1 A New Concept of Sampling Surfaces in Shell Theory S.V. Plotnikova and G.M. Kulikov Speaker: Professor Gennady M. Kulikov Department of Applied Mathematics & Mechanics

2 Kinematic Description of Undeformed Shell Figure 1. Geometry of the shell Base Vectors of Midsurface and S-Surfaces e i - orthonormal vectors; A , k  - Lamé coefficients and principal curvatures of midsurface c  = 1+k   3 - shifter tensor at S-surfaces;  3 - transverse coordinates of S-surfaces (I = 1, 2, …, N) III (1)(2)  1,  2, …,  N - sampling surfaces (S-surfaces) r(  1,  2 ) - position vector of midsurface  R = r+  3 e 3 - position vectors of S-surfaces  I = 1, 2, …, N III

3 Kinematic Description of Deformed Shell Figure 2. Initial and current configurations of the shell Base Vectors of DeformedS-Surfaces Base Vectors of Deformed S-Surfaces  (  1,  2 ) - derivatives of 3D displacement vector at S-surfaces (I = 1, 2, …, N) I Position Vectors of Deformed S-Surfaces (3)(4) u (  1,  2 ) - displacement vectors of S-surfaces I = 1, 2, …, N I

4 Green-Lagrange Strain Tensor at S-Surfaces Linearized Strain-Displacement Relationships Representation for Displacement Vectors in Surface Frame (5)(6)(7)

5 Representation for Derivatives of Displacement Vectors Strain Parameters Strains of S-Surfaces Remark 1. Strains (10) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system (8)(9)(10)

6 Displacement Distribution in Thickness Direction Presentation for Derivatives of 3D Displacement Vector Strain Distribution in Thickness Direction Higher-Order Shell Theory L (  3 ) -Lagrange polynomials of degree N - 1(I = 1, 2, …, N) L (  3 ) - Lagrange polynomials of degree N - 1 (I = 1, 2, …, N) I (11)(12)(13)(14)

7 Stress Resultants Variational Equation Constitutive Equations Presentation for Stress Resultants Remark 2. It is possible to carry out exact integration in (19) using the n-point Gaussian quadrature rule with n = N+1 p i, p i - surface loads acting on bottom and top surfaces   (15)(16)(17)(18)(19)

8 Finite Element Formulation Displacement Interpolation Assumed Strain Interpolation Figure 3. Biunit square in (  1,  2 )-space mapped into the exact geometry four-node shell element in (x 1, x 2, x 3 )-space Figure 3. Biunit square in (  1,  2 )-space mapped into the exact geometry four-node shell element in (x 1, x 2, x 3 )-space N r (  1,  2 ) - bilinear shape functions   = (   - c  )/  - normalized coordinates (20)(21)

9 Variant U 3 (0) S 11 (–0.5) S 12 (–0.5) S 13 (0) S 33 (–0.5) N = – –1.066 N = – –1.013 N = – –1.000 Exact6.047– –1.000 Numerical Examples 1. Square Plate under Sinusoidal Loading Figure 4. Simply supported square plate with a = b =1, E = 10 7 and = 0.3 Table 1. Results for a thick square plate with a / h = 2

10 a / h N = 5 Exact Vlasov’s solution U 3 (0) S 11 (–0.5) S 12 (–0.5) S 13 (0) U 3 (0) S 11 (–0.5) S 12 (–0.5) S 13 (0) – – – – – – Table 2. Results for thick and thin square plates with five equally located S-surfaces Figure 5. Distribution of stresses S 13 and S 33 through the plate thickness: Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( ) Figure 5. Distribution of stresses S 13 and S 33 through the plate thickness: Vlasov’s solution ( ) and present higher-order shell theory for N = 7 ( )

11 2. Cylindrical Composite Shell under Sinusoidal Loading Figure 6. Simply supported cylindrical composite shell (modeled by 32  128 mesh) Table 3. Results for a thick cylindrical shell with R / h = 2 Variant U 3 (0) S 11 (0.5) S 22 (0.5) S 12 (–0.5) S 13 (0) S 23 (0) S 33 (0) N = – –1.674– N = – –2.123– N = – –1.981– N = – –2.063– Exact – –2.056–0.37

12 R / h N = 7 Exact Varadan-Bhaskar’s solution U 3 (0) S 22 (0.5) S 13 (0) S 23 (0) U 3 (0) S 22 (0.5) S 13 (0) S 23 (0) – – – – – –3.859 Table 4. Results for thick and thin cylindrical shells with seven S-surfaces Figure 7. Distribution of stresses S 33 through the shell thickness: exact solution ( ) and present higher-order shell theory for N = 7 ( ) Figure 7. Distribution of stresses S 33 through the shell thickness: exact solution ( ) and present higher-order shell theory for N = 7 ( )

13 Variant U 3 (0) S 11 (–0.5) S 11 (0.5) S 33 (–0.5) S 33 (0) N = –0.5882– N = –0.9770– N = –0.9978– Exact –1.000– Table 5. Results for a thick spherical shell with R / h = 2 3. Spherical Shell under Inner Pressure Figure 8. Spherical shell under inner pressure with R = 10,  = , E = 10 7 and = 0.3 (modeled by 64  1 mesh)

14 R / h N = 7 Exact Lamé’s solution U 3 (0) S 11 (–0.5) S 11 (0.5) S 33 (0) U 3 (0) S 11 (–0.5) S 11 (0.5) S 33 (0) – – – – – – Table 6. Results for thick and thin spherical shells with seven S-surfaces Figure 9. Distribution of stresses S 11 and S 33 through the shell thickness: Lamé’s solution ( ) and present higher-order shell theory for N = 7 ( ) Figure 9. Distribution of stresses S 11 and S 33 through the shell thickness: Lamé’s solution ( ) and present higher-order shell theory for N = 7 ( )

15 ConclusionsConclusions  A simple and efficient concept of S-surfaces inside the shell body has been proposed. This concept permits the use of 3D constitutive equations and leads for the sufficient number of S-surfaces to the numerically exact solutions of 3D elasticity problems for thick and thin shells  A new higher-order theory of shells has been developed which permits the use, in contrast with a classic shell theory, only displacement degrees of freedom  A robust exact geometry four-node solid-shell element has been built which allows the solution of 3D elasticity problems for thick and thin shells of arbitrary geometry

16 Thanks for your attention!