Assessment of shell theories for the static analysis of cross-ply laminated circular cylindrical shells

This paper examines the accuracy of classical shell theories (CST) according to Flugge, Sanders, Love and Donnell, with respect to the recently available three-dimensional elasticity solution, for cross-ply laminated circular cylindrical shells under static loads. Further, a study has also been made to examine to what extent incorporation of first order shear deformation (FSDT), in aforementioned shell theories, improves the results. In general, all the basic equations (for both CST and FSDT), of aforementioned shell theories, have been presented in a unified form using tracer coefficients. A Navier type solution has been used to analyse both a simply supported circular cylindrical shell of revolution and an all round simply supported circular cylindrical shell panel. A parametric study has been carried out keeping in view the lamination schemes and geometrical parameters of the shell. From the detailed comparisons of the results it has been shown that (i) Donnell's theory (CST and FSDT) could be in error for certain lamination schemes and geometrical parameters and (ii) improved results for stresses and displacements could be obtained by incorporating shear deformation on more accurate theory like Flugge (CST).     

薄壁空心球节点承载能力的非线性数值分析

现行网架规程中的焊接空心球承载力设计公式是以试验为基础的经验公式 ,对以稳定性控制的球壳 ,难以求得较为精确的屈曲临界点。应用对称旋转薄壳屈曲的非线性有限元理论 ,对空心球节点进行大位移弹塑性屈曲分析 ,得出了可供设计参考的空心球承载力数据表 ,并回归出适用于大直径空心球计算的近似公式 ,以确定空心球节点的失效状态及临界荷载。     

Analysis and optimization of laminated composite circular cylindrical shell subjected to compressive axial and transverse transient dynamic loads

Optimization is one of the important stages in the design process. In this paper the genetic algorithms method is applied for weight and transient dynamic response and two constraints including critical buckling loads and principle strains optimization of laminated composite cylindrical shells. The multi-objective function seeks the minimum structural weight and transient dynamic response. Nine design variables including material properties (fibre and matrix), volume fraction of fibre, fibre orientation and thickness of each layer are considered. In analytical solution, vibration of composite circular cylindrical shells are investigated based on the first-order shear deformation shell theory. The boundary conditions are assumed to be fully simply support. The dynamic response of the composite shells is studied under transverse impulse and axial compressive loads. The modal technique is used to develop the analytical solution of the composite cylindrical shell. The solution for the shell under the given loading conditions can be found using the convolution integrals. An example of simply supported laminated composite cylindrical shells is given to demonstrate the optimality of the solution obtained by the genetic algorithms technique. Results are shown that the weight coefficient of multi-objective function and the type of the constraints have considerable effect on the optimum weight and dynamic response.     

非对称荷载对大跨度非规则单层网壳结构性能的影响

以某大跨度非规则单层网壳为研究背景,应用ANSYS通用有限元分析软件对网壳结构进行了建模并重点模拟了风荷载、雪荷载、温度效应等多种非对称荷载作用,进行了静力分析、特征值屈曲分析以及非线性稳定分析。得到了网壳结构在15种典型荷载组合下的内力、位移结果,分析了非对称荷载对网壳结构静力性能的影响;考虑了几何非线性、材料非线性、初始缺陷,得到了网壳结构在非对称荷载作用下的稳定系数,分析了非对称荷载对网壳结构稳定性能的影响。强调指出非对称荷载作用在大跨度单层网壳结构设计中的重要性,为大跨度非规则单层网壳设计提供参考。     

Transient dynamic response of composite circular cylindrical shells under radial impulse load and axial compressive loads

Free and forced vibration of composite circular cylindrical shells are investigated based on the first love's approximation theory using the first-order shear deformation shell theory. The boundary conditions (BCs) are considered as clamped-free edges. The dynamic response of the composite shells is studied under transverse impulse and axial compressive loads. The axial compressive load was less than critical buckling loads. The modal technique is used to develop the analytical solution of the composite cylindrical shell. The solution for the shell under the given loading conditions can be found using the convolution integrals. The effect of fiber orientation, axial load, and some of the geometric parameters on the time response of the shells has been shown. The results show that dynamic responses are governed primarily by natural period of the structure. The accuracy of the analysis has been examined by comparing results with those available in the literature and experiments.     

