770 UC-NRI ENGIN. B 3 mi LIBRARY GIFT OF Mars ton Campbell, Jr li,ngmeermgf Library RETAINING -WALLS ; FQR EAUTH. THEORY AS DEVELOPED BY PROF. JACOB J. WEYRAUCH. EXPANDED AND SUPPLEMENTED BY PRACTICAL EXAMPLES, WITH NOTES ON LATER INVESTIGATIONS, BY MALVERD A. HOWE, C.E. NEW YORK : JOHN WILEY AND SONS, 15 ASTOR PLACE. 1886. Engineering Library Copyright, 1886, By JOHN WILEY AND SONS. NOTE. FOR the translation of Prof. Weyrauch's paper the writer is indebted to the labor of Prof. A. J. Du Bois, of the Sheffield Scientific School, .. Yale College, who had copies printed by the electric-pen process. However, only the leading equations of Prof. Weyrauch were given ; hence a great deal of labor has been devoted to expanding, verifying, and filling in the intermediate steps of the work, and this nucleus of the mathematical part alone has grown to about double the original quantity. For most of the historical notes acknowledgment is also due to Prof. Du Bois for his article in Journal of the Franklin Institute for December, 1879, in which he called the atten- tion of American engineers to the value of Weyrauch's theory. M. A. H. 834140 PREFACE. THE following theory of Prof. Weyrauch is presented for the use of the practical man, although, at the first glance, the array of mathematical formulae may cause him to smile at such a statement. However, a brief examination of the results and their practical applications should con- vince him that a "long- felt want" has been supplied, not only in the graphical constructions that can be performed anywhere with the simplest instruments, but by formulae so simple that substitutions can be made and solutions obtained in a very short time. The mathematical operations are somewhat tedious, and are now presented for the first time in English, fully ex- panded and verified, so as to be easily followed by those who are inclined to doubt the results or wish to satisfy themselves that they are correct. In fact, all that it is absolutely necessary for the prac- tical man to have is the contents of the Recapitulation of Formulae, in order to determine the earth-thrust and its direction for a wall not leaning backward. For walls leaning backward, Prof. Kankine's method has been combined with Weyrauch's. Walls having a curved profile and those with counter- forts have not met with the approval of American engin- vi PREFACE. cers, and vith good reason, as they can as yet be treated only by empirical formulae ; for these reasons they will not be considered. Some may question the accuracy of the theory for sur- face of earth inclined, i.e., for surcharged wall ; yet in any case Prof. Weyrauch's theory is to be preferred to any of those previously advanced. An attempt has been made to present the subject in a simple manner, and to show by a few examples the sim- plicity of the application of the formulae and construc- tions. The reader who does not care to follow the theory until he is persuaded of its practical value in application should turn at once to examples and discussions in Part II. THAYEK, SCHOOL OF CIVIL ENGINEERING, April, 1886. M. A. H. INTRODUCTION. OLDER THEORIES. RETAINING- WALLS were first treated in 1687, but until Coulomb's time no theories were advanced that are worthy of much notice, as they were for the most part founded upon mere assumptions, for which the reasons, if given at all, were statements unproved. In 1773 Coulomb founded a new school that assumed the earth-pressure to act normally to the wall, and to be induced by a prism of maximum thrust. As Coulomb's theory was the only one generally ac- cepted for some sixty or more years, a brief outline of the principal points will be given. According to Coulomb, the surface of rupture is a plane along which a prism of rupture tends to slide ; he also assumes the direction of the earth's thrust to be normal to the wall. The weight of the prism of thrust, G, and the reaction of the wall, E, are decomposed into forces respectively perpendicular and parallel to the surface of rupture. Then the difference of the horizontal components must represent the resistance to sliding of the prism of thrust. This resistance consists of friction, which is proportional to the normal pressure, and cohesion, which is propor- tional to the surface of sliding. Vlll INTRODUCTION. B C FIG. 0. Let /= the coefficient of friction. c = the coefficient of cohesion. y = the weight of a unit of volume of the earth. h = the height of the wall. x = the distance BO. E = the earth- thrust against the v all. The wall will be assumed as vertical, and the earth-sur- face as horizontal (as this is the only case discussed by Coulomb). Then from the above figure __ *(x +/A) Coulomb then reasons that there must be somewhere a surface of rupture corresponding to a prism that will ex- ert a maximum pressure against the wall; and he proceeds to differentiate the above expression with respect to x, and finds that E is a maximum when x = fh + Ji Vl +/"" 1 , INTROD TTCTION. ix an expression wholly independent of the coefficient of co- hesion. The point of application of E is shown to be at one third the height of the wall. These deductions are correct for this particular case, as Prof. Weyrauch's theory shows. But when either the earth or the wall is inclined, the direction of E is not normal to the wall, but makes an angle, (5, with the normal. The value of E, as given by Coulomb, depends directly upon the position of the surface of rupture, and changes in intensity but not in direction as the surface of rup- ture is assumed to change. In reality E is constant for any given wall and earth, and does not depend upon the position of the surface of rupture. See Recapitulation ; and notice that GO, the angle the surface of rupture makes with the vertical, does not occur in the equations for the value of E. Later writers have proved (?) Coulomb's prism of maxi- mum thrust to be limited by a plane which bisects the angle made by the natural slope of the earth and the vertical rear face of the wall. In the nineteenth century, Rankine, Levy, and Mohr have considered the conditions of the earth-particles, and arrive at their results by integration. Rebhahn (1871) and Winkler (1872) advanced theories founded on assumptions identical with those of Rankine. Rankine assumed the surface of rupture to be a plane, and that the direction of the earth's thrust is parallel to the top surface. As has been said before, Coulomb assumed the pressure X INTRODUCTION. of the earth to act normally to the wall. Since 1840 it has been customary to assume the earth-pressure to make an angle with the normal equal to the angle of repose. If this were true, a horizontal wall would be pressed, not ver- tically, but by forces acting at an angle with the vertical equal to the angle of repose, which is manifestly incor- rect. It will be seen that Prof. Weyrauch's theory is closely allied to Prof. Rankine's, but conclusively proves that the earth-pressure acts parallel to the top surface of the earth only in special cases. WEYRAUCH'S THEORY OF THE RETAINING -WALL/ PART FIRST. IN the following the earth is supposed without cohesion, and its pressure is determined independently of any arbi- trary assumptions as to direction of the earth-pressure, and with sole reference to the three necessary conditions of equilibrium. The single and only supposition, then, is as follows: That the forces upon any imaginary plane-section through the mass of earth have the same direction. This assumption lies afc the foundation of all theories of earth-pressure against retaining-walls. For those cases, therefore, to which the following discussion does not apply no complete or satisfactory theory is yet possible. In what follows, the ordinary assumption as to the direction of the earth-pressure will be proved to be incorrect, except for special cases. * Zeitsclirift fur Baukunde, Band I. Heft 2, 1878. 2 TUEOHY. OF THE RETAILING - WALL. I. GENERAL RELATIONS. Let the surface of the earth have any form, and the wall AB, Fig. 1, have any inclination. The earth-pres- sure makes any angle, tf, with the normal to the wall. Suppose through the point A the plane A C. Then the weight G of the prism ABCis held in equilibrium by the reaction of the wall, E, arid by the resultant, R, of all the forces acting upon A O. Now decompose E, G, and R into components parallel and normal to AC-, then for every unit in length of the wall, denoting by e, g, and r the lever-arms of E, G, and R respectively with reference to A, the sum of the forces parallel to A = 0, or (i) GENERAL RELATIONS. 3 the sum of the forces perpendicular to A C = 0, or the sum of moments about A = 0, or Gg + Ee-Jtr = (3) Equation (3) was first introduced by Prof. Weyrauch. Further, according to the theory of friction, if cp is the coefficient of friction for earth on earth, p / P P / -~ - tan (p or n * = tan (p. . . (4) Va V T~ Vi If now there is any plane for which P P l (Q + d) tan cp, . . . (5) P P this plane AC will be a plane of equilibrium, and jr^r-fr will be a maximum, or ? p p \ i /i \ . n j Jl-^ir^O '(*) This plane is designated as the "surface of rupture." From Fig. 1, for every position of A G, P = G cos GO, Q G sin GO, Substituting the above values of P, P l9 Q, and Q l in equation (5), it becomes G cos GO Usiu (GO -f- + <5) = [G sin 6? -J- ^cos (GO -f- <* + ^)] tan

