Nonlinear Bending of Strait Beams

Transcription

1 Nonlinar Bnding of Strait Bams CONTENTS Th Eulr-Brnoulli bam thory Th Timoshnko bam thory Govrning Equations Wak Forms Finit lmnt modls Computr Implmntation: calculation of lmnt matrics Numrical ampls Bams

2 Von Kármán NONLINEAR STRAINS Grn-Lagrang Strain Tnsor Componnts E E ij u u i j j i u u m i u u u 3 u Ordr-of-magnitud assumption u E O 3 m 3 ( ), ( ) u u O u j Bams

3 z, w NONLINEAR ANALYSIS OF EULER-BERNOULLI BEAMS F0 Bam cross sction q() z q() f() M 0 y M M N N L cw f V V Displacmnts and strain-displacmnt rlations u ( uz ) ˆ wˆ, 3 d u( z, ) uz, u 0, u w ( ) 3 d u 3 z u du d w d d d Nonlinar Problms (-D) : 3,

4 PRINCIPLE OF VIRTUAL DISPLACEMENTS for th Eulr-Brnoulli bams W I V dv du d w d d d d [( ) z( )] b a A b du d w [( ) N M ] d a d d d d 6 b b WE qwd f ud Qi i a a i Equilibrium quations dn f d M d 0, N q0 d d d d dad dn dm dv d f 0, V 0, N q 0 d d d d d Bams 4

5 NONLINEAR ANALYSIS OF EULER-BERNOULLI BEAMS Equilibrium quations dn f d M d 0, N q0 d d d d Strss rsultants in trms of dflction du d w du N = σ da = E + Ez da EA d d = + d d d A A du M = σ z da = E + Ez z da EI d d = d d A A V dn dm dv d f 0, V 0, N q 0 d d d d d dm d d w = = EI d d d Bams 5

6 NONLINEAR ANALYSIS OF EULER-BERNOULLI BEAMS Equilibrium quations in trms of displacmnts (u,w) d du EA f 0 d d d d d w d du EI EA q 0 d d d d d d zw, u, F F w( L) u( L) Clarly, transvrs load inducs both aial displacmnt u and transvrs displacmnt w. Bams 6

7 Wak forms 0 0 EULER-BERNOULLI BEAM THEORY dv N vf d v ( ) ( ) a Q v b Q 4 b a d du N EA d d (continud) b d v d w dv EI N v q d a d d d d dv dv v ( ) Q Q v ( ) Q Q a 3 b 5 6 d d Q V( ) a Q M( ) 3 a a Q V( ) 5 b Q M( ) 6 b Q N ( ) a Q N ( ) 4 b v u, v w b Bams 7

8 BEAM ELEMENT DEGREES OF FREEDOM Gnralizd displacmnts w ( ) a w ( ) 5 b u ( ) a u ( ) 4 b ( ) 3 a Gnralizd forcs ( ) 6 b Q V( ) a Q V( ) 5 b QN ( ) Q N ( ) a 4 b Q M( ) 3 a Q M( ) 6 b 8

9 FINITE ELEMENT APPROXIMATION Primary variabls (srv as th nodal variabls that must b continuous across lmnts),, uw θ = d 4 w ( ) ( ), u ( ) u ( ), j j j j j j n Hrmit cubic polynomials µ µ Á a a = 3 + Á = ( a ) µ a 3 µ µ 3 Á a a 3 = 3 " µ # Á a 4 = ( a ) a 9

10 HERMITE CUBIC INTERPOLATION FUNCTIONS i ( ) ( ) slop = 0 ( ) slop = slop = 0 ( 3 slop = 0 ( 4 slop = slop = 0 0

11 FINITE ELEMENT MODEL Finit Elmnt Equations 4 u ( ) u ( ), w ( ) ( ) j j j j j j [ K ] [ K ] { u} { F } 3 [ K ] [ K ] { } { F } b d d j b d d i i j K EA d, K, ij ij EA d a d d a d d d b d d i j b K EA d, F f ij d ( ) Q ( ) Q i i i a i b a d d d a b d d j b i d d i j K EI d EA d, ij a d d a d d d 4 d d F q d Q Q Q Q b i i ( ) ( ) i i i a i b a d d a Q b Q5 Q3 Q 6 Q Q

12 MEMBRANE LOCKING Mmbran strain du d d 0 Bam on rollr supports q ( ) du d d 0 du d d Rmdy mak d 0 to bhav lik a constant

13 SOLUTION OF NONLINEAR EQUATIONS Dirct Itration Non-Linar Finit Elmnt Modl [ K ( )] F assmbld [ KU ( )] U F Dirct Itration Mthod r th r Solution { U} at r itration is known and solv for{ U} r r [ K({ U} )]{ U} { F} F F C K(U 0 ) K(U ) K(U ) K(U)U F(U) U C - Convrgd solution U 0 - Initial guss solution U 0 U U U 3 U C U Nonlinar Problms: (-D) - 3

14 SOLUTION OF NONLINEAR EQUATIONS (continud) Dirct Itration Mthod Convrgnc Critrion r th r Solution { U} at r itration is known and solv for{ U} r r [ K({ U} )]{ U} { F} Possibl convrgnc NEQ I NEQ I U r I U U r I r I spcifid tolranc 4

