A model of composite laminated Reddy beam based on a modified couple-stress theory

In this paper, a new anisotropic constitutive relation based on a modified couple-stress theory is defined for composite laminated Reddy beam. The theory contains only one material length-scale parameter in each ply of the composite laminated beam. The example of a cross-ply simply supported beam subjected to transverse load q₀sin(πx/L) is presented. Numerical results show that the proposed beam model can capture the scale effect of the microstructure. The proposed model can be reduced to several models of the modified couple-stress theory by adopting the assumptions in Timoshenko beam, Bernoulli–Euler beam and material isotropy. It can be seen that the deflections and stresses obtained by the proposed beam model are smaller than those based on Timoshenko and Bernoulli–Euler beam assumptions.     

A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates

A C⁰-type global-local higher order theory including interlaminar stress continuity is proposed for the cross-ply laminated composite and sandwich plates in this paper, which is able to a priori satisfy the continuity conditions of transverse shear stresses at interfaces. Moreover, total number of unknowns involved in the model is independent of number of layers. Compared to other higher-order theories satisfying the continuity conditions of transverse shear stresses at interfaces, merit of the proposed model is that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To verify the present model, a C⁰ three-node triangular element is used for bending analysis of laminated composite and sandwich plates. It ought to be shown that all variables involved in present model are discretized by only using linear interpolation functions within an element. Numerical results show that the C⁰ plate element based on the present theory may accurately calculate transverse shear stresses without any postprocessing, and the present results agree well with those obtained from the C¹-type higher order theory. Compared with the C¹ plate bending element, the present finite element is simple, convenient to use and accurate enough.     

A C-type higher-order theory for bending analysis of laminated composite and sandwich plates

In this paper, a C⁰-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C¹ interpolation functions. To assess the developed theory, the C¹-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

A model of composite laminated Reddy plate based on new modified couple stress theory

Based on new modified couple stress theory a model for composite laminated Reddy plate is developed in first time. In this theory a new curvature tensor is defined for establishing the constitutive relations of laminated plate. The characterization of anisotropy is incorporated into higher-order laminated plate theories based on the modified couple stress theory by Yang et al. in 2002. The form of new curvature tensor is asymmetric, however it can result in same as the symmetric curvature tensor in the isotropic elasticity. The present model of thick plate can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model for cross-ply composite laminated Reddy plate of couple stress theory with one material’s length constant is used to demonstrate the scale effects. Numerical results show that the present plate model can capture the scale effects of microstructure. Additionally, the present model of thick plate model can be degenerated to the model of composite cross-ply laminated Kirchhoff plate and Mindlin plate of couple stress theory.     

增强型Reddy层合梁理论与热应力分析

为推广Reddy理论准确分析复合材料层合/夹层结构的热变形和应力, 通过使用横法向热变形和自由表面条件, 发展了增强型Reddy层合梁理论。虽然考虑了横法向应变, 增强型Reddy理论位移变量数与Reddy理论相同。用虚功原理推导了复合材料层合梁平衡方程, 并分析了简支复合材料层合/夹层梁热膨胀问题。数值结果表明, 增强型Reddy理论能准确分析复合材料层合/夹层结构热膨胀问题, 而Reddy理论分析热膨胀问题误差较大。     

A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis

This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C¹ continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories.     

Bending of cross-ply laminated composites: An accurate and efficient plate theory based upon models of Lekhnitskii and Ren

Bending laminated composites results in a distinctive zig-zag shaped deformation pattern, accordingly jumping transverse shear strains at layer interfaces, but continuous courses of transverse shear stresses there. An accurate representation of this laminate-specific mechanical behavior in terms of plate theories is challenging, even more if computational efficiency is aimed for. Here, an axiomatic equivalent single layer plate theory for cross-ply laminated composites is presented, which is based on the work of Lekhnitskii and Ren and delivers accurate deformation and stress prognoses at the cost of six solution variables. Fulfilling transverse stress continuity, the infinitesimal equilibrium equations are considered in order to derive an appropriate ansatz for the transverse shear stresses including the influence of all plane stress reduced stiffness components. However, the effect of the normal stress σ_zz is neglected, and deflection w is assumed constant across the plate thickness. The equilibrium equations and corresponding boundary conditions of the plate theory are derived by application of the principle of virtual displacements. Numerical results for symmetrical and non-symmetrical composites as well as for typical sandwich plates obtained by the present theory show good agreement with corresponding exact elasticity solutions given by Pagano, even for thick plates.     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient FE model and Least Square Error method for accurate calculation of transverse shear stresses in composites and sandwich laminates

Accurate evaluation of transverse stresses in laminated composites and sandwich plates using 2D FE models involves cumbersome post-processing techniques. In this paper a simple and efficient method has been proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in composites and sandwich laminates by using a displacement based C⁰ FE model (2D) derived from Refined Higher Order Shear Deformation Theory (RHSDT) and a Least Square Error (LSE) method. The C⁰ FE model satisfies the inter-laminar shear stress continuity conditions at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C¹ continuity associated with the above plate theory (RHSDT). The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for calculation of transverse stresses in a composite laminate. In the proposed method, the first two equations of equilibrium are utilized to compute the transverse shear stress variation through the thickness of a laminated plate whereas the third equation of equilibrium gives the normal stress variation. Accuracy of the proposed method is demonstrated in the numerical examples through comparison of the present results with those obtained from different models based on higher order shear deformation theory (HSDT) and 3D elasticity solutions.     