^ _ 3 ^ g . n = 8) ttCi? Now since ? = J^ . dco . ky, equation (7e) reduces to CP -SOS sin (a ++a, +. + ) 3 cos ( + a.) + which reduces to o __ cos ( -r >) sin (<}> + <*-+ a +S) cos (a -s-v^-r , fi " o ,-:,, a. ,..,_, ^_ 2sin(a _|_ 5)cos( ^ ) _(_ w)siu (^_|_ a) _ ( _ a+6)+cos2(< | ( _|_ w) j- W C > Since sin (99 -j- GL> + ^ + #) = sin (^? -J- GL?) cos (a: -j- ) cos 2 sin (tf-|-tf) cos (cp-^-Go) cos + cos 2 (), or sin 2 (cp-{-G>) cos 3 (or 2 sin 2 (a +3) cos 2 + sin 2 (a-{-d) cos 2 (^-j-r^) -f- cos 9 ) cos 2 (a+d) cos 2 or sin 2 (^-J-cj) cos 9 (a-\-d) -{- cos 9 () cos 2 (ar-[-^), or cos 9 (tf+tf) [sin 2 (^+co) + cos 2 GENERAL RELATIONS. which equals cos 2 (a-\-d), and equation (8e) becomes, after dividing by cos (&-{-$) and factoring, from which sin 2G cos (a+6) - which being substituted in equation (7) gives _ " cos V-. . (io) FIG. 2. And, since the sum of the horizontal components of E, G, and R must be equal to 0, or Fig. 2, and = R cos (GO-\- cp), cos 8 THEORY OF THE RETAINING-WALL. which becomes, after substituting the value of E from equation (10), ..... (11) Let AD, Fig. 2, be the natural slope of the ground. From C let fall the perpendicular CH, and draw CJ, mak- ing the angle (a+d) with CH ; then A T AJ= S1n cos (a+6) FIG. 2. The expression for AJ is obtained in the following man- ner (Fig. 2) : ), AH &si : 6'^T :: sin (or+tf) : cos Cffs'm sn cos(a+<5) sin (

