over the base, whatever it may be, 
 must lead us to a line of resistance lying well outside the 
 middle third, and to tensions amounting to something 
 between 5 and 10 tons per square foot. 
 
HIGH MASONRY DAM DESIGN 179 
 
 > 
 
 From these investigations the author concludes as 
 follows : 
 
 (1) The current theory of the stability of dams is both 
 theoretically and experimentally erroneous, because : 
 
 (a) Theory shows that the vertical and not the hori- 
 zontal sections are the critical sections. 
 
 (b) Experiment shows that a dam first gives by tension 
 of the vertical sections near the tail. 
 
 (2) An accepted form of cross-section is shown to be 
 stable as far as the horizontal sections are concerned, but 
 unstable by applying the same conditions of stability 
 to the vertical sections. 
 
 (3) The distribution of shear over the base must be 
 more nearly parabolic than uniform, but as no reversal 
 of the statements follows in passing from the former to 
 the latter extreme hypothesis, it is not unreasonable to 
 assume the former distribution will describe fairly closely 
 the facts until we have greater knowledge. 
 
 (4) In future it is held that in the first place masonry 
 dams must be investigated for the stability of their vertical 
 sections. If this be done it is believed that most existing 
 dams will be found to fail, if the criteria of stability 
 usually adopted for their horizontal sections be accepted. 
 This failure can be met in two ways : 
 
 (a) By a modification of the customary cross-section. 
 It is probable that a cross-section like that of the Vyrnwy 
 dam would give better results than more usual forms. 
 
 (b) By a frank acceptance that masonry, if carefully 
 built, may be trusted to stand a definite amount of tensile 
 stress. It is perfectly idle to assert that it is absolutely 
 necessary that the line of resistance shall lie in the middle 
 
180 
 
 HIGH MASONRY DAM DESIGN 
 
 third for a horizontal treatment, when it lies well out- 
 side the middle third for at least half the dam for a 
 vertical treatment. 
 
 Immediately upon the publication of the preceding 
 results, Sir Benjamin Baker undertook some experiments 
 of a like nature.* The models employed by him were 
 
 Radius =Infinity 
 
 __.*._ 
 
 117175 
 Vyrnwy Dam 
 
 FlG. 29. 
 
 made of ordinary jelly however, and included not only 
 the transverse section of the dam itself, but the rock upon 
 which it rested as well. It is shown in the figure. 
 
 The horizontal and vertical lines drawn on the sides 
 of the model were for the purpose of detecting any dis- 
 tortion that might result through the application of 
 
 * Vol. 162, page 120. Minutes of Proceedings of the Institution of 
 Civil Engineers. 
 
HIGH MASQNRY DAM DESIGN 
 
 181 
 
 pressure. These pressures were applied against both 
 the upstream face and the floor of the reservoir, as it 
 was believed that, while according to the theory of the 
 middle third there could be no tension in the heel, never- 
 theless for the case of reservoir full, fairly severe tension 
 in the masonry might thus be caused. 
 
 The experiments indicated that the distribution of 
 shearing stress in the plane of the base, i.e., where the 
 
 FIG. 30. 
 
 dam met the rock, was more nearly uniform than para- 
 bolic, and that the strain extended into the rock for a 
 distance equal to about half the height of the dam before 
 it became undetectable. To solve the complete problem, 
 therefore, it would be necessary to consider the elasticity 
 of the rock on which the dam rested. Partially as a result 
 of these and the previous experiments, it may be pointed 
 out in passing, the proposed increase in elevation of the 
 Assouan dam, whereby the capacity of the reservoir 
 
182 HIGH MASONRY DAM DESIGN 
 
 would have been considerably augmented, was indefinitely 
 postponed. 
 
 The new feature in Atcherley's analysis is that, even 
 though the condition of " no tension in a horizontal 
 joint " is satisfied, dangerous tensions may be shown to 
 exist across vertical planes.* In connection with this 
 consider, for example, a section of the dam ABC, which 
 is triangular in profile, and construct EEC so that the 
 ordinates represent the variation of the unit normal 
 stress over the horizontal joint BC. 
 
 Taking a vertical section IK in which / locates the 
 centroid, the forces to the left are the upward pressure 
 acting over BK, tending to cause rota- 
 tion in a clock-wise manner and thus 
 produce tension at K, and two counter- 
 acting forces tending to neutralize this 
 pressure : the weight of the portion BKI 
 and the horizontal shearing force acting 
 along BK. The resultant effect of all 
 three will be tension at K, provided the 
 FIG. 31. rotation is right-handed, with a conse- 
 
 quent splitting along the vertical plane IK. 
 
 In view of the fact that the horizontal shear is present 
 as a factor, it is necessary to determine its distribution, 
 and this Prof. W. C. Unwin undertook to do.t Instead 
 however, of accepting the distribution in accordance with 
 Atcherley's assumptions, an analysis was attempted by 
 
 *" Engineering," Vol. 79, page 414. 
 
 t" Engineering," Vol. 79, page 513. "Note on the Theory of Un- 
 symmetrical Masonry Dams," by W. C. Unwin. 
 
HIGH MASONRY DAM DESIGN 
 i 
 
 183 
 
 which the shear might be actually calculated, and in 
 doing so attention was called to the fact that the accepted 
 theory of dam design is incomplete in just that feature, 
 since it fails to consider the rate of change in the hori- 
 zontal shear. 
 
 In any analysis the fundamental assumption must be 
 made that a masonry dam is a homogeneous-elastic solid, 
 and, while it is not absolutely essential that no tension 
 exist at any point in the cross-section, yet it seems desir- 
 able that there should be none at the upstream face of 
 horizontal joints. 
 
 It may be said therefore, that for a more exact analysis 
 the problem resolves itself into one of the determination 
 of shear on horizontal planes, and Prof. Unwin suggests 
 as follows, a method of procedure by which this may be 
 accomplished : 
 
 If, as in the figure, we assume a dam of triangular 
 section, in which AB is some 
 horizontal joint, other than 
 the base, and C its centroid, 
 then Q will represent the water 
 thrust, P the weight of ma- 
 sonry, and R their resultant. 
 
 In agreement with the or- 
 dinary theory we may write 
 the well-known formula for 
 the unit normal pressure on a 
 horizontal joint, at any point x, measured from A, as 
 follows : 
 
 *_.. 
 
 *-$+*%&} 
 
 (5) 
 
184 
 
 HIGH MASONRY DAM DESIGN 
 
 For the horizontal shear we must proceed further. 
 Consider, therefore, the forces to the left of HK in Fig. 33 , 
 we have (i) the vertical pressure on AK, (2) the weight 
 of AHK, and (3) the shear acting along AK. It is 
 evident that the difference between (i) and (2) repre- 
 sents the total vertical shear on HK. 
 
 If, therefore, the figure ALMB represent, in masonry 
 units, the distribution of normal stress on A B, as given 
 by Eq. (i), then ALTH will, in like manner, represent 
 the above-mentioned total vertical shear on HK. 
 
 B' 
 
 FIG. 33- 
 
 FIG. 34- 
 
 Consider now a second section A 'B', a small distance 
 z above AB\ the total shear on HK' may then be found as 
 before. Denoting the former by 5, and the latter by 
 5', then S-S f equals the total shear on KK' ', which, 
 when divided by z t will give the intensity of vertical shear 
 at K, and consequently the intensity of horizontal shear 
 at the same point. 
 
 Since all the forces to the left of HK are now known, 
 the normal stress on that plane may be found, and from 
 it we may readily determine whether tension or com- 
 pression exists at K. 
 
 At the base these results would be much modified, 
 
HIGH M;\ONRY DAM DESIGN 
 
 185 
 
 because of the discontinuity of form, which, in the opinion 
 of Prof. Unwin, places the exact determination of the 
 stresses beyond the power of mathematics. The author 
 believes the effect of the rock into which the dam is built 
 is to reduce the variation of stress which would otherwise 
 exist. 
 
 In a subsequent paper,* giving a complete demon- 
 stration of the preceding analysis as applied to a masonry 
 dam of triangular cross-section, it is found that the dis- 
 tribution of shear on a plane horizontal joint may be 
 represented by a right triangle whose base is the length of 
 the joint and whose vertex is 
 perpendicularly below the down- 
 stream edge. The figure illus- 
 trates the variation of normal 
 stress and shear on AB\ the 
 lines of resistance for both 
 vertical and horizontal planes; 
 and the centers of gravity of 
 the sections above the successive 
 horizontal joints. 
 
 Consequently the total nor- 
 mal or shearing stress on any 
 part of AB is equal to the area 
 between that part and the line 
 
 of normal stress or the line of 
 
 FIG. 35. 
 
 shearing stress.. 
 
 If the upward reactions and the weights of the dam to 
 
 *" Engineering," Vol. 79, page 593. "Further Note on the Theory 
 of Unsymmetrical Masonry Dams." W. C. Unwin. 
 
186 HIGH MASONRY DAM DESIGN 
 
 the left of each vertical section be combined with the 
 shears T, acting along AB, the resultants will cut the 
 vertical sections at points shown on the line of resist- 
 ance for these vertical sections. As this line lies wholly 
 within the middle third, there can be no tension on any 
 vertical section. 
 
 The total compressive stress on any vertical section 
 at its lower edge will therefore be: 
 
 where T is the shear on the horizontal plane from the toe 
 to the vertical section taken, y the height of the vertical 
 section, and z the distance from the center of the vertical 
 section to the point of application of the resultant forces 
 on that section. 
 
 Near the upstream toe the plane on which the greater 
 principal stress acts is found to be vertical while near 
 the downstream toe it approaches the horizontal. The 
 stresses are all compressive and on the water face the 
 compressive stress is at all points equal to the water 
 pressure at that point. 
 
 The above analysis is simply an application to vertical 
 sections of the method now accepted as applicable to the 
 horizontal planes and is a possible solution, since the 
 distribution of shear is known. It differs from Atcherly's 
 method in the fact that the latter assumes the usual 
 distribution of normal stress, together with a parabolic 
 variation for the horizontal shear. This latter hypothesis 
 the author thinks inconsistent with the previous one. 
 
HIGH MASONRY DAM DESIGN 187 
 
 Further investigations ^y Prof. Unwin* on dams of 
 various sections lead to the following conclusions: 
 
 (1) For a rectangular dam the distribution of shearing 
 stress on horizontal planes may be represented by the 
 ordinates of a parabola. 
 
 (2) For a triangular dam, the distribution may be 
 represented by the ordinates of a triangle with the apex 
 below the downstream toe. 
 
 (3) For a dam with vertical upstream face and curved 
 downstream face the distribution may be represented 
 by a figure consisting of a parabola superposed on a 
 triangle. 
 
 (4) For a dam with rectangular base the distribution 
 is represented by a parabola. 
 
 Following the results of the experimental investigations 
 of Atcherley and Baker, several other papers of a like 
 nature appeared in the Minutes of Proceedings of the 
 Institute of Civil Engineers, Vol. 162. The first of 
 these to be considered here is that by Sir John Walter 
 Ottley and Arthur William Brightmore, entitled, "Ex- 
 perimental Investigations of the Stresses in Masonry Dams 
 subjected to Water- Pressure." 
 
 In presenting this paper, the authors drew attention 
 to the fact that until the publication of Mr. Atcherley's 
 results, the question of dam design had been accepted 
 as settled, and that his memoir had had the effect of 
 reopening the entire subject of the distribution of stress 
 in structures of this class. 
 
 *" Engineering, " Vol. 79, page 825. "On the Distribution of 
 Shearing Stress in Masonry Dams." Prof. W. C. Unwin. 
 
188 HIGH MASONRY DAN DESIGN 
 
 It was also pointed out that tension was found by him 
 to exist on vertical planes near the outer toe, whether 
 the distribution of shearing stress over the base was 
 assumed to be uniform or to vary according to the para- 
 bolic law. 
 
