UNiveitsrnr OF CALIFORNIA, LCrARTMENTOFCIVILENCSlNCEHIN, WORKS BY PROF. WILLIAM CAIN VAN NOSTRAND SCIENCE SERIES Price 50 cents each. No. 3. Practical Designing of Retaining= Walls. Seventh Edition, Thoroughly Prevised. No. 12. Theory of Voussoir Arches. Fourth Edition, Revised and Enlarged. No. 38. Maximum Stresses in Framed Bridges. New and Revised Edition. No. 42. Theory of Steel - Concrete Arches, and of Vaulted Structures. Fifth Edition, thoroughly Revised. No. 48. Theory of Solid and Braced Elastic Arches. Second Edition, Re- vised and Enlarged. Brief Course in the Calculus, I2mo. Cloth. 280 Pages Illustrated Net $1.75 PRACTICAL DESIGNING RETAINING WALLS, WITH APPENDICES ON STRESSES IN MASONRY DAMS BT PROFESSOR WILLIAM CAIN, A. M., C. E. UNIVERSITY OP NOR^H CAROLINA. MEM. AM. SOC. C. E. ILLUSTRATED. SEVENTH EDITION, THOROUGHLY REVISED. NEW YORK: D. VAN NOSTRAND COMPANY, 25 PARK PLACE 1914 '?/ V Enrincerinj Library Engineering Library COPYRIGHT, 1888, BY W. H. FARRINGTON. COPYRIGHT, 1910, BY D. VAN NOSTRAND COMPANY, COPYRIGHT, 1914, BY D. VAN NOSTRAND COMPANY. PREFACE TO SEVENTH EDITION IN the first six editions of this work, considerable space was given to the results of experiments on . model retaining-walls and rotating retaining-boards. As this part of the subject has been fully discussed by the writer in a paper entitled " Exper- iments on Retaining-waUs and Pressures on Tunnels,"* it was thought best to omit a detailed discussion of the experiments in this edition, particularly as an adequate interpretation requires the consideration of the theory of earth pressure when the earth is supposed endowed with both fric- tion and cohesion. More important still, the omission gives space for a more ade- quate treatment of the designing of walls of various types. The present work is divided into an Introduction, where the direction of the * Transactions Am. Soc. C.E., Vol. LXXII (1911). 8O0300 IV earth thrust receives careful attention, and four chapters, pertaining to reservoir walls and the theory of retaining-walls, developed both by the graphical and analytical methods and leading up, after a short discussion of experiments, to the practical designing of retaining walls. The design of five different types of retaining-walls is given in detail not only for a horizontal earth surface but likewise for the earth surface at the angle of repose. The tables, giving ratio of base to height, for the most familiar types of walls, should prove especially valuable to constructors. In the brief discussion of dams, the occasion is taken to develop certain well- known elementary principles that are com- mon to retaining-walls as well as dams. In subsequent chapters of this work a good deal of new matter is given for the first time; notably in the analytical theory of the retaining-wall, and in the graphical discussion of " the limiting plane " in Chapter II. The theory of the retaining- wall has been deduced, with the one assumption of a plane surface of rupture, from well-known mechanical laws; Cou- lomb's " wedge of maximum thrust " being incidentally proved in the course of the demonstration, but not assumed as a first principle. Appendices I, II and III on Masonry Dams, have been added, leading to the computation of the " Stresses in a Masonry Dam " on any plane not too near the base. The results, especially when taken in con- nection with the experiments on rubber dams made in England by Messrs. Wilson and Gore, are thought to be of the highest importance. The limits of this book preclude the consideration of the stresses due to tem- perature changes and " uplift " due to water pressure, subjects which are now engaging the serious attention of engineers. WM. CAIN. CHAPEL HILL, N. C., May 5, 1914. TABLE OF CONTENTS. PAGB INTRODUCTION 1 CHAPTER I. RESERVOIR-WALLS 17 CHAPTER II. THEORY OF RETAINING- WALLS. GRAPH- ICAL METHOD 34 CHAPTER III. THEORY OF RETAINING-WALLS. ANA- LYTICAL METHOD 78 CHAPTER IV. EXPERIMENTS. COMPARISON WITH THEORY. THE PRACTICAL DESIGNING OF RETAIN- ING-WALLS 114 vii Vlll APPENDIX I. DESIGN FOR A VERY HIGH MASONRY DAM. 140 APPENDIX II. STRESSES IN MASONRY DAMS 149 APPENDIX III. RELATIONS BETWEEN STRESSES AT ANY POINT OF A DAM. 168 PRACTICAL DESIGNING OF RETAINING-WALLS. INTRODUCTION. 1. THE retaining or revetment wall is generally a wall of masonry, intended to support the pressure of a mass of earth or other material possessing some frietional stability. In certain cases, however, as in dock-walls, the backing or filling as the material behind the wall is called is liable to become in part or wholly saturated with water, so that the subject of water-pressure has to be considered to complete the inves- tigation. In cases where the filling is de- posited behind the wall after it is built, the full pressure due to the pulverulent fresh earth or other backing is experienced ; and the wall is designed to meet such pressure, with a certain factor of safety, as near as \t can be ascertained. In time the earth becomes more or less consolidated by the settling due to gravity, vibrations, and rains, from the compressibility of the material, which thus brings into action those cohesive and chemical affinities which manufacture solid clays out of loosely aggregated mate- rials, and often causes the bank eventually even to shrink away from the wall intended to support it, when, of course, there will be no pressure exerted against the wall. 2. Where a wall is built to support the face of a cutting, the pressure may be nothing at first, but it would be very unwise to make the wall much thinner than in the preceding case ; for it is a well-known fact of observation, that incessant rains often saturate the ground of open cuttings to such an extent as to bring down masses of earth, whose surface of rupture is curved, being more or less vertical at the top and approaching a cycloid somewhat in section ; the surface of sliding being so lubricated by the water that the pressure exerted hori- zontally by this sliding mass is even greater than for dry pulverulent materials. It is, in fact, on this account, as well as from the force exerted by water in freezing, and from the disturbing influences caused by the passage of heavy trains, wagons, etc., which set up vibrations that lower the co-efficient of friction of the earth, and besides add considerably by their weight to the thrust of tie backing, that a factor of safety against overturning and sliding of the wall is introduced, which factor in practice gen- erally varies between two and three when the actual lateral pressure of the earth is considered. 3. It is stated that retain ing-walls in Canada require a greater thickness at the top to resist the action of frost than farther south where the frost does not penetrate the ground to so great a depth. Again, if the strata in a cutting dip towards the wall, with thin beds of clay, etc., interposed that may act as lubricants when wet, the press- ure against the wall may become enormous ; or if fresh earth-filling is deposited upon an inclined surface of rock, or other impervious material that may become slippery when the water penetrates and accumulates at its sur- face, the pressure may become much greater than that due to dry materials. It is found, too, that certain clays swell when exposed to the air with great force ; others, again, remain unchanged. In all such exceptional cases the engineer must use his best judg- ment after a careful study of the material he has to deal with. The theory and methods used in this book will not deal with such exceptional cases, but simply with dry or moist earth- filling supported by good masonry upon a firm foundation ; and it is believed the theory deduced will be of mate- rial assistance to any one who may have to deal with even very exceptional conditions, or, as in the case of military engineers, with the design of revetment-walls partly as a means of defence. 4. When a retaining-wall fails, it is not generally from not having sufficient section for dry backing properly laid (in layers horizontal or inclined downwards from the wall), but because the earth has been dumped in any fashion against the wall, and no 4 'weep holes" have been provided to let off the water that is sure in time of rains to saturate the bank. If to this is added bad masonry, and a yielding foundation, or one liable to be washed out, the final destruc- tion of the wall can be pretty confidently counted on. 5. The following little table of weights and angles of repose of various materials used in construction may prove of assistance, but in any actual case the engineer should determine them by actual experiment : Weight per Cubic Foot in Pounds. Angle of Repose. Water .... 62.4 Mud 102. 0-? Shingle, gravel 90-109-120 35-48 Clay . . . 120 140.450 Gravel and earth 126 Settled earth 120-137 21-37 Dry sand . 90 34 Damp sand 120-128 350.450- Marl . . . 100 Brick . . 90-135 _ Mortar . . 86-110 _ Brickwork 110 _ Masonry . 110-144 _ Sandstone . 130-157 _ Granite 164-172 - We may assume generally, as safe values for brickwork, 110 pounds per cubic foot ; and for walls, one-half ashlar and one-half rubble backing, of granite 142 pounds, and of sandstone 120 pounds per cubic foot, though the last two values are generally exceeded. For ordinary earth or sand filling the angle of repose can be taken at one and one-half base to one rise, or a slope of 3342' with weights per cubic foot varying from 100 to 130. It is always advisable, where practicable, to put a layer of shingle next the wall, and to consolidate the layers of the filling by punning or other means, so as to reduce the natural slope as much as possible. With a well-built wall, designed after methods to be given ; having a good foun- dation-course, larger than the body of the wall, to better distribute the pressure, and resist sliding, and backed as described ; with weeping holes near the bottom at intervals, there should be no fear of failure under ordinary conditions. 6. It would take us too far to enter into the history of the theory of the retaining- wall. On this point see an interesting article by Professor A. J. DuBois in the " Journal of the Franklin Institute " for December, 1879, on " A New Theory of the Retaining- Wall." In this work^three methods will be developed : the first, a graphical method that will make clear the foundations on which all the theory rests; the second, a purely ana- lytical method, and the third, a graphical solution founded on it. Only the two graph- ical methods are available where the earth surface is not plane. 7. In case a wall moves forward, how- ever little, or there is settling of the earth behind it, the earth generally rubs against the back of the wall, thus developing fric- tion. There are, however, certain inclina- tions of the back of the wall that will be specially examined in articles 28-31, for which the earth sooner breaks along some interior plane, in its mass, than along the wall, so that a certain wedge of earth will move with the wall as it overturns or tends to move. For all other cases, which include nearly all the cases in practice, there will be rubbing of the earth against the wall, so that the earth-thrust against 8 the wall must be assumed to make, with the normal to the wall, an angle equal to the co- efficient of friction of earth on wall, unless this is greater than for earth on earth, in which case any slight motion of the wall forward will carry with it a thin layer of earth, so that the rubbing surfaces are those of earth on earth. 8. These suppositions are found to agree with experiments. The old theory that assumed the earth- thrust as normal to the back of the wall, or, as in Kankine's theory, always parallel to the top slope, does not so agree, and, in fact, often gives, for walls at the limit of stability, the computed thrust as double that actually experienced. The true theory, therefore, includes all the fric- tion at the back of the wall that is capable of being exerted. This friction, combined with the normal component of the thrust, gives the resultant earth-thrust inclined below the normal to the back of the wall at the angle o f friction to this normal. 1 1 In Annales des Fonts et Chaussees for April, 1887, M. Siegler has given the results of some simple experiments proving the existence of a vertical component of the earth- 9. Rankine's assumption that the direc- tion of the earth-thrust is always parallel to the top slope applies only to the case of an imaginary incompressible earth, homogene- ous, made up of little grains, possessing the resistance to sliding over each other called friction, but without cohesion ; of in- definite extent, the top surface being plane ; the earth resting on an incompressible foun- dation, or one uniformly compressible, and thrust against the movable side of a box filled with sand, by actually measuring the increased friction at the bottom of the movable board, held in place, caused by this vertical com- ponent. The box was one foot square at the base ; and for successive heights of sand of one-third, two-thirds, and one foot, the vertical components of the thrust for earth level at top were 0.66 pound, 1.76 pounds, and 3.97 pounds, respec- tively. Similarly for a box, 0.5 x 0.8 feet, filled with sand, but having a movable bottom supported firmly on iron blocks, the force necessary to move the blocks under the sides and under the bottom was measured; and from this the relative weights of sand supported by the bottom and sides of the box was found to be as one to one, nearly, for a height of sand of 0.6 foot, and about two to one for a height of 1.18 foot, the total weights ascertained by the friction apparatus also checking out with the actual to within five per cent. Other experimenters have actually weighed the amounts held up by the sides and bottom, respectively. See Engineering News for May 15 and 29, 1886, also the issue for March 3, 1883, on "A Study of the Movement of Sand;" also see article 60 following. 10 being subjected to no external force but its own weight. For such a material, the only pressure which any portion of a plane parallel to the top slope of greatest declivity can have to sustain is the weight of material directly above it ; so that the pressure on the plane is everywhere uniform and vertical. If we now suppose a parallelopipedical particle, whose upper and lower surfaces are planes parallel to the top slope, and bounded on the other four sides by vertical planes, we see that the pressures on the upper and lower surfaces are vertical, and their differ- ence is equal, opposite to, and balanced by the weight of the particle. It follows that the pressures on the opposite vertical faces of the particle must balance each other independently, which can only happen when they act parallel to the top surface, in which case only are they directly opposed. The pressures, therefore, on the two vertical faces parallel to the line of greatest declivity will be horizontal ; and on the other two faces, parallel to the line of greatest de- clivity. This is Rankine's reasoning, and 11 it is sound for the material and conditions assumed. It is likewise applicable to a material of the same kind, only compressible * provided we suppose it deposited, as snow falls, everywhere to the same depth, on an absolutely incompressible, or a uniformly compressible, plane foundation, parallel to the ultimate top slope of the earth ; for then the compression is uniform throughout the mass, and does not affect the reasoning. But if we suppose, as usually happens, that the foundation is not uniform in compressi- bility, then the earth will tend to sink where it is most yielding. This sinking is resisted to a certain extent by the friction resulting from the thrust of the earth surrounding the falling mass, so that much of its weight is transmitted to the sides, as actually happens in the case of fresh earth deposited over drains, culverts, or tunnel linings which settle appreciably. In the case of a tunnel driven through old ground, most if not all the weight of the mass above it is trans- mitted to the sides ; at least, at first, before the timbering or masonry is got in. Again, if the mass of earth is of variable depth, 12 even on a firm foundation, the mass of greatest depth will sink most, thus trans- mitting some of its weight to the sides, so that throughout the entire mass the press- ure is nowhere the same at the same depth as assumed. The vertical pressure over a drain or small culvert crossing an ordinary road embankment is less, too, for another reason, where the embankment is highest. The earth-thrust on a vertical plane, parallel to the line of road, is horizontal for a sym- metrical section when the plane bisects that section. On combining this thrust with the weight of the material on either side, we see that the resultant load on the culvert is removed farther from the centre than if there was no horizontal thrust. It is on account of this tendency to equalize press- ure by aid of the friction resulting from the earth-thrust, that sand, when it can be con- fined, is one of the best foundations, whether in mass or in the form of sand piles. 10. In the case of earth deposited behind a retaining-wall on a good foundation, the settling of the earth will generally be greater than that of the wall, so that the earth rubs 13 against the wall, giving generally the direc- tion of the thrust no longer inclined, even approximately parallel to the top slope (except when the latter is at the angle of repose), but making with the normal to the back of the wall an angle downwards equal to the angle of friction. If the wall should settle more than the filling, the thrust would at first have a tendency to be raised above the normal. But if such a thrust, when combined with the weight of the wall, passes outside of the centre of the base of the wall, the top of the wall will move over slightly, the earth will get a grip on the wall in the other direction ; so that it is plainly impossible for the wall (for usual batters at least) to overturn or slide on its base, with- out this full friction, acting downwards at the back of the wall, being exerted. Hence the theory which supposes it is safe ; for although it is possible that the earth may make the effort at times to exert the full thrust given by Rankine's formula, yet this effort is suppressed instanter by the external force now introduced by the wall friction, which force was expressly excluded from 14 the Rankine theory. The exceptions to this rule will be noted in article 31. 