Higher-order Displacement Model Research Articles

Quasi-3D flexural analysis of laminated composite plates resting on elastic foundations using higher-order displacement model

This work examines the impact of transverse strains on the static, vibration, and buckling analysis of laminated composite plates that are symmetric and anti-symmetric and rest on elastic foundations. This laminated plate theory, which is called a quasi-3D plate theory since it considers the effects of both transverse shear and normal strains, is provided here. The significance of nonclassical influences on plate deformation, such as shear deformation and thickness stretching, is sufficiently highlighted by the current theory. The theory demonstrates realistic transverse shear stress distributions across the laminated plate's thickness and complies with the traction-free boundary conditions at the upper and bottom surfaces of the plate. Hamilton's principle provides the current theory's equations of motion. For simply supported edges, the laminated plates supported by elastic foundations are investigated. For the aim of verification, the numerical results of the displacements, stresses, frequencies, and buckling load derived using the present theory are, if possible, compared with established literature. The current results and those derived from the other hypotheses found in the literature show a good degree of consistency. Additionally, the distributions of the interlaminar stresses in thickness direction of laminated composite plates sitting on elastic foundations are demonstrated.

Dynamic Response of Thick Laminated Shells using Higher-Order Theory

In this paper, the dynamic characteristics of thick cross-ply laminated composite cylindrical shells are studied using a higher-order displacement model. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thickness, and abrupt discontinuity in the slope of the in- plane displacements at any interface. The effect of inplane and rotary inertia terms is included. The analysis is carried out using finite element approach. The governing equations derived using Lagrange's equations of motion are solved employing Newmark's direct integration technique for transient response analysis. The influences of various terms in the higher-order displacement field on the free vibrations, and transient dynamic response characteristics of cylindrical composite shells subjected to thermal and mechanical Loads are analysed.

Nonlinear frequency prediction of cutout borne multi-layered shallow doubly curved shell structures

This is the first time two different geometrical nonlinear strain–displacement relationships are adopted to model the laminated shell structure with the cutout to predict the eigenvalues considering a higher-order displacement model. The nonlinear solution for the natural frequency value is obtained using Picard's iterative method in association with the isoparametric finite element method. The present research mainly indicates the inevitability of Green-Lagrange's nonlinear strain in the framework of higher-order displacement kinematics instead of von-Karman's nonlinear strain including all of the nonlinear higher-order strain. The predicted nonlinear solution consistency is verified by solving a series of examples, as same as the source data. The influences of the cutout (shape and size) related data and the corresponding geometrical parameters of the multilayered structure are expressed in the final output form. Lastly, the inclusive inferences are highlighted to recommend the nonlinear model to analyze a flexible laminated structural component with and without cutout.

Zero delamination of smart composite laminated cross- ply using hsdt with zig zag function

Thermal analysis of smart composite laminated plate using higher order theory with zig zag function. To develop the analytical procedure and to investigate the thermal characteristics, the material is considered to be orthotropic under thermal load based on higher order displacement model with zig-zag function, without enforcing zero transverse shear stress on the top and bottom faces of the laminated plates. The related functions and derivations for equation of motion are obtained using the dynamic version of the principle of virtual work or Hamilton’s principle. The solutions are obtained by using Navier’s and numerical methods for anti-symmetric cross-ply with specific type of simply supported boundary conditions SS. Computer programs have been developed to find the stresses and deflections for various aspect ratios, side thickness ratios (a/h) and voltages.

The Effect of Higher Order Model on the Geometric Nonlinear Analysis of Antisymmetric Angle Ply Laminates

Geometric nonlinear analysis of simply supported antisymmetric angle-ply laminated composite plates are investigated based on first order and higher order displacement models with five degrees of freedom. Analytical formulations and solutions are developed based on Von-Karman nonlinear plate theory and Taylor's series expansion of displacement components. Equations of equilibrium are obtained using Principle of Minimum Potential Energy (PMPE) and closed form solutions using Navier's Solution technique. A four layered square plate is considered for the present study. Parametric studies are performed on both the models to study the behaviour of displacements and stresses in laminated composite plates. Comparative studies are performed on both the models and the effect of geometric nonlinearity is discussed.

