Higher-order Transverse Shear Deformation Research Articles

Porosity-dependent buckling analysis of elastically supported FGM sandwich plate via new tangent HSDT: A meshfree approach

In this paper, new tangent shape function-based higher-order transverse shear deformation theory (NTHSDT) is proposed to compute the buckling behavior of the elastically supported functionally graded material (FGM) sandwich plates under porous medium. The proposed theory is found to be variationally consistent and fulfills the zero traction boundary conditions on the bottom and top layer without a shear correction factor. The material properties are presumed to be graded in the thickness direction as characterized by a modified power law distribution in terms of volume fraction of constituents. The governing equations are derived using Hamilton’s Principle. A strong form of solution discretizes the governing equations by employing a thin plate spline radial basis function-based collocation (TSRBFC) method. The proposed theory is efficient, reliable, and is in close agreement with the results in the literature. Comparison studies show that the NTHSDT is more accurate than other plate theories and is simple in analyzing buckling behavior. A parametric study is done to examine the effects of grading index, porosity index, sandwich schemes, aspect ratio, side-to-length thickness ratio and foundation stiffness.

New HSDT for free vibration analysis of elastically supported porous bidirectional functionally graded sandwich plate using collocation method

The present paper deals with the free vibration response of elastically supported porous bidirectional functionally graded soft-core sandwich rectangular plate with proposed new tangent shape function-based higher-order transverse shear deformation theory. The theory is accomplished to uphold the continuity of transverse shear stresses and zero shear stress conditions on both the extreme surfaces of a plate. The bidirectional gradation varies through the thickness ( z-axis) and length ( x-axis) directions. The effective material properties are calculated via the modified power-law. The governing differential equations (GDEs) of motion are acquired via the Hamilton principle. The inverse multi-quadric radial basis function-based Collocation method is implemented to discretize the GDEs obtained by strong form formulation. The current theory and method’s accuracy and efficacy are validated by comparing the present results with existing results in the literature. Studies show that the results of the proposed theory are in close agreement with 3D approach. Numerical results are obtained with a different scheme of sandwich layers for free vibration analysis. Effects of grading index, porosity fraction, two parameters foundation, different scheme, and the span to thickness ratio on free vibration response have been discussed. New results for free vibration response in porous media are presented with different skin materials.

New transverse shear deformation theory for bending analysis of FGM plate under patch load

Two new higher order transverse shear deformation theories (NHSDTs) with five variables have been proposed for the analysis of Functionally Graded Material (FGM) plate. A Governing differential equation (GDE) of the FGM plate is developed using energy principle. Wendland radial basic function (RBF) based Meshfree method is implemented for discretizing the GDE. A MATLAB code is developed to obtain the desired results. The normalized deflection and stresses are obtained and compared with other published results. The effect of different types of load, span to thickness ratio, grading index on the response is studied. Some new results for patch loading are also obtained.

Assessment of higher order transverse shear deformation theories for modeling and buckling analysis of FGM plates using RBF based meshless approach

Purpose The purpose of this paper is to assess different five variables shear deformation plate theories for the buckling analysis of FGM plates. Design/methodology/approach Governing differential equations (GDEs) of the theories are derived by employing the Hamilton Principle. A polynomial radial basis function (RBF)-based Meshless method is used to discretize the GDEs, and a MATLAB code is developed to solve these discretize equations. Findings Numerical results are obtained for buckling loads. The results are compared with other available results for validation purpose. The effect of the span-to-thickness ratio and grading index is observed. It is observed that some theories underpredict the deflection for thick plates, while at the same time they seem to be in good agreement with other theories for thin plates. Originality/value This paper assesses the different theories with the same method to determine their applicability.

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.

Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions

This paper presents an analytical investigation on the nonlinear response of thick functionally graded doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, explicit relations of load-deflection curves for simply supported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the panels are analyzed and discussed. The novelty of this study results from accounting for higher order transverse shear deformation and panel-foundation interaction in analyzing nonlinear stability of thick functionally graded cylindrical and spherical panels.

