Free vibration analysis of a toroidal shell

The free vibration of a toroidal shell (curved pipe) is studied using DMV (Donnell-Mushtari-Vlasov) thin shell theory, Fourier's series and Galerkin's method. Some behaviors of the vibration of cylindrical shell, as a special case of toroidal shell, are discussed. Also, a 90-degree-bend curved pipe is studied numerically. Results obtained by the present method are compared to those obtained by finite element and other methods. DMV linear thin shell theory is employed for simplicity. However, the method applies to any other shell theories without difficulties.     

Simplified theoretical solution of circular toroidal shell with ribs under uniform external pressure

The toroidal shell with stiffened ribs is a new-style structure in ocean engineering especially in underwater engineering. This paper attempts to provide a simple theoretical method to obtain the stress solution of toroidal shell with ribs for its strength assessment. Firstly according to the structural property of toroidal shell with ribs and theory of curve-beam, a simple model for toroidal shell with ribs has been developed; then coupled with theory of thin-shell and elastic beam, its stress and deformation have been solved and can be expressed into analytic formulas; lastly by finite element method (FEM) and model experiment method, this simple theoretical solution has been verified to be reasonable and quite accurate. Thus this simple theoretical solution could be applied for analysis and design of pressure equipment in such toroidal structure type.     

Vibration of prestressed thin cylindrical shells conveying fluid

A general approach to modelling the vibration of prestressed thin cylindrical shells conveying fluid is presented. The steady flow of fluid is described by the classical potential flow theory, and the motion of the shell is represented by Sanders’ theory of thin shells. A strain–displacement relationship is deployed to derive the geometric stiffness matrix due to the initial stresses caused by hydrostatic pressure. Hydrodynamic pressure acting on the shell is developed through dynamic interfacial coupling conditions. The resulting equations governing the motion of the shell and fluid are solved by a finite element method. This model is subsequently used to investigate the small-vibration dynamic behaviour of prestressed thin cylindrical shells conveying fluid. It is validated by comparing the computed natural frequencies, within the linear region, with existing reported experimental results. The influence of initial tension, internal pressure, fluid flow velocity and the various geometric properties is also examined.     

Vibration analysis of toroidal panels using the differential quadrature method

The free vibration characteristics of linear elastic toroidal shell panels are determined. A solution based on the Mushtari–Vlasov–Donnell shell equations is developed using the Differential Quadrature Method. The work represents the first application of this method to problems in shell theory with variable coefficients in the governing equations. Numerical results are calculated using the method, and these are compared with results found using a Fourier series and a finite element solution.     

Natural frequencies and mode shapes of an orthotropic thin shell of revolution

A method is developed to determine the free vibration characteristics of an orthotropic thin shell of revolution of arbitrary meridian. A solution is given within the context of the Sanders–Budiansky shell theory and using the differential quadrature method (DQM). Numerical examples for frequencies and mode shapes are given for a complete toroidal shell. Both completely free shells, and shells with circumferential line supports are considered. Close agreement is observed in comparisons with previously published results and with results obtained using the finite element method. The paper ends with a set of appropriate conclusions.     

自平衡单元嵌入式网壳结构的预应力优化

自平衡单元嵌入式网壳结构中的自平衡单元由预应力提供刚度,采用广义逆法求解平衡矩阵,编制含线性不等式约束及线性等式约束的线性优化程序并结合整体可行预应力的概念来确定初始预应力的分布。对杂交结构平衡矩阵的形成、广义逆求解、线性约束优化方法的数学模型、约束矩阵的集成等关键技术给予详细的推导说明;同时,根据理论推导编写出相应的程序用于求解预应力值,并利用非线性有限元软件对分析结果进行验证。     

Comparison of elastic solutions for three-point bending of curved pipes

In this paper the linear elastic shell theory problem of the three-point bending of a curved pipe is considered. Such a loading arises in the industrial pipe ram bending process. Summaries are given for solutions to this problem based on the Mushtari-Vlasov-Donnel (MVD) and Sanders linear shell theories. Numerical results for displacements and stresses are obtained using the two shell theories, and these are compared with results from the finite element method (FEM). The present study gives practical information about the behavior of curved pipes subjected to ram bending. As well it provides information about the solution characteristics of thin-shell theories in toroidal coordinates.     

Coupled static and dynamic buckling of thin walled beam by spline finite element

For the coupled static and dynamic buckling of thin walled beam subjected to various forces, such as axial force, uniform bending moment, and bending moment due to concentrated and distributed lateral forces, the spline finite element method is employed to obtain the dynamic stiffness matrix. Second order effects of the axial force and moment are considered. A doubly symmetric cantilever beam with uniform cross-section is investigated. Extensive static and dynamic interaction diagrams are plotted. The effects of warping rigidity, torsional rigidity, axial tension and compression on moment buckling, moment on axial buckling compression, higher buckling modes are discussed in detail. The spline finite element method is proved to be very efficient for the present problem and many interaction diagrams can be plotted easily. Some new results are presented. The methodology is based on finite element formulation and therefore it can be easily extended to analyze structural frames.     

