NONLINEAR FINITE ELEMENT ANALYSIS OF PLATE BENDING

Rad: Chaptr 7 NONLINEAR FINITE ELEMENT ANALYSIS OF PLATE BENDING CONTENTS Govrning Equations of th First-Ordr Shar Dformation thor (FSDT) Finit lmnt modls of FSDT Shar and mmbran locking Computr implmntation Strss calculation Numrical Eampls JN Rdd Nonlinar Plat Bnding: 1

THE FIRST-ORDER SHEAR DEFORMATION THEORY Displacmnt Fild of th FSDT u (,, z,) t u(,,) t z (,,) t 1 u (,, z,) t v(,,) t z (,,) t u( zt,,,) wt (,,) 3 Nonlinar strains z z w dw d w z dw = d dw = + d ij 1 u u i j um um j i i j z Von Karman Nonlinar strains u 1 1 um um u1 1 u1 1 u 1 u3 1 1 1 1 1 1 1 JN Rdd Nonlinar Plat Bnding:

JN Rdd NONLINEAR STARINS OF THE FSDT Actual Nonlinar strains u 1 1 u 3 u 1 w 0 1 z z u 1 u 3 v 1 w 0 1 z z u u u1 u 3 3 u v w w z 0 1 z u u w u u w z z 1 3 0 3 0 z z, z z Virtual Nonlinar strains u w w z z v w w z z 0 1 0 1 u v w w w w z z 0 1 w w 0 0 z z, z z

PRINCIPLE OF VIRTUAL DISPLACEMENTS 0 = W W + W I E z Q M N W z z h I = h Ω + + + 0 1 0 1 ( ) ( ) ( z ) + + + K 0 1 0 s z z } 0 + K sz z dz dd h WE = ( ) ( ) h nn un + zn + ns us + zs + nzw dzds Γ 0 + Ω ( q kw) w dd} = N + M + N + M + N + M Ω 0 1 0 1 0 1 0 0 + Qz + Q z qw dd ( ) Q M N M M N N Mnn N ns M ns Nnn un + Nns us + Mnn n + Mns s + Qn w ds Γ Q n N nn JN Rdd Plats (Nonlinar): 4

THE FIRST-ORDER SHEAR DEFORMATION THEORY @Q + @Q + @ JN Rdd Equations of quilibrium + @ @N @N µ @w N µ N @w + N @M @M + @N + @N + N @w @w + @M + @M = I 0 @ u @t = I 0 @ v @t q = I 0 @ w @t Q = I @ Á @t Q = I @ Á @t z z w dw d Strss rsultants w z z dw = d dw = + d Z h= µ Q = K s ¾ z dz = K s A 55 Á + @w h= Z h= µ Q = K s ¾ z dz = K s A 44 Á + @w h= M = D 11 @Á + D 1 @Á ; @Á M = D 1 + D @Á µ @Á M = D 66 + @Á Nonlinar Plat Bnding: 5

THE FIRST-ORDER SHEAR DEFORMATION THEORY Wak forms( v u, v v, v w, v, v ) 1 3 4 5 Z µ @±u 0 = N + @±u I N + I 0 ±u @ u @t dd ±un n ds Z µ @±v 0 = N + @±v I N + I 0 ±v @ v @t dd ±vn ns ds Z µ @±w @w 0 = Q + N + N @w + @±w µ @w Q + N + N @w I + I 0 ±w @ w @t ±wq dd ±wq n ds Z µ @±Á 0 = M @ I Á + ±Á Q + I ±Á @t dd ±Á M n ds Z µ @±Á 0 = M @ I Á + ±Á Q + I ±Á @t dd ±Á M ns ds JN Rdd Nonlinar Plat Bnding: 6

