FINITE ELEMENT ANALYSIS OF STRESSES IN BEAM

Transcription

1 Dpartmnt of Civil and Structural Enginring Conrad Arnan Ribas FINITE EEMENT ANAYSIS OF STRESSES IN BEAM STRUCTURES Final projct Espoo, Suprvisor: Profssor Jukka Aalto Instructor: Profssor Jukka Aalto

2 ABSTRACT AATO UNIVERSITY SCHOOS OF TECHNOOGY PO Box 000, FI AATO ABSTRACT OF THE FINA PROJECT Author: Conrad Arnan Ribas Titl: Finit lmnt analsis of strsss in bam structurs School: Enginring Dpartmnt: Civil and Structural Enginring Profssorship: Structural mchanics Cod: Rak-54 Suprvisor: Jukka Aalto Instructor(s): Jukka Aalto Abstract: This Final projct dals with dtrmining of strsss in bam structurs. First, warping function basd on finit lmnt formulations for dtrmining shar strsss at bam cross-sctions undr biaxial bnding and twisting ar dvlopd and discrtid. Scond, finit lmnt programs for analsis of bams and thr-dimnsional frams ar dvlopd. Ths programs, which hav bn implmntd in MATAB nvironmnt, ar thn combind to a program sstm, which dtrmins fficintl th combind stat of strsss at an cross-sction of a bam or 3D fram. In ordr to compar th rsults of th prsntd approximat tchniqu to practicall xact ons, a program, basd on th finit prism mthod, for solving strsss in a simpl supportd prismatic bam has furthr bn dvlopd. Th rsults of th prsntd tchniqu and finit prism mthod ar compard using xampl problms. In gnral, closd thin-walld and solid cross-sctions giv rall accurat rsults, whras opn thinwalld cross-sctions do not. Som discrpancis in th shar strsss valus can b found usuall in th points of th bam whr an load starts or finishs. Composit crosssctions can b analsd appling this mthodolog. Finall, applications of th dvlopd program ar dmonstratd with som mor practical xampls. Dat: anguag: English Numbr of pags: 90 Kwords: Finit lmnt analsis, 3D fram, Brnoulli-Eulr kinmatics, warping function, combind strss stat, irrgular cross-sction gomtr, composit cross-sction

3 Finit lmnt analsis of strsss in bam structurs 2 TABE OF CONTENTS Abstract... Tabl of contnts... 2 Prfac Introduction Aims and objctivs of th rsarch Rsarch mthods Finit lmnt mthod Quadratic isoparamtric lmnts -D Biquadratic isoparamtric agrang lmnts 2-D Numrical intgration Mthod dvlopmnt Assumptions Fram analsis Finit lmnt quations of a bar lmnt Coordinat chang Global sstm Strss rsultants Strss distribution Cross-sctional proprtis Tnsion / Comprssion... 30

4 Finit lmnt analsis of strsss in bam structurs Biaxial bnding Shar cntr Torsion Gnral cas Biquadratic finit prism mthod Assumptions Mthod dscription Validation of th mthod Comparison btwn th two mthods Discussion of th rsults Dmonstrativ xampls Conclusions Acknowldgmnts Rfrncs Appndix... 89

5 Finit lmnt analsis of strsss in bam structurs 4 PREFACE Dtrmining of strsss in bam structurs is standard taching matrial in basic courss on mchanics of matrials and structural mchanics [], [2]. Howvr, thr ar two topics which ar not dalt with nough dpth at this lvl. Th first thing is torsion. Torsion of circular and thin-walld hollow shafts and fr torsion of thin-walld opn cross-sctions ar usuall considrd in ths txts carfull. Introduction to torsion of bams with gnral cross-sction is, howvr, don using so-calld mmbran analog, which is basd on strss function formulation of fr torsion of a cross-sction with gnral shap [3]. This mmbran analog has bn chosn in basic taching txts, bcaus it is an xcllnt tool in xplaining th bhaviour of shar strsss in a crosssction. Th mmbran analog as such cannot, howvr, giv dfinit numrical rsults to th shar strsss in a twistd cross-sction. In ordr to do this, on should b abl to solv th original problm of fr torsion. Diffrnt formulations of this problm ar prsntd in th litratur (for xampl [3], [5] and [6]). Solution of this problm basd on strss function formulation [3] and th finit lmnt mthod is wll known and dmonstratd in rfrnc [4]. This formulation is not, howvr, straightforward and as to implmnt, if th crosssction contains hols. In this contxt, th warping function formulation [6] is simplr and th corrsponding finit lmnt solution using linar triangular lmnts has first bn prsntd in rfrnc [7]. In this projct, th finit lmnt quations for solving th torsional problm ar basd on this warping function formulation and biquadratic isoparamtric lmnts [8]. Th scond thing, which is not tackld through in standard txts on structural mchanics, is shar strsss causd b bnding. Tpicall shar flux on an longitudinal sction btwn two parts of th bam can b dtrmind using simpl quilibrium considration. With th hlp of such shar fluxs avrag, but not xact, shar strsss on th cross-sction can furthr b dtrmind. Howvr, dtrmination of th shar strsss distribution causd b bnding on a cross-sction of an shap is possibl. Corrsponding formulations hav bn prsntd for xampl in rfrncs [3] and [6]. Finit lmnt solution basd on warping function formulation and using linar triangular lmnts has bn first prsntd in rfrnc [9]. In this thsis th finit lmnt quations for solving th bnding shar strsss ar basd on this warping function formulation and biquadratic isoparamtric lmnts [0], [].

6 Finit lmnt analsis of strsss in bam structurs 5 2 INTRODUCTION 2. AIMS AND OBJECTIVES OF THE RESEARCH Th goal in this final projct is to dvlop a finit lmnt basd program in MATAB nvironmnt, which solvs th strss rsultants of a bam structur and using ths, th combind stat of strsss at an point in this structur. Th program thus consists of two main parts. Th first part solvs th strss rsultants of th structur using cubic C continuous spac fram lmnts undr givn loading. Th scond part prforms strss analsis of th cross-sction b combining th ffcts of axial loading, torsion and biaxial bnding using th obtaind strss rsultants as data. 2.2 RESEARCH METHODS Th spac fram program uss tpical straight, cubic C continuous spac fram lmnts [2] of constant stiffnss. Th loading consists of linarl distributd axial and transvrs load and twisting momnt within lmnts and concntratd loads and momnts at th nods. Although th lmnts ar rathr simpl, th produc xact solution for th strss rsultants at th lmnt nods if th lmnts ar straight and uniform and th loading is just as dscribd. Th can also b usd in connction with modratl curvd bam structurs and structurs with variabl thicknss if th finit lmnt grid is chosn to b sufficintl dns. 0 Th program for calculating th cross-sctional strsss uss C continuous isoparamtric biquadratic quadrilatral lmnts with nin nods and numrical intgration. It calculats th ffcts of normal forc, shar forcs, twisting momnt and bnding momnts to th cross-sctional strsss and thus obtains th combind stat of strsss. In ordr to stud th fficinc of th dvlopd tchniqu, a program, basd on th finit prism mthod [4], [3], [4] and [5], has also bn dvlopd in this thsis. This program is basd on thr dimnsional lasticit quations and dos not hav th rstrictions of classical bam thor. Thus th rsults obtaind using this program giv good comparison data for studing th accurac of th nhancd bam thor of this projct. Th finit prism program also uss C 0 continuous isoparamtric biquadratic lmnts and numrical intgration. It can handl simpl supportd straight bams with arbitrar cross-sction. Th loading can b surfac load, which is picwis constant in axial dirction and quadraticall

7 Finit lmnt analsis of strsss in bam structurs 6 distributd in transvrs dirction, volum forc, which is picwis constant in axial dirction, or distributd nodal lin load, which is picwis constant in axial dirction. Thr simpl supportd xampl bams, with solid rctangular, opn U-shapd and hollow rctangular cross-sction ar considrd in th comparison stud. Each bam is loadd b narrow uniform rctangular surfac load acting on th scond half of th bam. Both cntric loading causing pur bnding and ccntric loading causing mixd torsion and bnding hav bn studid. Th strss rsults of th two mthods ar compard at quartr points and midpoint of th bam. Th final xampl problms of th thsis hav bn slctd to dmonstrat th applications of th dvlopd program. Th first problm considrs a circular bam in th horiontal plan loadd b uniform lin load along th axis of th bam. Th cross-sction is trapoidal and non-smmtric. Both torsion and bnding ar prsnt. Th scond xampl problm is a slightl curvd smmtric girdr in th vrtical plan loadd b uniform transvrs load in th smmtr plan. Th cross-sction is solid and T-formd, but to avoid strss singularitis at th cornrs, th hav bn smoothd using circls. Both axial loading and bnding is prsnt in this problm. Th third xampl problm considrs th application of th dvlopd mthod to a composit bam. Th cross-sction consists of concrt dck and stl U-shapd thin-walld plat. Both pur bnding and twisting cass ar studid sparatl.