) sin (a+ 6) , ~Sos (a+d~ ' GENERAL RELATIONS. which reduces to COS ( and hence, according to equation (9), G = Func. y = y A A CJ. .... (12) Also, if AK is perpendicular to CJ, CH Ic cos ((p-\-co) _ E and if JL is made equal to JC, then, since the perpendicu- lar from L upon CJ is equal to CH, ACJL _ CH _E ACJA~ AK~ G' or E=yACJL (13) If, finally, AM=AC, AM. CH or R = yAACM. (14) All these geometrical results may be summed up as fol- lows : Draw from the highest point C of the surface of rupture a line CJ, which makes with the normal CH to the natu- ral slope the angle a -\- d, or the angle which the earth- pressure makes with the horizontal ; then the A A CJ is 10 THEORY OF THE RETAINING-WALL. equal in area to the A ABC, the prism of rupture. Then lay off JL = JC and AM= AC and draw CL and CM ; then for every unit in length of the wall the following relations exist : Weight of prism of rupture, G = yACAJ\ J Earth-pressure upon wall, E = yACJL; >- (14a) Reaction of the surface of rupture, R = yACAM. ) The first two relations were first made known by Bebhahu in 1871, for tf = or cp. Since, now, G : E : R = AJ : JC : CA, . . . (15) it can be asserted that The weight of the prism of rupture and the reactions of the wall and of the surface of rupture are to each other as the three sides of the A A CJ. Thus far no assumption whatever has been made as to the value of the angle tf. This is determined by equation (3), which, in all theories following Coulomb's method, does not occur. PLANE EARTH-SURFACE INCLINED. 11 II. PLANE EARTH-SURFACE INCLINED ADOPT in this case the notation of Fig. 3, anct let E be first determined for any value of d. II =90 (e+6>) 111= 90 (a- e> FIG. 3. If A G is the surface of rupture, then AAEG AACJ\ or, since AB _sin II 277 ~ shTlTP In like manner, AJ = AC sin II sin III ' sin V snTVl' or But since A ABC = J^ 6V, ^<7sin/ = A7. ^ sin /sin// sin /F sin V sin /// '. (16) ; . . . (IGa) sin VI THEORY OF THE RETAININO-WALL. or, finally, sin (ar-f-Gtf)cos sin cos cos cos (a: s). Further, from Fig. 3, if BN is perpendicular to 'AD, AADB = ZAAJC+AJDC, or AD . BN= 2AJ . CH+JD . Off; and since and BN _BO _ OP ~CH ^W' ~JD AD . OD = JD (AJ+ AD), AD (AD-AO) = (AD- AJ) (AJ + AD), whence AO . AJ= AJ . AD. . . . (17) D B FIG. Upon this relation rests the well-known construction of Poncelet for the earth-pressure. Draw (Fig. 3') BN per- pendicular to the natural slope AD\ draw BO, making the same angle with ^^that^makcs with the horizontal, and PLANE EARTH-SURFACE INCLINED. 13 then determine the point /so that equation (17) is ful- filled, that is, make A J a mean proportional between A and AD ; then draw JC parallel to OB. Thus the surface of rupture AC is found, and use can now be made of the relations already deduced in I. In order to determine J (A, 0, and D being given), there are several methods, one of which is indicated in the figure. In all these constructions 6 is assumed. Now from equation (13), E % y TO* cos (a -f d), but CJ_ _ AD- AJ _ AD - V~AD~. AO _ BO ~ AD - AO ~ AD-AO ~~AO ~D J0-, tnen UJ = - iU = - From Fig. 3, A ^ r Let w = y __ then CJ = T -2?0 = --- AD l n* .1 -f n = ) AB _ sin (cp - t) AB ~ cos (a: + 6)' AD ~ cos (a -~e) ; and the multiplication of these equations gives = -- n ^ cos (a + 6) cos (a - f)' ^L=^/. cos (a -\- 6) * 14 THEORY OF THE RETAINING -WALL. and by substitution of SO and n in the value for CJ, and of CJ in that for E, rcos(-a)-| _ Zy_ r cos (0 - a) n ft Y L n-f-l J 2 cos (a + 8) L(n + l) cosaJ 2cos(a+) - ' For the special Ciise of the earth-surface parallel to the angle of repose, = cp, n = 0, and cos J '^ ' These formulae hold good for any value of d. But the angle d is determined by equation (3). In order to insert e and r in this formula, the points of application of E and R must be known. The angles d and G? are connected by the relations in (16&), in which there are no other unknown quantities. Since now d, according to the single assump- tion of Prof. Weyrauch's theory, is independent of the height, so also is co, and then for variable li equations (19) and (11) become E = Cl\ R = CJc\ dE = 2 CW, dR = 2 CJcdJc. Let x and z equal the distance of the point of application of E and R from A, respectively. Now considering the top as the origin or centre of moments, Z, R(k - x) = 2 and therefore x = $l and z $k. Now O must act through the centre of gravity of the A ABC, and it has been already proved that the points PLANE EARTH-SURFACE INCLINED. 15 of application of E and R are at distances $1 and -J& re- spectively above A ; hence (Fig. 3') ah = ed and hf=g = bd ah = ^Jc sin GO ^l sin a. Substituting these values in equation (3) and referring to equation (15), AB (CJ cos S AJ sin a) = AC (AC cos - AJ sin w), ... (22) or sin //(sin TV cos 8 sin Fsin a) = sin ///(sin VI cos - sin Fsin a>), (22a) or cos (e -f w) [cos ( + <>) cos 8 sin ( + w-|- a-f-S) sin a] = COS (a e) [COS (a -f- S) COS < Sin (< -f- w + a -f- 5) sin w]. . . (226) By means of the two equations (16&) and (22b) the two unknown quantities d and GO are completely determined. As soon as these are known, E can be found from equation (19) or (20). Also by the relations in equations (16) and (22), or (16#) and (22#), the surface of rupture and direc- tion of the earth-pressure may be determined, and can therefore be found by a graphical construction. 16 THEORY OF THE RETAINING-WALL. III. HORIZONTAL EARTH-SURFACE. FOR this most important practical case it is simply nec- essary to make in equation (19). The proper values of d and GO in this case are found from (16) and (2%b). Making s = in equation (22b), it becomes cos GO [cos (cp -j-G?) cos d sin (cp -j- GO -\- a -|- 6) sin a] cos a [cos (a -j- tf)cos

-j- G?) -f cos 2 a: sin GO sin d cos (

)sinG?] sin# p-\- GO] cos GO-\- cos 2 ** cos cos 2 a cos

, in the second to cos cp. If in the third cos 2 a = I sin 2 a, and in the fourth sin 2 a = 1 cos 2 a, equation (22c) be- comes + cos GO cos (cp + G?) cos d sin a: cos cos d sin 9) sin a cos a sin # cos cp + sin # sin 2 sin (^-j-^) cos co+sin d sin a? cos (^>-|-CL)) sin 2 : sin GO sin (5 1 cos (

