Nonlinear Plate Bending Research Articles

Thermal and thermomechanical postbuckling of FGM sandwich plates resting on elastic foundations with tangential edge constraints and temperature dependent properties

This paper investigates nonlinear bending and postbuckling behavior of functionally graded sandwich plates resting on elastic foundations and subjected to uniform external pressure, thermal loading and uniaxial compression in thermal environment. The material properties of both face sheets and core layer are assumed to be temperature-dependent, and effective material properties of FGM layers are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Formulations are based on first order shear deformation theory taking von Karman nonlinearity, initial geometrical imperfection, Pasternak type elastic foundations and tangential edge constraints into consideration. Approximate solutions are assumed and Galerkin procedure is applied to derive nonlinear relations of load and deflection. In thermal postbuckling analysis, an iteration algorithm is adopted to obtain buckling temperatures and postbuckling curves. The effects of material, geometry and foundation stiffness parameters, face sheet thickness to total thickness ratio, imperfection and degree of tangential restraint of edges on the nonlinear bending and postbuckling behavior of FGM sandwich plates are analyzed and discussed in detail.

Local continuum shape sensitivity with spatial gradient reconstruction for nonlinear analysis

Gradient-based optimization for large-scale, multidisciplinary design problems requires accurate and efficient sensitivity analysis to compute design derivatives. Presented here is a nonintrusive analytic sensitivity method, that is relatively easy to implement. Furthermore, it can be as accurate as conventional analytic sensitivity methods, which are intrusive and tend to be difficult, if not infeasible, to implement. The nonintrusive local continuum shape sensitivity method with spatial gradient reconstruction (SGR) is formulated for nonlinear systems. This is an extension of the formulation previously published for linear systems. SGR, a numerical technique used to approximate spatial derivatives, can be leveraged to implement the sensitivity method in a nonintrusive manner. The method is used to compute design derivatives for a variety of applications, including nonlinear static beam bending, nonlinear transient gust response of a 2-D beam structure, and nonlinear static bending of rectangular plates. To demonstrate that the method is nonintrusive, all analyses are conducted using black box solvers. One limiting requirement of the method is that it requires the converged Jacobian or tangent stiffness matrix as output from the analysis tool. For each example the design derivatives of the structural displacement response are verified with finite difference calculations.

Large deflection analysis of point supported super-elliptical plates

Nonlinear bending of super-elliptical plates of uniform thickness under uniform transverse pressure was investigated by the Ritz method. The material was assumed to be homogeneous and isotropic. The contribution of the boundary conditions at the point supports was introduced by the Lagrange multipliers. The solution was obtained by the Newton-Raphson method. The influence of the location of the point supports on the central deflection was highlighted by sensitivity analysis. An approximate relationship between the central deflection and the super-elliptical power was obtained using the method of least squares. The critical points where the maximum deflection may develop, and the influence of nonlinearity were highlighted. The nonlinearity was found to be sensitive to the aspect ratio. The accuracy of the algorithm was validated by comparing the central deflection with the solutions of elliptical and rectangular plates.

Geometrically nonlinear finite element analysis of sandwich plates using normal deformation theory

The geometrically nonlinear bending and vibration behavior of soft core sandwich plates is investigated here using higher order finite element model incorporating transverse shear and normal deformation. The geometric nonlinearity, based on von Kármán’s assumption is introduced and the nonlinear governing equations of motion are derived considering in-plane and rotary inertia. The nonlinear governing equation is solved by Newton–Raphson iteration technique for the nonlinear bending problem, whereas, the harmonic balance method is employed to obtain the frequency versus amplitude relationships for the large amplitude free and forced vibration of sandwich plates. The results obtained from the normal deformation theory are compared with the results of first-order and third-order shear deformation theories. Limited parametric study is conducted to examine the influences of span-to-thickness ratio and core-to-face sheet thickness ratio of soft core sandwich plates.

Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads

This paper presents an efficient Rayleigh–Ritz numerical model to predict the deformed shape and the multistable behaviour of free-standing square and rectangular 0/90 unsymmetric composite plates subjected to thermal and environmental loads. The out-of-plane displacement functions are generated through proper development of the analytical solution presented by Ashwell [34] for the nonlinear pure bending of isotropic plates; the in-plane displacement functions are formulated following a procedure similar to that employed by Galletly and Guest [32], starting from geometrical considerations about the actual deformations of the distorted structure. The resulting model is characterised by high-order displacement functions and by few unknown terms, it is appropriate to achieve sufficient accuracy, with high efficiency. The performance of the proposed approach is compared to that of some alternative models available in the literature, to FE simulations and to experimental results.

Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets

This paper studied the large amplitude vibration and the nonlinear bending of a sandwich plate with carbon nanotube-reinforced composite (CNTRC) face sheets resting on an elastic foundation in thermal environments. The material properties of CNTRC face sheets are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both CNTRC face sheets and homogeneous core layer are assumed to be temperature-dependent. A detailed parametric study is conducted to study the effects of nanotube volume fraction, core-to-face sheet thickness ratio, temperature change, foundation stiffness and in-plane boundary conditions on the nonlinear vibration characteristics and nonlinear bending behaviors of sandwich plates with functionally graded CNTRC face sheets. The results for the same plate with uniformly distributed CNTRC face sheets are also provided for comparison.

Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher-order shear deformation plate theory and general von Karman-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.

Nonlinear local bending of FGM sandwich plates

This paper investigates the nonlinear local bending of a sandwich plate consisting of two composite laminated face sheets and a graded core subjected to a lateral patch load. It is assumed that the material composition of the graded layer varies symmetrically along the thickness direction according to a power law distribution. The present analysis is based on the first order shear deformation plate theory and von Karman nonlinear kinematics, with the interaction between the loaded face sheet and the graded core being modeled as an elastic plate resting on a Vlasov-type elastic foundation. A perturbation technique and Galerkin method are used to determine the nonlinear local bending response. Numerical results show that compared with conventional sandwich plates with a homogeneous soft core, the use of a functionally graded core can effectively reduce both the local deformation and interfacial shear stresses. A parametric study is performed to show the influences of the volume fraction index, Young’s modulus ratio, thickness of the graded core, boundary condition, and load position.

Nonlinear Bending of the FGM Circular Plates with Temperature-Dependent Properties

Axisymmetric nonlinear bending of the functionally graded circular plates is investigated in the present work. The material properties of plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. Based on the classical nonlinear plate theory, the governing equations for the problem are derived, and then a shooting method is employed to numerically solve the equations. Effects of material constant, temperature- dependent properties and boundary conditions on the nonlinear bending behavior of the FGM plate are discussed in details.

Nonlinear Bending of Composite Circular Plates with Embedded Shape Memory Alloy Fibers

Based on Brinson’s one-dimensional thermo-mechanical constitutive relations of shape memory alloys and the theory of thin plates in the von Kármán sense, the response of bending of a uniform heated circular plate embedded with SMA fibers in the radial directions and subjected to a uniform distributed mechanical load is studied. The characteristic curves of the central deflection versus temperature rise of the circular plate with both clamped and simply supported boundary conditions are obtained. The numerical results show that, the recovery forces of the pre-strained SMA caused by the phase transformation from martensite to austenite can modify the bending deformation significantly. So, it can be concluded that the bending deformation can be adjusted effectively and actively by embedment of the SMA fibers into the circular plates

Geometrically Nonlinear Deformation of Circular Sandwich Plates under Transversely Non-Uniform Temperature Rise

Geometrically nonlinear bending and buckling of circular sandwich plates subjected to transversely non-uniform temperature rise is investigated in this paper. On the basis of sandwich plate theory, nonlinear equations governing the large thermal axis-symmetric deformations of circular sandwich plate in terms of the middle plane’s displacements are derived. Numerical solutions of the nonlinear boundary value problem are obtained by using the shooting method. Equilibrium paths and configurations for different boundary conditions and different values of materials and geometry parameters are illustrated. Numerical results show that the boundary conditions and the stiffness greatly effect critical buckling loads.

