A consistent beam element formulation considering shear lag effect

Transcription

1 OP Conferene Serie: aterial Siene and Engineering onitent beam element formulation onidering hear lag effet To ite thi artile: E Nouhi et al OP Conf. Ser.: ater. Si. Eng. View the artile online for update and enhanement. Thi ontent wa downloaded from P addre on 3//8 at 7:3

2 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// Conitent Beam Element Formulation Conidering Shear Lag Effet E Nouhi, Kurata, Buntara S an,3 and K Sugiama Department of rhiteture, College of Engineering, Nihon Univerit, - Nakagawara, Koriama, Fukuhima, , Japan. Department of rhiteture, raduate Shool of Engineering, Nihon Univerit, - Nakagawara, Koriama, Fukuhima, , Japan. buntara@arh.e.nihon-u.a.jp btrat. Thi paper preent general olution for beam element whih are derived analtiall b onidering the effet of hear lag phenomenon. The ma and tiffne matrie whih an be obtained from the general olution of diplaement an be aembled for general frame anali b uing finite element proedure.. ntrodution n tuding and evaluating the ultimate trength of teel frame truture, it i apparent that the behavior of teel frame truture at failure i full ontrolled b oneutive loal bukling failure of it member. Hene, to evaluate the loal bukling of eah member element, epeiall thin-walled beam, the tre and deformation at etion hould be onidered properl b taking into aount the effet of hear lag phenomenon. The laial theor of beam and thin-walled member wa unable to reflet the hear lag phenomenon ine it wa baed on the aumption of ro etion remain plane after deformation. The effet of hear lag reult in a ditribution of diret tree in the ro etion whih i different from that predited b the laial theor of beam. Therefore, the formulation of mot analtial and/or emi-empirial method involved man implified aumption. a reult, the an onl handle hear lag problem for a partiular imple geometr and annot be eail etended to truture with omple geometr. The energ approah ha been proven to be a imple and pratial method in hear lag anali []. Later, the hear lag phenomenon wa invetigated in bo-girder uing finite element []. ombination of finite element and tranfer matri tehnique are propoed b [3]. ntrodution of bai warping funtion in the finite element wa ued to eplain the hear lag phenomenon []. However, all of thee approahe are mainl uitable for a partiular tpe of ro etion and tati anali, thu it i not appliable for general appliation. When the hear lag phenomenon i to be onidered for olving dnami anali, the ma propert ha to be introdued and/or approimated. Thi i due to the hort of fundamental theorie whih an inlude the effet of hear lag diretl in the formulation [5]. Preent paper addree a onitent beam element formulation onidering hear lag phenomenon to deal with the diffiultie. 3 To whom an orrepondene hould be addreed. Publihed under liene b Ltd

3 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x///. eneral olution for beam with olid ro etion onidering hear lag n thi etion, ome fundamental equation of an elati beam with olid ro etion ling on a right-handed ae oordinate onvention onidering hear lag will be derived. Diplaement at an arbitrar point on the ro etion beam with member ai in the diretion i given b U. The tranveral diplaement of the arbitrar point in the and diretion are epreed b V and W, repetivel. The following aumption are made for the arbitrar point at the ro etion of a plane beam in deriving equation. U u β f (.a V V (, v( v, W W (, (.b where, U U (,, u u( and β β ( in whih, f f (, whih i defined a a nd order or higher funtion in diretion. Hene, train and diplaement relationhip an be given a, f f κ, γ γ (. W W W,, γ, γ whih are obtained from the aumption that the ro etion remain traight after deformation du dβ dv where,, κ and [ γ ] γ β. Here,,, are normal train in, and d d d diretion and γ, γ, γ are engineering hear train on -, - and - plane. t an be noted from equation (. that the diplaement U and V are independent of ai and - plane, repetivel. uming elati material following the Hooke law, the hear train γ an be related to the hear tre from the relationhip γ, hene the funtion f from equation (. an be obtained a, f ( d (.3 where, i the hear modulu of elatiit and γ i given b γ. Subtituting the funtion f into equation (.a and (., the normal train and tre equation are then given a, κ ( d (.a E σ E Eκ ( d (.b where, E i the normal modulu of elatiit B integrating the normal tre σ for the entire ro etional area, the following aial fore and bending moment an be obtained. E d E ( N σ dd (.5a E σ d Eκ ( dd (.5b where, d, S d d.

