Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Composite Structures in Cylindrical Bending

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Santosh Kapuria ◽  
Poonam Kumari
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Thickness Direction ◽  
Kantorovich Method ◽  
Cylindrical Bending ◽  
Extended Kantorovich Method ◽  
Elasticity Problems ◽  
Plane Direction ◽  
3D Elasticity ◽  
2D Elasticity

The extended Kantorovich method originally proposed by Kerr in the year 1968 for two-dimensional (2D) elasticity problems is further extended to the three-dimensional (3D) elasticity problem of a transversely loaded laminated angle-ply flat panel in cylindrical bending. The significant extensions made to the method in this study are (1) the application to the 3D elasticity problem involving an in-plane direction and a thickness direction instead of both in-plane directions in 2D elasticity problems, (2) the treatment of the nonhomogeneous boundary conditions encountered in the thickness direction, and (3) the use of a mixed variational principle to obtain the governing differential equations in both directions in terms of displacements as well as stresses. This approach not only ensures exact satisfaction of all boundary conditions and continuity conditions at the layer interfaces, but also guarantees the same order of accuracy for all displacement and stress components. The method eventually leads to a set of eight algebraic-ordinary differential equations in the in-plane direction and a similar set of equations in the thickness direction for each layer of the laminate. Exact closed form solutions are obtained for each system of equations. It is demonstrated that the iterative procedure converges very fast irrespective of whether or not the initial guess functions satisfy the boundary conditions. Comparisons of the present predictions with the available 3D exact solutions and 3D finite element solutions for laminated cross-ply and angle-ply composite panels under different boundary conditions show a close agreement between them.

Multiterm Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
Santosh Kapuria ◽  
Poonam Kumari
Keyword(s):  
Stress Field ◽  
Composite Laminates ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Elasticity Solution ◽  
Kantorovich Method ◽  
Extended Kantorovich Method ◽  
Single Term ◽  
3D Elasticity ◽  
Laminated Structures

In an article recently published in this journal, the powerful single-term extended Kantorovich method (EKM) originally proposed by Kerr in 1968 for two-dimensional (2D) elasticity problems was further extended by the authors to the three-dimensional (3D) elasticity solution for laminated plates. The single-term solution, however, failed to predict accurately the stress field near the boundaries; thus limiting its applicability. In this work, the method is generalized to the multiterm solution. The solution is developed using the Reissner-type mixed variational principle that ensures the same order of accuracy for displacements and stresses. An n-term solution generates a set of 8n algebraic-ordinary differential equations in the in-plane direction and a similar set in the thickness direction for each lamina, which are solved in close form. The problem of large eigenvalues associated with higher order terms is addressed. In addition to the composite laminates considered in the previous article, results are also presented for sandwich laminates, for which the inaccuracy in the single-term solution is even more prominent. It is shown that considering just one or two additional terms in the solution (n = 2 or 3) leads to a very accurate prediction and drastic improvement over the single-term solution (n = 1) for all entities including the stress field near the boundaries. This work will facilitate development of near-exact solutions of many important unresolved problems involving 3D elasticity, such as the free edge stresses in laminated structures under bending, tension and torsion.

Bending deflection and stress analyses in a thin functionally graded material skew plate with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined in-plane and uniform lateral loads using the extended Kantorovich method

2021 ◽  
pp. 095440622199068
Author(s):  
Ahmad Mamandi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Functionally Graded ◽  
Kantorovich Method ◽  
Different Boundary Conditions ◽  
Extended Kantorovich Method ◽  
Skew Plate ◽  
Pasternak Elastic Foundation ◽  
Graded Material

In this study, bending deflection and stress analyses have been conducted for a thin skew plate made of functionally graded material (FGM) with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined loads including uniform transverse load, normal and shear in-plane forces, and planar body forces. The Cartesian partial differential equation governing the bending deflection of the skew plate has been converted into a partial differential equation in oblique coordinates using the conversion relations. Then, by employing the variational principle and residual weighted Galerkin method and using the Extended Kantorovich Method (EKM), the equation has been converted to a set of linear differential equations in terms of two functions in the longitudinal and transverse directions of the oblique plate, and afterward, the equation has been solved using the iterative solution method. Different boundary conditions in a combined form of simply and clamped supports have been investigated and their effects on bending deflection and generated in-plane normal and shear stresses are discussed.

Experimental Investigation of the Effect of Strain Rate on the Compression Behavior of 3D E-Glass Fiber-Reinforced Composites

2012 ◽  
Vol 174-177 ◽  
pp. 1528-1532 ◽  
Author(s):  
Ling Yan Shen ◽  
Yong Chi Li ◽  
Zhi Hai Wang
Keyword(s):  
Strain Rate ◽  
Glass Fiber ◽  
Three Dimensional ◽  
Dynamic Mechanical Properties ◽  
Thickness Direction ◽  
Fiber Reinforced ◽  
Compression Behavior ◽  
Reinforced Composite ◽  
Glass Fiber Reinforced ◽  
Plane Direction

The MTS810 and SHPB are used to experimentally study the quasi-static and dynamic mechanical properties of three-dimensional E-glass fiber-reinforced composite. Stress-strain curves along the plane and thickness direction are obtained under varying strain rates, ranging from 10-3 to 103s-1. Experimental results show that strain rate has a significant effect on the material response. It is found that the compressive strength and the modulus increase with increasing strain rate, the failure strain, however, decreases slightly. But, the effect of strain rate in-plane and through thickness directions is different. A higher strain rate sensitive modulus and failure is found in the thickness direction, while a higher strain rate sensitive failure strength is found in the in-plane direction.

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