A solution of laminated cylindrical shells using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piece-wise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, the displacement functions remain to be unconstrained. This theory is very useful for the finite element analysis because it requests only C⁰ continuity for the assumed displacement fields. By comparing with three-dimensional elasticity theory for laminated orthotropic cylindrical shell, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for the thin and moderately thick shells.     

A quasi-conforming triangular laminated composite shell element based on a refined first-order theory

A quasi-conforming triangular laminated shell element based on a refined first-order shear deformation theory is presented. The Hu-Washizu variational principle, involving strain and displacement fields as variables, with stresses being considered as Lagrange multipliers, is used to develop the laminate composite shell element. Both strains and displacements are discretized in the element, while displacements alone are discretized at the boundary. The inter-element C ¹ continuity is satisfied a posteriori in a weak form. Due to the importance of rotations and shear deformation in the geometrically non-linear analyses of shells, 7 degrees of freedom per node are chosen, viz. three displacements, two first-derivatives in the in-plane directions of the out-of-plane displacement, and two transverse shear strains at each node. To consider the effect of transverse shear deformation on the global behavior of the laminated composite shell, the Reissner-Mindlin first-order theory, with shear correction factors of Chow and Whitney, is adopted. The transverse shear stresses are obtained through the integration of the 3-D equilibrium equations; and the warping induced by transverse shear is considered in the calculation of the in-plane stresses to improve their accuracy. Numerical examples show that the element has good convergence properties and leads to highly accurate stresses.     

层板弯曲与振动的近似分析

横向剪切变形对复合材料层合板弯曲与振动的影响甚大。在本文的近似分析中,假定板在弯曲时横向位移沿整个板厚为常量。横向剪切应变沿各层厚度方向也分别为常量,但各层不同。文中以特殊正交各向异性层合板为例,采用两种不同的方法建立了各层剪切应变间的关系,推演了层合板横向弯曲与振动的微分方程组及边界条件。算例表明,即使层合板的跨——厚比很小,用本文两种分析方案计算位移、应力及固有频率,都仍具有较高的精度。      

A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels

Based on the global-local superposition technique 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.], a global-local higher order laminated shell model is proposed for predicting both displacement and stress distributions through the thickness of laminated shells. This shell model satisfies transverse shear stress continuity conditions at interfaces as well as free surface conditions of transverse shear stresses. The merit of this model is that transverse shear stresses can be accurately predicted directly from constitutive equations without smoothing techniques. Cylindrical bending of laminated and sandwich shell panels is chosen to assess the present model wherein the results from several 2D laminated shell models and three-dimensional elasticity solution are available for comparison. In addition, thermal bending and thermal expansion of laminated cylindrical shell panels are also considered in this paper.     

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

ABSTRACT

We analyze static infinitesimal deformations of doubly curved shells using a third-order shear and normal deformable theory (TSNDT) and delineate effects of the curvilinear length/thickness ratio, a/h, radius of curvature/curvilinear length, R/a, and the ratio of the two principal radii on through-the-thickness stresses, strain energies of the in-plane and the transverse shear and normal deformations, and strain energies of stretching and bending deformations for loads that include uniform normal tractions on a major surface and equal and opposite tangential tractions on the two major surfaces. In the TSNDT the three displacement components at a point are represented as complete polynomials of degree three in the thickness coordinate. Advantages of the TSNDT include not needing a shear correction factor, allowing stresses for monolithic shells to be computed from the constitutive relation and the shell theory displacements, and considering general tractions on bounding surfaces. For laminated shells we use an equivalent single layer TSNDT and find the in-plane stresses from the constitutive relations and the transverse stresses with a one-step stress recovery scheme. The in-house developed finite element software is first verified by comparing displacements and stresses in the shell computed from it with those from either analytical or numerical solutions of the corresponding 3D problems. The strain energy of a spherical shell is found to approach that of a plate when R/a exceeds 10. For a thick clamped shell of aspect ratio 5 subjected to uniform normal traction on the outer surface, the in-plane and the transverse deformations contribute equally to the total strain energy for R/a greater than 5. However, for a cantilever shell of aspect ratio 5 subjected to equal and opposite uniform tangential tractions on the two major surfaces, the strain energy of in-plane deformations equals 95–98% of the total strain energy. Numerical results presented herein for several problems provide insights into different deformation modes, help designers decide when to consider effects of transverse deformations, and use the TSNDT for optimizing doubly curved shells.     

