A mesh-free method for analysis of the thermal and mechanical buckling of functionally graded cylindrical shell panels

The buckling response of functionally graded ceramic-metal cylindrical shell panels under axial compression and thermal load is presented here. The formulation is based on the first-order shear deformation shell theory and element-free kp-Ritz method. The material properties of shell panels are assumed to vary through their thickness direction according to a power-law distribution of the volume fraction of constituents. Approximations of the displacement field are expressed in terms of a set of mesh-free kernel particle functions. A stabilized conforming nodal integration approach is employed to estimate the bending stiffness, and the shear and membrane terms are evaluated using a direct nodal integration technique to eliminate membrane and shear locking for very thin shells. The mechanical and thermal buckling responses of functionally graded shell panels are investigated, and the influences of the volume fraction exponent, boundary conditions, and temperature distribution on their buckling strengths are also examined.     

Reissner-Mindlin板壳无网格法的闭锁与灵敏度分析及优化的研究

该文利用罚函数法施加边界条件,建立了Reissner-Mindlin板壳无网格法的离散形式,通过数值锁死试验,探讨了EFG法、RPIM以及基于节点积分的无网格法在解决Reissner-Mindlin板壳闭锁问题中所存在的优缺点。所得结果表明,基于匹配近似场和节点积分方案的无网格法在处理剪切闭锁问题时具有优越性。然后以SCNI-MLS无网格法为基础,对Reissner-Mindlin板壳结构的尺寸、形状和轮廓设计进行了统一的设计灵敏度分析,结合约束变尺度序列二次规划法,完成了SCNI-MLS无网格法壳结构优化设计的算例,算例结果验证了所建立灵敏度分析的精度和优化方法的可行性。     

A generalized displacement method for the finite element analysis of thin shells

The shear locking problem for the bilinear degenerated thick shell elements, when used in the context of thin shell structures, can be overcome by a generalized displacement method presented in this paper. The transverse shear energy in the degenerated thick shell elements is totally suppressed by introducing discrete Kirchhoff constraints in each element. The constrained variational problem based on the nodal displacement space is transformed into an unconstrained one based on a so-called generalized displacement subspace. It is shown that shear looking phenomena completely disappear and no degradation of results is observed as the ratio of thickness to span approaches zero.     

新型协同转动六节点三边形复合材料壳单元

为分析复合材料层合板壳结构,提出了一种协同转动六节点三边形复合材料曲壳单元。不同于现有的其它协同转动有限单元:1) 该单元中采用了增量可加的矢量型转动变量,因而在非线性增量求解过程中更新节点转动变量非常简单;2) 在计算应变能对局部节点变量的二阶偏微分时,微分的次序是可以交换的,并且通过链式微分计算应变能对整体节点变量的二阶偏微分时,微分的次序也是可以交换的,因此,得到的局部和整体坐标系下的切线刚度矩阵都是对称的;3) 在此有限单元公式中引入了混合公式法,以减轻膜闭锁和剪切闭锁的不利影响。对4个典型算例进行了分析,并与其他文献的结果进行对比,该文提出的单元的可靠性和计算效率得到了验证。     

Postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads

This paper presents a postbuckling analysis of functionally graded cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method. The formulation is developed to handle problems of small strains and moderate rotations, based on the first-order shear deformation shell theory and von Kármán strains. The effective material properties of the shells are assumed to be continuous along their thickness direction, and are obtained using a power-law distribution of the volume fractions of the constituents. The approximations of the two-dimensional displacement fields are expressed in terms of a set of mesh-free kernel particle functions. The system bending stiffness is evaluated using a stabilized conforming nodal integration method and the membrane and shear terms are estimated using direct nodal integration to eliminate shear locking. The postbuckling path is traced using a combination of the arc-length and mesh-free kp-Ritz methods. The proposed formulation is validated by comparing the results of the proposed method with those in the literature. The postbuckling responses of two types of functionally graded conical shells, one composed of Al/ZrO₂ and the other of SUS304/Si₃N₄, are investigated and the effects of volume fraction, boundary condition, and length-to-thickness ratio on postbuckling behavior are discussed in detail.     

A Continuum Based Thick Shell Element for Large Deformation Analysis of Layered Composites

A new continuum based thick shell model is presented for modeling orthotropic laminated shell structures undergoing large elastic deformations. An equivalent single-layer model involving seven nodal degrees of freedom is used. In that layered model, there are no restrictions on the number of layers, their thickness and their stacking sequence. The shell model accounts explicitly for the thickness change in the shell, as well as the normal stress and strain states through its thickness. Shear locking is avoided using an assumed natural strain formulation, while thickness locking is avoided using modified displacement interpolation functions. The performance of the layered shell element is tested using several linear and non-linear composite plate and shell problems involving anisotropic, angle and cross-ply laminates, cylindrical and spherical shells.     