-)- GO) cos 2 a: cos cp cos d -\- sin a: cos a cos ^? sin # Reducing and dividing by cos #, sin of cos a sin

sin (cp -\- GO) -\- cos 2 a cos GZ? cos ( cp -j- GJ Since cos GO sin (< 4- GO) sin GO cos = 0. 18 THEORY OF THE RETAINING- WALL. and sin GO sin (cp -f- GO) -f- cos GO cos (cp -f GO) = cos ), it becomes sin (a -j- GO) cos (a -f- $) cos a? = sin (

) sin sin -j- cos (^ -f- <*>) sin a cos d -f- cos (^> + a?) cos sin 20 THEORY OF THE RETAINING- WALL. and equation (24#) reduces to cos GO sin (a + G?) cos a cos $ cos oo sin (a -|- GO) sin a sin cos 3 a cos (

) sin (cp + G?) tan tf f ~ cos sin cos 2 (

) cos 2 oo = cos 2(q> -{- &?) cos 2 which equals 2 sin -J- [2&? sn -f (2(7? -f __ 2 sin ( q>) sin (2&? -j- cp) ' or and sin also, sn cos sin 9?, cos % sin 2(^> + GO) -J sin sin 2r , 3 cos 2 t sin a cos = - , and cos 3 a = 1- -J. / /v Hence, after multiplying by 2, equation (24^) reduces to sin 2a sin (2cc? -f- 00 s ^ u V cos 2a -J sin 2(q> + a?) -j- cos 2 | sin 2 < -J sin 2(<7? -f- GJ) -\- -J- sin 2G? - tan 6^00820: cos 2 (^+c 6 9) cos 2 () tan d sin 2a sin ca cos a> tan d 22 THEORY OF THE RETAINING-WALL. Now 2 tan dsin 2 a cos 2 GO = [since sin 2 a 1 cos 2 a] [cos 2 6? cos 2 a cos 2 , and sin 2(q> -\- GO) sin 2&? _ sin 2(^> + co) sin 2oj ~~2~~ ~~~ ~2~ 2 sin j- (2y + 2&? ^<^) c 2 ' = sin cp cos (2cj+ ^?), and tan d cos 2or cos 2 ((p -{- GJ) -\- 2 tan # cos 2 a cos 3 GO L cos 2 . . \ = ( by making cos a = - --- 1- -J J tan # cos 2a [cos 2 (q) -f <*>) cos 2 GO] + tan d cos 2 G?, or tan d cos 2# sin 2&? -- < s ^ I HORIZONTAL EARTH-SURFACE. 23 Also, cos 2 (cp -\- GO) tan d -\- tan d cos 2 GO = tan # [cos 2 (cp -j- GO) cos 2 GO] = sin

cos GO tan (J = tan d sin 2# [sin (

) sin G? cos G?] F sin2( -f- G9) sin 2G?~| = tan o sin &a = = tan (^ sin 2fx sin <^> cos (2G? -f~ cos (2G? -j- cp) + sin 9? [sin (2G? -j- 9?) cos 2ar + cos (2 G? -f- + ft) = 2 cos 2 w -sin ft [sin (2w+ft) cos 2a+cos (2w+ft)"sin 2a] - sin ft sin (2w-f ft)' By making sin 2a = 2 sin a cos a and cos 2a = I 2 sin 2 a in the numerator, and cos 2 or = 2 cos or cos <* 1 and sin 2a = 2 sin cos # in the denominator, this becomes tan 8 = sinft [sin (2o)-fft) 2 sin a cos a cos (2a>-|-ft) + cos (2a>+ft) 2sin 2 a] - sin ft cos (2) cos a+cos (^M+ft) sin a] - 2 sin ft cos (2u>-f ft) tan 2cos a w-2sinftcosa[sin(.2w+ft)cosa+cos(2o) + ft)sino] ' 24 THEORY OF THE RETAINING-WALL. which reduces to tan d = sin cp sin a sin (2&? -f cp -}- a) sin cp cos (2cj -\- cp) cos 2 G? sin

cos a sin sin ? sin %a a) siu (2o? -j- cp) sin 9? cos 2a ' Dividing by sin cp, clearing of fractions and dividing by sin a, also transposing, this becomes sin (2ft? + cp -j- a) sin (2oo -J- 9) jj -\- cp -{- a) sin

