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This Proof Is sent to you (or discussion only, and on the express understanding that it Is not to be used for any other purpose whatsjevor.--(''>'''' *''■• w nr Hh- (;fiatiiiiiin),.\ OTanaoiau ^orifty of (.''ivil (r'ni^inccrs. IN.:()l!l'il|!ATK.I> IHST. N.B. -Tliif* Snciftty, nH fi body, <1nrrt nnl linNl itself it'spimsilil.' f(W' Ihi' f;icts 4nit opinion'' slatc'il in any if iis |inl>lit'iiii"ns. ,. . Til h,' ,;,„l Tl,,ii: nieiitioiied in parlicidar. What is now wanted even nmie limn I'xperimenls. is a ihemy of llie subject which will enable us to interpiel their iisiilts in a iiiiieial way, and also to indicate the kind of experiments desirable. We .^hall there- fore attempt here to liive a trealmeiil of thi' ^ulljeel som.whal more adcqiiati' tliiin is usually liiunil in text books. A column diff.'rs from a beam or "inlei- in the iiatiiie of the foici's it is intended to resist, a lieam beim; used to resist forces transverse to its leni;lh. a column lo resist forces directed between its two ends. 01' course, the same mati'rial may be used to resist forces cvf both kinils at on(je. but for tin' sake of clearness we shall ni'iih'et tr.msverse forces ultoiicthcr in treating of the column. The Iradini; feature in the behaviour of a loadi'd column is that it bends, the deflection constantly increasini; with the load. Tiie stress which mi.'';t at lirst be .ihnost uniformly distributed in the I'olumn. becomes thus more and more uiie((nal at the luiter and inner edi;es a-^ the load inciease.s. With a tensile stre.s.s on the other hand. allhoiio|i a small flexure undoubtedly must result friun the same causes that jiive rise to flexuie ol' columns, the direction of the flexure is opposite, and thei'efore tends to produce! a more uniform distribution of .stress. The ilifrerenee between the two cases ujay lie illustrated by that between stable ami unstable e(piili- bri\im. The ei|uatioii which forms the basis of all iuvestifiations into the Uexure id' beams and coliimns is (1) where .M moment of external forces at a cross section of the beam or column, whose moment of inertia about an axis throusih its centre of jiravity at riiiht an'.:les to the |ilane of flexure is 1 ; H = radius of curvature of axis of beam or column at .same cross section . and li - = initial value of R when M — K = modidus of elasticity of material . By the axis of a column we here and hereafter mean the axis of fimire and not the neutral axis (wiiich for columns i;cnerally lies at a considerable distance luit^ide the column). It will be well hero to call attention to the assinn]itiiins on which (1 ) rests, because it has often been ajjplied to circumstances in which it tThis investiaalion coincides in certain parts with papers by Professors Ayrton ami I'errv, in the A'h(//«(T/- (Loiiiloii), Deceniber Kllli anil 'Jltli, 1881), and bv I'rolessor Krobn in the I'm.- Am. Sur. (".K.. ISST. 'This ei|uation is usually jiivcn as M K.r I! tliii-' a-~niiiiini (lie initial t'orin of the axis to be n slrai):ht line. For a b( Itin iifrh the assump- tion cannot be trtie in fact, it does >iot atlect the distril)Ulioii of stress in the beam. With a oolumn, however, it is of vital importanc shall see. is not true, ami iiiohI niislriulinii poncluKiims liiivi' bcoii dniwii iu (•0lisi'<|lH'li!'i'. Ki(Uiit.iiii ( 1 j MiplMisi's : Fii>t. that ivriy [ilaii,' s 'i-ticiii nf the biaiii or ciiliinin noruial tn lliu axis in lln' nnslraiii.il state remains jihine anil normal to the axis throughout. 