^ ^^J^ ^ ^.^^> IMAGE EVALUATION TEST TARGET (MT-3) ^ 1.0 !l lii|28 £ 1^ |2.0 112^ Wl 1.4 m l^gy^ Hiotographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. M580 (716)87^-4503 .> iiiiiliU> r<>r Hit* ftti-tit ainl oitliiiKiit alultil ill liny of ilH )>iil>lifiili<)iin, Tfi f'f rntif iti Mfiy. IIASONUV .MU'FIKS I'oll ItAILWAY I'llM'OSKS. By \V. Uki.i, IU\vh..\, MA, Ma K., M. (V\N Su,. (' K. In (li'uling willi llu' iipIi. iIh' writir IWls tliut ll will In' iiiw-isiiry I" proci'L'd c':irrl'\ill_v ; :!■' ii];iii_\ hjiniinci-s liiin' lIu' I i|nrs>ioii lliul I'nr ;iii.v puMsililu view (if llii' siilijcil sdiiii' iiul|]iiri(y um\ \i-iit |tii]H'i' liiH iin dr^iiv to propiiuiiil any iiiw iIhhij uI' tlu' aivli, nr in ailvm-iitc any i>ii<' in partiuulur; liiit nill rallii'i- uiaki' il his aim to i-ini |iiii'i' lin' ilu'oii^'-i which havi' obtaiiu'd tlir urrati'sl {'UirciRy with then; jirt ol aM'Citaiiiinn the I'limlitioiis ol' I'oiistniulioii ami loailiii<; to which tlii'V aiv a|i)ili- uablu, ami also In imlieat.' iliivelioiH in wliieh I'liitlwi' iiiM'stis.'atioii is still rL\(uired to iiict't the cast- of railway art'lii'^ carrvinv' lu-avv ritllini: louda. As it ol'teii liuii|)ciis, tho ilisi-ussion on this sulijii-i is lanioly dih- lo UliAumlcrstamlinjfs which havi' arisen, oniin; lo iho (lilVcrcnl jioinis ol view taken by diller. nl aiilhors, ami ilie ililleriiii; cnmlilions which llicy have in their niimls ami do not sntlicienlly e.\|ilairi. The |iliy-ical ami uiechanieal propiTties which the inalcrials are assumed lo |ici-c~s, |h,. melliod of dcsijin and eoiislrnclion. and ihc purpose of ihc arch :is regards the duty it has lo fulHII. are seldom iiivin wilh the chavness that such fundamental considerations should have. Ill must eases il is only by readina and le-rcadiu';. exainiiiimithecondition. a- expressed uiathematicall_ , ami iiotiu'.' eaicfoily some obscure hint oeeurrin.' in the course of the in^csti^atiuu. or even by relcrence to the exuinpl > liittid, that the assumed conditions can be inferred. The invcstii;atioii in diffeieut countries ha^ also been caiiyieil on without sufficient reference to what has been ascertained elsewhcii'. The older theories of ihe arch need not be reviewed here. The modern study of the subject dates iVuin the iulroiluetion of railways, and oriain itkixA with .Mery ami Moseley. who were the first to iuvestiijate the properties of the curve of pressure. (1) The various theories on the subject have now resolved themselves into two leading; ones, wliicli under more or less uiodilii'd forms have obtained by far the trreatesi eurreiiey. The first of these may be briefly stated a- follows: llu.' curve .if pressure must remaiu within the midille third of iheareh rina, in order thiit the stabilily of the structure he a.s; ned. The .second is based upon the principle of least resisiauee, ami .jaintains ihat the thrust (h'velopcd ai the key will not ctceed the le:ist amount e ipible of maintaining' eijuili- brium. The first of lb' se ha< meat iHirreney amonnsi Km;li..h authois. and being endorsed by Rankine. is often reliir -d to a- an axiom, On the t'ontincnt llie second is larireiy adopted as an improvement on older theories, and proinimnf anioui: its e.\p .nciiis are Uopuit and Sehcftbi. Hince Graphical .'italics has devel.iped so widely fioiu ilie foumlalioiis laid by Culniann ;uiil Levy, (i) it bus b.en almo.n universally aeloptcd as the best uiellioil for ihe inve-liiraiioii of the stability of the arch. It must be noted, however, thai so far as tlicarch iKconecrned it is only a method, and Joes nol afford a solution apart from more sreiieral prin- ciples. We have also the help of a laiyc amount of obsi'rvation ami experiment, bul these must be distineuiscd from each oilier. In observ- ing structures durinj: and after their ereclion all the conditions are prescut, but the dilficnlties olobservaliun ailmonish eorre-poudiiii; care ill the inf rences drawn. In makiri;; experiments with models, lhe.se diffieuliies can be largely removed, hut the m.iterial employed beinu usually diBerent, the ditteieiice in physical properiies must be allowed for. A eoiuplete theory ol the arch uiu-t apply to the finished structure fulfilling the purpos, s „f jt^ eoiiblruction ; but b.d'ore proceeding to examine llie Iheories referred lo. we may first menliou llie conditions which accord wilh ibe ordinary principles ol -lalics and liir which no Kpeeial Ihcoiy is ici|niied ; and wc must abo examine caicliilU Ihe physical and meehanieal pro|K'rtieb of the mati'tial wilh which »<• have to deal, as explained in works on Ihe snbjeel. The desii;ii, nielhod ol eonstruelion. and purpose nl'ilie aivh willi v>-.\k,-{ t,, the loudini; il has to carry, furnish lurlher eondilions which must be taken up in eotinee- tion with the theories theiuselvcs. There iiri' only two ways in wliicli n jK-rfict :itcli cnn bu constructed, mill tlii'f' .ti coiiverw of each (illicr ; cither the curve of the urch iiiusi he (li"hiila iir>' laiiiiliar iix the fnriiis ri'i|uireii, aeennliii!; ;\n tlie loiiil is uiiiliiruily ili>li'iliiileil uliin^ the eurve itself or aloii^ a Imriieiiiital line ; ami a eir- enlar iin h may also lieenuie ei|iiililir.ile I Ijy a Miitahh' ilistriliiitinii iif the InaiiiiiLt U|>iiii it, If llii'M' Mii'tlioils Were ailo|iteii, the eurve iif |iie<>nre wonhl eoiiieiile with the eeiitre of tlie arch I'in^ thriiiti;hnul, anil the ilinetioii nl the |iies-uii' woulil lie |)er|ienilienlar tii the joints. There wiiiilil he entile liarainny In Iweeii priiiei|ile ami |iriielie ■, anil the only eiimliliiiii iil'stahiiily wmilil he the ahsolnte |ires«nre ailniis.siliie in ueiMi'ihiiiee with the iiiateriul useil. Wle'ii it i^ the Inailiiii- that is )ii'e>erilieil, the liirin ri'i|iiireil liir the aieh is usually iliHieiilt of eon striieliiiii, I'uiii^ to the eoiiliiiual eliaii^e in llie railiiis of eiirvaliire : hu. iiotwithstaiiilin<{ tliix, cIoh' a|i|iroxiuiiitiniis to the true forms liavu heiu a'lo|>teil with eeonoiny mi ai|Ueiluets ;inileanils. 