. /rt sin (%(& -\- cp -j- a') sin cp cos 2 -- cos GO sin cos (2ft? -f- cp) [sin cp) sin sn a or sin (Sea -}- cp -\- a) sin (2&? -j- <^) sin cp cos 2^ sin (2a? -j- ^> -j-^) 2 cos or cos 2 -{- sin 9? 2 cos 2 a sin (2oi9 -}- cp -j- <*) _ cos (2G? -f- 9?) [sin (2Gt>+ 9*) sin ^ cos Since sin a 2 cos 2 cos 2. =0. HORIZONTAL EARTH-SURFACE. 25 this becomes sin (2cj -f~ cp -\- a) [sin (26? -f- -(-a?4-a)8in (co+cp) cos a cos 09 =0. 2 cos &? The formulae for GO, d, and E can now be found in the simplest manner. Equation (25) is satisfied for269-j- and likewise for a height h J t 2 jji j t a (7 representing the constant quantity. From equation (29*) E = C(H* 7i*); hence dE ' = 2CHdH2Ch dJi . Now let x equal the distance of the centre of pressure below the top of the wall, then Ex =20 f H H*dH- 2C f\*dk, or C(H* - Ii *)x = %CH 3 - f 6V, 2 H 3 - h s ' and if y = the distance from bottom, Equation (30) holds good when the earth-surface is loaded and the loading is equal to a distributed load of the height h . Still, even then, h is often so small that can o be substituted for it just as for unloaded earth-surface. In all cases d is determined by equation (28). 28 THEORY OF THE RETAINING-WALL. Instead of using equations (28) and (29), the following simple construction can be used : a E B c v FIG. 4. Draw (Pig. 4) AC and A D vertically and horizontally, each equal to h, also DF making the angle FDG = 45 ^ So with the horizontal. Through the points D and ^describe a circle whose centre lies in AD. Then draw GH parallel to AB, and through A the straight line HJ. Then JO is the direction of the earth-pressure upon the wall AB. If ^i^is made perpendicular to AB, and equal to AH, then the AABK gives the intensity and distribution of the earth-pressure, or E = yAABK. The proof of this construction is as follows : Conceive, in Fig. 4, JD and FG drawn, then A irn A P _ AG cos a ~ PH~ HG-[AGsm~a=PG]' in which AP represents the perpendicular let fall from A upon GH. HORIZONTAL EARTH-SURFACE. 29 but AG : AF :: AF : AD =h, AF therefore AG = -T = li tan 2 _. \ 2 Now ## GD sin or = ( J. G + AD) sin / \ = h sin a -\-li tan 2 (45 ^j sin a\ tan ^#(7 = A tan 2 (45-^ h sin a + h tan 2 ^45 - |) sin a-Ji tan 8 ^45 - |) sin r therefore tan AHG = ^-^ tan 2 ^45 - ?} = cot tan 2 (45 - %}. sin a \ 2/ \ Ji/ From Fig. 4, = cot AHG = tan or cot 2 f 45 - |] = tan (a+tf), or cos (a -f 6) 2 " L cos a J 2 cos (or + )' ^ ' a formula which holds good for all values of $, and which f or 8 = or GO gives results usually accepted in previous theories of retaining-walls. In order to find the proper values of 6 and GO, equations (16#) and (22b) must be used. In equation (22Z) replace sin (cp -f- GO -j- a -f d) by sin ( f cos (cp -|- Ce9) cos (^? -j" ^ + ^) si n ^ sin a: I + cos (a cp) cos (a -}- tf) cos cp = \ ~ cos (<* 9?) sin (9? -j- GO -f ) sin G;? cos 6 ( cos (a cp) cos (^-j* GO -\- a) sin (J sin a? ; ANGLE = ANGLE cp. 33 dividing by cos d and transposing, cos (a cp) cos (a -\- d) cos cp cos tf 4- cos (a cp) sin (cp 4- <& -\- a) sin &> 4~ cos (cp 4~ G&) cos ( 4~ ^) cos (a? 4- fi?) sin (< 4- co 4- or) sin # I 4- cos (^ + *) cos (

_ _ cos (a (cos a cos 5 sin a sin S) cos 6 cos S" = COS (a - <) COS ) sin a) sin(^fu > + c O sin a cos (a ) cos sin a cos(<#>+ "*) cos( +-|-a) -j- cos(a <#) sin w cos(<|>+w-(-a) and this equation fulfils the condition that the sum of the moments of Gf, E, and- R shall be zero. If equation (16#) is treated in a like manner, the result- ing equation will fulfil the condition that the sum of the forces parallel to the surface of rupture shall equal zero. Making s (p in equation (16), it reduces to sin (a 4- GO) cos (cp -\- a?) cos (a 4- #) sin (cp 4- of 4- &> 4" $) cos (4-Ge9-|-a')cos(a' )sin# B(a-90-- -^J^ v - =0; therefore . _ cos sin (a -j- <) sin (9? -f ^ + ) cos (a cp) ~ sin (a -\- GO) sin a + cos (^> + GO -\-a) cos (a' cp)' Setting both values of tan d equal to each other and clear- ing of fractions, the following expression is obtained: -f- cos a cos cp sin a sin (GO -j- a) cos (a -f- cp) cos a: sin a sin (&? -|- ) cos (GO -j- ) cos (GO -\- cp -}- a) sin a? sin a sin (GO -f- #) cos (a cp) sin (

) sin (cp -\- GO -f- a') -j- sina cos(^>-|-c) cos(^>-)-G?-}-a')cos(: cp) sii\(cp-\-Go-{-a) sin GO cos 3 (a cp) cos (

+ <*)], which becomes sin GO sin (GO -j- a) cos (cp -f- GO). Terms number 5 combined give cos(0+a>)cos( + <>+) +sin a sin (< 4- + )] wliich becomes cos (cp + GO -j- a) cos (cp -|- GO) cos (cp + GO). Terms number 4 combined give -j- cos cp cos (a -{-<# -f cos (a cp) sin GO cos oo -f a) cos GO i (CP + GO+ a}} Substituting for &?, 90 9>, this becomes cos cos (9? + 90 ^+a)sin( ^-|- 90-^) -f cos a cos (cp+ 90 ^>) sin(^+ 90-^+ a) [ + cos (-^) sin (90- ^) cos(^ + 90 As the second term reduces to zero, this becomes [cos a sin a cos (a (p) cos a sin

) (cos a sm 9 cos 9? sm a) to, or Fsin 2a . \ / x~| 7 cos ( #>) sin (9? a) J&9 sin %a sin ^(<^ a) ]) cos

