Proposed Method for Predicting the Elastic Shear Force Capacity of a Cracked Web Panel

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1 ntrnational Journal of Scinc and Enginring nvtigation vol., iu 19, Augut 013 SSN: Prood Mthod for Prdicting th Elatic Shar Forc Caacity of a Crackd Wb Panl Sbatian B. Mnd Dartmnt of Civil and Environmntal Enginring, Univrity of Rhod land, Kington, R, USA (mndb@my.uri.du) Abtract-Th growth of a diagonal fatigu ack in an -had tranvrly tiffnd tl lat girdr loadd undr rdominantly har may dgrad th har trngth of an individual wb anl. Analytical xrion wr drivd for th ridual har forc caacity of a ackd wb anl. Svral limit tat wr conidrd including har yilding, wb buckling, brittl fractur, and imnding ductil failur. Th xrion wr validatd with finit lmnt analy and mloyd to invtigat th govrning limit tat and aociatd trngth of variou wb anl gomtri. Wb buckling wa found to b th rdominant govrning limit tat for ralitically izd girdr. Th formulatd xrion may b mloyd in th dign and analyi of girdr conidring ribd diagonal fatigu ack configuration. Kyword- lat girdr; har caacity; wb anl; fatigu ack; ridual trngth; wb buckling; fractur; tability.. NTRODUCTON -had tl lat girdr form an intgral art of labgirdr highway bridg. Th ty of girdr ar contructd of individual lat wldd togthr to form an -had oction ( Fig. 1a). Th dimnion of th lat ar modifid uch that th aft dign i achivd whil matrial u i minimizd. t follow from convntional bam thory that th wb lat rimarily rit intrnal har forc whil th tnion and comrion flang lat rimarily rit intrnal bnding momnt [1]. A major conidration in th dign of a girdr i th avoidanc of har-bad limit tat including har yilding of th wb and wb local buckling []. Th har forc caacity of a girdr corronding to har yilding may b inad by nlarging th gro o-ctional ara of th wb. Similarly, th har trngth of a girdr corronding to wb local buckling may b inad by modifying th dimnion of th wb a wll a adjuting additional aramtr. t can b dducd from claical lat thory that th latic buckling trngth of a lat loadd undr ur har i maximizd if th lngth-to-width ratio of th lat aroach unity []. For thi raon, tranvr tiffnr lat ar gnrally wldd to th wb and flang at varying intrval along th lngth of th girdr, ffctivly ubdividing th wb into individual anl boundd by th flang and tiffnr lat ( Fig. 1b) []. n rgion of th girdr loadd undr rdominantly har, th wb anl ar ffctivly loadd undr ur har ( Fig. 1c) []. Hnc, th modification of th acing btwn th tiffnr lat dirctly influnc th buckling trngth of th wb anl. Th ongoing aag of vhicular traffic ovr a lab-girdr bridg induc ub-itical cyclic bnding momnt and har forc within th uorting girdr [3]. Th forc ar dirctly could to fluctuating bnding and har tr acting throughout a girdr. High concntration of fluctuating tr may form at inhrnt dicontinuiti in th girdr. Th dicontinuiti ar gnrally attributabl to flaw in th fillt wld conncting th wb to th flang and tiffnr lat, and may includ incomlt fuion, oroity, undrcutting, and artial ntration [4]. Th fluctuating tr concntration may in tim cau a r-xiting mioack to grow into a through-thickn maoack [5, 6]. Figur 1. (a) -had tl lat girdr o-ction, (b) girdr loadd undr rdominantly har and ubdividd into individual wb anl by tranvr tiffnr lat, (c) wb anl loadd undr ur har. n rgion of a girdr loadd undr rdominantly har, a fatigu ack may continu to roagat undr mixd-mod loading [7, 8]. n th mot gnral ca, a fatigu ack may initiat in a cornr of a wb anl at th junction of a flang and tranvr tiffnr lat ( Fig. a) [4, 9-14]. Primarily oning-mod loading (Mod ) comlmntd by low valu 1

2 of liding-mod loading (Mod ) may thn cau th ack to roagat in a traight lin diagonally through th wb anl at an angl nar to 45 rlativ to th flang ( Fig. b) [8, 15]. t can b forn that th continud growth of th ack may advrly affct th har trngth of th wb anl, and concurrntly th ovrall girdr, corronding to th limit tat of har yilding and local buckling. Furthrmor, th rnc of th ack introduc two additional limit tat including brittl fractur and imnding ductil failur [8]. Ovrall, th growth of a diagonal fatigu ack may bring about th rmatur occurrnc of latic limit tat in a girdr [16-]. Figur. (a) Girdr loadd undr rdominantly har with diagonal fatigu ack originating at a cornr of a wb anl, (b) diagonal fatigu ack within th wb anl dilaying th x-y and x -y axi ytm. Many tudi hav invtigatd th influnc of ack and othr dicontinuiti on th trngth and tability of lat-lik tructur through th u of analytical, numrical, and xrimntal mthod [3-36]. Far l rarch ha analyzd th ffct of through-thickn fatigu ack on th har trngth and tability of lat girdr and wb anl [37, 38]. Th bulk of rlatd rarch ha concntratd uon th influnc of larg dicontinuiti uch a hol, lot, and coing on th har trngth and tability of bam-lik tructur [39-48]. Th ot-buckling bhavior and aociatd tnion fild action of unackd wb anl ha bn thoroughly analyzd [49-51]. t may b uful for nginr to b abl to dign or othrwi analyz th latic har forc caacity of lat girdr wb anl for ribd diagonal fatigu ack configuration o a to avoid th occurrnc