'I'his is prolialily a|)|)rnximately true tor lionio^jeiuous beams or eolunins I'ormeil out of i>ii>' mwM of uniform material, as. for iu.stant'e. a Iol; of timl)er, a cast iron lolumn. or a rolleil joi.st, hut it is very douhlfnl how far it can lie relied on for a framework Imill n|i ofa numlicr of .separate pieces. Si'ciindly, that the value of K is constant tlnoU;;liout the cross sec- tion. This is approximately true iu most ca. sis. anil it can he shown that the variations of K oecurrin:: iu praelic latuiot larnely aft'.ct the strcnjith of iron columns,, though sonic writers have attaclml an exaa- geratJtl iniportauce to this point. Thirdly, that the fihres of the heam or column are all strained iu accordance with llooke's Law. As soon as the most slraiuod ti'irci pass the limit of elasiicily, ei|Uation (I) ceases to express the condition of ei|i\ilibriuni. and all conclusions bused ou it eea.sv^ to have an appli- cation. When therefore, as is sometimes done, a comparison is madj between the stress in the extreme libres of a beam as deduced from ciiuatiou ( I ), when .M is the moment cau.siun' fracture, and the direct tensile stress the .same fibres will bear in a testiii;; machine, it should cause no surprise that there is an enormous discrepancy. Kijuatiun ( 1 ) is ba.sed upon a snp|iosetl distribntioii of stress throuiihout the various parts of the cross section which is utterly wide of the truth when the moment of Hcxure is such as to cause rupture.* Ki|uatiou(l) lieini; llic equation of moments at any .sectuiii of a column. We have aUo an cipiation between the sum of the applied forces jiarallel to the a.xis and the sum of the internal stresses at the same cross .section wbich will be found tube: (-') P ■■- KVi (^ |{— j[ ,) "'•'^'"'^'^ Kv '1^'/, —p y, where/) is the total apjilied tijrce per unit of area of cross section, and V, i.s the distance of the neutral axis from the axis of liyinri- at the same section. In order to determine .\1 we must know the form of the bent column under a i;iven load, and in order to ascertain thai, eiiiiation 1. 1 ) irusi be solvi'il. Now the essential ditb'rence between a beam and a column us reuards this solution is. that ior a beam .M is independent of the deflection, while for ii column it depemis dircclly on the detleition. Iu other words, if ,i and // b,' reet;uii;nlar co-ordinates detinin;; the position ol any point nn (he axis of the column, .i beiui; measured parallel lo the leiii;th of the coliiimi. .M is for beams a fuiiilioii of .(only, while for columns it is a fnnetiou ofy. This .iitilely allev~ the ehiiraeter of the ditlerential ei|UU- tion to be .solved, and makes it i|uite inadmissible to assume, as is tVeipienlly done, that the deflections of a column will be iioverned by the same la'v is are deduced from ci|ualion (1 j for beams. Let AQB be the axis of a coluuin bendiuy under the action ofa load I*. Take Al> as axis of ,i , its middle point ,( ) ar- oriLrin. and let AH =i: /. Since equilibrium of the column as a whole reiniires that the forces act iu)^ on its two cruls should be reducible to two eipnil ami opposite forces 1', let the line of action of these forces be at a distance « from (t and we shall siiii|Mise it |iarallel lo A15. because it is found that the ]irincipal cH'ect in producing- flexure is due to the mean value of the eccentricities of the loads at the two ends. Of course « is small since the intention is suii|iosed to be to load the column centrally so far as jios- sible. Let (( be the cross section of the column at Q. and :P. Let (/,, be the initial value of ^f, for the unstrained column. —- Y let pa We shall suiiposc .(„ = .('. tliereby nciileetin'; the chi.stic shorteninir of the column. Let ,, -^ radius of ;.'yration at Q.-^ y I: then since ^ —""y (I us Ion- as U is very lar-e and similarly _ ^: ' •'(" pa (y + n). (1) becomes R dx- and since M = * For a full discussion of the iineslioii "'' the relation of Iran.svorse to direct sirenxth of iron, v. Kludes siir PKinpioi du ter et de I'ucier, by Mr. Considerc, I'uris, iHSIi. 8 writ/' fur tin-vity K' for '' , nml siipimsf K to )ir cmistaiif tliMii-jlumt h ■ " tlic Icliutli 111' till' niliiiiiii. /'<.. ^.ii|i|iH-r ill.' c'liliiiiiii t(i |p' iiiiil'iii III ill si'i'- tiiiii mill ill iiiuti'i'ial. fill 11' II 11 1/ , I- . , llirli .