Exact Three-dimensional Elasticity Solutions Research Articles

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature. Thus, natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies. This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes. The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number. The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures. To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the three-dimensional finite element method and the chosen higher-order models. By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models. Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

Development of the consistent second-order plate theory for transversely isotropic plates and its analytical assessment from the three-dimensional perspective

In this paper, the consistent second-order plate theory is developed for transversely isotropic plates. It is validated against the three-dimensional elasticity theory using a well-known benchmark problem of a simply-supported rectangular plate subjected to symmetric transverse sinusoidal loading. The choice of the benchmark problem is based on the fact that it allows for an exact three-dimensional elasticity solution to be derived in closed form. In this study, a closed-form solution based on Elliot’s displacement potentials for transversely isotropic solids is specifically derived for validation purposes. Its equivalence to other closed-form analytical solutions is established. Expanding the closed-form analytical solution into a power-law series with respect to the non-dimensionalised plate thickness enables a direct term-by-term comparison with the consistent second-order plate theory solution and provides a valuable mechanism to validate the consistent plate theory for transversely isotropic plates in a purely analytical form. The term-by term comparison reveals that the first terms of the above power-law series coincide exactly with the expressions of the consistent second-order plate theory. In addition to the analytical validation, a numerical validation using the finite element method is performed. A comparative analysis of several plate theories for transversely isotropic plates demonstrates that the consistent plate theory can predict displacements and stresses in thick transversely isotropic plates with very high degree of accuracy, such that even for very thick plates with a thickness-to-length ratio of 0.5, the deviation from the three-dimensional elasticity solution is less than 1%.

Assessment of the consistent second-order plate theory for isotropic plates from the perspective of the three-dimensional theory of elasticity

In this paper, the consistent second-order plate theory for isotropic plates is validated against the three-dimensional elasticity theory using a well-known benchmark problem of a simply-supported rectangular plate subjected to symmetric transverse sinusoidal loading. The choice of the benchmark problem is based on the fact that it allows for an exact three-dimensional elasticity solution to be derived in closed-form. In the paper, two equivalent closed-form solutions are employed for validation purposes, one of which is specifically derived for this study. Once the equivalence of the two closed-form analytical solutions is established, they are expanded into a power-law series with respect to the non-dimensionalised plate thickness. This enables a direct term-by-term comparison with the consistent second-order plate theory solution and provides a valuable mechanism to validate the consistent plate theory in purely analytical form. The term-by term comparison reveals that the first terms of the above power-law series coincide exactly with the expressions of the consistent second-order plate theory. In addition to the analytical validation, a parametric study is carried out with a view to establish the range of applicability of the consistent second-order plate theory in terms of the thickness-to-length ratio. It is demonstrated that the consistent plate theory can predict displacements and stresses in thick plates with very high degree of accuracy, such that even for very thick plates with thickness-to-length ratio of 1/2, the deviation from the three-dimensional elasticity solution is less than 1%.

Exact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section

Exact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section

A higher-order shear and normal deformation functionally graded plate model: some recent results

A higher-order shear and normal deformations plate theory is employed for stress analysis and free vibration of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. This idealization is required in order to obtain the closed-form solutions of some fundamental solid mechanics problems and also simplify the evaluation and development of numerical models of the structures made of FGMs. The material properties of FG plates such as Young’s moduli and material density are considered in this case to vary continuously in the thickness direction according to the volume fraction of constituents and mathematically modeled as exponential and power law functions. Poisson’s ratio is assumed to be constant. The effect of variation of material properties in terms of material grading index on the deformations, stresses, and natural frequency of FG plates is studied. The accuracy of the presented numerical solutions has been established with the solutions available of other models and the exact three-dimensional (3D) elasticity solutions.

An accurate higher order displacement model with shear and normal deformations effects for functionally graded plates

Functionally graded materials (FGMs) are the potential candidates under consideration for designing the first wall of fusion reactors with a view to make best use of potential properties of available materials under severe thermo-mechanical loading conditions. A higher order shear and normal deformations plate theory is employed for stress and free vibration analyses of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although FGMs are highly heterogeneous in nature, they are generally idealized as continua with mechanical properties changing smoothly with respect to spatial coordinates. The material properties of FG plates are assumed here to vary through thickness of plate in a continuous manner. Young's modulii and material densities are considered to be varying continuously in thickness direction according to volume fraction of constituents which are mathematically modeled here as exponential and power law functions. The effects of variation of material properties in terms of material gradation index on deformations, stresses and natural frequency of FG plates are investigated. The accuracy of present numerical solutions has been established with respect to exact three-dimensional (3D) elasticity solutions and the other models’ solutions available in literature.

