Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Asymmetric vibration and stability of circular plates

The asymmetric vibration and stability of circular and annular plates using the finite element method is discussed. The plate bending model consists of one-dimensional circular and annular ring segments using a Fourier series approach to model the problem asymmetries. Using displacement functions which are the exact solutions of the static plate bending equation, the stiffness coefficients corresponding to the 1st and nth harmonics are used in closed form. By assuming that the static displacement function closely represents the vibration and stability modes, the mass and stability coefficient matrices for an annular and circular element are also constructed for the 1st and nth harmonics. Several numerical examples are presented to demonstrate the efficiency and accuracy of the finite element model with that of classical methods.     

In-plane vibration of plates by continuous mass matrix method

The continuous mass matrix method derived for frameworks is extended to the analysis of in-plane vibration of plates. A continuous mass distribution which is the same as the actual mass distribution of the plate is considered over each rectangular finite element. Taking into account that the rigid body movement produces inertial forces in dynamic analysis for a rectangular plate element eight independent conditions are provided to satisfy eight independent freedoms. Each condition is obtained from an independent displacement distribution satisfying the equations of motion at any point of the element and not only at the nodes of the rectangle. The dynamic element stiffness matrix thus obtained is a function of the natural circular frequency. The limit of the dynamic element stiffness matrix when the value of the natural circular frequency tends to zero is the static, stress compatible element stiffness matrix. The analysis of plates under forcing forces is performed by modal analysis after the natural circular frequencies and the corresponding modal shapes have been obtained from the free vibrations, for all the forcing forces are assumed to be function of the same time variation. Otherwise one must recur to a numerical analysis. The effect of the sizes, number of the meshes, the additional static load on the plate and the rigidity of the boundaries on the vibration of the plate is discussed. Few example problems are solved in order to illustrate the above mentioned effects. The numerical results obtained by continuous mass matrix method are compared with those of consistent mass matrix method. The convergence in terms of the sizes of meshes and the limit of convergence are examined.     

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.     

Asymmetric buckling of bimodulus thick annular plates

An annular element with Lagrangian polynomials and trigonometric functions as shape functions is developed for asymmetric finite element stability analysis. The annular element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. Using the asymmetric finite element model, the asymmetric static buckling of bimodulus thick annular plates subjected to a combination of a pure bending stress and compressive normal stress is investigated. The obtained results of non-dimensional critical buckling coefficients are shown to be very accurate when compared with the exact solutions. The effects of various parameters on the buckling coefficients are studied. The bimodulus properties are shown to have significant influences on the buckling coefficients.     

Axisymmetric buckling of bimodulus thick circular plates

The static buckling of bimodulus thick circular and annular plates subjected to a combination of a pure bending stress and compressive stress is investigated. The thick finite element model, which includes the effect of transverse shear deformation, are created for axisymmetric buckling problems. The obtained results of buckling coefficient are compared with the exact solutions for ordinary thin plates. The accuracy of the finite element solutions are shown to be very good. The effects of various parameters on the buckling coefficients and neutral surfce locations are studied. The bimodulus properties are shown to have significant influences on the buckling coefficient.     

Stability analysis of skew orthotropic plates by the finite strip method

A stability analysis based on the Finite Strip Method is presented for skew orthotropic plates subjected to in-plane loadings. The straight sides of the plate are simply supported and the other two skewed sides are supported with any combination of fixed, free and simply supported boundaries. The plate is divided into strips, in contradistinction to elements in the Finite Element Method, and the displacement function is so chosen that it satisfies the boundary conditions and also the inter-strip compatibility conditions of an elemental strip. The energy expressions required to formulate the stiffness and stability coefficient matrices are formulated using smalldeflection theory. The buckling load intensity factor is evaluated for different aspect ratios of isotropic and orthotropic skew plates and the results of certain rectangular isotropic cases are compared with earlier investigations.     

Finite element analysis of the in-plane behaviour of annular disks

In-plane analysis of annular disks using the finite element method is presented. A semi-analytical, one-dimensional finite element model is developed using a Fourier series approach to account for the circumferential behaviour. Using displacement functions which are exact solutions of the two dimensional elasticity plane stress problem, the shape functions, stiffness matrices and mass matrices corresponding to the 0th, 1st and nth harmonics are derived. To show the utility of this new element, example probelms have been solved and compared with the exact solution. The present element can be readily coded into any general purpose finite element program.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.     

