IMAGE EVALUATION TEST TARGET (MT-S) k '.* ^ // V5^ A ^ b 1.0 1.1 |2.5 Ll:0 IL2| i u 2.2 1^ 1.6 Hiotogrephic ^Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 87I2-4503 CIHM/ICMH Microfiche Series. CIHM/ICIVIH Collection de microfiches. Canadian Innituta for Historical Microraproductlons / Instltut Canadian da microraproductlont hiatoriquat ;V Tachnical and Bibliographic Notaa/Notaa tachniquas at bibliograpliiquaa Tha Instituta has anamptad to obtain tha bast original copy availabia for filming. Faaturas of this copy which may ba bibliographically uniqua, which may altar any of tha imagas in tha raproduction, or which may significantly changa tha usual mathod of filming, ara chackad balow. □ Colourad covars/ Couvartura da coulaur r~~| Covars damagad/ D D D D Couvartura andommagAa Covars rastorad and/or laminctad/ Couvartura rastaurte at/ou palliculte Covar titia miasing/ La titra da couvartura manqua I — I Colourad mapa/ Cartas gtegraphiquas an coulaur Colourad ink (i.a. othar than blua or black)/ Encra da coulaur (i.a. autra qua blaua ou noira) I — I Colourad platas and/or illua»ratlons/ D Planchas at/ou illustrations an coulaur Bound with othar matarial/ Rail* avac d'autras documantt Tight binding may causa shadows or distortion along intarior margin/ La ro liura sarria paut cauaar da I'ombra ou da la dlstorsion la long da la marga intirioura Blank laavas addad during rastoration may appaar within tha taxt. Whanavar possibia. thasa hava baan omittad from filming/ II sa paut qua car;ainas pagaa bianchaa ajoutAas lors d'una raatauration apparaiasant dans la taxta. mac's, lorsqua cala itait possibia. caa pagaa n'ont pas iti filmias. Additional commants:/ Commantairas supplimantairas: L'Institut a microfilm* la maillaur axamplaira qu'il lui a iti possibia da sa procurar. Las details da cat eaamplaira qui sont paut-itra uniquas du point da vua bibliographiqua. qui psuvant modifiar una imaga raproduita, ou qui pauvant axigar una modification dans la mAthoda normala da filmaga sont indiquAs ci-dassous. □ Colourad pagaa/ Pagaa D Pagaa da coulaur Pa^aa damagad/ Pagaa andommagtes Pagas rastorad and/oi Pagaa rastaurtes at/ou pallicui^as Pagaa discolourad. stainad or foxai Pagas dicoiortes. tachatias ou piquies Pagas datachad/ Pagas ditachias Showthrough/ Transparanca Quality of p rin Qualiti inigala da I'imprassion Includas supplamantary matarii Comprand du material supplimantaira Only adition availabia/ Saula Mition disponibia |~~| Pa^aa damagad/ |~~1 Pagas rastorad and/or laminatad/ FT] Pagaa discolourad. stainad or foxad/ r~| Pagaa datachad/ FT] Showthrough/ n~| Quality of p rint varias/ r~*1 Includas supplamantary matarial/ r~| Only adition availabia/ Pages wholly or partially obscurad by errata slips, tissues, etc., hava been refilmed to ensure the best possible image/ Lee pagas totalement ou partiallement obscurcies par un feuillet d'arrata. una pelure, etc.. ont M filmtes A nouveau da fnqon i obtenir la mailleure image possible. This item is filmed at the reduction ratio checked below/ Ce document est film* au taux da reduction indiqu* ci-daaaous. 10X 14X 18X 22X 26X 30X J 12X 16X 20X 24X 28X 32X The copy filmed here has been reproduced thank* to the gerteroaity of: Library of the Public Archives of Canada The images appearing here are the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustrated impression. The last recorded frame on each microfiche shall contain the symbol —^^ (meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper ieft hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: L'exempiaire film* fut reproduit grAce A la gdn«rosit« de: La bibliothdque des Archives publiques du Canada Les images suivantcs ont «t« reproduites avec le plus grand so!n, compte tenu de la condition et de In nettet« de l'exempiaire