) cos (a or [1 sin cp sin (cp therefore tan * = 1 sin

2a) To find an expression for the sin d, clear equation (33) 38 THEORY OF THE RETAINING -WALL. of fractions and deduce tan d tan d sin cp sin (cp = sin cp cos (cp %a). Multiplying by cos tf, sin 6 sin d sin cp sin (cp % = - 2- ^-. -- '-* = tan o\ 1 -f- sm cp sin (cp *Za) Since HD is perpendicular ivAB, the earth -pressure has the direction GJ. Further, TTTJ_ sn a sn # cos q> ~ sin (a -[- d ^)~~ sin (a -|- d 9?) AD = - - - -. or, with reference to the value of FD. cos cp A A T>Tr COS (^ ~ a ) S1T1 ^ ? J ' AABK = -- if -- ^ --- r-, and since from equation sin (a -}- d cp) 2 (34) sin (a-\- d cp) cos (9? a) = sin or cos (a -j- o"), V. THE RELIABILITY OF PROF. WEYRAUCH'S THEORY. PROF. WEYRAUCH'S theory is based upon the single as- sumption that the surface of rupture is a plane, and is mathematically correct for that assumption. If the sur- face of rupture is a plane, then all the forces acting upon this plane must be parallel, and can be investigated by considering the equilibrium of the earth-elements, as was done by Rankine (1856), Levy, and Mohr, who proved mathematically that, for earth without cohesion, surface at any inclination and of unlimited extent, the assumption is absolutely correct. The only question that can be raised, then, is whether it holds good for earth-surface limited, as by a wall. The latest experiments to determine whether or not the surface of rupture is a plane were made in 1885 by M. L. Leygue, Ingenieur auxiliaire des Travaux de VEtat,* who proved to his own satisfaction that the surface was not a plane, but a curve. A brief outline of his experiments will be given, so that any one can form an opinion as to the reliability and worth of the above result or statement. In one end of a box, with glass sides, having a width of * Paper No. 98, Annales des Fonts et Chaussees, novembre, 1885 : "Nouvelle Recherche sur la poussee des terres et le profil de re- ve'tement le plus economique." 42 THEORY OF TEE RETAINING -WALL. O m .40 (1.312 ft.), a length of O m . 80 (2. 624 ft.), and a height of O m .73 (2.395 ft.), he placed a movable board, the board rotating about a knife-edge at the bottom as an axis. The movable board or plane was from O m .374 (1.227 ft.) to O m .397 wide, and from O m .20 (0.651 ft.) to O m .25 (0.820 ft.) high. This plane was placed in the position to be studied, and fine dry sand filled in behind to the height desired. In the sand were placed horizontal layers of fine plaster. As the layers of plaster would be quite distinct from the sand, any change in their position would be readily detected through the glass sides of the box. Hence, as the plane was rotated, the sand and plaster would change position and a certain amount would break away from the mass in the box, and the line marking the surface of rupture would be clearly defined by the points where the planes of plaster ceased to be horizontal. According to the above, M. Leygue made many experiments : the movable plane taking all positions possible and probable, and the surface of the sand making various angles with the horizontal, and in each and every case the line marking the surface of rupture was found to be a curve passing through the rear toe of the wall and convex to its rear face. And, further, he noticed that this line remained the same after the initial movement of the wall until the movement stopped, when the upper portion of the curve would begin to recede from the wall, and the sand would tend to take its natural slope. The result of these experiments would apparently over- throw Prof. Wey ranch's theory, or rather make it value- less, as it is founded upon the single assumption that the surface of rupture is a plane. But it appears to the writer, RELIABILITY OF WEYRATJCH'S THEORY. 43 as it must to any one who stops to consider the matter, that the experiments of M. Leygue were performed upon such a small scale that their results have little real value. The curve is but slightly convex, and, as the experimenter points out no means of ascertaining exactly its equation, but in- stead introduces a coefficient K, which he calls a coefficient of experience, it is certainly better to follow a single as- sumption that is, if not exactly true, sufficiently so for all practical purposes. Experience has shown that the angle of repose for all earths oscillates around the angle 33 within narrow limits, or has a slope of 3 horizontal to 2 vertical. Thus the slope of rock varies from \ to f , and that of dry sand is f. Navier observed the general facts that when earth is jarred or exposed to the air or changed in humidity or affected by frost, etc., it changes its qualities, the con- tiguous parts of the surface successively detach themselves, and the earth tends to take a slope that it would assume if cohesion did not exist, and that this slope approaches! but rarely surpasses it. Then it is sufficiently exact to say that earth newly moved and placed behind a rigid wall is in a state analogous to dry sand or the tan cp may be taken as constant and equal to 0.666 or cp= 33 40'. M. Leygue performed many experiments to determine how much the factor cohesion influenced the thrust of the earth, and came to the conclusion that, after the initial movement, the factor cohesion was practically zero. These experiments were, unfortunately, also on a small scale. But it has been generally conceded that cohesion is of but little moment in the deduction of earth thrust, as it rs.