Inter-laminar stresses in a laminated shallow shell panel

The evaluation of inter-laminar shear stresses in laminated shells using 2D finite element models involves cumbersome post-processing techniques. In this paper, a simple and efficient method has been proposed for accurate evaluation of transverse shear stresses in laminated composite shells by using a displacement-based C⁰ FE model derived from higher-order shear deformation theory (HSDT) and a least square error (LSE) method. In order to include the effect of three curvature terms in the strain components of composite shells, Sander’s approximations are followed. In this model, the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C¹ continuity in the FE implementation associated with the present shell theory (HSDT). The LSE method is applied at the post-processing stage, after in-plane stresses are calculated by using the present FE model based on HSDT. Thus, the proposed method is quite simple compared to the usual method of integrating the 3D equilibrium equations for the calculation of transverse stresses in laminated composite shells. The accuracy of the method is demonstrated in the numerical examples by comparison of the present results with those obtained from different models based on HSDT, exact analytical and 3D elasticity solutions.     

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.     

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.     

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.     

复合材料层合板壳热弹性动力响应

本文提出了一个复合材料层合扁壳的位移场。据此导出了在力载和热载作用下的板壳运动方程。该位移场满足层间位移和横向剪应力的连续性,较充分反映了横向剪切对层合板壳力学行为的影响,但位移场的变量和运动方程均为五个,与一阶剪切变形理论相同。文中以暂态响应为例考察了横向剪切变形对层合板壳动力响应的影响,显着改善了解的精度。      

Bending theory of laminated plate

An exact solution is presented for the problem of an infinite cantilever anisotropic plate with only a single plane of elastic symmetry parallel to the xy-plane and with elastic coefficients changing through the thickness, and the bending theory of anisotropic laminate plates is established. The solution is obtained in an inverse form and is then used to obtain an exact solution for the infinite cantilever anisotropic laminated plate. On the basis of this solution, an assumption about the in-plane displacements is made for a laminated plate of N layers, each of which possesses only a single plane of elastic symmetry parallel to the xy-plane. Under this assumption the in-plane displacements and transverse shear stresses are continuous between layers. Using the principle of minimum potential energy, the equilibrium equations and boundary conditions are obtained which are similar to the classical theory of laminated plate. The closed form solutions are compared with the exact solutions. The results show that the bending theory is in excellent agreement with the exact solution.     

Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.     

对称层板的一种解法

复合材料多层粘合板, 当它在几何上及弹性上都具有对中面的对称性时, 就不要考虑面内应力与弯、扭应力的耦合效应。这时, 其分析方法与均质各向异性板的分析方法相同。但必须考虑横向剪切变形的影响, 本文提出从设定横向剪切应力的分布出发, 导出一种解法。算例说明这种方法的精度较好。      

Analysis for buckling and vibrations of composite stiffened shells and plates

This paper deals with development of triangular finite element for buckling and vibration analysis of laminated composite stiffened shells. For the laminated shell, an equivalent layer shell theory is employed. The first-order shear deformation theory including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear stresses a correction of transverse shear stiffness is employed. Based on the equivalent layer theory with six degrees of freedom (three displacements and three rotations), a finite element that ensures C⁰ continuity of the displacement and rotation fields across inter-element boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Results of vibration and buckling analysis of stiffened plates and shells are discussed.     