film«, et en coR/ormit« avec les conditions du contrat d© filmage. Les exemplaires originaux dont la couverture en papier est imprim«e sent film«s en commen^ant par le premier plat et en terminant soit par la derni«re page qui comporte une empreinte d impression ou d'illustration, soit par le second plat, salon le cas. Tous les autres exemplaires originaux sent film«s en commen^ant par la premiere page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernlAre page qui comporte une telle empreinte. Un des symboies suivants apparaftra sur la dernidre image de cheque microfiche, selon le cas: le symbols — ^ signifie "A SUIVRE" le symbols ▼ signifie "FIN". Les cartas, planches, tableaux, etc., peuvent Atre fiimto a des taux de reduction diff Arents. Lorsque le document est trop grand pour dtre reproduit en un seul ciichA, ii est film« A partir de I'angle sup«rieur gauche, de gauche A droite, et de haut en bas, en prenant le nombre d'images nAcessaire. Les diagrammes suivants illustrent la mAthode. 1 6 ( All rit/htu reseroeil ) ADVANCE PROOF-(>"''K' / /" rerhion). Tliis iifdiif is sent (o viiii fur Jiscussiim imly, ami on tliu (-{priss iiii>lei>tiiiiilin,i; that it in not In lio iiscil tor ai>y ntliri' piirpiw^ wimt- I'ViT, — {■"•'(€ Sec. -17 of Ihr. ('o]iitilution). MaiKiiiini; Socioty of Hii'il (fngiiiccps. rNi;01lP0llATKI) IHHT. TRANSACTIONS. N.B — Tills Society, us ii boily, ilocs not liulit itself icsponsilile for the lac Is am' opinions stntoi in iiny of its piiblicntions. A NEW METHOD FOR THK DKSIGN OP RRTAIXIXfi WALLS. Ry W. Bki.i, Pawson, Ma.K., Assoc. .U. Ln.si. C. K To lip road Tlinrsilny, 5th ncpombcr, If !>.'). The mothud proposed is to take as a basis a wall which supports an uitliniitpil slope of loose material at its own angle of rest, and to bring all other eases into relation to this. The nrlvantagc of approaching the probl'in from this (ruection is, that llic t'ornniln> become so simple that the comparison lictwrcn clifforent tonus of walls can be made with facility ; ami from these a wall of any rc(|uired stability to meet exi.st- ing cnnditiiins can be conveniently and correctly dcdueed. Tel show at the outset that the basis proposed is not .so fir from natural conditions as is usually supposed, the circumstances which first directed the writer's attention to this method maybe briefly mentioned. .Along the Krasorand Tiiompsnn Rivers in British Columbia, on the line of the (Jan;idi;in Paeittc Uailway. there are stretches of some miles at different places where very long slopes of sand and gravel occur. Tliese slopes extind IVom near the edge I'f the river to a height of from 300 to .500 feet or more, where they reach the top of a tt^rraco. They arc usually more or less envered with grass and sage-brush, although some broken gaps also ncciir which are termed gravel slides. The general location of tip' railway along the face of these slopes isatabnut half the height between the liver and the top of the terrace above. Althou^^h this class nf country has in gcncrnl an irregular appearance, yet c careful cxaminaiion of these long slopes shows that they arc exceed- ingly uniform in their inclination where the m.itcrial is snnd and gravel ; and this inclination does not vary appreciably from 33° 41', or 1 J to 1, the well known angle ofrcpose for material of this kind. Thero also occur long natural slopes of rock debris in the more rooky part', which show with grei't uniforuiily an inilinatioii ,f 37° to 40°, or )^ to I. These are thus examples of natural " surfaces of e(|uilibrium '• such as are seldom met with on so large a scale. These .slopes, while maintaining thcirgctierpl inclination as the side of the main valley, are more or less fluted or corrugated on an inimonso scale, by the valleys of the side streams. In these