'I'heir ailvaiitaues (lisa|i|iear hnuever wlieu a vhan}.'u in the |Hisiiiiiii of the hiailin^ has to he allowed li>r ; and for railway |iur|Hises seuii-eirenlai, i'lli|jtii' and ne^'Uifiilal arelies are substituted. Altlioii^li this is the ease, a valimblii iui|iroveuieiit ill the siahility of the areli may he o'llaiiied by a snitiible distrihntion of the dead weight. To siuiplity eonstriietinn tin* elli|isi; is ofieii replairil hy eiieiihir ales, and tlieie aiu very elegant niellioils of discriliin;; these by means of a series of eentres, (it) to wliiuli the name of " ba?kel handle arches " lia- been given. When both the form of the .ireh and the liiadiiej are detined. the areli is suhji.'eted to eonditioiis iliveriiiDg from tlinse reijuired for [M*r- leit ei|iiilibrinui. This is nnfortnnately the ease we are called Ufion to eonsider in |iiaetiee, and for which a thenry has to be tiiiind. The eurve of |iri'ssine im li Ufier cuts the joints of the aieli riiii; at the centre, and a Heak |ioiiit ilevelii|M :it the liauneh known as the joint ol'rii|ilui'e where the eurve passi's nearest to the intrados. In considering a series of voiissoirs in siieh an ai'eli, we iiiii>t Hisi deteniiine limv tlie |ires.suie on the surface of a joint will he distributed when till leMillant of tl xtcrmil forces is kiionii. We will su| pose (lie lesiiltani known In pnsition and maijiiitnde. and fi>r slin|,lieily we will take It perpenilleiilar to the surface of llie joint. We further take the surf.iees at (lie joint to he in pt-rti-i't eontaei. hiii witlinnl euhestun, so il.at only lonipressioii can be resisted and not leiisioii. With re^iard 111 stnue and briik we have aUo some evidence that we arc dealinu with a material that is elastic and therefore also compressible. As boys we learii that a inarblu will bounce on a paviie.' stmie ; the amoiinl thai a brick cbiuiney will osi'illate in the wind eaiinoi he aitribiited M the nioit.ir alone; mid we have fuither the eorriilmrative cvid-'in-'e that stone expands witli heal, (l) Co eHicienls of i la«tieity have been (letemiined for t^l.iss and also for slate; but not for (he materials ordiii arilv used in constinelion and when under direet eompression, so far as the writer is aware. It is the e.xistenee of elasticity and not its anioniil howcvei , tliiit;.lfeets the i|Uc-tion ; and it is ilnuhtfnl whetlier any solids exist which are either absolutely ineonipressibh' or inliiiitely hard; Without t:oing fuillier into those physical pr";iertin of >toi id brick tiir which the experinieiital data are so nieaure. it i<4 emiii;:h that we have the rii;lit in infer thai they are clastic, and to apply to masonry the same Theory of Klastieity as to other mati rials. In ai 'rdance with this theoiy. the particles which are in a plane when llic body liai its n.itur.i! shape will leniain in a plane when the bo,iy heeoniesdeiiirined under conipiession. ihe new plane beiiiL' either parallel or incliniil to its ori^'iiial position. Apart Ironi this theoiy we have iin means of delciminiiiii- siderin^: this the only function of the mortar, (ti) and even in tlie ease of cenn lit the ailhesion is not eouuled uisjn. but is left to fonii pin of the uiai'Liin of .-afety. Some few make au exception of brickwork built ill cement, and eoiisiibr both tension and compression as occurrintr at the joints. T'his amounts to transferrin!.' brickwork from the conditions of nmsoniy to tliiw ol a uetdlie arch, and liniitiiiL' inasnnry to ^toiie- Work oiil) ; but then' are few who make surh a disiiiieliun betwieii stone and hriek. The only ninainiUK action eonsidered at the joiiii is fricthiii. which lenders an ini'liiiation ol the ri"-ultant compatible with stability when wilhiu the limits ol the uniile ol frictiini. It foUows then dineily from the tlicory ol elasticity that when the resultant pasw s throu^ih the centre of the joint, the presdurc la uui- foinily (li-lrilmt«d ovit \t« Mirfiiuo; mid iiU> that us it nioH3» from l\w oentrn tin' |iri'»»iir(' lioeninrs iiiii'i|IIhI, till nii riwIiiiiK a point ill one lliinl IVoin till' ei\nv llii' |)r.>«'irc' lirmlMi'i /.iTo .it tlir oili'-r I'll'^i', iiH uliowii ill Kill. 1. Tlii» is iii'i'i'KMirily ilio liniitiiii; |HMilioll III' till' ri'SllllUlll I'nr wllil'll till' wliolu jllillt c'lin Ik- ill i'oin|)ii's- llint llir I'urvo orim'siiiii' 'M\\M 111' williiii tlu' iiiidilli' tliinl 111' till' iircli riiii: i> lliiTi'l'in' ii|iiiviilriil to iiiiiiii tuiiiinK tliiit eviTJ ji itit ill tlir iirrli iiiiist Im' in i'niii|irc«Hiiiii over itsri.tiii' >iirliii'i'. This corri's- ]hiiiiU with II niiirli lii'_'liir ili'iiri'i' oC ^tiihilily tliiiti Hinny iiri'lici iii'tiiully liiivo; iiml \Vi' iii:iy tlii'irl'iiri' fm , liillow tlic I'Ifi'Oti |>ioiliii'i'il III llii' joiiil :ii (111' I'i'Milliinl |iii«i'i oiil iil the iiiidilli' third iiiid ii|i|iiii:ii'lii's llii' iiiludns. Tin' j'>iiit will no loM'.^"r U) I'litiri'ly in I'linipn'Maioii, Iml lli' |iri-iiMe ol'ln'iii- inn tension. (Kii;. "J. I Sirielly s|ii'iikiiin the outer ed(ie nill ci|ieii Imt only liy an iinioiiiit that i- entirely iii!i|.|ireei ddi'. It is e:i|ialile nl' riil culntion, beiim |irn|ioriiiiinil In the enui|iii'ssiiiii of ihe stone at ill inner ed'je ; Imt the eoTii|iiessiliiliiy of stone is .so inliiiitisinial thai (iraL'tieiilly it eHiiin.t be measured; ami the ' npeniiif; ' of the joint under siii'li eircuiiistani- - is tlierefori' entirely tlieiiretical. This iHisitiim of the resultant is then still eoiii|iatilile with stability. As the re.-ullBnt continues t4i anproneli llii^ el'.'e. the siineaetion e 'ntiniies in an iiierea-iiiL' ratio, until the (.ressure at the inner edu'e reaelii^s the ultimate stienutli of the iiiitei ial. w lien eriishiiiL' will take pliiee. rotation throuLdi a suiall aii,i;le will ensue, and the jo'iit will open visibly at the opposite edt;e. T'liis is ihe Inn' liiiiiliie.' position of the resultant in actual eases ; and for hard and sri;i| it iiniy b* vi'ry near the inlr.idos, '11. at llie le-iiltniit may be very near the edi^e of the joint, without eausiiij; llie failure ol the areli. is also held as proved by authors nhii base their eonehisioiis nii i'X|H'rinienl. Prof ( 'aiii shows by exper- iments with wooden inoihls tli.it at the jnini iif iiipiiire the eurvii iif pie.s-ure passed on an aveiMt'e at only one eii;liteeiitli of the width of the joint from Ihe edn'' willioiil rupture eie.iiiin.'