11. Weyranch's objections to taking the thrust inclined at the angle ' of friction to the normal are easily met. He says, Take a tunnel-arch ; and if we suppose the pressure, as we go up from either side, to make always the angle <' with the normal, we shall have at the crown two differently directed pressures : similarly for a horizon- tal wall with level-topped earth resting on it. If there is no relative motion, or ten- dency to motion, the thrust in the latter case is of course vertical, and in the former is probably vertical at the crown and in- clined elsewhere ; but if the arch or wall moves, and there is rubbing of the earth on the masonry, there is necessarily friction exerted ; so that the thrust at any point can have but one direction, making the angle ' with the normal. 12. Mr. Benjamin Baker, in his paper before the Institution of Civil Engineers, on the " Actual Lateral Pressure of Earth- work " (republished by Van Nostrand as " Science Series," No. 56), tested an old 15 theory (where the earth-thrust was assumed to act normal to the wall) by the results of experiments, and found the theoretical pressure often double the actual. In the discussion which followed, not a single engineer so much as alluded to a truer theory which assumes the true direction of the earth-thrust, and has been known and used, just across the channel, since the time of Poncelet. The writer tested this theory by many of the experiments recorded by Baker and some others, and found it to agree, within certain limits, remarkably well (see "Van Nostrand's Magazine " for February, 1882). These results have been carefully revised, and new experiments included, in the table given farther on, from which the reader can form a fair estimate of the theory as a working theory within certain limits that will be indicated. The reader is referred, however, to Mr. Baker's essay, not only for experiences under ordinary conditions, but for those exceptional cases which seem to defy all mathematical analysis. In fact, the engi- 16 neer almost invariably lias to assume che weights of earth and masonry, and angie of repose of the earth. Where there is water, the conditions one day may be very different from what they are the next, especially if the foundation is bad, as often happens ; in which case the wall will move over simply on account of the compres- sibility of the foundation, so that it has perhaps nothing like the estimated stability. For all such cases an allowance must be made over the results given for a firm foundation, etc., as to which no rule can be given. As water often saturates the filling, and perhaps gets under the wall, we must con- sider, in certain cases, water-pressure in connection with the thrust of the backing. Therefore, a short chapter on reservoir- walls, or dams, follows, in which many of the principles that must likewise apply to retaining-walls proper are given. CHAPTER I. RESERVOIR-WALLS. 13. THE design of reservoir- walls is a subject that has received the attention of many engineers and mathematicians ; but they are by no means agreed, except in a general way, upon the precise profile that is best to satisfy, as uniformly as possible, the requirements of strength and stability. We shall very briefly, and by the shortest means, point out the main principles of design of a dam that resists overturning or sliding by its weight alone, and is called a gravity dam, in contradistinction to one built on a curve that requires the aid of arch action to render it stable. Let Fig. 1 represent a slice of the dam contained between two vertical parallel planes one foot apart, and perpendicular to the faces. When the dam is large, a roadway is 18 generally built on top, so that the faces ks and gi are vertical or nearly so for some distance down ; after which the profile is designed to meet certain requirements, to be given presently. Let us suppose that the .rr.TTTrrrrrrrrnft dam has been properly designed down to the horizontal joint df/, and that the weight of the portion above df equals W v regard- ing the weight of a cubic foot of masonry as 1, and that its resultant cuts the joint df at the point o. 19 To design the part fabd below df by a rapid though tentative method, we must first assume the slopes db and fa corre- sponding to the depth dc; then compute the areas of the triangles bed and afe, and of the rectangle feed. The distances of the centres of gravity of these areas (which represent volumes) from the point b are re- spectively 6c, be + ^ae, and be -j- \ce. On multiplying each area by its correspond- ing arm from #, adding the products to W^bo -f- do), and dividing by the sum of W l (which equals the area of gMf) and the portion added fabd, we find the horizontal distance bm from b to where the resultant of the weight above joint ab cuts this joint. Its amount W is equal to the sum of the areas ( TF 3 + abdf), and we have only to combine W acting along the vertical through w, with the horizontal thrust H .rf the water acting on the face ksdb, to uud the resultant R on the joint, and the po*jt n where it cuts that joint. There is a vertical pressure of the water on the part sdb; but, as it adds to the stability, it is generally neglected, particu- 20 larly as the inner face is generally nearly vertical. 14. The horizontal pressure of the water H for the height 7i, by -known laws of mechanics, is equal to the area h x 1 mul- tiplied by the depth of its centre of gravity - below the surface of the water, and by the weight of a cubic foot of water w> where a cubic foot of masonry is taken as the unit. This pressure acts horizontally at ^h above the joint afr, so that its moment about the point n where the resultant R h h h s w rp, cuts the base db is h . - - w ~ TT- l M zoo moment of W about the same point is W X mn. As these two moments must be equal, we find the distance between the resultant pressures on joint ab for reservoir empty a'nd reservoir full, _ h*w = GTf' The above is substantially one of the methods adopted by Consulting Engineer A. Fteley in the design of the proposed Quaker Bridge Dam. See bis interesting report, and that of B. S. Church, 21 chief engineer, with many diagrams of existing dams of large proportions, in " Engineering News " for 1888, Jan. 7, 14, Feb. 4, 11; also the discus- sions by the editor in the numbers for Feb. 4 and 25, and March 3. 15. There are three well-known condi- tions, that must hold at any joint if the profiles Ja and ~di> have been designed correctly : 1st, The points m and n where the re- sultants for reservoir empty or full cut the base ab must lie within the middle third of the joint or base ab. 2d, The unit pressures of the masonry at the points a or b must not exceed a certain safe limit. 3d, No sliding must occur at any point. 16. The last condition is evident, and requires that H < Wf where / is the co- efficient of friction of masonry on masonry, the adhesion of the mortar being neglected. If (f> is the angle of repose of masonry on masonry, / = tan <, and we must always have, 22 that is, the resultant R must never make with the normal to the joint an angle greater than the angle of friction. In fact, in practice, we should employ some factor of safety as 2 or 3, so that 2H or 311 should always be less than Wf. This third con- dition is of supreme importance at the foundation joints of dock-walls, which fail (wnen they fail at all) by sliding from the insufficient friction afforded by the wet foundation. For ordinary retaining- walls, too, the foundation should, when practi- cable, be inclined, so that R shall make a small angle with the normal to the base. In all cases, deep foundations are to be preferred, as the earth in front of the wall resists the tendency to slide appreciably. 17. We shall now proceed to give a reason for the first condition above, and likewise deduce a formula to ascertain the unit stresses at the points a and b. If we decompose the resultant R at the point w, distant u = an from a (Fig. 1), into its two components H and TP, the former is resisted by the friction of the joint, and will be neglected in computing 23 the stresses at a and fr, though it doubtless affects them in some unknown manner. The remaining force W, acting vertically at w, must necessarily cause greater press- ure at the nearest edge than elsewhere on the joint, at least when the angle at a is not too acute, and the dam is a monolithic structure. For large dams built of stones in cement, it is likely that there will be greater pressure at the middle of the base than in a monolithic structure where the resistance to shearing or sliding along ver- tical planes is much greater than in a wall made up of many blocks, particularly if they are laid dry. But it is probably best, until experiment can speak more decisively on the point, to assume the pressure great- est at the toe nearest the resultant, and as given by the following theory : Call I = length of joint ab u = an = distance from R to near- est toe ; then if we suppose applied at the centre of the joint two vertical opposed forces, each equal to IF, it does not affect equilibrium. We can now suppose the force W acting 24 downwards at the centre to be the resultant W of a uniformly distributed stress p 1 = , shown by the little arrows just below joint ab; and that the remaining forces TF, one at the centre and one at n, acting in oppo- site directions, and constituting a couple, whose moment is W (^l u), cause a uni- formly increasing stress, as in ordinary flexure (shown by the little arrows below the first), whose intensity at a or b is by known laws, The total stress p at the nearest toe a is therefore the sum of p l and j? 2 , and is com- pressive. The stress at b is of course p l p^ where this is not minus indicating tension, unless the joint can stand the tension required. If we call u' the distance from n to the farthest toe, i.e. u' = n&, we have the mo- 25 ment of the two weights W=W(u' $) . On substituting this value for W(^l u) in the value for p 2 above, we find for the unit stress at b the identical equation (1) above, provided we replace u by u' ; so that the equation is general, and applies to either toe, if we only substitute for u the distance of the resultant from that toe. The stress is distributed, as shown by the lower set of arrows in Fig. 1, where there is only com- pression on the joint as should always obtain. The stress is thus uniformly in- creasing from the right to the left. If the limit of elasticity is nowhere exceeded, it follows that a plane joint before strain will remain a plane joint after strain, as must undoubtedly be the rule for single rectangu- lar blocks. Referring to equation (1), we see that if we replace u by u' = f Z, that the stress at b is zero, from which point it increases uni- formly to a, where its intensity, for u = JJ, W is p = 2, or twice the mean. For greater i values of u' than f Z, the stress at 6 becomes tensile, which is not desirable ; hence the 26 reason for condition 1 above, that the re- sultant should lie within the middle third of the joint. If the joint cannot resist tension at all, and R strikes outside the middle third, the joint will bear compression only over a length 3., and the maximum intensity at W a is now 2. This is evident, if we treat 3u 3u = I' as the length of joint, and substi- tute this value for I in formula (1). There is now no pressure at the distance 3u = I' from the left toe by the previous reasoning for the original joint /, and to the right of that point the joint will open, or tend to open. It is evident for full security that the resultant should strike within the mid- dle third some distance to allow for con- tingencies. 18. Having computed the unit pressures at the nearest toes for reservoir full or empty, condition 2 requires that these pressures do not exceed certain limits : in case they do, the lower profiles have to be re- vised, and the computation above repeated, until all the conditions are satisfied. 27 In the proposed design for Quaker Bridge "Dam, maximum pressures per square foot at the toes, at the base, were limited to 30,828 Ibs. at the back, and 33,206 Ibs. at the face; these pressures diminishing gradually to one-half to within about 100 feet from the top, the total height of dam from the foundation being 265 feet ; the argument being that the lower parts could stand more pressure than the upper parts shortly after construction, on account of the cement there attaining a greater strength. Be- sides, for this unprecedented height of dam, to keep the lower pressures within more usual limits "it would be necessary to spread the lower parts in an impracticable manner, and to incline the slopes to an extent incompatible with strength." It is evident that by this method of design there is no fixed rule by which any two computers could arrive at the same profile, having given the upper part empirically, sufficient in section to carry a roadway, and to resist the additional stresses due to the shock of waves and ice, at a time, too, when the mortar is not fully set. Such a rule is most easily introduced by requiring a certain factor of safety against over- turning, and, moreover, that the factor of safety against sliding along any plane shall not fall below a certain amount. It is suggested, however, that the factors of safety should increase from the foundation upwards, to make the section equally strong everywhere against overturning, when 28 allowance is made for the effects of wind and wave action, floating bodies, the expansive force of ice, or perhaps the malicious use of dynamite. If this is admitted, it would add one more con- dition (4) to the three previously stated, and would secure greater uniformity in design. See Appendix. As to the unit pressure test (condition 2), it must be observed, that we know little or nothing as to what limit to impose; for not only is the stress all dead load (which would allow of higher unit stresses), but the unit resistance of masonry in great bulk is undoubtedly much greater than in small masses (not to speak of tests on small ^specimens as a criterion), since the shearing off 'which follows, or is an incident to, crushing can hardly occur in the interior of a large mass of masonry. 19. We shall find in the end, that, for different forms of retaining-walls to sustain earth, that a factor of safety of about 2.5 against overturning is highly desirable, and that it will generally satisfy the middle third limit. In such walls this factor must be introduced to provide against an actual increase of the earth-thrust, due to water, freezing, accidental loads, and above all to the tremors caused by passing trains or 29 vehicles (if these are not considered sepa- rately), which it is well-known have caused, by increased weight, and the increased pressure due to lowering the natural slope, a gradual leaning and destruction of walls of considerable stability for usual loads. In a very high dam this is different : the pressure rarely changes but little, ex- cept on the upper portions ; so that, if such conditions were to hold indefinitely, the limiting unit stresses should control the lower profile more than a factor of safety against overturning. But, as pointed out by the editor of " Engineering News " (in the issues above referred to), a dam oq which the fate of a city may ultimately depend should be designed, as far as pos- sible, to resist earthquakes also. For that contingency, there is a reason for the factor of safety against overturning and sliding being as great as possible throughout ; and by putting the gravity dam in the arch form, convex up stream, the resistance to earthquake and other shocks is enormously increased. 