A state space finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for the stress analysis of cross-ply laminated composite plates. The method is based on a mixed variational principle that includes the variations of both displacements and stresses. Finite element approximation is introduced only for the in-plane variations of displacements and stresses, while the through-thickness distributions of them are obtained by using the method of state equation. Numerical tests show that the results obtained approach the analytical three-dimensional solutions. Moreover, the use of the recursive formulation of the state equation leads to the solution of an algebra equation system whose order does not depend on the number of material layers of the laminate. Compared with the traditional finite element method, the new solution always provides continuous distributions of both displacements and transverse stresses across material interfaces.     

Three-dimensional finite strip analysis of laminated panels

In this paper, a combined finite strip and state space approach is introduced to obtain three-dimensional solutions of laminated composite plates with simply supported ends. The finite strip method is used to present in-plane displacement and stress components, while the through-thickness components are obtained by using the method of state equation. The method can replace the traditional three-dimensional finite element solutions for structures that have regular geometric plans and simple boundary conditions, where a full three-dimensional finite element analysis is very often both extravagant and unnecessary. The new method provides results that show good agreement with available benchmark problems having different material compositions, thickness and boundary conditions. The new method provides a three-dimensional solution for laminated plates, while the advantages of using the traditional finite strip method are fully taken. This solution also yields a continuous transverse stress field across material interfaces that normally is not achievable by other numerical modelling of laminates, such as the traditional finite element method.     

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.     

Finite element analysis of bimodulus composite stiffened thin shells of revolution

This paper is a sequel to the work published by the first and third authors[l] on stiffened laminated shells of revolution made of unimodular materials (materials having identical properties in tension and compression). A finite element analysis of laminated bimodulus composite thin shells of revolution, reinforced by laminated bimodulus composite stiffeners is reported herein. A 48 dot doubly curved quadrilateral laminated anisotropic shell of revolution finite element and it's two compatible 16 dof stiffener finite elements namely: (i) a laminated anisotropic parallel circle stiffener element (PCSE) and (ii) a laminated anisotropic meridional stiffener element (MSE) have been used iteratively.The constitutive relationship of each layer is assumed to depend on whether the fiberdirection strain is tensile or compressive. The true state of strain or stress is realized when the locations of the neutral surfaces in the shell and the stiffeners remain unaltered (to a specified accuracy) between two successive iterations. The solutions for static loading of a stiffened plate, a stiffened cylindrical shell. and a stiffened spherical shell, all made of bimodulus composite materials, have been presented.     

Thermal stress analysis of laminated composite plates and shells

In this paper, Semiloof shell finite element formulation has been extended to thermal stress analysis of laminated plates and shells. The accuracy of the formulation has been verified using sample problems available in the literature. Thermal stresses in cross-ply and angle-ply laminated plates and shells subjected to thermal gradients across the thickness are presented for different boundary conditions, taking into account the temperature dependence of the material properties. The behaviour of laminates under thermal load is found to be different from that under mechanical loads in certain respects.     

A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates

In view of the increasing interest in using composite materials for aerospace structures, the analysis of laminated composite plates becomes essential. A three-dimensional eight-node hybrid stress finite element method is developed for the analysis of laminated plates. The hybrid stress model is based on the modified complementary energy principle and takes into account the transverse shear deformation effects. The displacement field is interpolated through shape functions and nodal displacements. All three displacement components are assumed to vary linearly through the thickness of each lamina. The stress field is interpolated through assumed stress polynomials with 55 stress parameters for each lamina. All six stresses are included and satisfy the homogeneous equilibrium equations. The validity of the hybrid stress finite element model is determined by comparing the predicted numerical results with the existing three-dimensional elasticity solutions. Excellent accuracy and fast convergence are observed in the numerical results.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

An equilibrium method for prediction of transverse shear stresses in a thick laminated plate

First two equations of equilibrium are utilized to compute the transverse shear stress variation through thickness of a thick laminated plate after in-plane stresses have been computed using an assumed quadratic displacement triangular element based on transverse inextensibility and layerwise constant shear angle theory (LCST). Centroid of the triangle is the point of exceptional accuracy for transverse shear stresses. Numerical results indicate close agreement with elasticity theory. An interesting comparison between the present theory and that based on assumed stress hybrid finite element approach suggests that the latter does not satisfy the condition of free normal traction at the edge. Comparison with numerical results obtained by using constant shear angle theory suggests that LCST is close to the elasticity solution while the CST is closer to classical (CLT) solution. It is also demonstrated that the reduced integration gives faster convergence when the present theory is applied to a thin plate.     