8 Finit lmnt analsis of strsss in bam structurs 7 3 FINITE EEMENT METHOD In ordr to solv th lastic problm, th finit lmnt mthod will b usd with modlling and discrtiation of th objct undr stud. On- and two-dimnsional lmnts ar ndd, so th basics of both ar going to b dscribd [6]. In all cass, th sort of lmnt is going to b th sam, which is quadratic in on dimnsion and biquadratic agrang in two dimnsions, to obtain mor accurat rsults. Morovr, th us of isoparamtric lmnts is important to b abl to modl cross-sctions of an shap. Furthrmor, numrical intgration [6] for solving on- and two-dimnsional intgrals is xplaind in connction with th application of th finit lmnt mthod. 3. QUADRATIC ISOPARAMETRIC EEMENTS -D A quadratic lin lmnt with thr quall spacs nods in natural coordinats, ξ, is considrd (Fig. 3.). Th shap functions of this lmnt ar obtaind using quadratic agrang polnomials as ( ξ ξ2)( ξ ξ3) ( ξ 0)( ξ ) N ( ξ) = = = ξ( ξ), ( ξ ξ2)( ξ ξ3) ( 0)( ) 2 ( ξ ξ)( ξ ξ3) [ ξ ( )]( ξ ) N2 ( ξ ) = = = ( ξ)( + ξ), (3.) ( ξ2 ξ)( ξ2 ξ3) [0 ( )](0 ) ( ξ ξ)( ξ ξ2) [ ξ ( )]( ξ 0) N3 ( ξ ) = = = ( ). ( ξ ξ )( ξ ξ ) [ ( )]( 0) 2 ξ + ξ Fig. 3.: Shap functions of a thr nod quadratic lin lmnt in natural coordinat ξ Drivativs N, dn / dξ of th -dimnsional shap functions (3.) with rspct to th i ξ natural coordinats ξ ar i

9 Finit lmnt analsis of strsss in bam structurs 8 N N N, ξ 2, ξ 3, ξ ( ξ ) = ( 2 ξ ), 2 ( ξ) = 2 ξ, ( ξ ) = (+ 2 ξ ). 2 (3.2) In Fig. 3.2 (a), a thr nod quadratic isoparamtric lin lmnt is prsntd b a lin sgmnt of lngth 2 in th natural ξ lin and in Fig. 3.2 (b) b a curvd lin sgmnt in th phsical x,-plan. Fig. 3.2: A thr nod quadratic isoparamtric lin lmnt (a) in natural coordinat ξ and (b) in phsical coordinats x, Th shap of th lmnt in th phsical plan is xprssd b position vctor r( ξ ) = x( ξ) i+ ( ξ) j, (3.3) whr th functions x( ξ ) and ( ξ ) ar 3 x( ξ) = N ( ξ) x, i= 3 ( ξ) = N ( ξ). i= i i i i (3.4) Diffrntial vctor btwn points ξ and ξ + dξ in th phsical plan is dr dx d dr = dξ = i+ j dξ = ( x, ξ i +, ξ j ) dξ. (3.5) dξ dξ dξ Th lngth of this diffrntial vctor is 2 2 ds = x, ξ +, ξdξ (3.6)

10 Finit lmnt analsis of strsss in bam structurs 9 and it is th lngth of a diffrntial lin lmnt corrsponding to diffrntial chang dξ of th natural coordinat. Th drivativs of th coordinats functions x( ξ ) and ( ξ ) quation (3.6) ar obtaind using formulas in 3 x, ( ξ) = N ( ξ) x, ξ i= 3 i= i, ξ, ( ξ) = N ( ξ). ξ i, ξ i i (3.7) 3.2 BIQUADRATIC ISOPARAMETRIC AGRANGE EEMENTS 2-D Shap functions of a biquadratic quadrilatral agrangian lmnt in natural co-ordinats ξ and η (Fig. 3.3) can b obtaind with th hlp of shap functions (3.) using th formula N3( i ) + j( ξη, ) = Ni ( ξ) N j( η), i, j =,2,3. (3.8) Fig. 3.3: Thr shap functions of a nin nod biquadratic quadrilatral agrang lmnt in natural co-ordinatsξ and η Thrfor, on obtains

11 Finit lmnt analsis of strsss in bam structurs 0 N( ξη, ) = N( ξ) N( η) = ξ( ξη ) ( η), 4 N2( ξ, η) = N2( ξ) N( η) = ( ξ)( + ξ) η( η), 2 N3( ξη, ) = N3( ξ) N( η) = ξ( + ξη ) ( η), 4 N4( ξ, η) = N( ξ) N2( η) = ξ( ξ)( η)( + η), 2 tc. (3.9) Partial drivativs N, N / ξ and N, N / η of th 2-dimnsional shap functions (3.9) ar i ξ i iη i N N, ξ N, η ( ξη, ) = ξ( ξ)( 2 η), ( ξη, ) = ( 2 ξη ) ( η), 4 4 N ( ξ, η) = ( ξ)( + ξ)( 2 η), 2, 2, (, ) ( ), η ξ ξη = ξη η 2 N3, ξ( ξη, ) = (+ 2 ξη ) ( η), N3, η( ξη, ) = ξ( + ξ)( 2 η), 4 4 N4, (, N4, (, ) ( 2 )( )( ), η ξη) = ξ( ξ) η, ξ ξη = ξ η + η 2 tc. tc. (3.0) Fig. 3.4: A nin nod quadratic isoparamtric quadrilatral lmnt (a) in natural coordinat, (b) in phsical coordinats x, ξ η and

12 Finit lmnt analsis of strsss in bam structurs In Fig. 3.4 (a), a nin nod quadrilatral agrang isoparamtric lmnt is prsntd b a squar of sid lngths 2 in th natural ξ, η plan and in Fig. 3.4 (b) b a curvd quadrilatral in th phsical x,-plan. Th shap of th lmnt in th phsical plan is xprssd b position vctor r( ξ, η) = x( ξ, η) i+ ( ξ, η) j, (3.) whr th functions x( ξ, η ) and ( ξ, η ) ar 9 x( ξη, ) = N ( ξη, ) x, i= 9 ( ξη, ) = N ( ξη, ). i= i i i i (3.2) Diffrntial vctor btwn points ξ, η and ξ + dξη, in th phsical plan is r x dr = dξ = i+ j dξ = ( x, ξ i +, ξ j ) dξ. (3.3a) ξ ξ ξ and a diffrntial vctor btwn points ξ, η and ξ, η + dη in th phsical plan is r x dr2 = dη = i+ j dη = ( x, η i +, η j ) dξ. (3.3b) η η η Th diffrntial ara of th lmnt in th phsical plan corrsponding to a diffrntial squar dξdη in th natural plan is x, ξ, ξ 0 x, ξ, ξ da= dr dr2i k = x, η, η 0 dξdη = dξdη. (3.4) x, η, η 0 0 This rsult is usuall xprssd as da = dt J dξdη, (3.5) whr dt J is dtrminant of so-calld Jacobian matrix x ξ ξ x, ξ, ξ J = x x, η,. (3.6) η η η Th partial drivativs of th functions x( ξ, η ) and ( ξ, η) in formula (3.2) ar obtaind using formulas