20. We have now given the general prin- 30 ciples that should guide in the design of dams, which likewise apply in the design 6f retaining-walls proper, where, however, the height is rarely sufficient to call for much, if any, change of profile, and the maximum pressures are usually far within safe limits when a proper factor of safety against overturning or sliding has been introduced, which satisfies likewise the con dition that the resultant shall cut the base within the middle third. We of course have, as stated before, the direction of the earth-thrust inclined below the normal to the wall at the angle of friction ; otherwise, the methods above are applicable when the value of that earth-thrust has been deter- mined. For dock or river walls, saturated with water, The buoyant and lubricating effect of the water must be considered. If we suppose the filling of gravel, the water surrounding each stone allows free- dom of motion ; but the weight of the solid stones of the filling must now be taken less than when in air, by the weight of an equal volume of water, or at the rate of 62.4 Ibs. per cubic foot (or say 64 for salt 31 water), and the earth-thrust then found for the angle of repose of stone lubricated with water. Thus, if the weight of the solid stone be 150.4 Ibs. per cubic foot, and the voids are thirty per cent, the weight of solid stone in water is 88 Ibs. per cubic foot, and that of the filling 88 x .70 = 61.6 Ibs. in water, although it was 105 in air. If the wall is founded on a porous stratum, the weight of the masonry is sim- ilarly reduced by 62.4 Ibs. per cubic foot, or say one-half ordinarily ; but if the foundation is rock or good clay, "there is no more reason why the water should get under the wall than it should creep through any stratum of a well-constructed masonry or puddle-dam," as Mr. Baker has ob- served. If the water cannot get in behind the wall, the water in front only assists the stability. It has been previously observed that sliding is principally to be guarded against in dock-walls and others similarly situated, which can only be done by a sufficient 32 weight of masonry irrespective of its shape, unless the foundation is inclined, which even in the case of piling has been effected Fig. 2 by driving the piles obliquely, of course as nearly at right angles to the resultant pressure as is practicable. Fig. 2 represents a wall with a curved batter, in brickwork with radiating courses, 33 that might be used for a quay or river-wall, or a sea-wall, as ships ean come closer to the brink than in the case of a straight batter ; besides, for sea-walls it resists the action of the waves better. The centre of gravity can be found by dividing the cross section up into approximate rectilinear figures, and proceeding as in finding the position of W in Fig. 1. Its position is a little farther back than for a straight batter, which adds to its stability. But it is difficult to construct, the joints at the back are often thicker than is advisable, iuA-;. there is probably no ultimate economy ir its use. 34 CHAPTER II. THEORY OF RETAINING- WALLS. Graphical Method. 21. IN the theory of earth-pressure that follows, we shall consider the earth as a homogeneous, compressible mass, made up of particles possessing the resistance to sliding over each other called friction, but without cohesion. This is a much simpler definition than the one that Rankine's theory calls for (see Art. 9), and is more true to nature; the only approximation, in fact, consisting in neglecting cohesion, if we consider a homogeneous earth like drj sand. Let Fig. 3 represent a vertical section of a retaining-wall ABCD, backed by earth, whose length perpendicular to the plane of the paper is unity. 35 Assumption. We assume that the earth behind the wall, whether the top surface is a plane or not, has a tendency to slide along some plane surface of rupture as Al, A2, . . . . . 3 No proof is given of this assumption, so that it can only be tested by experiment ; but for the present we shall adopt it. In connection with the hypothesis of a, plane surface of rupture,, we shall use only one principle of mechanics relative to the 36 stability of a granular mass, first stated by Rankine as follows : It is necessary to the stabiliij of a granular mass, that the direction of the pressure between the portions into lohich it is divided by any plane should not, at any point, make ivith the normal to that plane an angle exceeding the angle of repose. This principle will alone enable us to ascertain the earth-thrust against any plane without resorting to a special principle, like Coulomb's " wedge of maximum thrust," which last, however, will be in- cidentally demonstrated as a consequence, of the above law. 22. In Fig. 3, let us consider the triangular prisms CMO, CAl, . . . , as regards sliding down their bases A0 t Al,... . If AF is the natural slope of the earth, the tendency of the prism CAP to slide along AF is exactly balanced by friction, as is well known. But if we consider other possible planes of rupture, lying above AF, as ylO, Al, . . . , we see, unless the wall offers a resistance, that sliding 37 along some one of these planes must occur: so that, the earth exerts an active thrust against the wall, which must be resisted by it ; otherwise, overturning or sliding would ensue. In case the wall is subjected to a thrust from left to right, as /rom earth, water, etc., acting on BD, and this thrust is sufficient to more than counterbalance the active thrust of the earth to the right of the wall, it will bring in the passive resistance of the earth to sliding up some plane as A'2, and the surface of rupture will now resist motion upwards, in place of downwards as hitherto. In the first case, of active thrust, where the prism is just on the point of moving down the plane, we know by mechanics that the resultant pressure on the plane of rupture makes an angle < cf friction of earth on earth with the normal to that plane and directed belotv the normal ; in the second case, of passive thrust, the direction of the pressure lies above or nearer the horizontal than the normal, and makes the angle < with the latter. 38 23. In the first case, w.here the wall receives only the active thrust of the prism of maximum thrust, let us call G (Fig. 3) the weight in pounds of this prism ; S the resultant pressure on the surface of rupture, making an angle (f> with the normal to that plane below the normal ; and E the resultant earth-pressure on the wall, which (except for cases to be noted in Art. 31) makes an angle cf>' of friction of earth on wall with the normal to the wall below the normal, unless (' > (/>, in which case a thin layer of earth will go with the wall, in case of relative motion, and this layer rubbing against the remaining earth will only cause the friction of earth on earth, and E will only be directed at an angle $ below the normal ; supposing always that the tendency to relative motion corresponds to the earth moving down along the back of the wall AC, as in settling from its compressibility, or as in case of an incipient rotation of the wall forward, from a greater pressure on the outer toe or a slight unequal compres- sion of the foundation. It remains to find the position of the true 39 plane of rupture. As preliminary to this, we note from Fig. 3 an expeditious way of finding the direction of S on any trial plane of rupture, as A\. Thus calling w the angle that Al makes with the vertical Al, the straight line making an angle (< -f- w ) with any horizontal, as DCI, below that horizontal, is parallel to S, since any line inclined at an angle w below the horizontal is perpendicular to Al, and S is inclined at an angle < below that normal. In laying off the equal angles, it is convenient to use a common radius, AH, to describe the arcs having A and I respectively as centres, and to take chord distances of the arcs < and to, and lay them off on the arc with / as a centre, as shown. For any other trial plane, as A2, we have simply to lay off the corresponding value of w below the angle (f> as before. 24. We shall now refer to Fig. 4, to illustrate the general method to follow to find the earth-thrust E in pounds. Here the wall, one foot long perpendicular to the plane of the paper, is shown in section BACD, the earth sloping at an angle from 40 some point on the top of the wall to the point marked 2, where it is horizontal. This is called a surcharged \vall, the earth 41 lying above the horizontal , plane of the top of the wall being called the surcharge. Extend the line AC of the inner face to 0, where it intersects the top slope of the earth ; the possible prisms of rupture are then .101, .i02, .403, . . . , and we shall now proceed to reduce these areas to equiv- alent triangles having the same base A2. Draw the parallels 00', IT', 33', . . . , to line A'2 to intersection with a perpendicular to A"2, passing through the point 2. Then the triangle .402 is equivalent to the triangle .40'2, and Al2 to A\ f '2, so that triangle AO'l' is equivalent to .401. Similarly .426 is equivalent to triangle A'2 6' A, having the same base, A'2, and vertices in a line parallel to this base, giving the same altitude. Thus the area .4026.4 is replaced by AQ'6'A ; and the weight of the corresponding prism, if we call e the weight per cubic foot of earth, is \A2 x 0'6' X e. Similarly the weight of .4024 is A2 x 0'4' X e ; so that if we use O'l', 0'2, 0'3', . . . , to represent the weights of the successive prisms .401, ^402, ^403> . . . , on the force diagram given below,, we have simply to multiply the value of J, 42 given by construction, by %e,A'2 to find its true value in pounds. We next lay off the successive values of (< + w), as in Fig. 3. Thus, with any convenient radius, as .40, we describe an arc, ogdf, and call the intersections with -41, A'2, . . . , a r a 2 , a 3 , . . . , respectively. Next, through point g on the arc in the vertical through A, draw vertical and hori- zontal lines, and describe an arc, bss v . . . with the same radius ; then draw gs, making the angle < below the horizontal gb (by making chord bs = chord fd) , and lay off with dividers, chords ss^ ss 2 , S6' 3 , . . . , equal to chords ga r ga^ ga^ ... It is evident now that lines gs r gs 2 , gs 3 , . . . , make the angles < with the normals to the successive planes ^41,^42,^.3, . . . , and thus give the direction of the >S's corresponding to those planes. We now lay off with dividers on the vertical line gA the distances gg r gg^ . . . equal respectively to OT, 0'2, 0'3', . . . , and draw through the points g v g. 2 , g s , parallels to the direction of E (drawn as before explained) to intersection with the 43 lines f/Sj,, respectively. Next, on the vertical #3., lay off gg r gg. 2 , . . . , equal to the bases 01, 02, of the supposed prisms of rupture lying to the right of Ag, and through their extremities draw g^V, g. 2 2', . . . , parallel to top slope to intersection 1', 2', . . . , with the directions of the resultants first found. The greatest of these lines cc x , to scale, represents the actual thrust OD ^40 ; and we have only to multiply it by \ep, where p is the perpendicular let fall from A on the top slope 010 produced, to scale, to get the pressure in pounds, if desired. 53 Now, if the direction of the pressure on the Wall AC cannot be taken as usual, inclined below the normal to AC, at an angle <, it is (Art. 7) because, in case of motion, the earth does not rub against the wall sufficiently to develop the required friction, whence it must follow that the earth breaks along some plane as .44, Ao, . . . , to the left of Ag, where the thrust is inclined at the angle < to its normal ; so that this plane is a veritable plane of rupture, and its position can be found as usual on assuming the direction of the thrust on AQ as parallel to the top slope. In case such a plane exists between .40 and AC, the earth below it, if the wall moves, will go with the wall ; further, it is evident that the thrust against the vertical plane AO, due to the wedge of rupture on the left, must exactly equal the thrust first found corresponding to the wedge of rupture on the right, otherwise equilibrium will be impossible. To ascertain the position of this plane of rupture on the left, that we shall hereafter call the limiting plane, most accurately, it 54 is weK to magnify the lines representing the forces as much as the limits of the drawing will admit of. We have consequently divided the top surface, 06', into a number of equal parts, of which the first eight are only one-fourth the length of the corresponding parts to the right of ^40. By laying off the loads gg^ gg 2 , . . . , however, to a scale four times as large as just used, we have the lengths gg^ gg. 2 , . . . , exactly four times the lengths 01, 02, . . . , along the surface to the left of 0, so that the old lettering applies again. We now produce the lines A\, .42, . . . , to intersection n v n 2 , . . . , with the arc gn (it is obvious that the top slope, 0(7, should best be drawn, in the first instance, through g, for accurately fixing the positions of n v n 2 , . . .) ; then lay off, below the horizontal, the angle clgm = < ; and from m, the inter- section of gm with the semicircle cL46, lay off the arcs mm v mw. 2 , . . . , equal to gn v gn 2 , . . . : so that the lines gm r gm 2 , o . . , all make angles equal to < with the normals to the corresponding planes Al, A2,... 55 Next, on drawing through g v g v . . . , lines parallel to the assumed direction of the thrust on AO, to intersection with the corresponding lines gm v gm^ . . . , the greatest of the intercepts (<7 5 5 nearly) represents, to the scale of loads, the thrust on the plane ^40 ; and this length should exactly equal four times the length cc 7 representing the thrust from the right, as we find to be the case. The plane of rupture to the left of the vertical through A thus coincides nearly with ^45, which is marked "limit" on the drawing. [On a larger drawing, for c/> = 33 42' and the top sloping at 25, the limiting plane was found to make an angle of 15 to 16 (see a more accurate determination in Art. 41) to the left of the vertical Ag, and to lie slightly below ,45, as this drawing would indicate.] If we lay off along the lines parallel to top slope, through g v g^ . . . , the true thrusts, gjv g. 2 t 2 , . . . , gj v . . . , the directions of gt r gt^ . . . , gt n , . . . , of the true thrusts on the planes A\, ^12, . . . , ^47, . . . , all necessarily lie above the first assumed directions ; so tha f , the 5G actual thrusts on all planes other than Ao (which we shall regard as the plane of rupture, for convenience), lying above or below ylo, make less angles with the normals to those planes than the angle of friction, just as we found in Art. 25. The conditions of stability of Art. 21 are thus satisfied in the present case ; but it is evident that this is no longer so if we lower the direction of the thrust on ^fo, which lessens the horizontal component of the thrust from the right, since intersections like 6', 7', in the right diagram move towards the vertical Ag, though the reverse obtains for the diagram to the left, which of itself indicates some absurdity. If, now we combine the new thrust on AQ from ?-he right (which has a less horizontal compo- nent than before) with the wedges of earth lying to the left of AS, it is readily seen that the directions of some of the resultants, as gt^ . . . , will fall below their first positions, and will thus make greater angles with the normals to their planes than the laws of stability will admit of ; so that any lowering of the first assumed position, 57 parallel to the top slope, of the thrust on ./40, is impossible. We thus reduce to an absurdity every other case but the one assumed, which is therefore true ; so that the proposition enunciated at the beginning of this article is demonstrated. We see, therefore, that we cannot, as before, assume the direction of the thrust on the wall, AU, as having the direction gm c , making the angle (/> with the normal to AJC) and find the wedge of maximum thrust corresponding ; but that its true direction, gt c , is found by combining the thrust found on AO, acting parallel to top slope, with the weight of the wedge of earth, QAC, between the wall and the vertical plane AO ; otherwise, if the left diagram is constructed, we find its direction and amount in a similar manner to that used in finding the direction, etc., of gL, . . . , by laying off on gA (produced if necessary) (K7 X 4 ; from the end of this line we draw a parallel to the top slope (K) to intersection t c , with the vertical through t & . The line gt c to the last scale used mui- 58 tiplied by %ep (where p is the perpendiculai from A on 0(7 to the scale used in laying off 06 y ) gives the thrust E against the wall in pounds. It is laid off in position by drawing a line parallel to gt e through a point on AU, %AC above -4, as previously enunciated. 29 (' <). In case this construction gives a thrust on the wall which makes a greater angle with its normal than the co-efficient of friction, <' of wall on earth, (f) f being less than <, then it is correct to assume the direction of E as making this angle <' with the normal, and proceed as in Fig. 4 to find the thrust. In the preceding 'article, no trial-thrust on the vertical plane A\J was assumed to lie nearer the horizontal than the top slope, as there was no reason for considering such exceptions to the usual direction in a mass of unlimited extent. Now, however, the wall requires the thrust on AO to lie nearer the horizontal than 0(? does, in which case the horizontal component will be increased (since intersections like 7', 8', move away from the vertical J#), and the thrusts on all planes Al, A2, . . . , lying to the left of Ag, will be raised above 59 their previous positions, gt l9 gt^ . . . ; so that the thrusts on all the planes now make less angles than with the normals to those planes, so that the conditions for stability of "the granular mass " are assured. 30. The "limiting plane," corresponding to the plane of rupture on the left, can be found by a different construction from that given above. Thus, having found the line cc 7 representing the maximum thrust from the earth to the right of -40, multiply by 4, say, and combine with the successive wedges of earth lying to the left of ^40, on magni- fying the lines OT, 02, . . . , in the same proportion, thus giving the lines gt v gt^ . . . , for the direction of the thrusts on the planes A1,A2, . . . ; these all lie above the directions gm r #m 2 , . . . , making the. angles (f> with the normals to the planes, except for the limiting plane, where gt & and gm b nearly coincide, as they should exactly if A5 was the limiting plane. The lowest relative position of gt with respect to gm is, of course, the one selected. It is evident, though, that the construction for the wedge 60 of greatest thrust to the left of Jig gives a more accurate evaluation of the thrust than the one to the right ; so that we can preferably use the left construction not only for getting the limiting plane, but for finding the thrust on any wall lying below the limiting plane. It is evident, from what precedes, that the double construction of Art. 28 applies only when the thrust on AQ is parallel to the top slope ; for the moment it is lowered, there results several planes of rupture to the left of ^40, which is impossible. Even if we attempt the left construction, we have seen besides that the resulting thrust on ^40 is greater than by the construction on the right. In case the face of the wall, AC, lies above the "limiting plane," as found before, we evaluate the thrust on it, as in Fig. 4, by assuming its direction to make an angle with the normal equal to or to when ' < (f). Thus, if the inner face of the wall had the position y!2, to the left of -40, the direction of the thrust on it would now be gm^ in place of gt^ as before, and 61 the conditions of stability of the granular mass will be found to be everywhere verified as in Fig. 4 (see Art. 25). 