Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings

This paper describes the underlying theory, and a general-purpose computer program, VIPASA, for determining the critical buckling stresses or natural frequencies of vibration of thin prismatic structures, consisting of a series of plates rigidly connected together along longitudinal edges. Each plate may be either isotropic or anisotropic and may carry a basic stress system consisting of longitudinal and transverse direct stress combined with shear. The structure is assumed to be subjected to a “dead load” system which does not cause buckling; in addition a “live load” system, defined in magnitude by a single load factor, may be applied and the value of the load factor at buckling is determined. Alternatively the natural frequencies of vibration of the structure when subjected to the dead load system are determined. Any number of critical load factors or natural frequencies can be obtained. The theory is based upon the assumption that all modes are sinusoidal, in the sense that all three components of displacement vary sinusoidally along any longitudinal line, but phase differences are incorporated to allow for the effects of anisotropy and shear. Apart from this assumption no further approximations are made other than those inherent in thin plate theory.     

Buckling and vibration of stiffened plates subjected to partial edge loading

The buckling and vibration characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element method. The initial stresses are obtained considering the pre-buckling conditions. Buckling loads and vibration frequencies are determined for different plate aspect ratios, edge conditions and different partial non-uniform edge loading cases. The non-uniform loading may also be caused due to the supports on the edges. The analysis presented determines the stresses all over the region for different kinds of loading and edge conditions. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The vibration characteristics are discussed and the results are compared with those available in the literature. Buckling results show that the stiffened plate is less susceptible to buckling for position of loading near the supported edges and near the position of stiffeners as well.     

Efficient multi-level substructuring with constraints for buckling and vibration analysis of prismatic plate assemblies

The use of substructuring in the buckling and vibration analysis of large structures permits very substantial improvements in computational efficiency. The exact multi-level substructuring capability of the widely used computer program VICONOPT for the analysis and optimum design of prismatic plate assemblies has been extended by the inclusion of new theory, presented in this paper, which permits constraints on any of the internal or external nodes of substructures. The computational savings by using substructuring in this way are shown to be typically 50–70% compared with previous VICONOPT solutions. The theory is applicable to any method or computer code for structures whose buckling or vibration modes combine responses of different half-wavelengths, with VICONOPT being used as an example.     

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.     

A high-order shear element for nonlinear vibration analysis of composite layered plates and shells

This paper introduces a family of high-order facetted shell elements for linear and nonlinear stress and vibration analysis of composite layered plate and shell structures. Engineering slope angles are employed in element equations, and transverse stresses are expanded over the thickness. The lateral deflection is modelled by conforming or non-conforming Hermitian shape functions, within rectangular or paralellogrammic and triangular elements. Nonlinear terms associated with geometrical nonlinearity are also derived using a practical approach based upon the actual components of strain. A finite element programming package was designed employing the newly developed elements. Several case studies have been investigated and package results were compared with existing theoretical and/or experimental results. It has been proved that the developed elements can lead to accurate estimations of natural frequencies. The effect of fibre angles on natural frequencies has also been investigated with some case studies, and the results proved that the package can be a useful tool for the design optimization of composite layered plate and shell structures.     

Stream surface strip element method for simulation of the three-dimensional deformations of plate and strip rolling

A new method—the stream surface strip element method (SSSEM)—for simulating the three-dimensional deformations of plate and strip rolling process is proposed. The rolling deformation zone is divided into a number of stream surface (curved surface) strip elements along metal flow traces, and the stream surface strip elements are mapped into the corresponding plane strip elements for analysis and computation. The longitudinal distributions of the lateral displacement and the altitudinal displacement of metal are constructed respectively to be a quartic curve and a quadratic curve, the transverse distributions of them are expressed as the third-power spline function, and the altitudinal distributions of them are fitted to be a quadratic curve. Based on the flow theory of plastic mechanics, the three-dimensional deformations and stresses of the deformation zone are analyzed and formulated. Compared with the streamline strip element method, the SSSEM considers the uneven distributions of stresses and deformations along altitudinal direction, and realizes an accurate three-dimensional analysis and computation. The simulation examples indicate that the method and the model of this paper are in accord with facts, and provide a new reliable engineering-computation method for the three-dimensional mechanics simulation of plate and strip rolling process.     

柔性制管过程的样条有限条仿真

提出弯管过程样条有限条分析方法,建立基于更新拉格朗日方法(U.L.法)的弹塑性大变形样条有限条法的理论模型.采用该方法分析柔性制管过程变形带材的位移场、应变场和应力场.并应用可视化技术有效、直观、快速地显示模拟结果,对实际生产具有重要意义.     

