Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

Finite deformation plate theory and large rigid-body motions

A refined non-linear first-order theory of multilayered anisotropic plates undergoing finite deformations is elaborated. The effects of the transverse shear and transverse normal strains, and laminated anisotropic material response are included. On the basis of this theory, a simple and efficient finite element model in conjunction with the total Lagrangian formulation and Newton-Raphson method is developed. The precise representation of large rigid-body motions in the displacement patterns of the proposed plate elements is also considered. This consideration requires the development of the strain-displacement equations of the finite deformation plate theory with regard to their consistency with the arbitrarily large rigid-body motions. The fundamental unknowns consist of six displacements and 11 strains of the face planes of the plate, and 11 stress resultants. The element characteristic arrays are obtained by using the Hu-Washizu mixed variational principle. To demonstrate the accuracy and efficiency of this formulation and compare its performance with other non-linear finite element models reported in the literature, extensive numerical studies are presented.     

带旋转自由度C^0类任意四边形板（壳）单元

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｉｌｉｎ板弯曲理论和Ｖｏｎ－Ｋａｒｍａｎ大挠度理论,采用单元域内和边界位移插值一致性的概念,将四节点等参弯曲单元与Ａｌｌｍａｎ膜变形二次插值模式相结合,对层合板壳的大挠度分析提供了一种实用的带旋转自由度的四节点Ｃ＾０类板单元。大量算例表明：该单元对板壳结构的线性强度、稳定性和后屈曲分析都表现出良好的收敛性和足够的工程精度。     

Buckling and post-buckling of laminated composite plates

This paper addresses the buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with Green–Lagrange non-linear strain–displacement relationships. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strain conditions at the top and bottom surfaces of the plate in von Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model.     

梁板壳的几何大变形--从近似的非线性理论到有限变形理论

对梁板壳的线性理论、近似几何非线性理论与有限变形理论作了比较,介绍了有限转动理论,指出了应用有限变形理论求解梁板壳的大变形问题的高效率、高精度的巨大优越性。     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

板壳有限变形精确理论及有限元列式

以位移型退化壳理论为基础,提出了板壳模型的有限变形精确理论.该理论引入转动伪矢量概念,充分发挥刚性线段的运动学模型和有限转动矩阵的作用,精确表达非线性位移-应变关系,抛弃以往的小位移、小应变、小转动增量乃至小加载步长等各种简化假设,并严格遵循能量共轭关系建立有限元平衡方程,能够可靠和有效地用于板壳结构的大位移、大转动分析.算例表明,该文列式在弹性范围内具明显的平方收敛效率,并且计算结果与加载步长无关.由于不受小加载步长、小转动增量等的限制,实现了板壳几何非线性问题的大步长加载 .     

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.     

Enhanced first-order theory based on mixed formulation and transverse normal effect

An accurate prediction of displacements and stresses for laminated and sandwich plates is presented using an enhanced first-order plate theory based on the mixed variational theorem (EFSDTM) developed in this paper. In the mixed formulation, transverse shear stresses based on an efficient higher-order plate theory (EHOPT) developed by Cho and Parmerter [Cho, M., Parmerter, R.R., 1993. Efficient higher-order composite plate theory for general lamination configurations. AIAA Journal 31, 1299–1306] are utilized and modified to satisfy prescribed lateral conditions, and displacements are assumed to be those of a first-order shear deformation theory (FSDT). Relationships between the modified EHOPT and the FSDT are systematically derived via both the mixed variational theorem and the least-square approximation of difference between in-plane stresses including the transverse normal stress effect. It is shown that the transverse normal stress effect should be considered in predicting the in-plane stresses when the Poisson effect is dominant. The developed EFSDTM preserves the computational advantage of the classical FSDT while allowing for important local through-the-thickness variations of displacements and stresses through the recovery procedure. The accuracy and efficiency of the present theory are assessed by comparing its results with various plate models as well as the three-dimensional exact solutions for thick laminated and sandwich plates.     

