Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

On applications of generalized functions to the analysis of Euler–Bernoulli beam–columns with jump discontinuities

In this article some applications of the distribution theory of Schwarz to the analysis of beam–columns with various jump discontinuities are offered. The governing differential equation of an Euler–Bernoulli beam–column with jump discontinuities in flexural stiffness, displacement, and rotation, and under an axial force at the point of discontinuities, is obtained in the space of generalized functions. The auxiliary beam–column method is introduced. Using this method, instead of solving the differential equation of the beam–column in the space of generalized functions, another differential equation can be solved in the space of classical functions. Some examples of beam–columns and columns with various jump discontinuities are solved. Deflections of beam–columns and buckling loads for columns with jump discontinuities are calculated using the Laplace transform method in the space of generalized functions.     

Vibratory characteristics of flexural non-uniform Euler–Bernoulli beams carrying an arbitrary number of spring–mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring–mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler–Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring–mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring–mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.     

Concise buckling, vibration and static analysis of structures which include stayed columns

The substitute columns previously used¹ to give exact elastic critical loads of individual stayed columns also give their exact member equations, which represent them in stiffness matrix analyses, very economically. Subject to appropriate assumptions, such member equations apply to the classical buckling and vibration analyses emphasised herein, and to static analysis. A typical example indicates that the substitute columns give the member equations with 6% of the effort involved in a standard sub-structure analysis of the original column, or 2% if the column is symmetric about its centre. Moreover, the substitute columns give helpful insights into the behaviour of structures which include stayed columns.     

The effect of pretwisting on the buckling behaviour of slender columns

The equations governing the buckling of a pretwisted column are derived as a subset of the general stability equations for a spatial rod. Numerically exact solutions to these fourth-order differential equations are obtained by determining the analytic forms of the buckled mode shape equations, and calculating the buckling loads and mode shape coefficients using a simple iterative technique. Plots are provided to detail the effect of pretwist on the buckling strength for both statically determinate and indeterminate configurations. These plots indicate that, for statically indeterminate configurations, a pretwisted column may be more than twice as strong as the corresponding prismatic column for the first buckling mode. It is also shown that, generally, the pretwisted buckled mode shapes strongly resemble their prismatic counterparts, and may be treated as effectively planar.     

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.     

Concise buckling analysis of stayed columns

The columns considered consist of a uniform central core with N identical stay frames equally spaced around it. Every stay frame usually involves short arms, which each have one end connected directly to the core and the other end connected by taut stays to the ends of the core and/or the ends of the two adjacent arms of the frame. This paper examines the critical buckling of such columns analytically and draws many general conclusions, the most crucial of which are confirmed by check calculations. The main conclusions include the following.Buckling can be divided into two types, which can be represented exactly by two simple substitute columns whose complexity is independent of N. Also, buckling is impossible if usual prestressing procedures are used and the column is stable under prestress alone and when fully loaded. Thus both substitute columns must apparently be used for each of these two load cases. However, these four substitute column calculations can often be reduced to one, or even to a single simpler substitute plane frame calculation. Finally, minimum weight design procedures should use N = 3, because the lightest design for N > 3 is always equalled or bettered by the optimum N = 3 design.     

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.     

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

A study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory

A thick shell theory is used to calculate the critical load of plastic buckling of axially compressed cylindrical shells. The buckling equations are derived with the principle of virtual work on the basis of a transverse shear deformable displacement field. The deformation theory of plasticity is used for constitutive equations. To fit the uniaxial stress–strain curve, the Ramberg–Osgood equation is used. In the numerical examples special attention is paid to the dependence of the buckling mode on the ratios of radius to thickness R/h and length to radius L/R. This dependence divides the (R/h,L/R)-plane into simply connected regions each of which corresponds to a buckling mode. These regions form a “buckling mode map”.     

A twelfth-order theory of antisymmetric bending of isotropic plates

In this paper the transverse shear and normal strain and stress effects on antisymmetric bending of isotropic plates are considered. A set of twelfth-order partial differential governing equations as well as a set of fourth-order ordinary differential equations for ƒ(z) and φ(z), which represent the transverse shear and normal effects, are derived from a mixed variational theorem. There exists coupling between the partial differential equations and the ordinary ones. In the homogeneous solutions for the former, besides an interior solution contribution, there exist two types of edge-zone solution contributions. One of them is similar to the edge-zone solution in the Reissner—Mindlin theory. The other one is an edge-zone solution consisting of a pair of conjugate functions. Two sample examples are calculated using the present theory. In the former the present two-dimensional theory obtains the three-dimensional exact solution. The latter gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. The numerical results still approximate to exact solutions.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

A numerical investigation of the sinusoidal model for elastic layers in line contact

Distributions of normal stresses and surface deformations, induced when an elastic layer of finite thickness is indented by a frictionless rough rigid flat or cylindrical indenter, are calculated numerically. It is assumed that the punch has a sinusoidal roughness superimposed on its nominal profile. Two cases will be examined, namely when the elastic layer is either bonded to a rigid backing or resting on a frictionless rigid backing (unbonded). Chebyshev polynomials of the first kind T_n(x) are utilized to model both the unknown pressure and the given deformation over the contact area. The governing elasticity equation is thereby reduced to a finite set of linear equations and hence a complete solution is found. The present numerical method is simple, accurate and valid in the full range of Poisson's ratio 0 v 0.5. Moreover, a set of semi-analytical solutions for the contact pressure is obtained for both thin unbonded and bonded elastic layers. The numerical results compared favourably with the asymptotic solutions. The effects of the layer thickness, layer compressibility and roughness amplitude parameters on the contact stresses and deformations are considered.     

Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko's beam theory

The static, dynamic, and buckling behavior of partial interaction composite members is investigated in this paper by taking into account for the influences of rotary inertia and shear deformations. The governing differential equations obtained are very comprehensive, covering and extending the current models for the problems that are based on Euler–Bernoulli beam theory. The analytical solutions of the deflection are then found for the beam with uniformly distributing load under common boundary conditions. The free vibration and buckling behavior are also studied and the analytical expressions of the frequencies of the simply supported beam are obtained explicitly, as are the buckling loads. For other boundary conditions, the eigen-equations are transcendental and thus some numerical examples are presented to demonstrate the effects of the shear deformation and rotary inertia on the resonant frequencies and buckling loads.     

Free vibration of SDOF systems with arbitrary time-varying coefficients

A new analytical approach for determining the exact solutions for free vibration of single-degree-of-freedom (SDOF) systems with non-periodically time-varying coefficients (mass and stiffness) is presented herein. In this paper, the function for describing the variation of mass of a SDOF system with time is an arbitrary one, and the variation of the stiffness is expressed as a functional relation with the mass function and vice versa. Using appropriate functional transform, the governing differential equation for the title problem is reduced to a Bessel’s equation or other analytically solvable equations. Exact solutions for free vibration of SDOF systems with non-periodically varying coefficients are obtained for six important cases. In order to simplify the free vibration analysis of a SDOF system with multi-step time-varying coefficients, the fundamental solutions that satisfy the normalization conditions are constructed based on the exact solutions derived. It is more convenient to determine the displacement response of the SDOF system by using the fundamental solutions and a recurrence formula developed in this paper. Numerical example shows that the proposed procedure is a simple, efficient and exact method.     

The analytical solution of the buckling of composite truncated conical shells under combined external pressure and axial compression?

The objective of this research is determining the buckling load of composite truncated conical shells under external loading by theoretical and numerical methods. The boundary conditions are assumed to be clamped. At first, basic equations and stability relations of conical shells were derived. The analysis is carried out using Donnel-type stability equations for thin cross-ply conical shells. By applying Galerkin??s method, these equations are converted to a system of ordinary time dependent differential equations. Ritz method is employed for finding the dynamic stability load. Finally, the critical static and dynamic buckling loads and the corresponding wave numbers have been found analytically. Then comparison of results is considered. Results of analytical calculations are compared with numerical results and with other researchers?? analytical results. The effects of geometric parameters, the cone semi-vertex angle, number of layers and material of fibers on buckling loads are discussed.     

The collapse of steel model columns with unequal flanges. I. Perfect

The paper is concerned with the collapse behaviour of initially straight pin-ended steel columns having unequal flanges. A modified Shanley model column with unequal flanges is presented and analysed, together with the corresponding modified Calladine construction. Each column flange is assumed to have a uniform state of stress and strain, and to behave elastically until fully yielding in compression or tension—thus local buckling is not included. However, column buckling is fully allowed for, and the complete collapse behaviour of various model column designs is investigated, the parameters varied being the normalised column slenderness, and the flange area ratio. The behaviour of the model columns is shown to agree well with that of corresponding real columns, i.e., columns of uniform cross-section that deform at all points within the volume. Compressive yield of the smaller flange is an expected source of failure, but the results show that tensile yield of the smaller flange can be equally important. Violent collapse behaviour is shown to be possible whether the smaller flange yields in compression or tension. This is particularly so at intermediate slenderness. The results apply most directly to unsymmetrical I-sections, but are also relevant to T-sections, and to wide stiffened compression panels. The Calladine construction greatly elucidates behaviour.     

Three-dimensional static solutions of rectangular plates by variant differential quadrature method

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.     

Application of Trefftz theory in thin-plate buckling with in-plane pre-buckling deformations

The paper presents the application of Trefftz initial stress theory in determining the elastic buckling load of rectangular thin plates under in-plane stress resultants, with allowance for in-plane pre-buckling deformations. Exact buckling solutions for simply supported plates are obtained and compared with earlier results furnished by Ziegler (1983, Ing.-Arch.53, 61 [1]). Although somewhat different, due to the use of different theories, both sets of results show that the buckling loads may be reduced when pre-buckling deformations are accounted for. This observation is contrary to that for column buckling where it has been shown that the allowance for pre-buckling shortening results in a higher buckling load. This reduction in plate-buckling load is significant especially for plates with large thickness-to-span ratios and subject to a combination of tensile and compressive in-plane stress resultants that increase pre-buckling deformations.