31. Summary. For all cases of top slope, when the inner face of the wall is battered, we first find the limiting plane by the construction of Art. 28 ; then when the inner face of the wall makes a less angle with the vertical than the limiting plane does (as is nearly always the case in practice, unless the surface of the earth slopes at or near the angle of repose, in which case the limiting plane is at or very near the vertical), we assume the direction of the thrust on it as making the angle < or <' (for <' < <) with the normal, and proceed as in Art. 23, et seq. ; but, if the face of the wall lies below the limiting plane, we proceed as in Art. 28, or if f f> we .may have to proceed as in Art. 29, to find the true thrust. If the wall leans backward, there is no need to find the limiting plane, as the usual construction applies. For earth level at top, the limiting plane is inclined to the left of the vertical equally with the plane of rupture to the right ; as 62 the top slope increases, it approaches the vertical, and coincides with it for the surface sloping at the angle of repose. Remark. It is found from the con- struction to the right of Ag, in Fig. 5, for planes of rupture lying 7 to 14 above the one corresponding to the greatest thrust, that the thrust is less only by from 6 to 16 per cent, though it more rapidly diminishes as the assumed plane of rupture nears the vertical. It must not be inferred, then, particularly for steep surface slopes, that a considerable divergence between the theoretical and actual surfaces of rupture will invalidate the theory, if the object is simply to get the thrust within a few per cent of the truth, particularly as the theory neglects cohesion. In fact, for a surface slope equal to the angle of repose, the plane of rupture is parallel to the surface ; but a plane lying much nearer the vertical will give nearly the same thrust. 32, In this connection, it may be well to describe an experiment made by Lieut.-Col. Aude in 1848, and repeated subsequently by Gen. Ardant, M. Curie, and M. Gobin, on a peculiar retaining-wall 63 made of a triangular block or frame, in which the inner face was inclined to the horizontal at the angle of repose of the sand backing, when, of course, by the usual assumption as to E making an angle of

= 45, 66 and the angle of friction on the ground-surface is only 5, that the slope of the embankment would change to 32 15'. For the ground-friction angle 7 7' 20" there would be exact equilibrium; so that, generally, there need be no fear from spreading of embankments due to this cause, as the amount of friction required is very small. 34. We have now given methods for finding the thrust against a retaining-wall, which simply resists this active thrust of the earth, for the usual cases of a surcharged wall and earth-level at the top, to which may be added the case of earth sloping downwards from the top of the wall to the rear, for which the construction is evident. It now remains to find the passive resistance of the earth to sliding up some inclined plane due to an active thrust of the wall from left to right (Fig. 4), caused by water, earth, or any other agency acting against the wall on the left. Now (Art. 22) we lay off the angle bgs (Fig. 4) above bg, and then, from the new position of s, lay off arcs ss r ss 2 , . . . , below s equal to ga v ga 2 , . . . , as before, giving the direction of gs r gs. 2 , . . . , inclined at anle above the normal to the 67 corresponding planes ^41, ^42, . . . The construction then proceeds as before, only it is now the least of the resistances, , acts upwards and the resistance of the wall downwards. The thrust E is now inclined at the angle (f> above the normal to A C and nob below s,s formerly. The active thrust is of course the only one exerted, unless the wall tends to slide, so that the consideration of the passive resist- ance is of small practical value. In case of a heavy structure resting on a foundation, we can replace the total weight by that of earth, and estimate the active thrust exerted pgainst 68 a vertical plane just below the foundation, for the full weight of the supposed earth, by the method to be given in the next article. The earth to one side of this vertical plane can be conceived to exert a passive thrust, which may be estimated as explained, and should exceed the active thrust for a stable foundation. This method, though, of estimating the stability of a foundation, while doubtless on the safe side, is otherwise illusory, as any one who has seen a heavy locomotive move at great speed along a narrow embankment must admit. The mass, by its friction, rapidly and safely transmits and distributes the weight over the ground, without exerting any horizontal thrust at the side slopes, which are perfectly stable. 35. Underground Pressures. To find the unit pressure at a depth x below the surface of a large mass of earth, level at top, of indefinite extent, and resting upon a uniformly compressible foundation, every where at the same depth (see Art. 9), we proceed as follows : Let Fig. 6 represent a slice of the earth contained between two 69 vertical planes one unit apart, and bounded on one side by the horizontal plane (K7, at a depth x below the surface, on the left by the vertical plane ^10, whose depth is Ax, and below by the plane AC ; the planes AQ, 0(7, and AC being supposed perpendicular to the plane of the paper. Let the length -40 = Ait 1 , and the length OU = w.Ax. The plane AC will be con- 70 sidered to take successively the positions .41, A2, . . . ; so that if we divide AO = Ao? into ten equal parts, as shown, and lay off similar equal parts along (H7, as AC varies in position, n will take the successive values 0.1, 0.2, . . . Calling e the weight per cubic foot of earth, the weight of the prism of earth resting verti- cally over (X7 is represented generally by e.x.n. Asc, which, being directly proportional to n, we can lay off on the vertical OA the lengths 01, 02, . . . , to represent the suc- cessive values of w, or the vertical loads sustained by the horizontal bases 01, 02, . . . , of the successive prisms considered. When the length Aa? is very small, we can neglect the weight of the small prism of thrust, .40(7, in comparison with the weight of the vertical prism above it, without appreciable error, and ultimately find the position of the plane AC, which gives the true thrust against Ati, by previous methods. Thus, draw the quadrants shown with A and as centres, and AO as a radius ; note the intersections a v a 2 , . . . , of the lines ,41, A2, . . . , with the arc OD ; next, 71 construct angle Cos = the angle of repose of the earth, and arcs ss l = p ss^ = 0. 2 , . . . ; so that each of the lines Os r Os 2 , . . . , next drawn, make the angle (f) with the normals to the corresponding planes Al, A2, . . . , and thus represent the direction of the resistances offered by these planes in turn regarded as planes of rupture. On drawing horizontals through the points of division 1, 2, . . . , on AO to intersection 1", 2", . . . , with the cor- responding directions Os v Os 2 , . . . , we note, that, if the thrust on .40 is taken as horizontal (Art. 9), the lines 11", 22", . . . , represent the horizontal thrusts caused by the weights resting on the suc- cessive prisms .401, .402, . . . , treated as successive wedges of rupture. The greatest of these 7T 77 represents the actual thrust on AO ; for if we assert that any other, as 4I 77 , represents the actual thrust, to get the corresponding thrusts on all the planes .41, .42, . . . , in direction and amount, we must lay off a length equal to W along each of the horizontals 11", 22", . . . , produced if necessary, and through the 72 extremities draw lines to 0, which thus represent in amount and direction the thrusts on the corresponding planes. But since 14 77 is less than 7T 7 , this construction will give a thrust on the plane ^47, lying below the position 07", and thus making a greater angle than c with its normal, which is inconsistent with the laws of stability of a granular mass. Hence, any other thrust than the maximum, as given by the above construction, is impossible. The length of 77^ to scale is 0.52, which we must now multiply by ex&x to get the total horizontal thrust on the plane JU in pounds. On dividing this thrust (0.52 ex&x} by the area pressed = 1 X A#, we get the unit pressure on a vertical plane at a depth x below the surface equal to 0.52e.x, which is called "the intensity of pressure," at a depth x. As we neglected the weight of the prism AO C. we must conceive A# to diminish indefinitely, so that the error tends indefinitely towards zero, and the approximate intensity of pressure on JI) = A# approaches indefinitely that at the point 0. 73 By analysis we shall show hereafter that the plane of rupture, Al in this case, bisects the angle between the natural slope and the vertical. In this case we have taken = 18 26', and the resulting intensity (0.52ex) is found to be exactly that given by the usual formula, ex tan 2 (45 - Y The intensity at any point of a vertical plane thus varies directly with x. The total amount on a vertical plane of depth x from the surface is then rCx 2 Cxdx = (where C = 0.52e in the present z case), and its resultant is at a depth z equal to the limit of the sum of the moments of the pressures (Cxdx) on the elementary areas dx x 1, taken about the top surface, divided by the total pressure, or , =f * + f = f. Also, ~ = e f X 0.52x = ^ X line represent- Z Z i ing thrust, if old construction is used. These are precisely the conclusions derived from previous constructions. In case the* top surface is sloping, a similar construction applies, only ~QC must now be drawn parallel to the top slope, and the pressure on OA must be assumed to act 74 parallel to this direction. The construction is similar to that given for Fig. 5 (on neglecting the weight of the wedge of thrust as above), either to the right or left of the vertical Ag, only as the weight of the prisms vertically above 01, 02, (Fig. 5) is now represented by ex cos i (where i is the inclination of the top slope to the horizontalVwe must multiply the length of the line ~cc' (Fig. 5) to scale, by ex cos i, to get the intensity of the pressure at the depth x, since the lengths n alone were laid off to represent the loads 99*i 99% - - i as ' lu Fig. 6, and the resulting thrust cc' must now be magnified ex A# cos i times to get the thrust in pounds on the plane A# X 1. As A# approaches zero indefinitely, the approximate intensity ex AX cos i '- - cc , on the area &x X 1, ap- A3? proaches that at the depth x (ex cos i. cc') as near as we please. It must be observed that ZO in Fig. 5 must be taken equal to unity in this construction, and the same scale used in laying off the distances along the top slope 10. 75 36. If the earth to the right of AO, in Fig. 6, does not experience the similar active thrust of earth to the left of J.O, but only the passive resistance of a tunnel lining, etc., of an underground structure, the conditions are changed if this lining gives in consequence of its elasticity ; for the wedge of thrust, J.OC, cannot move to the left without developing friction along the surface 0(7, therefore the pressure on this surface must no longer be taken as. vertical, but as inclined at a direction 1C', making an angle < with the vertical (Fig. 6). The load on any supposed wedge of_ thrust, as .A 04, is now represented by 04', the thrust on H) by ^Vand the pressure on the plane A4 by 04". The greatest of the lines, I'l", 2'2", . . . , now represents the true thrust, and it is readily found to be 4 / 4 // = .33 to scale j so that the intensity of the thrust on a square foot at the depth x is now 0.33e.r, or one-third the intensity on the horizontal plane 0(7. Mr. Baker ("Science Series,' 7 No. 56) found for a heading, driven for the Campden-Hill Tunnel, at a depth of 44 feet from the 76 surface, the angle of repose of the over- lying clay, sand, and ballast, heavily charged with water, being only 18 26' as .assumed above, that the relative deflec- tions of the timbering in the roof and sides indicated that the vertical and horizontal intensities of pressure were in the ratio of 3.5 to 1, which is very near what we obtain by the last construction. The first construction indicates a ratio of only 2 iol. In most cases, a portion of the weight of "the earth abovo the tunnel is transferred "to the sides (Art. 9), though here it was thought that "the full weight of the ground took effect upon the settings." We have now carefully examined the conditions of interior equilibrium of a mass of earth, and ascertained the thrusts ex- erted, whether in the interior or against a retaining- wall j and we see that the graphi- cal method is capable of handling, with qual ease, any case that ordinarily pre- sents itself. The results, of course, agree with the analytical method, founded on the same hypotheses j but as it is often more 77 convenient to calculate the thrust, even when a graphical method is afterwards used for testing the stability of the wall, we shall now proceed to deduce formulas for evaluating it. 78 CHAPTER III. THEORY OF RETAINING- WALLS. Analytical Method. 37. As in the preceding chapter, we shall assume a plane surface of rapture, and regard the mass as subject only to the laws of gravity and friction al stability stated in Art. 21. In Fig. 7, let AFPQEC represent a cross-section of the earth-filling, taken at right angles to the inner face of the wall A.F. We shall consider the conditions of equilibrium of a prism of this earth con- tained between two parallel planes, per- pendicular to the inner face of the wall,, and one unit apart, regarding the wall AF as resisting the tendency of the earth to slide down some plane, as AC, passing through its inner toe. Call G the weight of the prism of earth 79 in pounds, directed vertically j E, the earth- thrust against the wall AF T directed at an angle f of friction of earth on wall when ' < <, or of when ' > ^ below the normal to the inner face of the wall (Art. 7); and $ the reaction of the plane AC, inclined at an angle < (the angle of repose of earth) below the correspond- ing normal, since the prism is supposed to be on the point of moving down the plane 80 A C. These three forces are in equilibrium when E and S act towards and G acts downwards. Call the angle that AC makes with the horizontal y, and the angle FAC, ft. On drawing the parallelogram of forces as shown, we have, since E and 6r are pro- portional to the sines of the opposite angles in the triangle ONL, E _ sin ONL ~G ~~ sin NLO It is easily seen from the figure that ONL y t, and that NLO = < -f- ft + <' ; hence the above general relation becomes, a ~ sin (< + ' + ft) Now, if we conceive the plane A C, always passing through the point A, to vary its position, that value of E, corresponding to the greatest value obtained by the con- struction above, is the thrust actually ex- erted against the wall ; for, if A C is the plane of rupture corresponding to this greatest trial thrust, any less value of the 81 resistance of the wall E will cause 8 to make an angle greater than < with the normal to AC, which (Art. 21) is inconsist- ent with the law of stability of a granular mass (also see Art. 25) : hence the least thrust consistent with equilibrium corre- sponds to the greatest value of E thus ob- tained 5 and this is the actual active thrust exerted against the wall, when the wall simply resists the tendency to overturning or sliding on its base, caused by the ten- dency of the prism of rupture to descend. If there is a thrust exerted on the wall to- wards the earth, from any external force' acting on the left of the wall j from left to right 5 then, if this be supposed to increase gradually, the act ire thrust of the earth on the right is first overcome j then, as the ex- ternal force increases, the directions of S, on all planes as AC y approach the normals to those planes, pass them, and finally the full passive resistance of some prism of earth to sliding upwards along its base is brought into play. The greatest force E, as regards sliding up the base of some prism, which can be exerted is that corresponding to the least of the trial forces, E, obtained by supposing the position of the plane A C to vary, for S lying above the normal to A C at an angle < for each plane ; for if we suppose A C to represent the corresponding plane of rupture, if the external force, equal to E, and acting from left to right, is increased, it necessarily causes the direc- tion of S to make a greater angle than < with the corresponding normal, which is inconsistent with equilibrium (Art. 21). In this chapter we shall only consider the passive resistance of the wall to over- turning or sliding caused by the active thrust of the earth tending to descend, which is all that is required in estimating the stability of retaining-walls. 