of latic limit tat in btwn bridg inction riod. Th objctiv of thi rarch wa to driv analytical xrion for th latic har forc caacity of a ackd wb anl corronding to th limit tat of har yilding of th wb, wb local buckling, brittl fractur, and imnding ductil failur. Thi rarch concntratd uon latic limit tat, and th otbuckling bhavior and tnion fild action of ackd wb anl wa not conidrd. Svral aumtion wr mad with rgard to th formulation of th caacity xrion. t wa aumd that th ortion of lat girdr that th xrion ar alicabl to ar loadd undr rdominantly har ( Fig. a). Th lat girdr itlf wa aumd to b an -had tl lat girdr. Th individual lat of th girdr wr aumd to b contructd of high-trngth low-alloy tructural tl []. A uch, th tl wa imlid to b homognou and bhav a a linar iotroic latic matrial [5]. Alo, th lat wr aumd to b ufficintly thin nough for lan tr condition to rdominat. n accordanc with convntional dign rocdur, th tr fild wr drivd rlativ to th dfind coordinat axi ytm and comard to th matrial trngth [1, ]. Th wb anl itlf wa aumd to hav a dth, d w, and a width,, dignating th acing btwn th tiffnr lat ( Fig. b). Th diagonal fatigu ack of lngth a wa aumd to b through-thickn and loadd rimarily by Mod loading with minimal influnc from Mod loading. A uch, th ack wa aumd to roagat in a comarativly traight lin from a cornr of th wb anl at an angl θ 45 rlativ to th flang ( Fig. b) [8]. Finit lmnt (FE) analy wr rformd uing th gnral FE oftwar ABAQUS to validat th caacity xrion. Uon validation, th caacity xrion wr lottd a function of ack lngth for variou wb anl gomtri. Concluion wr drawn rgarding th diffrnt caaciti and limit tat aociatd with th variou wb anl gomtri. Finally, th alicability of th har forc caacity xrion wa dmontratd with a imlifid lat girdr dign roblm.. WEB LOCAL BUCKLNG A. Aroximation of ntrnal Shar Str Ditribution Within Crackd Wb Panl t wa rviouly acknowldgd that a wb anl locatd in a rgion of a girdr loadd undr rdominantly har i itlf ffctivly loadd undr ur har []. Thi configuration i maniftd in th form of uniform har tr, τ, acting along th rimtr of th wb anl ( Fig. 1c). Th rimtr har tr i dirctly linkd to and aroximatly qual to th intrnal har tr within th wb anl. Alo, τ i dirctly could to th xtrnal har forc, V, acting at that articular rgion of th girdr ( Fig. 1b). Th ditribution of intrnal har tr along th dth of th wb i narly uniform, and thu τ may b aroximatd a th magnitud of V dividd by th gro o-ctional ara of th wb, xrd a [] V V xy (1), Aw dwtw whr A w i th gro o-ctional ara of th wb and t w i th thickn of th wb. Th imortanc of τ li in it dirct link to th intrnal har tr, which itlf ha a dirct influnc uon th latic buckling trngth of th wb anl [53]. Th rnc of a diagonal ack originating at a cornr of a wb anl ( Fig. b) may rv to influnc th magnitud ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

3 and ditribution of th intrnal har tr, thu affcting th buckling trngth of th wb anl. Thi roc may b lucidatd in two t. Firt, th intrnal har tr fild, τ c, within th ackd wb anl i dvlod bad uon τ ( Fig. 3a). Scond, τ c influnc th buckling trngth of th wb anl ( Fig. 3b). t can thu b forn that th formulation of an xrion for th latic har forc caacity of a ackd wb anl aociatd with th limit tat of wb local buckling rquir an xrion for τ c. Th har tr fild around th inclind cntral ack ( Fig. 4b) i dtrmind by urimoing th tr fild of thr ditinct ca [8]. Th firt ca conit of an infinit lat without th ack loadd by far-fild har tr qual in magnitud to τ ( Fig. 5a). Th cond ca conit of an infinit lat with th inclind cntral ack loadd by ackfac normal tr, σ y ( Fig. 5b). Th third ca conit of an infinit lat with th inclind cntral ack loadd by ack-fac har tr, τ xy ( Fig. 5c). Th magnitud of σ y and τ xy ar obtaind from th tat of tr of an lmnt within th firt ca that ha bn tranformd to th x -y axi ytm from th x-y axi ytm. Figur 3. (a) ntrnal har tr influncd by rimtr har tr, (b) itical magnitud of rimtr har tr corronding to wb buckling a influncd by intrnal har tr. Th intrnal har tr fild, τ c, within a ackd wb anl may b aroximatly dtrmind uing laticity thory. Th gomtrical boundary condition aociatd with th actual configuration of a diagonal ack originating at a cornr of th wb anl ar inxorably comlx, and a clod-form olution i difficult to obtain ( Fig. b) [5]. Howvr, two obrvation ar mad rgarding th gomtrical boundary condition of th ack and wb anl. Firt, both nd of th ack ar ffctivly rtraind from oning. Scond, th rimtr of th wb anl formd by th flang and tiffnr lat i much mor rigid than th wb. A a conqunc, th diagonal ack may b aroximatd a lying within th wb lat and boundd by an outid margin of additional wb lat rrnting th rigidity of th flang and tiffnr lat, thu rolving th original thr-dimnional configuration into a two-dimnional configuration ( Fig. 4a). A furthr aroximation may thn b mad in auming that th rviou configuration i narly quivalnt to a diagonal ack lying within an infinit lat. Th roblm i thu rducd to that of dtrmining th har tr fild around a cntral ack inclind at an angl θ 45 and loadd by far-fild har tr qual in magnitud to τ ( Fig. 4b). Figur 