Finit Elmnt Modls of Th First-ordr Plat Thor (FSDT) (Continud) Finit Elmnt Approimation u(; ; t) = Á (; ; t) = mx u j (t)ã j (; ); v(; ; t) = j=1 w(; ; t) = mx v j (t)ã j (; ) j=1 nx w j (t)ã j (; ) j=1 px Sj 1 (t)ã j (; ); Á (; ; t) = j=1 px Sj (t)ã j (; ) j=1 Finit Elmnt Modl 6 4 M 11 0 0 0 0 0 M 0 0 0 0 0 M 33 0 0 0 0 0 M 44 0 0 0 0 0 M 55 3 8 >< 7 5 >: Äu Äv Äw ÄS ÄS 9 >= >; K 11 K 1 K 13 K 14 K 15 3 8 9 u K 1 K K 3 K 4 K 5 >< v >= + 6 K 31 K 3 K 33 K 34 K 35 7 w 4 K 41 K 4 K 43 K 44 K 45 5 >: S >; K 51 K 5 K 53 K 54 K 55 S = 8 >< >: F 1 F F 3 F 4 F 5 9 >= >; JN Rdd Nonlinar Plat Bnding: 7

Full Dicrtizd Modl and Itrativ Schm Full Discrtizd Finit Elmnt Modl [ ^K] s+1 f g s+1 = f ^F g s;s+1 [ ^K] s+1 = [K] s+1 + a 3 [M] s+1 f ^F g s;s+1 = ff g s+1 + [M] s+1 (a 3 f g s + a 4 f g _ s + a 5 f g Ä s ) a 3 = ( t) ; a 4 = t ; a 5 = 1 1 Acclrations and Vlocitis f Ä g s+1 = a 3 (f g s+1 f g s ) a 4 f _ g a 5 f Ä g s f _ g s+1 = f _ g s + a f Ä g s + a 1 f Ä g s+1 whr a 1 = t and a = (1 ) t. JN Rdd Nwton-Raphson Itrativ Schm f g s+1 r+1 = [ ^K T ] r 1 s+1 f ^Rg r ; [ ^K T ] r " @f ^Rg @f g # s+1 r Nonlinar Plat Bnding: 8

Stiffnss Cofficints (tpical) Z Kij 11 = Z Kij 1 = K 13 ij = 1 Z µ @Ãi A 11 µ @Ãi A 1 @Ã i @Ã j @Ã i + A 66 @Ãj + A @Ãi 66 µ @w 0 A 11 @Ã j @Ã j @' j + A @w 0 1 dd dd = Kji 1 @' j JN Rdd @Ãi + A 66 Z µ Kij @Ãi = A 66 Kij 3 = 1 Z @Ã i Z Kij 31 = @Ãi + A 66 @' i + @' i µ @w0 µ @w 0 A 1 µ @w0 µ @w 0 A 11 @' j + @w 0 @' j @Ãj + A @Ãi @Ã j @' j + A @w 0 @' j + @w 0 @' j # dd dd @' j # dd @Ãj + A @w 0 @Ãj 66 µ @w 0 @Ãj A 66 + A @w 0 @Ãj 1 # dd Nonlinar Plat Bnding: 9

Tangnt Stiffnss Cofficints (tpical) R T R K F u v w S S 5 n( ) i 1 3 4 1 5 ij =, i = ik k i, i i, i i, i i, i i, = = = = i = i j = 1 k= 1 K T K F K ( ) 5 n( ) 5 n( ) ik ij = ik k i = k + ij j = 1 k= 1 = 1 k= 1 j T = K = ( K ), T = K = ( K ) 1 1 1 T T T = K = ( K ), T = K = ( K ) 4 4 4 T 5 5 5 T K K T K w K 5 n( ) 1 n 13 13 ik 13 ik 13 ij = k + ij = k + ij = 1 k= 1 wj k= 1 wj (1) () () 1 i w j w j = A 11 A1 + Ω (1) () () i w j w j + A66 + dd + K = K + K = K = T 13 13 13 31 ij ij ij ji JN Rdd Plats (Nonlinar): 10 13 ij

Tangnt Stiffnss Cofficints (tpical) 5 n( ) 3 n( ) 31 3 33 33 33 K ik 33 Kik Kik K ik Tij = Kij + k = Kij + uk + vk + wk = 1 k= 1 w j k 1 wj wj w = j () () () () () () () () 33 i j i j u i j i j u = Kij + A11 + A1 + A 66 + dd Ω () () () () () () () () i j i j v i j i j v + A 1 + A + A66 + dd Ω () () () () () () () () w i j w i j w w i j i j + A11 A A66 + + + Ω () () () () () () () () 1 w i j w i j w w i j i j + ( A1 + A6 6 ) dd + + + JN Rdd Plats (Nonlinar): 11