 Considering a transverse section of a dam, the authors 
 argued that, whatever the distribution of shear over the 
 base might be, it must follow some other law near the top, 
 since the conditions in these higher levels are radically 
 different from those existing in the lower, where the dam 
 is fixed to the foundation, and where the water pressure 
 ceases abruptly. 
 
 The investigation was therefore undertaken, at least 
 in part, to determine the distribution of shear on horizontal 
 planes in the higher levels of the dam and to see how it 
 varied from that at the base; and it might be stated here 
 that it was found to be uniform in the latter plane but to 
 vary uniformly from zero at the heel to a maximum at 
 the toe in the higher levels, the change from the one 
 condition to the other being gradual. It will be shown 
 that it is near the inner toe rather than near the outer 
 toe that tension may be anticipated. 
 
 The model dams were triangular in section, made from 
 a kind of modeling clay called " plasticine," and so pro- 
 portioned that the resultant pressure on the base cut 
 that plane at the downstream extremity of the middle 
 third. 
 
 For purposes of observation the sections were placed 
 between vertical sides of plate glass, upon which vertical 
 and horizontal lines had been etched, corresponding to 
 similar lines on the model, so that any displacement in 
 
HIGH MASONRY DAM DESIGN 189 
 
 & 
 
 the latter might fre noted by comparison with the former. 
 Pressure was applied, by means of a thin rubber bag con- 
 taining water which was made to fit the frame. Though 
 the water was allowed to act over a period of 33 days, 
 after the elapse of one week a crack was noticed at the 
 upstream toe, running downward and at an angle of 
 about 45. At the end of the longer period an examination 
 showed that in the neighborhood of the base the dis- 
 placement of the vertical lines was such as to make them 
 all about equally inclined, thus indicating a uniform 
 intensity of shear on that section, while in the higher 
 levels and near the outer portion of the dam the lines 
 became more inclined as the elevation increased, indicating 
 that the intensity of shear increased also as the top was 
 approached. 
 
 Turning to the horizontal lines in the model for the 
 purpose of discovering the method of distribution of 
 normal stress, it was found that they were curves at the 
 base, sloping downward from the inner toe to a point 
 about two-thirds the distance to the outer toe, then re- 
 maining fairly level until almost reaching the down- 
 stream face, when they finally bent up slightly. In the 
 higher levels, however, these lines gradually developed a 
 uniform slope running from the inner to the outer toe. 
 
 An investigation of the shearing stresses on vertical 
 planes requires that, to draw the line representing the 
 intensity of normal reaction at the base the following 
 facts must be considered: 
 
 (1) The total normal reaction equals the weight of the 
 dam. 
 
 (2) Since the resultant pressure on the base acts at 
 
190 
 
 HIGH MASONRY DAM DESIGN 
 
 one-third the width from the outer toe, the moment of the 
 reaction stresses about this point must be zero. 
 
 (3) The intensity of the reaction at the outer toe 
 must equal the intensity of the shearing stress in the vertical 
 plane multiplied by the ratio of the height to the base of 
 the dam. 
 
 FIG. 36. 
 
 Referring to the figure: AB represents the base of the 
 dam, and BC twice the average intensity of normal stress 
 on AB. AC is then drawn; consequently ABC represents 
 the total normal stress on A B, or the weight of the structure. 
 
HIGH MASONRY DAM DESIGN 191 
 
 i 
 
 If AE, on the other hand, represents the actual intensity 
 of normal reaction over AB, then for (i) to hold true the 
 area Y must equal the areas (x + z) and if (2) is to hold, 
 the moments of x, y, and z, about D (equal to \AB from 
 B), must be zero; also for (3) to be satisfied, BE must 
 equal the limiting value of shearing stress in a vertical 
 plane near the toe, multiplied by the height and divided 
 by the base of the dam. 
 
 From these considerations AE may be fitted in by 
 trial till it is found to satisfy all of the above conditions. 
 
 Dividing the cross-sections into vertical strips i inch 
 wide we may properly consider the equilibrium of each such 
 strip. Evidently the difference between the weight of 
 each strip and the normal reaction on the base is equal 
 to the difference in shear on the two adjacent vertical 
 planes, and if in the figure these weights be plotted upward 
 from AE, the curve FE will result. Furthermore, both 
 the curves for " total shear on vertical planes " and 
 " average intensity of shear on vertical planes " may now 
 be drawn, whereupon it is evident to what extent the 
 average intensity of shear on vertical planes varies, and 
 how it compares with the average intensity on the 
 base. 
 
 Since the shear on horizontal and vertical planes at 
 any one point is equal, and the shear on the base is practi- 
 cally constant, it follows that above the base the shear on 
 horizontal or vertical planes is small near the heel while 
 in the outer half above the base it increases as the outer 
 edge is approached; in fact it increases from zero at the 
 heel to a maximum at the toe. These facts show that the 
 shearing stresses to be provided for are those existing 
 
192 HIGH MASONRY DAM DESIGN 
 
 in the higher levels and near the toe, and not those at the 
 base. 
 
 In considering the effect of shear on the base, neglecting 
 the " fixing " at that level, we may assume that the re- 
 action stress and that due to the weight of a strip, is constant 
 over each inch. They then act at the middle of each 
 strip; and, taking these points successively as centers, 
 the difference of the moments of the horizontal pressures 
 on the vertical sides of the strip, it is evident, will equal 
 the sum of the shearing stresses on the same vertical sides 
 multiplied by \ inch. 
 
 This makes possible the determination of the moment 
 of the horizontal pressures on each vertical strip. 
 
 The horizontal shear on each inch of base being the 
 difference between the horizontal pressures acting on the 
 two vertical sides, the latter may be determined as soon 
 as their points of application are given. As these points 
 are known for the innermost and outermost strip, an easy 
 curve may be drawn which will approximately locate the 
 other points and thus give the desired heights. From 
 these results it may be shown that the shearing stress on 
 the base increases from practically zero at the inner toe 
 to a point near the center of the base and then remains 
 fairly constant. 
 
 The modification of this distribution, due to the fixing 
 of the dam to its base, must, on the other hand, be con- 
 sidered. The water tends to cause a maximum pressure 
 and displacement at the inner face, which diminishes to 
 zero at the outer. As the dam is fixed, this displacement 
 is prevented, thus inducing corresponding shears, and the 
 effect of this conflicting condition, with that previously 
 
HIGH MASONRY DAM DESIGN 193 
 
 j 
 
 shown to exist, causes a"*nearly uniform shear over the 
 base. 
 
 Further evidence of uniform shear on the base was 
 obtained as follows: The models, after being subjected 
 to water pressure, showed cracks which appeared at the 
 inner toe, the angles which these made with the horizontal 
 steadily diminishing as the base was decreased in width 
 from a maximum of 45 for the widest base used to 25 
 for the narrowest. 
 
 The variation of these inclinations corresponded closely 
 with the computed directions, on the assumption that the 
 shear was uniform over the base and the experiments 
 therefore strongly support the inference that shear over 
 the base is uniformly distributed. 
 
 It was shown by means of the models that there are 
 tensile stresses on other than horizontal planes passing 
 through the inner toe. The models indicated this by 
 cracking, even when the back was sloped away from the 
 vertical so as to cause vertical pressure and hence com- 
 pression on the upstream face. 
 
 The impossibility of tension on vertical planes near 
 the outer toe may be shown by means of the following 
 equation for principal stress: 
 
 r r ' r ' \r i r / *t\rr ** / / \ 
 
 where compressions are plus and tensions are minus. 
 When pp' > q 2 at any point, there can be no tension at 
 that point, since under the above conditions both principal 
 stresses will be compression and hence stresses on all other 
 planes passing through that poinf will be compression 
 
194 HIGH MASONRY DAM DESIGN 
 
 also. This condition may be shown to exist near the outer 
 toe, and hence no tension can act across any vertical 
 plane in that position. 
 
 - For example consider the equilibrium of a wedge of 
 unit length cut off by a vertical plane near the toe. 
 
 p' = intensity of pressure normal to a vertical plane 
 
 at base. 
 
 p = intensity of reaction normal to the base. 
 q = intensity of shearing stress. 
 
 Then p'h = qb or, 
 
 (8) 
 
 The weight of the particle is negligible because it varies 
 with h 2 . 
 
 Since the resultant stress must be parallel to the outer 
 face it follows that, 
 
 (9) 
 
 Multiplying (8) by (9) there results 
 
 pp' =q 2 at outer toe. But p has been shown to in- 
 crease for some distance from outer toe and the point of 
 application of p f becomes relatively lower as the inner 
 toe is approached and since the average pressure is con- 
 stant it follows that p increases as the distance from the 
 outer toe increases and hence in the vicinity of the outer 
 toe pp f is greater than <f and consequently there can be 
 no tension in that neighborhood. This was checked by 
 the behavior of the models. 
 
HIGH MASONRY DAM DESIGN 195 
 
 P 
 
 It is a fact that in dam work the normal stress is the 
 only one specified, whereas the absolute maximum is about 
 50 per cent greater. 
 
 The conclusions reached from this set of experiments 
 follow : 
 
 (1) If a masonry dam be designed on the assumption 
 that the stresses on the base are uniformly varying and 
 that the stresses are parallel to the resultant force acting 
 on the base, the actual normal and shearing stresses on 
 both horizontal and vertical planes would be less than 
 those provided for. 
 
 (2) There can be no tension on any planes near the 
 outer toe. 
 
 (3) There will be tension on certain planes other than 
 the horizontal near the inner toe, and the maximum 
 intensity of such tension in the foundation being generally 
 equal to the average intensity of shearing stress on the 
 base, and the inclination of its plane of action being about 
 45; and its maximum intensity in the dam above the base 
 about J the above amount and acting on a plane less in- 
 clined to the horizontal. 
 
 The investigation undertaken by Mr. Hill * for " The 
 Determination of the Stresses on any Small Element of 
 Mass in a Masonry Dam," are on the other hand purely 
 analytical in character, being directed toward a solution 
 of (i) the vertical, (2) horizontal, and (3) tangential shearing 
 forces acting on the faces and along the edges of such an 
 element. 
 
 * Minutes of Proceedings of the Inst. of C. E., 72. 
 
196 HIGH MASONRY DAM DESIGN 
 
 In this analysis, there is first expressed a perfectly 
 general formula for C (the distance of the load point 
 from the center of the joint), and two other general formulae 
 for the pressures p\ and p2 in terms of the total load and 
 C from its above value, where pi is the minimum and 
 p 2 the maximum pressure. For the pressure p at any 
 point x on the joint of length b the following equation 
 is used: 
 
 (10) 
 
 Up to this point the analysis is identical with the general 
 procedure of investigation, which assumes that the hori- 
 zontal pressures are proportional to the vertical, and 
 does not analyze the shear. 
 
 Citing Prof. Unwin, the author states tnat the former 
 " suggested that the shearing stress at any point might 
 be found by considering the difference between the total 
 net vertical reactions (between that point and either face) 
 along two horizontal planes a unit's distance apart, and 
 has applied the principle by the use of algebraical methods." 
 Mr. Hill, on the contrary, employs the calculus to obtain 
 more rigorous results. 
 
 The procedure follows: Consider any point distant x 
 from the inner toe and on the lower of two horizontal 
 planes, a unit's distance apart. The total vertical reaction 
 
 is then I pdx. Subtracting the weight of masonry resting 
 JQ 
 
 on this portion of the horizontal joint, and denoting the 
 difference by r we have an expression for the " net 
 vertical reaction." If this value of r be differentiated 
 
HIGH MASONRY DAM DESIGN 197 
 
 ' jit 
 
 with respect to K, the distance between the two hori- 
 zontal planes, the change in the reaction will be obtained, 
 and this change or difference is the vertical shearing 
 stress at the point located by x. It is also, therefore, the 
 horizontal shear at the same point, which we may denote 
 byq. 
 
