HIGHERORDER THEORIES


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1 HIGHERORDER THEORIES THIRDORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1
2 ThirdOrder Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) + zá x (x y t) c 1 z 3 Á x 0 µ v(x y z t) v 0 (x y t) + zá y (x y t) c 1 z 3 Á y 0 w(x y z t) w 0 (x y t) φ x Transverse Shear Strains ½ yz ½ (0) yz xz (0) yz xz xz ½ (0) yz (0) xz ½ Áy 0 Á x 0 ¹ Q 44 ¹Q 45 + z 2 ½ (2) yz ½ (2) yz (2) xz Transverse Shear Stresses ³ yz (0) + z 2 yz (2) ³ yz (0) + z 2 yz (2) (2) xz c 2 3c 1 ( ³ ) Á y 0 c 2 Áx 0 + ¹ Q 45 + ¹Q 44 ³ xz (0) + z 2 xz (2) ³ xz (0) + z 2 xz (2) (u,w) ( u 0, w 0 ) w 0 x xz (x y h 2 ) 0 yz(x y h 2 ) 0! c 2 4h 2 2
3 ThirdOrder Shear Deformation Plate Theory (TSDT) xy xy ¹ Q x ¹ Q 2 P xx + c 1 " 2 + c 1 I 0 Äu 0 + J 1 Ä Áx c 1 I Äw 0 I 0 Äv 0 + J 1Áy Äw 0 c 1 I 0 Nxx + 0 xy 0 + N 0 + P xy P yy 2 + q I0 Äw 0 c 2 1I 2 Äw 0 2 Äw 0 0 I 3 Á Ä x + J4 Á Ä y M ¹ M ¹ xy ¹ M xy ¹ M yy ¹ Q x J 1 Äu 0 + K 2 Ä Áx c 1 J Äw 0 ¹ Q y J 1 Äv 0 + K 2 Ä Áy c 1 J Äw 0 ¹M M c 1 P ¹ Q Q c 2 R I i Z h 2 h 2 ½ (z) i dz (i ) 3
4 ThirdOrder Shear Deformation Plate Theory (TSDT) (Continued) HigherOrder Stress Resultants 8 < : P xx P yy P xy 9 Z h 2 h 2 8 < : xx yy xy 9 z3 dz ½ Rx R y Z h2 h2 ½ yz xz z 2 dz Primary Variables : u n u s w Á n Á s Secondary Variables : N nn N ns ¹ Vn P nn ¹ Mnn ¹ Mns ¹V n c 1 xy n x + c 1 " µ I 3 Äu 0 + J 4 Ä Áx c 1 I Äw 0 n x " yy n y µ + I 3 Äv 0 + J 4 Ä Áy c 1 I Äw 0 n y # P(w 0 ) + ¹Q x n x + ¹Q y n y + P(w0 ) + c ns µ N 0 + N 0 n x + N xy + 0 yy n y 4
5 Bending of a symmetric crossply (0/90/90/0) laminate under uniformly distributed load Deflection, w _ D Elasticity Solution CLPT FSDT TSDT E psi (7 Gpa) E 1 25E 2, G 12 G E 2 G E 2, ν at x0 and xa v 0 w 0 0 _ φ y N xx M xx 0 y a SS 1 at y0 and yb u 0 w 0 0 _ φ x N yy M yy 0 b x a/h SS1 Figure Boundary Conditions J.N. Reddy 5
6 Bending of a symmetric crossply (0/90/90/0) laminate under uniformly distributed load 0.50 c ess coo d ate, / CLPT (E) FSDT (E) FSDT (C) TSDT (E) TSDT (C) (E): equilibriumderived (C): constitutivelyderived Stress, σ _ xz (0,b/2,z) J.N. Reddy 6
7 Bending of a symmetric crossply (0/90/90/0) laminate under uniformly distributed load 0.50 c ess coo d ate, / CLPT (E) FSDT (E) FSDT (C) TSDT (E) TSDT (C) (E): equilibriumderived (C): constitutivelyderived Stress, σ _ yz (a/2,0,z) J.N. Reddy Thirdorder Laminate Theory 7
8 LAYERWISE THEORY z y Equilibrium of Interlaminar Stresses x z kth layer th layer σ zx k+1 x σ zx σ zy σ zz σ zx σ zy σ zz σ zy σ zz σ zz σ zy σ zx k J.N. Reddy Layerwise Theory 8
