Fourier p-elements for curved beam vibrations

Several Fourier p-elements for in-plane vibration of thin and thick curved beams are presented. Fourier trigonometric functions are used as enriching functions to avoid the ill-conditioning problems associated with high order polynomials. The element matrices are analytically integrated in closed form. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly improved. Furthermore, the elements with enriching shape functions can avoid membrane and shear locking. The vibration of a thin ring, whose exact solutions are available, is analyzed by the present elements. The present elements can compute accurately high natural modes as the higher mode shapes synchronize with the Fourier functions nicely. The free vibration analysis of a number of hinged circular arches with various subtended angles and the tapered cantilever arches having uniform and non-uniform cross-section is carried out as numerical examples. The condition numbers for polynomial p-elements and Fourier p-elements are compared to show the superior numerical stability of the method.     

Two-dimensional viscoelastic vibration by analytic Fourier p-elements

Three new Fourier p-elements of rectangular, skew and trapezoidal shapes are given analytically for plane viscoelastic vibration problems. The natural frequencies of the plane viscoelastic structures with complex Young’s modulus are computed by a complex eigenvalue solver. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly increased. Since trigonometric functions are used as enriching functions instead of polynomials in the proposal elements, the ill-conditioning problems associated with polynomials of higher degree in the traditional p-version finite element method are avoided. The two mapped plane coordinates in the Jacobian are uncoupled for trapezoidal elements whose element matrices can then be integrated analytically. A triangle can easily be divided into three trapezoids. Therefore, any plane viscoelastic problem with polygonal shape can be analyzed by a combination of rectangular and trapezoidal elements. Numerical examples show that the convergence of the present elements is very fast with respect to the number of trigonometric terms. The natural frequencies of several polygonal viscoelastic plates subject to in-plane vibration are presented.     

Natural frequencies of square plates with reinforced central holes subjected to inplane loads

The natural frequencies and the elastic buckling loads of square plates, containing reinforced square holes and subjected to inplane loads, are determined using the finite element method. The reinforcing beams which are located at the four edges of the hole are either square or rectangular in cross section.

To determine the appropriate out-of-plane stiffness and mass matrices, a rectangular non-conforming bending element is employed. This element has three degrees of freedom at each of its four corner nodes. The finite element method is also used to calculate the inplane stress distribution prior to buckling. This inplane element is rectangular and is based on strain assumptions. The element also includes the inplane rotation as well as the two translational displacements as the three degrees of freedom at each of its four corner nodes. To model the behaviour of the reinforcement, a three-dimensional exact beam element is used. For this element, the degrees of freedom for out-of-plane displacement are consistent with the non-conforming bending element, and the inplane degrees of freedom are consistent with the inplane strain-based element.

In the present paper the natural frequencies and the corresponding modes of vibration are investigated when the reinforced plates are subjected to uniform uniaxial, biaxial and shear loads. The variation of the natural frequencies with various sizes of reinforcement is obtained for plates subjected to a range of magnitude of inplane loading.     

Strain-based finite element for the natural frequencies of cylindrical shells

The free-vibration natural frequencies of complete cylindrical shells, having different end conditions, and cylindrical panels have been investigated. The finite element method is used in which a strain-based cylindrical shell element is employed. This element satisfies the exact representation of rigid-body modes of displacements and all components of strains are assumed to be independent insofar as it is allowed by the compatibility equations. The element has only external degrees of freedom, five at each corner node, to avoid difficulties associated with higher-order elements. The stiffness and mass matrices are obtained using numerical integrations by the deployment of Gauss-Legendre quadrature points.     

Structural characteristics of Hagia Sophia: I—A finite element formulation for static analysis

The historic Hagia Sophia in Istanbul, which held the record as the world's largest domed building for some 800 years, is analysed with a finite element formulation, including the effects of thickness shear deformations and the term z/R$z/R$, to understand its structural behaviour under the action of static loading. The structure, including all essential elements of the system, is modelled by using the same curved trapezoidal finite element with 40 degrees of freedom. Its structural behaviour and its structural load carrying system are demonstrated and the results are compared with those obtained earlier and also with those observed at the structure.     

