WIXLIB

JN Reddy - 1Lecture Notes on NONLINEAR FEMMEEN 673Nonlinear Finite Element AnalysisNonlinear Bending of Strait BeamsRead: Chapter 5JN ReddyCONTENTS The Euler-Bernoulli beam theory The Timoshenko beam theory Governing Equations Weak Forms Finite element models Computer Implementation:calculation of elementmatrices Numerical examplesBeams 1

JN Reddy - 2Lecture Notes on NONLINEAR FEMTHE EULER-BERNOULLI BEAM THEORY(development of governing equations)q( x )z, wzf (x )xxudw dxDeformed BeamJN ReddyUndeformed Beam dwdxEuler-BernoulliBeam Theory (EBT)Straightness,inextensibility, andnormality2

JN Reddy - 3Lecture Notes on NONLINEAR FEMKinematics of Deformation in theEuler-Bernoulli Beam Theory (EBT) dwdxDisplacement fieldzu (u z x ) eˆ 1 w eˆ 3 ,uzwx zσ zzzyxσ zyσ zxσ yxdwdxdw x dxu1 ( x , z ) u zσxzσxyσxxσ yy σ yzu2 0,dwdxu3 ( x , z ) w( x )Notation for stress componentsJN Reddy3

JN Reddy - 4Lecture Notes on NONLINEAR FEMVon Kármán NONLINEAR STRAINS Green-Lagrange Strain Tensor Components1 ui u j 1 um um Eij 2 x i x j2 x j xi u u u1 1 u1 12 2 12 3 Exx 2 x x x1 x111222 Order-of-magnitude assumption u3 O( ) x1 u1 O( ), x1 u1 1 u3 Exx xx 2 x x21JN Reddy1Beams 4

JN Reddy - 5Lecture Notes on NONLINEAR FEMNONLINEAR ANALYSIS OFEULER-BERNOULLI BEAMSz, wBeam cross sectionq(x) f(x)zMMMyx NN cf wLV V Displacements and strain-displacement relationsdwˆˆu (u z x ) e1 w e3 , x dxdw, u2 0, u3 w( x )u1 ( x , z ) u zdxq(x)F002 u1 1 u3 du 1 dw d 2w, 2 2 xx z2 dx dxdx x x 2Nonlinear Problems (1-D) : 5

JN Reddy - 6Lecture Notes on NONLINEAR FEMNONLINEAR ANALYSIS OFEULER-BERNOULLI BEAMSEquilibrium equationsdNdMdVd dw f 0, V 0, q 0 Ndxdxdx dx dx dN f 0,dxd2 Md dw q 0 N2dxdx dx Stress resultants in terms of deflection222 du du dwdwdw 11 E dA 2 2 σ xx dA N EA Ez 2dxdxdxdxdx AA M σ xxAV JN Reddy 2 du dwd2w d2w 1 z dA z dA E 2 EI Ez 22dxdxdxdx A dMd d2w EI dx dxdx 2 Beams 6

JN Reddy - 7Lecture Notes on NONLINEAR FEMNONLINEAR ANALYSIS OFEULER-BERNOULLI BEAMS Equilibrium equations in terms of displacements(u,w)2 d du 1 dw EA 2 f 0 dx dx dx 2 du d 2 d 2w d dwdw1 EIEA q 0 2 2 2 dx dx dx dx dx dx z,wx,uFF w( L ) u( L ) Clearly, transverse load induces both axialdisplacement u and transverse displacement w.JN ReddyBeams 7

JN Reddy - 8Lecture Notes on NONLINEAR FEMEULER-BERNOULLI BEAM THEORY(continued) Weak forms0 xbxa xb dN v1 f dx dxxa dv1 N v1 f dx v1 ( x a )Q1 v1 ( xb )Q4 dx xbxa0 xbxa xbxa dv1 N v1 f dx v1 ( x a )[ N ( x a )] v1 ( xb )N ( xb ) dx 2 du dw 12 N EA dx dx d2 M d dw v2 q dx N dx 2dx dx d 2v d 2w dv dw dv2 dv2 22 EI N22 v2q dx v2 ( x a )Q2 dx Q3 v2 ( xb )Q5 dx Q6 dxdxdxdx xaxbQ2e V ( x a )Q3e M ( x a )Q N ( x a )e1Q5e V ( xb )21Q6e M ( xb )Q4e N ( xb )heJN ReddyBeams 8

JN Reddy - 9Lecture Notes on NONLINEAR FEMBEAM ELEMENT DEGREES OFFREEDOMGeneralized displacements 2 w( x a ) 5 w( xb ) 4 u( xb ) 1 u( x a ) 3 ( x a ) 12he 6 ( xb )Generalized forcesQ2 V ( x a )Q5 V ( xb )Q4 N ( xb )Q1 N ( x a )Q3 M ( x a )JN Reddy1he2Q6 M ( xb )9