4 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// B ubtitution of andκ from equation (.5a and (.5b into (.b, reult in the following normal tre equation. N σ σ (.6 Here, σ i the effet of hear lag in the normal tre whih i given a follow. E E E σ ( dd ( dd ( d (.7 The dnami equilibrium equation at an arbitrar point an be given a follow. σ X ρ U & with γ (.8 B integration, the hear tre in the equation (.8 an be given a, e σ X ρ U && bd (.9 b U where, X i the bod fore, ρ i the material denit, U& i the aeleration with t in time, t b b( i width of ro etion a a funtion of, e i defined a a ditane to the outermot fiber of ro etion where [ ]. e Dnami governing differential equation for the anali of a beam b onidering firt order olution of normal treσ whih orrepond to diplaement u,v and defletion β are given b, N q ρ u&, Q m ρ & β, Q q ρ v& (. ( d where, (, Q d d, q Xd X, q Yd and m Xd. The olution of member fore an be obtained b integrating the equation (. to reah the following equation. N N ( ( ( q ρu&& d, Q Q( ( q ρv&& Q( ( q ρv&& dd ρ && βd d (. Defining differene of hear tree in the following form, QS QS Q QS ( S S (. b b b b where, QS b and QS. Here, firt moment of area from and or to e of the ro b etion are defined b S e bd, S e bd and S e bd. Subtituting equation (. into equation (.5a and (.5b, reulted in, E N E ( q ρv&& Sdd b (.3 E Eκ ( q ρv&& Sdd b 3

5 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// from where, and κ an be obtained a follow. du N ( q ρv&& Sdd d E b (. dβ κ ( q ρv&& Sdd d E b Finall, the diplaement u, β and v of a beam onidering hear lag phenomenon an be obtained a, N k u d ( q v d u( ρ & (.5a E k β d ( ρ β ( q v& d (.5b E k k v dd Q( ( q ρ v& dd β ( v( (.5 E S S where, k Sdd, k Sdd, k and k k. b b b b t an be hown that for a olid retangular beam etion with dimenion ofb h, the value of normal and bending hear lag oeffiient k., k. 3, k 5. and k. omputed are agree with the ommon hear lag oeffiient reported elewhere. B uing the equation in (. whih are baed on the linear firt order approimation, the dnami governing equation, i.e. ma matri, tiffne matri and loading vetor, of a beam element onidering hear lag effet whih onneting beam nodal diplaement with diplaement along the member ai an be derived epliitl b uing the diplaement given in equation (.5... Cantilever beam eample Cantilever beam, hown in figure, are ubjeted to onentrated load P at the free end and ubjeted to ditributed load q are onidered. t i noted that the tranvere vertial diplaement v reult for both ae are different with the laial beam theor. t i alo hown that the differene between tree omputed b the preent method and the laial beam theor i quite ignifiant, i.e. the effet of hear lag mut be onidered in engineering deign. E 5 N/mm P N q N/mm mm 5 mm 5 mm 5 mm v ( mm ( mm Conidering hear lag effet.5 Claial beam theor.6 Retangular beam defletion ( onentrated load v ( mm ( mm Conidering hear lag effet. Claial beam theor σ ( Pa 5. 5 Retangular beam defletion ( ditributed load Normal tre ditribution Figure. Retangular etion antilever beam eample. ( mm -5-5 ( mm f( Funtion f(

6 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// 3. eneral olution for thin-walled beam onidering hear lag n thi etion, the fundamental equation of an elati plane thin-walled beam either open or loe etion whih ling on a right-handed,, Carteian and an auiliarς, oordinate tem b onidering hear lag will be derived. n figure, point O i the entroid of a thin-walled beam ro etion and ling along the oordinate ai. Thi point O will beome the origin of the and oordinate ae. i i defined along the enter line of the thikne t of the urvature thin-walled beam, from where ai n whih i drawn normal to thi urvature line i defined a t ( whih i a funtion of. Diplaement at an arbitrar point P(,, on the ro etion of thin-walled beam are given b U, V, W in the,, diretion and U, V, W in the,, ς diretion. Here, the diplaement u in the diretion i referred to a point where. The tranveral diplaement v, w in the, diretion are originated at point C whih i the hear enter of the ro etion. ς o,u t r w p, U V W v r ς V W p Figure. Cro etion of a thin-walled beam. The train and diplaement relationhip are given a follow. and γ (3. ( ς r where, the radiu urvature of the thin-walled beam ro etion i given b r r(. The diplaement of the point P (,, an be aumed a follow. U u β β f, V v ( φ and W w ( φ (3. From figure, the diplaement an then be epreed in term of,, ς oordinate tem, U U u β β f (3.3a V V o W in v o w in rς φ (3.3b W V in W o v in w o r φ (3.3 where, r ( in ( o and r ( o ( in ς S. The following aumption an be aumed for an arbitrar point at the ro etion of the thinwalled beam. γ (3. 5