5. (a) Ca 1: infinit lat without ack loadd by far-fild har tr, (b) Ca : infinit lat with diagonal ack loadd by ack-fac normal tr, (c) Ca 3: infinit lat with diagonal ack loadd by ackfac har tr. Conidring th firt ca with rct to th x-y axi ytm, an infinitimal lmnt at any location within th infinit lat i undr a tat of ur har qual in magnitud to τ ( Fig. 6a). Rotation of th lmnt by an angl θ to th x -y axi ytm rult in th tranformd normal and har tr σ y and τ xy ( Fig. 6b) givn by [1]: y in( ) () xy co( ) (3) Figur 6. (a) nfinitmal lmnt undr a tat of ur har, (b) rotatd lmnt with tranformd normal and har tr. Figur 4. (a) Two-dimnional wb anl configuration, (b) infinit lat wb anl configuration. Th normal and har tr ar aumd to act uon th ack-fac of th cond and third ca, rctivly ( ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

4 Fig. 5b, c). Th tr fild for ach of th two ca i nxt dtrmind. n accordanc with laticity thory, th lan tr fild i xrd in trm of th Airy tr function, F(x,y), a [5] F x y F y x F xy (4) xy Th Airy tr function mut atify th alicabl boundary condition and th govrning biharmonic quation of lan laticity givn by [5] F F F F 0 (5) 4 4 x x y y uch that th quilibrium quation and Bltrami-Michll comatibility quation ar idntically atifid. Th rnc of a ack introduc local diturbanc in th tr fild which comlicat th dtrmination of a uitabl tr function. Th tr fild around a ack may thn b dtrmind uing a ubt of th comlx otntial mthod calld th Wtrgaard function mthod [54]. n thi mthod, th Airy tr function i xrd in trm of th Wtrgaard function, Z(ζ), for Mod loading a [8] F RZ ymz (6) and for Mod loading a [8] F whr R Z y mz y R Z (7) dz Z d d Z Z d d Z Z (8) d Th trm Z and Z ar th Wtrgaard function for Mod and Mod loading, rctivly. Alo, ζ i th comlx variabl ζ = x + iy. Th tr fild for th cond and third ca ar dtrmind by mloying th Wtrgaard tr function aociatd with ack-fac normal tr and ack-fac har tr, rctivly, givn by [55, 56] Z 1 y a Z 1 xy a (9) (10) t i notd that th tr function ar givn with rct to th x -y axi ytm. Subtituting (6) into (4) rult in th comlt tr fild for th cond ca rlativ to th x -y axi ytm, givn by [8] x, c ( x, y ) R Z y, c ( x, y ) R Z y mz y m Z (11) c ( x, y ) y R Z n a imilar mannr, ubtituting (7) into (4) rult in th comlt tr fild for th third ca rlativ to th x -y axi ytm, givn by [8] x, c3 ( x, y ) m Z y, c3 ( x, y ) y R Z y R Z (1) c3 ( x, y ) R Z y m Z Th ummation of (11) and (1) rult in th urimod tr fild obtaind from th cond and third ca, givn by x, c ( x, y ) x, c x, c3 y, c ( x, y ) y, c y, c3 (13) c ( x, y ) c c3 Th urimod har tr fild, τ ca, obtaind from th cond and third ca i tranformd to th x-y axi ytm from th x -y axi ytm by ubtituting (13) into th following tranformation xrion [1]:, c, c x, y) x y in( ) xy c co( ) (14) ca (, whr th coordinat of th x -y axi ytm ar xrd in trm of th coordinat of th x-y axi ytm a x yin x co (15) y y co xin (16) Finally, th comlt har tr fild around th inclind cntral ack with rct to th x-y axi ( Fig. 4b) i obtaind by urimoing (14) with th har tr fild, τ, of th firt ca: c( x, y) ca (17) Th full xanion of (17) i quit lngthy and conit of ral and imaginary trm. Howvr, (17) i gratly imlifid if θ 45, thu rducing th xandd form of (17) to th following: in( ) a (18) ix y x y a c ( x, y) in( ) ym 3/ Rolving th imaginary trm rult in whr, in( ) y a k k ( k k ) (19) 1 1 c ( x, y) 3/ k1 k k x y x a y a x y a (0) x y a (1) Equation (19) rrnt th comlt har tr fild around th cntral ack inclind at θ 45 and lying within th infinit lat loadd by th far-fild har tr τ ( ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

5 Fig. 4b). n accordanc with th aformntiond aroximation, th har tr fild obtaind from thi configuration i aroximatly quivalnt to th intrnal har tr fild obtaind from th actual configuration of a ackd wb anl with th diagonal ack originating at a cornr of th wb anl ( Fig. 3a). mortantly, it i notd that (19) i in art a function of τ, and i thrfor dirctly could to V by way of (1). Clo xamination of (19) rval that th ditribution of τ c within a ackd wb anl i quit comlx. Magnitud of har tr much gratr than τ ar cially rvalnt in th local rgion around th ack. Th ditribution of τ c may b aroximatd by ubdividing th wb anl into qual-izd rctangular lmnt. Th total numbr of lmnt, N, ar arrangd uch that th numbr of lmnt row, r, ar qual to th numbr of lmnt column, c. Each lmnt i dignatd by it row numbr, r, and column numbr, c, rlativ to th origin of th ack ( Fig. 7a). Alo, ach lmnt i loadd undr ur har dignatd by τ -rc. Th magnitud of τ -rc for ach lmnt i dvlod bad uon τ c. Scifically, th magnitud of τ -rc for ach lmnt i th valu of τ c at th cntral oint of th lmnt ( Fig. 7b). t can b forn that a gratr numbr of lmnt rult in a highr dgr of accuracy in dibing th actual ditribution of τ c. Figur 7. (a) Wb anl ubdividd into rctangular lmnt dilaying row and column numbr, (b) individual anl lmnt loadd undr ur har. Th magnitud of τ -rc for any lmnt i xrd in trm of τ c by tting x and y qual to th coordinat of th cntral oint of th lmnt, x rc and y rc, xrd a x c ar co o rc () r y d r d ac in w w o rc (3) c