A Higher Order Shear Deformation Model for Bending Analysis of Functionally Graded Plates

In this paper, the static response of simply supported functionally graded plates subjected to a transverse uniform load and resting on an elastic foundation is examined by using a new higher order displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the new higher order shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.

Dynamic Response of Strain Rate Dependent Glass/Epoxy Composite Beams Using Finite Difference Method

This paper deals with a numerical analysis of the transient response of composite beams with strain rate dependent mechanical properties by use of a finite difference method. The equations of motion based on Timoshenko beam theory are derived. The geometric nonlinearity effects are taken into account with von Karman large deflection theory. The finite difference method in conjunction with Newmark average acceleration method is applied to solve the differential equations. A modified progressive damage model which accounts for strain rate effects is developed based on the material property degradation rules and modified Hashin-type failure criteria and added to the finite difference model. The components of the model are implemented into a computer code in Mathematica 6. Glass/epoxy laminated composite beams with constant and strain rate dependent mechanical properties under dynamic load are analyzed. Effects of strain rate on dynamic response of the beam for various stacking sequences, load and boundary conditions are investigated. orthotropic beam which incorporates the effects of transverse shear and normal deformations. Obst and Rakesh (2) studied the nonlinear static and transient response of laminated beams using a one-dimensional finite element formulation based on higher-order displacement model which accounts for geometric nonlinearities and a parabolic shear strain distribution through the thickness. Marur and Kant (3) used higher-order shear-deformable refined theories, based on isoparametric elements, for transient dynamic analysis of symmetric and un-symmetric sandwich and composite beams. Khdeir (4) developed an analytical solution of the classical, first- and third-order laminate beam to study the transient response of antisymmetric cross-ply laminated beams with generalized boundary conditions and for arbitrary loadings. Kant, et al. (5) reported an analytical solution to the natural frequency analysis of composites and sandwich beams based on a higher order refined theory. Gong and Lam (6) investigated the transient response of layered composite beams subjected to underwater shock involving the effects of structural damping and stiffness. None of these literatures considers the effects of strain rate. In this paper, a macro-mechanical approach using finite difference method and progressive damage modeling algorithm which accounts for geometric nonlinearity effects, transverse shear strain effects and the effects of strain rate is presented. Coupled equations of motion for a laminated composite beam based on Timoshenko beam theory are reduced to ordinary differential equations in time domain using finite difference approximations for displacements. Newmark time integration scheme in association with Newton-Raphson iteration method are applied to solve the system of equations. The variations of mechanical properties due to strain rate and failure are taken into account using empirical relations and sudden material property degradation rules, respectively. A computer code in Mathematica 6 is developed to implement the numerical procedures. The effects of strain rate on transient response of composite beams with various stacking sequences, loadings and boundary conditions are presented and discussed.

A higher-order displacement model for stress concentration problems in general lamination configurations

Abstract The higher-order displacement model has been proposed to analyze the stress concentration problems in angle-ply laminates under different loads. The proposed model a priori satisfies the transverse shear stress continuity conditions as well as free surface conditions. The total number of degrees of freedom in this model does not depend on the layer number of laminates. Thus, computations of the proposed model are simple and efficient. It is important that the proposed model is extended to solve the stress concentration problems in laminates with arbitrary stacking sequences. Based on the proposed model, a three-node triangular element has been presented. To circumvent the requirement of C 1 continuity at element interfaces, the refined element method has been employed. Numerical results show that the proposed model is suitable for the free-edge problems in the cross-ply laminates as well as the angle-ply laminates. After ascertaining the accuracy of present model, the model is extended to analyze the free-corner problems in laminates subjected to thermal loads. Numerical results imply that the analytical approach proposed by Becker et al. seem to underestimate the peak stresses in the vicinity of free corner in laminates.

Higher-order Closed-form Solutions for Free Vibration of Laminated Composite and Sandwich Shells

Closed-form formulations of two-dimensional (2D) higher-order shear deformation theories (HOSTs) for the free vibration analysis of simply supported cross-ply laminated composite and sandwich doubly curved shells are presented. The formulation includes the Sander’s theory for doubly curved shells. Two of the HOSTs account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of transverse shear deformation. In these developments a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique and by solving the eigenvalue equations. Numerical results are presented for the natural frequencies of laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plates and shells are compared with the available 3D elasticity and analytical solutions and it is believed that the solutions for sandwich laminates using various higher-order displacement models will serve as benchmark in future.