-■'- - ;''; =- - K- (//!„) if. I'- fl.i~ il'jl il'ii,, ,lr' -/,.-■ 'I'll -^(livr this ii|iiMtiiiii Wf iiiiHi kiiinv wliiK ' ■''" is ill t/riiis mI'.i- ny in il.i iitlui' Mdiils. wliiit llio iiiisliiiiiiiMl riiriii 1,1' (li,. .,x|^ j^ I, |„,|||^ ^.|,., JPdsnl lliiit the I'lijllliill is iliiliiiliv a- ■^..iviiht as tl •iliiiun niiiilil inii-i iiriiiaiiularliii-i' |ii iiiiit. till' actual lurm nl' tlir a \:- wil' lie an irir'jiiliu' ciiiv.' il,'|iailili- viiv littif li the line Ali, \\\. jijiajl i,sMiiii,. that tiK' cili'w is thiTUrvf i,fsii„.> ,y„ = ,„ <•„, '^''-- This ass,„„,,ti,i„ c-a„ bo justified by tlic laft that the I'liivcwhativi • ils fi)nii, can be ox|)i'.'ssi'J byuquatinii ^„ to an iiilinitr s.iiis pidceeiiini; hv sines ami ciisiiies of multiples ol' ' 1)1' wlliell •>,, L'lis -j Ilia V he eiiMsiileieil the lirst teiin. It' this be (lime, it will be fiuiml that in iiiiliuaiv eases the ert'ect nl'tlin successive terms of the sorii'sim the resiiltinu solution nipiilly decreiises. We have then ' /-■', uiul ,/.,-■ il,' - K' ( = -'*„ 'ji «o« I ,'/ ^ ") - >\, j: ws The solution of this eijUiitioii with the coiiilitioiis that ii-v when / / . X - — 01' - .- is: |{( I'ore proeeeilint; to obtain I'loiii tlii einiation the stresses in the cohiiiiii, it will be well briefly to point out the relation it bears to the theory of flexure uiven in most text books, ami known by the name of Killer's theory Iroin the name ol' its author, who iiiiblislied il in the middle of last eenlury. In that theory ,i ami " are iieeleeted. so that the differential eijiiatimi is •' = — K .'/. 'I'he direet solution iifthis is// - A ens /:.r l!sin/.M'. ami the terminal conditions reipiire kl — ami I! 11, while A is indelerminate. In other words /i — "■ K ( )•. :i"d /I liavine that value ,i/ may be aiiytliiirj. The same result ma \ be deilneid I'iniii (.'! ). for if „ and .'., vanish, // niilsl always be zero e^eeiil when K/ --:- -, and in that ease /:/ is all imleteriiii- l I'OS •> mite fraction for which an arbitrary constant A may be snbstituled. This theorv i- nut only ineniisi^tenl with l',u 1. , which iniLdit be exiiccted friiiii its assiiiiiiii',; ideal comlilions that cannot be realized u iiraetice, but in the fiirni uivcn above, which is that usually published, it evidentlv contain- -oinc other serious error ; for while it is i|uite conceivable that iimlcr the ideal conditions supposed, no flexure can occur until /I reaelii s a certain limit, it is not conceivable that the Hexui-e. uh. II il do ■- ncriir. >iioiild b, iiiil. p< nil. Ill of the load, so that the same lo-,d which first eausi - llexiire will bend the column double. The error lies in lle',;leelil|e the elastic -llMltmilli; of t lie coliimii. Ill the theory here uiveii this is peiini>silile. but in Killer's theory. <: and •',. beiiiLi neglected, a ditt'ereiit order of small i|iiiiiitities is dealt with iimi the contraction of the eoluniii lieeoines important. If this be taken into account, and /> i- tiie load for which flexure lii-t beeomes po.ssible. or -■ E ( '' ) . then the deflection .i^ for a eiveii load y; per .sij. in. urcaU'r tliiiu y),, i^ .ii'vcii by '' W /' ' It must also be observed that althoii'di by Kider's theory flexure can- not take place until /I rciclies the value /-,. it does not follow that it will tlieii occur. For every value of /i ..ileal er than /«,, there will be two positions of ei|iiilibriiiin for the cobiiim. one straii;lit and one curved. The forni.r will be iiiistalile, and the elioiil of the arc of the bent column will be shorter than tli..' straight coliiiun under the .same load by uu uuiuunt w ■ III. Til.' diir.r.iir.' ill ivsiiltK lirfw.di Kulci'x tlicorv ami .qimtioi. C.\)\» Hdikiii^ilv >li<'Wi> I'.V oU'iviiiutluit il.'ithiT .. ..r.