whr r and c ar xrd in trm of N a and, r c N (4) o (5) Alo, θ in (15), (16), and (19) i modifid to tak into account ngativ valu of x: xrc xrc o (6) xrc Thu, th valu of τ -rc for any lmnt i dtrmind by ubtituting r, c, (4), and (5) into () and (3), and introducing th rult into (19), imlicitly xrd a rc c x rc, yrc (7) t i obrvd that inc τ -rc i a function of τ c, it i thrfor alo a function of τ. Altogthr, th varying valu of τ -rc for ach lmnt rrnt an aroximation of th intrnal har tr ditribution, τ c, within th ackd wb anl. B. Buckling Strngth of Crackd Wb Panl A clod-form olution for th latic buckling trngth of th ackd wb anl i difficult to obtain uing claical lat thory [, 53]. Thi i rimarily du to th non-uniform intrnal tr fild caud by th rnc of th ack, a wll a th comlx gomtrical boundary condition introducd by th ack. Th Rayligh-Ritz nrgy mthod mloy th rincil of tationary otntial nrgy to aroximat th buckling trngth whn comlx boundary condition ar rnt [53]. n thi mthod, th buckld ha of th ackd wb anl i aumd to tak on a form dibd by an out-of-lan dilacmnt function, w(x, y ), xrd in trm of th x - y axi ytm with th origin at th cntral oint of th wb anl ( Fig. 8). Th dilacmnt function atifi all or mot of th gomtrical boundary condition and includ an arbitrary t of variabl, A i, which control th ha of th dilacmnt function, in th form n w( x", y") A f ( x", y") (8) i1 i i whr n i th numbr of dgr of frdom of th dilacmnt function. Th total otntial nrgy, П, of th anl i thn formulatd, xrd a [53] UdV T u ds (9) V S i i whr U i th train nrgy dnity function, V i th volum of th anl, T i ar th alid urfac traction, u i ar th dilacmnt caud by th traction, and S i th urfac ovr which th traction ar alid. t i vidnt that th formulation of П ntially coul w(x, y ) with th xtrnally alid tr, τ ( Fig. 8). Th variation of П with rct to A i i thn t to zro, hown a [53] 0 (30) A i t follow that th itical xtrnal tr, τ, nabling thi quilibrium i an initial timation of th buckling trngth of th ackd wb anl [53]. ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

6 x y w ( x ", y ") A co co (33) d w Th formulation of П i dvlod by ubtituting (33) and τ into (9), imlicitly xrd a [53] Figur 8. Crackd wb anl with xtrnally alid rimtr har tr and corronding out-of-lan dilacmnt function. Th initial magnitud of τ may b furthr dvlod by mloying th wb anl lmnt and th rviouly drivd aroximat intrnal har tr ditribution, τ -rc. Scifically, th ovrall initial ackd wb anl buckling trngth i aumd to b uniformly ditributd among ach of th wb anl lmnt. Th trngth of ach lmnt i furthr aumd to b dgradd by th ratio of τ to τ -rc. n thi way, th buckling trngth of ach lmnt, τ,-rc, i qual to th roduct of th ratio of τ to τ -rc and th initial ackd wb anl buckling trngth, dividd by th total numbr of lmnt, xrd a (31), rc N rc Each lmnt i thu aumd to buckl whn τ -rc qual or xcd τ,-rc. Th corronding itical valu of τ for ach lmnt may thn b dtrmind. t follow that th final timation of th buckling trngth of th ackd wb anl, τ, i th um of th itical valu of τ. An xrion for w(x, y ) for th ovrall ackd wb anl i firt rquird for th formulation of П and th dtrmination of τ. Th gomtrical boundary condition concrning th dg of th wb anl ar conrvativly aumd to b imly uortd []: w, y" 0 w, y " 0 d w w x", 0 d w x", w 0 (3) Th aroximation i mad that w(x, y ) for th ackd wb anl i narly idntical to w(x, y ) for an unackd wb anl. Th gomtrical boundary condition concrning th diagonal ack ar thrfor nglctd, and w(x, y ) i formulatd bad only uon th boundary condition givn by (3). n nc, th aroximation i mad that th initial buckling trngth of th ackd wb anl i dndnt only uon th xtrnal har tr, with w(x, y ) rmaining idntical to w(x, y ) for th unackd wb anl. A unilatral dilacmnt function of th following form atifi th boundary condition givn by (3): D whr, and, U d w d w d w d w U dx" dy" W dx" dy" w w x" y" w w w (1 ) x" y" x" y" (34) (35) w w W (36) x" y" Alo, th trm D i th wb anl rigidity givn by [] 3 Et D w (37) 1 1 whr E i th modulu of laticity and ν i Poion ratio. Th initial ackd wb anl buckling trngth i thn dtrmind by introducing (34) into (30), olving for τ, and dividing th rult by t w, giving d w d w D (38) d w t w d w U dx" dy" W dx" dy" whr th arbitrary trm A in (33) vanih in accordanc with calculu of variation [53]. Th initial ackd wb anl buckling trngth i furthr rfind by dtrmining th buckling trngth of th wb anl lmnt and umming th aociatd itical valu of τ. Th xtrnal tr alid to ach lmnt i th rviouly drivd ur har tr τ -rc givn by (7) ( Fig. 7b). Equation (7) may b rwrittn to b xlicitly xrd in trm of τ a ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