. He also ealeiiliiles that ill the bridL'e ol Neiiilly the thru I at the eiowii nimlil pii-s wilhiii less than two iiiehe- IVoni the edi;i' without enishini; tlie aniterial. l7) Wc must thereloii' lomlud" that in the ease of actual niiisoiiry, the curve iif pressure may approaeh I'stremely near either to the inlrailos or the exiradus without rupture nceurrin'.' in tin' arch. This sh iws the iireat difficulty nl ilcdiicinu a theory fnunexistiiiL' structures, as so little can be inferreil from the bare fuel tbiit a structure remains slandinir. Iji't us now lake tin aei pinyiiii: outline to represent a portiiiii uf the ai'cb-rinf; in a full areli, either semiM-iieiilar or elliptic. We will take it in the iisiiil way to represent by its area the weii;lit of a uuit of width in the direction ot the axis ; and the loading may also he sIkuvii as an 'area represent iUL' its avera'_'e amount |«riinil of width. We will also ennsiiler tlie Inailinu; as plaeeii syinmetrically on eaeli side of the key. and a half areh will therefore be suthcient for the (igure. Koi " the hirins of loadiie.' wi' have onlimiriU t .stated that the eiirve ul pres-ure will resemble a parabola. It is easier to determine its u'einral form than to determine its posiiioii in the aieli riug ; but if^ueli a eiiivi' is plaerd within an areh riiiir of either eir- culer or elliplie from, it will appiuaeh tlie iiitiailosat toupposiie points it the hauinhes. ami at ihose points Ji will be parallel le ihe lainient at the intrados. This ilctirmiiies the posilioiis ot lln^ joints ol' rupture, one of tthieh is shown at CO : and if we ulsn lake a verlicai joint AJi at tile key, these are the only one- wliieh ibi' eiirvi- of pres-iir.' will necesbarily cut at rii^lit aiiL'le-. 'i'lie luomeiils nl the forces must there- fore be taken Willi refiii'iiee to the portion .HUM': as to liivide the arch rinu' at any mher point wniild iiitrodiiee a radial toree representing friction. We have iheii tliiee loii'es In imiintain ei|iiilibriuin. 'I'he liiriist '/' ii horizontal on aecnunl ot the symnieiry. I'he line /, It' is either a ver tical thrnugb the centre of gravity of the area r'pr.'sentiii'.; the areh riioj; itself belwei'ii the joint of rupture and ihe key ; or it is the resultant of the weight of the aieli liiig. anil the pressures upon its extrados between llie -aiiie pninis. When thesi> pressures are all vertical. /. H' will al-n be Vfrtiial ami may be leniied the G.-aiifi/ liiif When the joint of riipl lire is known in position, it lieeniiies kiinwii in position dileetiiiii and aniniint : and the resislanee tf also becomes kiinwii in direelion Tlnis ot the iliiee forces two are kiiowii in direction only ; but to maintain i'>|iiililiriuin tbev must intersect in a point at A, and this may be termed the ( 'ui'ililinu uf iutasuftiini. The position of the joint of rupture muM correspond with this couJitiou. ric 1 ii u|)|iaiuiit. Wi' liiivi' nil I'xt I'liiil MiiMiii'iit iinidiiui'd liy tlio loud; Imt •'.) i'i|uivali'Ut iiioiiii'iit of i'i'>i?i' iiii'i' Ik di'vi'lopi'd in nil iiri'li riii|; i'.< it would bu in a ilii'tiil rill. Ili'iii'ii iiiilll llii' iHiintH of it)i|ili('ution A' iind J euu Ini lU'ti'rniiiK'il. lliiM'i|iiiition lii'twi'i'ii tliu imiuiuiitn of 7'iind IK ruuiaiiii iii'li'liriiiiiijlr. and ur niiiiioi ihi'rufor" priH'i'i'd I'itlii'r to un uiiulytiL'al ur II u'rajiliii'iil Miliitlnn. Tliin uliuiidantly ucrouiilH liir llii' ureat variety of i>|iiiiiiiiisaiiii>n^Ml luillior*. All tin." tliiiiriui* iit llii' ari'li > iii bu riaaiti lii'd ill uri'iirdai wllli tin' poailioiis llii'y attribute to lbi'w< pointa. In inntlii'iniiliial uoiks ii is ufii'U voiy diflicult lo discover tbe reaanuH fur llii'i'lioii'i' niaili' . iiimI in woi ks ti'i'utiii}> the nubjei't ^graphically, while tlii'iu is ani|ili' i'X|ilan:ilioii iif tliu iiiiilhnd of drawiu); tin' .'urvu of prca- nnri', tlii'ii' i» iiMi illy a lai'k of i'Ii'.hui'iih a^i lo tlio |iiinri|>lt'a upon whiuh the piHiiiiiii iif llii'si' detirinihlnii |K>inlii ileiH'udt. AlthoUKh an author uiay givi' ^oml ri'iminn liir tlieir position in uiiu caM.', ho uiay hi; too himty in ni'ikin^; n general application of the result to all cases. It ia for this reason ihai some llieorius have fallen iolo discredit, although thiy unihiuhti'dly have "onie giiod fuutures in theiu, and may be i|uit« applieable iiniler eerlain eoiiditioiis. The direct di'leiinination of the piMition of thiise points has been iittein]!*' d lit aliont ilie s^iine time by Scheftler and hupuit, based upon the principle of least resistance, but applied in very different ways. This principle is due lo Mosiley (8; and may lie briefly stated thus: the thrust deVelo|Hid nt tin' key will not be greater ihiin the ainoiiDt which is jusi suffii'ient to maintain cquilihriuui. As it is evident thai the least amouiil of thrust will ecrrespimd with tlie eo-iiicidence uf AT with .1, and ■/ with I), .Moseley lakes the curve uf pressure as laiiKeDl lo the extiados and iiitiadosat the key and joint of rupture resfiectivcly. ■''^elieHler (U| u'eiieializes llie same \ iew, and takes a curve uf pressure iHiigeni III till' I'xtiadiis ami inlr^idos uf the arch rin^ for all oii.ses ul In^idini.', wliethersynimelricai or not. >Snch a curve corresponds obviously to the CMse of a siruetnie in which hoththeai'cb rin^ and abutments are everywhi'i n tlie point of {.'iviiiu: way. If the abulments are about to yiehl.aiiil llir 'ym\lr CD and .I/tare nlmul loiipi'ii, we have iherec|uircd conilitioiis. anil lailine will take place in the way sii iifieii represented in l< xt IsHiks. Altliiiii^'li ii can be slmwii that the curve will only v is taken from Claudel : (Vi) While the :o'e)i remaiii' mi its centres the pressure in the arch riii^ is zero tliroiiulii'iit ; ami as the eenlres ale eased, the prcssiiies liiiind in the tiiiished anil aii' ileveli'|K'il. I'lese pressures iherelore must pass throuuh all the liiti i nieiliate values Iroiii zero upward; and in acuer dance with the principle of least resislanee, he infers that when ouue tht Icaul amonnls necessary lor stability are reached, there is no further iraMiii why llie pressures shonhl continue to increase. In tin: uiuve- nient nliichueciirs when the eenlres are struck, there is roiuelbiuL' fixed and C' rtalii which Jepeiiils niily upon the curve of the arch, and also an amount ot unecrtainty ili'|H'ndiii» up'Hi the nature of the materials and the nictlinils nf innstrm tioii. su that the uncertainty finally rcinainiiif: is reiluced to sach narrow limits as to have no further iutereat from a practical point ol view. Wliile the arch reniaint on its centres the curve of i'i|nillhriuni isintircly beneath ih arch, and is determined by the intersections i,i the verticali- through the successive centre! of gravity, with the production of thi' j'linl lines ( I'l I < This may be called the curve ol static ei|niiibriuir. and correspunils u ith an absence of pressur,- in tin- arch rim*. This curve serves merely tu indicate the point in the arch urouml which rotation would tend to take place. As a sn:all tliiust develops at tin' key. this curve rises toward the arch and indicates more ilislinctly the position of the point of rotation O. When the thrust incirases siiftieiently to make the curve of pressure pass thrnuuh the point /'. ii is evident that it is sufficient to maintain equdi- briuui. There is therefme no reason why the curve should pass this •Ad the eurvi^i'f pressuieisjyinmetrical fur symmelriial loadin);, and the puints A' ami J iin|ily ilii,-itiuii a« well aspusilioii, tin y are mathematicAlly tquivalelit ill till- loliipU'li' anil te six ileti lllllliilltj puilits <.