\, Imsiinvtiiiitr viiluo. I.i.w.vo'i' siimll. >i|UalMiii (It) slmw^ tliiil // is iiiHiiit.' Ini- tin' viiluu K/ -. wlinv Kul'l's tlnHiv til>t ilildWS tlixinvtn liri'i.llM' |K.H«il)li'. It ^1„,„|,1 „1m, 1 1.S.IV..I that U'l- .pidinary i.rujK.i-ti.piis III' I'oliiiiiii.- /., ,i|i|.r.m.lii>s tlir liinil «\' |.la^li.■ily "I' tli.- loliiinii. s«i tliiit at niaiiiiiry wnikiii- liMiU no II.Mir.' \M>iiltl o\vr iicuui- if KiiUt's tluM.ry iippliL-d. Koi- iiii^laiKr, wilii wiMii-lii inni liaviiii; an .lastic limit <>\' lli ton-, llic Iniatli ola ...liiinii W'nil.l liav tn !»• moiv ilian KMI tiln.,^ ils radius nf ^ynifii.n ill Midn tliat aii> llrMiiv sinmlil ii.riii' with a lini.l of 1-' Kms JUT w|. ill., an.l MH' lini.s u< Ihh.I with a iuaili.rti tuns |»T r..|. in. \Vi! Iiavi' iriv.'ii >.> iniuli >liaiv U< Kiii.rH lli/oiy unly l.imnsf it lia.s Ij.'oii h.. wid.'ly Mpivad in tr.alis.'s on I'liiiiin'tiim; 'r, i! In- coniv's m'.vs...aiy to distiiifiuisli bi'lwi'«'ii tlio viirioiiH ways in wliicli th« .•iidsol' tlir loliiniii may !»■ .■oiiMraini'd siiico tlic valm- of" dcpinids uiioti the iiatiii-i' ol til.' roiir-liaiiit. i. Til.' iniU may n'>t in liiiijjox in wliitdi tin- iViftion is iiiNufliuiout to (iivviiit idlalion. In this u is indupfiidimt ol'y*. ii. The iikIs may hi' riiziilly attacht'd to HximI ohjouts, in which oast- „ i^ a (hti'iiiiiiial.' rnnctioii ol' /». ( Klat-i'iidcil columns art; anothiii' vai'iciv which coin-sj.onds mechanically ni'illici' with i or ii, and prc- M'lils '^I'cal .Icimnts of iim-irlaiiity. As it docs not represent any column used in pracliee wc shall not consider it here. The same remaiks apply to roiind-eiided colnmns). riic ends may he riiiidly attached to .dijects which themselves in positi.ni as the load varies. This is ii very common case and is that of the .striit> of a riveltcd bridfic Imt we shall not deal with it, as it woiilil re.(iiii'c a Vi'i-y len;itliy invcstiiiatioii. We only call utt.'ii- tion to the I'act that the condition- vary IVoin tho.sc of ease ii, and that ii rivell.'d hridi;i' sinit has in conseiiueiice less strength than a similar strut whos. ends an- really Hxed. Not only does it cndiirc Hccondary stresses arising from the he|ldin^ moments ajiplied to its ends by the booms of the brid-c. hut its axis is thereby thiowii out of line, by which its strenuth is impaired for heariiiL' the dir.et load. We can refer students who wish to pursue the subject (which is one of i;rcat coin- ].lexily 1 to Dr. Winkler's treatise. • Theoric der bruckcn," Vienna. 18H(|. Consideriici ease i, we must lii'st remark that a hinged strut diffeis in no way from a lixed-end strut so hnn; as the friction of the hinues is below its limit in- value, or if tliecii'i'iimslances ar.' >ueh that the Hexurc takes place ill some othir liliin,' than that noiinal to the axes of the hiic'cs. I'resiimiii'j that rotation of the liinues ;ictiially occurs, let /t be the radius of the hiieje. >' , the di.stance of till' centre of the hinue from the axis of ilie eohiniii. ,. the aiiiilc of trictioii ; then II '-II Ii "I" "■ where the si.jii of A sill o di'|ieiids on the values (d'ually lie much Lireater than »,,. In eiiiiatiou i:\) it is p.Tmi-sibl.' to expand in powers of /,■/ and iieulect the hi'jher powers becaiiM' /.■/ y ?'_ and must always be small for .-iich viilues of/, as Would i stiliite a sale working; load. This iiiav I'.' veriticd by calculatiiiii the values of// direct from ci|ua- tioii {.'•'•) for certain cases, and c parinir them with the vahius given by the approximation. If this expansion be made, wi' obtain (4) //^" " " (1- -7-) <•■!« ''"'C t- '„ ^li — — ^ cns-y ./" '\i\ when .r^o.y^ii • ''" '' (^>< •■Jj) HO that the central deflection is '^^^ ) ('^ t- '_", ) Let ;■ he the distance of the innermost or most strained fibres of the column from the axis and / the stress )ie;- sip in. in them, then, so long lis llook<' s Law holds, the moment ol' resistance of the column may be exiiressed by "-- {/'—/>) which must therefore bo c((ual to ay (^ -(-n). Therefon ,. ' ( /-—J') ^ J> (,'/ + ")• /■ /■ is greatest where // is liieatest. /.