7 C 1 (39) rc rc whr C rc i a contant dirctly drivd from (19) and artially dndnt uon r, c, x rc, and y rc of a givn lmnt, xrd a C rc in( ) y a k k k ( 1 3 / 1 k k ) 1 (40) Th itical valu of τ corronding to buckling of ach lmnt i dtrmind by tting τ -rc = τ,-rc, introducing (38) and (39) into (31), and olving for τ, giving, rc w D dw NtwC rc 1 (41) W dx" dy" dw d dw U dx" dy" Finally, th aroximat buckling trngth of th ackd wb anl i dtrmind by taking th um of τ,-rc for all of th lmnt, xrd a N N r1 c1, rc (4) Th trm τ i th har yild trngth of th wb anl tl givn by [] (43) 3 whr σ i th yild trngth of th wb anl tl. Stting τ = τ, ubtituting (4) into (1), and olving for V rult in th har forc caacity of a ackd wb anl corronding to th limit tat of wb local buckling: V d t (44) w w. SHEAR ELDNG Th growth of th diagonal fatigu ack originating at a cornr of th wb anl rduc th gro o-ctional ara of th anl ( Fig. 9). From (1) it can b dducd that th rduction in gro o-ctional ara rv to da th har forc caacity corronding to har yilding. Th rducd wb anl dth, d wc, abov th ack ti i givn by d wc d ain (45) w Subtituting (45) for d w and τ for τ in (1), and olving for V rult in V t d ain (46) w w Th intrnal har tr along d wc achiv har yilding firt at th ack ti and lat at th rimtr of th wb anl. Th itical valu of τ corronding to V thu occur whn τ qual th har yild trngth of th wb anl tl, givn by V. (47) Figur 9. Crackd wb anl with rducd wb anl dth. BRTTLE FRACTURE AND MPENDNG DUCTLE FALURE Th roagation of th diagonal fatigu ack originating at a cornr of th wb anl ( Fig. b) i accomanid by a corronding ina in th tr intnity factor, K, at th ack ti. Thi ina in K introduc two additional limit tat including brittl fractur and imnding ductil failur. Brittl fractur i charactrizd by th uddn rutur of th wb anl and occur whn th fatigu ack grow to a itical lngth and K qual th itical tr intnity factor, K c, of th wb anl tl [8, 57]. Altrnativly, imnding ductil failur i charactrizd by th growth of th latic rgion around th ack ti to a itical iz and occur whn th ack lngth and K ach qual a itical magnitud [57]. n accordanc with linar latic fractur mchanic (LEFM), th thortical form of K i givn by [8] K o a (48) whr σ o i th far-fild Mod or tr acting uon th ack. A rviouly mntiond, th diagonal ack hown in Fig. b i loadd rimarily by Mod loading with minimal influnc from Mod loading whn θ 45. Emloying th rincil of uroition, th total tr intnity factor, K T, i th um of th tr intnity factor drivd from th Mod and loading, xrd by [8] K T K K (49) whr K and K ar th tr intnity factor aociatd with Mod and loading, rctivly. Th magnitud of K and K for th diagonal ack may b aroximatd by auming onc again that th actual configuration of th diagonally ackd wb anl ( Fig. b) i narly quivalnt to th configuration of a diagonal ack inclind at an angl θ 45 and lying within an infinit lat loadd by far-fild har tr qual in magnitud to τ ( Fig. 4b). Th Mod and tr intnity factor ar drivd from th tat of tr of an lmnt in th rviouly dibd configuration ( Fig. 4b) that ha bn tranformd to th x - ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

8 y axi ytm from th x-y axi ytm ( Fig. 6). Th normal tr, σ y, and har tr, τ xy, of th tranformd lmnt ar xrd by () and (3), rctivly. Th magnitud of σ y and τ xy rrnt th far-fild normal and har tr, rctivly, acting uon th diagonal ack. Subtituting () and (3) into (48) rult in xrion for K and K, hown a K K in( ) a (50) co( ) a (51) Undr lan train condition, Mod brittl fractur occur whn K xcd th Mod fractur toughn, K c, of th wb anl tl [57]. Similarly, Mod brittl fractur occur whn K xcd th Mod fractur toughn, K c, of th wb anl tl [57]. Givn that th ack i concurrntly ubjctd to Mod and loading, crtain itical combination of K and K may roduc mixd-mod fractur. Th intraction btwn K and K with rct to K c and K c may b dibd by a iml llitical modl [8], xrd a K K c K K c 1 (5) Str intnity corrction factor may b ud to modify K and K in (5) to account for th finit dimnion of th wb anl [58]. Howvr, th lngth of th diagonal ack i xctd to rmain mall in comarion to th wb anl rior to an latic limit tat bing attaind, and thu th corrction factor i nglctd. Th rimtr har tr caacity of th ackd wb anl corronding to th limit tat of brittl fractur i dtrmind by introducing (50) and (51) into (5) and olving for τ : a K c in K c K c K in c K c (53) t i notd that th u of K c and K c in (50) i conrvativ inc lan tr condition ar rumd to rdominat throughout th wb anl, and th actual itical tr intnity factor ar gratr than K c and K c [58]. t follow that th ubtitution of (53) into (44) rult in th har forc caacity of a ackd wb anl corronding to th limit tat of brittl fractur. Th latic rgion around th ack ti a inducd by th Mod and loading mut rmain mall in ordr for K and K to rmain valid [57]. Th growth of th latic rgion byond a itical iz rndr th fractur toughn charactrization of th wb anl tl inalicabl. Elato-latic fractur mchanic (EPFM) mut thn b mloyd to dib th imnding ductil failur [57]. A nw limit tat may b otulatd corronding to a itical latic rgion iz indicating th aroximat tranition to a ductil failur mod (i.., th tranition from LEFM to EPFM). Th iz of th latic rgion around th ack ti a inducd by th mixd-mod loading may b dtrmind by urimoing th latic rgion obtaind from th Mod and loading. Scifically, th two latic radii along th longitudinal dirction of th ack a obtaind from th Mod and loading may b dtrmind and urimod to obtain th mixd-mod latic radiu ( Fig. 10). Th itical magnitud of τ aociatd with a itical mixd-mod latic radiu indicating imnding ductil