模拟板带轧制三维变形的流面条元法

提出模拟板带轧制过程三维变形的一种新的数值方法——流面条元法。沿着金属的流动轨迹,将变形区划分为若干流面(曲面)条元,为方便分析和计算,又将其映射为平面条元。横向位移和高向位移的纵向分布被分别构造为四次曲线和二次曲线,其横向分布均用三次样条插值函数表示,高向分布均用二次曲线拟合。根据塑性力学流动理论,分析推导了变形区三维变形和应力的数学模型。与曾经提出的流线条元法相比,考虑了应力与变形沿高向的不均匀分布,实现了精确的三维分析和计算。仿真实例表明,该方法和模型符合实际,为板带轧制过程的三维力学仿真提供了一个新的实用工程数值方法。     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

STUDY OF THE 3-DIMENSIONAL STRESSES OF COLD STRIP ROLLING UNDER CENTRE WAVE

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Generalized buckling analysis of laminated plates with random material properties using stochastic finite elements

A generalized layer-wise stochastic finite element formulation is developed for the buckling analysis of both homogeneous and laminated plates with random material properties. The pre-buckled stresses are considered in the derivation of geometric stiffness matrix and the effect of variation in these stresses on the mean and coefficient of variation of buckling strength is studied. The mean buckling strength of plates under uniform stress assumption exactly matches with those reported in the literature. However, it is shown that the actual mean buckling strength of plates can be significantly different based on the pre-buckled stress analysis which depends on boundary constraints, principal material directions, aspect and thickness ratios of plates. The statistics of buckling strength is determined using a Taylor series expansion based mean centered first order perturbation technique. The stochastic finite element solutions obtained using layer-wise plate theory is also validated with analytical solutions presented in this paper. Parametric studies are conducted for different aspect ratios, ply orientations and boundary conditions.     

板带失稳临界应力的参数化计算方法研究

平直度是衡量板带质量好坏的重要指标,板带失稳临界载荷是是研究平直度的关键参数。基于弹性力学板带失稳理论,通过参数化有限元计算方法求出板带在轧制过程中产生中波和边波两种浪形时的失稳临界载荷,从而求出板带翘曲临界应力系数kcr。由kcr得到发生中波和边波时的失稳临界应力公式。得到一种弹性力学板带失稳理论和参数化有限元计算方法相结合的快速求解板带失稳临界应力的方法。运用该方法可以通过残余应力快速的对带钢平直度做出准确判断。     

Nonlinear dynamic behavior and instability of slender frames with semi-rigid connections

The free and forced nonlinear vibrations of slender frames with semi-rigid connections are studied in this work. Special attention is given to the influence of static pre-load on the natural frequencies and mode shapes, nonlinear frequency–amplitude relations, and resonance curves. An efficient nonlinear finite element program for buckling and vibration analysis of slender elastic frames with semi-rigid connections is developed. The equilibrium paths are obtained by continuation techniques, in combination with the Newton-Raphson method. The ordinary differential equations of motion of the discretized frame are solved by the Newmark implicit numerical integration method using adaptive time-step strategies. Three structural systems with important practical applications are analyzed: an L-frame, a shallow arch, and a pitched-roof frame. The results highlight the importance of the static pre-load and the stiffness of the semi-rigid connections on the buckling and vibration characteristics of these structures.     

Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses

Natural frequencies and buckling stresses of angle-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Numerical results are compared with those of the published existing theories. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. The present global higher-order approximate theories can predict the natural frequencies, buckling stresses and modal stresses of thick multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers.     

Buckling analysis of a laminated composite plate with delaminations using the enhanced assumed strain solid shell element

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the buckling analysis of a laminated composite plate with delaminations. The object is to study the buckling behavior of structures with delaminations using the enhanced assumed strain (EAS) solid shell element with 5, 7 and 9 parameters. The EAS three-dimensional finite element formulation presented in this paper is free from shear locking and leads to accurate results for distorted element shapes. The developed FE model is used to study the effects of some parameters in the buckling load, such as the stacking sequences, delamination size, aspect ratio, width-to-thickness ratio. The feasibility of the proposed method is confirmed by numerical examples. Results show that using hexahedral solid-shell finite element in the buckling analysis is more efficient than using the enhanced solid finite element.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

高温带钢的局部宽度内压屈曲及后屈曲分析

根据工业生产中最主要8种带钢浪形(即板形缺陷)的浪形函数和其对应的带钢纵向不均匀伸长率的横向分布,抽象出产生此浪形的局部内压屈曲过程的位移函数和端边界外力条件,然后分别运用弹性薄板小位移理论和大位移理论建立了屈曲和后屈曲变形模型,并使用能量法进行了求解。在过去对常温态带钢此类问题研究的基础上,对高温态带钢的此类屈曲变形进行理论分析和数值计算,获得了各种工况下的屈曲条件和瓢曲生成路径。     

带轴锥齿轮超声研齿振动系统的设计与试验

为了满足带轴锥齿轮超声研磨加工的谐振频率要求,利用波动理论对不同结构的3种带轴锥齿轮的超声变幅系统进行谐振设计,建立了统一的谐振频率方程,并对位移放大系数和最大应力进行了比较。运用有限元方法,对由复合变幅杆和换能器组成的超声激励-变幅振动系统进行了模态分析和谐响应分析。最后对超声振动系统进行了阻抗分析和频率测试。试验结果表明：实际超声振动系统的谐振频率为23.269kHz,与设计频率误差约1.17%,试验结果与有限元分析和理论分析结果基本吻合,验证了理论设计的准确性。     

Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.