Bending analysis of composite laminated plates using a partially hybrid stress element with interlaminar continuity

A partially hybrid stress element for modelling composite laminated plates is developed based on the state space in which only the displacement components and the transverse stress components are assumed to be independent. This formulation satisfies exactly the interlaminar continuity requirements and the surface traction free conditions. It also combines the advantages of both the conventional displacement elements and the fully hybrid stress elements. Numerical examples are illustrated to investigate the accuracy, convergence and shear locking sensitivity of the present method. The nonlinear distributions of the normal and transverse stresses along the thickness direction are also especially studied.     

Analysis of laminated shells of revolution with laminated stiffeners using a doubly curved quadrilateral finite element

A finite element analysis of laminated shells of revolution reinforced with laminated stifieners is described here-in. A doubly curved quadrilateral laminated anisotropic shell of revolution finite element of 48 d.o.f. is used in conjunction with two stiffener elements of 16 d.o.f. namely: (i) A laminated anisotropic parallel circle stiffener element (PCSE); (ii) A laminated anisotropic meridional stiffener element (MSE).These stifiener elements are formulated under line member assumptions as degenerate cases of the quadrilateral shell element to achieve compatibility all along the shell-stifiener junction lines. The solutions to the problem of a stiffened cantilever cylindrical shell are used to check the correctness of the present program while it's capability is shown through the prediction of the behavior of an eccentrically stiffened laminated hyperboloidal shell.     

Transverse shear stiffness of laminated anisotropic shells

The additional constitutive equations required by transverse shear deformation theory of anisotropic heterogeneous shells are derived without the usual assumption of thickness distribution for either transverse shear stresses or strains. The derivation is based on Taylor series expansions about a generic point for stress resultants and couples which identically satisfy plate equilibrium equations. These equations give the in-surface stress resultants and couples in terms of the transverse shear stress resultants at the point and arbitrary constants, which may be interpreted as redundant “forces”. Starting from these expressions, we derive statically correct expressions (in terms of the transverse shear stress resultants and redundants) of the following variables: (1) in-surface stresses, using the stretching-bending constitutive equations and the Kirchhoff distributions of in-surface strains, (2) transverse shear stresses, by integration in the normal direction of the three-dimensional equilibrium equations, and (3) the area density of transverse shear strain energy, by integration in the normal direction of the corresponding volumetric density. Finally, by applying Castigliano's theorem of least work, the shear strain energy is minimized with respect to the redundants, thereby leading to the desired constitutive equations. Corresponding transverse shear stiffnesses are presented for several laminated walls, and reasonable agreement is obtained between transverse shear deformation plate theory using these stiffnesses and exact three-dimensional elasticity solutions for the problem of cylindrical bending of a plate.     

Analysis of laminated shells with laminated stiffeners using rectangular shell finite elements

A finite element analysis of laminated shells reinforced with laminated stiffeners is described in this paper. A rectangular laminated anisotropic shallow thin shell finite element of 48 d.o.f. is used in conjunction with a laminated anisotropic curved beam and shell stiffening finite element having 16 d.o.f. Compatibility between the shell and the stiffener is maintained all along their junction line. Some problems of symmetrically stiffened isotropic plates and shells have been solved to evaluate the performance of the present method. Behaviour of an eccentrically stiffened laminated cantilever cylindrical shell has been predicted to show the ability of the present program. General shells amenable to rectangular meshes can also be solved in a similar manner.     

A hybrid-stress finite-element formulation for thick multilayer laminates

A hybrid-stress-based finite-element formulation for thick multilayer laminates is presented and evaluated. Particular attention is given to the through-thickness distributions assumed for both stress and displacement components; high order through-thickness distributions are assumed within each layer in the present formulation. Attention is restricted to cylindrical bending of cross-ply laminates and a two-dimensional plane-strain element is derived. Results obtained for through-thickness distributions of stress and displacement are compared with the elasticity solutions for various numbers of layers and increasing thickness to span ratios. Good correlation between computed and elasticity solutions and the need for the inclusion of high-order through-thickness distributions are shown.     