Tangnt Stiffnss Cofficints (tpical) () () 33 33 u w v 1 w i j Tij = Kij + A11 A1 ( 1 66 ) + A A dd Ω + + + () () v w u 1 w i j + A A1 ( A1 A66 ) dd + + + + Ω () () () () u v w w 1 w w + A66 + i j i j + ( A1 A66 ) dd Ω + + + () () () () 33 i j i j () () Tij = A55 + A44 + ki j dd Ω () () () () () () () () i j i j i j i j + N + N + N + N Ω () () () () w w i j i j + ( A1 + A66 ) + () () () () w w i j w w i j + A11 + A66 + A66 A dd + JN Rdd 1

Shar and Mmbran Locking (Rvisit) Shar Locking Us rducd intgration to valuat all shar stiffnsss (i.., all K ij that contain transvrs shar trms) Mmbran Locking Us rducd intgration to valuat all mmbran stiffnsss (i.., all K ij that contain von Kármán nonlinar trms) JN Rdd Nonlinar Plat Bnding: 13

SOME ELEMENTS OF THE PLATE PROGRAM C Rad matrial proprtis and gnrat plat matrial stiffnsss READ(IN,*)E1,E,G1,G13,G3,PR1 READ(IN,*)CF,SCF,THI PR1=(E/E1)*PR1 WRITE(IT,45) E1,E,G1,G13,G3,PR1,PR1, CF,SCF,THI Q11 = E1/(1-PR1*PR1) Q1 = (PR1*E)/(1-PR1*PR1) Q = E/(1-PR1*PR1) Q44 = G3 Q55 = G13 Q66 = G1 A11 = THI*Q11 A1 = THI*Q1 A = THI*Q A44 = THI*Q44*SCF A55 = THI*Q55*SCF A66 = THI*Q66 Error calculation, convrgnc chck, and othr nonlinar analsis aspcts rmain th sam as in D A must b ij and Dij transfrrd to th subroutin ELMATRCSD through common block. D11 = THI*THI*THI*Q11/1.0 D1 = THI*THI*THI*Q1/1.0 D = THI*THI*THI*Q/1.0 D66 = THI*THI*THI*Q66/1.0 JN Rdd Plats (Nonlinar): 14

SOME ELEMENTS OF THE PLATE PROGRAM Initializ (bfor th do-loops on numrical intgration): K, F, T ij i ij Dfin (insid th do-loops on numrical intgration). For ampl, th linar stiffnsss cofficints ar [F0=F0+DQ(NL); DQ(NL) arra of load incrmnts ] DO I=1,NPE ELF3(I)=ELF3(I)+F0*SFL(I)*CNST DO J=1,NPE S00=SFL(I)*SFL(J)*CNST S11=GDSFL(1,I)*GDSFL(1,J)*CNST S=GDSFL(,I)*GDSFL(,J)*CNST S1=GDSFL(1,I)*GDSFL(,J)*CNST S1=GDSFL(,I)*GDSFL(1,J)*CNST ELK11(I,J)=ELK11(I,J)+A11*S11+A66*S ELK1(I,J)=ELK1(I,J)+A1*S1+A66*S1 ELK1(I,J)=ELK1(I,J)+A1*S1+A66*S1 ELK(I,J)=ELK(I,J)+A66*S11+A*S ELK33(I,J)=ELK33(I,J)+CF*S00 ELK44(I,J)=ELK44(I,J)+D11*S11+D66*S ELK45(I,J)=ELK45(I,J)+D1*S1+D66*S1 ELK54(I,J)=ELK54(I,J)+D1*S1+D66*S1 ELK55(I,J)=ELK55(I,J)+D66*S11+D*S ENDDO ENDDO Similarl, dfin th shar and nonlinar cofficints in th rducd intgration loop; comput th rsidual vctor and tangnt stiffnss cofficints. JN Rdd Plats (Nonlinar): 15