13 Finit lmnt analsis of strsss in bam structurs ξ ξη = i, ξ ξη i ξ ξη = i, ξ ξη i i= i= x, (, ) N (, ) x,, (, ) N (, ), 9 9 η ξη = i, η ξη i η ξη = i, η ξη i i= i= x, (, ) N (, ) x,, (, ) N (, ). (3.7) atr, partial drivativs of th shap functions N i with rspct to th phsical coordinats x and in trms of th corrsponding drivativs with rspct to th natural coordinats ξ and η ar ndd. Using th chain rul of diffrntiations on can writ Ni x Ni N i = + ξ ξ x ξ Ni x Ni Ni = + η η x η or with our notation N = x, N +, N N = x, N +, N i, ξ ξ i, x ξ i, i, η η i, x η i, (3.8) or N i, ξ N ix, = J. N i, η N i, (3.9) Invrting this rlation givs N ix, N i, ξ = J, N i, N i, η (3.20) whr J is th invrs of th Jacobian matrix. Th 2 2 Jacobian matrix J can asil b invrtd and th rsult is,, η ξ J = dt x, η x,, (3.2) J ξ whr th Jacobian dtrminant is x, ξ, ξ dt J = = x, ξ, η x, η, ξ. (3.22) x,, η η Rlations (3.20) can now b writtn in final form as, (,,, Nix= η Niξ ξ Ni, η ), dt J, (,,, Ni= x η Niξ + x ξ Ni, η ). dt J (3.23)

14 Finit lmnt analsis of strsss in bam structurs NUMERICA INTEGRATION If tpical two dimnsional problms ar anald using th finit lmnt mthod, following lin and surfac intgrals ovr th lmnt ar ndd () s I = F s ds, I2 = F2 ( x, ) da. (3.24) A Ths intgrals can b xprssd in natural coordinats, b using quations (3.6) and (3.5) and changing th intgration limits + I = f ( ξ ) dξ, I f ( ξ, η) dξdη =, (3.25) whr 2 2 ξ ξ 2 2 f ( ξ ) = F ( ξ) x, ( ξ) +, ( ξ), f ( ξ, η) = F [ x( ξ, η), ( ξ, η)]dt J ( ξ, η). (3.26) Intgrands (3.26) ar, in gnral cas, so complicatd functions, that analtical intgration is impossibl. Thrfor numrical intgration is ndd. Approximat valus of intgrals (3.26) ar obtaind numricall using formulas n k k= I w f ( ξ ), I2 w f2( ξ, η ), (3.27) k n k= k k k whr n and n ar th numbr of intgration points of th lin lmnt and th surfac lmnt, rspctivl, w k is th wight and ξ k and η k ar th coordinats of th intgration point k. Ths quantitis for on and two dimnsional intgration ar listd in Tabls 3. and 3.2 rspctivl.

15 Finit lmnt analsis of strsss in bam structurs 4 Tabl 3.: Intgration wights, coordinats of intgration points and maximum intgr xponnt p,, p whichh allows xact intgration of powr ξ, for on dimnsional Gauss numrical intgration ovr intrval ξ. n Figur k wk ξ p k 2 0 ξ ξ ξ /9 8/9 5/9 / 3 +/ 3 3/ /5 3 5 Tabl 3.2: Intgration wights, coordinats of intgration points and maximum intgr xponnt p,, p p whichh allows xact intgrationn of powr trm ξ η, for two dimnsional Gauss numrical intgrationn ovr squar ξ, η. n Figur k w k ξk η k p / +/ / +/ / / +/ +/ /8 40 /8 25 /8 40 /8 64 /8 40 /8 25 /8 40 /8 25 / /5 3 /5 3 /5 3 /5 3 /5 3 / / 5 3/ 5 3/ 5 3/ 5 3/ 5 3/ 5 5

16 Finit lmnt analsis of strsss in bam structurs 5 4 METHOD DEVEOPMENT In ordr to achiv th strss distribution, first of all an analsis of th fram must b ralisd to obtain strss rsultants that act in ach cross-sction, and latr, th strss valus ar calculatd with this information. 4. ASSUMPTIONS Whn doing th analsis, som hpothsis will b takn into account: matrial is lastic and isotropic and th analsis will b linar gomtricall and matriall. Cross-sctions can b mad of diffrnt matrials and th fram can b composd of diffrnt cross-sctions. Thus, plasticit, ffct of joints and buckling will not b considrd. Th fram can b loadd in diffrnt was: distributd axial forcs, transvrs forcs and twisting momnts; and concntratd nodal forcs and momnts. 4.2 FRAME ANAYSIS Th first us of th finit lmnt mthod is mad with th fram analsis. Th ntir fram must b discrtid into svral bar lmnts. Each lmnt provids a st of quations to form th gnral quation st and solv th problm [7], [8] FINITE EEMENT EQUATIONS OF A BAR EEMENT Th basic unknown functions of a bar lmnt undr tnsion/comprssion, torsion and biaxial bnding ar axial displacmnt, ux ( ), dflctions vx ( ) and wx ( ) and angl of twist ϕ ( x). Th axial displacmnt and angl of twist within an lmnt ar approximatd using t 0 C continuous linar finit lmnt approximations ux ˆ( ) = N( xu ) + N( xu ), l l 2 2 ˆ ϕ ( x) = N ( x) ϕ + N ( x) ϕ, l l t t 2 t2 (4.) whr N = ξ, N = ξ, (4.2) l l 2 and

17 Finit lmnt analsis of strsss in bam structurs 6 ξ = x x. (4.3) Th dflctions within an lmnt ar approximatd using C c continuous cubic finit lmnt approximations c c c vx ˆ( ) = N ( x) v + N ( x) ϕ + N ( x) v + N ( x) ϕ, c wx ˆ( ) = N ( x) w N ( x) ϕ + N ( x) w c c c 4 2 N ( x) ϕ c, 4 2 (4.4) whr 2 3 c 2 N = 3ξ + 2 ξ, N = 3 c 2 3 c 2 ( ξ 2ξ + ξ ), N = 3 ξ 2 ξ, N = ξ c ( + ξ 3 ). (4.5) Fig. 4.: (a) Numbring of dgrs of frdom, (b) gnralid displacmnts and (c) gnralid forcs Numbring of th dgrs of frdom of th bam lmnt is shown in Fig. 4.a, th gnralid displacmntss a, i =,,2 (s Fig. 4.b) ar th u, v, w ( i =,2) and rotations ϕ, ϕ, ϕ, ( i =,2) and th corrsponding gnralid forcs i i i i ti i i nodal displacmnts

18 Finit lmnt analsis of strsss in bam structurs 7 F, i =,,2 (s Fig. 4.c) ar th nodal forcs U, V, W ( i =,2) and momnts i i i i Mti, Mi, Mi, ( i=,2). In th following, rlationships btwn gnralid forcs and gnralid displacmnts of th bam lmnts will b drivd. Th approximations (4.) and (4.4) of th basic unknowns can b writtn as ux ˆ( ) vx ˆ( ) uˆ ( x) = N ( x) a (4.6) wx ˆ( ) ˆ ϕt ( x) 4 whr l l N N c c c c 0 N N2 0 N N4 N = (4.7) c c c c 0 0 N 0 N N3 0 N4 0 l l N N and a T = [ u v w ϕt ϕ ϕ u2 v2 w2 ϕt2 ϕ2 ϕ2] 2 a 2 a. (4.8) Th approximations of th axial strain, twist pr unit lngth and curvaturs ar duˆ l l ˆ ε x( x) ( x) = N, x( x) u + N2, x( x) u2, dx d ˆ ϕ ˆ( ) t ( ) l, ( ) l θ x x = N x x ϕt + N2, x ( x ) ϕt2, dx 2 dwˆ c c c c ˆ κ( x) ( x) = N 2, xx( x) w + N2, xx( x) ϕ N3, xx( x) w2 + N4, xx( x) ϕ2, dx 2 dvˆ c c c ˆ κ( x) ( x) = N 2, xx( x) v N2, xx( x) ϕ N3, xx( xv ) 2 N c 4, xx ( xϕ ) 2 dx (4.9) and th can b writtn as ˆ ( ) ( ) Ε x = B x a, (4.0)