 If q be integrated with respect to x, between the limits 
 of x and b, the resulting expression will give the entire 
 horizontal shear between such limits on the joints in 
 question. Represent this by Qx. 
 
 To find the horizontal pressure intensity, we have 
 but to consider the above integration. This shear must 
 be resisted by the material along the vertical section at x. 
 Similarly the total shear on a plane a differential distance 
 below the last must be resisted by the vertical section 
 at x, differing in height from the former by dh. Conse- 
 quently the differential of Qx with respect to h=p' will 
 represent the horizontal pressure intensity at point x. 
 These expressions for p, p f and q therefore give respectively 
 the values of the vertical pressure intensity, horizontal 
 pressure intensity, and shearing force acting on a unit 
 element of mass. 
 
 Cain * presents a treatment of this matter, which, 
 while presenting no new features, is strictly arithmetical 
 in character, and in that respect at least differs from the 
 preceding. Its purpose, as Hill's, is to determine the 
 amount and distribution of stress at any point in a masonry 
 
 * Wm. Cain, M. Am. Soc. C. E., Trans. Am. Soc. C. E., Vol. 64, 
 page 208. 
 
198 HIGH MASONRY DAM DESIGN 
 
 dam, on the assumption that the law of the trapezoid 
 represents the variation of pressure on horizontal joints. 
 
 The analysis finally establishes formulae for (i) the 
 normal unit stress at any point in a horizontal joint, (2) 
 the normal unit stress on a vertical plane at any point 
 of a horizontal joint, (3) the unit shear on either horizontal 
 or vertical planes at any point of a horizontal joint, and 
 at the same time indicates the method of determining the 
 maximum and minimum normal stresses and the planes 
 on which they act. 
 
 The solutions are only approximate, but the results are 
 found to be close enough for the purpose. 
 
 Before proceeding it may be advisable to review certain 
 features involved in a consideration of the stresses in a 
 masonry dam which Prof. Cain presents in a very satis- 
 factory manner. 
 
 i. It will be evident from an examination of the figure 
 that the intensities of shear on two planes at right angles 
 
 to each other are equal. For, in 
 the elementary cube under consider- 
 ation, the weight may be neglected, 
 - since it is an infinitesimal of the third 
 order, while the opposing normal 
 forces balance as the cube is reduced 
 in size. 
 
 For equilibrium then, q-a-a = 
 q' -a-a, or q = q f and, because each side is a differential 
 quantity, it may be assumed that the values q and q' 
 represent the average unit shear on the respective faces. 
 As a consequence they are equal to the shear at any 
 point, for example A, of the particle. 
 
HIGH MASONRY DAM DESIGN 
 
 199 
 
 2. In a triangular element of the dam, Fig. 38, at the 
 down-stream edge, and of unit's length, the forces acting 
 are those shown. Because it is an element we may neglect 
 the weight, and therefore, if p r is the normal intensity of 
 stress on a vertical plane, p the normal intensity of stress 
 on a horizontal plane, and q the shear intensity, for 
 
 for 
 
 then 
 
 or, 
 
 or pq~r and q=ptd.n(f). . (n) 
 o 
 
 I.H = o,p'a=qb or p' = q -. . . (12) 
 
 p f =p tan 2 0, 
 
 PP 
 
 (13) 
 
 FIG. 38. 
 
 FIG. 39. 
 
 3. The same analysis may be applied to an element 
 at the inner face, Fig. 39, where <$>' is the inclination to the 
 vertical; but, for the reservoir full, the intensity of water 
 pressure, horizontal or vertical, at c, and in this case 
 represented by w, must be taken into account. 
 
200 HIGH MASONRY DAM DESIGN 
 
 Under these circumstances, 
 
 pb=qa + wb, . . . , . . (14) 
 and 
 
 p'a=qb + wa, . . . ... (15) 
 
 (16) 
 and 
 
 p'=q tan <f>' + w ...... (17) 
 
 When, as is usually the case, the vertical component 
 of water pressure acting along the back is neglected, the 
 above equations become, 
 
 p = qcot<j>', ...... (18) 
 
 p'=q tan <f>' + w, ..... (19) 
 
 ' y ...... (20) 
 
 and 
 
 p f = tan 2 (/>' + w ....... (21) 
 
 4. If an element at the down 
 
 X 
 /^v stream face be again considered, since 
 
 tne shear on the outer face DC is 
 
 
 A^ q zero, that on a plane AD perpen- 
 \P dicular to DC, must be zero also, and 
 
 hence the stress d on A D is wholly 
 FIG ' 4 - normal. 
 
 The total pressure on AD is therefore, 
 
 f-AD=f-bcos<f>. (22) 
 
HIGH MASONRY DAM DESIGN 
 
 201 
 
 The vertical component of this is /-6cos 2 <, because 
 
 or, 
 
 pb=f-bcos 2 <f>, 
 
 P 
 
 p sec 2 <, 
 
 (23) 
 
 (24) 
 
 which is the maximum intensity of normal stress at the 
 outer face. 
 
 FIG. 41. 
 
 5. To determine the planes of principal stress, i.e., 
 planes upon which the stress is wholly normal, and also 
 the intensity of that stress, we may assume the conditions 
 indicated in the figure. 
 
 The total stress on c then is jc ; its vertical component 
 jc cos 6 =fb, and its horizontal component fc sin 6 =/a. 
 
 When 2V = o and 
 
 , . . (25) 
 
 -p = qcot 0, . (26) 
 
202 HIGH MASONRY DAM DESIGN 
 
 The difference of these two equations gives, 
 
 p-p'=q (cot 0- tan 0) =q . (27) 
 
 2 tan 20 
 
 This equation gives a plane upon which there is none 
 but normal stress. 
 
 To determine/, multiply equation (25) by (26). 
 
 (f~P)(f-p')=q 2 , ..'..- (29) 
 whence, 
 
 q*)}- . . -(30) 
 
 This will give two values of /, which correspond to the 
 two principal planes of stress, the stress being compressive 
 when / is positive, and tensile when / is negative. There 
 can be no tension when pp'^.q 2 . 
 
 Determination of the vertical unit stress at any point 
 of a horizontal plane joint: From the law of the trapezoid, 
 we have the pressures at the upstream and downstream 
 toes represented respectively as follows: 
 
 W, ..... (31) 
 
 jr 
 
 W. . . (32) 
 
 The resultant is supposed to act within the middle 
 third. If x f represent any point along EB, measured 
 from E, then /?, the pressure at x, is given by, 
 
 iv, ..'. . . (33) 
 
HIGH MASONRY DAM DESIGN 
 i 
 
 while the total noHnal stress from E to xf is, by integration, 
 
 (34) 
 
 To find the unit shear on vertical or horizontal planes, 
 we have but to consider a slice of dam between two hori- 
 zontal joints one foot apart, extending from the inner to 
 the outer face, a distance x along the lower joint. (The 
 back is supposed to slope .02 feet for each foot in height). 
 
 "I" a--Q-nr 
 
 a?,! I 
 
 FIG. 42. 
 
 |p 
 FIG. 43- 
 
 The vertical forces acting are: 
 
 (1) A uniformly varying stress on the upper joint 
 acting downward. 
 
 (2) The same on the lower joint acting upward. 
 
 (3) The weight of the strip. 
 
 (4) The shear on the vertical face at x. 
 For equilibiium, 
 
 (35) 
 
 P f and P may be obtained as indicated in the previous 
 demonstration. 
 
/ 
 
 204 HIGH MASONRY DAM DESIGN 
 
 The above value of q\ is the average unit shear at the 
 depth taken, but a similar value q^ may be determined 
 at a depth one foot below. Under these circumstances 
 
 is the average of the two, and may be said to be 
 approximately equal to the shear at the depth of the joint 
 between the two slices. 
 
 Q' 
 
 '/ . 
 
 X 0.01 
 
 X -H 
 
 M 
 G - 
 N 
 
 p. 
 
 07+0.01 
 
 FIG. 44. 
 
 To find the normal unit stress on a vertical plane, a 
 similar section to that just used may be employed; but the 
 horizontal components are now to be equated for equilib- 
 rium. 
 
 Let h = the horizontal water pressure at the assumed 
 
 depth. 
 
 Q' = total shear on upper face. 
 0= total shear on lower face. 
 p' = average normal stress. 
 
 q\ and q^ = the intensities of horizontal shear at the points 
 indicated. 
 
 Q' and Q may be found by integrating q\ and q^ be- 
 tween the proper limits. 
 
 '-'..-. f-h + ff-Q. . .... . (36) 
 
 This value of p f is assumed as the average intensity 
 on the vertical plane and as the unit intensity on the 
 
HIGH MASONRY DAM DESIGN 205 
 
 '> 
 
 vertical plane at "a point midway between the two hori- 
 zontal planes. 
 
 Three general formulae may be written for p, q, and p f 
 which, it has been suggested, be put in the following form: 
 
 p = a + bx, (37) 
 
 (38) 
 (39) 
 
 \ 
 
I 
 
 'Jfc 
 
 APPENDIX I 
 
 DERIVATION OF CANTILEVER EQUATIONS 
 
 THE expression for the moment of inertia, J, of the hori- 
 zontal cross-section of the cantilever contained between 
 two vertical radial planes of the arched dam, and the 
 two fundamental equations for deflection represented by 
 A and n , will now be derived. Th derivation of the 
 expressions for these last two quantities for the special 
 case of an arched dam of rectangular vertical cross-section 
 will also be indicated. 
 
 Derivation of the expression for the moment of inertia, I, 
 of the horizontal cross-section of the cantilever at the 
 distance x, below the given origin, or Eq. (i) of page 146: 
 
 
 3 6R n ( 2 R n -Bx) 
 
 Referring to the nomenclature of page 145, and to 
 Fig. 27, it is evident that l=Bx. (Cf. Fig. 45 and Fig. 46.) 
 
 If we assume two vertical radial planes to intersect 
 the dam, i foot apart at the up-stream face, the plan of 
 the section to be considered will be indicated by the shaded 
 area in Fig. 45. It will be sufficiently exact, however, to 
 substitute the chords for the curves themselves as limiting 
 this area. Furthermore, where the up-stream face of the 
 
 section under consideration is battered, the average value, 
 
 207 
 
HIGH MASONRY DAM DESIGN 
 
 between the two successive load points, may be assumed 
 for the up-stream radius, /?,, as local variation in this respect 
 
 By >-, 
 
 FIG. 45. 
 
 * 
 
 has been treated as negligible in the derivation of the 
 general expression for 7. 
 
 The resulting expression will then be transformed by 
 the substitution of like terms into one corresponding to 
 
 CentroiJ 
 
 !__+_* 
 
 FIG. 46. 
 
 the nomenclature of Fig. 45, which wifl be the desired 
 
 Eq. (i). 
 
 The reciprocal of Eq. (i) will then be modified by 
 factoring its denominator, in order to render it integrable 
 in further operations where it may occur. 
 
APPENDIX I 209 
 
 I 
 
 The general expression "for the moment of inertia, /, of 
 Fig. 46, is 
 
 I=fXY*dY. ...... (a) 
 
 in which 
 
 Substituting this value of X in Eq. (a) and designating 
 the limits of integration as 
 
 f/F.+aFV 
 
 + 
 
 and 
 
 V= 2 ' 
 s 
 
 there results: 
 
 /-\ I ^ : 
 _/-T 
 
 F-F, r*5<Ka.^_ 
 
 Performing the above indicated integrations, 
 
 rt rpr'w 
 
 _ _ 
 
 F-F ' M 
 
 Completing the above, gives 
 
 -3(F-F 1 )[(F 1 +2F) 
 
 *-(2F 1 +F)*]\, 
 
210 HIGH MASONRY DAM DESIGN 
 
 which may be reduced to 
 
 T _ *- \->- 1 / J\ 
 
 ^ / T^ - 7~*\ * \^*'/ 
 
 To transform Expression (d) into the form of Eq. (i), 
 substitute in (d), i for F, i for Fi, and 5^ for /, 
 as indicated in Fig. 45 . There will then result 
 
 j 
 
 3 6R n ( 2 R n -Bx) 
 which is Eq. (i) of page 207.^ 
 
 To render -= integrable by factoring. 
 