REARRANGE THE ELEMENT COEFFICIENTS II=1 DO 0 I=1,NPE ELF(II) =ELF1(I) ELF(II+1)=ELF(I) ELF(II+)=ELF3(I) ELF(II+3)=ELF4(I) ELF(II+4)=ELF5(I) JJ=1 DO 10 J=1,NPE ELK(II,JJ) = ELK11(I,J) ELK(II,JJ+1) = ELK1(I,J) ELK(II,JJ+) = ELK13(I,J) ELK(II,JJ+3) = ELK14(I,J) ELK(II,JJ+4) = ELK15(I,J) ELK(II+1,JJ) = ELK1(I,J) ELK(II+,JJ) = ELK31(I,J) ELK(II+3,JJ) = ELK41(I,J) ELK(II+4,JJ) = ELK51(I,J) ELK(II+1,JJ+1) = ELK(I,J) ELK(II+1,JJ+) = ELK3(I,J) ELK(II+1,JJ+3) = ELK4(I,J) ELK(II+1,JJ+4) = ELK5(I,J) ELK(II+,JJ+1) = ELK3(I,J) ELK(II+3,JJ+1) = ELK4(I,J) ELK(II+4,JJ+1) = ELK5(I,J) ELK(II+,JJ+) = ELK33(I,J) ELK(II+,JJ+3) = ELK34(I,J) ELK(II+,JJ+4) = ELK35(I,J) ELK(II+3,JJ+) = ELK43(I,J) ELK(II+4,JJ+) = ELK53(I,J) ELK(II+3,JJ+3) = ELK44(I,J) ELK(II+3,JJ+4) = ELK45(I,J) ELK(II+4,JJ+3) = ELK54(I,J) ELK(II+4,JJ+4) = ELK55(I,J) 10 JJ=NDF*J+1 0 II=NDF*I+1 Th sam applis to th tangnt stiffnss cofficints JN Rdd Plats (Nonlinar): 16

Post-Computation of Strss Componnts Q Q 0 11 1 z Q55 0 z Q1 Q 0, z 0 Q 55 z 0 0 Q 66 E E E Q, Q, Q, Q G, Q G, Q G, 1 1 11 1 111 111 111 66 1 44 3 55 13 u 1 w w v 1 w z z, z w u v ww JN Rdd Plats (Nonlinar): 17

Post-Computation of Strss Componnts SUBROUTINE STRESS (NPE,NDF,NGPR,ELXY,ELU) IMPLICIT REAL*8 (A-H,O-Z) COMMON/STR/Q11,Q,Q1,Q44,Q55,Q66,THI COMMON/SHP/SF(9),GDSF(,9) DIMENSION GAUSSPT(4,4),ELXY(9,),ELU(45) DATA GAUSSPT/4*0.0D0,-.5773507D0,.5773507D0,*0.0D0, 1 -.77459667D0,0.0D0,.77459667D0,0.0D0,-.86113631D0, -.33998104D0,.33998104D0,0.86113631D0/ C DO 40 NI=1,NGPR DO 40 NJ=1,NGPR XI=GAUSSPT(NI,NGPR) ETA=GAUSSPT(NJ,NGPR) CALL INTERPLND (NPE,XI,ETA,DET,ELXY) X=0.0 Y=0.0 JN Rdd Plats (Nonlinar): 18

Initializ hr Subroutin STRESS (continud) DO 0 I=1,NPE L=(I-1)*NDF+1 X=X+SF(I)*ELXY(I,1) Y=Y+SF(I)*ELXY(I,) DUX=DUX+GDSF(1,I)*ELU(L) DUY=DUY+GDSF(,I)*ELU(L) DVX=DVX+GDSF(1,I)*ELU(L+1) DVY=DVY+GDSF(,I)*ELU(L+1) DWX=DWX+GDSF(1,I)*ELU(L+) DWY=DWY+GDSF(,I)*ELU(L+) PHIX=PHIX+SF(I)*ELU(L+3) PHIY=PHIY+SF(I)*ELU(L+4) DPXX=DPXX+GDSF(1,I)*ELU(L+3) DPXY=DPXY+GDSF(,I)*ELU(L+3) DPYX=DPYX+GDSF(1,I)*ELU(L+4) 0 DPYY=DPYY+GDSF(,I)*ELU(L+4) PHIX=0.0 PHIY=0.0 DUX=0.0 DUY=0.0 DVX=0.0 DVY=0.0 DWX=0.0 DWY=0.0 DPXX=0.0 DPXY=0.0 DPYX=0.0 DPYY=0.0 JN Rdd Plats (Nonlinar): 19

