This paper presents an analytical solution to the static analysis of functionally graded plates (FGPs) by using a new trigonometric higher-order theory in which the stretching effect is included. The ...governing equations and boundary conditions of FGPs are derived by employing the principle of virtual work. Navier-type solution is obtained for FGPs subjected to transverse bi-sinusoidal load for simply supported boundary conditions. Benchmark results for the displacements and stresses of geometrically different plates are obtained. The results are compared with 3D exact solution and with other higher-order shear deformation theories, and the superiority of the present theory can be noticed.
► A new higher order shear deformation theory for composite sandwich plates and shells is developed. ► Parabolic distribution of the transverse shear strains through the thickness is accounted. ► ...Therefore no shear correction factors are needed. ► The governing equations and boundary conditions are derived by employing the principle of virtual work. ► Exact solutions for static and dynamic results of cylindrical and spherical shells and plates are presented.
A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “
m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.
A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “
m”, whose value is determined ...so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.
A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the ...plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.
The closed-form solution of a generalized hybrid type quasi-3D higher order shear deformation theory (HSDT) for the bending analysis of functionally graded shells is presented. From the generalized ...quasi-3D HSDT (which involves the shear strain functions “f(ζ)” and “g(ζ)” and therefore their parameters to be selected “m” and “n”, respectively), infinite six unknowns' hybrid shear deformation theories with thickness stretching effect included, can be derived and solved in a closed-from. The generalized governing equations are also “m” and “n” parameter dependent. Navier-type closed-form solution is obtained for functionally graded shells subjected to transverse load for simply supported boundary conditions. Numerical results of new optimized hybrid type quasi-3D HSDTs are compared with the first order shear deformation theory (FSDT), and other quasi-3D HSDTs. The key conclusions that emerge from the present numerical results suggest that: (a) all non-polynomial HSDTs should be optimized in order to improve the accuracy of those theories; (b) the optimization procedure in all the cases is, in general, beneficial in terms of accuracy of the non-polynomial hybrid type quasi-3D HSDT; (c) it is possible to gain accuracy by keeping the unknowns constant; (d) there is not unique quasi-3D HSDT which performs well in any particular example problems, i.e. there exists a problem dependency matter.
This paper presents an analytical solution for the buckling and free vibration analysis of laminated beams by using a refined and generalized shear deformation theory which includes the thickness ...expansion. The eigenvalue equation is derived by employing the Rayleigh quotient, and the Ritz method is used to approximate the displacement field. The functions used in the Ritz method are chosen as either a pure polynomial series or a hybrid polynomial-trigonometric series. The hybrid series is used due to the superior convergence and accuracy compared to conventional pure polynomial series for certain boundary conditions. The boundary conditions are taken into account using the penalty method. Convergence of the results is analyzed, and numerical results of the present theory are compared with other theories for validation. Nondimensional natural frequencies and critical buckling loads are obtained for a variety of stacking sequences. The effect of the normal deformation on the fundamental frequencies and critical buckling loads is also analyzed.
•3D exact solution for the static analysis of magneto-electro-elastic of shells is presented.•The shell unknowns are modelled analytically by the Navier technique.•The continuity conditions is ...warranted.•The correct load traction condition is considered at the top and bottom of the shell.•Benchmark problem for shell is proposed.
This paper presents an exact solution for the static analysis of magneto-electro-elastic simply supported shallow shells panels. The mechanical equations are derived via equilibrium elasticity relations. The electrical and magnetic governing equations are obtained by electrostatic and magnetostatic equilibrium relations. The shell displacements, electrical and magnetic potential functions are solved analytically by the Navier closed form solutions. The governing equations formulated in terms of thickness coordinate are solved semi-analytically by using the differential quadrature method. The Lagrange polynomials are employed as basis functions. The equations are discretized per each layer by the Chebyshev-Gauss-Lobatto grid distribution. The continuity conditions in the adjacent layers for mechanical displacement, transverse dielectric displacement, electric and magnetic scalar function and transverse magnetic induction are complied. The correct load traction condition is considered at the top and bottom of the shell. Numerical results for spherical, cylindrical and rectangular panels are reported. The results are in excellent agreement with other 3D elasticity solutions reported in the literature so a new benchmark problem for shell is proposed.
This paper presents an analytical solution of the linear buckling, free vibration and bending behavior of simple supported functionally graded sandwich plates subjected to transverse and axial ...mechanical loads. The used optimization strategy allows to express the transverse and in-plane displacement fields as a function of the n and m parameter, respectively, so the used Carrera's unified formulation (CUF) is also n and m parameters dependent. Principle of virtual displacement (PVD) is utilized to obtain the highly coupled differential equations. The solution is obtained via Navier-Type solution. Good agreements with quasi-3D solutions are found. The optimized parameters are used for solving the buckling problem of functionally graded sandwich plates with different side-to-thickness ratios. Numerical results for buckling are compared to different advanced theories since there isn’t 3D solution available in the literature. Overall, the presented results have a high accuracy to estimate the critical loads, modes and natural frequencies.
•An optimized hyperbolic unified higher order shear deformation theory is proposed.•Buckling, free vibration and bending behavior of plates is studied.•This unified theory is capable to model the thickness stretching effect.•Good accuracy is obtain compared with different solutions in the literature.•Bending and buckling results are remarkable when compared with well-known theories.•The case dependent problem of shear deformation theory is outlined.
•Three-dimensional solution of functionally graded shells is analyzed.•Differential Quadrature Method (DQM) is used.•Chebyshev-Gauss-Lobatto grid and Lagrange interpolation polynomials are ...used.•Shells are subjected to bi-sinusoidal and uniform distributed load.•Benchmark results are provided.
A numerical solution for the three-dimensional static analysis of functionally graded shells with constant curvature is presented. The solution is based on three-elasticity equations written in orthogonal curvilinear coordinates which are valid for spherical, cylindrical shell panels and rectangular plates. The equations in term of the mid-surface variables are solved using a summation of harmonics in term of Navier method which is valid only for simply supported structures. The equations in term of the thickness direction are solved numerically by the Differential Quadrature method (DQM) which permitted to easily calculate the approximate derivative of a function using a weighting sum of the functions evaluated in a certain grid. The layers of the structure are discretized separately by the Chebyshev-Gauss-Lobatto grid and Lagrange interpolation polynomials are considered as the basis functions. The inter-laminar continuity of transverse shear is imposed as part of the boundary conditions for the presented method. The boundary conditions of out-of-plane stresses at the top and the bottom due to the applied loads on the shell are also considered for the analysis, as a result this method can predict the correct behavior of through-the-thickness distribution of transverse stresses. This method permitted easily to discretize the material in term of the thickness direction and several types of single functionally graded layer and sandwich structures with functionally graded core are analyzed. Several shells subjected to bisinusoidal and uniform distributed load are analyzed. The results are compared with other three dimensional solutions proposed in the literature and accurate two dimensional models.
This paper presents a simplified first order shear deformation theory (FSDT) for laminated composite and sandwich plates. Unlike the existing FSDT, the present one has a novel displacement field ...which include undetermined integral terms and contains only four unknowns. Equations of motion and boundary conditions are derived from the Hamilton’s principle. Navier-type analytical solution is obtained in closed form and by solving the eigenvalue equation. The comparison of the present results with the available elasticity solutions and the results computed independently using the FSDTs available in the literature shows that this theory predicts the fundamental frequencies with good accurately. It can be concluded that the proposed theory is accurate and simple in solving the dynamic behavior of single and sandwich laminated composite plates.
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