Developing an element free method for higher order shear deformation analysis of plates

A procedure is presented to eliminate transverse shear locking in analysis of laminated composite plates using Element Free Galerkin (EFG) method based on higher-order transverse shear deformation theory (HSDT). In the procedure, derivatives of the transverse displacement are introduced as independent variables. Thus, a formulation requiring C⁰ continuity shape functions for approximation is proposed for higher-order transverse shear deformation analysis of plates. Shear locking is avoided considering reduced integration for shear stiffness matrix; a method which is implemented in FEM. Moving Least Squares (MLS) method is utilized for shape functions and the penalty method is implemented to impose approximate boundary conditions. The present solutions are verified with other higher-order shear deformable studies. Moreover, a comparison between the present solutions with those obtained by EFG procedure based on First-order transverse Shear Deformation Theory (FSDT) is performed.     

Bending analysis of thin annular sector plates using extended Kantorovich method

Highly accurate approximate closed-form solution is presented for bending of thin sector plates with clamped edges subjected to uniform and non-uniform loading using the extended Kantorovich method (EKM). Successive application of EKM converts the governing equation which is a forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of r and θ. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration, exact closed-form solutions are obtained for both ODE systems. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plate shows very good agreement with results of other analytical and numerical analyses.     

Approximate large-deflection analysis of simply supported rectangular plates under transverse loading using plate post-buckling solutions

This paper describes an approximate analytical method to determine the large-deflection behaviour of rectangular simply supported thin plates under transverse loading. The method is based on a simple trial function to describe the shape of the initial and total deflections of the plate, corresponding to the first buckling mode shape of the plate when subjected to uniform in-plane compression. Therefore, the large-deflection behaviour of the plate under transverse loading can be expressed as a function of the ratio of the pre- to post-buckling in-plane stiffness of the plate, which is given in literature for plates with various membrane boundary conditions of the plate edges. Due to the simplicity of the trial function flexural and membrane actions of the plate are decoupled. It was found that the approximate method gives good insight into the large-deflection behaviour. The results of the approximate method have been compared to the results of finite element solutions and solutions available in literature. For uniformly distributed loads the approximate method gives deflections which differ less than about 10% from the finite element simulation values.     

Analysis of flexural natural vibrations of thin‐walled box beams using higher order beam theory

In order to study the influence of high‐order shear deformations and shear lag on the dynamic characteristics of thin‐walled box beams (TWBBs), this paper takes the Hamilton principle as a basis to consider multiple factors such as high‐order shear deformations, shear lag, and cross section rotary inertia of the TWBBs. The vibration differential equations and natural boundary conditions of TWBBs are deduced. On the basis of eight examples of TWBBs with different boundary conditions and span–width ratios, analytical results of this paper are compared with those of the ANSYS finite element method. Both results are in good agreement with each other, and the validity of the calculation method is verified. The effects of higher order shear deformations and shear lag on natural vibration characteristics of TWBBs are analyzed. And some meaningful conclusions are drawn: This theory shows the capability to accurately describe both higher order deformations and shear lag; when the span–width ratio is small, neglecting higher order web deformations will produce a large calculation error; under the action of shear lag, the natural vibration frequency of TWBBs decreases greatly, which cannot be neglected; and both the high‐order shear deformations and shear lag effect increase with increasing mode order and increase with decreasing span–width ratio.     

Stability analysis of simply-supported rectangular plates under non-uniform uniaxial compression using rigorous and approximate plane stress solutions

The non-classical problem of buckling of a simply-supported rectangular plate due to various types of non-uniform compressive edge loads is analysed here. For each case, the elasticity solution for the internal in-plane stress field is obtained rigorously using a superposition of Airy's stress functions and also approximately using extended Kantorovich method. Subsequently, the convergent buckling loads are obtained using Galerkin's method. Results are presented to highlight the dependence of the total buckling load and the corresponding buckled shape on the edge load distribution, as well as to illustrate the applicability of the approximate plane stress solutions.     

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C₀ finite element formulation has been done to overcome the problem of C₁ continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.