Bending of laminated plates with mixed boundary conditions based on higher-order shear deformation theory

The bending of laminated plates is considered using higher-order transverse shear deformation theory. The principle of virtual work is used to derive a new set of seven governing equations and corresponding boundary conditions. These equations, combined with eighteen relationships between the resultant stress and displacement components, compose a system of first-order partial differential equations that is solved by the generalized differential quadrature method. Numerical results for laminated plates with a variety of mixed boundary conditions are calculated using the proposed method, and good agreement is found with the corresponding solutions obtained using ANSYS. Fiber-reinforced laminated composite materials are widely used in a variety of engineering fields, such as aerospace, civil, marine, mechanical, nuclear, and petrochemical engineering. Such materials are popular for industrial applications due to high strength-to-weight ratios, long fatigue life, good stealth characteristics, and enhanced corrosion resistance. A number of theories describing laminated composite plates exist in the literature. Classical plate theory is based on the Kirchhoff kinematic hypothesis that straight lines normal to the undeformed midsurface remain straight and normal to the middle surface after deformation and undergo no thickness stretching. Neglecting transverse shear effects, this theory produces unacceptable approximations in the analysis of even thin laminated plates and shells. Surveys of various classic shell theories can be found in [Naghdi 1956; Bert and Francis 1974; Bert and Chen 1978]. The development of plate theories with transverse shear effects has improved the accuracy of results considerably. The refined theories are of different orders, based on discretization of the transverse shear effects and the number of terms included in the assumed displacement field. Reissner [1945] was the first to develop a plate theory that included transverse shear deformation for static analysis. Mindlin [1951] then expanded Reissner’s theory for dynamic analysis. Both approaches rely on first-order shear deformation theory (FSDT). Reissner’s theory is stress-based, whereas Mindlin’s is displacement-based. These theories do not satisfy the condition of zero transverse shear stress at the top and bottom surfaces of the plate, and consider a uniform transverse shear stress distribution across the thickness of the plate. Therefore they require the use of a shear correction factor to increase the precision of the results. Later a set of theories, generally known as higher-order shear deformation theories (HSDT), has been developed by a number of researchers. Basset [1890] appears to have been the first researcher to suggest that displacement fields can be expanded in a power series of the thickness coordinate. The higher-order

Sandwich plates-free vibrations and damping analysis

A new analytical approach to the optimization analysis of stacking sequences of sandwich plate faces is presented. FRP faces are assumed to be constructed of 02°, 902°, and ±45° plies. A new set of design variables was succesfully used for the optimization analysis of natural flexural vibrations and modal loss factor of sandwich plates with FRP faces. In addition, the influence of the core thickness on optimization solutions was studied. Three variants of plate theories were considered, namely those of Love-Kirchhoff and first- and higher-order transverse shear deformation. To verify the results of the models proposed, the values of the natural vibrations were compared with the results obtained using the NISA II FEM package.

A refined theory of laminated shells with higher-order transverse shear deformation

Based on refined analysis, a theory of laminated composite shells with higher-order transverse shear deformation is presented. The continuity conditions of displacements and transverse shear stresses at layer interfaces and the conditions of zero transverse shear stresses on the surfaces of the shells are introduced to improve and simplify the displacement field. The number of the displacement unknowns and the order of the equilibrium equations are the same as in the first-order shear deformation theory, but the present theory can predict continuous parabolic transverse shear stresses. The closed-form solutions of simply-supported cross-ply shells are obtained and compared with the elasticity solutions and other theories' solutions. For both shallow and deep shells, the present solutions of displacements and in-plane stresses are very close to the elasticity solutions.

THERMOMECHANICAL POSTBUCKLING OF GEOMETRICALLY IMPERFECT FLAT AND CURVED PANELS TAKING INTO ACCOUNT TANGENTIAL EDGE CONSTRAINTS

A study of the effects of tangential edge constraints on the postbuckling response of flat and shallow curved panels subjected to thermal and mechanical loads is presented. The mechanical loads investigated are uniform compressive edge loads and transverse lateral pressure. The temperature fields considered are associated with spatially uniform heating over the thickness panel and a linear through-the-thickness temperature gradient. The structural model is based on a higher order transverse-shear deformation theory of shallow shells that incorporates the effects of geometric nonlinearities, initial geometric imperfections, and the tangential edge motion constraints. Simply supported panels are considered in which the tangential motion of the unloaded edges is either unrestrained, partially restrained, or fully restrained. Results are presented for three-layer panels made from transversely isotropic materials, and a number of conclusions about the implications of the previously mentioned effects are outlined.