eircunistanccs the railway cannot follow a natural eont«)nr without bringing iu an exces- sive amount of cnrvatur.;; but in making a series of cuttings and embankments on this side hill ground, a pr.ictical ditlieulty at one- arises, as the slope of both cutting and embankment would be parallel to the natural surface lor a vertical height of 200 or 300 feet both above and Uilow. As the writer had occasion a few years ago to make estimates for the completion on T> permanent basis of the railway Torks on 125 miles along these rivers, it was at once evident that the ooiistruction of retaining walls to support the slopes of Iwlh cuttings and embank- ments would be the most economical method to purau..'. In desigaing such walls for stability, the pressure which they have to withstand is practically the same as for an unlimited slope at the angle of repose, which shows that this condition is no more theoretical consideration. Till- ootnpli'xiiy tlint atlnclios I" llii; problem oC flu- sinliility ol' nliiiniii'^ walls uiiscs from tlic tnct lliat aiilliors iiavo iisualK' i-iim''i (liTPil till' C;is<' 111' sniiic ni'liitrary liinitii)!; Miifaec for tlu' carlli tn ln' Hupporlfd, iiistcail cil' tin' iiaiiiral Miilucc^or i(|uililii'i«in. Tin' sim- plification of till' liiriimlii' wliioli rcsullH in this oaHO is sniin'liin'H tnnclicil iipnii a< u tl'i'oiitirnl cori'llarv of no pfaetical value. Hut, lint eonsideiation of llie Mirlacc of piiuiiilniuni, iiiclini'il at its own ani;!'' of friction to the Imrizoii, liriii),'s ilu' ipu.-tioii into relalion t" tlio liyilro. static problem of walla for tlie i-nppoit of waler, Tlie simplicity in llioir case is due to tlie fact that lof them a .-uifaee of eipiilibrinni is always consiileroil. The use of walls to siMi|iort a liori/.onlal snifaci' of earth on a level with the crest of the wall itself (known on railways as grade walls) is 80 frci|UCnt in practice as tn require speei:il consideration, bnd t3 deal with this case satisfactorily certain arbitrary siniplitications have to be introduced. But lor walls with any considerable surcharfje, which come ne.irer to the natural ronditions as already explained, the best method will be to dcsii;n tirst « wall of ei|uilibriuni which is at the limit of stability when supporlin;^ an unlimited surface at ihe angle of repose j then to allow the difference between this and the actual sur- charge as a margin of safety ; and if necessary in addition, to increase the wall ilsi'lf for greater stability. The wall of eiiuilibrium can be calculated without any a.ssuniplion I'rsimplilieation of the actual enndi- tions; and the margin or iiiereasc for stability which there is to count upon becomcr quite distinct. It is jiiLssible al-o in dealing with a surface of cquilibriuin, to calculate ihc stability of dry masonry walls, or even a heavy layer of rip-rap for the support of earthwork. The term " loose nnilerial " is preferable to " earth," as it avoids conveying tip impression that tliere may bj cohesion in the material ; as any cohesion at oiici vitiates the theory. Cohesion, too, may not always be on the side of safety, especially in Ihe case of water-soaked material, which by itself may possess a certain amount of cohesion, and may yet exert an imieased pressure ou the w.ill. The t 'rm " loose maierlal " is al-o the most general, and may be taken to include sand and gr.ivcl, loose rock, or even corn stored in tin elevator. It is also essential to liistiiiguish clearly between the modes iti which a ret.iining wall may fail, in order to know definitely the cnnditions under which the various formula' are ajiplicable. The.se are ; — (A) Failure by overturn around the outer edge of the base. This is the mode nf biilure which is generally Itiken for granted ; but authors seblom point out what i.s really iniplioj by this condition. It uieaus physically that the surface mi which the wall stands is incom- jiiessible, inelastic, and in fact ab.