Ball joint behavior in a double layer grid by dynamic model updating

Owing to uncertainties occurring in double layer grids with ball joint system during assembly and erection stages, the behavior of a separate ball joint does not represent the actual behavior of the joint in the structure. In the present work, the behavior of a ball joint system under actual conditions in a double layer grid has been determined by means of the inverse problem method. A double layer grid with ball joint system was constructed from the components which are generally utilized in practice. Modal testing was carried out on the grid in free support condition and its frequency response functions were measured at appropriate degrees of freedom. Frequencies of twelve vibration modes of the grid were experimentally obtained in the frequency range of 0–100 Hz. A suitable finite element model of the grid was prepared in which the behavior of the jointing system was modeled with an equivalent beam element at the end of each member. By performing finite element model updating for the grid via minimizing the differences between eight experimental and analytical natural frequencies, geometric section properties of the equivalent beam element were determined. The updated model presented very good estimates of the actual twelve natural frequencies of the grid and analytical frequency response functions from the updated model were in agreement with those obtained from the experiment. Using only natural frequencies of the grid, a reasonably accurate updated model was obtained.     

Strain based finite element for the vibration of cylindrical panels with openings

This paper investigates the effect of cut-outs on the dynamic behaviour of cylindrical panels, using a newly developed strain based finite element. The developed element is based on assumed strains and has only five necessary degrees of freedom at each corner node. The displacement fields of the element satisfy the exact requirement of rigid body displacements. The efficiency of the element is tested by applying it to the calculation of natural frequencies of shells. Investigations are first carried out to test the convergence of the element and to establish the mesh size to be used. The element is further applied to analyse cylindrical panels with cut-outs. Simply supported as well as clamped panels on their edges were considered and the effects of the size and location of the openings on the natural frequencies are investigated. The subspace iteration technique which is shown to have an economising effect, is used to obtain the natural frequencies and the associated modes of vibration.     

Natural frequencies of plates with square holes when subjected to in-plane uniaxial, biaxial or shear loading

The finite element method is used to determine the natural frequencies of flat square plates containing eccentrically located square holes. The plates are subjected to in-plane uniaxial, biaxial or uniformly distributed shear along the four outer edges. These edges are either simply supported or clamped.

In order to evaluate the stiffness and mass matrices, the non-conforming rectangular displacement element is used to model the out-of-plane behaviour of the plate. The in-plane stress distribution within the plate, required in the analysis, is determined by using a rectangular finite element having the only essential degrees of freedom at each of the four corner nodes. The element displacement functions are based on assumed strains and satisfy the exact requirements of strain free rigid-body modes of displacements.

The variation of the natural frequency with the size and location of the hole is first investigated in the absence of any in-plane stresses. This analysis is then repeated for a series of values of the applied in-plane stresses. When uniform shear is applied, tension and compression zones are produced in the plate and hence the effect of locating the hole in each of the regions is also investigated. The values of the applied in-plane stresses ranged up to the point that would cause the plates to buckle. In this way a comprehensive set of results can be obtained.     

Numerical simulation model of vibration responses of rectangular plates embedded with piezoelectric actuators

A numerical simulation model for random large amplitude vibration control of composite plate using piezoelectric material is presented. The H_∞ control design is employed to suppress the large amplitude vibrations of composites plates under random loading. The numerical simulation model is developed and based on the finite element method. The finite element governing equation includes fully coupled structural and electrical nodal degrees of freedom, and consider the von Karman large amplitude vibration. The modal reduction method using the structural modes is adopted to reduce the finite element equations into a set of modal equations with fewer degrees of freedom. The modal equations are then employed for controller design and time domain simulation. In the simulations without control, the value of the linear mode to the nonlinear deflection is quantified; and the minimum number of linear modes needed for accurate model is obtained. In the simulations with control, it is shown that the truncated modes, which are neglected in the control design, deteriorate the controller performance. Generally, the vibration reduction level is not monotonically increasing with the size of the piezoelectric actuator. The optimal piezoelectric actuator size depends on the excitation level. For higher excitation level, optimal actuator size is larger. The H_∞ controller based on the linear finite element formulation gives better vibration reduction for small amplitude vibration, but it still gives reasonable performance for large amplitude vibration provided that the piezoelectric actuator is big and powerful enough.     