 From the above equation we have 
 
 By factoring ^ 2 ~~D~^+ 6 (^) m tne denominator of 
 
 (e) y into the typical factors, (x+d)(x+b), there results, 
 after reduction, the following integrable form : 
 
APPENDIX I 211 
 
 ill 
 
 Development of the equations for Deflection in the Cantilever 
 for EA a and E$ n : 
 
 Eqs. (9) and (10) of page 150 are the fundamental 
 differential expressions from which are derived Eqs. (13) 
 and (14), for the cantilever. 
 
 As there stated, the expression for I and that for 
 M are employed by substituting them in Eqs. (9) and 
 (10) and integrating. These operations are indicated 
 below. 
 
 V 
 
 Derivation of E n . 
 Eq. (9) of page 150 is 
 
 Eq. (n) of page 150 is 
 
 Eq. (/) of p. 210, Appendix, may be written 
 
 The general integral of Eq. (9) may be written 
 
 (g) 
 
 Substituting the expressions for M n and j in Eq. (g) and 
 reducing, gives 
 
212 
 
 HIGH MASONRY DAM DESIGN 
 
 -gyv 
 
 t dx 
 
 * - (3 - V 3 ) 
 
 
 +(Pidi+P 2 d 2 + 
 
 * 2 [* - (3 + V?)'] [* ~ (3 
 
 *[* - (3 + V3)] [x - (3 - V 3 ) n ] 
 
 +P n 
 
 .To integrate Eq. (h) with respect to #, between the 
 limits c^ n+1 and c^ ra , it will be necessary, first, to determine 
 the integrals for the four fractions containing dx in the 
 numerator of each, and the denominators as written above, 
 functions of descending powers of x, beginning with x 3 , 
 as a factor for the first denominator. 
 
 These separate integrations may be accomplished by 
 expanding each fraction into a series of partial fractions 
 by the method of undetermined coefficients and then 
 integrating each term of the series. This will result in 
 Eqs. (f), (/), (k), and (/). These equations will serve for 
 the derivation of E0 n , as well as for EA n . 
 
 According to the theorem of undetermined coefficients, 
 there may be written for the first fraction of Eq. (h) 
 expanding in ascending powers of x and distinguishing 
 
213 
 
 B, the coefficient, b^ a vertical letter from B, the batter, 
 an inclined letter, 
 
 dx _Adx Jbdx Cdx 
 
 1 5" I o~ 
 
 X X 2 X 3 
 
 *-(3 + V 3 )^||*-(3- 
 
 Edx 
 
 
 
 Clearing of fractions, collecting the terms in the second 
 member involving like powers of x, equating coefficients 
 of like powers of x in the two members and solving, gives 
 
 - 
 
 The integral of this first fraction may thence be written 
 
 doc 
 
 S_IB\* r +i dx i/B 
 
 3W . * 6 
 
 -MrT 
 
 B 
 
214 HIGH MASONRY DAM DESIGN 
 
 The definite integral of which is 
 
 r d +i dx 
 
 [/- R 1 /R 
 
 *- (3 + vfrg \\x - (3 - v 3) n 
 JD J L n 
 
 $B^ d B^(d d ^ R2f/72 
 
 "1 s T~t A -^Oe J I f T~> 1 _7 J 
 
 The second fraction of Eq. (/&), by the method of unde- 
 termined coefficients, may be expanded, thus: 
 
 dx Adx Edx 
 
 "1 " 
 
 
 ? z? 
 
 whence there follow the values : 
 
 36 
 
 3U ' 
 
APPENDIX I 215 
 
 4* 
 The integral of the second fraction may then be 
 
 written : 
 
 r d n+i dx 
 
 / r R i r R 
 
 Jd n y?\x - (3 -f-V^) w 1 1 x - (3 - v^)-^ 
 
 r* ^n +1 f* ^"n -4- 1 
 
 . I(J/^ ^ + l\Rj J J 
 
 , - n+l 
 
 36 
 
 The definite integral of which is 
 
 , d n+1 
 d n 
 
 i-( 3+ VJ)f 
 
216 HIGH MASONRY DAM DESIGN 
 
 Similarly, for the third fraction of Eq. (h), 
 dx Adx 
 
 Edx Cdx 
 
 Proceeding as before, there result, 
 
 12 
 
 The integral of this third fraction follows 
 dx 
 
 r d n+i 
 
 J*. X\X- 
 
 - 
 
 VJ-i/gy f 
 \RjJ da 
 
 12 
 
The definite integral of which is: 
 
 APPENDIX I 217 
 
 Df 
 
 dx 
 
 - + log + - l log 
 
 & U & e 
 
 I2 7? 
 
 And for the fourth fraction of Eq. (h), 
 
 dx 
 
 W 
 
 Adx Edx 
 
 = H 
 
 From which : 
 
 L 
 
 The integral of this fourth fraction is 
 
 B 
 
218 HIGH MASONRY DAM DESIGN 
 
 The definite integral of which is : 
 doc 
 
 L 
 
 The integration of Eq. (/&) may now readily be accom- 
 plished by simply substituting in that equation the right- 
 hand members of Eqs. (i), (/), (k), and (/), above, for 
 their corresponding left-hand members, as they occur in 
 Eq. (fe). 
 
 By performing the indicated operations, collecting 
 terms, and writing, in place of the Napierian logarithm, 
 the conversion factor, 2.30259, so that common logarithms 
 may be used in computations, there results Eq. (13), 
 for EA n , as finally written on page 153. (Note that l n 
 has been substituted for Bd n and l n+1 for Bd n+1 , in this 
 expression.) 
 
 Derivation of E0 n . 
 
 Eq. (10), of page 150, is: 
 
 , Mdx x 
 
 The general integral of which may be written : 
 
 E CdB = Cjdoc. . . . . . . (m} 
 
APPENDIX I 219 
 
 i 
 
 Substituting .(/), - from page 211 and the expression 
 
 for M n , Eq. (n), in the above Eq. (m), and collecting terms, 
 gives : 
 
 (Pl +P 
 
 [* - (3 + V3)|" ] [* - (3 - VJ)|] 
 
 * - (3 + VI)|"] [* - C3 - vD 
 
 Obviously, the definite integrals of Eqs. (i), (/), and 
 (k), as developed, are directly applicable to Eq. (n), above. 
 
 Substituting the right-hand members of Eqs. (i), (/), 
 and (), for their corresponding left-hand members, as 
 they occur, in Eq. (n), performing the indicated operations, 
 collecting terms, together with substitution of the con- 
 version factor, 2.30259, and l n+l and /, as explained in 
 the treatment of Eq. (h), above, will result in Eq. (14), 
 for Ed n , as finally written on page 153. 
 
 Deflection equations, for Arched Dam of rectangular, vertical 
 cross-section. 
 
 Derivation of ElA n . 
 
 The moment of inertia, being a constant, becomes a 
 factor with E on the left side of the equation. 
 
220 HIGH MASONRY DAM DESIGN 
 
 As before, Eq. (9) applies, or, as d n = na, 
 
 Whence 
 
 X(w + l)a f*(n + l)a 
 
 dA n = I M(x-nd)dx. t - . (p) 
 
 i Jna 
 
 Substituting the value for M n from Eq. (15) of page 167 
 and performing the integrations indicated and collecting 
 terms, Eq. (16) results directly. 
 
 Derivation of EIO n . 
 
 Eq. (17) of page 167 may be similarly derived by means 
 of Eq. (10), which may be written 
 
 El C l} de n = ( Mdx, 
 
 X(+l)a /~(n+l)a 
 
 dB n = j 
 I JtM 
 
 into which the value of M n , from Eq. (15), is substituted 
 and integrations performed, terms collected, etc., as before. 
 
APPENDIX II 
 
 MOVEMENTS AND STRESSES IN AN ARCH SUBJECTED TO 
 A UNIFORM, RADIAL LOAD 
 
 With Derivation of Eq. (8}, of page 149, for Arch Crown Deflection. 
 
 The nomenclature of page 145, together with designa- 
 tions of Fig. 47 and such other as may be immediately per- 
 tinent, will obtain in the following discussion which is 
 largely adapted from a discussion by the late R. Shirreffs.* 
 
 In Fig. 47, line 1-2 '-3' represents one-half of the axis 
 of a segmental arch ring in its unloaded position. For 
 convenience of reference the line $ f -0' may be assumed as 
 vertical and passing through the crown of the arch. The 
 abutment, or skewback supporting the arch, may be assumed 
 to be in the line 1-4-0'. <, then, is one-half the central 
 angle of the arch span, and r n the radius of the axis. 
 
 In elucidating the various analytic expressions for 
 effects of loading both as to stressing and deflecting the 
 arch, certain changes in position are imagined. 
 
 These are, in order, as follows : 
 
 Beginning with the unloaded arch, one-half of which is 
 shown in Fig. 47, and assuming it either to be a portion of 
 a closed ring or to rest upon frictionless abutments, a radial 
 loading of intensity q n , will produce a shortening of the 
 
 * Trans. Am. Soc. C.E., Vol. LIII, p. 163. 
 
 221 
 
222 
 
 HIGH MASONRY DAM DESIGN 
 
 original length and a reduction of the radius length by a 
 proportionate amount, /. The half arch will be moved 
 into a new position 4-5-6. It will be in equilibrium under 
 only the axial thrust q n r n . 
 
 Secondly, assume the arch severed at the crown and 
 the right half, removed, replaced by a crown thrust = q n r n 
 
 FIG. 47. 
 
 and the remaining half arch moved outward until Point 4 
 again coincides with Point i. The perpendicularity of the 
 axis at the skewback remains unchanged. 
 
 The center of the arch will be at o instead of 0' ; Point 6 
 will be in the position of Point 3, at the horizontal distance 
 k' from the vertical through the middle of the span or 
 crown of the arch, and at the vertical distance A c below 
 
APPENDIX II 223 
 
 I 
 
 the original crown -ef the arch. The crown joint will still 
 be vertical. 
 
 Thirdly, in order to restore the integrity of the arch, 
 under its loading, the crown thrust, q n r n , must be so di- 
 minished that under the combined action of this diminished 
 thrust and the loads on the half arch, the curved beam 
 1-2-3, now considered fixed at the abutment, shall be 
 deflected through the horizontal distance k' and as the 
 original crown thrust, q n r n , just holds the arch in equilibrium 
 against the action of the loads, a force H f applied at the 
 crown and equal to the necessary diminution of q n r n , H' 
 acting therefore to the right, will cause a movement iden- 
 tical with that through k' '. 
 
 Fourthly, the crown joint, which will have been de- 
 flected through an angle 0, by this movement, must again 
 be made vertical in its new position. This can be accom- 
 plished only by the application of a moment, M e . 
 
 Fifthly, the total movement of the arch at any point 
 will be obtained by combining the movement resulting from 
 axial stress with those movements produced by the force 
 H' and the moment M c . 
 
 The above considerations will next be treated analy- 
 tically. 
 
 Derivation of expression A c , under axial thrust. 
 
 Assume the arch ring to be of thickness = / and a depth 
 (normal to the plane of the paper in Fig. 47) of i. 
 
 Let X = shortening of the half length of arch shown in 
 
 Fig. 47. 
 
 L = curved length of arch from abutment to crown. 
 / = area of radial vertical cross-section of arch ring. 
 E = modulus of elasticity. 
 