FREE VIBRATION OF FUNCTIONALLY GRADED PLATES WITH A HIGHER-ORDER SHEAR AND NORMAL DEFORMATION THEORY

Free vibration analysis of functionally graded elastic, rectangular, and simply supported (diaphragm) plates is presented based on a higher-order shear and normal deformation theory (HOSNT). Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with mechanical properties changing smoothly with respect to the spatial coordinates. The material properties of functionally graded (FG) plates are assumed here to be varying through the thickness of the plate in a continuous manner. The Poisson ratios of the FG plates are assumed to be constant, but their Young's modulii and densities vary continuously in the thickness direction according to the volume fraction of constituents which is mathematically modeled as a power law function. The equations of motion are derived using Hamilton's principle for the FG plates on the basis of a HOSNT assuming varying material properties. Numerical solutions are obtained by the use of Navier solution method. The accuracy of the numerical solutions is first established through comparison with the exact three-dimensional (3D) elasticity solutions and the present solutions are then compared with available solutions of other models.

Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory

This paper introduces a generalized 5 degrees of freedom (DOF) higher-order shear deformation theory (HSDT) to study the bending and free vibration of plates and shells, which may be used to create other HSDTs. It also introduces a new HSDT for shells that is more accurate than many available HSDTs despite having the same 5DOF, and which is also able to reproduce the well-known Soldatos’ HSDT as special case. The governing equations and boundary conditions of the generalized formulation are derived by employing the principle of virtual work. These equations are solved via Navier-type closed-form solutions. Static and dynamic results are presented for plates and cylindrical and spherical shells with simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. Results from the new and well-known HSDTs introduced and reproduced based on the present generalized 5DOF HSDT are compared with the exact three-dimensional elasticity solution. The present new HSDT for plates and shells is found to be more accurate than the well-known HSDTs developed by other authors, for analyzing the static and free vibration of isotropic and multilayered composite plates and shells.

Benchmark solution for transverse vibration of annular Reddy plates

The exact closed-form solution for freely vibrating annular thick plates is presented on the basis of the Reddy's higher-order shear deformation plate theory. Several combinations of classical boundary conditions including, free, soft simply supported, hard simply supported and clamped, are applied at the inner and outer edges of annular plates. Hamiltonian and minimum potential energy principles are employed to derive the equations of dynamic equilibrium and natural boundary conditions of the plate. To validate the accuracy and effectiveness of the present formulation, direct comparisons are made between our results and those from the literature. In parametric studies, the first eight natural frequencies of annular plates with different combinations of boundary conditions are tabulated for various values of the inner–outer radius ratios and thickness–radius ratios. A set of three-dimensional vibration mode shapes along with their corresponding contour plots are presented to illustrate transverse displacements of annular thick plates. Since the results of the present exact solution are the closest to those of exact three-dimensional elasticity solution, they can serve as a reliable benchmark for checking the accuracy of future numerical and analytical solutions.

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.

Three-dimensional Elasticity Solution for Uniformly Loaded Cross-ply Laminates and Sandwich Plates

This article establishes the bending problem of cross-ply laminated plates using the three-dimensional elasticity equations as well as the technique based on the state space concept. It presents a wide variety of results for the symmetric and antisymmetric analyses of rectangular multilayer plates subjected to a sinusoidally/uniformly distributed load (SDL/UDL). In addition, this study provides a strong mathematical tool allowing one to determine, in an exact and unified manner, the state of stress and displacement of cross-ply laminated composites and sandwich plates. The well-known results (given in Pagano ((1970), Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composite Materials, 4(1): 20—24.) and Pagano and Hatfield ((1972), Elastic Behaviour of Multilayered Bidirectional Composites, AIAA Journal, 10(12): 931—933.) due to the exact three-dimensional elasticity solution are in fact special cases of the present technique.