19 Finit lmnt analsis of strsss in bam structurs 8 whr ˆ ε x( x) ˆ ˆ( θ x ) Ε( x) = ˆ κ ( x) ˆ κ ( x) 4 (4.) and B 42 l l N, x N2, x l l N, x N2, x 0 0 = c c c c 0 0 N, xx 0 N2, xx N3, xx 0 N4, xx 0 c c c c 0 N, xx N2, xx 0 N3, xx N4, xx (4.2) In th following, th finit lmnt quations of a bam lmnt ar drivd using th principl of virtual work. In this contxt, th virtual gnralid displacmnts δu and th virtual gnralid strains δε ar approximatd similarl to δux ˆ( ) δvx ˆ( ) δuˆ( x) = N ( x) δa (4.3) δ wx ˆ( ) δϕˆ t ( x) and δε x( x) ˆ δθ ( x) δε ( x) = B ( x) δa, (4.4) δκ ( x) δκ ( x) whr th virtual gnralid displacmnts ar δ a T = = [ u v w t u2 v2 w2 t2 2 2] δ a 2 δa δ δ δ δϕ δϕ δϕ δ δ δ δϕ δϕ δϕ. (4.5) Now, th principl of virtual work to lmnt is usd. That is

20 Finit lmnt analsis of strsss in bam structurs 9 δw δw + δw =, (4.6) int xt 0 whr th intrnal virtual work of th lmnt is Ε Σ, (4.7) T δwint = ( Nδεx + Mtδθ + M δκ + Mδκ) dx= δ dx whr δε is th vctor of virtual gnralid strains and N( x) M t ( x) Σ ( x) = M ( x) M ( x) 4 (4.8) is th vctor of gnralid strsss. Th rlation btwn th gnralid strsss and gnralid strains can b writtn as Σ= D( Ε Ε 0), (4.9) whr ( EA) ( GI) 0 0 t = 0 0 ( EI) ( EI) 0 0 ( EI) ( EI) D (4.20) is th gnralid strss-strain matrix, whr its trms ar dscribd in chaptr 4.3. and 4.3.4, and Ε 0 ε x0 θ 0 = κ 0 κ 0 (4.2) is th vctor of initial gnralid strains. B insrting xprssion (4.9) of gnralid strsss into th xprssion (4.7) of intrnal virtual work, on gts

21 Finit lmnt analsis of strsss in bam structurs 20 δw = δε DΕdx+ δε DΕ dx. (4.22) T T int 0 For th xtrnal virtual work of th lmnt, δw = ( q δu+ q δv+ q δw+ mδϕ ) dx xt x t + Uδu( x ) + Vδv( x ) + Wδw( x ) + M δϕ ( x ) + M δϕ ( x ) + M δϕ ( x ) t t + U δu( x ) + Vδv( x ) + Wδw( x ) + M δϕ ( x ) + M δϕ ( x ) + M δϕ ( x ), t2 t (4.23) whr qx ( x ), q ( x ) and q ( x) ar axial and transvrs distributd loads and mt ( x ) is distributd twisting momnt load. Now, th total virtual work of th lmnt can b obtaind as δw δw + δw int xt = δε DΕdx + δε DΕ dx + δu qdx T T T 0 + Uδu( x ) + Vδv( x ) + Wδw( x ) + M δϕ ( x ) + M δϕ ( x ) + M δϕ ( x ) t t + U δu( x ) + Vδv( x ) + Wδw( x ) + M δϕ ( x ) + M δϕ ( x ) + M δϕ ( x ) t2 t (4.24) whr qx q q m q = (4.25) is th vctor of distributd gnralid loads. Basd on th principl of virtual work, th total virtual work (4.24) should disappar with arbitrar virtual gnralid displacmnts δu ( x). Substituting th approximations (4.9), (4.0), (4.2) and (4.3) into th xprssion (4.24) and taking furthr into account that ( δux) = δu = δa, ( δ vx) = δv = δa2, ( δ wx) = δw = δa3, δϕt( x) = δϕt = δa4, δϕ( x) = δϕ = δa5, ( δϕ x) = δϕ = δa6, (4.26) ( δux2) = δu2 = δa7, ( δ vx2) = δv2 = δa8, ( δ wx2) = δw2 = δa9, δϕt( x2) = δϕt2 = δa0, δϕ( x2) = δϕ2 = δa, ( δϕ x2) = δϕ2 = δa2

22 Finit lmnt analsis of strsss in bam structurs 2 th xprssion (4.24) gts th form δwˆ = δa ( B DB dxa + B DΕ dx+ N qdx+ F ). (4.27) T T T T 0 Th principl of virtual work δ W = 0 with arbitrar virtual gnralid displacmnts rsults to th lmnt quation of th bam lmnt δa now F = K a R, (4.28) whr F T = [ U V W Mt M M U2 V2 W2 Mt2 M 2 M2] F 2 F, (4.29) is th gnralid nodal forc vctor, K = B DB dx T (4.30) is th lmnt stiffnss matrix and R T T = N q dx + B DΕ 0 dx (4.3) is th lmnt load vctor. In th gnral cas, whn th gnralid strss-strain matrix D, th initial gnralid strain vctor Ε 0 and distributd load vctor q ar functions of x, th lmnt stiffnss matrix and load vctor can b obtaind using numrical intgration. Th intgration variabl is first changd to ξ rsulting to T K = B ( ξ ) D( ξ) B ( ξ) dξ (4.32) 0 and T T = [ ( ) ( ) + ( ) ( ) 0( )] 0 R N ξ q ξ B ξ D ξ Ε ξ dξ (4.33)

23 Finit lmnt analsis of strsss in bam structurs 22 Numrical intgration is thn applid rsulting to n T K = wk B ( k) D( k) B ( k) k = (4.34) and n T T = wk [ ( k) ( k) + ( k) ( k) 0( k)] k = R N q B D Ε, (4.35) whr th notation ( k) = ( ξ k ) for th valu of a function at intgration point k is usd. Hr th problm is rstrictd to th cas in which th gnralid strss strain matrix D and th gnralid initial strain vctor Ε 0 ar constants, D and Ε 0, within th lmnt. Th distributd load q is supposd to b a linar function of x and can b writtn as q( ξ ) = q ( ξ) + q ξ. (4.36) 2 It is as to s that th lmnt stiffnss matrix and th lmnt load vctor gt th form of T K = B ( ξ) D B ( ξ) dξ 0 (4.37) and = ( ξ) T ( ξ) dξ + ξ T ( ξ) dξ T 2 + ( ξ) dξ R N q N q B D Ε (4.38) Th intgrations in quations (4.37) and (4.38) can b prformd analticall rsulting to final xprssions (4.39) for th lmnt stiffnss matrix and (4.40) for th lmnt load vctor.

24 Finit lmnt analsis of strsss in bam structurs 23 ( EA) ( EA) ( EI) 2( EI ) 6( EI ) 6( ) 2( ) 2( ) 6( ) EI EI EI EI 6( EI) ( EI ) 2( EI ) 6( EI ) 6( EI) 2( EI) 2( EI) 6( EI ) 6( EI) ( GI) t ( GI) t ( EI) 6( EI ) 4( EI) 4( EI ) 6( EI ) 6( EI) 2( EI) 2( EI) ( EI) 6( EI ) 4( EI ) 4( ) 6( ) 6( EI) EI EI 2( EI ) 2( EI ) K = ( EA) ( EA) ( ) 2( EI) 6( EI) 6( ) 2( ) 2( EI ) 6( EI ) EI EI EI 6( EI) ( EI) 2( EI ) 6( EI) 6( EI ) 2( EI) 2( EI ) 6( EI) 6( EI ) ( GI) t ( GI) t ( EI) 6( EI ) 2( EI) 2( EI ) 6( EI) 6( EI) 4( EI) 4( EI) ( EI) 6( EI ) 2( EI ) 2( EI ) 6( ) 4( ) 6( EI) EI EI 4( EI ) (4.39) R 2qx + q 2 x ( EA ) ε x0 6 7q+ 3q q+ 3q m + m 2 ( GI t ) θ0 6 q q ( EI) κ0 ( EI) κ q q ( EI) κ0 + ( EI) κ = qx + 2qx2 + ( EA ) ε x0 6 3q + 7q2 20 3q + 7q2 20 m + 2m2 + ( GI t ) θ0 6 q q ( EI) κ0 + ( EI) κ q q2 2 + ( EI) κ0 ( EI) κ (4.40)

