A New Quasi-3D Model for Functionally Graded Plates

This article investigates the static behavior of functionally graded plate under mechanical loads by using a new quasi 3D model. The theory is designated as fifth-order shear and normal deformation theory (FOSNDT). Properties of functionally graded material are graded across the transverse direction by using the rule of mixture i.e. power-law. The effect of thickness stretching is considered to develop the present theory. In this theory, axial and transverse displacement components respectively involve fifth-order and fourth-order shape functions to evaluate shear and normal strains. The theory involves nine unknowns. Zero transverse shear stress conditions are satisfied by employing constitutive relations. Analytical solutions are obtained by implementing the double Fourier series technique. The results predicted by the FOSNDT are compared with existing results. It is pointed out that the present theory is helpful for accurate structural analysis of isotropic and functionally graded plates compared to other plate models

A refined shear deformation theory for bending analysis of isotropic and orthotropic plates under various loading conditions

In this paper, a refined trigonometric shear deformation theory is applied for the bending analysis of isotropic and orthotropic plates under the various loading conditions. The two unknown variables are involved in the present theory. The present theory satisfies the shear stress free condition at top and bottom surface of the plates without using shear correction factors. The governing equations and boundary conditions are obtained by using the principle of virtual work. A closed form solution is obtained using Navier Solution Scheme. A simply supported isotropic and orthotropic plate subjected to sinusoidally distributed, uniformly distributed and linearly varying loads are considered for the detailed numerical study. The results obtained using present theory are compared with previously published results.

Cylindrical bending of orthotropic plate strip based on nth-order plate theory

In this paper, cylindrical bending of orthotropic plates is presented using nth-order plate theory. Classical plate theory and parabolic shear deformation theory of Reddy can be considered as special cases of present theory. The theory accounts for realistic variation of the transverse shear stress through the thickness of plate and satisfy the traction free conditions at top and bottom surfaces of the plate. The number of unknown variables in the present theory is same as that of first order shear deformation theory. The theory is variationally consistent. The use of shear correction factors which are problem dependent and are normally associated with first order shear deformation theory is avoided in the present theory. The governing equations and associated boundary conditions are derived by the principle of virtual work. Navier solution technique is employed for the simply supported plates. The program has been developed in FORTRAN. The displacement and stresses of a simply supported plate infinitely long in y-direction under sinusoidally distributed load are calculated to demonstrate the accuracy and efficiency of the present theory

Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theory

A trigonometric plate theory is assessed for the static bending analysis of plates resting on Winkler elastic foundation. The theory considers the effects of transverse shear and normal strains. The theory accounts for realistic variation of the transverse shear stress through the thickness and satisfies the traction free conditions at the top and bottom surfaces of the plate without using shear correction factors. The governing equations of equilibrium and the associated boundary conditions of the theory are obtained using the principle of virtual work. A closed-form solution is obtained using double trigonometric series. The numerical results are obtained for flexure of simply supported plates subjected to various static loadings. The displacements and stresses are obtained for three different values of foundation modulus. The numerical results are also generated using higher order shear deformation theory of Reddy, first order shear deformation theory of Mindlin, and classical plate theory for the comparison of the present results

Cylindrical bending of orthotropic plate strip based on nth-order plate theory

In this paper, cylindrical bending of orthotropic plates is presented using nth-order plate theory. Classical plate theory and parabolic shear deformation theory of Reddy can be considered as special cases of present theory. The theory accounts for realistic variation of the transverse shear stress through the thickness of plate and satisfy the traction free conditions at top and bottom surfaces of the plate. The number of unknown variables in the present theory is same as that of first order shear deformation theory. The theory is variationally consistent. The use of shear correction factors which are problem dependent and are normally associated with first order shear deformation theory is avoided in the present theory. The governing equations and associated boundary conditions are derived by the principle of virtual work. Navier solution technique is employed for the simply supported plates. The program has been developed in FORTRAN. The displacement and stresses of a simply supported plate infinitely long in y-direction under sinusoidally distributed load are calculated to demonstrate the accuracy and efficiency of the present theory

Buckling analysis of thick plates using refined trigonometric shear deformation theory

In this paper, a refined trigonometric shear deformation plate theory is applied for the buckling analysis of thick isotropic square and rectangular plates. The theory involves only two unknowns, as against three in first order shear deformation theory and other higher order theories. The theory involves sinusoidal function in the in-plane displacement. The transverse displacement involves bending and shear components. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. A simply supported isotropic rectangular plate subjected to uniaxial and biaxial compression is considered for the detailed numerical study. Results of critical buckling load for simply supported isotropic rectangular plates are compared with those of other refined theories

Comparison of various refined beam theories for the bending and free vibration analysis of thick beams

In this paper, unified shear deformation theory is used to analyze simply supported thick isotropic beams for the transverse displacement, axial bending stress, transverse shear stress and natural frequencies. This theory enables the selection of different in-plane displacement components to represent shear deformation effect. The numbers of unknowns are same as that of first order shear deformation theory. The governing differential equations and boundary conditions are obtained by using the principle of virtual work. The results of displacement, stresses, natural bending and thickness shear mode frequencies for simply supported thick isotropic beams are presented and discussed critically with those of exact solution and other higher order theories. The study shows that, while the transverse displacement and the axial stress are best predicted by the models 1 through 5 whereas models 1 and 2 are overpredicts the transverse shear stress. The model 4 predicts the exact dynamic shear correction factor (π2/12 = 0.822) whereas model 1 overpredicts the same

Single variable refined beam theories for the bending, buckling and free vibration of homogenous beams

In this paper, single variable beam theories taking into account effect of transverse shear deformation are developed and applied for the bending, buckling and free vibration analysis of thick isotropic beams. The most important feature of the present beam theories is that unlike any other higher order theory, the proposed class of theories contains only one unknown variable and does not require shear correction factor. The displacement field of the present theories is built upon the classical beam theory. The theories account for parabolic distribution of transverse shear stress using constitutive relations, satisfying the traction free conditions at top and bottom surfaces of the beam. Governing differential equation and boundary conditions of these theories are obtained using the principle of virtual work. Results obtained for the displacements, stresses, fundamental frequencies and critical buckling loads of simply supported isotropic solid beams are compared with those obtained by other theories to validate the accuracy of the present theories

Analytical solutions for the hygro-thermo-mechanical bending of FG beams using a new fifth order shear and normal deformation theory

Anew analytical solution is presented for functionally graded (FG) beams to investigate the bending behaviour under the hygro-thermo-mechanical loading using a new fifth order shear and normal deformation theory (FOSNDT). The material properties of the FG beam are varied along the thickness direction according to the power law index. In the present theory, a polynomial shape function is expanded up to fifth-order in terms of thickness coordinate to consider the effects of transverse shear and normal deformations. The present theory is free from the shear correction factor. Using the Navier’s solution technique the closed-form solution is obtained for simply supported FG beams. All the results are presented in non-dimensional form and validated it by developing the classical beam theory (CBT), first order shear deformation theory (FSDT by Mindlin) and third order shear deformation theory (TSDT by Reddy) considering the hygro-thermo-mechanical loading effects which is mostly missing in the literature. It is noticed that the presented FOSNDT is very simple and accurate to predict the bending behaviour of FG beams under linear and non-linear hygro-thermo-mechanical loadings