屋面系统对单层球面网壳稳定性及地震响应的影响研究

单层球面网壳的静力稳定性及抗震性能已经取得较为系统的研究成果,但在传统的研究方法中,学者们只考虑结构部分,忽略了屋面板、吊顶、隔墙等非结构构件的影响;其中,屋面系统作为一类非结构构件,其平面刚度较大、在动力荷载下的阻尼也较大的特点会在一定程度上影响结构的受力特性。因此,选取考虑屋面系统的单层球面网壳作为研究对象,开展屋面系统对于单层球面网壳弹塑性稳定性及抗震性能的影响研究。采用有限元分析软件ABAQUS分别建立带屋面系统单层球面网壳和传统单层球壳的数值模型;通过弧长法分别对两种模型进行均布荷载作用下的非线性数值模拟,对比分析屋面系统对单层球面网壳弹塑性稳定承载力的影响;对考虑屋面系统的网壳结构进行不同地震荷载作用下的时程动力分析,采用基于响应的分析方法,考察屋面系统对于网壳结构抗震性能的影响,得到屋面系统会显著提高网壳整体刚度、增强结构抗震能力的结论;本研究可为以后网壳结构的理论研究及工程实践提供一定的参考。     

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.     

薄壁钢梁稳定性计算的争议及其解决

在规范修订过程中,对钢梁的整体稳定性计算,有学者对《钢结构设计规范》(GBJ17—88)所依据的临界弯矩计算公式提出了质疑,并提出了新的计算公式。本文对规范所依据的公式和新提出的公式所依据的理论进行了对比,对新公式所依据的理论提出了疑问。为避免薄壁构件理论引入的各种假设带来思辨上难以解决的争议,本文采用ANSYS通用有限元程序的三维板壳单元SHELL63进行了6组共36根梁的整体失稳分析。有限元分析结果表明,采用板稳定理论求得的梁临界弯矩与《钢结构设计规范》(GBJ17—88)所依据的公式符合较好,而与新提出的公式比较相差较大。本文根据薄壁梁弯扭稳定理论,推导了考虑非线性正应变和剪应变的梁失稳过程应变能的变化公式,从而解释了考虑非线性剪应变的理论与ANSYS分析结果不符的原因,并对传统的稳定理论得到的结果进行了肯定。     

Buckling of laminated skew plates

A general high precision triangular plate bending finite element has been extended to the buckling analysis of laminated skew plates. This procedure involves development of the transformation matrix between global and local degrees of freedom for nodes lying on the skew edges and suitable transformation of the element matrices. The accuracy of the present formulation has been verified against literature values. New results are obtained for antisymmetric angle-ply and cross-ply laminated skew plates. In this analysis, the critical buckling loads for different skew angles with various lamination parameters, such as number of layers, fibre orientation angle, different boundary (simply supported, clamped) and loading (uniaxial, biaxial) conditions, have been presented.     

Instability of cracked CFRP composite cylindrical shells under combined loading

Numerical analysis of cracked composite cylindrical shells under combined loading is carried out to study the effect of crack size and orientation on the buckling behavior of laminated composite cylindrical shells. The interaction buckling curves of cracked laminated composite cylinders subject to different combinations of axial compression, torsion, internal pressure and external pressure are obtained, using the finite element method. In general, the internal pressure increases the critical buckling load of the CFRP cylindrical shells while torsion and external pressure decrease it. Numerical analyses show that axial crack has the most detrimental effect on the buckling load of a cylindrical shell while for cylindrical shells under combined external pressure and axial load, the global buckling shape is insensitive to the crack length and crack orientation.     

复合圆柱形壳在径向冲击荷载和轴压荷载作用下的瞬时动力响应

基于首次逼近理论,利用第一阶剪切变形理论对圆柱形壳的自由和强迫振动进行分析。边界条件（BCs）考虑为悬臂状态。分析复合壳体在横向冲击和轴向压力作用下的动力响应（轴压荷载小于临界屈曲荷载）,同时对复合柱形壳进行了建模分析。利用卷积积分对给定荷载状况下的壳体进行分析。揭示纤维方向、轴向荷载及一些几何参数对壳体时间响应的影响。结果表明：动力响应主要由结构的自振周期所控制。     

Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell

This paper presents a finite element formulation of Koiter's initial post-buckling theory using a multi-mode approach. Initial post-buckling theory provides direct information about the imperfection sensitivity of a structure under compressive loading, and is also the basis of a nonlinear reduced order model. The objective of the present work is to illustrate the capability of the implementation for buckling analysis of shell structures including modal interaction. A coupled mode initial post-buckling analysis for a composite cylindrical shell under axial compression, including the effect of a nonlinear pre-buckling state, has been carried out using a small number of representative modes. For small imperfection amplitudes the limit-point buckling loads obtained with the reduced order model compare reasonably well with full model nonlinear analysis, illustrating that a fast prediction of the coupled mode response of imperfect shells is possible using the approach proposed.     

