NONLINEAR ANALYSIS OF PLATE BENDING

NONLINEAR 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 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 Nonlinar Plat Bnding:

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 Plats (Nonlinar): 4

THE FIRST-ORDER SHEAR DEFORMATION THEORY @Q @ @Q @ @ @ 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 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 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 >= >; Nonlinar Plat Bnding: 7

Full Dicrtizd Modl and Itrativ Schm Full Discrtizd Finit Elmnt Modl [ ^K] s1 f g s1 = f ^F g s;s1 [ ^K] s1 = [K] s1 a 3 [M] s1 f ^F g s;s1 = ff g s1 [M] s1 (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 s1 = a 3 (f g s1 f g s ) a 4 f _ g a 5 f Ä g s f _ g s1 = f _ g s a f Ä g s a 1 f Ä g s1 whr a 1 = t and a = (1 ) t. Nwton-Raphson Itrativ Schm f g s1 r1 = [ ^K T ] r 1 s1 f ^Rg r ; [ ^K T ] r " @f ^Rg @f g # s1 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 @ @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 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 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 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) Nonlinar Plat Bnding: 13

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 Plats (Nonlinar): 14

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 Plat bnding: 15

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). Plat bnding: 16

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 Plat bnding: 17

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. Plat bnding: 18

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 Nonlinar Plat Bnding: 19

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 ) Plats (Nonlinar): 0

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) Nonlinar Plat Bnding: 1

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 Nonlinar Plat Bnding:

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 Nonlinar Plat Bnding: 4