Static Analysis of Laminated Composite Plates Using n-Order Shear Deformation Theory and a Meshless Global Collocation Method

In this article, an n-order shear deformation theory is used to analyze the static characteristic of laminated composite plates. The third-order theory of Reddy can be considered as a special case of present n-order theory (n = 3). Governing equations and boundary conditions expressed in terms of strong form based on the present n-order theory are discretized by a meshless global collocation method. Maximum deflection and stress of the simply-supported laminated plate under sinusoidal load are compared with available published results, which demonstrate the accuracy and efficiency of present n-order theory.     

Further results concerning the refined theory of anisotropic laminated composite plates

A simple refined discrete-layer theory of anisotropic laminated composite plates is substantiated. The theory is based on the assumption of a piecewise linear variation of the in-plane displacement components and of the constancy of the transverse displacement throughout the thickness of the laminate. This plate model incorporates transverse shear deformation, dynamic and thermal effects as well as the geometrical non-linearities and fulfills the continuity conditions for the displacement components and transverse shear stresses at the interfaces between laminae. As it is shown in the paper, the refinement implying the fulfillment of continuity conditions is not accompanied by an increase of the number of independent unknown functions, as implied in the standard first order transverse shear deformation theory. It is also shown that the within the framework of the linearized static counterpart of the theory, several theorems analogous to the ones in the 3-D elasticity theory could be established. These concern the energetic theorems, Betti's reciprocity theorem, the uniqueness theorem for the solutions of boundary-value problems of elastic composite plates, etc. Finally, comparative remarks on the present and standard first order transverse shear deformation theories are made and pertinent conclusions about its usefulness and further developments are outlined.     

A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates

In the present paper, a n-order model for functionally graded and composite sandwich plate is developed. This model uses the n-order polynomial term to represent the displacement field. Zero transverse shear stress boundary conditions at the top and bottom of the plate are satisfied. The third-order theory of Reddy [8] can be considered as a special case of present n-order theory. Natural frequencies of the functionally graded and composite plates with various side-to-thickness ratios, material properties are computed by present n-order theory with different n values. The results are compared with available published results which demonstrate the accuracy and efficiency of present n-order theory.     

A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of f_w = q₀ sin (πx/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory.     

A general doubly curved laminate shell theory

The aim of the work summarized in this paper is the theoretical modelling of laminated composite shells of arbitrary shape, in order to: improve the calculus of shear stresses, and especially to avoid shear correction factors; account for continuity conditions of displacements and transverse shear stresses at layer interfaces as well as compatibility conditions of transverse shear stresses on the bonding surfaces; estimate the relevance of shear refinements, of membrane refinements and of the transverse stretching beyond standard kinematics; and to analyse the sensitivity to edge effects for significant boundary conditions.     

Analysis of laminated composite plates using a higher-order shear deformation theory

A higher-order shear deformation theory is used to analyse laminated anisotropic composite plates for deflections, stresses, natural frequencies and buckling loads. The theory accounts for parabolic distribution of the transverse shear stresses, and requires no shear correction coefficients. A displacement finite element model of the theory is developed, and applications of the element to bending, Vibration and stability of laminated plates are discussed. The present solutions are compared with those obtained using the classical plate theory and the three-dimensional elasticity theory.     

Refined global–local higher‐order theory and finite element for laminated plates

Based on completely three‐dimensional elasticity theory, a refined global–local higher‐order theory is presented as enhanced version of the classical global–local theory proposed by Li and Liu (Int. J. Numer. Meth. Engng. 1997; 40 :1197–1212), in which the effect of transverse normal deformation is enhanced. Compared with the previous higher‐order theory, the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature. However, the previous higher‐order theory will encounter difficulty for this problem. A refined three‐noded triangular element satisfied the requirement of C¹ weak‐continuity conditions in the inter‐element is also presented. The results of numerical examples of moderately thick laminated plates and even thick plates with span/thickness ratios L/h = 2 are given to show that in‐plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique. In order to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. Copyright © 2006 John Wiley & Sons, Ltd.     

Effects of Boundary Conditions in Laminated Composite Plates Using Higher Order Shear Deformation Theory

This paper extends the applicability of a modified higher order shear deformation theory to accurately determine the in-plane and transverse shear stress distributions in an orthotropic laminated composite plate subjected to different boundary conditions. A simpler, two-dimensional, shear deformable, plate theory accompanied with an appropriate set of through-thickness variations, is used to accurately predict transverse shear stresses. A finite element code was developed based on a higher order shear deformation theory to study the effects of boundary conditions on the behavior of thin-to-thick anisotropic laminated composite plates. The code was verified against three dimensional elasticity results. The study also compared the stresses and deformation results of higher order theory with those obtained using commercial software such as LUSAS, ANSYS and ALGOR. The commercial software are heavily used by designers to design various components/products made of composites. Various combinations of fixed, clamped and simply supported boundary conditions were used to verify a large class of anticipated applications. Results obtained from software are in good agreement for some cases and significantly differ for others. It was found that LUSAS and ANSYS yield better results for transverse deflection and in-plane stresses. But for transverse shear stresses, it is highly dependent on boundary conditions.     

A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates

In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n = 3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.     

Finite element analysis of laminated composite plates using zeroth-order shear deformation theory

In this paper a generalized finite element model is developed for static and dynamic analyses of laminated composite plates using zeroth-order shear deformation theory (ZSDT). The theory ensures the parabolic distribution of transverse shear stresses across the plate thickness. A four-noded plate element is considered in this model and the generalized nodal variables are expressed using Lagrangian linear interpolation functions and Hermitian cubic interpolation functions. The solutions of the finite element model have been compared with the existing solutions for symmetric and antisymmetric laminated composite plates. The comparison confirms that the ZSDT can be efficiently used for finite element analysis of both thin and thick plates with high accuracy.