Dynamics and stability of plane trusses with gusset plates

The effect of gusset plates on free and forced vibration and stability analyses of plane trusses is investigated. The gusset plates are considered to be finite joints possessing mass and rotational flexibility. The bars of the truss are assumed to be elastic Bernoulli-Euler beams with distributed mass. Axial deformation of the bars and the effect of a constant axial force on the bending stiffness are taken into account. On the basis of these assumptions element stiffness matrices are constructed and presented in detail. The general formulation and solution of stability and free and forced vibration problems of trusses is discussed. Examples are presented in detail which demonstrate the effect of the gusset plates on the behavior of trusses under static or dynamic loads.     

Perspectives on layout and topology design

This paper deals with elastic stability of annular plates under uniform radial loads applied at its edges; emphasis is set on the analysis of nonaxisymmetric buckling forms. In order to increase the buckling strength a circular support is laid to the plate. Three loading types are considered. The stability equation is put together and integrated by separating the variables. Such a location of the support for which the buckling strength is maximal is sought. Calculations are carried out for plates with clamped edges. Received September 29, 2000     

Vibration of annular plates partially treated with unconstrained damping layer

Mass and stiffness matrices of an annular element consisting of base plate and unconstrained damping layer have been derived assuming a modal solution to the equation of motion of the plate. The complex eigen equations have been solved for frequencies and loss factors by an extension of the simultaneous iteration technique. Frequencies of unlayered plates and loss factors of fully layered plates have been compared with those available in literature. The effect of staggering of layers on the vibration characteristics of the plate has been investigated. It is found that when the plate is partially layered starting from its inner edge the loss factors will be higher for all boundary conditions and modes than when the plate is fully layered provided the mass of the damping layer is kept the same in both cases.     

Asymmetric vibration and dynamic stability of bimodulus thick annular plates

The free vibration and dynamic stability problems of asymmetric bimodulus thick annular plates are studied. The annular element with Lagrangian polynomials and trigonometric functions as shape function is developed. The element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. The dynamic stability of an annular plate subjected to a combination of a pure dynamic bending and a uniform dynamic extensional stress in the plane of the plate is investigated. The non-axisymmetric modes are shown to have significant effects in the annular bimodulus plates.     

Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps

This investigation deals with the axisymmetric static and dynamic buckling of a cylindricaliy orthotropic truncated shallow conical cap with clamped edge. The cases of conical caps with a free central circular hole and with a hole plugged by a rigid central mass have been considered. The governing equations are formulated in terms of normal displacement w and stress function Ψ. The orthogonal point collection method is used for spatial discretisation and the Newmark-β scheme is used for time-marching. Analysis has been carried out for a uniformly distributed conservative load normal to the undeformed surface and a central axial ring load at the hole. Dynamic load is taken as a step function load. The influence of orthotropic parameter β and annular ratio on the buckling loads has been investigated. New results for static and dynamic buckling loads have been presented for the isotropic and orthotropic truncated conical caps. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.     

A semi-numerical approach to the buckling and post-buckling solution of plate

In this semi-numerical approach to buckling of plates, a combination of polynomial and trigonometric functions are used as displacement functions in the Rayleigh-Ritz method. It is shown that a variety of loading and boundary conditions can be handled using simple variation of the trigonometric function proposed here. A two-dimensional plate buckling problem is therefore reduced to selecting one of the set of trigonometric function shown. The buckling coefficient values are then computed as eigenvalues of the stiffness and geometric matrix pair. These values compare well with available analytical and numerical approach solutions. The approach can also be extended to post buckling analysis using the eigenvectors found.     

功能梯度材料圆板的非线性热振动及屈曲

采用弹性理论建立了功能梯度材料板的静力平衡方程,利用静力平衡方程确定了功能梯度材料板的中性面位置,在此基础上推导出了功能梯度材料板在均匀温度场中的非线性振动及屈曲微分方程组,求得了功能梯度材料圆板的非线性振动及屈曲的近似解,讨论分析了中性面位置、梯度指数、温度等因素对功能梯度材料圆板非线性振动及屈曲的影响.把该方法计算结果与有限元计算结果进行了比较,验证了该方法的计算结果是可靠的.算例分析表明,中性面位置对均匀温度场中功能梯度材料圆板的非线性振动及屈曲有一定影响.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

Free vibrations and stability analysis of laminated composite plates and shells with hybrid/mixed formulation

Modern theory for applications of laminated plates and shells calls for detailed study of the effect of large spatial rotations on the geometric stiffness for stability analysis as well as inertia operators for vibrations. These two issues are carefully examined here in conjunction with the recently developed mixed finite element formulation for plates and shells with low-order displacement/strain interpolations. An extensive set of stability and vibration problems has been solved to demonstrate the effectiveness and general utilities of the formulation described for laminated plate and shells with arbitrary geometry.