Free vibration analysis of thick laminated plates using a mixed finite element

Abstract

A mixed finite element scheme based on assumed local high‐order displacements is proposed for the free vibration of thick laminated plates. The effects of transverse shear deformation, transverse normal stress and rotary inertia are considered in the formulation. Cross‐ply laminates with simple supports and angle‐ply laminates with clamped edges are presented as examples. The three dimensional elasticity solutions of cross‐ply laminates with simple supports are used to assess the accuracy of the present scheme. The effects of the span‐to‐thickness, aspect and material anisotropy ratio on the fundamental natural frequency are investigated. The present results are compared with the results in the published literature, and agree closely with the 3‐D elasticity solutions.     

A mixed finite element method for bending of plates

A simple mixed finite element method is developed. The finite element is a rectangular triangle and rectangle. In the element the deflections are assumed to be simple four-element polynomials, bending moments, M_x and M_y with a partially linear distribution, and a constant, M_xy, expressed in terms of the node deflections. The element matrix is of the order of 8 × 8. It is derived in a common engineering way. The unknowns are the deflections at the nodes and mid-diagonal, the two moments at the end of the diagonal and the two moments on the cathetus. The results obtained by this method show good convergence and an improvement in the accuracy of the moments as well as in the deflections, compared with results obtained by similar methods, such as those of Herrmann.     

Triangular finite element for analysis of thick laminated plates

The development of a general triangular C⁰ element, based on an assumed quadratic displacement potential energy approach, is presented for the analysis of arbitrarily laminated thick plates. The element formulation assumes transverse inextensibility and layerwise constant shear-angle. Convergence of transverse displacement, moments and stresses, the effects of two different Gauss quadrature schemes and comparison of the present solutions with the available analytical/finite-element results also form a part of the investigation. Furthermore, numerical results indicate close agreement between the LCST (layerwise constant shear-angle theory) and the three-dimensional elasticity theory with the length (or width) to thickness ratio as low as 4. Detailed comparison of the LCST-based finite-element solutions with those based on the CST (constant shear-angle theory) and the CLT (classical lamination theory) clearly demonstrates the superiority of the former over the latter two, especially in the prediction of the distribution of the in-plane displacements and stresses through the laminate thickness. This paper also introduces a new non-dimensionalized parameter, Δθ*, which is shown to be a very useful measure for classification of the laminated plates and the suitability of different plate theories over various ranges of length-to-thickness ratio.     

A 3D cylindrical finite element model for thick curved beam stress analysis

A finite element model is proposed to perform stress analysis for thick curved beams and panels subjected to various types of loadings. The model has 18 nodes in a three‐dimensional cylindrical co‐ordinates system. Three stress components on radial surface (σ_rr, τ_rθ, and τ_rz) and three displacement components (u_r, u_θ, and u_z) are used as nodal degrees of freedom. Therefore, the continuity condition for both stresses and displacements is achieved in the radial direction. Formulation of nodal shape functions and equilibrium equations are based on three‐dimensional elasticity theory and a minimum potential energy method. The accuracy of the method is verified with the standard test problems and exact solutions from the theory of elasticity. The model shows no locking phenomena. Convergence is investigated and the application to layered composite panel is illustrated. Copyright © 2003 John Wiley & Sons, Ltd.     

Higher-order finite element modelling of laminated composite plates

P-version finite elements based on higher-order theory are developed for the two-dimensional modelling of general bending and cylindrical bending of thin-to-thick laminated composite plates. In the case of general laminated plate elements, three displacement fields are used. In the special case of cylindrically bent laminated plate elements, two displacement fields are needed. In each case the displacement is expressed as the product of two functions—one in terms of out-of-plane co-ordinates alone and the other in terms of in-plane co-ordinates. The shape functions used to build the displacement fields are based on integral of Legendre' polynomials. The quality and performance of the elements are evaluated in terms of convergence characteristics of displacements and stresses. The predicted response quantities are compared with those available in the published literature based on analytical as well as conventional finite element models.     

Sub-region mixed finite element analysis of V-notched plates

In this paper the eigenproblem of elastic plates with V-notches is studied in terms of the complex potentials of elasticity. The variation of the eigenvalues as functions of notch angle is discussed. The phenomenon of bifurcation in the curves of higher-order eigenvalues is discovered and the concept of critical angle is proposed. Furthermore, a singular stress element, according to the stress field around notch-tips, is developed to account for notch-tip singularity. Moreover, conventional regular displacement elements are used outside the singular stress element, and then the basic finite element equations can be established based on the sub-region mixed energy principle. In two numerical examples, the stress intensity factors K _I and K _II of the notched specimens with various opening angles are evaluated, satisfactory accuracy can be obtained with very coarse meshes.     