JN Reddy - 10Lecture Notes on NONLINEAR FEMFINITE ELEMENT APPROXIMATIONPrimary variables (serve as the nodal variables that must bedwcontinuous across elements)u, w, θ dx4w( x ) j j ( x ),j 1JN Reddynu( x ) u j j ( x ),j 1Hermite cubic polynomials¶2¶3µµx ¡ xax ¡ xaÁe1 1 ¡ 3 2heheµ¶2x ¡ xaÁe2 ¡(x ¡ xa ) 1 ¡heµµ¶2¶3x ¡ xax ¡ xaÁe3 3¡2hehe#"µ¶2x ¡ xax ¡ xaÁe4 ¡(x ¡ xa )¡hehe10

JN Reddy - 11Lecture Notes on NONLINEAR FEMHERMITE CUBIC INTERPOLATIONFUNCTIONS i ( x ) 1 ( x )1slope 0 2 ( x )xxhe 3 ( x )slope 0JN Reddyxslope 0he 4 ( x )1heslope 1xxslope 1slope 0hexx11

JN Reddy - 12Lecture Notes on NONLINEAR FEMFINITE ELEMENT MODEL Finite Element Equations2Q1e1Q5eQ3e e2Q4ej 1 5e [ K 11 ] [ K 12 ] { F 1 } 3e{u} 1 e 1 [ K 21 ] [ K 22 ] { } { F 2 } xbxbd i d jdw d i d j11121K ij EAdx , K ij 2 EAdx ,xaxdx dx dxdx dxaxbxbdw d i d j211K ij EAdx , Fi f i dx i ( x a )Q1 i ( xb )Q4xaxadx dx dx222xbxb d dd i d jdwj22i K ij EIdx EA dx , 22 xaxa dx dx dxdx dxxb d i d i 2 Fi q i dx i ( x a )Q2 i ( xb )Q5 Q3 Q dx dx 6xaxaJN ReddyQ6e24u( x ) u j j ( x ), w( x ) j j ( x )j 1Q2exb2 e6 e412

JN Reddy - 13Lecture Notes on NONLINEAR FEMMEMBRANE LOCKINGBeam on roller supportsMembrane straindu 1 dw 2 dx dx20xxq (x )du 1 dw 0 xx 2 0 dx dx2 du1 dw 2 dx dx2 Remedy dw make to behave like a constant dx 2 JN Reddy13

JN Reddy - 14Lecture Notes on NONLINEAR FEMSOLUTION OF NONLINEAR EQUATIONSDirect IterationNon-Linear Finite Element Model[K e ( e )] e F e assembled [K (U )] U F Direct Iteration MethodSolution {U }r at r th iteration is known and solve for{U }r 1[K ({U }r )]{U }r 1 {F }FFCK(U1)K(U0) JN ReddyK(U)U F(U) K(U2)UC - ConvergedsolutionU0 - Initial guesssolutionU0 U1 U2 U3 UCUNonlinear Problems: (1-D) - 14

JN Reddy - 15Lecture Notes on NONLINEAR FEMSOLUTION OF NONLINEAR EQUATIONS(continued)Direct Iteration MethodSolution {U }r at r th iteration is known and solve for{U }r 1[K ({U }r )]{U }r 1 {F }Convergence CriterionNEQ Possible convergenceU r U r 1 I 1NEQI I 1JN ReddyIr 1 2UI 2 specified tolerance15

JN Reddy - 16Lecture Notes on NONLINEAR FEMSOLUTION OF NONLINEAR EQUATIONSNewton’s Iteration MethodTaylor’s seriesResidual,{R} [K ({U }r )]{U }r 1 {F }rrr2 1 R (U r 1 U r )2 2 2! U R {R(U r 1 )} {R(U r )} (U r 1 U r ) U r R {R(U r )} (U r 1 U r ) O( U )2 , U U U r 1 U rRequiring the residual {R}r 1 to be zero at the r 1st iteration, we have[K tan ({U }r )]{ U } {R}r {F }r [K (U r )]r {U }rThe tangent matrix at the element level is tanK ij JN Reddy Ri j j 2 1n p 1K ip p Fi 16

JN Reddy - 17Lecture Notes on NONLINEAR FEMSOLUTION OF NONLINEAR EQUATIONSNewton’s Iteration (continued)Tij n 2 2n K ip F K TK ippi ijpij j j 1 p 1 1 p 1j[T ({ }r )]{ } {F }r [K ( r )]r { }r , { }r 1 { }r { } Ri FFCT( 2)T( 0)T( 1) δ δ 12 0 1 δ 1 0K( ) F R( ) C - Convergedsolution 0 - Initial guesssolution C 3 2 δ 2 0 Nonlinear Problems: (1-D) - 17