224 HIGH MASONRY DAM DESIGN: 
 
 Then 
 
 ^-- = intensity of stress in ring. . . . (i) 
 
 -= strain due to stress. . ..... (2) 
 
 j --/ 
 
 Whence X= n ' . . . . (3) 
 
 Since the ratio of the shortening of the arch to its 
 original length is equal to the ratio of the change in length 
 /', of the radius to its original length, 
 
 L = 7 a 
 Whence 
 
 /=^f (4) 
 
 Combining Eqs. (3) and (4) gives 
 
 And as k r =/ sin $ n (see Fig. 47), 
 
 y-sffl*; .... (5) 
 
 and as A c =/(i -cos n ), from Eq. (40), there follows, in 
 this connection: 
 
 (6) 
 
ABPfeNDIX II 225 
 
 > 
 
 At a distance of </> degrees from the crown there will obtain 
 
 A^ : A c = fa < : <t> n , 
 therefore 
 
 Derivation of expression for H f , or diminution of stress 
 q n r n . 
 
 The slight reduction in the compression of the arch ring 
 due to diminishing q n r n by the amount H f is neglected in 
 the following. 
 
 The general expression for the differential deflection 
 ds, of a beam is 
 
 Mxdx 
 
 in which x is referred to any point in the beam's axis, 
 I ds is the deflection with respect to that point and M 
 is the bending moment about that point. (See Point 2 of 
 Fig. 47-) 
 
 In this case 
 
 M=H'r n (i cos <), 
 
 x = 2r n sin , sufficiently close. 
 2 
 
 dx = r n dcj>, 
 
 J 3 
 I--2k 
 
 12 
 
 Substituting these last expressions in the general expres- 
 sion for ds, results in 
 
 ds' = * (i cos 0) sin -d^>. 
 
 2 
 
226 HIGH MASONRY DAM DESIGN 
 
 But the horizontal component of ds' is dk', 
 
 since (Fig. 47) ds', approaching zero, sensibly coincides in 
 direction with the line 3-7, the tangent to the arc ds'. 
 Therefore 
 
 *dt. .... (7) 
 
 Integrating Eq. (7) between the limits <f> n and o and 
 equating the result with the right-hand member of Eq. (5), 
 for k', gives 
 
 2 Sin * 
 
 . 
 12 3 n + sin 4> n cos re - 4 sin 
 
 (8) 
 
 Derivation of expression for moment M c , or the moment, 
 the effect of which is to render the crown joint vertical, 
 in position. 
 
 The moment M c must cause the same angular deflection, 
 j8, in the entire beam as the force H f . 
 
 The general equation for angular deflection is 
 
 x 
 
 x and M are again taken with reference to the same point, 
 which may be any point at <, r n in beam. (See Point 2.) 
 From the foregoing considerations, substitute for . 
 
 M =M C =H'r n (i -cos 0), 
 
 12 
 
APPENDIX II 227 
 
 i 
 
 in the above expression for $0, whence 
 
 dp= ^^-(i-cos 
 
 Integrating both expressions for dp between the limits 
 of <f> = 4> n and = o and equating the results give : 
 
 M.=H'r *-*** +*. (9) 
 
 <t>n 
 
 Substituting the value of H'r n from Eq. (8) above in 
 Eq. (9) gives for M c , 
 
 M = gn4 2 /<An-sin <j> n \ 2 sin M / , 
 
 12 \ <t> n /30 n +sin <f> n cos (j) n -4 sin <J> n ' 
 
 M c is opposite in effect as to resulting deflection, to that 
 produced by H f . 
 
 The deflection of the arch under the combined action of 
 H f and M c . 
 
 The object of this phase of the analysis is to determine 
 the combined movement in a radial direction, due to H' 
 and M c , for this radial deflection is the arch deflection 
 which, in a curved dam will produce stresses in the vertical, 
 or cantilever beams. 
 
 The general expression for the deflection of a beam 
 again serves, i.e., 
 
 ds = Mx m- 
 
 The reference point or origin now will be some other 
 point than 2, of Fig. 47, such as #, at any angle co with the 
 radius through Point 2. 
 
228 HIGH MASONRY DAM DESIGN 
 
 The two moments considered with respect to Point x 
 
 will be 
 
 H'r n [i -cos O + co)] and M c . 
 
 The resulting moment, etc., will therefore be 
 M=#V[i-cos (0 + co)]-M c . 
 
 CO 
 
 X = 2T n Sin . 
 2 
 
 I 3 
 
 dx=r n du and J= . 
 12 
 
 Substituting these in the general expression for ds above 
 gives 
 
 '* ={H'r n [i -cos (* + )] -M c ] sin -du. 
 
 2 
 
 Let the radial component of this movement ds'^ be 
 represented by dD'+. 
 Then 
 
 therefore 
 
 dD=\H'r n V -cos (0 + co)] -Me! sin 
 
 Integrating this last expression for dD' , between the 
 limits co = (< 4>)=7 and co=o and substituting J/V n and 
 M c , as written in Eqs. (8) and (10) respectively, gives 
 for D', 
 
 in which C has the general value given in Eq. (na), or 
 
APPENDIX II 229 
 
 ~ \2 sin n/ " v . sin $, . N 
 
 G = (i cos 7) H (27 sin 27) 
 
 L 4> n 2 
 
 COS <p / \ I / ' 
 
 - (COS 27 l) -r- ( -r 
 
 When 0=o and therefore 7 = </>, the value of C is as 
 given in Eq. (u&), or 
 
 *?(i -cos n ) +Kcos 2 n -i) 
 
 sin n 
 
 and for the semicircle, since < n =-, 
 
 2 
 
 
 (nc) 
 
 To combine the effect of the axial thrust or A c with the 
 crown deflection as given in Eq. (116) and Eq. (n), there 
 should be noted the following : 
 
 The last term in the numerator of Eq. (nfr) involves 
 2 0. This may be written in terms of <, for 
 
 J(cos 20 rt -'i) =(cos 2 n -i), 
 
 or 2 
 
 also, the coefficient of C in Eq. (n), -^f- is the same as 
 
 hl n 
 
 the coefficient of (i cos n ) in Eq. (6) for A c . Hence the 
 combined trigonometric function of < n designated as CC C , 
 Fig. 28, which is the coefficient plotted as a curve, referred 
 to before, for the common factor of Eqs. (n ) and (6), 
 
230 HIGH MASONRY DAM DESIGN 
 
 or 2 
 
 viz., *?- This combination is written as Eq. (8) on 
 El n 
 
 page 149 for D e . 
 
 Derivation of the moment M $, at any point, <j> degrees 
 from the crown. 
 
 The moment is HV n (i -cos 0) -M c . 
 
 Substituting the values of H'r n and M c as written in 
 Eqs. (8) and (10), and reducing give 
 
 (12) 
 
 sin 
 
 M becomes o, or there is a point of contra flexure in the 
 curved beam used as an arch under a uniform radial load, 
 
 when 
 
 sin n 
 cos < = , 
 
 or in the semicircle when 
 
 cos 0=-, 
 
 7T 
 
 or at about 50 from the crown. 
 
 The stresses of tension or compression produced by M^ 
 at any point in the arch may be combined with the axial 
 stress q n r n , whence the resultant stress at that point may 
 be obtained. The force H' t except in very flat arches, may 
 usually be neglected in getting the resultant stresses. 
 
APPENDIX III 
 
 CROSS-SECTIONS OF EXISTING MASONRY DAMS 
 
 THIS selection of cross-sections of notable masonry 
 dams, following Table XIV, is arranged in chronological 
 order to illustrate the evolution of the masonry dam, 
 the design of which has been largely a matter of following 
 precedent, in many cases. 
 
 The series begins with a few of the heavy, Spanish 
 type and continues through the French designs, such as 
 those of de Sazilly and Delocre, to the types of present-day 
 construction, as illustrated by foreign and American 
 (United States) examples. 
 
 With respect to the design of de Sazilly, reference to 
 whom was made early in Chapter I, it should be stated 
 that his analysis resulted in a cross-section that was 
 necessarily stepped, and Delocre suggested substituting 
 curved faces in place of the steps, thereby effecting an 
 appreciable saving of material in construction. 
 
 Attention is directed to the fact that all of the sixty- 
 one sections could not be shown to the same scale; but 
 each cross-section is sufficiently dimensioned so that com- 
 parisons are possible. 
 
 The various methods of designating batters are also 
 apparent. 
 
 The later, overfall types only, have been grouped 
 separately. 
 
 231 
 
232 HIGH MASONRY DAM DESIGN 
 
 Table XIV contains a page index of the cross-sections, 
 together with other data concerning each, for convenience 
 of reference. 
 
 In the table, under " Type of Section," 
 
 G= gravity section (depending on gravity); 
 A = arch section (depending on arch action) ; 
 
 and under " Type of Service," 
 
 N.O. = non-overflow; 
 S. = spill way dam. 
 
 A dash, in the column, headed " Radius, etc.," denotes 
 a straight alignment of dam. 
 
APPENDIX III 
 
 233 
 
 > S 
 
 t I jj 
 
 X Q 
 
 9 I 
 
 W o 
 
 S| 
 
 ign 
 or 
 lder 
 
 De 
 Bu 
 
 s 
 
 Hi 
 
 S p< ''.>b^^^S'^-S^a^'"'' 3 a3S^^S > 
 to fe <u^ w & ^3 g, 4 " -g g3 a 2-0 > SStS'S aS^ -O^S 
 
 <L)Q.><UC! <u wO' Q odhS' Q O^ (i; aji2 l ^7^te3 S 13 
 
 "^ 5 ^ .* S .8 a 3 'oOcSi' 3 *u <l) ^ OT fe2o^M K^S 
 . pq Jo co (3 O<3 O co H "^ "~ ' ' ' "~ """ 
 
 
 8 8 
 
 
 
 fe pc, 
 
 8 1 
 
 sSs 
 
 g 9 
 
 H 
 
 
 
 W * jy * W 
 
 fO t^ N 
 t-00 O> 
 
 00 00 00 
 
 S 
 
 Ov 00 00 
 
 OOOOOO. .OO OOO . O 
 
 O O O O < OO O O O OOO O O 
 
 \o ^o ^* 
 
 O f*5 
 <5 * 
 
 : 
 
 
 t 1 
 
 ' 
 
 (U<u.2J J O O O'rC 
 
 pq pq pq pq pq pq o 
 
 O 
 
 U O 
 
234 
 
 HIGH MASONRY DAM DESIGN 
 
 
 Remarks. 
 
 N. Y. City Water-supply. Cyclop, 
 masonry; concrete face blocks. 
 Chemnitz Water Supply. Rubble 
 
 masonry. 
 U. S. Reclamation Service. 
 Remscheid Water Supply. 
 Near Pittsfield. Expansion joints 
 similar to those of Kensico Dam. 
 Near St. Etienne. Delocre de- 
 sign. Is watertight dam. 
 Rubble and cut stone. 
 Partially failed in 1885. Rubble. 
 Foundation, soft rock. Buttressed, 
 and in 1896 earth filled on face. 
 
 Poilo^in rRRr flnnrl ti ft. nvpr rrfist. 
 
 Repaired in 1883-7. New profile 
 not found. 
 Failed. Rubble masonry. 
 Almeria. Two dams. Rubble with 
 cut stone facing. 
 At foot of Indian Lake, N. Y. 
 Thermophones installed to detect 
 internal temperature changes. 
 Tar and asphalt waterproofing in 
 
 up-stream face. Drainage system. 
 Rubble. Has had 15 ft. depth of 
 
 water on crest. 
 For Denver Water Supply. Gran- 
 ite in Portland cement mortar. 
 
 Cyclopean masonry. 
 Straight for half its lehgth. Rest 
 arched. 
 Three-coat, pitch waterproofing in 
 
 up-stream race. 
 New York City Supply. Spillway 
 at one end. 
 
 
 
 
 '$ 
 
 
 03 
 
 
 
 
 
 Location. 
 
 New York . 
 Germany. . 
 
 w G 'Sb .2 G .! 
 
 o ^ m <i < 
 
 i .j|.S a 
 
 I 
 
 U 
 
 o s 
 
 ! 
 
 Continued 
 
 Designers 
 or 
 Builders. 
 