A C0 three-node shell element for non-linear structural analysis

A C⁰ three-node shell finite element well suited to non-linear calculations is proposed. The element is based on Mindlin kinematics and the degenerated solid approach. Linear Lagrange functions are used for geometry and displacement interpolations. The formulation is made in the natural material frame. A strain interpolation avoids shear locking and an intermediate material frame related to the element sides is introduced in order to fix nodal transverse shear strain components. The modifications of strain interpolations concern both the non-linear and linear parts of strain and are taken into account in ail calculations, among others in the expression of the initial stress stiffness matrix. A single set of integration points on the normal at the centre of gravity is sufficient, which is very interesting for numerical efficiency especially in the case of non-linear analyses.     

新型协同转动3节点三边形壳单元

发展了一种新型3节点三边形壳单元。计算单元在局部坐标系下的节点变量时,通过采用协同转动法,预先扣除节点整体变量中的刚体转动成分,从而简化了单元的计算公式。不同于现有的其他协同转动单元,在该单元中采用了增量可以直接累加的矢量型转动变量,单元的切线刚度矩阵可以通过直接计算能量泛函对节点变量的二阶偏微分得到,且对节点变量的偏微分次序是可以互换的,因而在局部和整体坐标系下都得到了对称的单元切线刚度矩阵。为消除单元中可能出现的闭锁现象,引入了MacNeal提出的线积分法,分别用沿单元边线方向的膜应变和剪切应变构造新的假定应变场。最后,通过对几个产生了大位移与大转角变形的板壳问题进行分析,检验了该单元的可靠性、计算精度和计算效率。     

Geometric nonlinear analysis of plates and cylindrical shells via a linearly conforming radial point interpolation method

In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander’s nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton–Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.     

Machine locking of degenerated thin shell elements

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena. This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.     

Development of geometrically exact new shell elements based on general curvilinear co‐ordinates

In the present study first‐order shear deformable shell finite elements based on general curvilinear co‐ordinates are proposed. For the development of the present shell elements, a partial mixed variational functional with independently assumed strains is provided in order to avoid the severe locking troubles known as transverse shear and membrane lockings. Bubble functions are included in the shape function of displacement to improve the performance of the developed element. The proposed assumed strain four‐ and nine‐node elements based on the general tensor shell theory provide an efficient linkage framework for shell surface modelling and finite element analysis. In the several benchmark problems, the present shell elements with exact geometric representations demonstrate their performance compared to previously reported results. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient co‐rotational formulation for curved triangular shell element

A 6‐node curved triangular shell element formulation based on a co‐rotational framework is proposed to solve large‐displacement and large‐rotation problems, in which part of the rigid‐body translations and all rigid‐body rotations in the global co‐ordinate system are excluded in calculating the element strain energy. Thus, an element‐independent formulation is achieved. Besides three translational displacement variables, two components of the mid‐surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out‐of‐plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co‐rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Shear‐flexible subdivision shells

We present a new shell model and an accompanying discretisation scheme that is suitable for thin and thick shells. The deformed configuration of the shell is parameterised using the mid‐surface position vector and an additional shear vector for describing the out‐of‐plane shear deformations. In the limit of vanishing thickness, the shear vector is identically zero and the Kirchhoff–Love model is recovered. Importantly, there are no compatibility constraints to be satisfied by the shape functions used for discretising the mid‐surface and the shear vector. The mid‐surface has to be interpolated with smooth C¹‐continuous shape functions, whereas the shear vector can be interpolated with C⁰‐continuous shape functions. In the present paper, the mid‐surface as well as the shear vector are interpolated with smooth subdivision shape functions. The resulting finite elements are suitable for thin and thick shells and do not exhibit shear locking. The good performance of the proposed formulation is demonstrated with a number of linear and geometrically non‐linear plate and shell examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of cylindrical shell panels. A meshless solution

The Meshless Analog Equation Method, a purely meshless method, is applied to the static analysis of cylindrical shell panels. The method is based on the concept of the analog equation of Katsikadelis, which converts the three governing partial differential equations in terms of displacements into three substitute equations, two of second order and one fourth order, under fictitious sources. The fictitious sources are represented by series of radial basis functions of multiquadric type. Thus the substitute equations can be directly integrated. This integration allows the representation of the sought solution by new radial basis functions, which approximate accurately not only the displacements but also their derivatives involved in the governing equations. This permits a strong formulation of the problem. Thus, inserting the approximate solution in the differential equations and in the associated boundary conditions and collocating at a predefined set of mesh-free nodal points, a system of linear equations is obtained, which gives the expansion coefficients of radial basis functions series that represent the solution. The minimization of the total potential of the shell results in the optimal choice of the shape parameter of the radial basis functions. The method is illustrated by analyzing several shell panels. The studied examples demonstrate the efficiency and the accuracy of the presented method.     