薄壁槽形截面压杆弹性整体屈曲数值计算研究

为研究薄壁槽钢整体屈曲下的数值算法的适用性,以4种不同杆端约束下的薄壁槽形截面压杆为研究对象,根据广义梁理论确定其发生弹性整体屈曲的有效长度及临界荷载,与开口薄壁理想压杆的线性及非线性解析解进行对比,得到一致的临界荷载。利用ABAQUS有限元分析软件建立壳体有限元模型,采用杆端截面形心集中加载,依据约束情况确定杆端约束,以及根据有效长度施加不同初始缺陷的计算模型,开展特征值算法和隐式弧长法数值计算分析。计算结果表明:当开口薄壁压杆的有效长度足够大时,临界荷载的特征值数值解与理论解一致,弧长法计算得到的非线性解误差相对较大。当有效长度不够大时,弧长法计算得到的极限荷载与理论解基本一致,且弧长法可有效追踪后屈曲路径。     

Transient vibrations of laminated composite cylindrical shells exposed to underwater shock waves

The problem of the transient vibration of an elastic laminated composite cylindrical shell with infinite length exposed to an underwater shock wave is solved approximately. The linear acoustic plane wave assumption and Sanders thin shell theory are adopted. The reflected-afterflow virtual-source (RAVS) procedure is used to model the fluid–structure interaction involved during the underwater shock event. For the validity of the present analysis, the response of a laminated cylindrical shell under step plane wave is first analyzed and compared with the numerical solution available in the literature. Detailed numerical results for the transient responses of the shells under an exponentially decaying underwater shock wave are presented, and the influences of fiber angle, shell radius and thickness upon the dimensionless radial velocity, mid-surface strain, 0th mode radial displacement and 1st mode radial velocity of the shells, are investigated.     

Influence of geometric imperfections on the load capacity of orthotropic stiffened and composite shells of revolution with arbitrary meridians and boundary conditions

A stiffness matrix for an element of a shell of revolution has been derived, considering arbitrary load distributions and initial geometric imperfections. This element-stiffness matrix is based on the transfer-matrix method and describes the whole section of a shell of revolution between two rings in modal coordinates (a so-called super-element). The modal coordinates here are circumferential Fourier members, thus reducing the partial differential equations to ordinary ones.

Several stability analyses investigating the sensitivity of composite shells to different geometric imperfection shapes were carried out. The influence of the load distribution and boundary conditions in combination with geometric imperfections was analysed by different modellings of a hypothetical Jupe Avant shell of the ARIANE 5 rocket.     

Buckling strength of the cylindrical shell and tank subjected to axially compressive loads

This paper aims to develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Both geometrically perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluate buckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the parametric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases. These results are different from those in the case of columns. The buckling strength of the perfect tank placed on an extremely soft foundation and a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Convenient and sufficiently accurate design equations and charts used for estimating buckling strength are provided.     

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C₀ finite element formulation has been done to overcome the problem of C₁ continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.     

Buckling analysis of a cooling tower shell with measured and theoretically-modelled imperfections

Geometrical imperfections were measured using photogrammetric techniques on an existing reinforced concrete cooling tower shell. The imperfections, related to the radii of such a real shell, were used as input data to create a real shape of the cooling tower. Numerical analysis was carried out for three models: (P) perfect shell of revolution, (M) shell with measured imperfections, (T) shell with a theoretical imperfection corresponding to the primary buckling mode under dead load. The buckling analysis was related to the linearized eigenvalue problem of elastic shells. The shell midsurface was approximated by eight-node quadrilateral isoparametric finite elements. Computations were carried out using the ANKA computer code. Critical values of the load parameter enable confirmation of a partial correlation between existing imperfections and buckling modes under dead load. The most disadvantageous direction of the wind load application on the real shell was found, in order to evaluate the decrease in the load-carrying capacity of the cooling tower shell against buckling. Theoretically modelled imperfections give rather unrealistic values of buckling loads of the real shell.     

Buckling and postbuckling of cylindrical shells with one end pinned and the other end free

Using finite element analysis, this paper examines the linear bifurcation buckling loads, and nonlinear collapse loads, of cylindrical shells with one end pinned and the other end free, under a variety of axial and pressure load combinations. The pinned end is formulated so as to provide no axial restraint. For the bifurcation analysis, loads are related back to the classical solutions for cylinder buckling loads, to explain the very low values found for this set of boundary conditions.The nonlinear analysis includes both imperfections and material plasticity. In this analysis, it is found that cylindrical shells with pinned-free boundary conditions are notably imperfection insensitive, and for a range of geometries are able to reach collapse loads significantly greater than their bifurcation load. For other geometries, collapse loads very close to the bifurcation load are found. This unusual imperfection insensitivity for a cylindrical shell is explained in terms of the large flexibility engendered by the pinned-free boundary conditions and the oval buckling mode.