( . at the middle of the column. ~ ' " ■<• '* , Substitute from (4) lor y . ,i and write b lor - — and c f.jr ^(^,^). Wehavcthen';: (,/^)=y { ^-^ ^ "■ !( r) ^ } Th«< reason for introdueinjr r into tho vhIuph of h nn.l <• w tlint .. nnd . , iiiv lik.lv t(. vuiv loimlilv with th.' si/.r C.I tl. .Ii.lni.. W.' Imv,. WTi. Il.at 111.' priniiiiiil lunt nf " ^r.^iicnillv will ii'i>"' •'""' *''•''•'■'"" "'"' bn ,,n.|H.rti..nul to tlf riulius ..t thf l.inj.v. u.mI .',. i^ lik.'ly ti. iiktoum- wi.litli.. loM-th ..!■ tl,...M.luiim»ilh r.>|«.'l I., wliioli tlu' .liiun.!.'!- is K.'Morull.v tix.'.l. Al liny nil.' '*^: "ili i""'<' i"i"l>' '"' iii'l'l"'"''"' "'' the ililni'nHii.ii> ..liiny piiiliciiliir ...Uiinii llmii ..„ Miii|.ly. It.'i.nmi.Ji.u III. I, Tins.,/' /' 1 1 ( ' ) ('' ' '■'' (- ) ) ( Tliin is a .|iiailratif (.. ..I.lain /. in I. nn- ..f,/'. I'lit A an. I < l..•i"^; -*liiiill. Wc may, a,-, a tiisl a|.|pn.xiniali.iii. writ.' / \\>r f in tli.- last t.'ini m-mlc the braek.l. tin- »'.(.iuiiiilt tlif nniue assuiMiiti.iii as lielori' with ifi;a>il to th.' initial lorni oC the axis bo uiaJe, tht'.Me onilitioiiM .^ivc Id hi orexpandiniiin powtTiof/iVasbufon'"-- — " '• — '',/.''/- -".--,- Tliuuiaximum biiulin.j: nionicnlis in this caso at thcfiid of tht) coluimi and oi|nal to /) ./. Pi.ici'i'.lini; as hi't'on !.' .'.iiiat.' y. " t.i '' (/ — p) wiMibtain a i-.'snit ot'i'Xactiv the sanu^tiiriii as (5), Imt in this case A - 2 .»„ - — or nearly '^ — and 12— T early ,L I'lie 1 •> - central delleetion will 1)l I'liund to he " _ - '>„ /.•' /- or nearly --ir(T;) It iiiiLihl, liowi'Ver. iierha]is he a fairer assuniiilioii for a fix.'d-end column to sniiji.ise tin: initial lorni of the axis to h.' tli.' r.'versed curve)/, — (I I COS " ' I, haviie^ tin' sain.' e.'iitral or.linat.' .V, hut taii"ent t.i ill.' line Al! at holh cn.ls. On this assuiuptioii ./will hu , <>" /-, /,■■• l'\ found t.i hi' eiiiial to -j; M I i , :; ) and the ec|Uation to the curve of tli.' axis will he /.■■-' I' v:!.^ jilvini: a e.'ntral d.tlietion due to tli.' L.a.l of '>„ V_^^-or nearly ^^ ^^ Till' relati.in hetweeii/' and /. is of th.' form of (5) as before, only that now /- h— and f _1_ - -- nearly ,(,— The followiiiLr eonelusions maybe drawn from the investigation which lias r.'snlted in this formula. First, the aetiial strenuth of any eolnmii (l.'|i.'nils partly on known facts with veuar.l to its .lini.'ii^ions. nialerial. etc.. ami partly .m acci- dental eirenmstanees which can iirither h.' preiliete.l n.ir .ihs.'rved. This is trill' ill some m.asiiri' of all constructions. Imt with eoluuiiis these aeeid.'iital eiieninstaiiees do n.it m.'fely require corrections in the calculations ol'stri'iijith that would otherwi.se In: rue, hut are ^■overninjj; factors in the calculations them.s.lv.s. It toll.iws tluii that any formula suidi as (5). siiplHisiiij; th.' constants di't.'riuin.'d. slmuld express, not the stren,!:th whieli may h.' expected from a ))artienlar column ..f .'.'rtain .limensions aii.l mat. 'rial, hut the miiihinnii strength which ex|H'rimi'nts .show may fairly lie expi'cted with ordinary care and .skill in manufacture. Tli(^ strenirth ot a iiiimber .if c.ilnmns apparently alike and loaded alik.' may lu' exjHcte.l to vary widely without apparent cause, and a formiilii for |iraitieal use should express the minimum and not the av.'ra,ue of the results given hy a vcy wide and numerous range of experiments. Srmn.ll,,. , x,,. tM.M„t- |„. nippli,,^ „,. ,l,.„n„.»iori of p,,! num. '' ' '" '-M-'f'l I" «i>.M.il,.r.m r..iiliHw|„„ „,,,,|iHl t.. thf dwer- ' """" "' 'I''' '"'i'i»"t« in urij li.iii.tilH -iich ,,« (5i. Th.. n-a^nm »..rH,l« has,. I,.,,, f,ill.v i.„liml .lr..a,|y. Tl „s„„.t. .,„• o„ly piM|,..rK I,.. ,|,.„.niiinr.