38. We shall now express the value of G for the earth-profile shown in Fig. 8 taken to represent the general case, and proceed to find the maximum value of E, for different trial-planes, which represents the actual thrust exerted against a stable wall. We shall suppose the true plane of rupture to intersect the part EY of the profile ; the line EY is then produced to 83 B, so that the area of the triangle ABC is equal to that of the polygon AFPQRC, which can be effected by ordinary geomet- rical means. The point B therefore does not change, as we suppose the position of C to vary between E and Y. . s Let us drop the perpendicular AT from. A upon B Y, and designating by e the weight per cubic foot of earth, we have a = &.AT.BC. For future convenience we have desig- nated, in Fig. 8, the angle that A C makes with the vertical &?, and the angle that the inner face of the wall AF makes with the 84 vertical a; so that the angle /3 of (1) is now replaced by (co -f- ') if the wall leans forward, or by (GO a) if the wall leans backwards. In Fig. 8, let us draw the line CT, mak- ing the angle ACI = ( -f- ' -f /j) = (<-]- \ and the whole angle A C!=(GJ +<}>+<}> '+a'), it follows that .the angle HCI = NBO (cf>' -\- a) as marked, if the wall leans for- wards; otherwise HCI=NBO=('a) y since .ACT is then equal to (Go-\-+' a), as previously observed. To reduce (4) to a simpler form, we remark that AT.BD represents double the area of the triangle ABD, and can be re- placed by AD.BN = AD.BO cos OBN ; which gives ATBD.BO = acos OBNl--): oif \OD) ' = ie. cos OBN (- (a y)2 . . . (5). Now, from similar triangles, BOD, CID, TtO C*T we have = , which, substituted in the above expression, we have, noting that (a 4/ab) = (a x) ZZ), the very sim- ple formula, 87 E= \e. cos (<' + a) CI- . . . (6). It is to be remarked, that, if the wall leans backwards, cos (<' -}- a) is to be replaced in this formula by cos (<' ex) ; further, if we lay off IL = 1C on the line IA f and draw a line from L to C, the thrust E is exactly represented by the area of the triangle ICL multiplied by e, the weight per cubic foot of the earth. 39. This simple conclusion has been previously reached, in an entirely different manner, by Weyrauch (see "Van Nostrand's Magazine" for April, 1880, p. 270), who states that Rebhahn in 1871 found a similar result, assuming, however, that ' = 0. or ' (f> (for the special cases of earth-level at top, or sloping at the angle of repose, I infer). Recurring now to the fact, that for the true plane of rupture we found x = A I = and that angle NBO =(<'-far) or ($' a), according as the wall leans forwards or backwards, we have the following simple construction to find the plane of rupture 88 and earth-thrust E, as given by Weyrauch in 1878, for a uniform slope and wall lean- ing forward. Having found the point B on the pro- longation of the line RY, which it is thought will be intersected by the plane of rupture, so that area ABE rr area AFPQR, we next draw J50, making the angle NBO with the normal to the line of natural slope AD, equal to (' -f- a) or (' a), accord- ing as the inner face of the wall lies to the left or to the right of the vertical through A (replace <'by < whenever <'><) j then erect a perpendicular at to AD to inter- section M, with the semicircle described upon AD as a diameter, and lay off AI = chord AM, since AI y'A O.AD > next, draw 1C parallel to OB to intersection C with the top slope, whence A C will be the plane of rupture if the point C falls upon JtY as assumed j otherwise another plane, as YZ, will have to be assumed as con- taining the point (7, and the construction effected as before. Having found C in this manner, E can be computed from (1), since Cr = b A T.BC 89 is now known : or by measuring CI to scale, E can be found directly from (6) This graphical construction is more rapid and accurate in working than the methods of the preceding chapter, and is superior to Poncelet's construction, in taking less space to effect. In surcharged walls, the point B will generally lie to the right of A F. Thus, in Fig. 4 the upper line 26 is extended to the left ; from a line is then drawn parallel to A2 to intersection 0' with the line 26 extended. The point 0' thus found corre- sponds to the point B of Fig. 8. 40. The construction is true whether the earth-surface slopes upwards or down- wards from the top of the wall. In the latter case, if the surface, say BD, falls upon the line BO, the construction fails ; but a formula given farther on gives the value of E. If the surface BD falls below BO, it is easily seen, on drawing a figure, that all the previous equations hold, and we reach the same conclusion as before, A J= ^/AD.A j only as AO now is larger than AD, the 90 semicircle must be described upon A as a diameter, and the perpendicular to the point M erected at D ; or A Jean be calcu- lated if preferred. If the points 0, J, and D are near together, it will be best to compute BC from BC BD.--, since the terms in the right member can be measured to scale. 4i. Position of the Limiting Plane. In Fig. 9 ; let BD represent the earth-surface, uniformly sloping at the angle i to the horizontal, of an unlimited mass of earth (Art. 9), in which the pressure on a verti- cal plane, AB, can be taken as parallel to the surface BD. Let AD represent the line of natural slope ; it is required to find the position of the plane of rupture A C, corresponding to the thrust E, acting above the horizontal at the angle i, and of course balancing the opposed thrust of the earth to the left of AB. On referring to Fig. 7, it is seen that equation (1) holds on replacing the de- nominator of the right member by sin i)- Therefore, in Fig. 8, the angle 91 ACI must now be laid off equal to (fi+ ), whence the line CI falls below CH, and BO below BN, both being in- clined to these normals at the same angle ; ^-j-=i-fO=ti With this exception, the above demon- stration holds throughout, and we reach the following construction to find the point C. From B draw BO, making the angle i below the normal BNio AD, or preferably making the angle ( i) with the vertical AB, to intersection with AD. From draw OM perpendicular to AD to intersec- tion Mj with the semicircle described upon 92 AD as a diameter ; lay off AI along AD, equal to chord AM, and from I draw a parallel to BO to intersection C with the top slope BD. The plane A Cis the plane of rupture, or the limiting plane of Art. 28, which see. If the inner face of the wall lies below AC, then (Art. 28) the thrust=ie. cos i. C/ 2 on AB is computed, and, regarded as acting parallel to BD, from left to right, is combined with the weight of the earth and wall to the right of AB to find the true resultant on the base of the wall. If the wall lies between AB and A C, the constructions of Arts. 37 and 38 are used. To be as accurate as possible in these, as in all constructions, true straight edges on both ruler and triangle are imperative. Lay off all angles, including right angles, by aid of a beam compass to a large radius, say ten inches, using a table of chords (except for the right an^le) and an accurate linear scale. With all care, the angles BAG thus found can scarcely be counted on to nearer than ten minutes, which, however, is sufficiently accurate. 93 In the table below will be found, for various inclinations i, the values of the angle BAG that the limiting plane makes with the vertical ; also the co-efficient K (see Art. 42), or the thrust on AB^e cos i 6T 2 , when AB and e are both taken as unity, made out for earth which naturally takes a slope of one and a half to one, or whose angle of repose is 33 42'. The value of K agrees fairly well with calculation, the last figure not differing more than one or two, at the outside, from the results of Art. 47. From the construction we see that as i ap- proaches indefinitely, B A C tends to zero and E approaches the limit \e cos . AB 2 , as given by analysis. The increase of thrust is very rapid from *=30 to t=^ = 33 42'. i J5 1C C 15 20 25 2634 30 33 42' BAG 2809' 26 24 21 50' 19 10' 16 14 40' 11 10' E .143 .145 .149 .,57 .172 .194 .207 .244 .416 94 42. Uniform Top Slope; Formula for Earth-thrust. When the upper surface of the earth slopes uniformly at the angle i to the horizontal, it is easy to deduce from what precedes a general formula for the thrust exerted by it. Fig. 10 represents a Fig. 10 retaining-wall leaning towards the earth. We shall first deduce a formula for this case, when it will be observed, as we pro- ceed, that the same formula holds, when the wall leans forward/ on simply changing a to ( a). In this case, we note from Fig. 10 the following values for the angles. 95 NBO = V a, ABO = < + <', AOB 90 (<' a), ADB = i, Finally, designate by Z the length .AJ? from the inner toe to where the inner face of the wall intersects the top slope, and by h its corresponding vertical projection. From formula (5) we deduce, remem- bering that OD = (al), E = e. BO* ~ a . s OBN. . . (7). La b J We can now write the [ ] as follows : It a Vab __ 1 \a 1 a-b Placew = Na P. to find its value in .terms N of the functions of known angles, we have from the triangles AOB and ABD by the law of sines, 96 AO __ sin ( + $') AB _ sin ( i) AB cos (V a)' AD ~ cos (or + i)' On multiplying these two equations to- gether, and extracting the square root, we find, _ ,J / sn (<> + <) sn (0- ~ \A> \ cos (^-a) cos ' Again, from the triangle BOA, we have, cos ( f a) Substituting these values in (7), and putting cos OBN= cos (<' a) for this case, and we have finally, n + 1 / 2 cos (<' -J- <') sin (< i) N cos (<' + a') cos (a i) which we obtain from the old values by simply changing a to ( a]. Ifc is to be observed, for all cases, when <' > (j> that we must replace ' in all the formulas by <. These formulas are identical with those of Bresse ("Cours de Mecanique Appliqu6e," Vol. I. 3d ed ) and Weyrauch, for the case of the wall leaning forward, the only cases examined by them. Bresse uses the 98 Poncelet method for the general case, which leads to Poncelet's celebrated con- struction. The routes pursued by these authors is different from that given above, the method of Weyrauch, in particular, being much more complicated 5 still, all three methods lead to precisely the same formula, so that it must be considered as established beyond question. Weyrauch, too, in subsequent reductions, follows Rankine as to the direction of the earth-thrust against the wall, whereas Bresse takes it as above. The'case of the lt limiting plane n is not considered by either. 43. The case where the top surface slopes downwards to the rear is very rarely met with in practice. The previous formula apply though directly on simply changing i to ( i), since it is seen that angle ADB - (<{>+i) and angle ABD = 90 -j- (ai), A . AB . i , sin (-\-i] and the ratio is now equal to {. AD cos(tf i) 44. Earth Level at Top; Back of Watt Vertical. For the earth level at top, back of wall vertical, and <' = < as usually taken, the formula (11) takes a very simple 99 form. Here we have <* 0, '- <, ^=0, whence. -j sin 2d> sin d> * cos and - For <' 0, which corresponds to a per- fectly smooth watt, or otherwise may refer to the direction of the pressure on a ver- tical plane in a mass of earth of indefinite extent, level on top (Art. 9), we have, when a and i =0 , n = sin < and, 17 , 1 sin < eh^ 1 -j- sin < 2 ). ... (16). 9 / 9 The equality of the two co-efficients o in (16) is easily verified from the known formula, 1 cos a? tan 2 J (x) = cos x by putting (90 <) for x in both members. Referring to Fig. 7, and regarding AF 100 vertical, the top surface horizontal, and ' = 0, we note that G=~. h 2 tan ft and p E - h 2 tan ft tan (y ft), in which y = 90 ft. Now, this result must agree with the right member of (16), which is only possible when ft = / 45 f- \ or 2/3 = , whence n=0 and, E=^.df. . . (17), ft as found in a different manner in Art. 41. This simple formula can likewise be de- duced directly from equation (1) of Art. 37, referring to Fig. 7, E- sin (y T ~cos (ft -f ) ~2~' we find for the trial thrust -p _ sin ft cos < e/i a " sin (2 < + />) ~2 cos < eft 2 sin 2 < cot /? -j- cos 2 2 Now, by the reasoning of Art. 37, the true thrust is the greatest value the above expression can have, as fi varies, and its greatest value corresponds to /?=90 < j for then cot ft is least, and E greatest, since cot ft is in the denominator. On substi- tuting this value a simple reduction gives E = J cos < . eh 2 as found above in (17). Since we have just found, for this case, that ft = 90 , it follows that the surf ace of rupture coincides wiih the natural slope. The value of E from equation (1) in this case assumes the form X oo, since G becomes infinite for an indefinitely sloping surface j but on reducing to the form above 102 we easily see the limit that E approaches indefinitely, which is its true value. The construction of Art. 39 fails for this case, but the one of Art. 41 leads directly to (17). 46. Pressure of Fluids. The general formula (9) above is true, no matter how small the angle of repose becomes, and must approach indefinitely the expression for the pressure of liquids, as and ' tend towards zero j so that at the limit, for 4> = <' = i = 0, we have the normal thrust of a liquid whose weight per cubic foot is e, E | eft cos a = 1 eh 2 sec a . . . (18), a well-known formula. By Art. 44 we see that for < = 0, 2/? = 90, or the plane of rupture approaches an inclination of 45 as < approaches zero indefinitely . 47. Mankinds Formula for the Earth-thrust on a Vertical Plane, in an Indefinite Mass, sloping uniformly. In Art. 9 we have stated the conditions that such a mass must satisfy in order that the pressure on a vertical plane, whose intersection with the top slope is a horizontal line, may be parallel to the line of greatest declivity. 103 Also in Art. 28 we have seen, that, when the wall face lies below the limiting plane, this direction of the thrust is the true one on a vertical plane, passing through the inner toe of the wall. We have a= 0, <' = i. and I = h, which gives in formula (9), \~*f-T/ 2 cos i where, sin ( -f i) sin (< _ V sin 2 cos 2 i cos 2 sin 2 i cos i _ V cos 2 i cos 2 cos i Whence, p _ cos 2 cos i eh 2 (cos i + Vcos 2 i cos 2 <) 2 ' "~2~* Now, since we can write, cos 2 < = (cos i + Vcos 2 1 cos 2 <^>) X (cos i Vcos 2 i cos 2 <) the above value becomes, on striking out 104 the common factor, (cos HVcos 2 i cos 2 cos which is Rankine's well-known formula for earth pressure. Now since Rankine's formula was framed without the use of any assumption, as that of a plane of rupture, and is accepted as correct for the case in question, it follows, that, when the pressure is assumed to be parallel to the surface, the assumption that the surface of rupture is a plane will give correct results, and can be safely used in the graphical method which is absolutely dependent on it. It will be observed that formulae (16) and (17) can be deduced directly from (19) by making i = and i = < respectively. Rankine has given a simple graphical con- struction of the last fraction in (19) in his " Civil Engineering/ 7 which saves labor in computing. 48. Unit Pressures on a Vertical Plane at Depth x below a uniformly Sloping Sur- face, the Direction of the Pressure being 105 taken Parallel to the Line of Greatest De- clivity. As in Art. 35 we shall consider a wedge of thrust of infinitesimal dimen- sions, of which the left face A B (Fig. 10) is vertical, and the upper surface paral- lel to the top slope. The weight of the vertical prism that rests upon any trial base as BC is, e . BC . cos i . x . = AT . BC . ex I A B (Fig. 8); so that neglecting the weight of the infinitely small wedge ABC we get the value of E from equa- tion (1) of Art. 37 by simply replacing G by this value. Equation (2) of Art. 38 is thus replaced by F -^L AT BC CI - AB .AT. BC Tr which is exactly that given in Art. 38 mul- tiplied by the constant 2x1 AB. All the subsequent reductions, therefore, hold if we simply put h=AB in the final equations, and multiply the result by 2x/AB. Hence divide (19) by AB = h and change the coefficient eh/ 2 to ex, to find the intensity of the pressure, E + AB, at a depth x; and on integrating this expression, mul- tiplied by dx, between the limits o and h, 106 we are at once conducted to (19), which is thus .proved true by the method of integration of the effects of earth particles, which is independent of the assumption of a plane surface of "upture ^xcending to the surface. Precisely the scum, conclusions hold for a vertical tvall, or one leaning forwards, ivlien E is assumed to maJce the angle <' or with the normal to the wall, since G is simply replaced as before by the weight of the vertical prism for a uniform top slope, and ultimately we replace h 2 by 2x in the general formula (11) to get the intensity of pressure in the direction given, at the depth x from the surface, so that on integrating as before we deduce (11) without the necessity of con- sidering the surface of rupture as extend- ing to the surface. The graphical method, using this hypothesis, should again give good results. It is possible though, in this case, that the influence of the wall friction may have some effect in deflecting the weights of the vertical prisms from a ver- tical line ; for, when it is so transmitted, the usual direction of the pressure is parallel 107 to the surface (Art. 9). For walls leaning backwards the prisms do not rest vertically over the bases of the prisms of thrust, and the theory would seem to be inapplicable ; so that the formulae for this case, (8) and (9), have to rest upon the unproved hypo- thesis of a plane surface of rupture extend- ing to the surface, and may depart consid- erably from the truth. We conclude, that, except for the cases for which -Rankine's formula is applicable, the plane surface of rupture is still an unproved hypothesis. 