Geometrically non-linear transient analysis of laminated composite and sandwich shells with a refined theory and C finite elements

A C⁰ continuous finite element formulation of a higher order shear deformation theory is presented for predicting the linear and geometrically non-linear, in the sense of von Karman, transient responses of composite and sandwich laminated shells. The displacement model accounts for the non-linear cubic variation of the tangential displacement components through the thickness of the shell and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and includes effects due to rotary inertia terms. Numerical results for central transverse deflection and stresses are presented for composite and sandwich laminated shells with various boundary conditions subjected to different types of loads and are compared with the results from other sources. Some new results are also included for future reference.     

Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

An asymptotic variational formulation for dynamic analysis of multilayered anisotropic plates

A multilevel variational formulation for dynamic analysis of multilayered anisotropic plates is developed within the framework of three-dimensional elasticity. By means of asymptotic expansions the Hellinger-Reissner functional for the elastodynamic problem is decomposed into a series of functionals with which a computational model can be constructed. In the formulation multiple time scales are introduced so that the secular terms can be eliminated systematically in obtaining a uniform expansion leading to valid asymptotic solution. Modifications to the approximation of various orders are determined by considering the solvability conditions of the higher-order equations. The model is adaptive, when combined with the finite element method, it has many appealing features, including that the displacements and transverse stresses may be interpolated independently, that the nodal degree-of-freedom (DOF) at each level is less than that of Kirchhoff plates, and that the mass and stiffness matrices generated at the leading-order level are always used at subsequent levels. Above all, the solution is three-dimensional in effect yet requires only two-dimensional interpolation. The through-thickness variations of the field variables are determined analytically with no need of interpolating in the thickness direction.     

A finite element study of the effect of the angle between two intersecting cylindrical shells subjected to internal pressure

The results of a study of the critical region of two intersecting cylindrical shells due to internal pressure loading are presented as a function of the angle between the two axes. The investigation was performed using the thin shell element of a three-dimensional finite element program. Three models with the angle between the axes of the two cylindrical shells equal to 30, 60, and 90°, were analyzed. In all of the three models, the diameter ratio of the two shells was 0.5283; the diameter to thickness ratio of the larger or main shell was 44.76, while the same ratio for the smaller or attached shell was 15.487. The results of these analyses show that the stresses in the critical intersection region are least when the two axes are perpendicular to each other; for other angular configurations, the stresses increase as the acute angle between the two axes decrease. This effect of inclination for pressure loading is just opposite of the effect found by authors [1] for external moments. In that case, for in-plane as well as out-ofplane moments, the stresses are larger for normally intersecting shells as compared to other angular configurations.     

Singular stresses in a finite region of two dissimilar viscoelastic materials with traction-free edges

Traditional finite element analyses of the stress state in regions with dissimilar viscoelastic materials are incapable of correctly resolving the stress state because of the unbounded nature of the stresses. A hybrid formulation is developed utilizing the exact solution for the stress and displacement fields based on the eigenfunction expansion method under general loading. The region has two dissimilar viscoelastic material wedges with perfect bonding, and is not limited to a particular geometric configuration. The solution method is based on the principle of work in conjunction with the use of Laplace transformation to eliminate time dependency. The strength of the singularity is obtained in the time space without resorting to approximate Laplace inversion techniques. However, the intensification of the stress components is obtained by employing an approximate inversion technique.     

Laminated composite structures by continuum-based shell elements with transverse deformation

In this paper a first-order shear/fourth-order transverse deformation theory of laminated composite shells is presented. A nonlinear continuum-based (degenerated 3D) finite element model with a strain/stress enhancement technique is developed in such a way that the nonzero surface traction boundary conditions and the interlaminar shear stress continuity conditions are all satisfied identically. Analytical integration through the shell thickness is performed. The resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the shell laminate characteristics in the normal direction can be evaluated precisely and the computational cost of the overall analysis is reduced. The numerical results are compared with analytical solutions and other finite element solutions to demonstrate the effectiveness of the theory and the computational procedure developed herein.