25 Finit lmnt analsis of strsss in bam structurs COORDINATE CHANGE Whn assmbling th global stiffnss matrix and th global load vctor, it is compulsor that lmntar componnts of both ar xprssd in th sam coordinat sstm, which is arbitrar and will b calld th global coordinat sstm. Th dgrs of frdom and lmnt quations of a thr dimnsional fram lmnt in lmnt coordinat sstm x,, (s Fig. 4.2) ar idntical to thos of a bam lmnt undr axial loading, twisting and biaxial bnding (s Eq. 4.28). Th lmnt quations in lmnt coordinat sstm can thus b writtn as F = K a R, (4.4) whr K and R ar th lmnt stiffnss matrix and th lmnt load vctor of a bam lmnt undr axial loading, twisting and biaxial bnding. j k i Global coordinat sstm x n l Elmnt coordinat m sstm 2 x Fig. 4.2: Global coordinat sstm and lmnt coordinat sstm Essntial part of forming th lmnt quations of a thr dimnsional fram lmnt is finding out th coordinat chang btwn th global coordinat sstm and th lmnt coordinat sstm (Fig. 4.2). Th bas vctors l, m, n of th lmnt coordinat sstm in th global coordinat sstm ar writtn as

26 Finit lmnt analsis of strsss in bam structurs 25 l = l i+ l j+ l k, x m= m i+ m j+ m k, x n= n i+ n j+ n k. x (4.42) In th following, th componnts of ths bas vctors will b chosn in a rasonabl wa. Bcaus th bas vctor l is coinciding to th axis of th lmnt, it is obtaind for its componnts x2 x 2 2 lx =, l = l =, (4.43) whr = ( x x ) + ( ) + ( ) (4.44) is th lngth of th lmnt. Th componnts of bas vctors m and n can b slctd using diffrnt was. Hr, th global axis has bn chosn to b vrtical and dircting downwards. Consquntl, th x and axs ar in a horiontal plan. Bcaus bams in civil nginring fram structurs usuall consist of vrtical sid plans and ar oftn smmtric with rspct to a vrtical plan, a rasonabl choic could b to dmand that th x ', plan should b vrtical. This mans, that axis should li in a horiontal plan or th componnt n of th bas vctor n should vanish. Bcaus th lngth of th bas vctor n is on and th bas vctors n and l ar orthogonal, on gts n n + n = n =± n. (4.45) x x and nl nl + nl = nl ± n l = n = 2 i x x 0 x x x 0 x. (4.46) 2 2 lx + l l Thus w hav for th componnts of th bas vctor n l l n =, n =, n = 0. (4.47) x x lx + l lx + l Th bas vctor m can b obtaind as a cross-product

27 Finit lmnt analsis of strsss in bam structurs 26 m = n l = ( n i+ n j) ( l i+ l j+ l k) = n l i n l j+ ( n l n l ) k. (4.48) x x x x x Thus its componnts ar m = n l, m = n l, m = n l n l. (4.49) x x x x Equations (4.43), (4.47) and (4.49) in this ordr, can now b usd for calculating th componnts of th bas vctors l, m and n. In cas that th fram contains vrtical bams, bas vctor n of ths bams must b obtaind in a diffrnt wa. It has bn dcidd that in this cas, vctor n is going to b th sam as its countrpart in th adjacnt non-vrtical bam. Vctor m can b obtaind using Eq. (4.49). A tpical displacmnt vctor u and rotation vctor ϕ in lmnt coordinats and global coordinats ar writtn as u = u l+ v m+ w n = ui+ vj+ wk (4.50) and ϕ = ϕ l+ ϕ m+ ϕ n= ϕ i+ ϕ j+ ϕ k (4.5) x x Multipling quations (4.50) and (4.5) b bas vctors l, m and n rsults to u = lxu+ lv+ lw ϕ x = lxϕx + lϕ + lϕ v = mxu+ mv+ mw, ϕ = mxϕx + mϕ + mϕ. w = nxu+ nv+ nw ϕ = nxϕx + nϕ + nϕ (4.52) Appling ths rlations to th nodal dflctions and rotations at both nods of th lmnt rsults to quation a = T a, (4.53) whr a a a =, a = (4.54) a 2 a 2

28 Finit lmnt analsis of strsss in bam structurs 27 ar th lmnt nodal displacmnts dgrs of frdom of in lmnt coordinats and global coordinats, rspctivl, and lx l l mx m m nx n n lx l l mx m m nx n n T = lx l l mx m m nx n n lx l l mx m m nx n n (4.55) is th corrsponding transformation matrix. Th xprssion of th virtual work of a thr dimnsional fram lmnt in lmnt coordinat sstm is ˆ T δw = δ a ( K a + R + F ). (4.56) Using quation (4.53) for both ral and virtual displacmnt dgrs of frdom, th xprssion (4.56) of th virtual work of th lmnt can b xprssd in global coordinats ˆ T δw = δa ( K a + R + F ), (4.57) whr T K = T K T (4.58) and T R = T R (4.59) ar th stiffnss matrix and load vctor, rspctivl, in global coordinat sstm. Ths ar th final lmnt stiffnss matrix and lmnt load vctor of a thr dimnsional fram lmnt.

29 Finit lmnt analsis of strsss in bam structurs GOBA SYSTEM Onc lmnt stiffnss matrics (Eq. 4.39) and lmnt load vctors (Eq. 4.40) ar assmbld using th corrsponding lmnt transformation matrix (Eq. 4.55), th quation to solv is Ka = R, (4.60) whr K is th gnralid stiffnss matrix, a is th gnralid displacmnt vctor and R th gnralid load vctor STRESS RESUTANTS i Aftr th gnralid displacmnts a, i =,,2 of th lmnt hav bn dtrmind, i th corrsponding gnralid forcs F, i =,,2 can b calculatd using th lmnt stiffnss quation (4.28). In trms of ths gnralid forcs th strss rsultants at th lmnt nods ar N = F, Q = F, Q = F, M = F, M = F, M = F, 2 3 t N = F, Q = F, Q = F, M = F, M = F, M = F t (4.6) It can b shown that, in connction with lmnts of uniform bnding stiffnss (constant D ), ths nodal valus ar xact. Using ths valus and standard quilibrium considration it is possibl to construct strss rsultant distributions within ach lmnt, which ar also xact. Hr, howvr, it is sufficint with ths lmnt nodal valus bcaus it is possibl to choos so dns finit lmnt grid that b conncting ths nodal valus b a straight lin on is abl to dscrib th strss rsultant distribution of an bam with sufficint accurac. 4.3 STRESS DISTRIBUTION Using th suprposition principl, strss valus in an cross-sction can b obtaind from th sum of strss valus du to ach strss rsultant sparatl. Som basic cross-sctional proprtis ar ndd to solv th problm, which ar going to b calculatd first [9].

30 Finit lmnt analsis of strsss in bam structurs CROSS-SECTIONA PROPERTIES Tpical cross-sctional proprtis of a composit bam ar dfind as follows. Wightd ara of th cross-sction is ( EA) = EdA. (4.62) A Wightd first momnts of ara ar ( ES) = EdA, (E S) = EdA. (4.63) A A Bnding stiffnss s or wightd scond momnts of ara ar. (4.64) 2 2 = = = A A A ( EI) E da, ( EI) E da, ( EI) EdA Th coordinats of th cntr of wightd ara of th cross-sction ar obtaind from C = ( ES) ( ES), C. ( EA) = ( EA) (4.65) Th cntr of cross-sction of a composit bam is this cntr C of wightd ara. Th origin of th cross-sctional coordinat sstm x,, is locatd in this point and th axis of th bam is th locus of ths cntr points. If th gomtr of th cross-sction is complicatd, on possibilit of dtrmining th crosssctional proprtis (4.62), (4.63) and (4.64), is to us finit lmnt mthodolog. Th cross-sction is dividd into N finit lmnts. Th procdur is dscribd hr brifl in connction with th wightd product momnt (Eq. 4.64c). On can writ N N (4.66) A = A = ( EI) = EdA = E da = ( EI), whr E is th modulus of lasticit and ( EI ) is th wightd product momnt of lmnt. Thus th wightd product momnt ( EI is obtaind as a sum of th wightd product )