49. Pointof Application of the Thrust; Uni- form Slope. We have the normal compo- nent of the thrust on a wall, by (9) whether the wall inclines forward or backward or is vertical, expressed by the relation, E 1 = (9) X cos # = cP, c being constant whence the thrust on the area dl X 1 is nearly dEi = 2cm, and the distance from where the inner face of the wall interesects the top slope to the centre of pressure is equal to the limit of the sum of the elementary 108 pressures multiplied by tlieir arms, divided by the total pressure, or, f *J I hence the centre of pressure on the wall is 1/4 vertically above the base. 50. Surcharge uniformly distributed. If the filling of height li has a horizontal surface upon which a uniform load of any kind rests, replace its weight by that of an equivalent quantity of earth, giving the total load the same, and call the height of the reduced load h'. The total pressure on the vertical wall of height his now by (11), E = Ke ((h + h') 2 h 1 -} = Keh (h -f 2k 1 ), whence, dE = Ke.2 (h -f h') dh ; and the distance of the centre of pressure from the top of the wall downwards is, 2 f J o '*') hdh h (h + 2h') 3\ or from the base of the wall upwards, 109 Ji- h 2 -f 37* + 07* 1 + ll + 27/7 a It is more than probable that the theory for this case will prove illusory in practice, and will give a large excess of pressure ; so that, most frequently, such surcharged loads are ordinarily allowed for by a large factor of safety, particularly where the earth is bound by cross-ties, stringers, etc., or the surcharge is not free to move later- ally as well as vertically. Surcharge . !O(a). In the case of sea walls, the backing is saturated with water at high tide, up to a certain level BF, fig. 10 (a), so that it is well to ignore the friction at the back of 110 the wall on BC and for additional safety it will be neglected on the portion AB. Call the weight of the backing per cubic foot above BF, e 1 and the angle of repose <#>!. The corresponding quantities for the saturated backing below J?jF will be desig- nated by 62 and ' with the normal to the wall, we have, from E = Keh 2 , Ei = E cos <' = K cos <}>'.eh 2 - or putting, K\ = K cos ' we have, EI = K\ eh 2 also, E% E sin ' = EI tan <'. It is understood in these formulae, that, when ' > <, we must replace ' by . If the inner face of the wall makes an angle a with the vertical, we have the thrust acting at a distance cl = ch sec a from the inner toe of the wall, where c = ^ by theory for a uniform slope ; there- fore, the moment M of the thrust about the inner toe of the wall is EI cl } since the mo- 113 ment of E^ is zero j or putting for abbre- viation, m = c K\ sec a we have, M E\ ch sec a = c K\ sec a . eh* meh*. In subsequent investigations it is well to recall that h represents the vertical height from the inner toe of the wall to where the line of the inner face pierces the top surface of the earth backing, and that e represents the weight per cubic foot of earth. CHAPTER IV EXPERIMENTAL METHODS. COMPARISON WITH THEORY. THE PRACTICAL DESIGNING OF RETAINING WALLS 52. Many experiments have been re- corded pertaining both to retaining-walls proper and to rotating retaining-boards. Where the backing is of dry sand, possess- ing little or no cohesion, the results, for the retaining-walls proper, agree fairly well with the theory advanced in this book, which includes all the wall friction that can be exerted, especially where the walls were several feet in height; but they do not agree with the Rankine theory, in which the direction of the pressure on a vertical plane is always assumed parallel to the earth surface. 53. The results for some of the exper- iments on model walls at the limit of stabil- ity are given in the adjoining table. 115 ;] 1 1 M g-33 0-i.g | |5.S8lH! f S 1 ^ " 0) ,, ex 1 i s " 1 ^^I-S *o w ^ js "^ 5" D a S S 3 * O O i-lMi^l =S|^5|S- -0- O O O O O O5 CO CO ^O *O iO CO CO CO CO CO TJH CO CO r *0 3^^ ^3 QJ co > fe ^ H S "! o3 53 fl fllllSll - OOOOiOOO v's^l^as i" -e- o be g w> ItfllllAi 8 CO 00000,S} t I .s s^ -^^s ^^-5fl^?- liis^l^l ^ o B 5 2 ** ^ ,, O iO X O5 (M C^l (M (M 05 T-H (M CD CD | S* 8 1-1 3 | (NOCO COOOO X^'o'cJ S 0^ M) * "S !? ^ | ^^-g.s^ lls all- * 1-1 i-i O O O i-t H s -c .s ii t * 1 -*' iM !N rH II J 11 "83 ifljflfi I' . ^5 ^ t> S Authority. | . 6 . . . . riltiti W.-1HOOOO z-3 P> B'*M a *s \*- . 9 "S^i^ajoStn iifiiiifi irJPl^ 6 fc r-t C^ CO * ^> CD l> ^^oj'oj ^o;r3 slilllP^ 116 The walls were all vertical walls of rectangular cross-section, except the last two, which were peculiar wooden triangular frames whose inner faces made angles 27 30' and^55 respectively with th<; vertical. In No. 6, the face coincided with the " limiting plane " (Arts. 28 and 41) and in No. 7 was below it. in either case, the thrust was first found on the vertical plane through the foot of the inner face and this was combined with the weight of the earth over the face and the weight of the frame to find the resultant on the base (see Art. 32). Wall No. 1 was of pitch-pine blocks, backed by macadam screenings, the level surface of which was 3 inches below the top of the wall. Wall No. 5, of brick in Portland cement, was a sur- charged one; the level upper surface of the sur- charge being 4.26 ft. above the top of the wall, the surcharge extending entirely over the top of the wall at 45 to the horizontal. In the other walls, the earth surface was level with the top of the wall. Wall No. 2 was of bricks laid in wet sand; No. 3, of wood, and No. 4 was of wood coated on the back with sand. 54. Elaborate experiments on rotating retaining-boards, backed by sand, have been made by Leygue (" Annales des Fonts et Chausse'es," Nov., 1885), Darwin and others, which, in the earlier editions of this work, were given in detail. They are omitted here, since they have been 117 discussed by the writer very fully in an article entitled " Experiments on Retain- ing- walls and Pressures on Tunnels."* The conclusion was drawn that the results can be harmonized with theory by includ- ing the influence of cohesion. The dis- cussion involved a complete theory, mainly graphical, of earth pressure, where the earth is supposed endowed with both friction and cohesion. As regards the experiments of Leygue on rotating-boards, it was found that, assuming an adhesion or cohesion, of only about 1 Ib. per sq.ft., for the dry sand used, the experiments were in harmony with theory; but that the results differ essentially from the usual theory where cohesion is neglected. The discrepancies were proved to be due entirely to the small size of the models used and it is suggested that in future, walls of 6 feet and upwards in height be experimented on, where the influence of a cohesion of only 1 Ib. per sq.ft. is very small and can be neglected in the analysis. * Transactions Am. Soc. C.E., Vol. LXXII, p. 403 (1911). 118 55. Center of Pressure. Leygue, in the experiments on retaining-boards, found the moment of the earth thrust about the toe and also determined the surface of rupture. Using the corresponding wedge of rupture, the writer computed the thrust and its normal component. On dividing the mo- ment given by the latter, the quotient gives the distance of the center of pressure of the earth thrust from the base. It was found to lie, as an average for all the experiments, at 0.34 height of the board in contact with the filling for dry sand and 0.405 height for millet seed. For sand, the values varied, for a vertical wall from 0.319 per earth surface horizontal to 0.346 for the surface sloping at the angle of repose. For boards leaning towards the earth, when tan a (Fig. 10, p. 94) was + i the variation was from 0.296 to 0.337; for tan = + from 0.325 to 0.375. For boards leaning from the earth, tan a = f , the variation was from 0.352 to 0.363. These results are approximate, for al- though the exact prisms of rupture were used, the chord of necessity replaced the true curved line of rupture in the construe- 119 tion of Fig. 3, p. 35, and cohesion was neg- lected. The effect of cohesion is to lower the center of pressure; so that doubtless for sand absolutely devoid of cohesion, the atios should be larger. However, from lack of more complete observations on large models, the theoretical value will be used in the computations below. 56. The center of pressure for a sur- charged wall of the type shown by Fig. 4, p. 40, only with the back vertical and the each surface extending from C, the top of the inner face, at the angle of repose, <=3341', to the level surface 2-6, has been found by the writer * for various ratios of ti to /?, where h= height of wall, h' = height of surcharge above the top of wall and c= vertical distance from foot h' c h' e h h 00 0.333 1.00 0.364 2.00 . 353 0.75 0.364 1.50 0.356 0.50 0.364 1.25 0.360 1.11 0.362 0.00 0.333 * Trans. Am. Soc. C.E., Vol. LXXII, p. 410. 120 of wall to the center of pressure, divided by the height of the wall. In finding the values of c, h' was taken as 10 feet and the earth thrusts on walls of heights, 1, 2, 3, . . .,20 ft., were found by the construction of Fig. 8 (see pp. 88-89). By subtraction, the earth thrust on each foot of wall was obtained, and by taking moments about convenient points, the centers of pressure for heights of wall varying from 5 to 20 feet were easily obtained and c computed, as given in the table. 57. From a discussion of all the exper- iments, the conclusion was drawn that the sliding-wedge theory, involving wall fric- tion, is a practical one for the design of walls backed by granular materials and subjected to a static load. Often, however, in practical design, vibration due to a moving load has to be allowed for; also the effect of heavy rains. Both these influences tend generally to lower the coefficient of friction and add to the weight of the filling. To allow for these in- fluences, in designing, the normal com- ponent Ei of the earth thrust will alone be 121 multiplied by a factor of safety (7, the fric- tion Ei tan ' due to rains and vibration, as well as for an increase in the thrust. A factor of safety (7=3.5 is suggested for walls 6 ft. high, decreasing to 3 for walls 10 ft. and up- wards. For walls 50 ft. high and upwards, or for lower walls with a high surcharge, this factor may possibly be still further de- creased, since before the embankment is finished, the cohesive and chemical actions in the earth have doubtless consolidated it to such an extent that the actual thrust is much less than the computed one when cohesion is neglected. In any case, the true thrust E (not multiplied by any factor) when combined with the weight of the wall, must give a resultant that will pierce the base within its middle third, since it is desirable that pressure should be exerted over the whole base. If this does not obtain for a certain type of wall, the base should be made wider. If heavy loads, as railway trains, pass 122 over the surface of the filling, near a re- taining-wall, the weight of the load should be replaced by an equal weight of earth and the earth thrust determined as in Art. 50 or by aid of the construction of Fig. 8, p. 83, or that of Fig. 4, p. 40. With an earth foundation, a footing of masonry, projecting beyond the wall, should be built of such width that the true re- sultant on the base should pass near its center. This should totally prevent the increased leaning with time sometimes observed. Lastly, to ensure against slid- ing, the base should be inclined. 58. General Formula for Stability of Retaining-walls against Overturning. Let Fig. 11 represent a wall A BCD, whose length perpendicular to the plane of the paper is unity and whose exterior and interior faces and diagonal AC, make angles with the vertical equal to ft, a and w respectively. Let W denote the weight of the wall and g the horizontal distance from its line of action to the outer toe A ; also call <7 the factor by which it is necessary to multiply the normal thrust Kieh 2 , leaving the friction fKieh* at the back 123 of the wall constant, in order that the resultant on the base may pass through the outer toe. Here /=tan ' (when Fig. 11. <'> 4>, replace <' by ) and the quantities h, t, e, w, i, and ' have the meanings given in Art. 52. Taking moments around A, we have, Wg+fK 1 eh*tcosa = , we find, tan 2 w-f r i tan w 2Ki(f cos a -'=, (whence /=tan <'=f), c= and ' =K cos 0* * The computation of K, for some of the types, by formulas, being very long, the graphical method of Art. 39 can be substituted for it. Thus in Fig. 8, let e =1 and lay off h = vertical projection of AF =1 foot (say to a scale 10 inches to 1 foot) and draw from B, now coinciding with F , a hori- zontal line to represent the earth surface; then exactly as indicated on p. 88, locate the points O, I, C, H. The thrust E = Keh* =K = \CI.CH.: Ki **K cos . When i=tf>, AD GO , AI ~ oo and 127 cos 2 =0.346. Whence, for e/w=%, t =0.541; e/w =, t = 0.583. 61. Type 2. Vertical Back. Front face battered at 2 inches to the foot. =0, tan = j-, 0=9 28'. The formula reduces to, -2Ki+-t&n* 0. 10 3 AC=AD; hence the point 7 can be taken any- where on AD. With i =0 and C located as before .-. as above, Ki = \CI.CH cos 0. In type 5, the earth pressure on the wall was taken as making the angle with its normal. The assumption was only intended for usual batters of leaning walls, say a<10, for which it is practically correct. For large values of a, the assumption is not to be made, the error increas- ing with the angle a. 128 For i = 0, as above, Ki =0.109. .'. e/w=\, * 2 +0.097Z =0.144+0.0093, /. t =0.346. e/w =i * 2 +0.116* =0.174+0.0093, .'. t= 0.374. For t=0, /d =0.346, Art. 60. /. e/w=l t =0.548; e/w=$, t =0.589. 62. T?/pe 3. 5o^ /aces battered 2 inches to the Foot. On replacing sin a by ( sin a), tan a by (-tana) in the general formula, and noting that here, tan 2 =tan 2 /3, tan 2 w+-2 J ft: 1 (/ cos a + o- sin )+tan a\ e r tan o> =2/^i a| sec tan a (/cos + 0- sin a) . Formulas (13) and (14) p. 97, give, for = 0' =33 41', i =0, a =9 28'; n =0.8434, #,=0.143. For e/w=l, tan 2 w+0.387, tan co =0.157. .-. tan co =0.247.-. t =tan co+tan a =0.414. 129 For e/w-i t t an 2 co-f 0.430, tan co =0.189. /. tan w =0.270.'. t= 0.437. 63. Type 3 continued. Let i = 0. In this case, the " limiting plane " of Art. 28 concides with the vertical AO of Fig. 5, p. 50. Since the inner face of the wall AB, Fig. 5, lies below it, the thrust on AO = T=%e cos , AO 2 (acting parallel to BO) must now be combined with the weight of the earth A BO to find the re- sultant on AB. Taking, as before, the vertical height h of AB =1, we find AO =1.111 and for e=l, !T =0.513. On combining graphically, this thrust on AO, making the angle < with the horizontal with the weight of ABO(e = 1), we find the resultant thrust on AB = 0.570 and that it makes an angle 32 03' with the normal to A B. We have to substitute in the general formula / = tan 32 03' =0.626; also the normal component of the thrust = 0.570 X cos 32 03' =0.483. As this corresponds to the assumed height of AB=h=l and e = l, it is the value of K\. Whence substituting Ki =0.483, / =0.626* * Formulas have been derived by the writer for 130 in the formula of Art. 62, we have, for e/w=l, tan 2 co+0.881 tan co =0.535. .'. tan =0.414, t =0.580. Fore/w=-, tan 2 + 1.0024 tan co =0.642. .-. tan co =0.439. /. t =0.606. 64. Type 4. Front Face Vertical, Inner Face Battered 2 Inches to the Foot. The moment formula differs from that of Art. 62 only in the addition of the term ( | tan 2 a) to the right member. Hence, Ki and / for any value of i, but the worlj is too long to be given here. The results for t = will be stated. From the formula, [-( -i)] *>- -tan 45 + -a tan* compute e. In this instance, e =48 34'. The thrust on the wall AB makes with the normal to the wall, the angle, 7 =90 -(e +) =90 -58 02' =31 58'; whence / -tan 31 58' =0.624. The value of K\ is now given by the formula, _cos 7 cos (0 a) tan a 2 cos ( +e) cos a' which for 7 =31 58'; e =48 34', =33 41', a =928', reduces to K t =0.483, as found graphically. 131 we at once derive, when i=0:K l =0.143 (Art. 62), for e/w = f, tan 2 co+0.387 taa = 0.148. /. tan co =0.237 /. =tan co+ tan a =0.403; for e/w =, tan 2 co+0.430 tan co =0.180. .-. tan co =0.260 /. =0.260+0.167=0.427. When i=, as in Art. 63, K^ =0.483, /= 0.626, and the moment formula just quoted reduces to: e/w=l, tan 2 co+0.881 tan co =0.526; v /. tan co =0.408, t =tan co+tan a. =0.574/ e/w = f,tan 2 co + 1.024 tan co =0.633. .-. tan co =0.434, #=0.601. 65. Type 5. Leaning Wall. Front Face Battered 2 Inches to the Foot, Rear Face Parallel to the Front Face. The formulas of p. 96 are now applicable for computing K! =K cos 0. For <=' =33 41', i=0, =9 28', we derive n =0.7544, K, =0.081. Putting a =/3 =9 28', the moment formula is , tan 2 \e 1 2Ki(/cos a 0.167,;part of the base only is in bearing. The ratio a/t will be counted positive or negative according as the resultant on the base meets it to the left or to the right of its center. It will be observed, for cases one and three of type 4, that a/t> 0.167. In the first case, increase t from 0.403 to 0.417; in the second case, from 0.427 to 0.432. These values are inserted in the table in parentheses. The resultant on the base, in each case, will then cut the base \t from the outer toe.* ' * If the resultant on the base of the wall for the actual thrust (v =1) is to pass \ base from the outer toe, then for the leaning wall shown 134 67. In the last column of the table, is given the angle that the true resultant on the base makes with the normal to the base. This should not exceed the angle <' of friction of masonry on earth or sliding will occur. The factor of safety against tan $' sliding will be at least. - - and if pos- tan0 sible this factor should not exceed two. The average angles of friction of masonry on dry clay, dry earth and firm sand or gravel, are 27, 30, 35 respectively, but on wet clay, 11 to 18 has been given. Hence it is not always possible, for reasonable thicknesses of wall, to ensure a factor of safety of 2 against sliding. In such cases, the base should be inclined, so that the re- sultant on it, should make an angle with in Fig. 11, the moment formula, deduced in a similar manner to that of Art. 58, is as follows: tan 2 w-f | 4Ki(f cos a -sin a) -f tan /3 tan u = 2Ki[3c sec a +2 tan a (/ cos a -sin a)] +tan (tan a+tan 0). The formula is adapted to the case where the inner face of the wall leans away from the earth, by replacing sin a by (sin a) and tan a by (tan a). 