A nonlinear finite element computer program for thin-walled members

A nonlinear finite element computer program has been developed to analyse thin-walled metal structural members. The program has the ability to handle both geometrical and material nonlinearities so that the post-buckling behaviour and ultimate strength of members can be predicted. A bending-membrane rectangular element with six degrees of freedom at each node forms the basic type of element used in the program. Marguerre's shallow shell theory is adopted for the strain-displacement relationships and hence the bifurcation point at buckling can be bypassed by providing an initial inperfection. The finite element formulation is based on the total Lagrange coordinate system and the flow theory of plasticity. Explicitly shown in the paper is the formation of the tangent stiffness matrix and the tridiagonal block form of solution procedure. Two problems of a square tube and a channel section beam subjected to pure bending were tested and found to be in close agreement with previous theoretical work.     

Geometric non-linear analysis of thin plate structures using the harmonic coupled finite strip method

The paper describes a modified finite strip method embracing the harmonic coupled Fourier series treatment. The well known uncoupled formulation, first developed in the context of thin plate bending analysis, represents a semi-analytical finite element process. As far as linear analysis is concerned, it takes advantage of the orthogonality properties of harmonic functions in the stiffness matrix formulation. However in the case of the geometric stiffness matrix calculation, the integral expressions contain the products of trigonometric functions with higher-order exponents, and here the orthogonality characteristics are no longer valid. All harmonics are coupled, and the stiffness matrix order and bandwidth are proportional to the number of harmonics used.     

Static and vibration analysis of orthotropic toroidal shells of variable thickness by differential quadrature

In this study the static and free vibration characteristics of linear elastic orthotropic toroidal shells of variable thickness are considered. A solution based on the Sanders-Budiansky shell equations is developed. The semi-analytical differential quadrature method in which Fourier series are written in the circumferential direction is adopted. This approach reduces the computational work to a series of one-dimensional problems. A novelty in the solution concerns the use of power series as trial functions in a domain exhibiting cyclic periodicity. Using the developed theory numerical results are determined for two separate applications. The results obtained are compared with results from the finite element method, and conclusions are drawn.     

基于等参元理论的钢筋混凝土任意截面的刚度计算

根据有限单元法中的等参元理论,把笛卡尔坐标系下的任意形状截面变换为自然坐标系下的规则形状单元,导出自然坐标系下的钢筋混凝土截面的应变场和刚度矩阵表达式,使用高斯积分法计算截面刚度矩阵各个元素,从而建立计算任意形状截面刚度的通用计算模型。计算过程简单、算法可移植性好、精度较高。用所提出方法对钢筋混凝土双向偏压截面进行非线性全过程分析,计算结果与试验结果吻合良好。     

Nonlinear inelastic response history analysis of steel frame structures using plastic-zone method

A beam–column element formulation and solution procedure for nonlinear inelastic analysis of planar steel frame structures under dynamic loadings is presented. The spread of plasticity is considered by tracing the uniaxial stress–strain relationship of each fiber on the cross section of sub-elements. An elastic perfectly-plastic material model with linear strain hardening is employed for deriving a nonlinear elemental stiffness matrix, which directly takes into account geometric nonlinearity and gradual yielding. A solution procedure based on the combination of the Hilber–Hughes–Taylor method and the Newton–Raphson method is proposed for solving nonlinear equations of motion. The nonlinear inelastic time-history responses predicted by the proposed program compare well with those given by the commercial finite element package known as ABAQUS. Shaking table tests of a two-story steel frame were carried out with an aim to clarify the inelastic behavior of the frame subjected to earthquakes generated by the proposed program. A more practical analysis method for seismic design can be developed by comparing it with the presented frames for verification.     

Static and dynamic loading of an ovaloid toroidal tank

In this study, two problems of static and dynamic loading for an ovaloid toroidal tank are considered. In the first problem, a linear elastic three-dimensional finite element analysis is carried out to determine the stress concentration in a pressurized ovaloid toroidal tank with a nozzle. For validation, comparison is made with previous results for a cylindrical shell with a nozzle. An analysis is carried out to indicate the influence of the nozzle location on the stress concentration. Results are provided for three ovaloid toroidal tanks, to indicate the variation of the stress concentration with nozzle geometry. In the second problem, a non-linear elastic–plastic shell-theory finite element analysis is carried out to determine the displacements arising in an ovaloid toroidal tank due to impact by a flat-nosed projectile. For validation, comparison is made with previous results for a spherical shell impacted by a projectile. Results are provided for three ovaloid toroidal tanks, impacted at the extrados or the crown, and these results are compared with results for equivalent shells of related geometry. Conclusions are drawn, noting characteristics of ovaloid toroidal tanks relevant to their design.     