 American . . . 
 German. . . . 
 
 G ' G ' 
 
 cs JH oj . a 
 3 S <3 a S> G G ! 
 
 -:j a 
 5 ll || | 
 
 > a) t* p a & 
 
 H Ul Q^ G G p 
 
 American . . . 
 
 American. . . 
 
 V *< 
 
 American. . . 
 
 J 
 
 g-a? 
 
 
 H- 
 
 
 oo 
 
 ^ 
 
 00 O 1-1 
 
 1 
 
 R 
 
 ^w& 
 
 M 
 
 
 
 
 
 
 
 rj 
 
 
 
 
 ,-.00 
 
 
 
 
 
 J 
 
 m 
 
 gl 
 
 O\ 00 
 
 "'"oo o\ oo oo oo oo c 
 
 - %k O^H- 
 
 3 oo o oo ^ a 
 
 1 
 
 | 
 
 & S 
 
 | 
 
 
 cl ^ 
 
 
 
 
 
 
 
 
 H 
 
 ^ G 
 
 O 
 O Os 
 
 a oo 
 
 N O\ ! N O <N O 
 O OO ' 00 OO 00 00 C 
 
 O O " '(NO 
 
 3 00 ' 'MO 
 D 00 O\ O\ 
 
 
 
 o\ 
 
 00 
 
 
 
 00 O 00* 
 
 00 O\ 00 
 
 Ok 
 
 00 
 
 
 CQ 
 
 
 
 
 
 
 
 
 
 l a li 
 
 
 
 O 00 
 
 1 M M ^ 
 
 1 -t oo oil 
 
 [Sill 
 
 I 
 
 
 
 S) O\ 1 
 
 3 M 1 
 
 1 
 
 
 
 
 
 OT 
 
 
 
 
 
 
 
 o 
 
 1 
 u CO 
 
 
 
 OO^" O OOO C 
 
 ) 00 00 
 
 - 
 
 
 
 00 
 
 
 
 
 o. 
 
 >, G 
 
 
 
 
 
 
 
 
 
 1 
 
 O 
 
 OOO O OOO C 
 
 5 OO OO O 
 
 o 
 
 O 
 
 OO O 
 
 O 
 
 
 w <u 
 
 
 
 
 
 
 
 
 
 
 || l 
 
 10 to 
 
 (M N 
 
 \O >O . ro * * fO 
 vO 
 
 ^ -cj- Tt 1/5 O 1/5 
 ^ N 04 N (N 
 
 1 
 
 10 
 
 O) 
 
 (MM <S 
 
 N 
 
 
 Name. 
 
 Croton Falls. . . 
 Einsiedel 
 
 4) . . ... 
 
 ^ jH ' ' <U W 
 
 &H | lii - 
 
 Swfe fe ooo & 
 
 1 1:1 |1 1 
 
 3 KW 5M M 
 
 \ \ 
 
 C 
 
 O 
 
 3 
 
 Lake Cheesman 
 
 -1 1 
 
 3 o 
 
 3% % 
 
 1 
 
 s 
 
 o 
 & 
 
 (U 
 
 z 
 
APPENDIX III 
 
 235 
 
 N.Y.City Supply. (Contract Draw- 
 ing.) Built with block parapet. 
 North Platte Reclam. Project, U. S. 
 Hydraulic lime concrete. Rubble 
 facing. 
 
 S* r (U **K jj 4_| ^ S^ *> S. 
 
 +1% * S g J2 c .'> -^Su- 
 gg | :? 1 'I o. If 4$S =3 I 3Ss G S 5 ^ S 
 
 Sif- JB.d 1 J8 ? ** oS o * 1^-c^-S, ^^a 
 
 lfi$ i J i I! !i if* JJH *l! 
 
 ^ S J 1 ^ P 1 ^ s. %?, ^- 8 * J - 1 C 'SI 
 
 *J|| l| | * illl* JL - s fe-|lllll . |3-S 
 
 j t> co feco Z O pq>5 ^ Ort ZQ^pi_i^< <j co 
 
 * Polygonal in Plan. t Under construction, 1916. J 300 to deeoest foundation. 
 
 
 .... ... 
 
 : 1 : : : : : : : : :| : 
 
 J 1 i : ^ : : : s 
 
 5 -g'S :TJ M "3-5 
 < rt u ' <uc ' bx 't>rttn 
 $, nj ort t"CC^w 
 
 ; So >S few ^coco^Sz 
 
 ^ I 
 
 O rt 
 bn t-i 
 ^ fe 
 
 1 1 
 
 New York . . . 
 
 Wyoming . . . 
 So. India 
 
 : - : j : i -a -t 
 
 s c a si ^ > 
 
 l! 1 1 II I J 
 
 American. . . 
 
 American . . . 
 English 
 
 ; c d -a a a da * ai;;*da 
 
 r<*^ O O _CiO O O OO *C rj^ oS^^-^OO 
 
 11 'S I'i 's 'S 'g'g i 11 i 11111 
 
 Sg.66 g aafia ^ |g alaa^ ea 
 
 fo co <J <3 fo < <3 <J <!< W feW <OcocoW 
 
 American . . . 
 French 
 
 O 10 
 
 00 * M 10 10 t 
 
 OOvOO t^ OO Ovi 
 M (M HI ro > 
 
 -H 
 
 OO> 00 IOO OvOOOOlOO 
 O t-O M NO OOvMi-roO^t 
 
 8 2 
 
 
 -* o o 
 
 M H OV 
 
 Ov O 00 
 
 MM M oo OOO N 10 1^00 M t^ro lOTj-MoOoOiO^f 
 OOOv M oo 10 M Ov O OvOO O> OO> OvOOvt^OOOM 
 OOt^ Ov 00 OOOv 00 Ov 0000 00 0000 OOOvt^oOOOOiOi 
 
 a oo" 
 
 
 00 
 Ov Ovoo 
 
 t^oo O 00 lOO 00 O OOO 00 OOO OvOoOI>OOOvM 
 OOt^ Ov 00 OOOi 00 Ov 0000 00 0000 OOCvC^OOoOOOOi 
 
 o | 
 
 o\ ; 
 
 
 
 
 1 
 
 <N O <M O 
 PO* 5- O 1 H? 
 
 M 
 
 ro N o> o O 
 
 Ml M > I 10 * 1 1 1 
 
 N 1 L 1 o^ -t 1 1 1 
 
 M O 
 
 1 ? 
 
 
 
 
 d 
 
 co' ,_; 
 
 00 
 
 oo o o oo o o o o oo oooo o 
 
 55 fcfc Z Z 2 Z m * K tt KKXZ* 01 
 
 
 O < O 
 
 OO O O O< O O O< O OO OOOOOOO 
 
 O < 
 
 \O Ov M 
 
 Nt-t-OO 0000 OvV 
 
 OO I> Ovoo Nt^ON^trOfO 
 
 <t 00 
 
 C* <N 
 
 MCS N (N ON CS M NM N MCS NMCMNNMM M N 
 
 Olive Bridge. . . 
 
 Pathfinder 
 Periyar 
 
 ^ii! si J ill 
 
 O3 o pj <D ji o ft 3^ rt <u & .tni-.cJr^>>K";^ 
 PH PH PH CO COCO CO CO COCO H HH H P |> f> > l> |S 
 
 ca 
 
 I 
 .is ^ 
 
236 
 
 HIGH MASONRY DAM DESIGN 
 
 CROSS-SECTIONS OF EXISTING DAMS 
 
 (Scales are as indicated by dimensions) 
 
 61.30 
 
 ALMANZA 
 
 Spain 
 Constr. prior to 1586 
 
 T 
 
 \ 
 
 VAL DE INFIERNO 
 
 Spain 
 Constr. 1785-1791 
 
APPENDIX III 
 
 237 
 
 Radius = Infinity- 
 Plan Pdlyyonal 
 
 16.69 
 
 f 
 
 30.39 
 
 44.08 
 
 153.50 
 
 115.20' 
 PUENTES 
 
 Guadalentin River, Spain 
 Constr. 1785-1791. Failed 1802 
 
 GROS Bois 
 
 River Brenne, France 
 Constr. 1830-1838 
 
238 
 
 HIGH MASONRY DAM DESIGN 
 
 Radius=158 ft. 
 
 ZOLA 
 
 Nea Aix, France 
 Constr., 1843 
 
 -Rad.=Inftnity- 
 
 69.00 
 
 J 
 
 SETTONS 
 
 Yonne River, France 
 Constr. 1855-1858 
 
APPENDIX III 
 
 239 
 
 Radius=1312.4 
 
 118.11 
 
 TERNAY 
 
 Ternay River, France 
 (Bouvier design) 
 
 Constr. 1861-1867 
 
 161.03 
 
 FURENS 
 
 Furens River, France 
 (Delocre design) 
 
 Constr. 1862-1866 
 
240 
 
 HIGH MASONRY DAM DESIGN 
 
 -Radius =Infiniej> 
 
 n0.23'/c-19.95-'-j 88,39- 
 
 116.79'^ ! 
 
 Profile of new 
 part of Habra dam 
 differs from old, but 
 new profile has not 
 been found. 
 
 HABRA 
 
 Habra River, A Igiers 
 (Old profile) 
 
 Constr. 1865-1871 
 
 Failed 1881 
 Repaired 1883-1887 
 
 U Radius= 
 
 1325' 
 
 151.91 
 
 =. 126,96- 
 
 BAN 
 
 Ban River, France 
 Constr. 1866-1870 
 
APPENDIX III 
 
 241 
 
 BOYDS CORNERS 
 
 West Branch of Croton River, New York, U. S. A. 
 Constr. 1866-1873 
 
 U Radius= 
 
 im* 
 
 27.88' /- 63,96- 
 
 GlLEPPE 
 
 Belgium 
 Constr. 1870-1875 
 
242 
 
 HIGH MASONRY DAM DESIGN 
 
 Radius=4iO' 
 
 84.29' 
 
 162.08 
 
 -451.59 
 
 VlLLAR 
 
 Lozoya River, Spain 
 Constr. 1870-1878 
 
 nterforts are IG.l'widt 
 and about 55 'centres. 
 
 PONT 
 
 ArmaQon River, France 
 Constr. 1878-1881 
 
APPRNDIX III 
 
 243 
 
 'Radius.^ Infinity 
 
 1.97. 
 
 13.12' 
 
 29.52' 
 
 50.0C' 
 64.03' 
 
 BOUZEY 
 
 Aviere River, France 
 Constr. 1879-1880. Overturned, 1895 
 
 BOUZEY 
 
 Large scale profile showing reinforcement of 1888-1889 
 Overturned in 1895, 580 feet going out 
 
244 
 
 HIGH MASONRY DAM DESIGN 
 
 . Radius = Infinity 
 
 59.05 
 
 92.22' 
 
 114.83 
 
 02.69!' 
 
 HAM iz 
 
 Algiers 
 Constr. 1880-1885 
 
 U Radius = 210' 
 
 HlJAR 
 
 Two such dams across the Martin River, Spain 
 Constructed j ^ 
 
APPENDIX III 
 
 245 
 
 187.8 
 
 134.52 ' 
 
 GRAN CHEURFAS 
 Sig River, Algiers 
 Constr. 1882-1884 
 
 Radius = Infinity 
 
 66.56' 
 MOUCHE 
 
 Mouche River, France 
 Constr. 1885-1890 
 
 Seven temperature cracks appeared during winter following completion, and 
 in the summer temperature change bowed the structure slightly. 
 
