Evaluation of some multilayered, shear-deformable plate elements

Recently, the author has developed an improved bidimensional transverse shear deformation theory for multilayered anisotropic plates which accounts for piecewise linear distribution across the thickness of the in-plane displacements u and v, and allows the contact conditions at the interfaces between the layers to be satisfied.

Based on this refined theory and on a Mindlin's-type transverse shear deformation plate theory developed by Whitney and Pagano J. appl. Mech. 37, 1031 (1970), several triangular and quadrilateral multilayered anisotropic plate elements which include extension, bending and transverse shear deformation states have been developed by making use of the displacement formulation in conjunction with the principle of virtual work.

In order to show the accuracy and the relative merits of the developed finite elements, results are presented for the sample problems of the bending and free undamped vibrations of a three-layered, symmetric cross-ply square plate that is simply-supported on all edges.

Excepting the conventional triangular and quadrilateral elements with linear shape functions, fast convergence to the respective analytical solutions for the global response (transverse displacements and fundamental flexural frequencies) is observed for all the elements tested. The rectangular finite element developed on the basis of the refined plate theory proposed by the author is also very efficient to model the warpage of the cross-section and to predict accurate values of the flexural stress at the interfaces. The finite elements developed on the basis of the Whitney and Pagano theory fail in this respect.     

Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.     

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

The mixed method for the biharmonic problem introduced in (Behrens and Guzmán, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.     

Finite element analysis of a laminated composite plate subjected to circularly distributed central impact loading

A two dimensional finite element analysis has been made for a fiber-reinforced composite laminate subjected to circularly distributed impact load which results, for example, from impacting the plate with a blunt-ended projectile. A finite element displacement model which includes the effects of transverse shear deformation and rotary inertia was used along with Hamilton's principle to derive the finite element matrices. Newmark's direct integration technique was used to integrate with respect to time. The interaction force between the projectile and the plate was calculated by using the Hertzian law of contact. Results for laminate deformations are shown to compare quantitatively with experimental results. Numerical values for stresses in the plate were calculated.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

Triangular thick plate elements based on a hybrid-Trefftz approach

The paper presents a family of triangular, thick plate elements derived using the hybrid-Trefftz approach. Exact solutions of the governing thick plate equations are used as interpolations for the internal element displacements. An immediate benefit of this approach is that the locking problem is avoided a priori. Independent interpolations are used to describe the displacement and rotations on the element boundaries. The element formulation is based on a modified hybrid-stress principle, leading to a standard stiffness formulation. This enables the elements to be readily implemented into existing finite element schemes. A number of examples are considered to demonstrate the accuracy achieved by the elements.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.