failur may thn b olvd for. Figur 10. Platic rgion around th ack ti. Th nar-ti tr fild i xrd in trm of K a [8], K 3 co 1 in in x r K 3 co 1 in in (54) y r xy K 3 in co co r Similarly, th nar-ti tr fild i xrd in trm of K a [8], K 3 in co co x r K 3 y in co co (55) r K 3 co 1 in in xy r t i notd that th nar-ti tr fild ar formulatd in trm of olar coordinat with th origin at th ack ti. Th variabl r i th radiu and θ i th angl of r with rct to th longitudinal dirction of th ack ( Fig. 10). Th mixdmod radiu of th latic rgion, r, i dtrmind by mloying th Mi yild itrion for lan tr, xrd a [] 1 1 (56) whr σ 1 and σ ar th rincial tr givn by [1] 1, x y x y xy (57) Subtituting (54) and (55) into (57), introducing th rult into (56), tting θ = 0, and olving for r rult in th latic radii along th longitudinal dirction of th ack aociatd with Mod and loading: ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

9 r r,, K (58) 3K (59) Th uroition of r, and r, rquir that r i th gratr valu of (58) or (59). Givn that th diagonal fatigu ack i inclind at an angl θ 45, th Mod loading rdominat and th contribution of r, i urdd uch that r = r,. Subtituting (50) into (58) and olving for τ rult in ain ar, (60) whr r, i a rdfind itical latic radiu along th longitudinal dirction of th ack corronding to th tranition from LEFM to EPFM. t follow that th ubtitution of (60) into (44) rult in th har forc caacity of a ackd wb anl corronding to th limit tat of imnding ductil failur. V. FNTE ELEMENT ANALSES A. FE Modl Ovrviw A total of four trial wb anl wr modld with ABAQUS 6.11 and mloyd to validat th har forc caacity xrion givn by (4), (53), and (60). Th width and dth for all four trial anl wr t a contant with = 10 cm and d w = 17 cm, rctivly. Each anl wa modld with a diffrnt wb anl thickn, t w, ranging from 0.15 cm to 1.0 cm. A through-thickn diagonal ack wa modld in ach anl with θ = 45. Th lngth of th ack, a, in ach anl wa t to rang from 10 cm to 70 cm. in 10 cm inmnt. Alo, th anl wr modld with th matrial rorti of high-trngth low-alloy tructural tl. Th gomtrical and matrial rorti of th trial wb anl ar litd in Tabl and, rctivly. Th trial wb anl with t w = 0.15 cm and t w = 0.30 cm (WP-A and WP-B) wr urly thortical for th uro of invtigating a broadr rang of otntial latic limit tat. Th anl wr mhd uing 4-nod hll lmnt dd at 0.50 cm. Th mh wa dd at 0.05 cm nar th ack ti. Alo, th ack in ach anl wa modld by aigning a am to a ingl-lin artition. For tr analy, th inad rigidity along ach anl rimtr caud by th flang and tiffnr lat wa accountd for by modling an outid margin of additional wb lat, rolving th actual thr-dimnional anl configuration into a two-dimnional configuration ( Fig. 11a). Th width of th outid margin, m, wa aroximatd by tting th gro o-ctional ara of th margin qual to th gro o-ctional ara of th flang lat and olving for m, rulting in b t f f m (61) t w whr b f i th flang width and t f i th flang thickn. Th flang thickn wa aumd to b twic th wb anl thickn uch that t f = t w, thu tranforming (61) into m = b f. Furthrmor, th flang width wa aumd to b contant with b f = 35 cm. For buckling analy, th outid margin of additional wb lat wa rmovd ( Fig. 11b). Th ack wa lightly hiftd away from th cornr of th wb anl to allow th ack to bhav a a cntral ack whil rmaining nar to it original oition rlativ to th anl rimtr. Th rimtr of ach anl wa t to hav imly uortd boundary condition rtraining out-of-lan and inlan movmnt whil allowing id-to-id movmnt ( Fig. 11a, b). Each trial wb anl wa thn loadd undr ur har by alying hll dg load along th rimtr of th anl margin ( Fig. 11a, b). Th rulting caaciti aociatd with har yilding of th wb, wb local buckling, brittl fractur, and imnding ductil failur wr thn numrically calculatd for ach trial wb anl and comard to th caaciti obtaind from th analytical xrion. TABLE. TRAL WEB PANEL GEOMETRCAL PROPERTES Trial Wb Panl Thickn, t w (cm) WP-A 0.15 WP-B 0.30 WP-C 0.60 WP-D 1.0 TABLE. TRAL WEB PANEL MATERAL PROPERTES Mod Mod Modulu of ild Poion fractur fractur laticity, E trngth, ratio, ν toughn, K (GPa) σ (MPa) c toughn, K c (MPa m 1/ ) (MPa m 1/ ) 00 a 0.3 a 345 a 98 b 74 c a. Gnral rorti of high-trngth low-alloy tructural tl []. b. Rrntativ fractur toughn of high-trngth alloy tl [57]. c. Auming Kc 0.75Kc [8]. Figur 11. (a) Trial wb anl configuration for tr analy, (b) trial anl configuration for buckling analy. B. Wb Local Buckling Th trial wb anl WP-C wa firt ud to validat th xrion for th intrnal har tr ditribution givn by ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