塔式起重机结构动态分析的两种有限元模型及比较

本文分别应用完整的杆系有限元模型和等效有限元模型对塔式起重机结构进行了动力分析，指出使用合理的等效单元代替复杂的杆系框架结构可以大大减少单元和自由度数，在使计算简化的同时保持了良好的分析精度，对于同类型塔机的动力分析具有很好的参考价值。     

Triangular higher-order element for better prediction of stress resultants and stresses in plated and shell structures

Although they furnish accurate displacements, conventional displacement-based lower order finite elements fail to predict accurate stress resultants and stresses in certain classes of plate and shell problems that involve free edges, steep stress gradients and singularities. In order to tackle such problems, a triangular higher-order shell element based on the nodal basis approach has been developed. The nodes of the element are located at optimal points and its more superior shape functions derived from orthogonal Proriol polynomials. To illustrate the improved performance of the higher-order element as compared to commonly used lower order shell elements in predicting the variations of stress resultants and stresses, three example problems involving a simply supported skew plate, a corner supported square plate, and a clamped cylindrical shell are solved. The stress resultants and the stresses furnished by the higher-order element for the problems considered are found to be accurate with the satisfaction of the natural boundary conditions and devoid of any oscillations. When compared to lower order elements, the higher-order element requires a simple mesh design and lesser degrees of freedom resulting in a considerable reduction in the computational effort, especially for large scale nonlinear analysis.     

基于2.5维有限元法分析横观各向同性地基上列车运行引起的地面振动

 根据2.5维有限元原理,从横观各向同性土体弹性本构方程出发,推导出横观各向同性土体2.5维有限元弹性波动方程。轨道简化成铺设在横观各向同性地基上的Euler梁。在轨道方向的垂直截面上,将轨道结构和地基进行有限元离散,采用八节点单元,每个节点包含3个自由度,从而使复杂的三维问题转化为平面应变问题,再通过FFT逆变换转换为沿轨道方向三维空间中的时域振动响应。分别基于上海和北京地区的典型土质参数,进行场地动力响应分析。结果表明：列车运行引起的北京地区场地的加速度响应幅值及位移响应幅值均分别大于上海地区场地的加速度响应幅值及位移响应幅值;随着距轨道中心处距离的增加,两地区土体的动力响应频率均衰减到简谐荷载的自振频率周围,反映出横观各向同性土体的滤波作用;应力波在软土层中传播所消耗的能量大于在硬土层中传播;列车运行引起的地面振动衰减曲线会出现反弹增大的现象,其反弹特性及出现的位置与地基土体参数及列车运行速度密切相关。     

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Derivation

This paper provides the first detailed presentation of the derivation for a newly proposed method which can be used for the decomposition of the stability buckling modes of a single-branched, open cross-section, thin-walled member into pure buckling modes. Thin-walled members are generally thought to have three pure buckling modes (or types): global, distortional, and local. However, in an analysis the member may have hundreds or even thousands of buckling modes, as general purpose models employing shell or plate elements in a finite element or finite strip model require large numbers of degrees of freedom, and result in large numbers of buckling modes. Decomposition of these numerous buckling modes into the three buckling types is typically done by visual inspection of the mode shapes, an arbitrary and inefficient process at best. Classification into the buckling types is important, not only for better understanding the behavior of thin-walled members, but also for design, as the different buckling types have different post-buckling and collapse responses. The recently developed generalized beam theory provides an alternative method from general purpose finite element and finite strip analyses that includes a means to focus on buckling modes which are consistent with the commonly understood buckling types. In this paper, the fundamental mechanical assumptions of the generalized beam theory are identified and then used to constrain a general purpose finite strip analysis to specific buckling types, in this case global and distortional buckling. The constrained finite strip model provides a means to perform both modal identification relevant to the buckling types, and model reduction as the number of degrees of freedom required in the problem can be reduced extensively. Application and examples of the derivation presented here are provided in a companion paper.     