Finite elements based on a first-order shear deformation moderate rotation shell theory with applications to the analysis of composite structures

The first-order shear deformation moderate rotation shell theory of Schmidt and Reddy [R. Schmidt and J. N. Reddy, J. Appl. Mech. 55, 611–617 (1988)] is used as a basis for the development of finite element models for the analysis of the static, geometrically non-linear response of anisotropic and laminated structures. The incremental, total Lagrangian formulation of the theory is developed, and numerical solutions are obtained by using the isoparametric Lagrangian 9-node and Serendipity 8-node shell finite elements. Various integration schemes (full, selective reduced, and uniformly reduced integration) are applied in order to detect and to overcome the effects of shear and membrane locking on the predicted structural response. A number of sample problems of isotropic, orthotropic, and multi-layered structures are presented to show the accuracy of the present theory. The von Kármán-type first-order shear deformation shell theory and continuum 2D theory are used for comparative analyses.     

Novel shear deformation model for moderately thick and deep laminated composite conoidal shell

This article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C¹ continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.     

Non-linear static-dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain-displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

Non-linear analysis of functionally graded fiber reinforced composite laminated plates,Part I: Theory and solutions

This paper deals with the large amplitude vibration, non-linear bending and postbuckling of fiber reinforced composite laminated plates resting on an elastic foundation in hygrothermal environments. Two kinds of fiber reinforced laminated plates, namely, uniformly distributed and functionally graded reinforcements, are considered. The material properties of fiber reinforced laminated plates are estimated through a micromechanical model and are assumed to be temperature-dependent and moisture-dependent. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the hygrothermal effect. A two-step perturbation technique is employed to determine the non-linear to linear frequency ratios of plate vibration, the load-deflection and load-bending moment curves of plate bending, and postbuckling equilibrium paths of laminated plates.     

Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.     

Axisymmetric static and dynamic buckling of laminated thick truncated conical cap

Axisymmetric buckling analysis is presented for moderately thick laminated shallow, truncated conical caps under transverse load. Buckling under uniformly distributed loads and ring loads applied statically or as step function loads is considered. Marguerre-type, first-order shear deformation shallow shell theory is formulated in terms of transverse deflection w, the rotation ψ of the normal to the mid-surface and the stress function Φ. The governing equations are solved by the orthogonal point collocation method. Truncated conical caps with a circular opening, which is either free or plugged by a rigid central mass, have been analysed for clamped and simple supports with movable and immovable edge conditions. Typical numerical results are presented illustrating the effect of various parameters.     

考虑横法向热应变的C~0型Reddy板理论和三角形板单元

考虑横法向热变形,建议了C0型Reddy理论,并用于分析复合材料层合/夹层板热膨胀问题。虽然考虑了横法向热应变,但不增加额外的位移变量。此理论位移场不含有横向位移一阶导数,构造有限元时仅需C0插值函数。基于这一模型,运用虚位移原理推导了复合材料板平衡方程以及构造了6节点三角形板单元,并分析了简支复合材料层合/夹层板的热膨胀问题。数值结果表明,建立的模型能准确分析复合材料层合/夹层板热膨胀问题,而忽略横法向热应变的理论分析热膨胀问题误差较大。     

NONLINEAR ANALYSIS OF DYNAMIC STABILITY FOR LAMINATED CIRCULAR CYLINDRICAL THICK SHELLS WITH ENDS ELASTICALLY SUPPORTED AGAINST ROTATION

Based on Timoshenko-Mindlin kinematic hypotheses and Hamilton'sprinciple,a dynamic non-linear theory for general laminated circular cylindrical shellswith transverse shear deformation is developed.A multi-mode solution for periodic in-plane loads is formulated for the non-linear dynamic stability of an anti-symmetricangle-ply cylinder with its ends elastically restrained against rotation.The resultedequations in terms of time function are solved by the incremental harmonic balancemethod.     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

Influences of large deformation and rotary inertia on wave propagation in piezoelectric cylindrically laminated shells in thermal environment

Influences of large deformation (geometrical non-linear) and rotary inertia on wave propagation in a long, piezoelectric cylindrically laminated shell in thermal environment is presented in this paper. Nonlinear dynamic governing equations of piezoelectric cylindrically laminated shells are derived by means of Hamilton’s principle. The wave propagation modes are obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in piezoelectric cylindrically laminated shells are relates to the large deformation, rotary inertia and thermal environment of the piezoelectric cylindrically laminated shells. The effect of large deformation, rotary inertia and thermal load on wave propagation in the piezoelectric cylindrically laminated shells is discussed by comparing with the result from the small deformation (geometrical linear shell theory). This method may be used to investigate wave propagation in various laminated material, layers numbers and thickness of piezoelectric cylindrically laminated shells under large deformation. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.