Dynamic stiffness analysis of toroidal shells

The free vibration of a toroidal shell is studied using the dynamic stiffness method. The dynamic stiffness method eliminates both spatial discretization error and mesh generation. Moreover, with a finite number of degrees of freedom, the dynamic stiffness method can predict an infinite number of natural frequencies. The dynamic behavior of the toroidal shell is modeled by DMV (Donnell-Mushtari-Vlasov) linear thin shell theory in the present paper. However, the procedure can be adapted to be used with any other linear thin shell theory without difficulty. Since a close form solution of toroidal shell using DMV theory is not (yet) possible, in order to obtain the desired dynamic stiffness matrix, a finite number of Fourier's series terms are taken in the circumferential direction and the unknown longitudinal displacements are then solved from the reduced governing equations exactly. The solution obtained from the dynamic stiffness method can be regarded as semi-analytical due to the Fourier approximation. With the dynamic stiffness matrices in hands, a toroidal shell with different boundary conditions and connections (to other toroidal shells) can be analyzed. This paper presents the procedure and assumption made in order to obtain the dynamic stiffness matrix of a toroidal shell in harmonic oscillation. Also some numerical examples will be given and discussed.     

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.     

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.     

曲线顶管的三维力学模型理论分析与应用

大口径急曲线顶管施工技术已经在上海地区得到成功的应用,但曲线顶管施工对周围土体稳定性影响和管体结构受力、位移的空间变化有待深入研究和探讨。本文试图从经典弹性力学基本原理出发,利用壳体理论和温克尔假设建立曲线顶管与土体相互作用的三维力学模型,以模拟曲线顶管在软土地层中的施工力学性态。给出了曲线顶管纵向和法向位移的理论解,并进而给出了结构的内力和地层抗力。与具体工程实测位移比较表明,位移的理论解和实测值有相当的一致性,本文的理论结果是合理的。     

Numerical solution of curved pipes submitted to in-plane loading conditions

An alternative formulation to current meshes dealing with finite shell elements is presented to solve the problem of stress analysis of curved pipes subjected to in-plane bending forces. The solution is based on finite curved elements, where displacements are defined from a total set of trigonometric functions or a fifth-order polynomial, combined with Fourier series. Global shell displacements are achieved through the one associated with curved arch bending and the other referred to the toroidal thin-walled shell distortion. Beam-type displacement and in-plane rotation are uncoupled and separately formulated, using trigonometric shape functions, as in Timoshenko or Mindlin beam theory. To build up the solution, a simple deformation model was adopted, based on the semi-membrane concept of the doubly curved shells behaviour. Several studies are presented and compared with experimental and numerical analyses reported by other authors.     

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.     

面内加载时弯管的数值计算方法

对平面内弯矩作用下的弯管的应力分析提出了一种新的计算方法,以替代现行的有限元网格法。新方法的提出基于有限弯曲单元,其位移由一系列三角函数或者五次多项式,与傅里叶级数共同计算而得。整体的壳位移由两类位移组合而成,一个是拱的弯曲位移,另一个则是环形的薄壁壳体的翘曲位移。梁模型的位移和平面内转角是不耦合的,分别应用三角函数,如Timoshenko或Mindlin梁理论计算得到。基于双曲线壳体的半膜单元概念,提出了一个简化的变形模型用于此计算方法。研究成果也与已有的试验和数值分析结果进行了对比。     

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.     

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.     

Geometrically nonlinear numerical procedure for Saint-Venant's problem for a symmetrical elastic shell with a closed contour and circular axis

An original numerical procedure of treating Saint-Venant's problem for toroidal-like shells under pressure and bending moment is suggested. Its peculiarity consists in that the loading and displacement parameters are separated into two groups: the first one describes the ring-like transverse deformation of the shell cross section, while the second the beam-like axial deformation of the latter. One of them is considered separately in the sequential iteration procedure while the other is reckoned to be known. After each step of calculation, the parameters considered to be known are made more exact.A lot of the well-known problems widely presented in the literature are considered. Among them are: (1) Karman's problem, which explains the ovalization of the cross section and the enlarged flexibility of a pipe bend as compared to a straight pipe under bending; (2) the so-called pressure reduction effect when the internal pressure prevents ovalization, which is a demonstration of the influence of the geometrical nonlinearity even at small displacements; (3) Brazier's effect, which deals with nonlinearly enlarged ovalization of an initially straight elastic pipe with an increase in the bending moment.The main advantage of the method is that it allows analyzing a toroidal shell of arbitrary cross section with a variable wall thickness. As an example, a toroidal shell with two long symmetrical axial cracks is considered, where cracks are modeled as the jumps of the contour angular displacement whose values are related to the crack depth.     

