Higher-order Displacement Model Research Articles

Nonlinear static and transient finite element analysis of laminated beams

A one-dimensional finite element formulation based on a higher-order displacement model has been developed. The model accounts for geometric nonlinearities, a parabolic shear strain distribution through the thickness, and satisfies the shear stress free boundary conditions at the upper and lower surfaces of the beam. The model also accounts for the bend-stretch, shear-stretch and bend-twist couplings inherent to generally laminated beams. The lateral strains are assumed to be nonzero and retained in the formulation. The model has been applied to the nonlinear static and transient analysis, free vibration analysis, and impact analysis of laminated beams. For impact problems, nonlinear effects may be important and the inplane stresses developed due to the geometric nonlinearity may be greater than those due to initial stresses.

Estimation of interlaminar stresses in fibre reinforced composite cylindrical shells

A C 0 finite element formulation for flexure-membrane coupling behavior of symmetric and asymmetric laminated cylindrical shells based on a higher-order displacement model is presented. This theory incorporates a realistic nonlinear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficient/s. The discrete element chosen is a nine-noded quadrilateral with nine degrees of freedom per node. The solutions are obtained through two formulations: (1) the geometrically thin shell formulation, based on the assumption that the ratio of thickness to radius of the shell is very much less than unity, and (2) the geometrically thick shell formulation, in which ( h/ R) 2⪡1. In these formulations, the in-plane stresses are obtained via constitutive relations. Reliable estimates of interlaminar stresses from equilibrium equations are obtained. A finite difference scheme maintaining the continuity of interlaminar stresses across the shell thickness is developed and used. The results obtained are compared with available elasticity, closed-form and other finite element solutions.

Higher-order theories for symmetric and unsymmetric fiber reinforced composite beams with C 0 finite elements

A simple C 0 isoparametric finite element formulation based on a set of higher-order displacement models for the analysis of symmetric and asymmetric multilayered composite and sandwich beams subjected to sinusoidal loading is presented. These theories do not require the usual shear correction coefficients which are generally associated with the Timoshenko theory. The four-noded Lagrangian cubic element with kinematic models having four, five and six degrees of freedom per node is used. A computer algorithm is developed which incorporates realistic prediction of transverse interlaminar stresses from equilibrium equations. By comparing the results obtained with the elasticity solution and the CPT (classical laminated plate theory) it is shown that the present higher-order theories give a much better approximation to the behaviour of laminated composite beams, both thick and thin. In addition numerical results for unsymmetric sandwich beams are presented which may serve as benchmark for future investigations.

Vibrations of unsymmetrically laminated plates analyzed by using a higher order theory with a C° finite element formulation

Recently developed shear deformation theory is used to analyze vibrations of laminated composite and sandwich plates in conjunction with a C° isoparametric finite element formulation. The present theory is based on a higher order displacement model and the three-dimensional Hooke's laws for plate material, giving rise to a more realistic representation of the cross-sectional deformation. The theory does not require the usual shear correction coefficients generally associated with Reissner-Mindlin theories. A special mass lumping procedure is used in the dynamic equilibrium equations. The numerical examples presented are compared with 3-D elasticity/analytical and Mindlin's plate solutions, and it is demonstrated that the present model predicts the frequencies more accurately when compared with the first order shear deformation theories and classical plate theories.

Refined theories for composite and sandwich beams with C0 finite elements

Refined higher-order displacement models for the behaviour of symmetric and unsymmetric laminated composite beams based on C0 finite element discretization are presented. These theories incorporate a more realistic non-linear variation of longitudinal displacements through the beam thickness, thus eliminating the use of shear correction coefficient(s). The discrete element considered is a four-noded cubic with kinematic models having three, four and five degrees of freedom per node. The computer program developed incorporates the realistic prediction of interlaminar stresses from equilibrium equations. The present results, when compared with the elasticity solutions, show excellent agreement. In addition numerical results for sandwich beams are presented for future reference.

Higher-order theories for composite and sandwich cylindrical shells with C0 finite element

A higher-order displacement model for the behaviour of symmetric and unsymmetric laminated composite and sandwich cylindrical shells based on C 0 finite element discretization is presented. Two theories, namely, (1) Geometrically Thin Shell Theory, based on the assumption that the ratio of the shell thickness to radius ( h/R) is less than unity, and (2) Geometrically Thick Shell Theory, in which ( h/ R) 2 ⪡ 1, are developed. These theories incorporate a more realistic non-linear variation of longitudinal displacements through the shell thickness and thus eliminate the use of shear correction coefficients. The influence of ( h/R) for a thick shell is studied and the results are compared with those of geometrically thin shell theory and other available results.

Dynamics of laminated composite plates with a higher order theory and finite element discretization

A simple isoparametric finite element formulation based on a higher-order displacement model for dynamic analysis of multi-layer symmetric composite plates is presented with an explicit time marching scheme. As is well-known, the Kirchhoff plate theory is inadequate to describe the propagation of waves in plates. The first-order shear deformable Mindlin plate theory hitherto considered was adequate for determining responses to dynamic loads. The present higher-order theory which is more accurate than the Mindlin theory is applied, herein, for the evaluation of plate response to different types of dynamic loads. A special mass lumping scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. Several numerical examples are presented and compared with results from other sources.