7 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// B ubtituting equation (3., the funtion f in equation (3.3a an be olved a follow. f ( γ γ o γ in ς r d χ f ( (3.5 ( ( ( dv dw dφ n whih, γ β, γ β, χ and r ( r d ς ς are defined. d d d ordingl, the average normal diplaement of the thin-walled beam whih an be epreed b uing u d i given a, U U u β β χ F (3.6 where, F ( ( ( r dd γ γ o γ in ς. The normal train i then given b, u β β χ F (3.7 Hene, the normal tre σ for an elati material following Hooke law an be obtained b, σ E Eu Eβ Eβ Eχ EF (3.8 The firt order olution of normal treσ in term of aial fore and moment of the beam an be epreed b, N σ (3.9 where, the internal fore and etion propertie are given a follow: N σ d ; σ d ; σ d ; σ d ; d ; S d ; S d ; S d ; d ; d ; d ; d and d. The dnami equilibrium equation of an arbitrar point on the ro etion i given a below. σ X ς & ρ U ( r ς ς (3. Subtitution of equation (3.9 into equation (3., the hear tre an be olved a, σ ( ς r d X ( ς r d ρ U & ( ς r d (,, ς (3. where, (,, ς i the indeterminate hear tre. The bod fore X X ( and applied moment per unit length term are m, m, m, the firt order approimation of hear tre an be given a, g~ Q (3. i i i,,,, ~ ~, ~ ~ (, ~ ~ (, ~ ~ χ Q Q Q 3 Q Q T g g g g g3 g g g ( where, Q. 6

8 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// Herewith, all the funtion in equation (3. are given epliitl a: Ω g ~ ( ς r rd ; S S S S g ( ; g 3 ( ; g ( S d df g ~ Q g~ φ ( ( Q d df g ~ Q g~ φ π ( π ( Q ; [ ] ; [ ],, d d ( ς r ; ( π ;,, d d e ς r ; S ( ( ς r rd ; S e ( rd S e ( rd Ω r ( ς r rd ; γ ( w β and γ ( v β. Furthermore, F in the equation (3.8 an be olved a, F a i Q i i (3.3 Herewith, all the funtion in equation (3.3 are given epliitl a: a a( ; a a ( ; a 3 a ( ; a a ( ; a ( ˆ ~ ( ~ i i ( i ( gi( C( Cˆ( gi3( π ( S( Sˆ( ; g ~ ~ ( ~ ( ~ ( g3 π g g3( π ; ~ i ( gi( ς r rd ; ˆ i ( i ( d ; C( o ( ς r rd ; C ˆ( C( d ; S( ( ς r rd in ; S ˆ ( S( d. The differential governing equation of -derivative diplaement an alo be given a, N u ( k χ Q k3q kt k E (3.a ˆ ˆ ˆ ˆ β, β (3.b ( ( ( χ k Q k Q k T 3 E k k (3. v β ( kq kq (3.d w β ( kq kq (3.e where, ˆ and ˆ are defined b ˆ E ( χ kq k3q kt k and ˆ E ( χ k3q k33q k3t k3. The hear lag oeffiient are then an be given a follow. ~ k jk λ jakd ( j, k,,3, and k ~ mn g mn ( m, n, (3.5 here, λ, λ 3, λ, a a(, a a (, a 3 a (, a a (, g ~ g~ (, g ~ g~ (, g ~ ~ (, g ~ ~ (. g g 7