> 44 THEORY OF THE RETAINING -WALL. usually small and very variable, and if omitted the wall will have a little greater stability. Accordingly, Prof. Weyrauch has omitted cohesion in his theory. In conclusion, then, it is seen that Prof. Weyrauch's theory is the theory of to-day, being founded upon a single assump- tion, which is, for all practical purposes, sufficiently near the actual fact. In using the formulae, the only variable that must be as- sumed is , and that, from what precedes and according to Mr. Trautwine (mentioned farther on), is practically a con- stant quantity, and is equal to 33 40'. g> is rarely less than 33 40', and therefore it is always safe to use that value for it. Prof. Weyrauch's theory will be more fully appreciated and its accuracy, superiority, and simplicity acknowledged after examining the following articles, some of which will be mentioned in Part II. : AnnalesdesP outset Chaussees, mai, 1882 (Pa per No. 24): "Note sur la brochure de M. Benjamin Baker, sur la poussee laterale des remblais," par M. J. Curie. A brief review of older theories and a comparison of results as ob- tained by them and a theory previously advanced by M. Curie. Discussion of examples given by Mr. Baker. Annales des Fonts et Chaussees, novembre, 1885 (Paper No. 98) : " Nouvelle Recherche sur la poussee des terres et le profil de revetement le plus economique," par M. L. Leygue. A very complete article, giving M. Leygue's experiments to determine the form of the surface of rupture, the thrust of earth, the point of application, etc. Also, a comparison of different theories, old and new. RELIABILITY OF WEYRAUCH'S THEORY. 45 " The Actual Lateral Pressure of Earthwork," by Ben- jamin Baker, C.E. (Van Nostrand's Science Series, No. 56). A statement of Mr. Baker's experience on the under- ground railways of London, and discussions of numerous examples that appear to him to be antagonistic to theory. " Surcharged and Different Forms of Retaining- walls," by James S. Tate, C.E. (Van Nostrand's Science Series, No. 7). An analytical discussion considering only the overturning stability. Tables of thicknesses. " Practical Designing of Retaining-walls," by Arthur Jacob, A.B. (Van Nostrand's Science Series, No. 3). An article giving several methods of proportioning walls ac- cording to the older theories. Tables of thicknesses. Van Nost rand's Magazine, Feb., 1882. "Earth-pres- sure," by Prof. Wm. Cain, C.E. "Prof. Rankine's Civil Engineering." A complete analytical treatment of the entire subject of retaining- walls. 46 THEORY OF THE RETAINING -WALL. RECAPITULATION OF FORMULAE. Inclined earth-surface, plane : - i/ sln K cos (a + d) cos (a - __ } 1 sin ? sin ? 2 RECAPITULATION OF FORMULA. 47 Horizontal earth-surface: =45-|; .... ...... (26) sin cp sin 2a , . tan tf = - - r*- ; ...... (27) 1 sin cp cos 2a tan' 45 - f) If PL 0, then d = 0, and If a (45 ~ } = GO, then S = cp, and \ 4 1 tan S ,n (45 + f) (28) E = tan' (45 - |) , ft-fi ! (29) \ 2 1 2 cos a -- tf tanar 7*V , , " - ' ' If the surface is loaded, substitute IT + ^ /a foi'^ 2 ? or con- sider ^ to be the height of the earth increased by the height of an amount of earth weighing as much as the applied load. PART SECOND.-APPLICATIONS. EXAMPLES. BEFORE giving the solutions of the following examples, it may be well to say that earth exerts the greatest pressure as earth when it is perfectly dry, or has the minimum angle of repose. Hence cp must be taken at its minimum value in order to obtain satisfactory results from the for- mulae. Walls having vertical backs often sustain great pressures due to frost, and, therefore, such walls should have a factor of experience introduced, depending upon the location and structure of the mass to be retained. A wall is stable when the resultant of all the forces cuts the base, but still it may fail by sliding or bulging. Ex- perience and theory prove that if the resultant cuts the base within the middle third, the wall is perfectly stable and will not yield eithe*r by sliding or bulging, and also that the wall has a safety factor of at least 2. The above supposes the wall to be well built and to have a foundation sunk well below the surface of the ground. The foundation should be stepped down from the base of the wall so as to distribute the pressure over more sur- face. Weep-holes should always be left in the wall to permit the water to escape from behind. One weep-hole three or four inches wide and the depth of a course of masonry is APPLICATIONS. 49 generally sufficient to every three or four yards of masonry front of the wall. When the backing is clean sand, the weep-holes will allow all water to escape ; but when it is composed of clay, which retains a great amount of water, a vertical layer of stones or coarse gravel should be placed next to the wall to act as a drain. The back of the wall should have a batter of at least 1 inch in a foot, in order that the frost may partially spend its force in lifting the earth rather than against the wall. The masonry should be well bonded together, and no smooth beds allowed. The resultant of all forces upon the wall should cut the base within the middle third. Many of these statements will be verified in the follow- ing examples. NOMENCLATURE. Height of wall H Thickness at base b Thickness at top V Batter in inches per foot of Hon front face. . . d Weight per cubic foot W Total weight of wall G Angle of repose of earth