A geometrically nonlinear thick plate bending element based on mixed formulation and discrete collocation constraints

In recent years there are many plate bending elements that emerged for solving both thin and thick plates. The main features of these elements are that they are based on mix formulation interpolation with discrete collocation constraints. These elements passed the patch test for mix formulation and performed well for linear analysis of thin and thick plates. In this paper a member of this family of elements, namely, the Discrete Reissner-Mindlin (DRM) is further extended and developed to analyze both thin and thick plates with geometric nonlinearity. The Von K<TEX>$\acute{a}$</TEX>rm<TEX>$\acute{a}$</TEX>n's large displacement plate theory based on Lagrangian coordinate system is used. The Hu-Washizu variational principle is employed to formulate the stiffness matrix of the geometrically Nonlinear Discrete Reissner-Mindlin (NDRM). An iterative-incremental procedure is implemented to solve the nonlinear equations. The element is then tested for plates with simply supported and clamped edges under uniformly distributed transverse loads. The results obtained using the geometrically NDRM element is then compared with the results of available analytical solutions. It has been observed that the NDRM results agreed well with the analytical solutions results. Therefore, it is concluded that the NDRM element is both reliable and efficient in analyzing thin and thick plates with geometric non-linearity.

Trefftz Finite Element Method and Its Applications

This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lamé-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article.

Cublic spline solutions of axisymmetrical nonlinear bending and buckling of circular sandwich plates

Cubic B-spline taken as trial function, the nonlinear bending of a circular sandwich plate was calculated by the method of point collocation. The support could be elastic. A sandwich plate was assumed to be Reissner model. The formulae were developed for the calculation of a circular sandwich plate subjected to polynomial distributed loads, uniformly distributed moments, radial pressure or radial prestress along the edge and their combination. Buckling load was calculated for the first time by nonlinear theory. Under action of uniformly distributed loads, results were compared with that obtained by the power series method. Excellences of the program written by the spline collocation method are wide convergent range, high precision and universal.

Application of the R-function method to nonlinear vibrations of thin plates of arbitrary shape

In this paper free large-amplitude flexural vibrations of thin plates with various planforms and boundary conditions are studied by the R-function method. This method is based on the joint application of the R-function theory and variational methods. The main feature of the R-function theory is the possibility to present all geometric information given in the boundary value problem in analytical form, which allows one to seek a solution in the form of some formula called the solution structure. A method of constructing the solution structures for the given nonlinear vibration plate bending problems is developed. Numerical examples of large-amplitude flexural vibrations of thin plates with arbitrary shapes and boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits and advantages.

On the finite displacement analysis of quadrangular plates

In this paper, a numerical method for the linear and geometrically non-linear static analysis of thin plates is presented. The method begins with the elasticity equations pertaining to strain components, stresses, displacement components, strain energy and work due to externally applied loads. The plate geometry is defined by a quadrangular boundary with four straight edges and the natural coordinates in conjunction with the Cartesian coordinates are used to map the geometry. The matrix equation of equilibrium is derived using the work–energy principle with the displacement fields expressed by algebraic polynomials, the coefficients of which are then manipulated to satisfy the kinematic boundary conditions. To validate the results from the present method, square plates having all sides fully fixed and all sides simply supported without in-plane movement are analysed. Comparison is made for the uniformly loaded square plate with the results obtained by Levy who solved the non-linear plate bending problem using the Th.von Ka’ rma’ n’ s equations. Rhombic plates are examined and numerical results corresponding to these cases are presented in this paper. Very good comparison of the results regarding deflection and bending stresses with other sources available in the literature is found.