-idutely resisting. This may be closely true for a foundation on rock or otbir very hard matirial. But if the surface has any cla-tieity, it is ea-^y to show from the theorem of the " central third," that the resultant of the riaetion against the base will have its point of application at two-tliiids of the width from the back of the wall, instead id' at the outer edge. With ordinary forms of walls, this will dcerease the .-•ti-bility to abnut one-hall'; mid it is a question how far this si ould be taken into aci-ouut for walls standing on pile foundations or on timber platl'oriiis, wlure the elastieity may be quite appreciable. (B) Failure by sliding 'W the b.ise. ur al the level ul any course in the masonry. This i.-^ olteii rofernd to ; but is of less pr.ietieal import taiice, because the weight nf ordinary walls give> them a greater resist- ance by friction than by nvertnrn ; and if llie friction is found deficient, it is more eeoimuiieal to adopt some expedient for inereasing it, such as inclining the I'ouiidation courses against the direction of the pressure, ratbir than to increase the diinensiniis of the wall itself. (C) Failure by sliding al ;iu angle Uirough the mass. This can only occur when the wall itself is of loo.se material ; as, for exaiiiplo. loose rock supiiorling earlliwoik by its superior weight and greater angle of friction ; a ejse which we will eunsidor further on. In this paper, failure by oveiluin in the ordinary way will be con- sidered, unless otberrti.se stated. Wal/^il/./iurlhi'l III! iilih'ni'/nl ,:r/riit i,/ >.,„■<,■ m il.rlul ,il ///.' iiwilr of reiioin:. Lot A L reprosnilt (In; back of tho wull, an.l L M' tlin MirPiicB nf tlio ixmi iiiitc'rirtl I'xtcii'lini: lo an iiiiliiniti'il ilist.muu at it'* own iini;l<- HC rc'pnso 0. It isc^vi'lont (Imttlio prt^iir ■ ajiain-ii tin' Wiill is .liie entirely tn till' inyi'i' liotwv 1 tli.' liiiis LM' :,n.i AM .is \\v i-.,ii;iiniii,' iiiatu- rial hclnwtlio lino AM is alrca.ly in M|iii|ilj|'iniii wit!;.,; t tlu( itid of tlin wall. Ill (iiiiliii!: a tiiiiniiln I'lir (lie iin.ssuiv against llio wail, wc obtain the siiii|il('«t fifprcssinn by iiikinir tlio valuo of tlie liorizontal coinponoiit of tlic t( tal prcssuri!, Ustially .•ailed tbe horizontal tlinist ; ami tliis thrust isalsntln' iiiostrdiivi'incnt in tiiilinj; t ninomfiit of o>erturn. In onb'r t'l iivoiil iniitlirinaii<-al (lrt:iil. llio ]iriiof of tliu tbrinula is jrivCD in llu,' Appc'iiilix. A L'laplilual im-ilin,! nf irdiieial application is tlicru i^ivcn, linm which the prool is dcilucctl ; anil the siniplificitioiis which obtain ill till' cas' considered a'.-o thu< easily traced. The result ia also shown to coni's|iiind with ibrniulic wlii^h llankine j;ives for special cases. The iisiiitinL' liirmuli is us tbllow.s : — I-i't 7'= the hori/.ontul thrust, "• := the weight of the loose inatBriul supported, per unit of voUiiiie. p = the perpendicular di.stanci, A I"" from the foot of the '•vail tn the surface of the looso niati'rial ; '.'•., Ilir thiekiiess of the layer supported. Then T zz= }, ir /I ' It will be noti'd in the ]iroof that the iliniinatiou of other variables enables the folluwini,' statement to be ii.aile ; — When a wall supports an uiiliniitod >urf:ir ■ of lo.se luaterial at an aiifile 1)1' iep,se, the horizontal eoiiipouent of the pressure on the wall i- iinli peiiilent of ibe existence or iioii existence of friction between the loii.si' material an I tin- wall. It- nuoniil 'leponl* only upon the weight of the loi'se niateriiil per unit ol volume, .ind upon the |ierpendicular thii'kiiess of ilie lnyer supiiorted It is :i!