Subroutin STRESS (continud) EX0 = DUX +0.5*DWX*DWX EY0 = DVY +0.5*DWY*DWY EXY0= DUY+DVX +DWX*DWY EX1 = DPXX EY1 = DPYY EXY1=DPXY+DPYX EXZ = PHIX+DWX EYZ = PHIY+DWY C Writ th statmnts for strss componnts SXXT, SXXB, C tc. and writ thm out for ach Gauss point C *** our task *** WRITE(IT,50) X,Y,SXXT,SYYT,SXYT,SXZ,SYZ WRITE(IT,50) SXXB,SYYB,SXYB 40 CONTINUE RETURN 50 FORMAT (5X,8E1.4) END JN Rdd Plats (Nonlinar): 0

TYPICAL SIMPLY SUPPORT CONDITIONS for Pur Bnding cas CPT: FSDT: CPT: w w 0 w 0 Smmtr conditions: b b w w 0; FSDT: w 0 w CPT: w 0 FSDT: w 0 Computational domain w a a CPT: w 0 FSDT: w 0 w w CPT: 0 at 0; 0 at 0 FSDT: 0 at 0; 0 at 0 JN Rdd Plat bnding: 1

Th ffct of rducd intgration, thicknss, and msh rfinmnt on th linar cntr dflctions and strsss of a simpl supportd, isotropic (ν = 0.5) squar plat undr a uniform transvrs load of intnsit q 0. F full intgration M Mid intgration 1 1 4 4 linar linar linar quadratic Eactz a=h Intg. ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ 10 F 0.964 0.018.474 0.119 3.883 0.16 4.770 0.90 4.791 0.76 M 3.950 0.095 4.71 0.35 4.773 0.66 4.799 0.7 0 F 0.70 0.005 0.957 0.048.363 0.138 4.570 0.68 4.65 0.76 M 3.669 0.095 4.54 0.35 4.603 0.66 4.633 0.7 40 F 0.070 0.001 0.79 0.014 0.944 0.056 4.505 0.70 4.584 0.76 M 3.599 0.095 4.375 0.35 4.560 0.66 4.59 0.71 50 F 0.005 0.000 0.18 0.009 0.65 0.039 4.496 0.67 4.579 0.76 M 3.590 0.095 4.47 0.35 4.555 0.66 4.587 0.71 100 F 0.011 0.000 0.047 0.00 0.18 0.011 4.48 0.66 4.57 0.76 M 3.579 0.095 4.465 0.35 4.548 0.66 4.580 0.7 CPT(N) 5.643 0.60 4.857 0.74 4.643 0.76 4.570 0.76 CPT(C) 4.638 0.6 4.574 0.7 4.570 0.75 4.570 0.76 ¹w = weh 3 10 =q 0 a 4, ¹¾ = ¾ (A; A; h)h =q 0 a, A = 1 4 a (1 1 linar), 1 8 a ( linar), 1 a (4 4 linar), 0:0583a ( 16 quadratic). JN Rdd Plat bnding:

Gauss Point Locations (basd on rducd Intgration Gauss points) for Strss Computation ( a / ) ( a/ 83, b/ 8) ( 3a/ 83, b/ 8) ( a/ 8, b/ 8) ( 3a/ 8, b/ 8) ( b/ ) ( a / ) ( b/ ) Msh of 4-nod (linar) lmnts Msh of 9-nod (quadratic) lmnts b( 3 1) b( 3 1), = ( 0. 0583a, 0. 0583b) 8 3 8 3 JN Rdd Plat bnding: 3