An improved shear-membrane theory for multilayered shells

A new two-dimensional theory for the analysis of deep, doubly curved, multilayered shells is proposed. The theory is based on a kinematical approach in which the continuity conditions for displacements and shear stresses at layer interfaces and on the bounding surfaces of the shell are exactly satisfied, while, at the same time, refinements of the shear and membrane terms are taken into account, by means of trigonometric functions. The accuracy of the model is assessed in statics and dynamics through investigation of significant problems, for which an exact three-dimensional elasticity solution is known. In statics, the cylindrical bending of a cylindrical panel, of thickness h, and inner radius R, is examined. The results obtained are in good agreement with the analytical solution of Ren [Comput. Sci Technol. 29 (1987) 169–187], and prove that no shear-correction factors are requested. In dynamics, wave propagation in a three-layered elastic cylinder is explored. The accuracy of the theory is assessed by comparison with the exact three-dimensional elasticity solution of Armenakas [J. Acoust. Soc. Am. 49 (5, Part 2) (1971) 1511–1520]. The model is then applied to the exploration of the dispersive behavior of a viscoelastic cylinder.

A three-dimensional zig-zag theory for analysis of thick laminated beams

A modified zig-zag technical theory suitable for accurate stress and failure analysis of thick laminated and sandwich beams is developed and tested. The postulated displacement field allows for appropriate jumps in the strains, in order to meet the transverse shear and the transverse normal stress and stress gradient continuity conditions at the laminae interfaces. The free-shear and normal pressure loading conditions are also met. Piecewise third- and fourth-order approximations are assumed in the thickness direction for in-plane and transverse displacements, respectively. The theory is cast in such a way to allow for sublaminates stacking, as this gives an opportunity of refinement for capturing interlaminar stresses. Developments are limited to beams with a rectangular cross-section, in cylindrical bending under transverse sinusoidal loading. Test cases wherein the exact three-dimensional elasticity and other approximate solutions are available are solved in closed-form to show accuracy of the present approach. Based on this numerical investigation, the present theory appears to predict with accuracy the stress field of thick laminated beams with general lay-up.

Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations

A modified zig-zag technical theory, suitable for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested. An equivalent single-layer model and a multiple-layer model are implemented. The displacement field of both these models is postulated as to allow for appropriate jumps in the strains, so that the transverse shear and the transverse normal stress and stress gradient continuity at the interfaces are met. A third-order piecewise approximation for the in-plane displacement and a fourth-order piecewise approximation for the transverse displacement are assumed in the two models. Their predictive capability is investigated in sample cases wherein the exact three-dimensional elasticity and other approximate solutions are available. On the basis of this numerical investigation, they appear to predict accurately and efficiently the displacement and stress fields of composite beams with layers of different materials.

Deep Doubly Curved Multilayered Shell Theory

A two-dimensional theory is presented for the analysis of deep, doubly curved, multilayered shells, The theory is based on a kinematical approach in which the continuity conditions for displacements and shear stresses at layer interfaces and on the bounding surfaces of the shell are exactly satisfied. It also takes into account refinements of the shear and membrane terms, by means of trigonometric functions as proposed previously for the transverse shear. The accuracy of the proposed theory is assessed through investigation of significant problems for which an exact three-dimensional elasticity solution is known: first, the bending of a three-layered, laminated, symmetric cross-ply rectangular plate, simply supported along all edges, submitted to a double sinusoidal transverse loading, and second, the bending of a circular, cylindrical panel. Results obtained with the model are compared with those yielded by previous theories. The sensitivity of the model to edge effects, for a hard-clamped, free-edge cylindrical panel, is also examined.

Efficient linear transverse normal stress analysis of layered composite plates

Based on the first order shear deformation theory (FSDT) a method is developed for calculating the transverse normal stress (in thickness direction) in layered composite plate structures. Two steps are necessary. First, the transverse shear stress calculation, and second, relying on the results of the first step, the transverse normal stress evaluation. In the first step strain derivatives are substituted with transverse shear forces which in turn are obtained from the corresponding material law. This leads to a derivative-free process which is numerically more accurate than a pure equilibrium approach. As a second step the transverse normal stress is equilibrated to the derivatives of the transverse shear stresses with respect to the in-plane coordinates. As compared to the standard equilibrium approach, the proposed procedure reduces the order of differentiation by one. Thus, only quadratic shape functions are necessary for evaluating the required derivatives on the element level. Numerical examples for symmetric cross-ply and antisymmetric angle-ply laminates show that the exact three-dimensional elasticity solution is very closely approximated. This holds for thin as well as rather thick plates with a slenderness ratio down to five. In contrast to many recently established methods, either higher order lamination theories or layerwise theories, the approach is easily applicable to finite elements, since only C 0-continuity is necessary and the numerical effort is low.