JN Reddy - 18Lecture Notes on NONLINEAR FEMSummary of the N-R Method r ¡fR(f g(r¡1) )g[T (f g(r¡1) ]f gf gr f g(r¡1) f gComputation of tangent stiffness matrixRi 2 XX 1 p 1Tij Ã@Ri @ j Kip p ¡ Fi ! n11X@KipnXp 1JN Reddyp 1 @ jp 1p 111 Kij 1Kipup nX@ Kij 11Tij11 Kij nX4XP 1 2KiP P ¡ Fi 4X¡ 1 Kip up @ @ jP 1¡ 2 KiP P412X@KiPup P@uj@uj0 up P 14XP 10 PBeams 18

JN Reddy - 19Lecture Notes on NONLINEAR FEMTHE TIMOSHENKO BEAM THEORYq( x )z, wzx, uf (x )x Deformed Beamsdwdxdwdxφx uJN ReddydwdxUndeformed BeamEuler-BernoulliBeam Theory (EBT)Straightness,inextensibility, andnormalityTimoshenko BeamTheory (TBT)Straightness andinextensibility19

JN Reddy - 20Lecture Notes on NONLINEAR FEMKINEMATICS OF THE TIMOSHENKOBEAM THEORYDisplacement fieldu (u z x ) eˆ 1 w eˆ 3zuu1 ( x , z ) u( x ) z ( x ),u2 0,Exxu3 ( x , z ) w( x )zw2 u1 1 u3 xx 2 x1 x 2d xdu 1 dw 2 z dx dxdx2Exz 2 xz xzJN Reddy u1 u3 x3 x1dw x dxφ dwdxxzφConstitutiveEquations xx E xx , xz G xz20

JN Reddy - 21Lecture Notes on NONLINEAR FEMTIMOSHENKO BEAM THEORY (continued)Equilibrium EquationsdNdMdVd dw f 0, V 0, q 0 Ndx dx dxdxdxBeam Constitutive Equations22 du du ddwdw1 z x dA EA 12 N xx dA E dx 2 dx dx dxdx AA 2 du ddw z dA EI d xx z 12 M xx z dA E dx dxdx dx AA dw dw V K s xz dA GK s x dA GAK s x dx dx AJN Reddy A21

JN Reddy - 22Lecture Notes on NONLINEAR FEMWEAK FORMS OF TBTWeak Form of Eq. (1)0 xbxav1 uxb dN v1 f dx dxxa dv1 N v1 f dx dx v1 ( x a )[ N ( x a )] v1 ( xb )N ( xb ) xbxa dv1 N v1 f dx v1 ( x a )Q1 v1 ( xb )Q4 dx2 du dvdw 1 1 0 EAvfdx 2 1 xa dx dx dx v1 ( x a )Q1 v1 ( xb )Q4xbJN ReddyBeams 22

JN Reddy - 23Lecture Notes on NONLINEAR FEMWEAK FORMS OF TBT(continued)Weak Form of Eq. (2)v2 w d dw d dw 0 Nv2 GAK s x q dx xadx dx dx dx xb dv2dv2 dwdw dx GAKv qN 2sx xadx dxdx dx xb dw dw v2 GAK s x N dx dx xa xb dv2dv2 dwdw dx q0 NGAKv s x2 dx xa dxdxdx v2 (x a ) Q2 v2 (xb ) Q5 xb JN Reddy2 du dw N EA 12 dx dx 23

JN Reddy - 24Lecture Notes on NONLINEAR FEMWEAK FORMS OF TBT(continued)Weak Form of Eq. (3)v3 x d d dwx 0 v3 GAK s x dx EI xadx dx dx xbx b dv 3 EI d x GAK sv3 x dw dx v3 EI d x dx xadxdxdx x ax b dv d dwx3 0 GAK sv3 x dx EI xadx dx dx v3 (x a ) Q3 v3 (xb ) Q6 xb JN Reddy24

JN Reddy - 25Lecture Notes on NONLINEAR FEMFINITE ELEMENT MODELS OFTIMOSHENKO BEAMSFinite Element Approximationu m j 1u j (1)j (x ), w K 11 K 21 K 31 w11w11JN Reddy K 12 K 22 K 32 n j 1w j (2)j (x ), hew2s12hew3 j 1S j (3)j (x ) K 13 F 1 u K 23 w F 2 S K 33 F 3 w22pm n 2311m n 3s1s2hes22he2s3325

JN Reddy - 26Lecture Notes on NONLINEAR FEMSHEAR LOCKING IN TIMOSHENKO BEAMSdw(1) Thick beam experiences shear deformation, x dxdw(2) Shear deformation is negligible in thin beams, x dxLinear interpolation of both w, xw( x ) w1 1 ( x ) w2 2 ( x ), x ( x ) S1 1 ( x ) S2 2 ( x )w2w121S2S121heheIn the thin beam limit it is not possible for the element to realizethe requirementdw x JN Reddydx26