|| the curve. It wuiilil llieieriiie have le 1 la very hij^h uiiiei t„ take two ililt'eient puuitiuiis hetvveen A' and J. point mill Hike up ii |io..itinh in Iho intcrinr nf thn nrcli, im nuoh a piwiliiiii wniilil cdir. wimiiil with ,i i.'rc;iti'r iimiiiiiit nf ihriist at the key. Till' iiniiii II |i,.iiil; ill I.TMiiiii'il «!• kiiiiw ill wliiil ilirrtiiiii tin' iircli will iii.iM' ilihi. |i.|r:,i-,|| is fnr ,1 iiiMiiiriii 1,(1 mi>iip|,nitr.l. Thi'imth *'"'■• ''li I'lii'ii '»> 'III' lii'li t'MiKi.i iiiJM'isi'viil Iv till' iiri'iiliidri'li' Ullillll.l l> II, :| iTliIri'. 'I'lllli till' linillti .1 mill // will IpihI III iliflcrilH! till' iiimll ari's .Mimil /(/( I'liiri'^iminliiii; tn tin' laiiii' iiii'.'li' iil llni ci'iitri' /'. '{'Ill' riii-i: r III,. iitliiT IiiiII'mI'iIi,- ari'li li,itt,'v,r \>rv\ t< nctiial iiiiiiiiiii , liiii 11 |'„||.,ns iliai t||„ |H,iiii< .1 |,|„| /( ,|||,|,,,ri ii,i,i|ii,il pri's-iiii'v, ii< III,' aiiiiMiiil 111' i',iiii|ir('s4inii m. (In'^c |Hiiiit< i- I'viili'ntly piiilHirtiiuiul (,i III.' Ii„ri/„n|il ilisl mei's lliiiiiii:li whii'li ilii-y woulil umvi' ii!,^|ii'. tivi'ly. il'llii'V wit.' IVi'i- t,i il,i -ii. Tlii' i-ffi'i'l pm iui'uil \iy the ntliiT 111."' Ill' till' iiri'li is III iiitkintiiln till- jiiiiil AH in n viTlirul |Ki»itiiiii ; uiiil it tlilliiw'' 111 It tin- liiiri'/iiiitui I'linijiri'ssinn hi ,.ai'li point in proimilioniil t,. lli,, di'iTeiisi' in li'ii'.'lli nl' tin' liiirlzinilal priijeitiim'' "t till' I'liilii (Iruwii I'lnni l> to .( iinil II ruipi'ilinly . It ii ea-ily ^liuwii tliMt llii'. prnpiirliiiii in III,' ..,||„. IIS iln. niliii ,il'/i i i t„ /,. H'w,- nlm (Kiir. j.) iwii liiii'- .l/'aihl //y ill this lalin. ii riijiiiws tioiii iln- ufi'i'ptt'il tliLiny iil'i'la-lii'lly lliat ttn' ri'-iiltaiit nl'llii' wlinlc prissure iin All will pass tlirim._.|i ili,' I'.ntr.' ,.1' irr ivily nl' tin' tripciiuiii AP(^H. Till' piiini A' is tlnri'l'iir,' nnt at llic ri'iitie nf AH. Iiut alwayi « litllo uliovc it. I',M' till' liiiiiiiiii< la-c-. if h — i a< in the plate huiiHi iir slruijjlit ari'h. /!K~^l. ami il'/i -inlinily. HA' -J I. Thinn'iiafjniiig UJijiunii'H thai tin- pnini I) is riuiiliy lix,',i. nr in nthiT wnrdH llmt tim ulintnii'Mts 111-1' iiiiinnviiliji' ami iiiciiiiipii'-sibli' ; and hiri' wi' ini'et with the mii'i'ilaiiity iiii'iili'iiiiil :il th tsri. lithe point I) were In yield laterally, the piiiiit A' ttmilil riie in prii|iimiMn ; ami would appmaclias near to A as the "tremjih or the uiaterial allowed. The tiiiure euii'loy,,; in ih,' ileuion-tration ii liir a seirmenlal orcli ; bnt for a I'ull areli it ejii be shown hy similar eimniilerotions that tli« enrve of pressnie iiiiisl he lanifeiil t" the iniiadn-t where it meets it, and that thi' pniiit of emiiail ditiriniiie-i ihe position of the joint iiC rnplnii'. In the aeliiil mh the iiirve iiiiisl be sufficiently within llw intrailos at llie joint nl riipliiie to previiii erushiiij.'. He further shows ihiit the piHJiioii- which tie' joint of rupture can take are alway* between nil,' half anil lwo-tliii,ls of the rise, in the ease both iif liiu iK'nii circle an, I ,'llipsi'. The exail position dipcu'ls on the ratio of the thii'km'ss of the arch rini; In Ihe ris,. nf the arch, as this aflects the distance of the centre nl irravity from the key; but for the ordinary proportions Umiid in practice, the joint will be very nearly at half the rise. Dupuits conclusion is then that with unyieldinif abutiueuls the posi- tion of the curve of pressure will be d'terniined as follows : — The point A' will be at the centre of A Ii f.)r a semi-cirouiar or elliptical arch, ami at the upfur third of A H for a platebaiide. For intermediate forms it will lie pro|inrtioiialcly between thtjee limits For semi-circular mill elliptical arches the curve of pressure will be tangent to the intrado- at th,' joint of rupture, and the joint of rupture is to be determined by this iimdilion. For sejimentiil arches of le.«s aniplitnde than the arc lietwcen the joints of rupture in the full arch, the curve of pressure will pass tliroU'.;li the point D at the ."priiifrinL'. In cxaniininir this tlicory it is evident that the determination of the point A' would not be appreciably aft'.ctid if the point ./were to m.ivu out along the joint ol rupture; baton the otlnr hand it does depend entirely on the supposition that llie abiilments are unyielding. Fftlie abiitinenls L'ivc laterally iimler the elh'cl of the thrust, the joints of rupture will open and also ilnjoinl at the key. I he curve of jiressuie will then necessarily pass thrnuiih .1 am! l); but the value of the thrust will be coiisi,lerably less than for a curve passini; through /T and D. We see then that any yii'lding nf th,' ahuiiuenls '.'ives an in]ui,'diaiu reduction in the value ol the trust ; and therefor'' the abutments reijuirc greater -Ireiigth to resi-t yi.lilinj than to .stand after a slight motion has i:iki'ii plii'i'. This aicoiiuis for the r,'gaiiie,l stability of sirnctures in tthii'h a sli'jhl iimveim lit has ocurred We have also in this a ilistingiii.shins diffcr.ni'e l.ctwein th'- tln'orics nf IHipuit and Schi'ffler. As ex miplesof eases in which the condition of unyielding abutments becomes perfectly reallzeil. nc may nieiition a segmental arch springing from the solid rock, or i «erii's of arches all cijual and equally loaded ; but it will h,' practically fullillcd in any case in which the abutments themselves and their fouinlations are suttieicntly resisting. It cannot be too carefully noted, however, that Dupuii's -ensoniiig is bused ujHin the consideration of llie arch ring itself while sti.nding alone. For that eas,' then «,' must consider his eouclusious as es'.ab- lislicl. Hut III siip|i,isiii,_r t||;,t the curve of prcs.sure will remain tangent to the intrailos when the arch ring carries the weight of llie spandrel walls, backing and lilliuL'. Dunuit is certainly going beyomi the limits of what he has actually proved ; although he renders most valuable assistance to the consideration :if the subject by indicatiiiL' the true startinL' point ami the first steps to be taken before more com- plicated I'ondilMins can be exaniiiied with advantage. His method lor the eiiinpleted arch is to liiid by trial a new position for the joint of lupture which will fultill th,' eoiiditioii of intersection , the tangent at the joint and the thrust at the key having to intersect on the gravity 5 liiic Tliit niw imnitinii will In- InwiT tluiti tiir llu' iinli rin« Ituclf, iM iln- t'f ivilj lull' will he fiirilur Irniu tho key. It i» ii cniii|il(l<' ilii' wlmlo «lriirtiir-' Iwluro nmiiiv- iiii: till' iiiitii!". M> liny wllli'iin'iii dl' iln' uri'li riiitf »i'iiM nii.l.r vrry III rljiii ill ' iicliiiil ili'iilliiilinii (il° ilii' |iri">' i)|i|ily llir |iriiiri|<|.' <'l' Iim^i ri'si^- iiiiiri' I.I ill.' tiiiiilii'.l i.