9 Layerwise Laminate Theory Equilibrium Requirements < xx : yy xy < xx 6 : yy xy < xz : yz zz < xz : yz zz < xz : yz zz < xz : yz zz! < xz : yz " zz 6 < xz : yz " zz SingleLayer Theories < xx : yy xy 6 < xx : yy xy < xz : yz zz 6 < xz : yz zz 8 < " xx " yy : xy 9 8 < " xx " yy : xy 9 < xz yz : " zz < xz yz : " zz J.N. Reddy Layerwise Theory 9
10 Layerwise Kinematic Model NX u(x y z t) U I (x y t) I (z) I1 NX v(x y z t) V I (x y t) I (z) I1 MX w(x y z t) W I (x y t)ª I (z) I1 z Ith layer x z N I+1 I I 1 4 U N U I+1 U I U I 1 U 4 I+1 U I+1 U 3 U I 2 1 U I U 1 1 u I 1 U I 1 U I Φ I (z) J.N. Reddy Layerwise Theory 10
11 Layerwise Displacement Field, Governing Equations, and FEM Approximation NX u(x y z t) U I (x y t) I (z) I1 NX v(x y z t) V I (x y t) I (z) I1 MX w(x y z t) W I (x y t)ª I (z) I I xy I xy QI x N X I yy QI y J1 NX J1 I U 2 I V ~ Q I x ~ Q I y ~ Q I z + ~ N I + q b ± I1 + q t ± IM MX J1 ~I W 2 Finite element approximation p8 I3 I2 I1 U I (x y t) V I (x y t) W I (x y t) px U j I (t)ã j(x y) j1 px j1 V j I (t)ã j(x y) qx W j I (t)' j(x y) J.N. Reddy Layerwise Theory 11 j1
12 Layerwise Kinematic Model Conventional 3D Layerwise 2D + 1D Cubic serendipity element Linear Lagrange element (1a) (inplane) (1b) (through thickness) Quadratic serendipity element Quadratic Lagrange element (2a) (inplane) (through thickness) J.N. Reddy Layerwise Theory (2b) 12
13 Table: Comparison of the number of operations needed to form the element sti ness matrices for equivalent elements in the conventional 3D format and the layerwise 2D format. Full quadrature is used in all. Element Type y Multipli. Addition Assignments 1a (3D) 1,116, , ,000 1b (LWPT) 423, , ,000 2a (3D) 1,182, , ,000 2b (LWPT) 284, ,000 69,000 y Element 1a: 72 degrees of freedom, 24node 3D isoparametric hexahedron with cubic inplane interpolation and linear transverse interpolation. Element 1b: 72 degrees of freedom, E12{L1 layerwise element. Element 2a: 81 degrees of freedom, 27node 3D isoparametric hexahedron with quadratic interpolation in all three directions. Element 2b: 81 degrees of freedom, E9{Q1 layerwise element. J.N. Reddy Layerwise Theory 13
14 Layerwise Kinematic Model 3D modeling with 2D & 1D elements z z y y a 2 h x a 2 a 2 a 2 x 2D quadratic Lagrangian element three quadratic layers through the thickness E psi E 2 E psi G 12 0: psi G 13 G 23 0: psi º 12 º 13 º 23 0:25 J.N. Reddy u(x a2 z) u(a2 y z) 0 v(a2 y z) u(x a2 z) 0 w(x a z) u(a y z) 0
15 Validation of the Layerwise Theory J.N. Reddy Layerwise Theory 15
16 Verification of the Layerwise Theory J.N. Reddy Layerwise Theory 16
17 SUMMARY In this lecture, we have discussed the following topics: Thirdorder Shear Deformation Plate Theory Development of governing equations Numerical results Layerwise Laminate Theory Development of governing equations Numerical results J.N. Reddy Layerwise Theory 17
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