High-order free vibrations of doubly-curved sandwich panels with flexible core based on a refined three-layered theory

There are few reports on the free vibration of soft core doubly-curved sandwich shells. Previous studies are largely based on the equivalent single layer theories in which the natural frequencies are grossly overestimated. This study deals with the analytical free vibrations of doubly curved sandwich shell with flexible core based on a refined general-purpose sandwich panel theory. In this theory, equations of motion are formulated based on displacements and transverse stresses at the interfaces of the core. The first order shear deformation theory and assumptions of linear distribution of transverse normal stress and uniform shear stresses over the thickness of core (based on 3D-elasticity solution of weak core) are used in the present theory. In this model, the in-plane displacements take cubic polynomial distributions and the transverse displacement has a quadratic one thorough the core thickness. Hamilton’s principle is used to obtain the equations of motion. The obtained results are validated by the analytical and numerical results published in the literatures. Parametric study is also included to investigate the effects of radius of curvature, thickness and flexibility of core.     

基于混合分层理论的粘弹层合圆柱壳的阻尼振动和层间应力研究

应用混合分层理论,并在壳厚方向采用位移和应力插值函数推导出粘弹层合圆柱壳的动力学方程。计算了两层粘弹层合圆柱壳的振动频率和结构损耗因子,与Okazaki A的结果吻合良好。针对不同模量和厚度的粘弹性材料,计算出层合圆柱壳的层间横向应力的幅值。结果表明:过大的层间法向正应力是导致自由阻尼层合壳脱层破坏的主要因素,采用环向加强的粘弹性材料将有效地降低层间法向正应力的幅值。     

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.     

Delamination in composite plates: influence of shear deformability on interfacial debonding

An elastic interface model is introduced to investigate the effects of in-plane and out-plane shear stresses on interfacial debonding in laminated composite plates by means of the energy release rate concept. This is done by utilising an improved laminated plate model in which the Reissner–Mindlin kinematics type for each layers is coupled with an adhesion mechanism modelled by means of a linear interface model, acting in the opening and sliding failure mode directions. The problem is faced through an analytical solution procedure. Increasing the stiffnesses of the interface leads to restoring displacement continuity at the interface between layers and to recovering energy release rate components through the work performed by the singular stress field at the crack tip. In view of the great importance of shear deformation in laminated composite plates the effect of shear stresses on the mechanism of delamination are investigated pointing out new features which emerge from the interaction of normal and shear stresses acting on the transverse section near the crack tip. Several examples of mixed mode delamination schemes used in experimental applications are examined, showing the influence of transverse shear stresses in coupling with normal stresses on energy release rates determination.     

Bending,buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory

The bending, buckling and free vibration problems of non-homogeneous composite laminated cylindrical shells are considered. Hamilton–Reissner's mixed variational principle is used to deduce a consistent first-order theory of composite laminated cylindrical shells with non-homogeneous elastic properties. The governing equations with their required boundary conditions are derived without introducing any shear correction factors. Numerical results for the transverse deflections, stresses, natural frequencies and critical buckling loads are presented to show the advantages of this theory. The influences of the non-homogeneity and thickness ratio on the shell structural response are investigated. The study concludes that the inclusion of the non-homogeneity effect is required, even if it is weak, for predicting the actual structural response of the shells.     

Bending and transverse shear stresses in fiber-composite beams by the transformed-section method

A method to determine bending and transverse shear stress distributions in beams composed of fiber-composite layers is presented. The method is based on the transformed-section concept and takes into account the effects of Poisson's ratios and in-plane shear coupling. To demonstrate the feasibility of the transformed-section model, numerical solutions are presented for three rectangular and I-shaped cross-sections. It is shown that for bending moments the transformed-section model presented herein is entirely consistent with the lamination theory. The model can also be used to determine the transverse shear stresses, and it is shown that the accuracy of the results may be significantly affected when the effects of Poisson's ratios and in-plane shear are neglected in the analysis.