Linear elastic isoparametric spline finite strip analysis of perforated thin-walled structures

This paper describes the application of the isoparametric spline finite strip method to the linear elastic analysis of tri-dimensional perforated folded plate structures. The general theory of the isoparametric spline finite strip method is introduced. Kinematics assumptions and the procedure for combining in-plane (membrane) and bending effects are set out. Particular attention is paid to the procedure for rotating the stiffness matrix and load vector from local to global coordinates. The reliability of the method is demonstrated by comparisons with finely meshed finite element analysis results. Square stiffened perforated plates in compression and bending are analysed.     

板锥网壳结构的拟三层壳分析法

板锥网壳结构是一种受力性能合理 ,技术经济效益良好的新型空间结构形式。本文对板锥网壳结构提出了一种连续化的分析方法 ,根据其受力特点按照刚度等效的原理 ,采用弹性力学的基本方法推导出板锥网壳结构的等代壳体刚度 ,将其连续化为能共同作用的特殊形式的三层薄壳 ,按薄壳理论建立其位移法和混合法的基本微分方程。通过对微分方程的求解 ,计算其整体位移及结构内力。该法具有一定的精度 ,可宏观地了解结构的力学性能 ,并可用于板锥网壳结构的初步设计。     

Orthotropic von Karman plates using higher order elements

The nonlinear behaviour of orthotropic plates subject to transverse loading is studied in this paper using higher order finite elements. The formulation procedure for elements with up to 25 nodes is established and derivation of the tangential stiffness matrix based on the classical von Karman nonlinear theory is presented. These elements are derived by employing a high order polynomial displacement function to represent the in-plane and the bending displacements. By adopting separate variables for the transverse deflections and the rotations of the plate, the effect of transverse shear deformation can be included in modelling thin to moderately thick plates, thus enabling the effect of transverse shear on the nonlinear behaviour of plates to be included. The Newton-Raphson method for solving nonlinear equations is adopted to obtain the final state of equilibrium of the system. Several numerical examples are presented and the results are compared with available experimental data as well as analytical and numerical solutions obtaned by other conventional methods. Satisfactory agreement on the nonlinear performance of the plates as predicted by the present method is demonstrated.     

Free vibration analysis of inflatable beam made of orthotropic woven fabric

The free vibration of inflatable beams was studied using the dynamic stiffness method. A 3D Timoshenko beam with a homogeneous orthotropic woven fabric (OWF) was considered. Using the usual total Lagrangian form of the virtual work principle, the model took the geometric nonlinearities and the inflation pressure follower force effect into account. The nonlinear equilibrium equations were then linearized around the prestressed reference configuration. The exact dynamic stiffness matrix was developed by directly solving the governing differential equations of a 3D loaded inflatable beam in a free vibration. The effects of the inflation pressure, fabric mechanical properties and the boundary conditions on the natural frequencies and mode shapes of the inflatable beams were demonstrated. The proposed model was validated favorably through its comparison with a 3D thin shell finite element model and an isotropic fabric model found in the literature.     

组合肋壳几何非线性分析

提出了一种球面组合肋壳截面形式,基于非线性有限元理论,采用空间梁单元和等厚曲边壳单元,考虑空间结构的几何非线性影响,推导了T．L坐标系下组合肋壳非线性单元刚度矩阵。分别讨论了在不同矢跨比、不同荷载作用、不同边界条件下组合肋壳的极限承载力和失稳形态。由大量算例得到了非线性极限承载力与线性极限承载力的比值为0．66及不同情况下结构承载力变化趋势,组合肋壳的极限承载力比混凝土壳承载力提高20％。     

基于组合-分层壳元理论薄壁预应力混凝土结构的力学响应

基于组合-分层壳单元,建立薄壁预应力混凝土结构的计算模型,并对其力学行为进行分析。首先引入壳元理论,对薄壁结构进行壳元离散。并对壳元中的混凝土应用分层理论描述,对壳元中的钢筋单元采用大变形杆单元模拟。根据钢筋和混凝土在壳元内的位移协调条件,推导了整体转换矩阵后,将两者组合成一个单元,同时基于虚功原理推导了钢筋对组合壳元整体刚度矩阵的贡献。算例分析表明,本文方法的计算结果与已有的试验结果吻合良好,本文研究的组合壳元模型能适应钢筋的任意布置方式,能较全面地反映混凝土内钢筋的力学效应,数值计算稳定性良好,弥补了商用有限元软件非线性计算稳定性较差这一缺点。组合-分层壳元法为薄壁预应力混凝土结构提供了一种有效的分析方法。