135 its normal, much less than the probable angle of friction. In the tabular thick- nesses, no foundation slab was assumed, though one is always desirable and it should be constructed with the toe pro- jecting beyond the front face of the wall sufficiently to allow the resultant on the base to pass as near its center as is prac- ticable and thus distribute the pressure on the base more uniformly. For an actual wall, the unit pressure on the base (the " soil pressure ") should be computed by (1), Art. 15 and if too large, the foundation slab must be widened, so as not to subject the soil to a greater pressure than is accepted as safe. If a value of t is desired, for a value of e/w intermediate between f and -f,it can be found with substantial accuracy, by ordi- nary interpolation from the tabular values, assuming a linear variation. 68. On referring to the column of volumes (or areas of cross-sections for a length of wall unity) it will be observed that for level-topped earth, the types are economical in the order, 3, 5, 2, 4, 1. 13C . O 1 i-(i-l (M o o o o o o o o s^5$ CO CO >O >O co co to >o CO COO tO ti ^^ ssg? CO CO CO CO Tf< X Tt< X CO COCO CO Tf X * X * d c8 co co O O O (M J. CO O5 [ OiCC I-H Oi 1 + 1 + 1 o ^^ o o TfOCt^ 00 ~ O5 I-H Ci O TT OCI>CO - Olp* OiO I* Nw'5'5 .'* oc 10 oc o O C^ O (M I s 8 G ^H O a c3 -' a B . I a r< 138 Types 3 and 5 are nearly equal in volume, but the pressure on the base is better distributed in number 5. When the earth surface slopes at the angle of repose, the volumes increase in the order 3, 2, 4, 5, 1. The value of t is the width of the wall at the base, t f , the width at top, both for h=l. They likewise represent the ratios t/h, t'/h for any height of wall h. Thus for h=W ft., type 2, i=0, e/w =f, width at base = 3.46 ft., width at top = 1.80 ft. 69. Walls with projections at intervals, on the exterior or interior, are known as buttressed or counter -forted walls respectively. Fig. 13 shows a good form of buttressed BUTTRESS. wall, with the face in the form of arches, convex from the earth side. In designing 139. such walls, moments are taken about the outer toes of the buttresses. The great objection to counterforted walls in masonry not reinforced, is that the coun- terforts are apt to break away from the face wall; so that they have not found favor in America, in spite of the large economy shown. When reinforced con- crete is used, they present a very effective type of wall. 140 APPENDIX I. DESIGN FOB A VERY HIGH MASONRY DAM. ENGINEEKS are by no means agreed upon the proper profile to give high- masonry dams ; although the three conditions, that there shall be no tension at any horizontal joint, safe unit stresses every- where, and no possible sliding along any plane joint, seem to be generally accepted as essential to a good design. The writer suggests one more condition, that the factors of safety against overturning about any joint on the outer face shall increase gradually as we proceed upwards from the base, to allow for the proportionately greater influence, on the higher joints, of the effects of wind and wave action, ice, floating bodies, dynamite, or other accidental forces. The exact amount of increase must be largely a matter of judgment; but, if the principle is accepted, it can only resul' in making stromger dams. 141 The accompanying sketch, cf a dam 258 feet high to the surface of water (see also ' ' Engineering News" for June 23, 1888) satisfies the four condi- 258 tions named, and will be briefly described. The dam is of the same total height (265 feet) and volume (nearly) as the proposed Quaker-Bridge dam, and, for ease of comparison, is designed, as 142 was that dam, for masonry weighing 2| times as much as water. The dam is 24 feet wide at top, 38 feet wide, 50 feet below the surface of water (7 feet below the top), and 196.1 feet wide at the base. The up-stream face is vertical for the first 57 feet from the top, and then batters at the rate of 30 feet in 200 to the base. The outer face slopes uniformly from the top to 50 feet below the water surface, and then slopes uniformly to the base. The curves of pressure, for reservoir full or empty (the lines connecting the centres of pressure on the different horizon tol joints are here styled the curves of pressure), are found as hitherto ex- plained, and are seen to lie well within the middle third of the base, so that the horizontal joints under the static pressure are only subjected to compres- sion throughout their whole extent. Further, it was found by construction, that if a horizontal force be assumed as acting at the surface of water, of such intensity (29,375 pounds) as to cause the total resultant, on the joint 50 feet below the water level, to cut the joint one-third of its width from the outer face ; then if this same force, acting at the surface of water, is combined in turn with each of the other resultants on the lower horizontal joints, the new centres of pressure will still lie well within the middle third for the lower joints. To secure uniformity of results for all the joints, the width at the 50 feet level should be increased, although it is now much greater than ordinarily constructed. If, however, the effects of earthquake vibrations are to be guarded against, we cannot re- place them by the action of a single force acting at the surface, so that the increased width of the upper joints must be largely a matter of judgment. l The numbers to the right of the figure, in the form of a fraction, give for the corresponding joints, for the upper numbers, the factor against overturning, or the factor by which it is necessary to multiply the static horizontal thrust of the water to cause the total resultant to pass through the outer edge of the joint considered; and for the lower numbers, the ratio of the weight of masonry above a joint to the static thrust of water against it ; which is, in a certain sense, a factor of safety against sliding on a horizontal joint. These factors are seen to increase from the base upwards, so that the suggested fourth condition is satisfied. i It is stated in Engineering News for June 30, 1888, on the authority of Mr. Thomas C. Reefer, President Ameri- can Society of Civil Engineers, that "an ice bridge of about 90 feet span, between *wo fixed abutments, ex- panded so from a rise of temperature, as to rise 3 feet in the centre." If we regard the arch thus formed as free to turn atthe abutments and at the crown, we easily find for ice one foot thick, the horizontal thrust H exerted at the abutments, from the equation, 3H= ^ x ^, to be in pounds per square foot H= 21,094 pounds. Much higher pressures may possibly be expeiienced sometimes near the top of high dams in northern latitudes, and it seems only proper to include such contingencies in their design. 144 The unit stresses, in pounds per square foot, at the outar edgas of tha joints for reservoir full, and at the inner edges for reservoir empty, are given in columns 4 and 5 of the following table, being computed from the formula Depth of Joint below Water Weights Pressure Pressure Water Level. Pressure. of Masonry. at outer edge. at inner edge. 1. 2. 3. 4. 5. feet. Ibs. Ibs. 50 1,250 4,417 8,860 10,460 100 5,000 11,540 13,480 16.130 150 11,250 23,4*0 20,410 21,440 200 20,000 40,040 27,330 27,170 250 31,250 61,4-20 34,350 33,130 258 33,282 65,270 35,300 34.120 The numbers of columns 2 and 3 for one foot in length of the wall are expressed in weights of cubic feet of water, and must be multiplied by 62.5 to reduce to pounds. The unit pressures, although necessarily high, are still permissible. By spreading the lower part of the dam still more, these unit stresses would be theoretically diminished, though it is likely that in reality the pressures ab the positions of the old toes would not be very materially altered ; but the masonry being surrounded with other masonry could, most probably, stand a higher pressure. The unit pressures p given in columns 4 and 5 are not the maximum normal pressures at the faces. In Appendix III (e), it is proved that the maximum normal stress at a face acts parallel to that face on a plane at right angles to it and that its intensity is given by the formula, f=psec 2 ( where tan = co- efficient of friction of masonry on masonry) is the total friction that can be exerted by the plane AK. If we lay off angle NDE = (taken as 35 here) to intersection E with the parallel component BN, we have DN tan = EN, so that BE must be resisted by cohesion ; and the unit-shearing stress 7? W along the plane AK = _ . If, now, we produce AIL KE on to intersection C, with AB produced, we BC have the unit shear represented by , which is a AL/ maximum, for various planes passing through A, when C is farthest removed from B. On effecting this construction, then, for a series of planes passing through A, we quickly find the plane which will have to supply the maximum in- tensity of shear, or the plane of rupture, to lie near AK (there is very little difference for a series of planes lying near each other) ; and the shear per square foot required to resist sliding, in addition to the frictional resistance, to be about twenty-seven hundred and fifty pounds. To offer the greatest resistance to sliding, there should be no regular courses, and the stones should break joint verti- cally as well as horizontally, or the courses near the outer face should be curved so as to be approxi- mately normal to that face. For a retaining-wall of dry rubble, carelessly laid, we see that there is every probability of failure by sliding along some inclined plane. Here the stones must be carefully 148 interlocked to prevent sliding. For the reservoir- wall, where the best cement is used, and the joints are broken, there should be no fear of sliding when sufficient thickness is given to avoid tension. In the Habra dam, a hundred and sixteen feet high, this was not done ; and the dam broke along a plane, passing through the outer toe nearly, and making the angle of friction of masonry on masonry with the horizontal. It is well to note, tco, that friction alone will not prevent sliding along planes inclined not far from the horizontal as well as those below, so that a proper resistance to shear must be provided for in every dam. Possibly the weak point of many dams is in this very particular. The capacity of the dam in question to resist rotation about the toe of an inclined base may next be tried, and it will be found to be stable ; for the weight of masonry, as well as its arm," increases to counterbalance the increase of arm of the water- thrust. The dam thus satisfies all the conditions of stability ; and, although some of its dimensions may be changed with advantage perhaps, it yet suffices very well to point out the principles of design. See Engineering News for January 12, 1893 and May 9, 1907 for effects of expansion of ice. APPENDIX II. STRESSES IN MASONRY DAMS. 1 THE object of this investigation is to deter- mine the amounts and distribution of the stresses in a masonry dam, at points not too near the foundations, having assumed the usual " law of the trapezoid," that vertical unit pressures on horizontal planes vary uniformly from face to face. Experiment indicates that such vertical stresses increase pretty regularly in going from the inner to the outer face, for reservoir full, until we near the down-stream or outer face, where the stress gradually changes to a decreasing one, which decrease continues to the end of the horizontal rection. The law of the trapezoid is thus only approximately true over part of the section, but, as it gives an excess pressure where it attains a maximum, it errs on the safe side. 1 What follows in Appendices II and III was first given by the author in Trans. Am. Soc. C.E., Vol. LXIV, p. 208. 150 The profile of the dam selected is of the trian- gular type, with some additions at the top, but the method, used in determining the stresses is general and will apply to any type of profile. The final equations will give, at any (interior or exterior) point of the horizontal section considered, the vertical unit stress on the horizontal section, the normal stress on a vertical plane, and the unit shear on either horizontal or vertical planes. From these stresses, the maximum and minimum normal stresses, and the planes on which they act, can be determined, and ultimately, if desired, the stress on any assumed plane can be ascertained. The solution presented is approximate, which is justifiable, in view of the approximation involved in "the law of the trapezoid" used. The results, however, are practically correct, as will be evident from the checks applied, resulting from the exact theory given in Appendix III. The theory used, being simple, should be easily followed. Let Fig. 16 represent a slice of the dam con- tained between two vertical parallel planes, 1 ft. apart and perpendicular to the faces. The batter of OB is !!?-*!?; that of OE being _ 4 _ - . 200 1 200 1 The batter of the inner face was found by trial, so that the centers of pressure on horizontal sections, for reservoir empty, should nowhere pass more than a fraction of a foot outside the middle third of the section. The simple type of profile shown was adopted for ease of computation. For convenience in subsequent computations, the breadths, b = EB, of horizontal sections, corre- 151 spending to various depths, h, below the surface of the water in the reservoir, are given, all dimen- sions being in feet: /i = 199.0, ^==199.5, h = 200.0, h = 200.5, h = 201.0, 6 = 133.330; 6 = 133.665; 6 = 134.000; 6 = 134.335; 6 = 134.670. Take the weight of 1 cu. ft. of masonry equal to 1; then the weight of masonry above any , = a section is equal to the corresponding area in Fig. 16 above that section. The area of the por- tion above EOB is readily found to be 712, and its moment about the vertical, AO, is 11,603, the unit of length being the foot. In Fig. 16, D is 152 where the vertical through the center of gravity of the dam above the joint, EB, cuts that joint, and C is the center of pressure on that joint when the water pressure on EO is combined with the weight of masonry, W , above EB. As h varies, suppose each horizontal joint, in turn, marked similarly to the joint at 7i = 200, with the letters E, A, D, C, B; then, for any joint, on taking moments of the triangles, AOB, AOE, and the area above OB about A, we find (AB*-EA 2 )+ 11,603 AD Assuming that the masonry weighs 2^ times the water per cubic unit, then the weight of a cubic 2 foot of water is . It would entail but little extra 5 trouble here, where the inner face has a uniform batter throughout, to include the vertical conv ponent of the water pressure on the face, EO; but it will be neglected, as usual. The horizontal water pressure for the height, h, is thus, _ X __ = _ h 2 , and its moment about C is 525 Lh*X--h= Lh*. 5 3 15 Taking moments of W and water pressure about C, we have at once, 15 W 153 From the last two formulas, we derive the following results: h W AD DC 199 13978.335 40.49141 37.58483 200 14112.000 40.70316 37.79289 201 14246.335 40.91488 33.00089 A seven-place logarithmic table was used throughout, the aim in the computations being to get the seventh significant figure correct within one or two units. The necessity for this accuracy will be seen later. The distances EC and CB are now readily derived. ForA = 199, #C=82.05624, C5 = 51.27376; A=200, EC = 82.49605, CB = 51. 50395; h = 201, EC= 82.93577, CB= 51. 73423. On any plane, EB, the vertical unit pressure b 2 4b at E=pi= b 2 where b=EB, and W is the weight of masonry above the plane. This follows from the assumed "law of the trapezoid." From these formulas we derive: At h = 199, pi = 177.45483, p 2 =32.22542 ; h = 200, pi = 178.3855, p 2 =32.24139; h =201, pi = 179.3160, p 2 =32.25798. Call p the vertical unit stress at a distance, x' ' , from E; then p\~pi , r , b and the total stress on the base, x' ', is (i) To find the unit shear on vertical or horizontal planes, 1 consider a slice of the dam, bounded by 1 The writer desires here to acknowledge his indebted- ness to a recent paper on "Stresses in Masonry Dams," by Ernest Prescot Hill, M. Inst. C.E., published in Minutes of Proceedings, Inst. C.E., Vol. CLXXII, p. 134. Mr. Hill considers the case of a dam with a vertical inner face. By the aid of the calculus, he effects an exact solution, which leads to general formulas for shear and normal pressures on vertical planes. The principles at the base of his method, though somewhat disguised by the calculus notation, are essen- tially the simo as those used by the author. Mr. Hill ascribes to Professor W. C. Unwin the sugges- tion, "that the shearing stress at any point may be found by considering the difference between the total net vertical reactions [between that point and either face] along two horizontal planes at unit distance apart," and states that Prof. Unwin "has applied the principle to a triangular dam by the use of alge- braical methods." Dr. Unwin states (Proc. Inst. C.E., Vol. CLXXII, Part II, p. 161) that he ascertained after his papers were written, that by a different method, Levy had previously arrived at the same conclusions. 