|iii.'iiir.', i« i.i inti.i.|iiiv n liiru'n iiii.l nil wiry iiiii'iiiiil I.I' iiiic'iljiiiily. N'liw wlii'M ill.' II uily liii.' iimvcn I'lirllii'i ii ill.' k.y il i'< I'Viili III li.iiii llii'i ilili.iii III' iiil.'i'<«'i'| lull, .'iiliii iliiil ■ III' jiiiiii III' i'ii|iliiri' will tuki' ;i liiwi'r im^iiinii, III' ilial llu' |i.iiiil ./ will IIIIIM'IIMI tiillll lilt' ihll'.llliM til ni:lilllllill /. ./ Ill ri'jill :tllL'll"< III )' />, 'I'll ili'i'i.li' liitwwii Ihi'iii' uIii'IIihIm"! Mr iiiiiKt lii'Hi I'Xiiiiiiiii' llii' i|iii'iiliiiii at 111 lii.w till' |iriwiirt' I'luiu tlit' !.|i:iiiilri'l in Ira iiiiilii'il l.i tin' Iniok iil' till' urcli riiii!. UH il in |Hiiwilili> tluil wln-ii lln'iinli liiiL'.'rirrii'i iii wi'lf^lit till' liiii' /, II' III ly nil liiiiuir In' virliuil. A> u Ininiiiiu iliiirli.iii. iliu [iri'-.uii' limy lir I'Vi'rywIii'i'i' nl riulil iiiiiili'- li lli.' i'»lrii.|.ii . lint iliii u.iiiiiil inn I'liii Hilly Ik* ri'tilizi'il liy liking i'i.-.'|ili.iii:il |irt'riiiiti.iiiH willi iliv iiliji'ci III iiljiiiiiiiM'^ til' ' liyilriKlaiii' iircli, iiiiil Wi ruiiiinl tin icl'iiri' Mipliiwi' it III 111' till' ii-iiiil riiH.'. Amiiii. ilii' |iii««iiri' may nci uluii'^ iuuliiK'l liiiuH siniiliir In iIioh.' ifivcii l>y iIk' |iri'>-.iirr nrcartli. Tliin will Iju iIic ('U'lal)ililv ilrprii.h iiii I'liiiilili.iii'* tno Tar rciiiiivi'.l rrinii tlMi»i; wu liuvc Miiri' iii. Smiu' KnL'iiK'ir-. iiiaiiiluiii timl llii' preKHiiri' III inawiniy will ;iNii art similarly nlniii.' ini'liinil liiiu* of IViicliirv. Il it |jililu that this liny In' thu rasti tn '■inn.' exUillt, i'!i|ii'rially liiwiiril" thi.' Iiaumho ; lull I'lir tin; I'lntral imrtimi iil'llii' iircli, il i* HUlK.'ii'iil lij nihipl thi' unliiiaiy siippniitinii thai h viiii>Koir carrii'it Ihr pmlinii nl' ilic Inail vcrtiially iilmvi' il. A* ri'..'ariU ihu hiiuiic'hes it iii 'iii aliiinst invuriabk' rule to curry up the biickiii'.' tii thi> h'Vil 111' tl' joint ..(' rii|.liiri' a* liiiiml I'nr thi' arch riiiL'. hd'nrc thcocn lr.» a.i' Ktiu.'k Winn thi^t 11 iliinc the .iiicliim i^ iiiucli ~iiiipliliuil, Of the wh.ih' 111' till' niUMiijiy bclnw thai juiiii cm llicii hi'lak.ii as r.inninK in rculily 1 pirt nt'thc ahiitinint. Wc will a~Miini' tli.n llial it is niily the rcinainilcr nl' the s|iinilri'l which is jilihil alter the I'l'nt'riii'^ m reiiinveil . uihl tliin has the sanctinn nl' ih.' Ii.sl ciin^triictnrs. Wlieii ihe nrch is built in thi" WHy. nc are nni at lilnrty In -uppnsc thit tlic joint III' riiptiir.' will take up a lower imsitinii th;iii it hail in ih.' nii;h riiii:. aiiil »i' arc Icl't in the altcinitliv.' that mi the aililiiinii nl' the ■panillel'i the pninl ./ Illii\e» nut I'rnlll the iiilrailns. We <.'e at til.t >a IIIC I i nil' the a.ivanla'ie ileriveil t'rinii a.l.iptn.i.' this incthnil nl' eon ^jlriiciion This also seeiin the more proliable when wc lompiirc III.' Inrin nl'lhe >pHniliel with H rnnii nl' Inuilini: which will nuke iIicciiiac nl' pr,".siirc eiiinciilc with ihe centre line nl'lhe arch rioi; lhi"ii'jlioiit. This liirm can leailily lie hminl liy rever-inu llic onlinary ..'raphi.'al process The e.itire line ol' the arch riiej is assniiieil to he the curve nl' pi.'SKure, su.'.'.'s-ivi' laii'jent" In il are ilianii. ami a rnrce ilia'.'ram enn^irnclcil in which till' Miccessive parts rcpii'scnl ilu' wciu'lil^ that lia\i' In he ipplieil i.l.ini; the curve. 1 S.'c I iii'i'n.''s " Ai.'h.'-, ' Art. I III. 1 lly plnttinij these parts rrmn the intra. I. is as nr.linaies. ihe wciuht ol the arch rini! i.s taken iiiin ai int. ami tin' ar.'a iii.lieal.'.l l.\ ll r.litiites i-. the hiailini; ri.|iiiii'siti.in rurlher Iriini the ki'V. .he jnint nl rupture will imt .'haieje il- pn-ilinn. hiii the pniiit ./ will miive nut rrnin the intrailns. Thi- change hnwcvi r. will not aftect the cmisi.lcralinMs which l.'.l |.. ili.' ilctcrmiiiatinn nl the pOsitinn III K. Knr the structure .ninpl.tcil a- almv.' ilc-crib.'il. aid with iinyielilinn abutments, the liillnwini; uencial iiiclhinl ol' liinliin: the ih lerinininjj points is siiniicsi.'.l ,is harmonizini; with th.' best ili-eiis-imi nl' tlu' -uli|.i'i as niveii hy iliftorent authors: — The pninl A' will remain at the same positinn as in the arch niiL' For a s|. mi circular or elliptieul arch the joint of rupture will riinain at the same piisilioii ns in the arch rin;.'. anil the point ,/ will In Innnil by ilrauiinj 110111 /.where '/' anil 11' intersect, the line /../' ai rii;hl ans-'lcs lo Cl>. Similarly hir a -cL-menlal arch, th.- point ./ vviil b,: Jetermiiicil by ilrawiicj /,./ parallel tn its I'nrmcr po-ilinii in tlic anli To inaki' the pi.-iti.in of the pmni- K ami ./ ile|iciiilcnt iipnii the positinn 111' Ihe 'jravity line a- this m.iho.l J.ic-, i- much mure rca-nn abl.'thaii t.. wive them the auie arhilniiy pn-iiinn liir all i':ises |i also utturils a means 111' a-eertaiiiiii'_' the 1110-1 a.lvani c is ili-tril'iilimi ol wtiulil ill till' spaiulrcl. A- a rule the |Kiiiit ./ will he only a -hurt Jistaiiee out I'rom the iiitrailos ; hut in I'Xircmc cases, il there is a lai'fie ttiuount ol'loailinj; over the hauuehcs, the jiravity line may be -o far (i ll„- Mini iiiiii Ih.' .Ii tin: Xfll ;li,-*ii iiirl from iht! key tt» to lirm,' the point ./in iho iwnir* nrfOnnilfVoutuullj fven to llu' I'lir.iilna Wi' h'lVf hvn iimiimimik. Iiowi'ur. llml lli'' l"ali<>ll Illil.V luirll It lilllil lii'jtdii.l Hhii'h it i« nil Innmr liiii' Wli.n llir IniLJin^ i- - ■ liinlily ninr. iilr.iliil oM'i- lln' Iimimm'Iii- Ii in ninic llniii [.rnliiiM.' llml llir ilini- llnii 111 till' pnoiirr lKi'i>nii"< iiiiliniil I" llii' liink nl' lln' iinli rinir. 