JN Reddy - 27Lecture Notes on NONLINEAR FEMSHEAR LOCKING - REMEDYIn the thin beam limit, φ should become constant so that itmatches dw/dx. However, if φ is a constant then the bendingenergy becomes zero. If we can mimic the two states (constantand linear) in the formulation, we can overcome the problem.Numerical integration of the coefficients allows us to evaluateboth φ and dφ/dx as constants. The terms highlighted shouldbe evaluated using “reduced integration”.K ij22 K ij23K i33jJN Reddy xbxaxbxaxbxa(2) d i(2) d j . dx GAK sdx dx d i(2) (3)GAK s j dx K 32jidx d (3) d (3) j(3)(3)i EI GAK s i j dx dx dx 27

JN Reddy - 28Lecture Notes on NONLINEAR FEMGENERAL LOGIC IN A COMPUTER PROGRAMfor the nonlinear analysisNL 1,NLSLogic in theMAIN programIter 0F F FIter Iter 1NLS no. of load stepsInitialize global Kij, fiDO 1 to NTransfer global information(material properties, geometry and solution)to elementCALL ELKF to calculate Kij(N)and fi(n), and assemble to formglobal Kij and FiImpose boundary conditionsand solve the equationsError εnoYesJN ReddyyesIter ItmaxPrint SolutionNo Write amessageSTOPBeams 28

JN Reddy - 29Lecture Notes on NONLINEAR FEMCALCULATION OF BEAM PARAMETERSAND INITIALIZATIONSIF(MODEL.GE.2)THENCC Define the beam stiffness coefficients, EA, EI, GAKs, from theC geometric and material parameters read in the main programC (should be passed to this subroutine)CC Initialize arraysCDO 20 I 1,NPEELF1(I) 0.0ELF2(I) 0.0ELF3(I) 0.0DO 20 J 1,NPEELK11(I,J) 0.0ELK12(I,J) 0.0ELK13(I,J) 0.0ELK21(I,J) 0.0ELK22(I,J) 0.0ELK23(I,J) 0.0ELK31(I,J) 0.0ELK32(I,J) 0.0ELK33(I,J) 0.0Beams 29

JN Reddy - 30Lecture Notes on NONLINEAR FEMCALCULATION OF BEAM PARAMETERSAND INITIALIZATIONSCCCCCIF(NONLIN.GT.1)THENTAN12(I,J) 0.0TAN13(I,J) 0.0TAN22(I,J) 0.0TAN23(I,J) 0.0TAN32(I,J) 0.0TAN33(I,J) 0.0ENDIF20CONTINUEENDIFFull integration of the coefficientsDO 100 NI 1,NGPXI GAUSPT(NI,NGP)CALL INTERPLN1D(ELX,GJ,IEL,MODEL,NPE,XI)X ELX(1) 0.5*(1.0 XI)*ELCNST GJ*GAUSWT(NI,NGP)DEFINE AXX, BXX, CXX, DXX, FX, and so on as needed to definethe element force and stiffness coefficientsBeams 30

JN Reddy - 31Lecture Notes on NONLINEAR FEMCALCULATION OF ELEMENT MATRICES(see Box 5.2.2 of the textbook)CCThe EULER-BERNOULLI beam element (MODEL 2) - LINEARMODEL Type of physical problem 1, 2nd order eqn. in 1 variable 2, EBT 2, TBTIF(MODEL.EQ.2)THEN i iDO 50 I 1,NPEI0 2*I-1ELF1(I) ELF1(I) F0*FX*SFL(I)*CNSTELF2(I) ELF2(I) F0*QX*SFH(I0)*CNSTELF3(I) ELF3(I) F0*QX*SFH(I0 1)*CNSTd iDO 50 J 1,NPEd 2 iJ0 2*J-12dxdxS11 GDSFL(I)*GDSFL(J)*CNSTH22 GDDSFH(I0)*GDDSFH(J0)*CNSTH23 GDDSFH(I0)*GDDSFH(J0 1)*CNSTH32 GDDSFH(I0 1)*GDDSFH(J0)*CNSTH33 GDDSFH(I0 1)*GDDSFH(J0 1)*CNSTELK11(I,J) ELK11(I,J) AXX*S11K ij11 ELK22(I,J) ELK22(I,J) DXX*H22ELK23(I,J) ELK23(I,J) DXX*H23K ij22 ELK32(I,J) ELK32(I,J) DXX*H32ELK33(I,J) ELK33(I,J) DXX*H3350CONTINUEENDIFxbFi1 f (x ) i dxxaxbFi 2 q (x ) i dxxad i d j xa dx dx dx ,2xbd 2 i d j xa Dxx dx 2 dx 2 dxxbAxxBeams 31