Higher-order Closed-form Solutions for Free Vibration of Laminated Composite and Sandwich Shells

Closed-form formulations of 2-D higher-order shear deformation theories (HOSTs) for the free vibration analysis of simply supported cross-ply laminated composite and sandwich doubly curved shells are presented. The formulation includes the Sanders theory for doubly curved shells. Two of the HOSTs account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of transverse shear deformation. In these developments, a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique and by solving the eigenvalue equations. Numerical results are presented for the natural frequencies of laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plates and shells are compared with the available 3-D elasticity and analytical solutions and it is believed that the solutions for sandwich laminates using various higher-order displacement models will serve as the benchmark in future.

Comparative dynamic studies of thick laminated composite shells based on higher-order theories

Here, the dynamic response characteristics of thick cross-ply laminated composite cylindrical shells are studied using a higher-order displacement model. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thickness, and abrupt discontinuity in slope of the in-plane displacements at any interface. The effect of inplane and rotary inertia terms is included. The analysis is carried out using finite element approach. The influences of various terms in the higher-order displacement field on the free vibrations, and transient dynamic response characteristics of cylindrical composite shells subjected to thermal and mechanical loads are analyzed.

A finite element model for the analysis of stiffened laminated plates

A refined higher-order displacement model for the study of the behavior of concentrically and eccentrically stiffened laminated plates based on C 0 finite element discretization is presented. The model incorporates non-linear variations of longitudinal displacements through the thickness and hence eliminates the need to use shear correction coefficient(s). Transverse shear deformations are included in the formulation making the model applicable for both moderately thick and thin composite stiffened plates. The plate element used is a nine-noded isoparametric one with seven degrees of freedom at each node. The stiffener element is a three-noded isoparametric beam element with four degrees of freedom at each node. The stiffness of a stiffener is reflected at all nine nodes of the plate element in which it is placed. Accordingly, the stiffeners can be positioned anywhere within the plate element along lines of constant natural coordinates and need not necessarily be placed on nodal lines which gives a great flexibility in the choice of mesh size. The integrals are evaluated by a selectively reduced integration (SRI) technique with three and two Gauss quadrature rules for membrane-flexure and shear parts, respectively. The present formulation is checked for different examples of stiffened plates made of isotropic and fiber-reinforced composites, and results are compared with existing analytical and other finite element solutions.

Buckling and dynamic behaviour of laminated composite structures using a discrete higher-order displacement model

This paper deals with buckling and free vibrations of multilaminated structures of arbitrary geometry and lay-up using a single layer higher order shear deformation theory discrete model. This model is based on an eight-node C 0 serendipity finite element with 10 degrees of freedom per node to contemplate general applications. The present model is tested on the evaluation of buckling loads and free vibrations of multilaminated plates and shells. The effects of different number of layers, lamination angles, material anisotropy, and length or radius to thickness ratios are studied.

A Higher Order Finite Element Model for the Vibration Analysis of Laminated Beams

A higher order displacement model based on a cubic axial strain, cubic transverse shear strain and quadratic transverse normal strain across the thickness of the beam, to model exactly the warping of the cross section is proposed which maintains zero stress at the top and bottom of the beam with out the aid of any shear correction factor. Numerical experiments carried out clearly bring out the efficacy of this model over the first order theory for laminated beams.

A higher-order facet quadrilateral composite shell element

A C0 finite element formulation of flat faceted element based on a higher-order displacement model is presented for the analysis of general, thin-to-thick, fibre reinforced composite laminated plates and shells. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with five and nine degrees of freedom per node. A comparison of results is also made with the 2-D thin classical and 3-D exact analytical results, and finite element solutions with 9-noded first-order element. © 1997 John Wiley & Sons, Ltd.