9 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// Sine the internal fore N, Q, Q,,, an be determined from the firt order approimation of the dnami governing equation, the bimoment an be olved from, J d J km ( k m d E ( km J ( ( J k ( ( ρ && m Q km Q km m χ km k (3.6 m m ρ S v&& S w&& && φ ( p in, S ( rς o r in d, p ( rς r where, S ( rς r o d d && φ. From the above equation, the right hand ide of the eond order differential equation are known dφ value, thu the general olution of the -derivative twit χ, whih i defined b φ χ, an d be olved. The general olution of the χ an then be ued to derive the element ma and tiffne matrie in the finite element proedure.. Pipe beam eample onidering hear lag To eamine a loe etion eample of thin-walled beam, a olution for pipe beam hown in figure 3 i derived b following the ame proedure for obtaining the general olution of thin-walled beam, U U ς W r V P(,, ς ς U U P V V W Figure 3. Thin-walled irular beam The diplaement of the entroid ro etion of the pipe beam and the, oordinate tem an be epreed in term of rotational angle β φ, β, β a follow. U u β f, V v and W (. where, β β, β. B auming w, φ, ( r ς in and ( r ς o, the diplaement beome U U, V v and W. Hene, equation (. ield to the following equation. U U u β f, V vo and W vin (. The train and diplaement relationhip for pipe beam an further be epreed a,, ( r W and γ (.3 ς ( r ς Hene the funtion f from equation (. an be olved for, ( γ γ o ( r ς W d d ( r ς d f ( d (. 8

10 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// Subtitution of equation (. into equation (.3 reult in the following normal train and tre equation. ( u β o ( r ς d (.5a E σ ( E Eu Eβ o ( r ς d (.5b B integrating the normal tre σ for the entire ro etional area, the following aial fore and bending moment an be obtained. E N σ ( d Eu o ( r ς dd (.6a E σ ( ( ( d Eβ r ς in r ς d d o (.6b For pipe beam etion, σ, the firt order olution of normal tre σ beome, N σ (.7 The dnami equilibrium equation at an arbitrar point an be given a follow. σ ς X ρ U ( r & (.8 ς ς B integration, the hear tre in the equation (.8 ield to, Q t( r ς d ( t (.9 where, due to mmetri geometr of the ro etion in ai, from ( π ondition, the QS equation (.9 reulted in and t QS t at [ ] in whih, S ( π t r ς d and S π ( r t ς d are denoted. From whih, the following equation an be obtained a, E N σ d Eu rknq (. EQ EQ σ d Eβ g( d Eβ km t where, g ( ( S S o ( r ς d, k n g( d and k m tr g( d are given. t Finall, the diplaement u, β and v of the pipe beam onidering hear lag phenomenon an be obtained a, N rk n u d ( q v d u( E ρ & (.a k m β d ( q ρv& d β ( (.b E k k m v dd Q( ( q v dd ( v( E ρ & β (. 9

11 WCC/PCO OP Conf. Serie: aterial Siene and Engineering ( doi:.88/ x/// S where, k and km km k. t Thu, the hear lag oeffiient in the diplaement olution an be obtained a k n, k m, k. and k.. m.. Cantilever beam eample Cantilever beam, hown in figure, are ubjeted to onentrated load P at the free end and ubjeted to ditributed load q are onidered. t i noted that the tranvere vertial diplaement v reult for both ae are different with the laial beam theor. Thu, the effet of hear lag mut be onidered in engineering deign. r t v ( mm ( mm ( mm Conidering hear lag effet Claial beam theor Conidering hear lag effet Claial beam theor.3.3 Pipe beam defletion ( onentrated load Pipe beam defletion ( ditributed load Figure. Pipe beam etion antilever beam eample. v ( mm 5. Conluion Baed on the laial beam theor where the effet of hear lag i inluded in the formulation of diplaement, train and tree of olid, thin-walled and pipe beam onitentl without uing an kind of tre approimation funtion. The hear lag phenomenon i now an be onidered diretl for olving dnami problem in whih the ma and tiffne matrie an be derived onitentl. Referene [] Hadji-rgri J and Co H L 9 Diffuion of load into tiffened panel of varing etion Br. eron. Re. Counil Report. em. 969 Reiner E 96 nali of hear lag in bo beam b the priniple of minimum potential energ Q. ppl. ath [] alolm D J and Redwood R 97 Sehar Lag in tiffened bo-girder J. Strut. Div. SCE 96 ST7 3-5 offatt K R and Dowling P J 975 Shear lag in teel bo-girder bridge Strut. Engineer [3] Tear 996 Shear lag in the behavior of thinwalled bo bridge Comp. and Strut. 67- [] Prokić 996 New warping funtion for thin-walled beam : Theor J. Strut. Eng. SCE ST 37- Prokić 996 New warping funtion for thin-walled beam : Finite element method and appliation J. Strut. Eng. SCE ST 3-5 Prokić New finite element for anali of hear lag Comp. and Strut. 8 - [5] Kurata 9 eneral olution of beam onidering hear lag phenomenon (in Japanee Summarie of Teh. Paper of Reearh nnual eeting Nihon Univ. College of Eng. 9 -