246 
 
 HIGH MASONRY DAM DESIGN 
 
 BRIDGEPORT 
 
 Mill River, Connecticut, U. S. A. 
 Constr. 1886-1887 
 
 Radius =-213.3' 
 
 SWEETWATER 
 
 Sweetwater River, California, U. S. A, 
 
 (Schuyler design) 
 
 Constr. 1886-1888 
 Additions, 1895 
 
AP?EKDIX III 
 
 247 
 
 105' 
 
 -110- 
 
 BEETALOO 
 
 Australia 
 
 Constr. 1886-1889 
 
 -Rad.= Infinity 
 =160* 
 
 99.8' 
 
 TANSA 
 
 Tansa River, India 
 Constr. 1886-1892 
 
248 
 
 HIGH MASONRY DAM DESIGN 
 
 THIRLMERE 
 
 England 
 Constr. 1886-1893 
 
 262 
 
 SAN MATED 
 California, U. S. A. 
 Constr. 1887-1888 
 
APPENDIX III 
 
 249 
 
 159.93 
 
 CHARTRAIN OR TACHE 
 Tdche River, France 
 Constr. 1888-1892 
 
 (^Radius = .Infinity 
 
 88.75' 
 
 SODOM 
 East Branch of Croton River, New York, U.S.A. 
 
 Constr. 1888-1892 
 
250 
 
 HIGH MASONRY DAM DESIGN 
 
 16.95 
 
 ESCHBACH 
 
 Germany 
 Constr. 1889-1891 
 
 80.3^ 
 
 LAUCHENSEE 
 
 Germany 
 Constr. 1889-1895 
 
APPENDIX III 
 i 
 
 251 
 
 PERIYAR 
 
 Periyar River, India 
 Constr. 1889-1896 
 
 "jg {4 iRadius -isio.oo' 
 
 ElNSIEDEL 
 
 Near Chemnitz, Germany 
 Constr. 1890-1894 
 
252 
 
 HIGH MASONRY DAM DESIGN 
 
 Rad.= Infinity 
 
 iR.= Infinity 
 
 Tmcus 
 
 Titicus River, New York, U. S. A, 
 Constr. 1890-1895 
 
 NEW CROTON 
 
 (As built) 
 
 Croton River, New York, U. S. A, 
 Constr. 1892-1906 
 
APPENDIX III 
 
 253 
 
 [*-R.= Inggitr 
 
 ,U-25.-75->! 
 EL 416 1 _ _l 
 
 El. 300' Finished 
 
 Surface of Earth// 81.49' 
 
 Built under over- 
 nging cliff 
 
 WACHUSETT 
 
 Nashua River, Massachusetts, U. S. A. 
 Constr. 1896-1905 
 
 Rad.= Infinity 
 
 INDIAN RIVER 
 
 New York, U. S. A. 
 
 Constr. 1898 
 
254 
 
 HIGH MASONRY DAM DESIGN 
 
 uilt of Granite Rubble 
 
 K- 80.-36- ->i 
 
 i t 
 
 ASSUAN 
 
 Nile River, Egypt 
 Constr. 1898-1903. Raised in 1909 to 131', by increasing entire cross-section. 
 
 BAROSSA 
 
 Gawlor, South Australia 
 Constr. 1899-1903 
 
APPENDIX III 
 
 255 
 
 LAKE CHEESMAN 
 Colorado, U. S. A. 
 Constr. 1900-1904 
 
 KOMOTAU 
 
 Upper Franconia, 
 A ustria 
 
 Constr. 1900-1903 
 
256 
 
 HIGH MASONRY DAM DESIGN 
 I 
 
 [ Radius = Infinity- 
 
 \ Concrete 
 
 SPIER'S FALLS 
 
 New York, U.S.A. 
 
 Constr. 1900-1905 
 
 MERCEDES 
 
 Durango, Mexico 
 Constr. 1902-1905 
 
 80.38 ' 
 
APPENDIX III 
 
 257 
 
 URFT RIVER, NEAR 
 AACHEN 
 
 Germany 
 
 ''Prof. Otto Intze de- 
 sign) 
 
 Constr. 1901-1904 
 
 Salt River, Arizona, 
 U. S.A. 
 
 Constr. 1905-1911 
 
258 
 
 HIGH MASONRY DAM DESIGN 
 
 .. 337.S9 7 
 
 3^&EL e - v ^P-^ 
 
 K. 
 -114rlO- - 
 
 CROSS RIVER 
 
 Cross River, New York, U. S. A. 
 Constr. 1905-1909 
 
 SHOSHONE 
 Wyoming, U. S. A. 
 Constr. 1905-1910 
 
APPENDIX III 
 
 259 
 
 PATHFINDER 
 
 North Platte River, 
 Wyoming,JJ.S.A. 
 
 Constr. 1906-1910 
 
 CROTON FALLS 
 New York, U.S.A. 
 Constr. 1906-1911 
 
 7/t 
 
 _ Y EU55' 
 
 '--' 
 
260 
 
 HIGH MASONRY DAM DESIGN 
 
 OLIVE BRIDGE DAM, ASHOKAN RESERVOIR 
 
 New York, U.S.A. 
 
 Const. 1907-1914 
 
 (See page 96) 
 
 KENSICO 
 
 New York, U.S.A. 
 Constr. began 1912 
 
d^MElev.3215' 
 
 APPENDIX III 
 * 
 
 261 
 
 .Radius= 661.74 
 
 ARROWROCK 
 
 Boise River, Idaho, U. S. A. 
 Constr. 1912-1915 
 
262 
 
 HIGH MASONRY DAM DESIGN 
 
 El. 264' I 18 " 3 "! Reservoir-full 
 
 1 
 
 - -212,58 
 
 ELEPHANT BUTTE 
 
 Rio Grande, New Mexico, U. S A. 
 
 Constr. began 1912 
 
APPENDIX III 
 
 263 
 
 5'0' 
 
 Selected Materi'a 
 .rolled in 4"layers 
 
 Note:-Top of dam to / 
 slope )^"per ft. from 
 center to outsiue 
 edge of ro'ping 
 
 ^ 
 
 Granolithic surface 
 Overflow E1.1585'j 
 
 TYPICAL SECTIO'N OF DAM 
 
 'Expansion- joints may be required to be 
 built with or without moulded blocks 
 
 Steps not shown 
 
 Class A 
 
 r 
 
 HORIZONTAL SECTION AT 
 EXPANSION-JOINT AT EL1542 
 
 FARNHAM 
 
 Mill Brook, Massachusetts, U. S. A. 
 Completed 1912 
 
 Designed for ice- thrust of 10,000 pounds per linear foot of dam length 
 and uplift, intensity due to full head at up-stream edge of joints diminish- 
 ing uniformly to zero at three-quarters the distance to the down-stream 
 edge of joints. 
 
264 
 
 HIGH MASONRY DAM DESIGN 
 SPILLWAY DAMS 
 
 Radius = Infinity 
 
 Vyrnivy River, Wales 
 Constr. 1881-1888 
 
 Reservoir Sid 
 
 16.40' 11 
 
 65.60 
 
 BETWA 
 
 River Betwa, India 
 Constr. about 1888 
 
APPENDIX III 
 
 265 
 
 ,( Radius=300 fU 
 
 127.5 
 
 LA GRANGE OR TURLOCK 
 
 Tuolumne River, Southern California 
 Constr. 1890-1893 
 
 COLORADO 
 
 Texas, U. S. A. 
 
 Constr. 1891-1892 
 
266 
 
 HIGH MASONRY DAM DESIGN 
 
 SUDBURY 
 
 Stoney Brook Branch 
 of Sudbury River, 
 Massachusetts, 
 U. S. A. 
 
 Constr. 1894-1897 
 
 E1.900' 
 
 BIG BEND 
 
 Feather River, Cali- 
 fornia, U. S. A. 
 
 Constr. began in 
 1909 
 
APPENDIX III 
 
 267 
 
 ASHOKAN WASTE WEIR 
 New York, U.S.A. 
 Constr. 1910-1912 
 (See page 104) " 
 
 El. 1595 
 El. 1585', 
 
 17-l<i bare, 18" c. to c., 16'o"long 
 16-1)4 "bars, 18"c. to c., 25'o"long 
 W)-l)4"bars, 18"c, to c., 33'8"long 
 
 gelected Material \orr 
 rolled in 4'layers 
 
 ra.u$' 
 
 E1.1191J 
 
 SECTION O.F SPILLWAY 
 
 FARNHAM 
 
 Mill Brook, Massachusetts, U. S. A. 
 Completed 1912 
 
 ASHOKAN DIVIDING WEIR 
 New York, U. S. A. 
 Constr. 1912-1914 
 
 (See page 103) 
 
INDEX 
 
 PAGE 
 
 Amount and distribution of stress at any point in a masonry dam . . 197 
 
 Application of arch equation 161 
 
 Application of cantilever equations 1 54 
 
 Arch and cantilever actions, solution for distribution of loading 
 
 between 152 
 
 Arch and cantilever analysis, method of 143 
 
 Arch dam 134 
 
 Arch dam, limitations of the invesigation of 142 
 
 Arch dam of rectangular cross-section 167 
 
 Arch deflection equations 152 
 
 Arch equation, application of 161 
 
 Arch equations, table for 163 
 
 Arch, nomenclature for the 148 
 
 Arch section 136 
 
 Arch stresses 166 
 
 Arch subjected to uniform, radial load, movements and stresses in. . 221 
 Arched and straight gravity dams, which may be used to best advan- 
 tage 134 
 