'I'liiN i- ilitfic'iili h iliniiili^ ill iininiini, Imt in oiiii iinnl ii m- i'lli|iili'ii| urclirs it will lir i|niti' KIlHicirlllly :lll"»iil I'm liy illlipi.-iiii; tllMl nlli'll ./ Ii'lll'lll'l till' I'l'lllli' iiM'/', lllr lilli' /,./ llli'll liniillli - liinui'ill III llii' niili'i' liiM' III' llii' null till'.' In i ->'.:iiii'iiliil anli in liki' in iiini r, tlii' liri'iwilir III iIk' <|iiiiiiIiiI nill |iii<\iiil ./ rrmn |iu»-iiiu' llir nnli'i nl ('/'. It wiiiilil, liiiwi'M T. I» |>ri I'l riilili' I I iiii'ii:!-!' ilii> lliii'kui'4- III llii' .'iri'li tin^t lowiirils ilii' «|iriiii:iin;,iir 111 n ili.inliiiti- ilir w.inlil nillui^i Iirl, rktlier lli»ii in iMnnt ii|iiin iiiy im liiiniiiHi in lln |iri'<>i>nri> uIhivi> tin- jiilut III' inpliiri' III iin|iri llir piHiluiii il llir riirvi nl' |iri'KKiiii'. Snrli IHiMtiniin III'./ ui'i' lint liki'k III iii'i'iir I'mi ii'>> nnliii ir\ rnriii iil'npiiinlri'l. I/vt us imw i'iini|i:iii' llii'< inrllinil Willi tin- tliinrv lliiiulii' riiivi, nl' pri'iauri' iiiuii ri lliinl iirtlii> nili riiifi. 'I'liin thnir) i» I'll liirv I In Itniknii', mil lln' Wi iu-lit it' lii^ niillnnily him Icnilvd iiiiiii' lliuii it "li'iulil III ili'trr Inrtln r iiiM'^li^'iiinn. IliHaiiitii' iiii'iit Ik tliut till' -luliilit) III' uii ari.'li is m'iiiii' if lliii I'lirvn nl' |ir<'«ii liir a" In -uy that iiltliiiii):li iinlii's ImiM' 4|ihii| uinl ~lii| stiiml in uliiili ihr I'urti' liiM iH'yiinil till' iiiiilillr lliiiil, till' -ililiililv III -mil iirrlii'i ii I'itliir hnw pri'i'iuiuUM or inii-l Imvi' In in pr, nirioii'. wliili' iln' iiniiiiii wiii Iri'iili, (I3j Wu inii-l Hr>l I'liilriiviii' In ii'.rriliiin lln inilnri' nl' lln- aiili niul tliu coiKlilioiiHiir wliii'li I'liiikiiii' I" lri':iiiii'.'. Il i< In li' iiil'iTU'il Iriiii lliecu8i'> III' tali! K ii|> an I tlir i'xaiiipli"< Ini '^ivi'i. tlial In' i» rniiiiilorlii)i »rehii!« Inr whieli till' iiuioiini nl' tlir iiiniin'j Imil i.'iin Ihmii'uIui'Ii'iIIii vumpuriNini with tlii' Wriulit nl' llii' nlrnrtiin liiiH'. IVnl Win. Alliin in llii " Tlii'iiry III' Ai'i'lii'H,' wliiili Im iloM'iiliuM iin ln'iii.' an aniplilira (inii ami I'Xpliinaliiiii nl' llaiikiii's I'liapti'n nii tli' siiliji'i'i laki'i Inr jiruuit'il tlmt tlii.'* is llankini' > point nl' vim : " In all suri' ilin' In pii'-HJiii' loaiU an u»iiiilly xli^lil, lu'i'iiiisi' till' wriiflit nl' siirli pi-siii;; liiaiU is ;;riii'ially sumll i.'oiii|BH'il Willi llii' ni'ii;lit nl' llic uiili itsijl' ami ils iMirkin^v' (14) Tliis stati'iiii'iii aHiinls lliu Vi\ In liaukini' - I'Xpliiialinii nl'llii' anbji'i'l ll'lliis liail liiiii ilisliniilN pninliil mil in lii> works, a l.ii'^i' uiiiounl ol'ilisni— inn nnl inisiiiiili'r'lan liin: ininlii liair lu'i'ii avniilil. The snini' i Iplanalinti nl'llii' rair^- n| :ip|i|ii'aiinn nl lln' llinnry nl ihi- uiidillv ihird is iiivin in tin' artnli' on " Urill<;l'^ in tlic last nliliii i ol the Encyrlii/Kiiliii Hiiluiiiiiin "Tin' ina-mir) arrli liiHir- Iroin tin' auptr.'lini'tnir nl' nilnr liriili;i- in lln' riijinwini.' ii'spni : il iltpcnds lor its ^lability nu ihi' pri'si'm't' nl' i pi'miainiil loail "pi't'iall\ arrain^iil. uiid !io I'onsiili'Uilili' in anmunl thai tin- I'liaii'.'i's pnnliu'i'il in i|ii< ,\]r,x tinii mid uia^nlliiili' nl' llm slnssi s liv tlii' passiinj Inail am insi^nili (.'»»t. ' (\'<) 'I'liv tlii'my nl'tlii' iniiMIr lliiid i'iin'i's|»iiii|s ilnn li llin rav nl' ilm finisht'd strni'tiiri' nillimit apprn'ia'ili' nmviii'j Inad 'I'n liinl in at'i'm- duhi'L' with Ihi' nntlinil .~ii^';:i'sli'il aliovi' llin liinils nilliin nliicli tin' uruvily liiii' iiiiist kii'p Ml otdi'r llial lln' I'lirin slnnilil iiiiiain wiihiii lln- iniddli' lliiril ol' ilk' ai-i'li riui!. is nii'ii'ly a nialirr nl' i;i'iiim'lrii'al I'lnis liiii'tinii , and wi' liavn ihiis :\ inaily iiinaii- nl' iniiiparisnii liy »liii'li ii< Vi'ril'y till' ri'sult.s Take Inr I'sinipli' llin ra-i' nl' a -I'liii I'iniilar anli tif r.idiu.s i\ in nliicli ilie ai' li liiir lui - a nnil'inn lliii'kin'ss /, ami lln' jnint of rnpliM'r is ai ,111 - , Tin.' pns-ihli' liniiliii;: pn^iiimis nl' A' and ./ tor It inrvc rrnniinin^ ill llic iniddlt' ilnnl will l> '. lln' nppi r tliiid al tlir kiy ami tin InWi r third al llii' jniiil nl'iuptnir, nr lln' Ihwit third ul till' key uiid ''le iipji^r tliiid al llie Iniiii iirrii{>tiii'<'. Tin- ili-laiii'>' 1/ id' tliu tjiuviiy line rnni tin ki \ i-nrri ■.pnndiiii; In ilii'-e liiniH »il lie t lull 110 - Mid 1/ t I ' I. Ill .111 - Or iiunieriiallv. liir StUii-eircuUr an-ln's nl' III t'<. and lim It. iliaiin't.r. in wliiili llie lliiek ui:Bj( IS deteiniiurd liy ItniiKiin > I'mni'ili. I'n' liinils ivill he : — 10 It. arel 11.577 / and II. llii I r. Mil )';, aieh, II r)77 -■ and O.IJO.") c. A* it liippeii-. till- '.'ravil) line Inr -pan inls as jeiieially liiiill lias very unirlv llii.s pnsitinii in lull arelies and .i eorrespiniliiii; one in seuMiieiila| arche.s , and this ai euuiits liir the lUrreney nt' the theory as applied to the conditions supp ised. This ihenry al-n implies, a- «e liavi- s. en. that eVi ly jninl in the ."reh must be entirely in eonipression. This enrie- pmids with a inneli hinhei degree nl' slubility ihiiii is ueei s-ary under i|uie-ei'nt loads, but the uJvoeulesnl' till iheory lake it piaeiieally as a inaiL'in nl'sarety. Tliev enu^idur that if the eurve i- vvilhiii the middle thirl in the strurliiie beuriui; its nwn weij^ht, it will he -illTie;eiilly slubh' under any innviii;; loads il may be ealled uihui to earry. (lli) As the theory is fairly in accord with the I'mm wliieh the spandrel usually has, It may Ix' sufli- eiently near the truth as regiird- roal bridges; but it is taking ipiile toumueh lor granted in the ease of a relies carrying' heavy enjiiie Inads. It could only be br, night iutn lea-nnahle agieeineiit with sui.li eases liy loweiiii'.' the spiiii^iin: and inen asiiig the depth and weight nl' the spaiidiel. lill the iii,<\iiig In, id beeiine relatively small enmijh In iiegleit . but sueb a eniisirii.'iinii wmilil be ini'niisi-i, nt with leniniiiii Tn lia\e the curve nl' piessun wiiliin the iniildli ihinl nl' the areli riie.; is viiy deiiirable ncd nhould bu aimed nt; but it c.iuuot belaid down as a prinijpli- elinl ii iiiuki Im mi. I( In in rrrilliy nnly ii >|Hi'iiil I'liMi'nuiiuK iiniliT it iiiiiri' )(i'n< ml iiii'lliuil , mid it wkiiM tlii'rt'liiri' lie riinri' cnrrt'i'i III work ill till' I'liiiviTHi ilirii'iiiiii, miil in Hml llu' ilplli nl' ilin i|iiiiiilri'l mill till- nmmiiii nl' liiii'kiiiit ri'i|ulriMl in niuko tlii> I'lirvi' tiiki< iIh' ilr*iri'il tiiriii, im iii'iirl) ii> llii' niKc will iiilnw. Wi'iiiiy iiiiM |iriirriil t.i ■imiiilrr llu' uri'li nirryiiin ii livi' Inml »)iii iiiitrii'iillt |ilii>-i'il 'I'liix jiiiiil »!' iiiii\ ii'iliii'i' ill ilii> Mfiiiii nuv III iiii una ir|iri'M'iiiiii^ mi i'i|iiImi|i'Ii| iiiii"iiiit iiriiiii-iiiii')'. 