| \,y ,.x|,..iii„ri,i, wilhiii limits |i,r «lii,.|, H,„ ;"'"""' '^ I'''l''''.v.'la.n. I ,.nt;,,, ,„|v v..,v r.w ,.x,...nm,.„U l.uv.. hrii ii.a.1.. uli|„.t k„„l, (»,„. „„.||,.„| „r ,l..|,nniMiMt. ll.« '""""""••' """I'' !'■ I" '"••H-'iiv tJM. variaii f ,|,.H,.rii,M. f„r >:iN.u MKT,„i.hl,„| |,„„l. If,, I.,, ,1k. i„,,v„«. „r .|,.(l,.,-ti„M Cm- ,i„ ilHT.MHi. j,r lua.l /,, «,■ luu,. t;,i- a liinur.l ,„|,„„m .r ^ (- )" ami /, iiraiK He, ' ' ' I'Vir a li.\i(l-i mi cd tlio tiiur) ■.i.'i o..iuu.„ 0... .1... .^.con.Jl,yi«.lia.MiH,„|.i,.|, i. ,„.o|,ahly i',T j^; (^-ly a,„i/, ..((,,. 7"'"":";-'"" ^"■""-'""-.."-";^.,n„l,.Kl „lH. a,,,,|,..,l. |„ Mr. JWearon's .xpon „,s ,|>,,. up .11 .1. I „l„r.,.„a„.|y .l,. .Ma,„n,y .,t ,..v,H.nn„.,„. a.v n,:.U: .„. ,|,,- ;iii"!r""" "'"'""" ■" ""'"''"• • I"—" i-n- .Wunlin, ,,, ,lu. .x,K.riuu.nt. ,m.„ti .d ahnvv, i, vv.,,.1,1 ...... sai;- '" ''";■':, ': '"'■ *"'^' '"'"'""•^ "• ■''■>• "'"' '- »• ■+■' n nicn «■,. tak.. k to Ik. I ...,()0 .„„> ami ./ ,ii„M. „l daMi.ity i„ diivct str.,.N, 15 l„„s wind, „„.,..s „.,„l,l apply , i|d M....1, ami if. r„r,lu.r, wc allow th. "or .... load .o h.. hall th. ..lastio load, we 11 |„v.. „. the Wo.k II. lh«. p.T «,. in. „t a „,ii,i ,,,,,1 |,i„„,,j ,,„|,„,^,^ •? (tij /,- lip.sdd Ki.doo V , . ) "(;)-;■■■_' I'or hxrd-rod culnmiis i. would >,...„, that ,; mij,l I,,, pm at (K.;-. ""^'/; ''! ■•'•;""' """'""' ' "■^"•'-'-' n,i,.mn in lh>. p,.,. „, i,,' would then he ' ' (7 ) I, = lli.8(M) 1^- (-)■■]■•■ ' (i\'i \ ■ / (. IJII.IMIII \ ) ) ILese delen.MManoMs eanm,,. however. I„. n r„„nm.mle.l with any conhdenee o„ M...h a narrow ha^i.s of exp,.,- .„. They would .ive :"^'"'; ^'"'■'^'"'^ ' '^ ''"■ 1""^ -I"""- n la..vel^ ,„ s , ;,nes than' the '"'■'""'"••■ •'''''- """ i" ">- ■I'hei;,,., tl.u „u.y di.sa;:ree with '"■'■"■:" l:''^';"'" "" '' • '-■^-■'l - - .ondenon.! ,■ heeau.e present P™|'"''M."l..>or,he Mren;:,., „r a eohunn tor „rdinarv workinj, h,ads -My l,y „> nl.in,:,,,. Mreo^.h. whieh we regard as a laise jaineiple hut -tt e.un,.„n, -„n,dieaia,lepar.nreasi- m,w advoealed eouhl m,t K."I-...IV I- ...adenn.il a sntheien. exp.rnn, o.al l,„>i. i. ,„,n.d 'or it as well a> 111!! Ih. ..ivtieal jnslilieati.m, Theuu,ho,| elo.HN .sid l,v,n.,nee, w ,. ,„ u.ake theworkin.. ''""''"''■' "■^"■" '■"" i-'l "'•n,p,n,vasde,,.rn,inedhyexperi': .......slor:, .,n,ilar„.e.:un. andlh,.. i. l.eM plan availal.i;; hut it .....M MO, I,.. >npp„sed .ha. i. .enivs any.hu,. like a d.lini.e lind. of v.a.k,n. s.re..s,n .heeolnnn.. The ler.n -eripplin. loa.l ■ n.eans ditier- .'..t tl.M.u. with diller, nt exp,.rin.,.M.er.. ■■ Cripplin, ' oeeurs in va.ious ways wuh d.lhr, n. e„lun,ns, and the rati,, „f the .,i,,plin. h,a,l to the ela.t,eh,ad van,- v.ry ,„vatly with .litlerent p...|,ortioes „f eolumns l'u.ther, the .,re.ss in the inueriiiost Hhies „r a eoh.mn at its point „f ''yl...v .« ..ot the nitiu.ate ,t,ess of the Hhres. or any constant amou..t whatever. ninll;,. sineey (ur n.ore strietly y,) is a .-oettieient of /i \ ' j,, the 'lH.o.....ia.,,rof (5), the ,,,u.stioM is raised as to whether / sh„uld be u.adetheelas.,e lin.it of ,|,e material „, dnve, con,p,essi„n. and the ?'■'""" '""V '''"""" '""■'' *■""■ ^■^''"'' "'■/' - d'"Vfd f.-om (i>), o.. whe.lu.r./ should he ,na,h. the perudssihl,. worKi... .t,vs.s iu the extnnne hhn.s, and the workin, load ,, derive,l dnvetly hy (f,, ,ro,u it. Ihe two n,e,ho,ls w„ul,U,ve,lift,.r,.nt workin, loads which would aoree fbrshorteoh.mnsaMdd,ver,emo,ean,ln,oreas^j:^ increased, thefir.t ruethod ,ivin. ,l,e hi.her lead. The first seeL, however, the luore lojiieul plan, and luis k f .■11 tolJowed in the tentative formula) (6; aud (7). i. i By it tho wdrkinjf load U mado a doflnito fraction of thr loud »t which till' ciliiiuii, lis ,1 wluilr, rouw.s to be |N'rU'orl_v ilil^tio. Ai Wi' httVi' bflbiv Hiiid, till' tiiuruiii l)«twirii tUU riu!.