Moment modification factor of I-girder with trapezoidal-web-corrugations considering concentrated load height effects

Moment modification factors of the I-girder with trapezoidal web corrugations subjected to concentrated load, applied at different heights on the cross section and various end restraint conditions, are investigated. The moment modification factors are obtained by finite element buckling analyses. The new FEM program is developed by using beam elements and new general formulas of cross-section properties as well as a new warping constant of the I-girder with trapezoidal web corrugations. The theoretical results of moment modification factors are compared with commercial finite element software (ABAQUS) using shell elements. A series of finite element analyses with different corrugation profiles and lengths is performed. Through comparative numerical studies, the theoretical results are successfully verified. The moment modification factors from the SSRC Guide are compared with present FEM results. The new values of the variable C_b, the moment modification factor, in the SSRC Guide are proposed as the variable C_b,c to improve the accuracy of lateral-torsional buckling strength.     

Shell element of relative degree of freedom and its application on buckling analysis of thin-walled structures

Shell element of relative degrees of freedom (SERDF) is a special transformation of solid element. It can be used in finite element (FE) analysis of both thin and thick shell structures and the formulation is simpler than normal shell elements. Introduction of additional internal degrees of freedom will improve the calculation precision.FE formulation of SERDF in the buckling analysis is derived in this article. Different measures of tackling additional internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the validity of this element and the method.     

非对称开口截面薄壁圆形曲梁的非线性自由振动分析

为具有非对称开口截面的薄壁曲梁的非线性自由振动分析构建了一个有限元公式,由虚功原理推导出动能和势能,并考虑了弯曲-扭转耦合、扭曲和剪心位置等影响。有限元分析中,2结点薄壁曲线构件的形状函数采用三次多项式计算。每个结点具有7个自由度,其中包括扭曲自由度。采用直接迭代法计算非线性特征值。将计算结果与直线梁的计算结果相对比。同时归纳了具有多种半径和对角的曲梁的非线性自由振动分析结果。     

Dynamic stiffness analysis of laminated composite plates

An analysis is presented for the vibration and stability problem of composite laminated plates by using the dynamic stiffness matrix method. A dynamic stiffness matrix is formed by frequency dependent shape functions which are exact solutions of the governing differential equations. It eliminates spatial discretization error and is capable of predicting several natural modes by means of a small number of degrees of freedom. The natural frequencies and buckling loads of composite laminated plates are calculated numerically. The effects of the boundary conditions, the number of layers, the orthotropicity ratio, the side to thickness ratio, and the aspect ratio are studied. It is also illustrated that connected composite plate structures can be handled without difficulty by the present method.     

Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section

A finite element formulation is present for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies are derived by the virtual principle. The energy functional includes the effect of flexural–torsional coupling, the torsion warping and the shear center location. For finite element analysis, cubic polynomials are utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem has been solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.     

Eine Finite‐Elemente‐Lösung auf der Basis eines erweiterten eindimensionalen Querschnittselementes für die freie Torsion dünnwandiger prismatischer Stäbe

A finite element solution on the basis of an extended one‐dimensional cross‐section‐element for the Saint‐Venant torsion of thin‐walled prismatic beams. It is presented a finite element solution on basis of an extended one‐dimensional cross‐section‐element to the calculation of the warping function, the torsional properties and the shear stresses, dependent on it, for thin‐walled prismatic beams under Saint‐Venant torsion. The formulated finite two‐node‐element with six element degrees of freedom can capture through inclusion of the torsion around the element axis the linear term of the variance of warping function perpendicular to the element axis. Only the shear stresses of the ring shear flows in the closed section parts unchangeable over the wall thickness can be calculated with the simple two‐node‐element with two element degrees of freedom. The extended two‐node‐element supplies in addition also the shear stresses of the cut open cross‐section linearly changeable over the wall thickness.     

Free vibration analysis of horizontally curved steel I-girder bridges

Presented herein is a finite element formulation for free vibration analysis of horizontally curved steel I-girder bridges. Stiffness as well as mass matrices of the curved and the straight beam elements is formulated. Each node of both of them possesses seven degrees of freedom including the warping degree of freedom. The curved beam element is derived based on the Kang and Yoo's thin-walled curved beam theory in 1994. A computer program is developed to carry out free vibration analyses of the various bridges. Comparing with the frequencies using the general purpose program ABAQUS, the validity of the presented numerical formulation is shown. The numerical formulation is extensively applied to investigate free vibration characteristics of the bridges considering effects of the initial curvature, boundary condition, modeling method, and degrees of freedom of cross frame. Invaluable information which help practicing engineers better understand the vibration characteristics is provided.