Extended rotation-free shell triangles with transverse shear deformation effects

The paper extends recent work of the authors to include transverse shear effects on rotation-free triangular element for plates (O?ate and Zárate in Int J Numer Methods Eng 83(2):196–227, 2010). Two new shell triangular elements are presented, the EBST+ and the EBST+1. Transverse shear deformation effects are important for thick shells, as well when the shell is laminated or formed by composite material. The ingredients for the element formulation are: a Hu-Washizu type mixed functional and linear interpolation for the displacement field. In both elements presented a finite volume approach is used for computing the bending moments and the curvatures over a patch of elements. The nodal translational degrees of freedom of the original enhanced basic shell triangle (EBST) are extended with the two shear deformation angles via two different approaches. The first one uses a linear interpolation of the rotation angles inside the element (EBST+) and the second one assumes a constant field for the rotation angles (EBST+1). For the thin shell case the shear angles vanish and the new elements reproduce the good behaviour of the original thin EBST element. As a consequence the elements can reproduce the solutions for thick to thin shells situations without exhibiting shear locking. The numerical solution for the thick shell case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin EBST element. Examples of the good performance of the new rotation-free shell triangles are given.     

用于光滑/非光滑壳的稳定化新型协同转动4节点四边形壳单元

该文发展了一种适用于光滑壳和非光滑壳的新型协同转动4节点四边形壳单元。在单元中每个节点采用了3个平动自由度和2/3个矢量型转动自由度,每个光滑壳的节点或非光滑壳的非交界节点采用壳中性面法向矢量的2个最小分量作为矢量型转动变量,在非光滑壳中性面交界线上的节点采用3个矢量型转动变量,他们分别是节点定向矢量组中一个定向矢量的较小或最小分量和另一定向矢量的2个最小分量。在非线性增量求解过程中,这些矢量型转动变量可以采用简单的加法将增量累加到原矢量中直接进行更新,且采用了协同转动框架的单元在局部和整体坐标系下得到的切线刚度矩阵都是对称的,结构整体切线刚度矩阵可以采用一维线性存储,可节省大量的计算机存储资源和计算时间。为消除膜闭锁和剪切闭锁的不利影响,采用单点积分方案计算单元内力矢量和切线刚度矩阵,并借鉴Belytschko提出的物理稳定化零能模态控制法来消除可能出现的零能模态。通过对2个光滑壳和2个非光滑壳进行非线性分析,检验了单元的可靠性、计算效率与计算精度。     

Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis

A degenerated shell element with composite implicit time integration scheme is developed in the present paper to solve the geometric nonlinear large deformation and dynamics problems of shell structures. The degenerated shell element is established based on the eight‐node solid element, where the nodal forces, mass matrices, and stiffness matrices are firstly obtained upon virtual velocity principle and then translated to the shell element. The strain field is modified based on the mixed interpolation of tensorial components method to eliminate the shear locking, and the constitutive relation is modified to satisfy the shell assumptions. A simple and practical computational method for nonlinear dynamic response is developed by embedding the composite implicit time integration scheme into the degenerated shell element, where the composite scheme combines the trapezoidal rule with the three‐point backward Euler method. The developed approach can not only keep the momentum and energy conservation and decay the high frequency modes but also lead to a symmetrical stiffness matrix. Numerical results show that the developed degenerated shell element with the composite implicit time integration scheme is capable of solving the geometric nonlinear large deformation and dynamics problems of the shell structures with momentum and energy conservation and/or decay. Copyright © 2015 John Wiley & Sons, Ltd.     

A general high‐order finite element formulation for shells at large strains and finite rotations

For hyperelastic shells with finite rotations and large strains a p‐finite element formulation is presented accommodating general kinematic assumptions, interpolation polynomials and particularly general three‐dimensional hyperelastic constitutive laws. This goal is achieved by hierarchical, high‐order shell models. The tangent stiffness matrices for the hierarchical shell models are derived by computer algebra. Both non‐hierarchical, nodal as well as hierarchical element shape functions are admissible. Numerical experiments show the high‐order formulation to be less prone to locking effects. Copyright © 2003 John Wiley & Sons, Ltd.     

THE FOUR-NODE C0 SHELL ELEMENT REFORMULATED

A reformulated four-node shell element, based on the analysis of the moment redistribution mechanism development by C⁰ plate bending and shell elements, is presented. The moment redistribution mechanism of a finite shell element model is shown to be predominantly activated by the membrane flexural action of the shell. This action is triggered through the membrane strain components which participate in the moment equilibrium equations of the finite element assembly system. An equivalent elastic foundation action, along with the activation of the in-plane twisting stiffness of the shell, may also contribute to the moment redistribution mechanism of the finite shell element model. The proposed shell element formulation aims at retaining the non-spurious contribution of the transverse shear/membrane strain energy to the flexural behaviour of the shell, through the activation of the moment redistribution mechanism. Yet, any potentially spurious, whether locking or kinematic, mechanism is rejected. In warped configurations, the element activates appropriate coupling mechanisms of the bending terms to nodal translations. The so-obtained reformulated four-node shell element exhibits an excellent behaviour without experiencing any locking phenomena or zero-energy modes, while its formulation is kept simple, based on physical considerations. The proposed formulation performs equally well in flat as well as in warped shell element applications.