31 Finit lmnt analsis of strsss in bam structurs 30 momnts ( EI ) of th lmnts. B changing th intgration variabls on gts, for th wightd product momnt ( EI ), = = A J (4.67) ( EI) E da E ( ξ, η) ( ξ, η)dt ( ξ, η) dξdη and appling numrical intgration furthr n ( EI) w E ( k) ( k)dt J ( k), (4.68) k= k whr th notation ( k) = ( ξ, η ) for th valu of a function at intgration point k is usd. This k k is th final formula for th wightd product momnt ( EI) of lmnt. In ordr to us this formula on must first calculat th coordinats of th intgration points k ( ), k ( )(Eq. 3.2), th drivativs, ξ ( k),, ξ ( k),, η ( k),, ξ ( k) (Eq. 3.7) and Jacobian dtrminants dt J ( k) (Eq. 3.22) at th intgration points. Th rst of proprtis into finit lmnt trms can b obtaind in a similar mannr TENSION / COMPRESSION In bam thor, a bam undr tnsion or comprssion onl producs normal strss in th axial dirction, σ x. Using Hook s law, this strss is σ = Eε, (4.69) x x whr ε x is th axial strain of th bam. Using th dfinition of th axial forc N x and (Eq. 4.62) on obtains, (4.70) N = σ da= Eε da= ( EA) ε x x x x A A which rsults to

32 Finit lmnt analsis of strsss in bam structurs 3 N x ε x =. (4.7) ( EA) Substituting (Eq. 4.7) into (Eq. 4.69) givs th standard xprssion of th normal strss EN x σ x =. (4.72) ( EA) Elmnt nodal valus of this, σ, ar obtaind simpl using xi ENx σ xi = i =,..., m, (4.73) ( EA) whr E is th modulus of lasticit of th lmnt and m is th numbr of nods in th lmnt BIAXIA BENDING Fig. 4.3: Non-smmtric cross-sction of a bam Considr pur biaxial bnding of a straight uniform bam with non-smmtric cross-sction (Fig. 4.3). A part of th bam will b considrd, whr it is assumd that th rotations ϕ ( x) and ϕ ( x) ar quadratic function of th axial coordinat x. Thus th curvaturs κ ( x) = ϕ ( x) and κ ( x) = ϕ ( x) of th bam ar linar and thir drivativs κ ( x) and κ ( x), which ar dnotd k and k, ar constants. For th axial displacmnt of th bam, th xprssion (4.74) is assumd

33 Finit lmnt analsis of strsss in bam structurs 32 ux (,, ) = ϕ ( x ) ϕ ( x ) + kψ (, ) + kψ (, ), (4.74) whr th functions ψ (, ) and ψ (, ) ar calld warping functions. In this cas, th axial strain of th bam is u εx = ϕ ( x) ϕ ( x) = κ( x) + κ( x). (4.75) x Bcaus in bam thor it is assumd that th transvrs strss componnts σ and disappar, on gts, basd on th gnralid Hook s law, for th transvrs strains of th bam th rsults ε = ε νε = νκ ( x) νκ ( x). (4.76) x σ Using th strain and displacmnt rlations, on gts, for th transvrs displacmnts v and w, diffrntial quations v w = ε = νκ ( + κ), = ε = νκ ( + κ). (4.77) In ths quations th dpndnc of th curvatur on coordinat x has bn lft out for clarit. In th following drivation it is assumd that th Poisson s ratio v of th cross-sction is constant. This assumption naturall holds if th bam is mad of on matrial. In a composit bam, howvr, th Poisson s ratios of th matrials of th cross-sction must b qual. Hr it is xpctd that th drivd quations could, howvr, b usd in composit bams, if th Poisson s ratios of th matrials do not dviat too much. Ths quations can b solvd for th transvrs displacmnts rsulting to 2 2 vx (,, ) = v0 ( x) νκ ( x) ( ) νκ ( x ), wx (,, ) = w0 ( x) νκ ( x ) νκ ( x) ( ). 2 (4.78) Th xprssions (4.74) and (4.78) now form th displacmnt assumption of th bam. For th shar strains on obtains

34 Finit lmnt analsis of strsss in bam structurs 33 ψ 2 2 ψ γ x = k ν ( ) + k ν, 2 ψ ψ 2 2 γ x = k ν ( ) + k ν. 2 (4.79) For th normal strss on obtains σ = Eε = E( κ + κ ) = 2 G( + ν)( κ + κ ) (4.80) x x and for th shar strsss ψ 2 2 ψ τx Gγ x = kg ν ( ) + kg ν, 2 ψ ψ 2 2 τx Gγ x = kg ν ( ) + kg ν. 2 (4.8) Th axial quilibrium quation of th strss componnts, if volum forcs ar omittd, is σ τ x x τx + + = 0. (4.82) x Using th xprssions (4.80) and (4.8) and th rlation E = 2 G( + ν ), (4.83) th quilibrium quation (4.82) gts first th form ψ 2 2 ψ kg ν ( ) + kg ν 2 ψ 2 2 ψ + kg ν ( ) + kg ν 2 + k 2 G( + ν) + k 2 G( + ν) } = 0 in A (4.84) and furthr

35 Finit lmnt analsis of strsss in bam structurs 34 ψ 2 2 ψ k G ν ( ) + G ν + 2 G( + ν) 2 ψ 2 2 ψ + k G ν ( ) G ν 2 G( ν) 0 in A = 2 (4.85) Equation (4.85) holds if th cofficints of constants k and k disappar. Thus, two sparat diffrntial quations for th warping functions ψ (, ) and ψ (, ) ar obtaind ψ 2 2 ψ G ν ( ) + G ν + 2 G( + ν) = 0 2 in A, ψ 2 2 ψ G ν ( ) + G ν + 2 G( + ν) = 0 2 in A. (4.86a) (4.86b) Th boundar condition of th shar strsss on th unloadd boundar of th cross-sction is n τ + nτ = 0. (4.87) x x Using th xprssions (4.8) th boundar condition gts first th form ψ 2 2 ψ n kg ν ( ) + kg ν 2 ψ ψ n kg ν ( ) + kg ν = 0 on s, 2 (4.88) and furthr ψ 2 2 ψ k ng ν ( ) + ng ν 2 ψ ψ k ng ν + ng ν ( ) = 0 on s. 2 (4.89)

36 Finit lmnt analsis of strsss in bam structurs 35 Equation (4.89) holds if th cofficints of constants k and k disappar. Thus, two sparat boundar condition quations for th warping functions ψ (, ) and ψ (, ) ar obtaind ψ 2 2 ψ ng ν ( ) + ng ν = 0 on s, 2 ψ ψ 2 2 ng ν + ng ν ( ) = 0 on s. 2 (4.90a) (4.90b) Equations (4.86a) and (4.90a), and (4.86b) and (4.90b) form boundar valu problms for solving th warping functions ψ (, ) and ψ (, ), rspctivl. Th can b solvd using th finit lmnt mthod. On th othr hand, using th dfinition of th bnding momnts on obtains 2 = σ x = κ + κ = κ + κ A A 2 = σ x = κ + κ = κ + κ A A M da E( ) da ( EI) ( EI), M da E( ) da ( EI) ( EI), (4.9) Equations (4.9) hold also whn th cross-sction is non-homognous. Ths can b solvd for th curvaturs rsulting to ( EI ) M ( EI ) M ( EI) M ( EI ) M κ =, =. ( EI ) ( EI ) ( EI) ( EI) ( EI) ( EI) κ 2 2 (4.92) Diffrntiating ths quations with rspct to x and using th quilibrium quations Q M = and = givs Q M k ( EI) Q ( EI) Q ( EI) Q ( EI) Q =, =. ( EI) ( EI) ( EI) ( EI) ( EI) ( EI) k 2 2 (4.93) Equations (4.92) and (4.93) can b usd for dtrmining th curvaturs and thir drivativs, whn th bnding momnts and sharing forcs of th cross-sction ar known.