Trefftz Finite and Boundary Element Method

Part 1: finite element technique shape functions and element stiffness matrix brief historical background basic relationships in engineering problems modified variational principles the concept of T-complete solution comparison of T-elements with conventional finite elements comparison of T-elements with boundary elements. Part 2: potential problems - introduction statement of the problem T-complete functions assumed fields generation of element matrix equation rank condition special purpose functions sensitivity to mesh distortion orthotropic case the Helmholtz equation HT-element with boundary traction frame frameless T-elements. Part 3: linear elastostatics - introduction linear theory of elasticity assumed fields in plane elasticity T-complete functions variational formulations element stiffness equation special-purpose elements p-extension approach three-dimensional elasticity numerical examples. Part 4: thin plates - introduction thin plate theory assumed field T-complete functions and particular solutions variational formulations for plate bending generation of element stiffness matrix p-method elements special purpose functions Extension to thin plates on elastic foundation Two alternative plate bending p-elements Numerical examples and assessment. Chapter 5 - Thick Plates - Introduction Basic equations for Reissner-Mindlin plate theory Assumed fields and particular solution Variational formulation for HT thick plate elements Implementation of the new family of HT elements A 12 DOF quadrilateral element free of shear locking Extension to thick plates on elastic foundation Sensitivity to mesh distortion Numerical assessment. Chapter 6 - Transient Heat Conduction - Introduction Elements of heat conduction Time step formula Element matrix formulations T-complete functions and particular solutions Numerical examples. Chapter 7 - Geometrically Nonlinear Analysis of Plate Bending Problems - Introduction Basic equations of nonlinear thin plate bending Assumed fields and Trefftz functions Particular solutions Modified variational principle Element matrix Iterative scheme Extension to post-buckling thin plates on elastic foundation Geometrically nonlinear analysis of thick plates Numerical examples. Chapter 8 - Elastoplasticity - Introduction Time discretization Basic relations Assumed fields Constraints on the approximation functions Finite element equilibrium and compatibility equations Finite element equations Finite element governing system. Chapter 9 - Dynamics of Plate Bending Problems - Introduction Basic equations Time-stepping formulation Numerical examples. Chapter 10 - Trefftz Boundary Element Method - Introduction Potential problems Plane elasticity Thin plate bending Moderately thick plates.

Nonlinear bending of inhomogeneous plates

This paper presents a study on inhomogeneous plates comprised of transversely isotropic materials. The properties of the materials are gradient along their thickness direction and include laminates as a special case. Based on the first-order shear deformation theory, a new set of nonlinear equations in the sense of the von Karman large deflection is presented and reformulated by use of displacement and stress potential functions. In an illustrative example, these equations associated with various boundary conditions are then solved for a rectangular inhomogeneous plate by using the mixed Fourier series technique.

The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates

The Hadamard and SJT product of matrices are two types of special matrix product. The latter was first defined by Chen. In this study, they are applied to the differential quadrature (DQ) solution of geometrically nonlinear bending of isotropic and orthotropic rectangular plates. By using the Hadamard product, the nonlinear formulations are greatly simplified, while the SJT product approach minimizes the effort to evaluate the Jacobian derivative matrix in the Newton–Raphson method for solving the resultant nonlinear formulations. In addition, the coupled nonlinear formulations for the present problems can easily be decoupled by means of the Hadamard and SJT product. Therefore, the size of the simultaneous nonlinear algebraic equations is reduced by two-thirds and the computing effort and storage requirements are greatly alleviated. Two recent approaches applying the multiple boundary conditions are employed in the present DQ nonlinear computations. The solution accuracies are significantly improved in comparison to the previously given by Bert et al. The numerical results and detailed solution procedures are provided to demonstrate the superb efficiency, accuracy and simplicity of the new approaches in applying DQ method for nonlinear computations.

A pseudo-layered, elastic-plastic, flat-shell finite element

A three-node, C 0-type, layered flat-shell finite element is developed for the analysis of large elastic-plastic deformations in plate and shell structures. The system equations are derived by using virtual work principle and the updated Lagrangian formulation. Material is assumed to be isotropic and rate insensitive obeying J 2-flow rule. The displacement field assumption of the MIN3 plate bending element is employed. A layered structure is used to model through-the-thickness distribution of elastic and plastic zones. The finite element results for three nonlinear plate bending problems are compared with experimental results to verify the accuracy of the formulation.