1 Exact 3D Stress Analysis of Laminated Composite Plates and Shells by Sampling Surfaces Method G.M. Kulikov and S.V. Plotnikova Speaker: Gennady Kulikov Department of Applied Mathematics & Mechanics

2 Kinematic Description of Undeformed Shell (1)(2)(3) Figure 1. Geometry of laminated shell Base Vectors of Midsurface and SaS Indices: n = 1, 2, …, N; i n = 1, 2, …, I n ; m n = 2, 3, …, I n -1 N - number of layers; I n - number of SaS of the nth layer r(  1,  2 ) - position vector of midsurface  ; R (n)i - position vectors of SaS of the nth layer e i - orthonormal vectors; A , k  - Lamé coefficients and principal curvatures of midsurface c  = 1+k   3 - components of shifter tensor at SaS (n)i(n)i(n)i(n)i n (n)i(n)i(n)i(n)i n n  (n)1,  (n)2, …,  (n)I - sampling surfaces (SaS)  (n)i - thickness coordinates of SaS  [n-1],  [n] - thickness coordinates of interfaces n 33 3 n

3 Kinematic Description of Deformed Shell (4)(5)(6) Figure 2. Initial and current configurations of shell Base Vectors of DeformedSaS Base Vectors of Deformed SaS Position Vectors of Deformed SaS u (  1,  2 ) - displacement vectors of SaS (n)i(n)i(n)i(n)i n  (  1,  2 ) - derivatives of 3D displacement vector at SaS (n)i(n)i(n)i(n)i n

4 Green-Lagrange Strain Tensor at SaS Linearized Strain-Displacement Relationships Presentation of Displacement Vectors of SaS (7)(8)(9)

5 Presentation of Derivatives of Displacement Vectors of SaS Strain Parameters Component Form of Strains of SaS Remark. Strains (12) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through Kulikov and Carrera (2008) (10)(11)(12)

6 Displacement Distribution in Thickness Direction Presentation of Derivatives of 3D Displacement Vector Strain Distribution in Thickness Direction Higher-Order Layer-Wise Shell Formulation (13)(14)(15)(16) L (  3 ) -Lagrange polynomials of degree I n - 1 L (  3 ) - Lagrange polynomials of degree I n - 1 (n)i(n)i(n)i(n)i n

7 Stress Resultants Variational Equation Constitutive Equations Presentation of Stress Resultants (17)(18)(19)(20)(21) C ijkm - components of material tensor of the nth layer (n)(n)(n)(n) p i, p i - surface loads acting on bottom and top surfaces  and   [0] [N][N][N][N] [N][N][N][N]

8 Numerical Examples 1. Simply Supported Sandwich Plate under Sinusoidal Loading Analytical solution Figure 3. Sandwich plate Table 1. Results for thick square sandwich plate with a / h = 2 InInInIn S 11 (-0.5) S 11 (0.5) S 22 (-0.5) S 22 (0.5) S 12 (-0.5) S 12 (0.5) S 13 (0) S 23 (0) S 33 (0.5) Pagano

9 Figure 5. Accuracy of satisfying the boundary conditions  i (-h / 2) and  i (h / 2)on the bottom (  ) and top (  ) surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4, where  i = lg  S i3 – S i3  Figure 5. Accuracy of satisfying the boundary conditions  i (-h / 2) and  i (h / 2) on the bottom (  ) and top (  ) surfaces of the sandwich plate: (a) a / h = 2 and (b) a / h = 4, where  i = lg  S i3 – S i3  3D Figure 4. Distribution of transverse shear stresses S 13 and S 23 through the thickness of the sandwich plate for I 1 = I 2 = I 3 =7: present analysis ( ) and Pagano’s solution (  )

10 ̃̃ InInInIn U 3 (0) S 11 (0.5) S 22 (0.5) S 12 (0.5) S 13 (0.145) S 23 (0.125) S 13 (-0.280) S 23 (-0.280) D Savoia Table 2. Results for square (b = a) two-layer angle-ply plate with h 1 = h 2 = h / 2, a / h = 4 and stacking sequence [-15  / 15  ] 2. Antisymmetric Angle-Ply Plate under Sinusoidal Loading Analytical solution

11  ̃̃  Table 3. Results for rectangular (b = 3a) two-layer angle-ply plate with h 1 = h 2 = h / 2, a / h = 4 and stacking sequence [-15  / 15  ] Figure 6. Distribution of transverse shear stresses S 13 and S 23 through the thickness of square unsymmetric two-layer angle-ply plate for I 1 = I 2 = 7 ̃  InInInIn U 3 (0) S 11 (0.5) S 22 (0.5) S 12 (0.5) S 13 (0.175) S 23 (0.145) S 13 (-0.270) S 23 (-0.270) D Savoia

12 Analytical solution 3. Cylindrical Composite Shell under Sinusoidal Loading Table 4. Results for thick two-ply cylindrical shell with R / h = 2 and stacking sequence [0  /90  ] Figure 7. Simply supported cylindrical composite shell with L / R = 4 InInInIn U 3 (0) S 11 (-0.5) S 11 (0.5) S 22 (-0.5) S 22 (0.5) S 12 (-0.5) S 12 (0.5) S 13 (-0.25) S 23 (0.25) S 33 (0.25) Varadan

13 Figure 9. Distribution of transverse shear stresses S 13 and S 23 through the thickness of three-ply cylindrical shell with stacking sequence [90  / 0  / 90  ] for I 1 = I 2 = I 3 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution (  ) Figure 8. Distribution of transverse shear stresses S 13 and S 23 through the thickness of two-ply cylindrical shell with stacking sequence [0  / 90  ] for I 1 = I 2 = 9: present analysis ( ) and Varadan-Bhaskar ‘s 3D solution (  )

14 ConclusionsConclusions  A simple and efficient method of SaS inside the shell body has been proposed. This method permits the use of 3D constitutive equations and leads to exact 3D solutions of elasticity for thick and thin laminated plates and shells with a prescribed accuracy  A new higher-order layer-wise theory of shells has been developed through the use of only displacement degrees of freedom, i.e., displacements of SaS. This is straightforward for finite element developments

15 Thanks for your attention!