-o iiidepuudeut of the inclina- tion of the vva'l piovideil that the tliiekiies-' of the biyer supported is uiieliaiii^eil . Its peint of applie.it ion is at me third of the vertical lieif;lil of the wall. Thus in the f ■llowini; tiL:nre<, in whieh the thiekne>s p of the layer supporled i- ihc s:iiiie, the .ib.-oliit ■ vabii; of the thrust /'will be the sa'ue liir the v.irious ineliii.ilion.s iif the back of the wall ; but the p liut ol' applieatinn will be at a vari.ible height above the /oi/'/xk/o/ plane through .\, and the inoiiiont of overturn will increase aeeordiiiiily. The identity of tlio formula hero found, with the ordinary forinuln for the hydroHtatio pressure of a liquid, is apparent. We find thnr.-- foro in this ease, thut the horizontal thrust due to the pressure of any .liiB^ of loose material is tho same as that of a liquid which him the same weight per cubic fool, provided that the ihic'cooss of the layer supported is the same ns the vertical depth of the li(iuid. The ouly physical difference is that tho loose material must havo a surface of unlimited extent, while the pressure of the liquid is the same with a limited surfaoo area. Wtilli of A'qi'llibrlum. We may now proceed to invostisate the most economieal forms of " walls of equilibrium," or walls which are on tho point of ovortuining when supporting loose material of unlimited extent at its an.'lo of repose. It is evident that a trianjjular form is tho most economical (As ALC in Figs. 21 and 22.) In tho case of a reotanpuh.r wall for an mfinite surchirge, in the series rccomiuended by Poncelet, the base is 0.934 of the height, while tlio stability is only J.86, which shows at once how far such a section is from being economical. If in Figs 21 and 22 we adopt any given face bnttor (angle S), and proceed to calculate the rec|uired back batter (angle <>), tho general formula is a quadratic in terms of Ian 0, which wo give with proof in the Appendix (IV). The reason of tho degree of complexity which this formula presents is that the thickness ,. of the layer supported by the wall is itself a variable as varies. The forms in Figs. 5 and (5 arc calculated by this general formula, and they represent two typical oases of sand and grovel suppovled by masonry walls ; — r j Firstly, on unlimited slope EF at IJ to 1 at a vortical height AB, above the point A, This is a very usual case in practice, as the point A may be the outer edge of foundation on side-hill ground, or the inner edge of a railway formation or road-bed, and the slope EK has to be supported at some point between B, and B„ . For comparison the height AB, is taken as 10 units, and the areas of the eross-scc- tions ore as ibllows : — i' Height AB, =10.00 Area ABiC, = 56.60 AB,C,' = 66.38 AB,C:<= 63.10 ABA =61, 64 \Bfi-. = 47.02 Secondly, the wall to be built on a natural surface GH at li to 1. and to support an unlimited slope EF at a height AB above A." Here EF may be taken to represent the slope of an embankment on side-hill ground. Height AB .- 10.00 Area ABC AiBC, -- se.no 47«>2 rig • In both these cases tho wall with a vertical back is the most econo- mical, and it also requires less excavation for its ibundation. The following table of triangular walls of equilibrium shows the values required for the face and back batters, in some of the more 4 ^ uHunl oa»o8. Tho rip-rnp is onnsiclercl lo ho in .<,,uili(,ii„ni around f.li(, ..utpr nlifo of the hii^o ,.s in tlu) -.Hf of inMmnr.v, uiwl llio n.-siitivoh^l^n iiiilicfttCH that tlic liiiok liattcv is on tlic siuiim sido of tliu voiticnl a<< tlm fuco but IT. Iso^ct'icH w.iils arc iniliiatnl l.y ||„. ,,|nnl)tv .if ili« two linttt r>. r.,n,/li;<,ns. - Aniil,. ,/, . 