A partial hybrid stress solid-shell element for the analysis of laminated composites

In this paper we introduce a low order partial hybrid stress solid-shell element based on the composite energy functional for the analysis of laminated composite structures. This solid-shell element has eight nodes with only displacement degrees of freedoms, and three-dimensional constitutive models can be directly employed in the present formulation without any additional treatment. The assumed interlaminar stress field provides very accurate interlaminar stress calculation through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of the laminate. The present solid-shell does not show the transverse shear, trapezoidal and thickness locking phenomenon, and passes both the membrane and the bending patch tests. To assess the present formulation’s accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, shell and laminated composite analyses are investigated and the results are compared with those available in the literature. The numerical results show the accuracy of the presented solid-shell element for the analysis of laminated composites.     

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.     

Hybrid-interface element for thick laminated composite plates

A novel 27-node three-dimensional hexahedral hybrid-interface finite element (FE) model has been presented to analyze laminated composite plates and sandwich plates using the minimum potential energy principle. Fundamental elasticity relationship between components of stress, strain and displacement fields are maintained throughout the elastic continuum as the transverse stress components have been invoked as nodal degrees of freedom. Continuity of the transverse stresses at lamina interface has been maintained. Each lamina is modeled by using hybrid-interface elements at the top and the bottom interfaces and conventional displacement based elements sandwiched between these interfaces. Results obtained from the present formulation have found to be in excellent agreement with the elasticity solutions for thin and thick composite cross-ply, angle-ply laminates, as well as sandwich plates. Additional results have also been presented on the variation of the transverse strains to highlight magnitude of discontinuity in these quantities due to difference in properties of face and core materials of sandwich plates. Present formulation can be used effectively to interface hybrid formulation that uses transverse stresses and displacements as degrees of freedom with conventional purely displacement based formulation for realistic estimates of the transverse stresses.     

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     

Three-dimensional finite-element analysis of laminated composites

A three-dimensional finite-element analysis treating the mechanical response of thick laminated composite plates in bending is presented. An isoparametric solid element with a cubic displacement expansion in planform and a linear variation through the thickness is used to model each layer of the laminate. The degrees-of-freedom of the element are retained at its boundaries so that interconnections between lamina with different fiber orientations can be made at their interfaces. An incore version of the conjugate gradient technique, which does not have bandwidth restrictions, is used to minimize the total potential energy of the system with the number of iterations to convergence being about one-fifth the total global degrees-of-freedom. Because a three-dimensional analysis is used, the effects of thickness-stretching, transverse shear, extension, and bending deformations are obtained. Comparisons with three-dimensional elasticity solutions are in excellent agreement and show the necessity of having individual elements for each layer when they have different fiber orientations and when the plates are thick.     

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.     

A mixed FSDT finite element for monoclinic laminated plates

A 4-node finite element for the analysis of laminated composite plates with monoclinic layers, as it occurs for example in piezoelectric applications, is developed. The element is built through the linked interpolation scheme proposed by Taylor and Auricchio [Int J Numer Meth Eng 1993;36:3057–66] and is a generalization of the element presented in [Auricchio F, Sacco E. A mixed-enhanced finite-element for the analysis of laminated composite plates. Int J Numer Meth Eng 1999;44:1481–1504]. Starting from a first-order shear deformation theory (FSDT), a mixed-enhanced variational formulation is considered. It includes as primary variables the resultant shear stresses as well as enhanced incompatible modes, which are introduced to improve in-plane deformations. Bubble functions for rotation degrees of freedom and functions linking transversal displacement to rotations are employed. The solvability of the variational formulation is proved whereas effectiveness and convergence of the proposed finite element are confirmed through several numerical applications. Finally, numerical results are compared with the corresponding analytical solutions as well as to other finite-element solutions.     

Refined three-dimensional finite strip model for analysis of thick composite sandwich plates

A refined three-dimensional finite strip model for stress analysis of thick hybrid composite sandwich plates is presented in the present article. The new model is based on a combination of two different strip elements: quadratic linear skin strip (QLSS), and modified quadratic cubic core strip (MQCCS). The applications of the proposed model for sandwich plates containing a local damage under membrane or bending loading are illustrated. Numerical results are compared to independent solutions and measured data and found to be in excellent agreement. A comprehensive discussion of the numerical aspects of the method is conducted to test the strips' characteristics regarding the spectral accuracy, finite element convergence and accuracy of numerical integrations.     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

A three-dimensional state space finite element solution for laminated composite cylindrical shells

On the basis of the theory of three-dimensional elasticity, this paper presents a state space finite element solution for stress analysis of cross-ply laminated composite shells. This is a continuation of the authors’ previously published work on laminated plates [Compos. Struct. 57 (1–4) (2002) 117; Comput. Methods Appl. Mech. Engrg. 191 (37–38) (2002) 4259]. Once again a state space formulation is introduced to solve for through-thickness stress distributions, while the traditional finite elements are used to approximate the in-surface variations of state variables. A three-dimensional laminated shell element is established in an arbitrary orthogonal curvilinear coordinate system, while the application of the element is shown by calculating stresses in laminated cylindrical shells. Compared with the traditional finite element method, the new solution provides accurate continuous through-thickness distributions of both displacements and transverse stresses.     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

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.     