Buckling behaviour of laminated composite structures using a discrete higher-order displacement model

A higher-order theory is used to develop a discrete finite element model for the linear buckling analysis of multilaminated composite plate-shell structures. This model is based on an eight node isoparametric element with 10 degrees of freedom per node. The geometric stiffness matrix is developed by taking into consideration the effects of the higher order terms on the initial in-plane and transverse shear stresses. The model is applied to study several cases that take into consideration different number of layers, lamination angles, length-to-thickness ratio, as well as symmetry and non-symmetry on the laminates. The accuracy of the present formulation is evaluated by comparing the present results with alternative solutions.

FINITE ELEMENT THERMAL STRESS ANALYSIS OF COMPOSITE LAMINATES USING A HIGHER-ORDER THEORY

A simple C0 isoparametric finite element formulation based on a higher-order displacement model for the analysis of symmetric and unsymmetric, composite, and sandwich laminates subjected to a thermal gradient across the thickness is presented. The displacement model accounts for the nonlinear distribution of in-plane displacement components through the plate thickness, and the theory requires no shear correction coefficients. The nine-noded quadratic Lagrangian two-dimensional element is used with five and nine degrees of freedom per node. The accuracy of the formulation is verified by analyzing sample problems available in literature. Numerical results are presented in nondimensional form for symmetric, antisymmetric, and cross-ply laminates, both thick and thin.

A finite element-difference computational model for stress analysis of layered composite cylindrical shells

A C 0 finite element space discretization procedure is employed in a general fibre-reinforced composite cylindrical shell theory based on a higher-order displacement model. The displacement model incorporates non-linear variation of tangential displacement components through the thickness of the shell. The use of a shear correction coefficient thus becomes redundant. The discrete element chosen is a nine-noded Lagrangian quadrilateral with seven degrees of freedom per node. Two formulations, one in which ( h/ R) ⪡ 1 and another in which ( h/ R) 2 ⪡ 1, are derived. After the nodal displacements are obtained from the global finite element analysis, the secondary quantities are determined element-wise. The planar lamina stresses are computed through the constitutive relations while the transverse shear stresses are estimated by making use of the equilibrium equations. A special finite difference scheme is developed to integrate the equilibrium equations with a view to estimate transverse/interlaminar stresses across the shell thickness. The transverse/interlaminar stresses computed by the above technique do maintain the continuity at the interface of two layers. The results obtained are compared with available elasticity, closed-form and other finite element solutions.

Different Numerical Techniques for the Estimation of Multiaxial Stresses in Symmetric/Unsymmetric Composite and Sandwich Beams with Refined Theories

A set of simple but efficient and accurate higher-order displacement models are used to evaluate the multiaxial stress behavior in symmetric and unsymmetric composite and sandwich laminates. These theories incorporate a more realistic non-linear variation of displacements through the beam thickness, thus eliminating the use of shear correction coefficient(s). Constitutive relations are used to evaluate inplane stresses. The computer program developed incorporates the realistic prediction of transverse stresses from equilibrium equations. The integration of the equilibrium equations is attempted through direct integration method, forward and central direct finite difference technique and a new approach called an exact curve fitting method. The versatility of the present higher-order theories is demonstrated by comparing the results with the available elasticity and other closed-form solutions for cross-ply and sandwich laminates. The results show that the exact curve fitting method gives good estimates of multiaxial stresses compared to finite difference and direct integration methods.

A general fibre-reinforced composite shell element based on a refined shear deformation theory

A refined shell theory has been developed for the analysis of isotropic, orthotropic and anisotropic fibre-reinforced laminated composite and sandwich shells. This theory is based on a higher-order displacement model and the three-dimensional Hooke's laws for shell material, giving rise to a more realistic representation of the cross-sectional deformation. The superparametric shell element with four-noded linear eight/nine-noded quadratic and twelve/sixteen-noded cubic, serendipity/ Lagrangian shape functions can be employed. In addition to the present higher-order shear deformation shell theory (HOST), a first-order shear deformation shell theory (FOST), following Reissner-Mindlin plate's formulation, is developed and the results are compared with the closed-form solutions (CFS). The parametric effects of the finite element mesh, radius-to-arc length ratio, arc length-to-thickness ratio, lamination scheme, Gaussian integration rule, and material anisotropy on the response of the laminated composite shells are investigated. Results are tabulated to provide an easy means for future comparisons by other investigators.