33 - H', o, 1 1 i„ l. \V,.i,^|,t ,„ of ,.arih = 100 lbs. per cnbli; fool, Wright ir of ni iscnrv, oto,, 1(17 or 12'> ll's., us sl.trii. CalrnlulLiI \,y iUr n.iirnii fonimiu in App.nilix (IV). Xiiiiirrmii \;iiii,;s /,„■ w'.i/i, ,,/ t:,j,(,i;i,iiitm, ' I nn g 'Pan f) ' Mdti rill (1/ Willi. MiiHiiiirij, Dry MllHIIIII-jf, JUji A''i/,. ir Katv baitiM Witiiiul ,'. = 0.083:{ ]■ =■ O.-JnOii 0.:ill)l Italic liultui', I.13il0 0,!I11'3 0.4SH5 (1.31111 Uitio of bast! of wall tn hciulit. 11.4558 \i.|tlcal ..| I o.;t8l(( i r= 0.5000 0.5:i(i3 I = l.li'lOO O.S3!l(i 0.3810 n.()iii)5 \'iTtii;»l - 0,!)a40 I . 1300 0.!(05(! 11.7385 0.0.382 0.4558 0.7620 0.5005 0.52u3 0.0700 0.1700 Tlio abovo lal.iu may l.r .'.Mrnduil by means of (lie i;.'neial loiniiila to Miciud,' any claw olloo,sc maloiial, .supiioit.'d by walls of any given weigbt pur cubic foot, Tbc foriniiln! lor walls with oitber the facu .ir tbc back vertical, d.du.vil from the .-.ncral firnmhi, arc also niv.Mi in the Appcndi.x. If in-ctcrrcd, ilic f.rn.s o|' sucb walN of (.(uilibrinm M:ay be found fiom the orijtinal fornmla '/'.-. S w /,- and the moment of stability, by means of direct trial and suouc.s,sivo approximations. We may also obtain a comparison of the jiresaures exerted by differ- ent cla.-ses of loose material, by means of a ratio botwoen thum. Let the two classes be defined as follows : — 7i= horizontal thrust i wi = weight; t^, = anj-l,; of repose 'Jl'^- " " w-,^ <■ <^. , •. i«i: cos- (f)j "'1 cos'^ if Xunierical example,-, <^i =.33 U' for saoi loose niek. 'I'lieii : — ri id 011 = 15^ lor 13 IS .\lso, it the .-tone weighs 107 lbs. p"r cubic foot, the broken stone (n' loose rock formed from it will have 40 percent, of void, and will weii,'h 100 lbs. l)er cubic foot, whteh i- the sam.; as for sand. Hence w., = ir, , and we have 71. - 7-.'2 7', '! he.se ratios, whi'thei alijebr.iied or nnmeiiral, will be closely ap- proximate for any consider.ilile lieiirht of snrebarije. (li'ulf W'lillx. or walls ^np|orlin^ Irose material limited by a boriz ontal surface at the level ot the cre-t of the wall, III re "e h;ive the loose material limited by an arbitrary surface; and the element of frietiim between it and tile wall i.s no liinn;cr elimin ated as in the ensi' of .i surface at rep 'se. Hence tlieie are two condi tions or a-.snin|ition- to be clearly disliiuui-lied : — (1) The existeiiec of frict'on between t!ic loose material and tho back of the wall, eijiiiv dent in amount lo the an^le of friction \^. In this case the bust solution is the ^Tapliical m\\ as shown in b'igure !."> in 5 llio Appendix, tlio Hurfnco lino being nmdo horiionlal. Tlio algobraio*! luetlidd iH oi.iuplicnt (1 ; nltlioujfli Nonio NJuiplifipntidn in ohuincil by nH«uiiiinj.' f "ip, tlio iinj;!.' oClricticin for tlic ninterial iticlf, wliivh i.s iisuiiliy II cloHo Hpproxiiiintion. (2) No I'rictiiin bptwecn die Ino^! ninleriiii and tlir wiill, Tliis i* olcnrly iin aHuuniptinii ; but it is '.'ciierdly employed in pniotiuo, k-eiiuw itjiivcH Ik ^reni.r value Icr the tlirunt ihon whdi tlio Irictiim io tnkeii inlii aoiniint, nnd llierefore leaves ii gieiitor niiiris'in on the side of stiiMlity. Tliisadditinnid sliibiliiy may hIs,. be taken to mil ke up for tlip effect of vihialion oceasiiini'd by eillior a train orwaKgcn wlioii- tlie wall supimiis a riiad-lied. Tlie grapliical and alt;ebraieMl m('tluiil, lor Ibix case are given in the Appendix (III), and nNo ilie prodl'nf the followinL' formula for tlie horiidnliil thrust T: - 7'=. 1 -Bin ((f) -e ) 1 +sin ((/) I e) (The sign of e will bo rcvcrwd if the inclination e is on the other side vortioal. Sec Fig. 20.) We can obtain an interesting comparison between the amount of pre.s.snie cccasionrd by the >anie kind of material in this otiseand in the original ca.so of an unlimited Mope at lepow, it we simplify the fl.rm- ulre to represent a wall with the back vertical. Let Ti = horizontal thrust for an unlimited slope at the angle of lejioge , as already found. y'l! = horizontal thrust for a horizontal surfaee. h = vertical height of the wall, 7'l, = J ir p- - J w h" cos" 10 li- T, 1 - .sin (^ _ 10 h- 1 i sin ~ •! tan- J (90 - (/,) Ti_ _ tan- J (90° - ,^) ^i cos-