Cylindrical bending of laminated cylindrical shells using a modified zig-zag theory

A relatively simple two-dimensional multilayered shell model is presented for predicting both global quantities and stress distributions across the thickness of multilayered thick shells, that is based on a third-order zig-zag approach. As for any zig-zag model, the layerwise kinematics is accounted for, with the stress continuity conditions at interfaces met a priori. Moreover, the shell model satisfies the zero transverse shear stress conditions at the upper and lower free surfaces of the shell, irrespective of the lay-up. By changing the parameters in the displacement model, some higher order shell models are obtained as particular cases. Although it potentially has a wide range of validity, application is limited to cylindrical shell panels in cylindrical bending, a lot of solutions of two-dimensional models based on rather different simplyfying assumptions and the exact three-dimensional elasticity solution being available for comparisons for this benchmark problem. The numerical investigation performed by the present shell model and by the shell models derived from it illustrates the effects of transverse shear modeling and the range of applicability of the simplyfying assumptions introduced. The implications of retaining only selected terms depending on the radius-to-thickness ratio are focused by comparing the present solutions to the exact one and to other two-dimensional solutions in literature based on rather different simplyfying assumptions.

Further investigation on the exact elasticity solution for anisotropic thick rectangular plates

The state equation governing the anisotropic response of monoclinic linear elastic material is deduced. An exact three-dimensional (3-D) elasticity solution is performed for simply-supported thick orthotropic rectangular plates subjected to arbitrary loading. For the first time, a sixth-order differential equation governing the transverse displacement is obtained and various expressions of the solution are provided. The results can be used to the plate of any kind of reduced material properties for example transversely isotropic or isotropic plates.

Constructing Exact Dynamic Elasticity Solutions for Axisymmetrically Deformed Plates From Classical Plate Theory Solutions

This paper addresses the question of how to assess the errors made when the exact three-dimensional linear elasticity solution for the axisymmetric dynamic deformation of an elastic plate is approximated by a solution inferred from the classical plate theory of Kirchhoff. Following the strategy used by Ladeve`ze and Simmonds for beams, the exact solution of a “nearby” three-dimensional problem, which differs from the original problem by the addition of incremental, computable body forces, face shears, and initial conditions—error increments, for short—is expressed in terms of the solution of a wave equation in which distance normal to the plate’s midplane plays the role of a time-like variable while the physical time itself enters only as a parameter. The error increments which, ultimately, can be computed in terms of the solution delivered by plate theory, can be regarded as an “engineering norm” because with them an engineer can decide if such a shift in the external data lies within acceptable bounds.

Eight-noded zig-zag element for deflection and stress analysis of plates with general lay-up

Abstract Based on the kinematics of a third-order zig-zag plate model accounting for the zero transverse shear stress conditions at upper and lower free surfaces irrespective for the lay-up, a eight-noded, 56 d.o.f. curvilinear plate element is formulated and tested. Nodal parameters are membrane displacements, transverse shear rotations, deflections, slopes and curvatures for corner nodes, membrane displacements and transverse shear rotations for mid-side nodes. Suitably defining the parameters appearing in the plate model, some higher-order plate elements, comprising previous zig-zag elements and some smeared-laminate plate elements, are obtained as particular cases. The benchmark problem is the cylindrical bending of simply-supported, symmetric and antisymmetric cross-ply plates under a sinusoidally distributed transverse loading. The body of numerical results is as follows. Closed-form solutions for the present plate model, together with the ones that can be particularized in it, are given that provide analytical comparison results. Finite element results by the present element and by the ones that are particularized in it are compared with closed-form solutions and with the exact three-dimensional elasticity solution, available for this benchmark problem. A great importance of stress-free conditions appears when plates are thick and unsymmetric. This justify development of present element accounting for unsymmetry in the lay-up. The quadratic approximation makes the present element able to accurately predict stresses by integrating local equilibrium equations. Inclusion of the layerwise kinematics of zig-zag models allows for satisfactory predictions of transverse shear stresses via constitutive equations. Use of C 2 approximation for deflections, instead of C 1 approximation required for obtaining conforming elements, appear to little improve accuracy.