155 horizontal planes at ft = 199 and A =200, the water face and a vertical plane, at a distance, x, from the inner face (Fig. 17), in equilibrium under the water pressure acting horizontally on its left face and the forces exerted by the other parts of the dam on the slice. These forces consist of the uniformly increasing stress, P' , on top, acting down; the uniformly increasing stress, P, on the bottom, acting up; a shear acting on the vertical plane P' FIG. I/. at the right, of average intensity qi per square foot, the weight of the body (x 0.01), besides the horizontal forces to be given later. The vertical component of the water pressure is here neglected, as usual. The origin for x is taken, here and in all subsequent work, at the level, ft =200, at the inner face. For equilibrium, the sum of the vertical com- ponents must be zero. Therefore, 5l = (x-0.01)4-P'-P. ... (2) 156 To find P', substitute in Equation (1), x' x -0.02, pt =32.22542, pi -pi = 145.22941, k = 133.330, giving P' =32.20364* + 0.5446238**- 0.6442906. For P, *'=*, p 2 = 32.24139, y>i-p 2 = 146.1441, and 6 = 134; therefore, P = 32.24139* +0.5453138* 2 . Substituting in Equation (2), we derive the average unit shear, g, = _ 0.6542906 - 0.96225* - 0.000690D* 2 . . (3) This value of q\ is strictly correct w en x^_ 0.02 It is slightly in error when 0<*<0.02. t=200 x +0.01 X 1 X + 0.02 FIG. 18. A similar investigation holds to obtain the average unit shear, 92 (Fig. 18), on a vertical plane, at a distance, *, from E, extending from the level, ft = 200, to the level, h=20l. We have, for equilibrium, P". (4) 157 We find P" by substituting in Equation (1), z' = (z+0.02), pz =32.25798, pi-p 2 = 147.05S02, and 6 = 134.67. P"=32.27982o: + 0.5459941x2 + 0.6453780. Substituting this, and the value pre- viously found for P, in Equation (4), we derive, q 2 = -0.6353780 +0.961 57:c-0.0006803z 2 . (5) This is strictly correct only when x>_0. The mean, ^(51+92), of these average sheare will be assumed as approximately equal to the inten- sity of shear at the point, G(x = EG), at the level, h = 200. Call q this intensity of shear on a ver- tical plane at G] therefore, q= -0.6448343 +0.96191* -0.0006856*'. (6) Checks. By Appendix HI (6) and (d), the exact value of q, at either face, =/> tan , where p = vertical unit normal stress at the face and ^ is the angle the face makes with the vertical. Thus, at the inner face,g= -32.24139X0.02 = -0.6448278, whereas Equation (6) gives for x = 0, q= 0.6448343. At the outer face, the exact value is, 178.3855 X0.65 = 115. 9506, whereas Equation (6) gives, for a; = 134, q = 115.9405. A still more searching test can be devised. It is a well-known principle that the intensity of shear at a point, on vertical or horizontal planes, is the same [Appendix III (a)]. Therefore, regard- ing Equation (6) as giving the horizontal unit 158 shear, at the level, h = 200, where b = 134 ft. ; the total shear, from face to face, on this level, is /*: I Jx :r=134 This should equal the total water pressure down to the same level, -=- (200) 2 = 8000. Formula (6) o thus gives practically exact results. In order to find the normal unit stress on a vertical plane, we shall assume that q\, given by Equation (3), equals the intensity of shear on a vertical or horizontal plane at the point, x, at h = 199. 5; and that qz, given by Equation (5), gives the shear intensity at x at h =200.5. This evidently supposes that the shear intensity in- vreases uniformly, vertically, from h = 199 to fc = 201. Consider a portion of the dam, Fig. 19, bounded by the water face; the plane, FM, at the level, h = 199.5, on which the total shear is Q', the plane EN, at the level 200-5, on which the total shear is Q, and the vertical plane, MN, 1 sq. ft. in area, on which the average normal stress is p' '. The water pressure on EF will be supposed to be exerted horizontally. It is equal to 80 units. Assuming, as stated, that 91 = intensity of hori- zontal shear at M, and 92 = the corresponding intensity at N, we have, taking the origin as before at O, r x rx '= I qidx; Q= I qi dx; Jo.oi J-o.oi 159 or, Q'= 0.006494794 -0.6542906z + 0.481 125*2 -0.00023z; Q = _ 0.00640186 - 0.6353780x + .480785*' x 8 - 0.0006803 o". o Checks. The total water pressure for h = 199.5 is -^(199.5) 2 = 7960.05 and for h = 200.5, -^-(200.5)2 o o = 8040.05. The first should equal Q', for x = 133.665, or 7959.22; the second should equal Q, Q' h = 199.J 200 ^ = 200.5 F an 0.0? X 1 r si x + 0.01 1 FIG. 19. for x = 134.335, or 8041.12. The slight differences tend to give confidence in the results. For equilibrium, the sum of the horizontal forces acting on EFMN, Fig. 19, must be zero; therefore, p'=80+Q'-0, .... (7) p' = 80.01 -0.0189* + 0.00034*2_o.00000323x. This average stress will now be assumed to be the intensity of the horizontal unit stress on vertical planes at h=200. 160 It will now be perceived why a seven-place table was necessary in the computations, the coefficients of x 2 and x 3 having only two or three significant figures in the final result. If the planes originally had been taken 0.1 ft. apart vertically, a ten-place table would have been required. Checks. The value of p' ', for x = 0, p' =80.012896, is the same as that given by Appendix III (d), 80+0.6448X0.02. When z = 134, the formula gives p' = 75.81, whereas the exact theory, Appen- dix III (6), gives p'=m2p = (0.65)2 XI 78.39 = 75.37. The difference is 0.44 at the outer face. For any other point, it might be assumed to vary with x, so that it could be corrected by substracting 44 -^-rx=0.0033x from the value of p' above. For JL54 ease of computation, the formula will be written, 7/ = 80.01 -0.02z + 0.00034x2 -0.0000l|-. (g) The first coefficient of x 3 cannot be counted on to the last two figures, hence we are permitted to change 323 to 333 in that coefficient. When z = 134, Equation (8) gives p' = 75.41, nearly the exact value. The three formulas for p, q, and p', at the level ^ = 200, are thus as follows: p =32.24 + 1.09063z; ; q= -0.64+ 0.962* -0.000686Z 2 ; p' = 80.01 - 0.02z +0.00034*2 _ o.OOOOl . 1G1 Since the weight per cubic foot of masonry was assumed as two and one-half times that of water, we must multiply the stresses given in Table I by ~n (62.5) = 156.25, to reduce to pounds per square foot; or by 1.085, to reduce to pounds per square inch. TABLE I. X 10 25 50 p 32.24 43.15 59.50 86.77 g -0.64 8.91 22.98 45.75 Max. / ! '. 80.01 80.02 80.02 82.06 79.66 94.67 79.11 128 . 85 Min. / 32.23 41 .11 44.48 37.03 8 for max. / . . 90 46' 77 06' 56 50' 42 36 X 75 100 134 P 114.04 141.30 178.39 q 67 . 65 88.70 115.95 p' 79.01 78.08 75.37 Max. / Min./ 166.40 26.64 203 . 85 15.52 253.71 B for max. / . . 37 44' 35 12' 33 01' In Table 1 the stresses are those experienced at the level, h = 200. p = vertical unit stress on a horizontal plane; q = shearing unit stress on horizontal or ver- tical planes; p'= horizontal unit stress on vertical planes; Max. /= maximum normal stress acting on a plane inclined to the horizontal at the angle, S, given on the last line; 102 Min. /= minimum normal stress acting on a plane perpendicular to the last. From max. / and min. /, with 6, the ellipse of stress can be drawn, and the stress in any direc- tion, with the plane on which it acts, can be ascertained. It will be observed that there is no tension exerted anywhere, and that the maximum com- pression is 253.71, or 275 Ibs. per square inch, which is exerted at the outer face, parallel to that face, upon a plane at right angles to the face. In Appendix III (e), the important formula, for the maximum normal intensity at the outer face, acting parallel to that face, is proved. In this 'instance, p = 178.39, tan < = 0.65, therefore = 33 01', whence /=253. 71. This' stress is unaccompanied with any conju- gate stress, perpendicular to the face. In the interior of the dam, where conjugate stresses prevail, the masonry is perhaps better able to withstand a certain compressive stress than at the face. The distribution of stresses, at the level, h = 200, is shown in Fig. 20, on the supposition that the base of the dam is a little below that level. The connection with the foundation mate- rially modifies this distribution; but Fig. 20 shows the distribution for sections, say, from 10 to 20 ft. above the base, up to the level h = 100, fairly well, on the basis of the trapezoid law. As has been 163 mentioned before, this law gives a pressure greater than the actual at the outer face. Since the batter of the inner face is very small, the results of Table I should agree approximately, except near the inner face, with those found by Mr. Hill in the paper referred to in the foot note. -rfrr r~n~ TT mTTTT 1-= =dd V -r^r -rrrrrr i 1 ! 1 i i i i i i | i i i r^K / \ FIG. 20. Substituting numerical values, Mr. Hill's formulas, for h =200, reduce to q = 0.9426x - 0.0005768x2, p'=80 0.0001289x2 0.0000009615x 3 ; giving: 104: X 10 25 50 75 100 134 q 9 36 1 23.2) 45.69 67.45 88.49 115.95 P' 80 79.99 79.93 79.56 78.87 77.75 75.38 On comparnig these formulas with those of the writer, it will be observed that the absolute term in the value of q and a consequent term of the first degree in x, in the value of p' ', are lacking in Mr. Hill's formulas. This results from taking the inner face as vertical. Although the coeffi- cients also differ, it is seen that the numerical values are very nearly the same. In Fig. 21 are shown, on a drawing of the dam, to scale, the lines of the centers of pressure for reservoir full and empty. To the right, and under the word "factors," are certain numbers, written in the form of frac- tions. For any joint, the upper number gives the factor against overturning, or the number by which it is necessary to multiply the water pressure down to the joint, to cause the total resultant to pass through the outer edge of the joint con- sidered. The lower numbers give the ratio of the weight of masonry above a joint to the water pressure corresponding. It is believed that these "factors" should in- crease from the base upward, to allow somewhat for earthquakes, expansion of ice in freezing, etc., since the effects of such accidental forces is pro- portionately greater on the upper joints. Stresses due to water infiltration are not included 105 here; neither are stresses due to temperature changes. The unit stresses, /, in pounds per square inch, acting parallel to the adjacent face, are as follows, FIG. 21. and refer to the outer edges of the joints, for reservoir full, and to the inner edges for reservoir empty : h f at Outer Edge, at Inner Edge. 50 85 58 100 136 133 150 204 180 200 275 228 166 The stresses, /, are normal pressures on planes perpendicular to the respective faces, and are the greatest stresses that can be experienced in the dam. In fact, they are greater than the true stresses, since the trapezoid law is not exact, particularly near the base, as before remarked. It would then seem that the dam, thus far, is safe, since the maximum unit stress is less than con- crete, even, is subjected to daily, in good practice. For an actual construction, the outer face should be curved, from near h = 50 to the top, as shown by the curved dotted line in Fig. 21. The subject of the stresses in masonry dams has caused a great deal of discussion among British engineers in the last two or three years. The subject was reopened by Mr. L. W. Atcherly and Professor Karl Pearson, 1 who gave the results of certain experiments which seemed to indicate considerable tension across vertical planes near the outer toe. The late Sir Benjamin Baker, Hon. M. Am. Soc. C. E., also published 2 the results of experiments on a model dam of stiff jelly, and very recently, the "Experimental Investigations" of Sir J. W. Ottley and Mr. A. W. Brightmore 3 on elastic dams of "plasticine" (a kind of modeling clay) and the experiments of Messrs. J. S. Wilson and W. Gore 4 on "India Rubber Models" have been presented. VMinutes of Proceedings, Inst. C. E., Vol. CLXII, p. 456. 2 Ibid., Vol. CLXII, p. 123. 3 Ibid., Vol. CLXXII, p. 89. < Ibid., Vol. CLXXII, p. 107. 1GT It is not the object of this paper to discuss these later experiments; but it may be remarked that they show very plainly that no tension exists near the outer toe, but that tension does exist at the inner toe, where the dam is joined to the foundation, and it has become a. serious matter how to deal with it. The influence of the founda- tion in modifying the distribution of the stresses at the base of the dam was found to be very great, causing the shear there to be more uniform than higher up, where the parabolic law, nearly as given by the formulas above, was found to hold. Also, above some undertermined plane, a small distance above the base, the usual "law of the trapezoid" was found to be approximately correct, leading to stresses on the safe side at the outer toe. This law leads to stresses at the outer toe of the base considerably in excess of the true ones. It was found, from the rubber models particu- larly, as theory indicates, that the greatest normal ' pressures are exerted at the down-stream face, for reservoir full, and they act in a direction parallel to that face. 1G8 APPENDIX III. RELATIONS BETWEEN STRESSED AT ANY POINT OF A DAM. (a) Consider a cube of masonry, Fig. 22, the edge of which has the length, a, bounded by ver- tical and horizontal planes and subjected to normal and shearing forces, caused by the action of the other parts of the dam. Since a will be supposed to di- minish indefinitely, the weight of the cube, which is proportional to a 3 , is an infinitesimal of the third order, and can be neglected in comparison with the normal forces, which vary as a 2 and are thus of the second order. Similarly, the average unit stresses exerted on the faces can be treated from the first as the unit stresses at any point, A, of tho cube. As a diminishes indefinitely, the oppositely directed 169 normal forces approach equality and balance independently; hence the couples formed by the shears on opposite faces must likewise approach equality; the one being right-handed, the other left-handed; therefore qaXa = q'aXa, or q=q'; hence, the intensities of shear at a point on two planes at right angles are equal. The relative directions of the shears on two planes at right angles are determined, as above, from the con- sideration that one resulting couple must be right- handed and the other left-handed. This applies also to Figs. 23 to 26. FIG. 23. (b) In Fig. 23, ABC is the right section of a prism at the outer face, with lateral faces one unit in length, perpendicular to the plane of the paper. Let AB be vertical; tan = m, a constant; p= normal intensity on a horizontal plane at C; p' = normal intensity on a vertical plane at C; g = shear intensity on horizontal or vertical planes at C. 170 The weight of the prism is %ab. Balancing vertical as well as horizontal com- ponents, we have, when a=AB and b = AC are very small, pb = qa + %ab, nearly; p'a = qb. Dividing the first equation by 6, the second by a, the limit, as a and 6 approach zero, gives exactly, p = q cot , therefore q = mp; p'=qtan, therefore p' = m 2 p, pp' = q~. These equations give the relations between p, q, and p' at the outer face. The same relations hold at the inner face, for reservoir empty, on replacing $ by 0', the angle the inner face makes with the vertical. For the remaining cases, the final limits will be written at once, since the complete process of deriving them is evident from the above. In fact, the weight of the prism, %ob, being of the second order, can be neglected in comparison, with qa, etc. 2 (c) For reservoir full, calling iv= h, the inten- o sity of water pressure, horizontally or vertically, at C, we have at the inner face, putting tan $' =n, Fig. 24, therefore p= q + w; 171 (d) If the vertical component of the water pressure is neglected, these equations reduce to therefore (e) Since the shear on the outer face is zerc, therefore, by (a), the shear on a plane, AD, Fig. 25, perpendicular to the outer face, is also zero, or the stress on AD is normal. Call / the intensity of such a stress at C. The total pressure on AZ)=/X AD=fb cos , and its vertical component is fb cos 2 , therefore balancing the vertical components, pb =*fb therefore 172 This is a most important formula for finding the maximum normal intensity at the outer fact-. It applies equally to the inner face for reservoir empty, on changing $ to <', the angle the inner face makes with the vertical. For either face, p is the vertical normal unit stress at the face con- sidered. (/) Principal Normal Stresses at Any Point in the Dam and the Planes on which they Act. In the prism, ABC, Fig, 26, let AB be one of the planes FIG. 26. on which the stress is normal. Let / be its inten- sity. The stress on the plane, AB, of unit length perpendicular to the plane of the paper, is thus fc\ its vertical component is /c cos =/&, and its horizontal component is fc sin B -=/a, 8 being the angle that A B makes with the horizontal Place the sum of the vertical forces acting on ABC equal to zero; also place the sum of hori- zontal forces equal to zero. 173 fb = pb + qa, therefore / p = q tan 0, fa = qb + p'a, therefore / p' = q cot 8 , The difference of the last two equations gives 1 tan 2 e The angles, (differing by 90), computed from this equation, give the directions of the planes, AB, on which the stress is entirely normal. From an equation above, we likewise have f-P tan 8 = -. 2 This gives directly the plane on which a given / acts. To deduce a formula for /, take the product of two equations above: V (p +p') 2 ~4(pp f - This equation gives the two values of / corre- sponding to the two planes mentioned; com- 174 pressive when / is positive, tensile when negative There can be no tension when pp' ">_ q z . A better form for computation is, THE VAN NOSTRAND SCIENCE SEKIES No. 47. LINKAGES: THE DIFFERENT FORMS and Uses of Articulated Links. By J. D. C. De Roos. No. 48. THEORY OF SOLID AND BRACED Elastic Arches. .By William Cain, C.E. Second edi- tion, revised and enlarged. No. 49. MOTION OF A SOLID IN A FLUID. By Thomas Craig, Ph.D. No. 50. DWELLING-HOUSES; THEIR SANT- tary Construction and Arrangements. By Prof. W. Hi. Corfield. No. 51. 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