Thin-walled thermoplastic pipes

Stressed skin and rib-reinforced shell structures optimize plastics pipe performance. Computation of engineered pipes is easy as they can be substituted for by orthotropic pipes having the same rigidity characteristics. The flexible pipe theory lends itself very well to the concept of thin walled plastics pipes predicting a 100-year design life. Practical experiences are studied as well as other dimensioning aspects, e.g. buckling, impact resistance and minimum wall thickness. Combinations of various construction principles and materials have different performance characteristics making market segmentation optimal.     

Static analysis of thin-walled structures using polynomial and exact solutions. Application to thin-walled pipes

A new finite element procedure for the static analysis of beam thin-walled structures of open or closed cross-section is presented. The method is a combined or mixed one, based on the superposition of beam and shell strains and displacements. An essential part of this work is the development of new shape functions (which are either ‘exact’, or polynomial) for the interpolation of the shell displacements.The above method is applied to the thin-walled curved pipe problem and compared to the von Karman approach. Excellent results are obtained, as well as a drastic reduction of the total number of nodal variables.     

The prediction of the elastic critical load of submerged elliptical cylindrical shell based on the vibro-acoustic model

Based on the vibro-acoustical model, an effective new approach to nondestructively predict the elastic critical hydrostatic pressure of a submerged elliptical cylindrical shell is presented in this paper. Based on the Goldenveizer–Novozhilov thin shell theory, the vibration equations considering hydrostatic pressures of outer fluid are written in the form of a matrix differential equation which is obtained by using the transfer matrix of the state vector of the shell. The fluid-loading term is represented as the form of Mathieu function. The data of the fundamental natural frequencies of the various elliptical cylindrical shells with different hydrostatic pressure and boundary conditions are obtained by solving the frequency equation using Lagrange interpolation method. The curve of the fundamental natural frequency squared versus hydrostatic pressure is drawn, which is approximately straight line. The elastic critical hydrostatic pressure is therefore obtained while the fundamental natural frequency is assumed to be zero according to the curve. The results obtained by the present approach show good agreement with published results.     

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.     

网状扁壳与带肋扁壳组合结构的拟三层壳分析法

本文对网状扁壳与带肋扁壳共同工作的组合结构(可简称组合网状扁壳),采用连续化的拟三层壳的计算模型,按弹性小挠度薄壳理论进行分析计算,推导建立了混合法的基本方程式。由于这种构造上的拟三层壳在一般情况下不存在中面,因而壳体的薄膜内力、弯矩与薄膜应变,弯曲应变是耦合的,存在一个耦合矩阵,使得基本方程式比单层光面的符氏扁壳方程要复杂得多。对于周边简支的组合网状扁壳可求得基本方程式的解析解。文中对三向、四向组合网状扁壳进行了详细讨论,并指出了在特定条件下,可退化为一个当量的各向同性单层扁壳。对于一般网状扁壳的拟壳分析法及带肋扁壳的拟壳分析法分别属于本文的两种特殊情况。文中附有计算例题。     

薄壁弯梁桥空间有限元分析

根据Novozhilor截锥壳几何方程及薄壁弯梁桥的特点 ,构造了一种线位移和转角各自独立插值的四边形截锥壳单元 ,并编制了用于计算薄壁弯梁桥的有限元程序。用该程序对一简支弯箱梁进行验算 ,理论值与试验值符合良好 ,从而验证了该模型是可靠的     

A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates

Using a differential quadrature (DQ) method, large amplitude free vibration analysis of laminated composite skew thin plates is presented. The governing equations are based on the thin plate theory (TPT) and the geometrical nonlinearity is modeled using Green's strain in conjunction with von Karman assumptions. To cause the impact due to nonlinear terms more significant, in-plane immovable simply supported, clamped and different combinations of them are considered. The effects of different parameters on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Some new results for laminated composite skew plates with different mixed boundary conditions are presented and are compared with those obtained using the first order shear deformation theory based DQ (FSDT-DQ) method. Excellent agreements exist between the solutions of the two approaches but with much lower computational efforts of the present DQ methodology with respect to FSDT-DQ method.     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels

In this present article, large amplitude free vibration behaviour of doubly curved composite shell panels have been analysed using the nonlinear finite element method. The nonlinear mathematical model is derived using Green Lagrange type geometric nonlinearity in the framework of higher order shear deformation theory. In addition to that all the nonlinear higher order terms are included in the mathematical model to achieve more general case. The nonlinear governing equation of free vibrated curved panel is derived based on Hamilton׳s principle and solved numerically by using the direct iterative method. The developed mathematical model has been validated by comparing the responses with those available numerical results. Finally, some new numerical experimentation (orthotropicity ratio, stacking sequence, thickness ratio, amplitude ratio and support conditions) have been carried out to show the significance and the efficacy of the proposed mathematical model.     

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。