REMARKS Th nin-nod lmnt givs virtuall th sam rsults for full (3 3 Gauss rul) and mid (3 3 and Gauss ruls for bnding and shar trms, rspctivl) intgrations. Howvr, th rsults obtaind using th mid intgration ar closst to th act solution. Full intgration givs lss accurat rsults than mid intgration, and th rror incrass with an incras in sid-to-thicknss ratio (a/h). This implis that mid intgration is ssntial for thin plats, spciall whn modld b lowr-ordr lmnts. Full intgration rsults in smallr rrors for quadratic lmnts and rfind mshs than for linar lmnts and/or coarsr mshs. JN Rdd Plat bnding: 4

u = w = = b Nonlinar Analsis of Simpl Supportd Plat (SS-1) 0 SS-1 v = w = = 0 Dflction vrsus load paramtr for simpl supportd (SS1) plat undr uniforml distributd load. 3.0 b.5 a a v = w = = 0 u = w = = 0 v = w = = b b 0 SS- u = w = = 0 Dflction, w/h.0 1.5 1.0 0.5 SS SS1 u = w = = a 0 a v = w = = 0 0.0 0 50 100 150 00 50 Load paramtr, P JN Rdd Nonlinar Plat Bnding: 5

Clampd Circular Plat undr UDL w 0 /h Dflction, 4.0 3.0.0 E = 10 6 psi, ν = 0.3 a = 100 in., h = 10 in. u = 0, φ at = 0 = 0 E = 10 6 psi, ν = 0.3 h = 10 in. 9 4 8 19 7 14 6 9 4 3 5 1 5 10 15 0 a = 100 in. v = u = w = = = 0 0 φ φ on th clampd dg v = 0, φ = 0 at = 0 1.0 Msh of 5-Q9 lmnts 0.0 0 0 40 60 80 100 10 Load paramtr, (q 0 a 4 /Eh 4 ) JN Rdd Plats (Nonlinar): 6

Simpl Supportd (SS) Orthotropic* Plat, 0 ( ) 0.50 0.40 Eprimntal [8] CLPT FSDT Gomtr and Matrial Proprtis a = b = 1 in, h = 0.138 in E 1 = 3 10 6 psi, E = 1.8 10 6 psi G 1 = G 3 = G 13 = 0.37 10 6 psi ν 1 = ν 3 = ν 13 = 0.3 0.30 0.0 0.10 Linar Nonlinar [8] Zaghloul, S. A. and Knnd, J. B., ``Nonlinar Bhavior of Smmtricall Laminatd Plats, Journal of Applid Mchanics, 4, 34-36, 1975. 0.00 0.0 0.4 0.8 1. 1.6.0 Prssur, q 0 (psi) JN Rdd Nonlinar Plat Bnding: 7

Dflction vs. load paramtr for plats undr uniforml distributd load w Dflction, 3.0.5.0 1.5 1.0 0.5 0.0 w qa 0 0 w =, P = h Eh SS-1 (FSDT) SS-1 (CPT) SS-3 (CPT) SS-3 (FSDT) 0 50 100 150 00 50 Load paramtr, P 4 4 σ Strsss, 4 0 16 1 8 4 0 a = Eh SS-1 (FSDT) SS-3 (FSDT) SS-1 (CPT) SS-3 (CPT) Mmbran strsss 0 50 100 150 00 50 Load paramtr, P JN Rdd Nonlinar Plat Bnding: 8

JN Rdd Cntr Dflction vs. Tim for a Simpl Supportd Isotropic Plat Undr Suddnl Applid Uniforml Distributd Prssur Load Cntr dflction, w0 (cm) 1.80 1.50 1.0 0.90 0.60 0.30 0.00-0.30-0.60-0.90 0 30 60 90 10 150 Tim, t (s) (ms) a = b = 43.8 cm, h = 0.635 cm, ρ ρ =.547 10-6 N-s /cm 4, E 1 = E = 7.031 10 5 N/cm, ν 1 = 0.5 q 0 = 4.88 10-4 N /cm, t = 0.005 s = 5ms Figur 13.4 1 (SS-) q0 (N/cm ) 10 8 6 4 0 0.0 0.4 0.8 1. 1.6 Dflction, w 0 (cm) q 0 ( t = 5.0 ms) q 0 ( t = 5.0 ms) 5q 0 ( t =.5 ms) 10q 0 ( t =.5 ms)

SUMMARY In this lctur w hav covrd th following topics: Govrning Equations of FSDT Finit lmnt modls of FSDT Tangnt stiffnss cofficints Shar and mmbran locking Programming aspcts (including strss computation) Numrical ampls JN Rdd Nonlinar Plat Bnding: 30