(ic limit ortlir cnliimii and itn failing point may vary rnoinmuHJy Inr ilirt'.r,Mi |iid|iortiMiw . wr ;,,■,• ilmt thi' Htrens;!)! of hinticil ro|iiiiiii.s may In' .'xp cticl hi I),, niiicli mmu'i' variiilil.' I'vi'ii than thai <>l' tjiosr with lixi'dcndH. Iiofiiiixf J( dt'|»'nilH nn two vuii ublo clinuntx a anil '„, wliil.' tin- Hxi'd i iid ifctK rid of". Fur » hinu.'d folilDlli iicc»iiit,.|y ccntonMl „„ — II mid .. — /, ,in „, li'inu' o|i|KMit,' in Niuii 111 .'„. mill's^ ihr hin-i' w,'iv v.i-y ^iiiail ind.vil. /, and <■ wnnld tliiii be iiciiutivi', Tlir nii'aiiiML: iiftlii^ i^ (hat tlir riicfiiin wiiiild nowi' riHe to itM limiting vuliu-, und tlic! column would bolmvofxttinly an if itn eudH wort) tixcd. Tlir moment iit tlic end of a flxnl I'olunin uiidir u load;) IH m-iii-ly p ' :^ | 1 f Jj^^, /i V j. , ,i,.il while /. ii within the clastic load, the uioment of frietinu in a hiie^e ut' onliiiaiy size will bo Hreuter than thin. When p i-iiiehe> sneh a maiinilude that tiie frietiiin in not sutlieient to re.siMt rotation, we may expeet the enliimii .suiblenly to Mpring into a ih!w poMition of ei|uilibrium. We HL'u, then, that aucordinut to formula (5) an areiiratejy ecnlered pinned coluiiiii will safely bear ax ^.Teat a load as if its ends were fixed, and would he niueli weaker il' it rested on knife eib.'es, ll is possiitle, however, that in structures the vibration which usually accompanie* increase of load may destroy the friction to a ureal extent, and. if so, it would largely diminish the .xlrenL-lh we nii';ht mherwise allow for in Iiinned columns well eeiilered, and lie-ides that "„ may in praclici' have H Very appieeial)le value. The above conclusions, which are strielly d 'lueed IVom lair iuvesli- ^atiou, ai.'ree in jicneral with the observatiiMH made by .Mr. Christie nu his experiments refencil (o above. Al Hrst siLdit it miuht .seem an objeelii i lo furniuLi (5) that when - becomes very Hunill /. does iml aiipr.iaili the value y' as a limit, but a definite i|Uiintity less than/. A little cimsideiation, however, will show that any tlieoiy whiib taki's into aeeount tiie variations from ideal conditions on which the Hexiire of columns really depends iiuf,'ht to };ivc that result. It must be remendiered that y' is not the safe load per st| ill. on a short eolunin uniformly loaded, but the stress per si|. in. in the extrenii' fibres, and this will always be Liieater than the averaije stress, except under the Ideal eniidil Ions of" and '\, lieinj.' zerii, which lead to Euler'H theory. The forninhc liltherlo used to express the shenuth of columns have k'cn cnqiirieal, that is, have been frann'il sn as to u'riee as nearlv as possible with the results of experiments. One of them has practically di.sjilaced all others, ami is known as (Jurdon's formula, haviuit been firnt made widely known by l'rof'e.s.sor Jicwis tiordon, althiiu>;h it wa.s put forward oili;lnally by Tredj;old. Uslii}; the same symbols as hitherto it is as folluws : n / . [and is sometimes jiiven in a less correct form as ; We see that for columns of similar section in wliieh '^ is the same /' this is praetieally the same fnrinula as (5). inr the seemid term in the denominator nf (5) eonld then be ydt rid of by usin^' a diflerent value for /, vii!. ; • , the safe lead ]ier sipiare inch on a short col- limn of the section in (piestlon. It may be reuarded as a confirmation of the truth of the ]irlnelples on whieli we liavr proceeded that we have arrived at a similar lornmla in the main to that which is usually accepted as the nearest possilil.^ expression of expcnmeutal results. The aureement id' (iordon's formula witli experimiTits on the destruc- tion of columns is. Imwever. only eoni]iarative. and. as we have pointed out, could only be cxjiected to lie so. It woidd ajipear n pn'nri that tho introduction of / _\ into (.Iordon's foruada, or some similar rorri'clinn, miulit t.i lie itii iniprnri-tni'nt. lH>(otii)a* hiiviiiu nil i'i|ii.il incline III uynitinii. tliiildnc with it ndiiiIIi'i ixti'i'iml iliiiliK trr wiiiilil liiivi' It hiulnT i'\»M\f liiiiil |icr N.