15 SOLUTION OF NONLINEAR EQUATIONS Nwton s Itration Mthod Taylor s sris r r r Rsidual, { R} [ K({ U} )]{ U} { F} r r r r r R r r R { RU ( )} { RU ( )} ( U U ) ( U U ) U! U r r r r R r r { RU ( )} ( U U ) O( U), U U U U r st Rquiring th rsidual { R} to b zro at th r itration, w hav r tan [ K ({ U} )]{ U} { R} { F} [ KU ( )] { U} Th tangnt matri at th lmnt lvl is r r r r r r n tan Ri Kij Kip p F i j j p 5

16 SOLUTION OF NONLINEAR EQUATIONS Nwton s Itration (continud) n n Ri Kip T K F K T j j p p j r r r r r r r [ T({ } )]{ } { F} [ K( )] { }, { } { } { } ij ip p i ij p ij F F C T( ) 0 T( 0 ) T( ) δ δ K( ) F R( ) C - Convrgd solution 0 - Initial guss solution C = 3 = δ + 0 = δ + 0 Nonlinar Problms: (-D) - 6

17 Summary of th N-R Mthod R i = T ij = X X = p= j [T (f g (r ) ]f g r = fr(f g (r ) )g f g r = f g (r ) + f± g Computation of tangnt stiffnss matri K ip p F i = = K ij + n X p= j K ip u p + 4X P = K ip up + K ip P F i 4X j K ip P T ij = K ij + = K ij + nx j u p + nx 0 u p + p= 4X P j 4X 0 P P = P Bams 7

18 THE TIMOSHENKO BEAM THEORY q ( ) z, w f( ) Dformd Bams z d, u d Undformd Bam Eulr-Brnoulli Bam Thory (EBT) Straightnss, intnsibility, and normality u φ d Timoshnko Bam Thory (TBT) Straightnss and intnsibility 8

19 KINEMATICS OF THE TIMOSHENKO BEAM THEORY Displacmnt fild u uz ) ˆ wˆ E u( z, ) u ( ) z( ), ( u 0, u( z, ) w ( ) 3 3 u u 3 du 3 z z z 3 E d z d d d u u d z z w u Constitutiv Equations E, G φ d zφ z z 9

20 TIMOSHENKO BEAM THEORY (continud) Equilibrium Equations dn dm dv d f 0, V 0, N q0 d d d d d Bam Constitutiv Equations A du d du N da E z da EA d d d d d A A du d d M z da E z z da EI d d d d A A V Ks z da GKs da GAK s d d A 0

21 WEAK FORMS OF TBT b 0 ( ) ( ) a b 4 0 a dv du EA w vf d v( ) Q v( ) Q a b 4 b a dv N vfd v Q v Q d d d d d b dv dv GAK s N vq d a d d d d v ( ) Q v ( ) Q a b 5 0 dv d EI GAK v d d d d v ( ) Q v ( ) Q b 3 s 3 a 3 a 3 3 b 6 Bams

22 FINITE ELEMENT MODELS OF TIMOSHENKO BEAMS Finit Elmnt Approimation m n p () () (3) jj jj jj j j j u u ( ), w w ( ), S ( ) 3 K K K F u 3 K K K w F S 3 K K K F w w s s w m= n= w w s s s m= n= 3

23 SHEAR LOCKING IN TIMOSHENKO BEAMS () Thick bam princs shar dformation, () Shar dformation is ngligibl in thin bams, Linar intrpolation of both w, d d w() w () w () () S () S (), w w S S In th thin bam limit it is not possibl for th lmnt to raliz th rquirmnt d 3

24 SHEAR LOCKING - REMEDY In th thin bam limit, φ should bcom constant so that it matchs /d. Howvr, if φ is a constant thn th bnding nrgy bcoms zro. If w can mimic th two stats (constant and linar) in th formulation, w can ovrcom th problm. Numrical intgration of th cofficints allows us to valuat both φ and dφ/d as constants. Th trms highlightd should b valuatd using rducd intgration. K ij () () b d d i j GAKs... d a d d () b 3 di ( 3) 3 ij GAKs j ji K d K a d (3) (3) b 33 d d i j (3) (3) Ki j EI GAKsi j d a d d 4

25 NUMERICAL EXAMPLES Pinnd-pinnd bam (EBT) w 0 Dflction, q 0 Pinnd-pinnd Clampd-clampd Load, q 0 q 0 Nonlinar Problms: (-D) - 5

26 Pinnd-pinnd bam (TBT) L = Lngth, H = Hight of th bam (in.) w L H = 00 q 0 L H = 50 L H = 0 q (psi.) Nonlinar Problms: (-D) - 6

27 Pinnd-pinnd bam (EBT, TBT).0 L/ H = 0(TBT) w 0.8 L/ H = 0(EBT) q0 L/ H = 50 (TBT,EBT) Dflction L/ H = 80 (TBT,EBT) L/ H = 00 (TBT,EBT) w = w(0.5 L) EH L 4 3 H = bam hight L = bam lngth Load (lb/in), q 0 Bams 7

28 Hingd-Hingd bam (EBT and TBT) Nondimnsional dflction w = weh 3 /ql EBT = Eulr Brnoulli bam thory TBT = Timoshnko bam thory TBT EBT L/H=0 TBT L/H=00 q Load, q 0 (psi) Nonlinar Problms: (-D) - 8

29 SUMMARY In this lctur w hav covrd th following topics: Drivd th govrning quations of th Eulr-Brnoulli bam thory Drivd th govrning quations of th Timoshnko bam thory Dvlopd Wak forms of EBT and TBT Dvlopd Finit lmnt modls of EBT and TBT Discussd mmbran locking (du to th gomtric nonlinarity) Discussd shar locking in Timoshnko bam finit lmnt Discussd ampls 9