as a radius ; draw GH parallel to BA from IT through A draw HJ', then through a point one third the length of AB above A draw E parallel to GJ, and this will give the direction of the thrust, and the angle made with the normal is found to be d = 27 13'. Now make AK perpendicular to AB and equal to AH; find the area of the triangle ABK, and multiply the re- sult by y 170, and obtain the value ofE= 3037.5 pounds. APPLICATIONS. 51 The next step is to find the centre of gravity of the wall ; to do this, make Ex and By equal to six feet ; Dm and Ac equal to two feet ; connect m with y and x with c, and the point of intersection, g, is the centre of gravity of the cross- section. The value of G = 15 X ^4"^ X 170 = 10200 /& pounds. Constructing the parallelogram of forces, R is found to cut the base within the middle third or q = 2.2 feet; therefore the wall is theoretically safe, so far as over- turning is concerned. In order that no one can say the wall has been favored by taking a large coefficient of friction, it will be taken as 0.4 ; it is well known that for rough limestone it is nearer 0.75. Then the wall offers a resistance to sliding equal to 10200 X 0.4 = 4080. pounds ; as this is much greater than E, it must be still greater than the horizontal component of E, and hence there is no danger from sliding. The plane upon which the wall is supposed to slide is at the bottom of the wall and on top of the foundation. To ascertain the values of E and d analytically : Equation (26), GO = 45 - f = 26; O 7* tan a = ^- = .18333; .-. = 1023 f . Equation (28), tan ( + *)= ;= 7704 ; . . a + d = 37 36' and 3 = 27 13'. Equation (89a), E = = 3 037. 5 Ibs 52 THEORY OF THE RETAINING-WALL. These results are identical with those obtained by the graphical method. To find the value of q the graphical method used above is preferred, as being much simpler titan the analytical method. EXAMPLE 2. Determine the dimensions of a trapezoidal wall built of dry, rough granite, having a vertical back and being 20 feet high, to safely retain the sides of a sand cut, the surface of the sand being level with the top of the wall. W = 165 Ibs. cp = 33 40' y = 100 Ibs. The graphical solution is given in Fig. 5. Let AB represent the back face of the wall, 20 ft. in height ; make AD = AB = h ; draw DP, making the angle ADF= 28 10' = GO = 45 - ?; pass the arc DFK through A D and F, the arc having its centre in AD ; draw BK, then the area of ABK X y E = 5740 Ibs. jE'acts normally to the wall at one third the height. The dimensions of the wall must now be determined by the process commonly called "cut and try." In this case suppose I' = 2 ft. and I = 8 ft., and therefore n i Q G = - - X 20 X 165 = 16500 Ibs. Now find the centre & of gravity g as in example 1, and draw the parallelogram of forces. R is found to cut the base in the middle third, and q about 2.8 ft. The coefficient of friction of granite on granite is at least 0.5, and hence the wall resists sliding at the base by 16500 X .5 = 8250 pounds, which is much greater than the thrust E. The triangle ABK represents the intensity of the thrust; and to find the thrust that must be resisted by friction at any height, all one has to APPLICATIONS. 53 do is to find the area of the triangle above the plane, and multiply it by y. Thus, it is seen, it is a very simple matter to find whether the wall will slip on any plane above the base. The above wall, then, is perfectly safe. Analytical Solution. a =0, G? = 45 -^ = equation (26), which gives a = 28 10'. Equation (28), tan (a -f 6) = ; .-. 6" = 0, and normally to the back face of the wall. Equation (29d), fi= .2867 1Q = 5734. pounds, which is the same as obtained graphically within 6 pounds. The dimensions are obtained by the above graphical method. From Trautwine's "Engineer's Pocket-book," 1885, p. 690, the above wall would take the proportions H = 20 ft., I - 20 X .389 = 7.78 ft., and V = 20 X .096 = 1.92 ft., and hence G = 1600 Ibs. The wall to be built of cut stone. For dry rubble, which the example calls for, his propor- tions are : H = 20 ft., b = 20 x .528 = 10.56 ft., V = 20 X .236 = 4.72 ft, and hence G = 25212 Ibs., or 8712 Ibs., or over 50 cu. ft. more masonry per lineal foot than is nec- essary. Such walls are no doubt safe, but involve a needless waste of material. Bear in mind, only well-laid walls are considered, and foundations are supposed to be immovable. A wall, if built properly, will be so bonded as to leave but few voids, and hence the average weight of the mate- rial used in construction may be taken in all practical cases with safety. 54 THEORY OF THE RETAINING-WALL. We infer, then, that Trautwine's table used above, giving the dimensions of retaining- walls, specifies much larger quantities of material than are absolutely necessary, and leads to waste of material. EXAMPLE 3. The same as Example 2 with a = 8, or a batter of 1.68 inches per foot in height. By the graphical method of Fig. 4, assuming H = 20, b = 8, V 2, it is found t\rdtE= 6328 Ibs., and that R cuts the base, making = .831*8 x .90133 - = - 8517 ' .- . a + 6 = 40 25' and 6 = 32 25'. .90133 (20) 2 Equation (20), - ~- ^ = * = 23600 Ibs. The wall is proportioned by graphical method above. These five examples illustrate the method of using the graphical constructions and the formulae. The graphical method seems to be the easier, and is fully accurate enough for all practical purposes. It has been noticed, no doubt, that cp has been taken throughout, excepting in the first example, to equal 33 40', or equivalent to a slope of about l to 1. Mr. Trau twine says, on p. 690 of his " Engineer's Pocket-book/' 1885 : "For practical purposes, ive may say that dry sand, gravel, and earths slope at 30 41' or 1|- to 1, as abundant experience on railroad embankments proves." This statement is reason- able, and for the majority of earths the angle is too small ; hence walls proportioned for cp = 33 40' will be on the safe side. In all that precedes it is supposed that there is no fric- tion between the earth and the wall, or, in other words, tf does not depend upon the structure of the wall for its value in any respect. 58 THEORY OF THE RETAINING- WALL. Now, it is plain that as soon as any movement of the wall takes place the friction existing between the wall and the earth has been overcome ; or if a coating of earth sticks to the wall, as is usual, the friction overcome is that of earth on earth ; if cp' represents the coefficient of fric- tion of earth and walls, then the direction of E must make an angle with the normal to the back face of the wall equal to at least cp'. To introduce cp' into Professor Wey- rauch's theory it is only necessary to find the value of d as given by his formulae, and see if it is greater or less than cp'; if it is less, use the value of

Average Ibs. per cu. ft. Alabaster 4-1 144 Brick best pressed 40 to 300 R 150 " conirnon linrd . . 125 " soft inferior 100 Chalk 20 to 30 S 156 49 6 to 102 Flint 162 g 166 Granite 300 to 1200 &3 & C 170 Gneiss ^ 168 Greenstone trnp fe 187 Hornblende black 203 Limestones and Marbles, ordinary .... Mortar hardened . . 250 to 1000 8| a m O> OJ j 104.4 [168 103 Quartz common !g 165 Sandstone 150 to 550 II 151 Shales . 3 QQ 162 Slate 400 to 800 175 Soapstone H- 170 VALUES OF y AND * 72 80 4 " shaken ^ o 82 92 tt " " rammed moderately .** 90 100 Gravel l| 90 106 Sand 5 co O. 90 - 106 UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine schedule: 25 cents on first day overdue 50 cents on fourth day overdue One dollar on seventh day overdue. DtC ENGINEERING 1947 27 1943 IBRARY LD 21-100m-12,'46(A2012sl6)4120 2406! JSnginec Library THE UNIVERSITY OF CALIFORNIA LIBRARY