37 Finit lmnt analsis of strsss in bam structurs 36 Wak form of th boundar valu problm of quations on ψ (4.86a) and (4.90a) is drivd in th following. Multipling both sids of quation (4.86a) b tst function intgrating ovr th domain A (cross-sction) givs δψ and ψ 2 2 ψ δψ G ν ( ) + G ν + 2 G( + ν ) da= 0 2 A (4.94) and ψ ψ δψ G ν ( ) + G ν da + δψ 2 G( + ν ) da = 0.(4.95) A A Appling intgration b parts givs furthr ψ 2 2 ψ δψ ng ν ( ) ng ds 2 + ν s (4.96) δψ ψ δψ 2 2 ψ G ν ( ) + G ν da + δψ 2 G( + ν) da = 0. 2 A A Basd on th boundar condition (4.90a), th first trm of quation (4.96) vanishs and it gts th form δψ ψ δψ ψ δψ δψ 2 A 2 2 G + da= Gδψ ( + ν ) + ν ( ) + ν da. (4.97) A This is th wak form of th boundar valu problm of diffrntial quations (4.86a) and (4.90a). Th wak form corrsponding to diffrntial quations (4.86b) and (4.90b) can b obtaind similarl. Th functions ψ and δψ can b xprssd using finit lmnt approximations M ψˆ = Nψ, δψˆ = Nδψ i i i i i= i= M. (4.98) B substituting ths into th wak form (4.97) givs

38 Finit lmnt analsis of strsss in bam structurs 37 M M N N i j N i δψ i G + i= j= A N j daψ 2 2 Ni Ni 2 G( + ν) Ni + G ν ( ) + Gν da = 0 2 A j (4.99) or M M δψ i Kijψ j Ri = 0, (4.00) i= j= whr K = G( N N + N N ) da, (4.0) ij i, j, i, j, A and 2 2 R = G 2( + ν) N + ν ( ) N + νn da, (4.02) i i i, i, 2 A whr tpical notation Ni, = Ni/ and N, = N / is usd for th drivativs of th i shap functions. Equation (4.00) holds with arbitrar nodal valus δψ i of th tst function, onl if th trms in parnthss will vanish with all valus of i. Thus w gt, corrsponding to th wak form (4.97), th finit lmnt quations i M Kijψ j = Ri, i=,, M. (4.03) j= Procding in th sam mannr with quations on ψ (4.86b) and (4.90b), on obtains similar finit lmnt quations M Kijψ j = Ri, i=,, M, (4.04) j= whr Ni 2 2 Ni Ri = G2( + ν) Ni + ν + ν ( ) da 2 A. (4.05)

39 Finit lmnt analsis of strsss in bam structurs 38 Equations (4.03) and (4.04) ar, in matrix form, Ka = R, Ka = R, (4.06) whr a ψ ψ =, a = (4.07) ψ M ψ M ar th vctors of unknown nodal valus of th warping functions. Th lmnts of th lmnt matrix K and lmnt vctors R and R corrsponding to th lmnts of th sstm matrix (4.0) and sstm vctors (4.02) and (4.05) ar (,,,, ) K = G N N + N N da (4.08) ij i j i j A and 2 2 R = G 2( + ν) N + ν ( ) N + νn da, (4.09a) i i i, i, 2 A 2 2 R = G 2( + ν) N + νn + ν ( ) N da. (4.09b) i i i, i, 2 A In ths xprssions, th shar modulus has bn assumd to b constant within th lmnt, which is a rasonabl assumption. Th Poisson s ratio should b constant within th whol cross-sction. B changing intgration variabl to natural coordinats and appling numrical intgration, xprssions (4.08) and (4.09) can b rducd to G n ij = k i, j, + i, j, k = K wg N ( k) N ( k) N ( k) N ( k) dt J ( k), (4.0) and

40 Finit lmnt analsis of strsss in bam structurs 39 R wg kn k k k N k kkn k k R wg kn k k k N k kkn k k n 2 2 i = k 2( + ν) ( ) i ( ) + ν ( ) ( ) i, ( ) + ν ( ) ( ) i, ( ) dt ( ) k = 2 J n 2 2 i = k 2( + ν) ( ) i ( ) + ν ( ) ( ) i, ( ) + ν ( ) ( ) i, ( ) dt ( ) k = 2 J, (4.) whr n is th numbr of intgration points. Ths quations can b solvd with proprtis dscribd bfor in chaptr 3. At this point, strsss du to biaxial bnding in a cross-sction can b obtaind. Substituting th curvatur of quation (4.92) into quation (4.80) givs th standard xprssion of th normal strss. Its nodal valus ar obtaind simpl using σ = E ( κ + κ ) i=,..., m, (4.2) xi i i whr curvaturs κ and κ can b found in (Eq. 4.92) and coordinats of th lmnt nods, rspctivl. i and i ar th and Substituting th drivativ of th curvatur of quation (4.93) into quation (4.8) givs improvd xprssions for th shar strsss. Th lmnt nodal valus of th shar strsss ar obtaind b first calculating thir valus at th intgration points and thn using 2 2 τx ( k) = kg ψ, ( k) ν ( ) ( ), ( ) ( ) ( ), 2 k k + kg ψ k ν k k 2 2 τx ( k) = kg ψ, ( k) ν ( k) ( k) + kg ψ, ( k) ν ( k) ( k). 2 (4.3) Th drivativ of th warping function at th intgration points ar calculatd using m m ψˆ ˆ, ( k) = Ni, ( k) ψ i, ψ, ( k) = Ni, ( k) ψ i, i= i=,,,. m m k = n (4.4) ψˆ ˆ, ( k) = Ni, ( k) ψ i, ψ, ( k) Ni, ( k) ψ = i i= i= Th lmnt nodal valus of th shar strsss τxi, τ xi, i=,, m ar finall obtaind b xtrapolating shar strsss at th intgration point τ ( k), τ ( k), k =,, n from th intgration points to th lmnt nods. This procdur is xplaind in th appndix. x x

41 Finit lmnt analsis of strsss in bam structurs SHEAR CENTRE Shar cntr S (s Fig. 4.3) is th point of action of th rsultant of th shar strsss causd b pur bnding. For dtrmining its coordinats S and S on can writ Q S+ Q S = ( τx+ τxda ), (4.5) A Substituting th shar strsss (4.8) into quation (4.5) first givs Q S + Q S = k( GJ) k( GJ) (4.6) ψ ψ whr ψ ψ 2 3 ( GJ ) ψ = G + ν ( + ) da, 2 A ψ ψ 2 3 ( GJ ) ψ = G + ν ( + ) da. 2 A (4.7) Substituting furthr th curvatur drivativs (4.93) into quations (4.6) givs ( EI) ( GJ ) + ( EI) ( GJ ) ( EI) ( GJ ) + ( EI) ( GJ ) Q + Q = Q + Q ψ ψ ψ ψ S S 2 2 ( EI) ( EI) ( EI) ( EI) ( EI) ( EI). (4.8) Bcaus ths quations must hold for all valus of th shar forcs th coordinats of th shar cntr th quations Q and Q, w gt for S ( EI) ( GJ ) + ( EI) ( GJ ) ( EI) ( GJ ) + ( EI) ( GJ ) =, =. ψ ψ ψ ψt 2 S 2 ( EI) ( EI) ( EI) ( EI) ( EI) ( EI) (4.9) Th quantitis ( GJ ) ψ and ( GJ ) ψ ar cross-sctional proprtis lik bnding stiffnss. In connction with isoparamtric lmnts th lmnt contributions of ( GJ ) ψ and ( GJ ) ψ ar calculatd using formulas n 2 3 ( GJ ) ˆ ˆ ψ = wkg ψ, ( k) ( k) ψ, ( k) ( k) + ν ( k) ( k) + ν( k) dt ( k), k = 2 2 J n 2 3 ( GJ) ˆ ˆ ψ = wg k ψ, ( k) k ( ) ψ, ( kk ) ( ) + νkk ( ) ( ) + νk ( ) dt ( k). k = 2 2 J (4.20)

42 Finit lmnt analsis of strsss in bam structurs 4 Quantitis ( GJ ) ψ and ( GJ ) ψ ar obtaind b summing thir lmnt contributions E E ψ = ψ ψ = ψ = = (4.2) ( GJ ) ( GJ ), ( GJ ) ( GJ ). Th product of transvrs xtrnal load and th distanc btwn its application point and th shar cntr must b includd into th fram analsis as an additional xtrnal twisting momnt TORSION Fig. 4.4: Twistd cross-sction Considr pur torsion of a straight uniform bam (Fig. 4.4). A part of th bam, whr it is assumd that th angl of twist ϕ ( x) is a linar function of th axial coordinat x, is t considrd. Thus rat of twist pr unit lngth of th bam θ = ϕ t is a constant. For th axial displacmnt of th bam, it can b assumd th xprssion ux (,, ) = θψ (, ), (4.22) t whr th function ψ (, ) is calld warping function. Furthr it is assumd that th t projction of th cross-sction in th, plan rotats lik a rigid plat. Thus w hav for th transvrs displacmnt componnts th xprssions vx (,, ) = ϕ ( x ), wx (,, ) = ϕ ( x ). (4.23) t t Exprssions (4.22) and (4.23) form th displacmnt assumptions of so-calld Saint Vnant s torsion or fr torsion.