'I'lir liiiiil nil lliilnuN ui'i'licx ii linn iilli'ii IIS ixii'iit nr pri'iilrr limn till' Mi'ipjlil iil'llir Ninii'tiirr ilv'ir. wliii'li iliiiw* ii^;iiiii Imw iiit'iitiii^Mirrli I'liii! s|iiiii jniiit nl' ru|iiiirii iiml till' ki\, »ill iiliiinsi iil«ii\s III' I'niilu'i I'lniii lilt' kry iliuii llii'i'i'iitii' iil'^nivity iil'ilii' ini'li riim iisill , uml ijiis l)i'iii;r thii'jsi', llii'ii'iisnniiitt )livrll In'liiri' ivill I Illiri'lN ll|i|i|il'illili' Tin- jnilll n| ril|illiri' will still liiivi' till' s.iiiii' |iiisitiiiii as III till' null riii|f. mill llir ti f'll uii'tliml h« )ii\'i!ii I'ur till' lull airli mil llii' siiiii'iiial anli will bi' llu' saiin'. Till' ^I'livil) liiii' Willi u ilisiriliiili'il liiail will bi' iiuuiiT tliv kiy tlimi ill till' iiiilnuili'il null, mill till pniiil ./ will ulsn Im' iii'iiri'l llii' iiilraijiii Willi an i'M'i"tuil. nr hIiIi u liia ibu iirrh riiij!. niHililyin'.' ibr Inriii nl'tlii' ari'b, ni ii' uiianging ibi woigbl in tbi' s|rainliils ill nrcli'i tn ki'i'p tin' i;ravity liiii' rurlliui' Iruni tlii^ key. Ifliir any |insiiiiiiis ,if tlie lint lumJ wliitli tlm airli lias lo cairy tlio gravity lini' is bmu'jbt still iicaior llii' key. tin' line /, ./will contiiiuu llln^ullt In ibu inii'uiluN iml thn Jnini nl' ruptiiri.' will lisu. In such I'Xtii'iiii' la-i's ii is pussihli' iliui till' p'liiit A' niiy iilsn risu at tin- key, I'Vi'ii with uiiyii'liliiii; abiitnu'iiis li appars bntli liiini vxpiiriuii'iita with uiiiili'Is. aiiil by I'liUipiirisnn with ilii' t'ni ui nl llic I'uiiiciilar pnlyguii, that till' rnrvi' nl' prissiiri' will rise iinniuiliiilily iinib'r any highly i.'ii till' hravii'st rnni'intriii il jnait passinu nvir thi' ai'ili ill. in nn any iitliei I'linsiilii.itinii \ ibiatinii nii till' I'.intr iiy lias mini' rtl'i'i't tnwaiils II. u biiiiiii'hi's. W'v ail' -lill ili'i'i'ihlcnt, linwi'iiT, ipiiii I'inpyriral fnrmiilui' I'nr ibu thu'kHHss «i ibc inwii Karly untlinrs siii;j.'i'sti'il in i'i|iiatii)n of a liiii'ar riinu , but tliisi liaii now bci-n r<'placi>il by llio fnruis / = C' v« or t = C -Jr I bi'iiig till' Ihickiii'ss r(i|Hiri'il, « tin' ipuii nf tlii' aiuli, anil /■ tlic railiu.^ al tilt' L'l'nwn. all in I'vel , ami Ca rnnslani. ilTl Tlif v.ilui's ninst ui'inriilly mlnpifl Inr '' uri' tlinsu i;ivi:ii by It mkiiiL' nml lliipiiit, whiuli are at- I'nllnws — itankini'. ( =: , (lli; r tiir a aiinili- arch. '"=,1117 (■ lur an uieb of a serius. Ihipuits. ((.'n iHli'iitits ri'ilni'L'il fur tuut i t -- O.illl ,» ' I'nr a full an.'h ( ~ 'J7 .3 fur .1 S''<.'nii'ntul aii'b Itiinkim'.s inlinilni'iinn nf tin' lailiii' nfi'iiivaliiii'al llm I'inwii insli'Mil nf till- s| all. is Imm'iI iipiin 11 rniiipaiisnn lii-lwi'i'ii till' ai'i-li anil an elasliu rill, but III' ili'iliii'i'- ilii' I'll 1 lUiii'iils I'rnni arliiul I'Xainpli's. Willi rt;garil In ihi' ^iratrr valin- in tbi' I'asi* nf au-hi-'s in si-rii's, bi' i-niisiiiori tliiit yii'liliii'.: is iiinii' liki'ly to laki' plan' wbiii a Inaili'il ari'li slunil^ btiwi'i'ii iwn uiilnaili'l niii'-. lliaii if il ■jinnil bi'iw II iibiilnii'nis l>upuit's fiirniiilai' aii' basi il iliii'illy mi very iinniirinis nxuinplus, a lur^'L' priipiirliiin b.iii); Hallway liriil<;i's. Ili' nli'rs m ly in ui'liial I'lin-trurti'in as jiisiifyini; ilu' Inwr I'n iHicirnt In! t'^ivi's I'nr si'^r||,(,||t,|| Hll'hrs I 1 s I Thu stalling pnint tiir all sinli linuiiilai' is thn M'lni I'liciilur arch nf diunic'li-r d . uml I'nr it a I'ninpariMin bi'twiM-n tin- I'nminlai' I'ari rt.'ailiiy bt inaili; Wi' biivi' Inr a -iii'.'li' anli : — Kankiiii' (-1121.../ Oupuit I - tfM •.'d Till' }.'ii.'ati.'r valii.' '^iveii by Uiipuit s tnrniiila is pmbably ilui: tu tbi; jiruatir pi'npiirii'iii nf Hallway anlns nn wliii'b il is basuil, ami puilmps also tn till' Inttir avirani' .-tr.ii'.'tli nf tin- stmii' iiscil in b'rani'i'. Ifwu I'nnsiili'r til.' -pun in ri'inain llir ^aini'. ami llm ari'li In rliaii'^e frniii a si'iiii-i'irt'l.' I'l .1 stiai'.'lil inb by passin.j il.rnii^li iln- inti'rmu.li^iti: 91'giiii'ntal fnrnis. wi- Knil ibat I'lipiiit wmilil 'jivn tbi- arch rin^ a iliuiininbi'il ibirkins-. Kankini' ■■ fninnla, on ibu cnntrary, wnuM uiako it I'liiitinnally imiias.', ami mi arrivin« al tbi' straiiibt arih tlii; ihic'kuuss wnulil 111' intiiiili.'. in in nlbrr wnrds a strai;.'bt arch Wuuld bi' tbi'orftically impussilili- This niay ivsiilt fnnii tbi' suppi^'itinn that till' joint lini's ail' alw.iys radial : and bis formula mi^'lit still bi' npplit'd tn thi' str.iiiibi airli by lakiiii; tlin pnini frmii wliiib tin- joints radialu as tilt' sn|i|insi'd I'fiitri', An inti'rnii-iliuli' fnrinula has bri-n priipn.si'd by Trautwint' in ibo lust I'diiimi nt bis I'oiki I bnok : — I = O.ilu vr 'is i- 0.20 (for fuut) 8 Kvr n aiiiiil'Oltrulnr nrch tlii* i> |iri«i'iiciilly llii' -hum' ii« KiiiiliiiK'''*. Willi lllf lllfliliiu I'lli,' l'l>ll l« lli«, lilll w'ilr. ,il II llllilllli' I'liikniii tci lllli >l iijlii iri'li ■rinri> i. ..llll 11.. ...I nrnul'.i III III lv-.|K.lllli; llll' r.ltliiM ll llil'ls lill MII'll Inriiiiilii', ii< llll V iJiHir ill |Miiii'i|i|i'. Tin y llll "IiIn In' ■:ili»l'iii'liirll) i'ii|ii|uiri'il by i|j\i>liiiK llll' i'ii,iiii|ili 4 nil ivIiii'litlii'V iil'i'lio 'I iiilu r|iii>»i'>, ill ui'u'iiriliiiii I' willi llii'lr |iiii|il,lllllll\ nl'lllr illilll Ill- Il i< null •^.ll'y I'l I'lillllllll ' In llll' rnlllnllllinll* Il IIIVi" nl |iri'--iir Hi"i|i.it|i|lll'J In llll' MiriiiiK fii>i.i i'.>ii-iili'i|.il .\.< HI' IhiH' iilri'inK Irnl ..i'i'Im .|i li lii'i', llic uliiiliiii'ii|.< iii|iiiri' ^nuiir -Inimili in ri'-i-l MihliiiL' lliui in Miiinl kIIiI' U »li>llll liinlllill ll:l< lilltiii |.|,||.,. , I uiiv' tin' |iii|hiiiiI< |.i II '.■ri'iili'r lIlllHl nilll uri'lllrr |.h.r.-III..H llilniulinlll llir ll'lli'lllh' Illilll ll il li>«ii In A llirniii'li llll. ii|ii'iiin..> III jiiilil. ill |li|. iiii'li MM... \\ llll .y iiil|irlrir:ll li'jili till' i:iiuli'.-l ii.«i.iiiiii... nil! |i,' iii|iiiriil hImh iIh' mli h rniii |4'lcl) Iniiili'il Jill iilllini|..:li llll' i-l|,.|.| III' |uirliil InirliiiL' iii:i) |in.«ilily bi' (iii'UliT. lliH i« .iliiii.si iii\iiiiiilily ltd III ilir iii.ir.'iii iil'-ali'ly U'llii- I'lirvi' nf |irri"iiii' slriki"" nui-nli' nl' iIh' mlilill.' lliinj nl llic bini', lliirc i» a IMirtiiiii III' till' iiliiiliiii'iil nliiiM' wi'!