Three-dimensional finite element analysis of cracked thick plates in bending

By using the finite element technique, stress intensity factors have been obtained for finite rectangular plates and the results have been given for various h/a, W/a and L/W ratios. By using a three-dimensional isoparametric element, the problem has been considered as a three-dimensional one and the variation of stress intensity factor across the plate thickness has been found to be nonlinear.     

A finite element model for Mindlin plates

In the general framework of Reissner-Mindlin theory, a plate model based on certain potential functions is discussed, together with its mechanical interpretation. A finite element implementation is also described and numerical results are reported.     

Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates

A class of mixed finite element methods for Reissner-Mindlin plates proposed by Arnold and Brezzi is considered. In these methods the shear energy term is split into two terms, leading to a partial selective reduced integration scheme. A parameter is involved in the splitting. In this paper an analysis of the behaviour of the approximate solution is performed in dependence of the parameter. Suggestions for a good choice of the parameter are also provided.     

Analysis of sandwich plates using a mixed finite element

The stress and displacement analysis of the thick sandwich plate is presented here by using an interlaminar stress mixed finite element based on local high-order deformable theory. The displacements of a sandwich plate are assumed to be high order polynomial functions layer-by-layer through the plate thickness. Since the interlaminar stresses at the interface between layers in this finite element scheme are regarded as primary variables, they can then be accurately determined. The accuracy of this finite element scheme is checked by comparing the present results with 3-D elasticity solutions of a simply supported sandwich plate. The response of a thick angle-ply, fiber-reinforced plastic (FRP) faced sandwich with fully simple supports, subjected to a sinusoidal distribution of transverse load is evaluated. The present finite element results are compared with results obtained from other finite element schemes.     

Physical shape functions in finite element analysis of moderately thick plates

A new rectangular finite element for moderately thick plates is presented. The element is based on the concept of physical shape functions, i.e. functions which contain in themselves physical and geometrical properties of the element. The analytic formulae of stiffness, geometric stiffness and mass matrices are presented for isotropic material. The element is free from locking and zero energy modes not corresponding to rigid body motions. Several examples are presented for static, initial stability and free vibration problems.     

A hierarchic 3D finite element for laminated composites

In this paper a new 3D multilayer element is presented for analysis of thick‐walled laminated composites. This element uses two steps to calculate the full stress tensor. In the first step the in‐plane stresses are computed from the material law using a displacement approximation, and then the transverse stresses are calculated from the 3D equilibrium equations. Since the 3D equilibrium equations require high‐order interpolation functions, a hierarchic interpolation of displacements is used. The new element is compared with existing ones, e.g. from MSC.MARC. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid finite element model for multilayered plates

We present a finite element model for multilayered plates, based on a primal-hybrid variational formulation. Namely, each layer is analyzed as it were a lonely structure, and the displacement continuity is imposed from one layer to the other by means of Lagrange multipliers. Then, a Mindlin-like displacement field is assumed for any layer; the resulting continuous problem is proven to be well-posed under rather general hypotheses. Finally, a finite element model is deduced, using a very simple scheme (piecewise linear approximation for the displacement components and piecewise constant Lagrange multipliers). The numerical results assess the good performance of the proposed model.     

A semi-analytical finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for laminated composite plates. The method is based on a mixed variational principle that involves both displacements and stresses. Finite element meshes are only used in the plane of plate, while the through thickness distributions of displacements and stresses are obtained using the method of state equations. Numerical results show that the rate of convergence of the new method is fast and the solutions can be very close to corresponding exact three-dimensional ones. The use of a recursive formulation of the state equations leads to an algebra equation system, from which solution are sought, whose dimension is independent of the numbers of layers of the plate considered.     

Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells

An elasto-plastic analysis of anisotropic plates and shells is undertaken by means of the finite element displacement method. A thick shell formulation accounting for shear deformation is considered, which is based on a degenerate three-dimensional continuum element. The accommodation of variable material properties, not only along the surface of the structure but also through the thickness, is made possible by a discrete layered approach. Although isoparametric elements of the Serendipity family give satisfactory solutions for thick and moderately thin shells the results exhibit ‘locking’ for an increasing ratio of span to thickness. To develop a numerical model which is applicable to thick or thin plates and shells, the nine-node Lagrangian element and the Heterosis element are also introduced into the present model. Plastic yielding is based on the Huber-Mises yield surface extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history. Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed.     

A triangular finite element for thin plates and shells

A finite element modelling technique which utilizes a triangular element with 45 degrees-of-freedom and seven-point integration has been tested for analysis of thin plate and shell structures. The element is based on the degenerate solid shell concept and the mixed formulation with assumed independent inplane and transverse shear strains. Numerical result indicates effectiveness of the present modelling technique which features combined use of elements with kinematic modes and those without kinematic modes in an attempt to eliminate both locking and spurious kinematic modes at global structural level.     

A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.     

A mixed finite element formulation for plate problems

A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.