43 Finit lmnt analsis of strsss in bam structurs 42 For th shar strains on obtains u v ψt ψt γ x + = θ ϕ t = θ, x u w ψt ψt γ x + = θ + ϕ t = θ + x (4.24) and th othr strain componnts ar ro. For th shar strsss on obtains ψt ψt τx Gγ x = Gθ, τx Gγ x = Gθ + (4.25) and th othr strss componnts ar ro. Th axial quilibrium quation of th strss componnts, if volum forcs ar omittd, is σ τ x x τx + + = 0. (4.26) x Using th xprssions (4.25) and noting that σ x = 0, th quilibrium quation (4.26) gts th form ψt ψt G + G + = 0 in A, (4.27) whr th ara A of th cross-sction has bn addd to rmark that this quation must hold at vr point on th cross-sction. Th boundar condition of th shar strsss on th unloadd boundar of th cross-sction is n τ + nτ = 0, (4.28) x x whr n and n ar th componnts of th unit normal vctor n of th boundar lin of th cross-sction (Fig. 4.4). Using th xprssions (4.25), it can b writtn as ψt ψt ng + ng + = 0 on s, (4.29)

44 Finit lmnt analsis of strsss in bam structurs 43 whr s has bn addd to rmark, that this quation must hold at vr point on th boundar lin of th cross-sction. Equations (4.27) and (4.29) form a simpl lliptic boundar valu problm for dtrmining th warping function ψ (, ). t With th hlp of Fig. 4.5, on obtains for th torqu M t = ( τx τx) da. (4.30) A Fig. 4.5: Dfining th torqu Substituting th xprssions (4.25) of th shar strsss into quation (4.30) givs M t = ( GI) θ, (4.3) t which rlats th torqu and twisting angl pr unit lngth. Hr ψ ψ GI = G + + da (4.32) t t 2 2 ( ) t ( ) A is th torsional stiffnss of Saint Vnant s or so calld fr torsion. Equations (4.3) and (4.32) appl also to th cas whr th cross-sction is non-homognous (composit bam). In th spcial cas of a homognous cross-sction, th torsional stiffnss is ( GI ) t = GI t, whr ψ ψ = + + (4.33) 2 2 ( t t It ) da A

45 Finit lmnt analsis of strsss in bam structurs 44 is th torsional constant. Th torsional stiffnss (4.32) or torsional constant (4.33) can b dtrmind aftr th warping function ψ (, ) has bn solvd from th boundar valu t problm of quations (4.27) and (4.29). Equation (4.3) can now b solvd for th twisting angl pr unit lngth rsulting to M t θ =. (4.34) ( GI) t Substituting this into quations (4.25) rsults to τ x GM t ψt GM t ψt =, τx = + ( GI) t ( GI) t. (4.35) Ths ar final xprssions of th shar strsss corrsponding to a givn torqu. In th cas of a homognous cross-sction ( GI ) t = GI t ths xprssions gt slightl simplr form τ x Mt ψt Mt ψt =, τx = + It I t. (4.36) Using th sam mthodolog as in chaptr for th cas of biaxial bnding, wak form of th boundar valu problm of quations (4.27) and (4.29) is δψ ψ δψ ψ δψ δψ t t t t t t G + da= G da (4.37) A A Th finit lmnt quations corrsponding to th wak form of quation (4.37) ar M Kijψ tj = Ri, (4.38) j= whr N N N N (4.39) i j i j Kij = G + da A and

46 Finit lmnt analsis of strsss in bam structurs 45 N N. (4.40) i i Ri = G da A Equations (4.38) in matrix form ar Ka = R. (4.4) Equations (4.39) and (4.40) ar xprssions of th lmnts of th sstm matrix K and sstm vctor R and ψ t a = (4.4) ψ tm M is th vctor of unknown nodal valus of th warping function. Th lmnts of th lmnt matrix matrix and sstm vctor ar K and lmnt vctor R corrsponding to th sstm K = G ( N N + N N ) da, (4.42) ij i, j, i, j, A and R = G ( N N ) da. (4.43) i i, i, A Final quations for calculating th lmnts of th lmnt matrix and lmnt vctor using finit lmnt mthodolog ar n ij = k i, j, + i, j, k = K w G [( N ( k) N ( k) N ( k) N ( k)]dt J ( k), (4.44) and n i = k i, i, k = R wg[ N ( k) ( k) N ( k) k ( )]dt J ( k). (4.45)

47 Finit lmnt analsis of strsss in bam structurs 46 In ths formulas, w is th wight of intgration point k, N, ( k ) and N, ( k ) ar th valus k i i of th shap function drivativs on nod i at intgration point k and n is th numbr of intgration points. Th torsional stiffnss of th cross-sction is obtaind as a sum of th torsional stiffnsss of th lmnts, or E ( GI) = ( GI ) (4.46) t = t whr GI G ψ ψ da. (4.47) 2 2 ( ) t = ( t, t, + + ) A Th torsional stiffnss (4.47) of lmnt can b calculatd using quation n 2 2 ( GI ) t = wkg [ ψt, ( k) ( k) ψt, ( k) ( k) + ( k) + ( k) ]dt ( k). k = J (4.48) In quations (4.47) and (4.48), th shar modulus of th lmnt G is assumd to b constant, which is a rasonabl assumption in connction with composit bams. Th shar strsss (4.35) at intgration point k of lmnt corrsponding to a givn torqu M t can b calculatd using τ GM GM ( k) = [ ψ ( k) ( k)], τ ( k) = [ ψ ( k) + ( k)]. (4.49) t t x t, x t, ( GI) t ( GI) t Th lmnt nodal valus of th shar strsss τxi, τ xi, i=,, m ar finall obtaind b xtrapolating shar strsss at th intgration point τ ( k), τ ( k), k =,, n from th intgration points to th lmnt nods. x x

48 Finit lmnt analsis of strsss in bam structurs GENERA CASE Finall, to handl th cas of a cross-sction undr tnsion/comprssion, biaxial bnding and torsion at th sam tim, on just has to add ach indpndnt rsult to ach othr. Thus, th normal strss is th sum of on obtaind from tnsion/comprssion and bnding momnts. It can b xprssd as EN x σ x = + E( κ+ κ), (4.50) ( EA) and its nodal valus as ENx σxi = + E ( κi + κi ). i =,..., m, (4.5) ( EA) On th othr hand, shar strsss ar th sum of th rsults obtaind from shar forcs and twisting momnts, and can b xprssd as ψ 2 2 ψ GM t ψ t τx = kg ν ( ) + kg ν +, 2 ( GI) t ψ ψ 2 2 GM t ψ t τx = kg ν ( ) + kg ν ( GI) t (4.52) Thir valus at intgration points ar 2 2 τx ( k) = kg ψ, ( k) ν ( k) ( k) + kg ψ, ( k) ν ( k) ( k) 2 GMt + [ ψ t, ( k) ( k)], ( GI) t 2 2 τx ( k) = kg ψ, ( k) ν ( k) ( k) + kg ψ, ( k) ν ( k) ( k) 2 GMt + [ ψ t, ( k) + ( k)]. ( GI) t (4.53) Th lmnt nodal valus of th shar strsss τxi, τ xi, i=,, m ar finall obtaind b xtrapolating shar strsss at th intgration point τ ( k), τ ( k), k =,, n from th intgration points to th lmnt nods. x x