»uni,d ho fur lins bocn thiit ilie wu'l will turn nround tlio outer cdso . f il,.. I,„h,., ,i,„| f„i| by „,,H.tli»},'. TIhh implies that the foun.latior, in iinyidlinj;, and Mso that tho friction on the baw) msiifficifiit to (.nvcnt thf wnll from pnsliinK lorwaid horizoutiilly. These M.od.'s of failure Imw bcm cxplnini'd ut tliooutwt iind indicated by tho letters (A) and ( H). We will now refer to the third mode of failure, in which the Hiipportiii!; nl«s^ may -ive wiy hy slidinK "t an angle on itself Tliis niny occur wIkii rip rap or loose rock \-> used for thosui.portofenrthwoik. In the region referred to, there were two prohlciim of this kind. (1) A >ide-hill cndmnkMient made of loose rock on the onlHide. (li) A euftinL' with the slope supported hy riprap. "9 These conditions are illustrated hy the figures, and the question is the thickness of rock embankment or riprap required. As the sup- porting materiid is itself loose, the support it gives will be due to its weight and friction only : and it will fail by sl.earinu' or slidini: along some line of least resistance. Tho position of this line in the mass involves the determination of a minimum under conditions which may be tlius stated : — I Let CBK he the mass which supports loose materiul with a surface BM' at the a.igle of repose. The line of least rcsistince OA, or line of shear along which tl, • mass will give way by :,lidi;,g. will be inclined at some angle \ to the horizon. Required 'lie value of \ which will give a minininm value to this rcsi.stanee ; imd the a.igle S which will give .sufficient thiekuiss to insure .stability. It is to be noted that the ihiekness /, of the layer suiiporteil varies iil.so wiili the an-le X. The simple expressi.)n alre;idy found for the horizonlnl thrust T of this layer enables a relation botiveen the angles g, X, and tho angles of friction (^ and i/r to lie luund, as given in the Appendix (V), For the most important praetieal iMse of loose rook supporting sand and gravel, the weight of the two kinds of material per cubic "foot is practiciilly the same ; and the stability depends entirely on the in- creased angle of friction in the loose rock, tho angles boinu' ^ = 33 = 41' 7 nnil t/f r 4[i : ii'>|nTtivi.ly. U vn' lunkv aim) ^=45« tlvrv willl. o(|iiilibiiiiin in III.' l.wrr [.urt nf tli.- iimi"H i'lW. up to tluMiKliintioii K'-Kf) , lull bImiv.i iIiIn (lie l |iiisli il i.ff liy ihr |lll^MHV Till re will I)' 11 viillli'Hr\ vv|ii,.|i yivi s ii lliiliiihiiin, ImwcviT, if 8 iHnrciiUT iliiiii 45® , i,iiili'(|iiilil)iiuiii will ilini ol,iiiin. Ill |iiiii't'(,'r, llii'ii|.|iii |iiirt iifdi. iriiiiiL'h' nil III' I'lipiliiT -Ir. ii'jtli"i('(l by iiililini,' till' iirrii Hlth'K, Fnr till' Nuppni'i ,,l ,1 li(iii/.,iiit.il MiiI'mT, lis iiiilir niii^ nl i-inliiiik riioiit Fiu;. 11, llii^ Mi'iitldii is Miffiiiii'iitiy I'l.wi', IIS llu' I'c'liilivi'ly liiwcost iilliMw inck, >vi'n wlidiil Iiiih to be nl.iftiiicil by I'xi'iiviulon, iiiukcs it iill'.wiibloto use iiti iiiiiMUiit cniisiilcriilily in cxncK.s nl' llii'oi'ct icol rtHiiiirt'iii('iif.«. Il ul-n umkis mm cnil'iinkiiKnt iniiciiciililL' on ^ti'op side bill nKPUinl, iinil w licri' ii iim-onivwill wcnlil bi' iniiili more cxpciiHivi'. Ill tlnM'ii-i) ofii I'nttin;;. Iml'. 1L'. tlo' I'l'Vei'M' if usually tlii' ciso, biifoimu llio voliiniu nC iiiu..oniy rujuiri'd is ho niiicli Irm tbim tlio iimount of loosu mok liiid on hh rip-iMp, tliat lU oiiliniiry pricrn tlic lliusnnry Is tlio niol'i' iiconomii'ul of tj,,. uv„. 