^lii -iliniiiil mil In' i ikcii iiiln iii'cniiiit. (Kii!. 7 "i 'I'll.' I'll'i'i'lur wiilili 'I I'li'li i'iiiir-|. \t ||||. ly iliri'i' tiiiii'K llll' ili>l mil' rrniii llll' i'iii\i' III llll' liiii'k nl llll' iilniniii'iii , mill llic I'lirvi' III |iri'iMiri' ^li.iilil Im ilriimi iiuniii, niiiilliiii; llii'»i'i;jlit nl'llii' ri'iimliiili'r. 'I'lii- |ii'i.i'uii|inii ii r<|ii'i'iiilly iiiii"<-iry ill llir ru'ii' nl' mifiuriitul iirclnH. Win ii nil lln' cii-i'i nl' liiiii|iii)j :ir iii>iii|i ml. llii< iibiitini'iit nill Ik' iiiiiliiiiliti'ijiy >tiili|i' wlii'ii ilm ie-iiltiii'_> iiirvii i^ silffit'ii'iilly »itliiii llll' l.,i-i' In |iri Villi i'Si'i'»-ivi' |iri'»-iiri'. I'l-nviili'd liin rminiliilinni urn lluiinii.jlily mhiihI. _ Il will lir ijiiili' in i'i'>-iiry In linllbli' llll' 'llilll>t I'lnlM llll' iili'li :|. n rii'iilin'll'li i| liy -nlllr ullllliiM. Ullll'U illlll'I'il lllis i^ iImIii' III ulJiiH lill |iii.il|ii||, nl' ll||. In III wlliill ;ir, linl cnlliiili II ll. Wlli'll Hli'lli'" liri' lillill Ml II ~i'l'ii'i, sniili'wihil iliir.'l'i'lll I'nliillliiiM^ ;in-i' Wlll'll llll' W|. infill nllly III III" >l nil'.. i> In III' :;lki'll illln .'II mil. lln' lil'i'in-ili'i' nil llll' (jii'i'.. i.1 vcriiral iiiil llll' II'. 1-1 II. 11' I.I i'rii>liin.: I- til.' nllly I'niHiili'i'iltiiiii. j.'iir uri'lii'i I'arnin;; .i iiiniii:; Iniiil. ii |iii'i- i» lli.. iliiiKt iliiliivnriilily >iliiiilt'il nlit'ti il in lii'tuii'ii u |..aili'il iiliilaii iitilna.lnl arvli. Till' ri'ai'tii III ul lln' jniiii> nrMi|iliiii' in lln' inn iiirli.n i. -lu'i' lively wln'ii ( imiiiiili'il witli llll' wi'i(;lii nl' Mil' |iii r. (ji\i' ilii' lliial leKillUiit. wliii'li ill u jiiiT !..iiri' |.ai~iiii; llii-iii.li H and <'. anil iIiitcI'ih'i' aNn In a hri'i'ly incu'ii-.-.l iliiii-l .Sii. I, i thrust ■ liilly '.'iviii in wnrk« .111 UrajjIiii'alSlitie.'-, In applying llie iirapliii-al nii'thn.l. a ili.ufficicot to koup the curve within the aroU rint:; hihI this ia u very inoderuto nuiount ooniparod with the limiting iiht;kMirrrii'ti<>n. The i uivo eorroMpomlin^; to suoh an ineliutHi thru!it is rniniit to ri'ie iintliT the Inml, us would he expected. The joint <»!' iiiptun- tfUiN III rail on the loaded side and to ri:}0 on the unloaded side , itiid ir the aieli is su coustrnctcd that the joint of rupture cannot take tt lower po'.ition, tlie value of the thrust ibr any ussuniea inelinv tioii e.iii Ih> round ul'ter a tew trials by e4|Uatiii^ the niouients on the I'lailnl >i(|i> only. So tur nn the writer has been able to diterniine, thti ellVern of Mich partial loading uiay be aelually -greater than with a runipli te load, e>p(eially as regards the ahntments. Althou|;li we may int'i-rtlie piieiai ipiiiiou to he that the results wouhl not difler largely ti'om thoM* idready obtained tor complete ht:ldin^^ the elVeets produced lihould not Ih- left to the margin of safety in the structure; and this part of the ^ubjcel therefore reqinres further investi^iition. RK^KHKNCES. (I.) t'tirrr uf' IVrssnir. lis proptM'tics wen* first ili-'ftisHciJ l.y Mery in K^27, i)iou;>l) lii*> ut>ik n'iiiiiiiii-il'vaiiou-< ineilioiU (2.) CulniaiHi- " l>ii* >;rapliHi'lie Statik." l.fvy. ** l.:i Sirtliiim- j:iuplitt(Uf," Pari-*, h.7-1. (it.) JiiisKtt h'tihUv iirrfi nr Jithi- ellitJ.ti' pro|HiM-i iln'-'e is iriven hy Mnranilii'ir, '''fmiti' ih- la C'ln-trin.'' lion ties INini.-. vl Vtatlue^,'" i-liap. .'!, pap'H It»8 tn HI ; i'li-in, 1h;4 h'l.r llie MietliiHls priiiHif'iil liv Mii'lial ainl Perrohcl, Hfcal-n Cluu.U'l, " AiiK'-iiitMiioir-i' th'^ IiipMiUMiM," An. H.')4 tMli lvIilio!i ; Pari-. 1^77. (I.) Kffdtisioii rf'^ttnnv fty /iti}t. Mehtidiifd l.y Stoni'v as iH't.urrinL' in tiittsniiry iiiclics; ''Theory uf Strain-* in (Jinlers' tke..('li. li*. Art 41-4 *.>b.-t'rvi'il in an experimental uruli tirl4'< feet -^paii liesi^ned \>y Kuinany tor (lie PonI tin l.tuvn-. Pan-; " Annult-' .li-" Punt^* el Chaii-"*'?!'"*," Hi;t>, 'ic ^enl^'^ln■, piij.M 10. .\Im' fitftl in MttVainh'Tr'- '• I' rain-," paj^e 2\1. lit.) Thfoii, itj Hlii.^tu-Htj uyjilifl to Mdstnuy- W.-II .-xplained \,\ Cnlluitiun, " Ht'-i-tioiif .li'- MaU'riatix," Art-J. 'Mi Ui tili. unl Art.'lii- UituiT." "in'l K.lilu.ii, pii^'f 4m2. I.im-lt'n, \Hji,. (T.) i,iinifiii;/ iiositiniis Iff' ffif ('nri't' of' Hie-imtrt' Si-h pa^e** "^i'l iimi 2H in " A piaeiieiil 'flii-ury n\' VouM.«oir Arches," liy Prof- Wni Cam ; Van Noh- intmlr' iM'ienei' Scrii'-. N<74. i'l'ilivil'lr i't' /.ftisf /tV.^■(^■^/H(f. (H.) Oriv'iiinlly pnl.li>lif.| l.y Mor)('|(>y in the *' Pliilif*it|tloial Mrtiia^on.* " for Ortnlu'r, ls;!'l. Si-i- iil-o lii- " Kni.'ine.'rin>r anil Anliiierinre, ' Art liS'i I'lu priority i-iaoii»Ml toi i'tiiil.nnl. caimot !«• t-!t»l)li-*lo'.l . \l' > " Tlo'orii 'l«r comparoil with experimenl, " Vou-i-oir Arche-^,'" l.y Prot. Wni. Cain; Van Nostran-iV Science Seric* No 42, fsT^. (U » Piipuit's Thi'nry. Pulili^heil originally in ilie ' Annale-* dc-' Ponis tt Clutn^'-t'CH, ' iHfiH. Iti a HepaiTte ti»rni a-* " f rail'.' Av IV'i^nilibre lif* Vtiuli-n;" iexl and Plales. Paris. Im70. (12.) Sniioiiary ul l>iipnit% work >!i\('n m Ciandi.-I. '' Aidc-ni-iinuirc de.-j In^C'iiifiuH," -titii'ii-* 870 to'.tlU- yili Kditiun. Thfo}}/ nj the MiiMIe T/nnl {\:\.i tiankitic, "Civil Kiijriiieering ;" An- 12:1 lu 141, and 270 to 2y>j. (11.) HankiiK-s clinptt-rM on tlie Arch antplilied and explaiiu'd l.y Prut Will- Allan. '* Tiiiurv uf Arclii.'f*,"'" Van Nostrand's Sijicnce Spri';-; No. 11, lt<74, (!;'),) '• Kucyidopii'tjiu Bntannica.' iiiiith Kdilion, 1"*7U Article*' Ijndgy^," ty Prof Flceiiiini/ .Ifidviri. (hi ) See foot nole lu Art. IV.*- in '• Orupiiical ij'atii;s"' by Piof. Uubuiii Ntw York. l>*-:t. Tfiuk 'I t'Sfi iij tlu. aruh rin>j ut the key. (17 ) riu-eqiiatiou ofiheforiu /— CV^va-* lir^t propo«'rd l.y J I Hurst, •■ UnildniL' Niu-," Ffl.. 27, lyu7 ; tlioujfh llio i-orri.-'iptjndinjf tbrni was u.liipted iiidopi>iid)nll\ >.«i tlo I'niilinent ^1^ ) Knr the dridjie- on wliu.li tlic-e lurniulti.' un- l.-a-ifd. -ee Kankiin's '•Civil KnLnnt-PMnu." An 290; and Dnpnii'^ " Tiaite." Cliapicr 7 and Plate 0. (19) 1jntf}hi'nl m<-thny Pn.f Clarke, "Oraplm- Staiu'-,'* Art. ■'»2. London, l'!7t> (20) Diitereiirc lit-tuei-n curve uf pre^:•luv and line of resiiilance lltii4 irutcil I'y IlnU'i?- under llie names " prcs-ure liiic" and " S'lpport line" lu lii-"Oiapliii'al Sirtti.'^' Arl- 1 7G ami Kig- 102- (21.) i'lisi/nimftiiritl linviintf. See Ci-'lliisnou-: ireiieral tlaMrrni, " Ii