|imri' iiirli. Tli« ri'niiltx "I' i'X|i<'riiiii'iitM iiImo Imvr nlinMrii tliiil (iiirilmi'M I'nrniiihi ri'i|uiri>i« It ilitri'I'i'lil Milllr III' I' till' ilifl'i'i'i III Ml liniii III' I'lilinilll III ili>iuiiiii;r II I'oliiliiM ll i" I'li'iir lliiil. iii'iiirilini: In jiintiiilu (.'i). » alvt'ti criwM Kfcduii A will .jIm' ilir urniti'-l -ti'fiiiitli lir-i wln'ii it m mimll IH |Hl>wili|c : sn ||y, \v|||.||/ ' !)" i- in «||i;|ll 41 |iiH4ili|i> \W tblTcfiil'l' ifivi' II tlllili' 111' lllr Villlli'n lif i / 1)' t'nr «l|UU HilnplH filfilll* 111' ITIIKKWllillll. !•' '1 r rni»> :< ''' ' 'i'i.'iiiiL' ■'iili'i. ' ' tliiikm-- lli-'''lii": - -... nii^iiliir |iiiiiit- I H. ilii clii .... 'riiiiiii^;li iliia'iiiiiil ;l. Si|iiiiri', ~i.|i' 2ii, III I wliicli iiiitin iiil I II I ili^i'i'liii^ I'lo-i'il vi'iilrali'ij lilting' iwn , -nli- oji|iiisili' niili "iiflliii'k- I iu«- 1/ I 111. llll llo .... Itini'dill;.' ll|ll'tl -nil'-' IHm/' A nr .t.\ 'ill' n;,/ :i.\ A A ■in' II '111 'In .... Tlllnli^ll illll|{i>lllll ""'; .ir ■' A .<' 1 4-./- ^^|, :i\ H A II- -I'/- :t.v Ml A 'iir- a ■J 1 A II-: :i 1:.'. AiikI'- liiir. -I'll' II. I 'riiruiiuli i....t nf Slllllll lilll'lilH'^" '/. . \ liiiiil- .Nil:-. 7 iiliil !• Ill'' ill,' illtilii.it, Inllii- nl' Iniill li|i iiil.ir,' niliiiiliis WItll f'liui' siili's hillinil mill with |\v,i |il:ii,il nml iwn liiilci'ij i','s|ii'ctivi'|y. It ii|i|ir:irs iliiiiliiliil. 11; wii- -;il.| lp, r,ii-.'. Ii,iw,'\, T linw lui' tin' Hi'Xiirc 111' liiiill ii|i ri.liiiiiii- Clin III' i'V|i,'ii;i| 1,1 Inlliiw til,' -mil,' lnw- lis tlimi! Ill' snliil I'liliniilis Till' tllllli' rnlllrailill'; I'nilllllnll I lll| ,r,'Ssilllls ill nil,' ill' twil I'i'slii'CtH. [t i- I'liiiiiiiniily sii|i|iiKiil tli:ii ,1 s,|ii,ii,' i',>liiiiiii is «,';ik,'st liir H.'xni'i' ill a |p|iiiii' liis.'i'tiii'j its siil,s lull >j ill,. iiii'iii nl' iii,'i'iia of tin' (.'inss si'i'liiiii is til,' sain,' I'lii'iill asli's. |||,' 1 ,'iit nf r.'si.stmji',' i;,|' ,, i;iv>'ii I'lMvatiiiv will 1,1' ill,' s;iiii,. \\,r ;ill plan,- ,,1' tli'Xiii'i', ami acri- ilnital lii'riiiiistaiir, - will ili'hiiiiiii,' ill,' |,Liiii' in wliii-li fli'Xiiri' lU'liially nri'iirs, l'',,i' ;i |>|iini' ilir,iiiuli a ,|ia'j,iiial ,,1' llii' -i|iiiir,v linwt'Vi'l ^ ) -■ is twi jl'i'al as r,il' a |,laii,' liisi-i'lin'j t.lir siili'.s. mill llii'ii r,irr llll' ,lia'.',,iial |,lmir is tlu' nii,' in wliirli lli,' inliiiiiii wniilil siHiiii'sl rcai'li Its limit nl' ilasticilv. I''ni- this |ilmii' a snliij s,|iiiiii. I'liluiiiii is tlH'i'i'liii'i' wi'akrr tlnin a stiljil rin-iilar niliiiiiii ni'(i|iiiil li'iii;tli and \vi'ii;iit. wliili', il' tlif |ilaiii' ,,!' Il,'\iii-,' llis.itiii^ lli,' siilcs nt' the H|iiarr Wi'ii' ainii,' ri',jiiri|,'i|. |||,. M|iiiii',' i',,|iiiiin wmiM lir tlm .stnnmci'. I, lii'iiiL; alliiiist til,' siiiiii' r,,i' l„,tli If |i,,ll,>w -ijiiai,- ami I'li'ciilal' slii'lls lie c'niii|ian'il, .similar riiiiai'ks ii|i|ilv, niily ill this rasf tlic i-irrli' has the ailvmita'ji' a|sn nl' a liii'L:rr nnliiis nt' 'jyvatimi nearly in tho nitin nt' toii-niiitlis. It' -,|iimv niliimiis ,,1' ,',|iial l,n;jili ami wi'i):lit ln' cinu- |iiii'i'il. ill wliii'h the mali'i'ial is in llii' liist nis,' ilistiilnit.'il in a iiiiil'iinii slii'li. In till' si'ciiiiil aliiii,j Iwn siilcs nl' the si|iiarc mily. ami in tlii' thinl ciiiiri'liliatril ill ill,' I'niir aiiiiiilai' |i,,iiits niily. ami takiii;; t'nr I'lii'li case till' imist iliira> ni'iililr |ilam' nl' fli'xui'i' t_ varies in lli,' thi't'i' I'lisi'.s as ('" i anil ( - ) as :!: :i: -J. slmwim; that ili,' last Inrm has a onu- ;!: t sidi'lalili' iiilvaiitai;,' in li,,th ri'sin'its nver the nil u'l's. anil I I'loseil si|iiiii'e is luiir,' ecnunmical than tl hat the I'uUy lilt liltlleeil nil twii hices.