Vibration and damping analysis of composite plates using finite elements with layerwise in-plane displacements

Finite element analysis is performed to study the effects of layerwise in-plane displacements on fundamental frequencies and specific damping capacity for composite laminated plates. The cross-ply and angle-ply composite laminated plates with simply-supported boundary conditions are considered. The strain energies of each stress component are computed to quantify the amount of the transverse shear deformations for thin and thick plates. The results show that the length-to-thickness ratios, cross-ply ratios, and fiber orientations have a great influence on the in-plane displacement responses. It is also shown that the layerwise in-plane displacements should be taken into account for the dynamic analysis of thick composite plate.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.     

Micromechanics of braided composites via multivariable FEM

A new method for micromechanical properties of three-dimensionally braided composite materials via homogenization theory and incompatible multivariable FEM is proposed in this paper. An incompatible displacement element and a hybrid stress element are developed to model the effective mechanical properties of three-dimensional textile composites. Some illustrative applications are presented for a typical class of four-step braided composites. Results of the hybrid stress element approach compare more favorably with the experimental data than other numerical methods widely used.     

Nonlinear transient responses of initially stressed composite plates

The nonlinear transient response of initially stressed composite plates is investigated using the finite element method. A nine-node isoparametric quadrilateral element is developed to model laminated plates under initial deformation and initial stress according to the Mindlin plate theory and von Karman large deflection assumptions. In the time integration, the Newmark constant acceleration method in conjunction with an efficient and accurate iteration scheme is used. Numerical results for deflections and bending moments for isotropic and laminated plates are obtained.     

A class of C finite elements for the static and dynamic analysis of laminated plates

A general higher-order deformation theory is developed to analyse the behaviour of an arbitrary laminated fibre-reinforced composite plate. Three-dimensional effects such as the warping of sections and the presence of interlaminar stress field components are taken into account assuming a power series expansion of displacements along the thickness. A class of C⁰ finite element models based on this theory is then developed for mono- and bi-dimensional elements. Applications of the models to bending and vibration of laminated plates are then discussed. The present solutions are compared with those obtained using the three-dimensional elasticity theory, classical laminate theory and other higher-order theories.     

Piecewise hierarchical p-version axisymmetric solid element for laminated composites

This paper presents a piecewise hierarchical p-version finite element formulation for laminated composites axisymmetric solids for linear static analysis. The element formulation incorporates higher order deformation theories and is in total agreement with the physics of deformation in laminated composites. The element geometry is defined by eight nodes located on the boundaries of the element. The lamina thicknesses are used to create a nine-node p-version configuration for each lamina of the element. The displacement approximation for the element is piecewise hierarchical and is developed by first establishing a hierarchical displacement approximation for the nine-node configuration of each lamina of the laminate and then imposing interlamina continuity condition of displacements at the interfaces between laminas. The hierarchical approximation functions and the corresponding nodal variables for each lamina are derived from the Lagrange family of interpolation functions and can be of arbitrary polynomial order p_c and ^kp_η in the ε and ^kη directions for a typical lamina k. The formulation ensures C⁰ continuity, i.e., continuity of displacement across interelement as well as interlamina boundaries.

The element properties are constructed by assembling individual lamina properties which are derived using the principle of virtual work and the hierarchical displacement approximation for the laminas. Transformation matrices, formed based on interlamina continuity conditions, are used to transform each lamina's degrees of freedom into the degrees of freedom for the laminate. Thus, each individual lamina stiffness matrix and equivalent load vector are transformed and then summed to establish the laminate stiffness matrix and equivalent load vector. There is no restriction on either the number of laminas or their lay-up pattern. Each lamina can be generally orthotropic and the material directions and the layer thickness may vary from point to point within each lamina.

Numerical examples are presented to demonstrate the effectiveness, modeling convenience, accuracy, and overall superiority of the present formulation for laminated composite axisymmetric solids and shells.     

Three dimensional interlaminar stress analysis at free edges of composite laminate

A composite “multilayer element” based on an assumed hybrid-stress finite element model is presented. The stress field in each layer is derived from an assumed in-plane strain such that it satisfies equilibrium conditions within each layer and the interlaminar traction reciprocity conditions. The interelement boundary displacement is assumed to enforce the interelement traction reciprocity. The complex three dimensional stress analysis can thus be easily performed for realistic composite laminates, especially for the computation of the interlaminar stresses at straight and/or curved free edges. Excellent agreement between the present computed results and referenced solutions shows the high accuracy and efficiency of this work.