'J'liis is proli;ibly tlh' reason tliiil tliiM pfobloiii bus not I'pceivi"! inoi'u 'itt ulloii ; iiltlioiii^li tlu'ii' are iimny cisoh in wliirli it is nl I'l'.icticiil viiliii'. AI'l'KNIUX. F>g IS frnipbiciiluictlioil (if iK'trrniiniiii,' tin' (liriist of liioci' iiiiiti'i iiil liniitcd by any iilmu' LM, iii-iiiii I tbt' b ,uk mI'u wiill A\,. Let (^ = u'lu'lo of rcjioM' : «'.., til,' aoL'li' of IVi,tion of ilio loosi' uiii- ttrial (in iisi'll, i|r =• anisic of Irii'ticn of tlic ](uw niahrinl a'jninst tl.c wall. € = ineliiiatifiii of bick of wall to tlo' vci'tioal. I" = weiglit ot loiiM' mat' i-iiil \w unit of voliiinc. Q = pressuri! iiorinal lo ilu' wnl. H = resultant prcssiiR' .-il ;iii an'^Io -^ to tbo wall. T = boriziiiital t'oiiiponuiil of 7.'. Tbo iTlaticiii i4' siirfiu'c and the wall, and to all dassi.'.s of loosu niati'rial. Draw AM at an nwAo tfy witb iIh' liorizontal. J)iaw A() at aiiaiiulr (^ \^ uilli AJj. I 'raw LK i.aialb'l to A.\l : and take OY a iiiuun picp-.-rtirinal iic- tweiu OK and OX : ai'd draw VX paidlcl to AM. Then XA will be tbe lino limitinj; tlic prism of inaxin iiiu pn-surc. Also tlio nciieiai i(|iiali'in giviii.' t\v value of tlii! pri'ssuiv is : — Q II- . , - , --= , ■ >ni (H+ /:; I 1/r) AY (;os \lr •! \ ' r ' 8 w •> PON ( ^ I «) 'AV' HU.I iiIno Mnco //cos ,^ = r^ „„,l ;- = /; ^,„, (^ , ^^ W(> Imvc '/' = ^ . CON* (^ ,) Xy' rir» Ih on the othcT .i.lonftho vortical, i.n «i„u will be uc^.tiw.) *or pro<.f of ,hU uMihod, ao.l tho nxultioK Kenoml equ.tion, loo tolliKnon, " .l/;,„»„y„, Appll,,i,^r," Book VIII, oli»p. IV. 1" ihow I.0W ««n..r.l thin i.u'llio.1 m, ihe following .d.litioH.I er "UiploM i,.»y t,c „ivt,,. I„ the Heoond ob.o the poi.it J i. found by umkmn U p,.ralM lo AH, in order that the triangle. ALH and AJJI may 1h! oi|Uii1 in arcB, for oonipcnxation. ^I — II. Fornu uiUimiUil nnr/ncp nt thi,„igU „/ repn»f. In thid case the following Bimplifioatious take place. As L.>I' is at the angle;of repose, .and therefore parallel to AM, the points K and Y co.oc.do with ; and the length AY becomes AO. Also, since AX coincides wHI. AM, the angle a becomes the angle 4> ; and the equal tion becomes ; — o6s4- = I • «n ( /3 ,. .^ ., ^) -ot cos ■^ 2 ' und '/• = ^ . cos' {y\r V () 7)A Jict p = the perpendicular distance AI'; then as p = sin (/9 4 <^ 1 1^ ) OA = cos (^ I f) OA, w« have finally, 7" = J »• p* NurU. Ill U: „;;i,u.'s " Civil lv,i;ill,.,Ti,lu',' ,, roniiiilac..nvs|»„uliii.r t.. tliLs IS ;.iv(Mi C.r-llio siK'ciftl caw uf a w:ill will, „ vortical back. In Articl.' is;!, I\-., tlic prcssm-o parallel u> tlic siirliici' is ilciiottMl liv /" and for a siirraco sio|,iiif; at the aiiLle of ropi.sc lie ijives (in our noti tioi.) /"= i w eos ^ ab' As /" .„. . y; ,„„i aB ,.os = ^,_ the fbrinuliB are eviilciitly iileiitieal. III. Shi-/iHi hii,-i;.ni,hll ; •lx/,,1- Grilili W.ilh. C"i,J!t;,„i ; i,n /ncHi.li l„'lin,n /..,«, y„„/, ,,.// „„,/ „■„//. With tliLs comlitioii, I'loii^' i,i,., „„owi. for a vertical wall, and Francals lor au ii.ollned wall, tliai .he lino A.\, linii.ini.' ilio prism of ma'xiumn, pressure, will bisect tlie aii^le LA.M. As the HOliCMl luclbod is also ai.plicabl,' to a lic,riz.int.il surfac,', we obtain the rornmlic at (inco by intro.liK'ii.i; the new ee^ndltions. r.iinn- tlie same nolati.,11 as beti.iv, an.l uiakin- ibc vertical height AB .— h, we have ill this ease f = o\ and (h,. iioruial pressure V theivfoiv eo' iucides with the resultant /.'. Hence /' - (^> — "' ,,„,; e .A Y^-' an : sin 4> vm ftf 0'-(l-^,sin^,^) .,n^^.. (:i„- -, 2 sin cos ^) tan ^ ('I'l.e negative si.n ,nn-( be used wlicn is on the left of the vertical M' 'I- lifTure ; and in this case also the negative value of the s,,n.ro '■'-of mn.t be taken ving- the equation fnr .,m 6.) When the back butter is vertical. = o and also ir or tan- II cos-