Thermomechanical buckling of laminated composite plates with higher-order transverse shear deformation

A higher-order displacement field is developed for the analysis of the thermomechanical buckling of composite plates subjected to thermal or mechanical load. The displacement field contains the same number of dependent variables as in the first-order shear deformation theory, but accounts for continuity of in-plane displacements and transverse shear stresses across each layer interface. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with other theory solutions. A coefficient θ including effects of transverse shear deformation on critical temperature and critical load is introduced.

Supersonic flutter of shear deformable laminated composite flat panels

Flutter-type instability of cross-ply shear deformable composite flat panels exposed to a supersonic flow is analyzed. The two-dimensional static aerodynamic approximation and a higher order transverse shear deformation theory for the structural plate model are used. Comparisons of the results obtained in this framework with those derived within their first order transverse shear deformation and classical (Kirchhoff) counterparts are presented and a number of conclusions concerning their range of applicability are outlined. The results underline the strong effect played by transverse shear deformation and transverse normal stress, as well as by the higher order effects on the flutter instability of panels made of advanced composite materials.

The effects of transverse shearing and anisotropy on vibration frequencies of laminated cylinders

Natural vibration frequencies of orthotropic and anisotropic simply supported circular cylinders are predicted by using a higher order transverse shear deformation theory. A comparison of natural vibration frequencies of laminated cylinders predicted by first order transverse shear deformation theory and the higher order theory indicates that an additional allowance for transverse shear deformation has a negligible effect on the predicted natural vibration frequencies associated with long wavelengths, but significantly reduces the natural vibration frequencies associated with short wavelengths. A parametric study of the effects of ply orientation on the natural vibration frequencies of laminated cylinders indicates that while stacking sequence affects natural vibration frequencies, cylinder geometry and mode shape are more important in accurately predicting transverse shear deformation effects. Transverse shearing effects are less important in predicting natural vibration frequencies associated with long wavelengths than in predicting axial compressive buckling loads. The effects of anisotropy are more important than the effects of transverse shear deformation for most strongly anisotropic laminated cylinders in predicting natural vibration frequencies. However, transverse shear deformation effects are important in predicting natural vibration frequencies associated with short wavelengths of thick-walled laminated cylinders. Neglecting either anisotropic effects or transverse-shear deformation effects leads to non-conservative errors in predicted natural vibration frequencies.

Higher-order shear-deformable two dimensional hierarchical beam elements for laminated composites

This paper presents a new two dimensional curved beam element formulation with higher-order transverse shear deformation for linear static analysis of laminated composites. The derived displacement approximation in the transverse direction can be of an arbitrary desired polynomial order p thereby permitting strains of at least order (p - 1). The element formulation provides displacement continuity across the interelement boundaries. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is presented for generally orthotropic and laminated composite materials. Numerical examples are presented to demonstrate the accuracy, simplicity of modelling, effectiveness, faster rate of convergence and overall superiority of the present formulation.

Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites

A new two-dimensional curved beam element formulation with higher-order transverse shear deformation for linear static analysis of laminated composites is presented. The displacement approximation in the transverse direction can be of arbitrary desired polynomial order p, thereby permitting strains of at least order ( p − 1). This is accomplished by introducing additional nodal degrees of freedom in the element displacement approximation that correspond to Lagrange interpolating polynomials in the transverse direction. The element has an important hierarchical property; the element stiffness matrix, generalized nodal displacement vector and the equivalent nodal load vector corresponding to a polynomial order p are a subset of those corresponding to the approximation order p + 1. The element formulation provides displacement continuity across the interelement boundaries, i.e. C 0 continuity is ensured automatically at the mating element boundaries. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is presented for generally orthotropic material behavior, where the material directions may not be parallel to the global axes, as well as for the laminated composites. The laminated composite material behavior is incorporated in the element formulation through numerical integration for each lamina. For each lamina, the material behavior may be generally orthotropic, and the material direction and lamina thicknesses may vary from point to point within the lamina. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation for laminated composite material behavior. Results obtained from the present formulation are also compared with the available analytical solutions.

Macromechanical behaviour of composite laminates

An improved higher order transverse shear deformation theory is employed to arrive at modified constitutive relations which can be used in the flexural, buckling and vibration analysis of laminated plates and shells. The strain energy for such systems is then expressed in terms of the displacements and the rotations for ready reference and use. Numerical values of vibration frequencies are obtained using this formulation employing Ritz's method of analysis. The results are compared with those available in the literature to validate the analysis presented.