A finite element model for a higher-order shear-deformable beam theory

The theory for a higher order shear-deformable beam model is first developed. It is based on a higher order displacement model and incorporates linear and quadratic variation of transverse normal strain and transverse shearing strain respectively through the beam thickness. The effects of the transverse normal and shear stresses are included in the definition of the material's constitutive law. The warping of the transverse normal cross-section of the beam is automatically incorporated in the mathematical model. The question of selecting a shear correction coefficient as in a first-order shear deformable Timoshenko theory does not arise. A linear two-noded finite element model of this theory is introduced and developed next. Both static and free vibration results of this theory are presented and compared with those of Euler and Timoshenko theories for various boundary and loading conditions.

Flexural analysis of laminated composites using refined higher-order [formula omitted] plate bending elements

A finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented here. The present higher-order theory incorporates linear variation of transverse normal strains and parabolic variation of transverse shear strains through the plate thickness, and as a result it does not require shear correction coefficients. A nine-noded Lagrangian parabolic isoparametric plate bending element is described. The applications of the element to bending of laminated plates with various loading, boundary conditions, and lamination types are discussed. The numerical evaluations also include the convergence study of the element used. The present solutions for deflections and stresses are compared with those obtained using the three-dimensional elasticity theory, closed-form solutions with another high-order shear deformation theory, and the Mindlin's theory. In addition, numerical results for a number of new problems, not available in the literature, are presented for future reference.

Finite element analysis of laminated composite plates using a higher-order displacement model

A C° continuous displacement finite element formulation of a higher-order theory for flexure of thick arbitrary laminated composite plates under transverse loads is presented. The displacement model accounts for non-linear and constant variation of in-plane and transverse displacement model eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with nine degrees-of-freedom per node. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form, the theory of elasticity and the finite element solutions with another higher-order displacement model by the same authors. A computer program has been developed which incorporates the realistic prediction of interlaminar stresses from equilibrium equations.

On a nonlinear theory of plates and shells including consistent and inconsistent kinematics and the finite element method

A nonlinear theory of plates and shells for thick and thin models accounting for consistent and inconsistent kinematical approximations is presented in general curvilinear tensorial form. The incremental shell governing equations for the finite element formulation of boundary value problems for finite strains and large deformations are derived employing the incremental virtual work principle defined in a Lagrangian description. Sample problems are investigated numerically employing a hyperelastic material model for which both general and specific forms are introduced. Finite element solutions of thick and thin plate and shell problems are obtained by employing lower and higher order displacement models and are compared with results deduced from application of the finite element method based on exact equations of continuum mechanics. It is found that the kinematical theories, which are based on one general formulation, adequately predict displacements and normal stresses. However, major discrepancies arise in the shear stress distributions at finite strains when exact and lower order theories are compared. Finally, it is noted that the consistent kinematical theory described by a cubic variation of in-plane and transverse displacements through the thickness coordinate yields the most accurate and meaningful solutions. Furthermore, it is shown that higher order kinematical theories become significant in the analysis of thick structures.

A refined higher-order generally orthotropic C 0 plate bending element

A finite element formulation for flexure of a generally orthotropic plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented here. This higher-order theory incorporates linear variation of transverse normal strain/stress and parabolic variation of transverse shear strains through the thickness of the plate. The nine-noded quadrilateral from the family of two-dimensional C 0 continuous isoparametric Lagrangian elements is then developed as a generally orthotropic higher-order element. The performance of this element is evaluated on square plates with different support conditions and under uniformly distributed and central point loads. The numerical results of the present formulation are compared with thin plate, elasticity and Mindlin/Reissner solutions. The effect of degree of orthotropy on the maximum bending moment location is examined for different loading and boundary conditions. The effect of directional orthotropy on the location of the maximum values for the various stress-resultants is also studied.

Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations

A simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant variation of transverse displacement through the plate thickness. Further, the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not imposed. The validity of the present development(s) is established through, numerical evaluations for deflections/stresses/stress-resultants and their comparisons with the available three-dimensional analyses/closed-form/other finite element solutions. Comparison of results from thin plate. Mindlin and present analyses with the exact three-dimensional analyses yields some important conclusions regarding the effects of the assumptions made in the CPT and Mindlin type theories. The comparative study further establishes the necessity of a higher-order shear deformable theory incorporating warping of the cross-section particularly for sandwich plates.

A higher order theory for free vibration analysis of circular cylindrical shells

The free, undamped vibration of an isotropic circular cylindrical shell is analysed with higher order displacement model, giving rise to a more realistic parabolic variation of transverse shear strains. The method accounts for in-plane inertia, rotary inertia, and shear deformation effects on the dynamic response of cylindrical shells. The frequencies obtained from the present analysis, shear deformation theory, and Flügge theory are compared with the exact elasticity results. It is found that the frequencies predicted by the present analysis are closer to exact values than those predicted by the shear deformation theory, especially for cases with shorter wavelengths.

Numerical analysis of thick plates

Bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory. The theory is based on a higher order displacement model and the three-dimensional Hooke's laws for plate material, giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness. It is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency, economy, reliability and accuracy in such applications

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.