 Arched dam investigation outlined 142 
 
 Ashokan reservoir, description of 75 
 
 Ashokan reservoir, storage capacity of 74 
 
 Atcherley's analysis on stresses in masonry dams 169 
 
 Austin, Pa., dam, cause of failure of 15 
 
 Austin, Texas, dam, cause of failure of 15 
 
 Baker, Sir Benjamin, experiments by, on dam stability 180 
 
 Bazin's experiments on sharp-crested weirs, vertical and inclined 
 
 up- and down-stream 126 
 
 Bouzey dam, France, cause of failure of 15 
 
 Buttress arch dam 135 
 
 Cain's consideration of stresses in masonry dams 197 
 
 Calculations for determining the theoretical cross-section of Olive 
 
 Bridge dam 77 
 
270 INDEX 
 
 Cantilever deflection equations 150, 151 
 
 Cantilever equations, application of 1 54 
 
 Cantilever equations, derivation of 207 
 
 Cantilever, nomenclature for the 145 
 
 Cantilever stresses 165 
 
 Capacity of Catskill aqueduct 74 
 
 Cataract dam 10 
 
 Catskill aqueduct, capacity of 74 
 
 Causes of failure of masonry dams 35 
 
 Center of pressure on submerged surface 22 
 
 Conclusions from experiments upon two model dams 173 
 
 Concrete and mortar, tensile strength of 27 
 
 Condition of stress in masonry dams, recent considerations of 169 
 
 Conditions for stability of masonry dams 35 
 
 Conditions for stability of vertical sections more critical than for 
 
 horizontal 177 
 
 Conditions under which it is not necessary to provide for ice pres- 
 sure 17 
 
 Constant angle dam 138 
 
 Cracks in masonry mass due to temperature changes 4, 97 
 
 Cross-section, equations for determining 54 
 
 Cross-sections, analytic work in designing and investigating should 
 
 be checked by the graphic method 73, 93 
 
 Cross-sections of dams, dimensions and conditions for five 12 
 
 Cross-sections of existing masonry dams 231 
 
 Crushing strength of masonry, limit of 2, 13 
 
 Dam foundations, effect of upward pressure in 5 
 
 Dam, pressure limits reached at toe and heel of 48, 49 
 
 Dam stability, experiments by Sir Benjamin Baker on 180 
 
 Dams of various sections, Unwin's conclusions on investigations on. 187 
 
 Dams, width of top of 43 
 
 Danger of neglect of large tension across vertical sections 173 
 
 Dangerous tensions existing across vertical planes 182 
 
 Deflection in the cantilever, development of equations for 211 
 
 Derivation of the expression for horizontal (cantilever) moment 
 
 of inertia 207 
 
 Description of Ashokan reservoir 75 
 
 Design of dams, formulae for 31 
 
 Determination of angle <t> n 161 
 
 Determination of maximum and minimum pressure in a masonry 
 
 joint ,. f .....,.,.,.. 25 
 
INDEX 271 
 
 PAGE 
 
 Determination of shear on horizontal planes , 183 
 
 Determination of stresses when down-stream face ceases to be linear, 
 
 graphical solution for 1 76 
 
 Development of equations for deflection in the cantilever 211 
 
 Differential expressions for flexure of a cantilever 1 50 
 
 Dimensions and conditions for five cross-sections of dams 12 
 
 Distribution of loading between arch and cantilever actions, solutions 
 
 for 152 
 
 Distribution of pressure in masonry joint subjected to external 
 
 forces 22 
 
 Distribution of shearing stress on horizontal joints 171, 183 
 
 Diversity of opinions regarding uplift and ice thrust 3 
 
 Doubtful value of factor of safety 41 
 
 Drainage channels to eliminate upward pressure 10 
 
 Drainage wells and galleries 9 
 
 Eccentrically loaded joints, variation of pressure in 24 
 
 Eccentricity in a dam, manner in which it is produced 24 
 
 Effect of upward pressure in dam foundation 5 
 
 Equation for deriving value of the safety factor, resultant at middle 
 
 third 36 
 
 Equation for determining tension in a joint 27 
 
 Equation for finding the length of any joint 45 
 
 Equation for ice conditions 59, 61, 63, 70, 115 
 
 Equation to determine depth of rectangular portion of dam 45, 46 
 
 Equations for determining a cross-section 54 
 
 Equations for determining the length of joints from the top down. . . 50 
 
 Equations for flood conditions 59, 60, 65, 108 
 
 Equations for spillway dams, illustration of use of 112 
 
 Equations for various conditions of overturning moment, spillway 
 
 dam 110-115 
 
 Excluding water from masonry dams 14 
 
 Existing masonry dams, series of. (Indexed.) 231 
 
 Experimental investigations of stresses in masonry dams. . 173, 180, 187 
 Experiments by Bazin on sharp-crested weirs, vertical and inclined 
 
 up- and down-stream 126 
 
 Experiments by Bazin on weirs of irregular shape 127 
 
 "Experiments upon two model dams, conclusions from 173 
 
 Extent and distribution of upward pressure 12 
 
 Factor of safety, doubtful value of 41 
 
 Factor of safety, modification due to uplift 37 
 
272 INDEX 
 
 INDEX 
 
 Failure of Austin, Pa., dam, cause of 15 
 
 Failure of Austin, Texas, dam, cause of 15 
 
 Failure of Bouzey dam, France, cause of 15 
 
 Falling sheet of water over spillway dam, shape of 120 
 
 Falling sheet of water, velocity and pressure heads in 123 
 
 Features involved in a consideration of stresses in masonry dams, 
 
 by Prof. Cain 198 
 
 JL' irst adoption of uplift and ice thrust in high masonry dam design . . 2 
 
 Flexure of a cantilever, differential expressions for 150 
 
 Flood and ice conditions, Olive Bridge dam 78 
 
 Flood conditions combined with ice pressure in design 91, 92, 93 
 
 Flood level calculations combined with ice pressure, method for .... 58 
 
 Formula for a condition of uniform intensity of stress over a joint . . 23 
 
 Formulae for design 53, 59 
 
 Formulae for design of dams, development of 31 
 
 Formulae for design of spillway dam 1 10-115 
 
 Formulae for investigation 69 
 
 Formulae for the determination of water pressure 21 
 
 Friction factor, Levy and Rankine's angles for 171 
 
 Friction of masonry surface modifies thickness of sheet of falling 
 
 water 132 
 
 Frictional and shearing resistance of a joint 41 
 
 General conditions of uplift, three 7 
 
 General formulae for imposed conditions of loading, Series F 65 
 
 Graphical solution for determination of stresses when down-stream 
 
 face ceases to be linear 176 
 
 Gravity section, as a choice of type 136 
 
 Gravity section, tabulation of results " 68 
 
 Guarding against ice thrust ' 16 
 
 Harrison's conclusions as to necessity for ice thrust provision 17 
 
 Hill's analysis of stresses in a masonry dam 195 
 
 Horizontal planes, determination of shear on 183 
 
 Horizontal planes passing through inner toe, tensile stresses on 
 
 other than 193 
 
 Ice pressure, conditions under which it is not necessary to provide 
 
 for 17 
 
 Ice pressure design, spillway dam 115 
 
 Ice thrust and upward pressure : l 
 
 Ice thrust, guarding against 16 
 
INDEX 273 
 
 PAGE 
 
 Impossibility of tension on vertical planes near the outer toe 193 
 
 Index of existing masonry dams 233 
 
 Influence of uplift and ice thrust on stability of masonry dams 3 
 
 Intercepting drainage system, theory of 9 
 
 Investigation, formulae for 69 
 
 Lateral strains, Poisson's ratio for 139 
 
 Levy and Rankine's angles for friction factor 171 
 
 Limit of crushing strength of masonry 2 
 
 Limitations of investigation of the arch dam 142 
 
 Loading, general formulae for imposed conditions of 65 
 
 "Loading," various conditions of, introduced into formulae 53 
 
 Manner in which water pressure is exerted against submerged sur- 
 face 20 
 
 Marklissa dam 8 
 
 Masonry dam, amount and distribution of stress at any point 
 
 1 in i95> 197 
 
 Masonry dam failures, precautions in design for preventing 35 
 
 Masonry dams, causes of failure 35 
 
 Masonry dams, excluding water from 14 
 
 Masonry dams, recent considerations of condition of stress in 169 
 
 Masonry dams, series of existing cross-sections of 231 
 
 Masonry dams, superelevation of top of 43 
 
 Masonry dams, water leakage through concrete of 4 
 
 Masonry dams, width of top of 43 
 
 Masonry joint subjected to external forces, distribution of pressure 
 
 in 22 
 
 Maximum and minimum pressure in a masonry joint, determi- 
 nation of 25 
 
 Method of arch and cantilever analysis 143 
 
 Method of determining pressure at joint by resultant couple 28 
 
 Methods for preventing masonry dam failures 35 
 
 Moment of inertia, derivation of the expression for horizontal 
 
 (cantilever) 207 
 
 Moments of pressure, resisting and overturning 39 
 
 Movements and stresses in arch subjected to uniform radial load ... 221 
 
 Nomenclature for masonry dam formulae 31 
 
 Nomenclature for the arch 148 
 
 Nomenclature for the cantilever 145 
 
274 INDEX 
 
 PAGE 
 
 " On some disregarded points in the stability of masonry dams," 
 
 Mr. Atcherley's paper 169 
 
 Olive Bridge Dam 74, 96 
 
 Olive Bridge dam, calculations for determining the theoretical cross- 
 section of 77 
 
 Olive Bridge dam, flood and ice conditions . 78 
 
 Ottley and Brightmore's investigations of stresses in masonry 
 
 dams ; 187, 195 
 
 Overfall or flood level design, formulae for spillway dam 108 
 
 Overturning moment, equations for various conditions of 108 
 
 Poisson's ratio for lateral strains 139 
 
 Precautions in design for preventing masonry dam failures 35 
 
 Pressures at toe and heel of dam attained 35, 48, 49 
 
 Rectangular cross-section, arch dam of 167 
 
 Rectangular portion of dam, equation to determine depth of 45, 46 
 
 Resisting and overturning moments of pressure 39 
 
 Safety factor, equation for deriving value of the 36 
 
 Safety factor, modification due to uplift 37 
 
 Shearing stress, on horizontal joints, distribution of 171, 183 
 
 Shearing stresses on vertical planes 189 
 
 Simultaneous equations, table for final 164 
 
 Spillway dam, design of 105 
 
 Spillway dam, ice pressure design formulas 115 
 
 Spillway dam, illustration of use of equation for 112 
 
 Spillway dam, overfall or flood level design of 108 
 
 Spillway dam, shape of falling sheet of water over 120 
 
 Stability of dam from standpoint of vertical sections must be first 
 
 consideration 174 
 
 Stability of masonry dams, influence of uplift and ice thrust on 3 
 
 Stability satisfactory for horizontal sections may be unsatisfactory 
 
 for vertical sections, conditions for 177 
 
 Storage capacity of Ashokan reservoir 74 
 
 Stress variation in masonry joints, no exact law for 23 
 
 Stresses in masonry dams, experimental investigations of. . 173, 180, 187 
 Stresses in masonry dams, features involved in a consideration of, 
 
 by Prof. Cain 198 
 
 Stresses on vertical planes, shearing 189 
 
 Stresses on vertical sections , , 175, 204 
 
INDEX 275 
 
 > 
 
 PAGE 
 
 Submerged surface, manner in which water pressure is exerted 
 
 against 20 
 
 Superelevation of top of masonry dams 43 
 
 Table for arch equations 163 
 
 Table for computation of " EZ>'s," 162 
 
 Table for computation of S' 2 and S 2 160 
 
 Table for final simultaneous equations 164 
 
 Table of coefficients for 6, in equations for D 156 
 
 Table of distances of origins above joints 156 
 
 Table of values for shapes of nappes 122 
 
 Tabulation of results of arch dam analysis , 165 
 
 Tabulation of results of gravity design 68, 94 
 
 Temperature changes, cracks in masonry mass due to 4, 97 
 
 Tensile strength of concrete and mortar 27 
 
 Tensile stresses on other than horizontal planes passing through inner 
 
 toe 193 
 
 Tension across vertical sections, danger of neglect of large 173 
 
 Tension existing across vertical planes, dangerous 182 
 
 Tension in a joint, equation for determining 27 
 
 Tension on vertical planes near the outer toe, impossibility of 193 
 
 Theory of dam stability, Atcherley's conclusions from investiga- 
 tions 179 
 
 Theory of intercepting drainage system 9 
 
 Thickness of sheet of falling water, friction of masonry surface 
 
 modifies 132 
 
 Treatment of uplift 6 
 
 Trigonometric function of < n 149 
 
 Uniform intensity of stress over a joint, formula for a condition of . . 23 
 
 Unwin's conclusions on investigations on dams of various sections. . 187 
 
 Unwin's " Theory of Unsymmetrical Masonry Dams '"' 182 
 
 Uplift and ice thrust, diversity of opinions regarding 3 
 
 Uplift and ice thrust in high masonry dam design, first adop- 
 tion of 2 
 
 Uplift, meaning of 4 
 
 Uplift, three general conditions of 7 
 
 Upward pressure and ice thrust i 
 
 Upward pressure, drainage channels to eliminate 10 
 
 Upward pressure, extent and distribution of 12 
 
 Upward pressure, variation of c to meet different conditions of. . . 72, 73 
 
 Upward pressure, various ways of caring for 8 
 
276 INDEX 
 
 PAGE 
 
 Value of the coefficient C in formula for discharge 130 
 
 Variation of c to meet different conditions of uplift 72, 73 
 
 Variation of pressure in eccentrically loaded joints 24 
 
 Various conditions of "loading" introduced into formulae 53 
 
 Velocity and pressure heads in falling sheet of water 123 
 
 Vertical planes, dangerous tensions existing across 182 
 
 Vertical planes, shearing stresses on 189 
 
 Vertical sections, stresses on 175, 204 
 
 Water leakage through concrete 4 
 
 Water pressure, formulas for the determination of 21 
 
 Weir or overfall dams 99 
 
 Weirs of irregular shape, experiments by Bazin on 127 
 
 Width of base of triangular dam for varying heights and uplift n 
 
 Width of top of dams 43 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 MAR J 1 1950, 
 APR 22V 
 FEB 17 
 
 JA 
 
 
 
 LD 21-100m-9/48(B399sl6)476 
 
YC 33131' 
 
 UNIVERSITY OF CALIFORNIA 
 
 DEPARTMENT OF CIVIL ENGINEERING 
 
 BERKELEY. CALIFORNIA 
 
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 Engineering- 
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 UNIVERSITY OF CALIFORNIA LIBRARY