10 (19). Th uniform rimtr har tr wa t to τ = 150 MPa, th ack angl t to θ = 45, and th ack lngth t to a = 40 cm. Plot of th intrnal har tr ditribution wr analytically dtrmind uing (19) for two arbitrary horizontal ath dignatd by τ c (x, 5 cm) and τ c (x, 40 cm) ( Fig. 1). dibing th mid-fild and far-fild har tr ditribution than in dibing th nar-fild ditribution. Nonthl, (19) i conrvativ for both ca dilayd in Fig. 13. All four trial wb anl wr nxt mloyd to dirctly validat th caacity xrion for wb local buckling givn by (4). Each trial anl wa loadd with a unit uniform rimtr har tr, τ = 1 MPa. A linar rturbation buckling analyi wa thn rformd on ach anl conidring th diffrnt ack lngth ranging from 10 cm to 70 cm. Th rulting Mod ignvalu wr dividd by t w to obtain th itical valu of τ aociatd with buckling. Th buckling caaciti of th trial wb anl a obtaind from (4) with N = 100 and th FE analy ar lottd in Fig. 14 a function of ack lngth. Figur 1. Location of horizontal ath in WP-C for which th intrnal har tr ditribution wr analytically and numrically obtaind. Th analytically dtrmind har tr ditribution wr comard to numrically calculatd ditribution by ating two nodal ath within WP-C in th location dignatd in Fig. 1. Th X data for th in-lan har tr wa thn rqutd for ach of th nodal ath. Th har tr ditribution a obtaind from (19) and th FE analy ar lottd in Fig. 13. Figur 13. Plot of intrnal har tr ditribution along th horizontal ath dignatd in Fig. 1 a obtaind from (19) and FEA. Th har yild trngth i tmorarily nglctd for th uro of dilaying th full xtnt of th tr ditribution. Th analytically and numrically dtrmind har tr ditribution xhibit a clor corrlation along y = 40 cm than along y = 5 cm. Thi uggt that (19) i mor accurat in Figur 14. Wb anl buckling caaciti a calculatd from (4) and FEA. t i vidnt that th analytically and numrically dtrmind wb buckling trngth dmontrat a rlativly clo corrlation for all ack lngth conidrd. Eq. (4) i lightly mor conrvativ for ack lngth u to 40 cm. Convrly, th FE rult ar lightly mor conrvativ for ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

11 ack lngth byond 40 cm. From (43) and Tabl, th har yild trngth i aroximatly τ = 00 MPa. n gnral, wb anl trngth xcding th har yild trngth ar govrnd by τ. Thi i th ca for th buckling trngth of WP-D for hortr ack lngth, a dmontratd in Fig. 14. C. Shar ilding Th har forc caacity of a ackd wb anl corronding to th limit tat of har yilding i givn by (46). Uon th advnt of har yilding, th rimtr har tr attain th har yild trngth. Givn that th trial wb anl ar loadd undr ur har by dfault, FE analy wr unncary to validat (47). D. Brittl Fractur and mnding Ductil Failur Th trial wb anl wr nxt mloyd to indirctly validat th caacity xrion aociatd with brittl fractur and imnding ductil failur givn by (53) and (60). Scifically, th trial anl wr ud to confirm th accuracy of K and th wb anl brittl fractur trngth corronding to Mod ack loading. Givn that θ = 45, K rmain ngligibl and th brittl fractur trngth corronding to Mod ack loading aroach τ =. FE analy wr thu unncary to validat K and th Mod fractur trngth. Th Mod fractur trngth i analytically dtrmind by olving (50) for τ and tting K = K c, rulting in K c (6) in( ) a Sinc (53) and (60) ar dirctly dndnt uon th accuracy of K, K, and th Mod and wb anl brittl fractur trngth, th validation of (6) rv to indirctly validat (53) and (60). Th wb anl brittl fractur trngth wa firt numrically calculatd for ach trial wb anl conidring Mod ack loading. Thi wa rformd by tting th hll dg load to a ram load configuration. Th magnitud of th hll dg load wa t to an arbitrary valu uch that K xcdd K c. Th total load tim riod wa t to 10 and th load tim inmntation t to 1. Th tr intnity factor wa obtaind by aigning a hitory outut rqut for K at th ack front for ach load tim inmnt. Load cal wr thn obtaind by dividing th load tim inmnt at which K quald K c by th total load tim riod. Latly, th load cal wr multilid by th final magnitud of th hll dg load to obtain th itical magnitud of rimtr har tr. Th Mod brittl fractur caaciti of th trial wb anl a obtaind from (6) and th FE analy ar lottd in Fig. 15 a function of ack lngth. Figur 15. Wb anl brittl fractur trngth a calculatd from (6) and FEA. t can b n that th FE rult corrlat vry wll with th analytical rult for all conidrd ack lngth. Th accuracy of K and (6) i thu confirmd, thrby roviding indirct validation of (53) and (60). V. RESULTS AND DSCUSSON Having validatd (4), (53), and (60) with th FE analy, th har tr caaciti and aociatd limit tat of th trial wb anl wr furthr invtigatd. Th wb anl trngth aociatd with brittl fractur and har yilding ar indndnt of t w, and ar lottd for all four trial anl in Fig. 16 a function of ack lngth. Convrly, th wb anl trngth aociatd with buckling and imnding ductil failur ar dndnt uon t w, and ar lottd for ach trial anl in Fig. 17 a function of ack lngth with N = 100 and r, = t w / 50 [58]. ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

12 Figur 16. Wb anl brittl fractur and har yilding caaciti for WP-A through WP-D. Figur 17. Wb anl buckling and imnding ductil failur caaciti for WP-A through WP-D. Th limit tat of brittl fractur and har yilding do not govrn th har tr caaciti of any of th trial wb anl. Wb buckling govrn th caacity of WP-A for all conidrd ack lngth. Alo, wb buckling govrn th caacity of WP- B for ack lngth u to a 19 cm, with imnding ductil failur govrning thraftr. Similarly, wb buckling govrn th caacity of WP-C for ack lngth u to a 3 cm, with imnding ductil failur govrning thraftr. mnding ductil failur govrn th caacity of WP-D for all conidrd ack lngth. t i notd that although imnding ductil failur i a limit tat in itlf and ignifi th invalidation of th brittl fractur limit tat, th har yilding and wb buckling limit tat may rmain valid. With thi in mind, it i obrvd that th wb buckling caaciti of th trial anl ar ignificantly dgradd with inad ack lngth. Th trial anl buckling ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D:

13 trngth ar dgradd by a much a 30% whn th ack lngth rach 70 cm. Th alicability of th har forc caacity xrion i finally dmontratd with a imlifid lat girdr dign roblm. n thi articular xaml, load and ritanc factor ar nglctd. An -had tranvrly tiffnd lat girdr i to b dignd to rit rdominantly har forc. Th girdr i rquird to hav an unfactord har forc caacity of V = 381 kn ( Fig. 1). Th gomtry and matrial rorti of WP-C ar initially lctd for th dign. From (1), th rquird rimtr har tr caacity of WP-C i τ = 50 MPa. From Fig. 17, th rovidd rimtr har tr caacity of WP-C i τ = 5 MPa, govrnd by th limit tat of wb buckling. Subtituting thi valu into (44) rult in th rovidd har forc caacity of WP-C, givn by V = 396 kn. Sinc V = 381 kn < V = 396 kn, th har forc caacity rovidd by WP-C i ufficint if th girdr rmain unackd. Th girdr i now rquird to maintain a ufficint caacity for a ribd diagonal fatigu ack configuration within a wb anl o a to avoid th occurrnc of latic limit tat in btwn bridg inction riod ( Fig. ). Th ribd ack lngth i aumd to b a = 50 cm and it angl of inclination i aumd to b θ 45. From Fig. 17, th rovidd rimtr har tr caacity whn a = 50 cm i τ = 10.7 MPa, govrnd by th limit tat of imnding ductil failur. Howvr, nglcting imnding ductil failur and conidring only th wb buckling limit tat rult in a rovidd rimtr har tr caacity of τ = 41.3 MPa. Subtituting thi valu into (44) rult in th rovidd har forc caacity of WP-C whn a = 50 cm, givn by V = 315 kn. Sinc V = 381 kn > V = 315 kn, th har trngth rovidd by WP-C i inufficint for th ribd ack configuration. naing th wb anl thickn of WP-C from t w = 0.6 cm to t w = 0.7 cm and mloying (4) with N = 100 rult in th modifid rimtr har tr caacity of τ = 56.3 MPa. Subtituting thi valu into (44) rult in th modifid har forc caacity of WP-C whn a = 50 cm, givn by V = 501 kn. Sinc V = 381 kn < V = 501 kn, th har trngth rovidd by WP-C with t w = 0.7 cm i ufficint for th ribd ack configuration. t i notd that thi dign mthod of trial-and-rror may b alid uing load and ritanc factor. V. CONCLUSONS Analytical xrion for th latic har forc caacity of a ackd wb anl corronding to th limit tat of har yilding of th wb, wb local buckling, brittl fractur, and imnding ductil failur wr formulatd. Th xrion wr cific to th ca of a through-thickn fatigu ack initiating and roagating from a cornr of a wb anl at th junction of a flang and tranvr tiffnr lat with θ 45. Th xrion wr validatd with FE analy and mloyd to invtigat th ridual ackd trngth of variou wb anl gomtri. Th govrning limit tat and aociatd trngth wr found to b largly dndnt uon th wb anl thickn and ack lngth. Ralitically roortiond lat girdr wr concludd to b govrnd by th wb buckling limit tat with imnding ductil failur bhaving a a condary limit tat. Ovrall, th caacity xrion may b alid in th dign and analyi of -had tranvrly tiffnd lat girdr loadd undr rdominantly har conidring ribd diagonal fatigu ack configuration with θ 45. Futur work could includ th rformanc of ulmntary FE analy to invtigat th influnc of th tranvr tiffnr lat acing. Alo, xrimntal tt could b rformd on full-cal or mall-cal r-ackd lat girdr to furthr validat th analytical xrion. Finally, advancd invtigation could b undrtakn to dvlo xrion conidring th ot-buckling bhavior and tnion fild action of ackd wb anl. REFERENCES [1] F.P. Br, E.R. Johnton, and J.T. DWolf, Mchanic of Matrial, 4 th d. Nw ork, N: McGraw-Hill, 006, ch. 4, 7, 8. [] C.G. Salmon, J.E. Johnon, and F.A. Malha, Stl Structur: Dign and Bhavior, 5 th d. Ur Saddl Rivr, NJ: Paron Prntic Hall, 009, ch., c. 6.14, 7.7, [3] R.M. Barkr and J.A. 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Vinon, Structural Mchanic: Th Bhavior of Plat and Shll. Nw ork, N: John Wily & Son, nc., 1974, ch. 5, c [54] H.M. Wtrgaard, Baring rur and ack, J. Alid Mch., vol. 6, no. 61, , [55] L.. Sdov, A Cour in Continuum Mchanic, J.R.M. Radok, Tran. Groningn, Nthrland: Woltr-Noordhoff Publihing, 197, c [56] T. Ftt, Str ntnity Factor, T-Str, Wight Function. Karlruh, Grmany: Univritätvrlag Karlruh, 008. [57] S.A. Mguid, Enginring Fractur Mchanic. Nw ork, N: Elvir Scinc Publihing Co., nc, 1989, ch. 3, 5, 6. [58] A. Shukla, Practical Fractur Mchanic in Dign, nd d. Nw ork, N: Marcl Dkkr, 005, ch. -4. Sbatian B. Mnd i a Doctoral candidat at th Dartmnt of Civil and Environmntal Enginring, Univrity of Rhod land, Kington, R. H arnd th B.Sc. dgr in civil nginring from Ro-Hulman ntitut of Tchnology, Trr Haut, N, in 009 and th M.Sc. dgr in civil nginring from Worctr Polytchnic ntitut, Worctr, MA, in 011. H ha rviouly workd a an ntry-lvl tructural nginr in Nw Havn, CT, and a a Rarch Aitant at Worctr Polytchnic ntitut. H ntrd th Univrity of Rhod land in 011 and i currntly rforming rarch and working a a Taching Aitant. ntrnational Journal of Scinc and Enginring nvtigation, Volum, u 19, Augut SSN: Par D: