A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint

In this paper, a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity. For various end boundary conditions, we transform the governing equation with varying coefficients to linear algebraic equations; then a characteristic equation in critical buckling loads will be obtained. Several examples of estimating buckling loads under typical end supports are discussed. By comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams, it can be found that our method has fast convergence and the obtained numerical results have high accuracy. Moreover, the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties. The effects of gradient parameters on the critical buckling loads are elucidated. Finally, we give an example to illustrate the enhancement of the load-carrying capacity of tapered beams for admissible shape profiles with constant volume or weight. The proposed method is of benefit to optimum design of beams against buckling in engineering applications.     

Navier-Stokes calculations with a coupled strongly implicit method—I: Finite-difference solutions

Stone's unconditionally stable, strongly implicit numerical method is extended to the 2 x 2 coupled vorticity-stream function form of the Navier-Stokes equations. The solution algorithm allows for complete coupling of the boundary conditions. Solution for arbitrary large time steps, and for cell Reynolds numbers much greater than two have been obtained. The method converges quite rapidly without adding artificial viscosity or the necessity for under-relaxation. This technique is used here to solve for a variety of internal and external flow problems. Moderate to large Reynolds numbers are considered for both separated and unseparated flows. The procedure is extended to higher-order splines in Part 2 of this study.     

Thermal stress analysis of skew plates by finite element method

Thermal stresses are induced in general due to nonuniform temperature distribution or due to the boundary restriction. Most of the work reported so far deals with either plates with edges clamped in plane of the plate or plates with stress free edges. While studying buckling or post-buckling problems, one should ideally analyse the plates with mixed in-plane boundary conditions. Hence, in the present analysis, thermal stress analysis of skew plates with mixed in-plane boundary conditions using finite element approach is attempted. In addition, the effect of in-plane boundary conditions on the thermal stresses is also discussed.     

Buckling of cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of unstiffened, isotropic, circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. For all sets of boundary conditions except one (SSI), the results show that the panel buckling loads monotonically approach the complete cylinder buckling load from above as the panel width is increased. Low buckling loads, sometimes less than half the complete cylinder buckling load, are found for simply supported panels with free in-plane edge displacements (SSI). The SSI buckling loads are below the complete cylinder load even for ‘360° panels’. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction also significantly increases the buckling load. Parametric studies are also performed to determine the effects of variations in panel length and thickness on the buckling loads.     

Elastic buckling of columns by initial parameter method

The method of initial parameters is shown to provide a unifying approach to the solution of elastic buckling problems of columns. The method is explained for constant stiffness columns under arbitrary boundary conditions and is extended to cover columns with changes in the cross section. Some illustrative examples are solved and the advantages of the method over other methods of analysis are discussed.     

Bending analysis of rectangular moderately thick plates using spline finite element method

A bending analysis of rectangular, moderately thick plates with general boundary conditions is presented using the spline element method. The cubic B spline interpolate functions are used to construct the field function of generalized displacements w, φ_itxand φ_ity. The spline finite element equations are derived based on the potential energy principle. For simplicity, the boundary conditions, which consist of three local spline points, are amended to fit specified boundary conditions. The shear effect is considered in the formulations. A number of numerical examples are described for rectangular, moderately thick plates. Since the cubic B spline interpolate functions have sufficient continuity and are piecewise polynomial, so the present numerical solutions show not only that the method gives accurate results, but also that the unified solutions of thick and thin plates can be directly obtained; the trouble with the so-called shear locking phenomenon does not occur here.     

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.     

Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier-Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r<d. We derive the discrete Navier-Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.     

Linear buckling analysis of thin-walled members combining the Generalised Beam Theory and the Finite Element Method

A finite element procedure to carry out linear buckling analysis of thin-walled members is developed on the basis of the existing Generalised Beam Theory (GBT) and constrained Finite Strip Method (cFSM). It allows designers to uncouple the buckling modes of a finite element model and, consequently, to calculate pure elastic buckling loads. The procedure can easily be applied to members with general boundary conditions subjected to compression or bending. The results obtained are rather accurate when compared to the values calculated via GBT and cFSM. As a consequence, it is demonstrated that linear buckling analyses can be performed with the Finite Element Method in a similar way as can be done with the existing GBT and cFSM procedures.     

Application of differential quadrature method to the delamination buckling of composite plates

The delamination buckling response of a composite panel containing through-the-width delamination is numerically modeled using a solution that is based on the differential quadrature method (DQM). The composite is modeled as a general one-dimensional beam–plate having a through-the -width delamination that can be at any arbitrary location through its thickness; hence, dividing the domain into four regions. The DQM is applied to each region and with the imposition of appropriate boundary conditions, the problem is transformed into a standard eigenvalue problem. Numerical results are presented, illustrating the stability and validity of the method. The results also demonstrate the efficiency of the method in treating this class of engineering problem.     

A fast immersed interface method for solving Stokes flows on irregular domains

We present a fast immersed interface method for solving the steady Stokes flows involving the rigid boundaries. The immersed rigid boundary is represented by a set of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundary, singular forces at the rigid boundary are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are approximated using the cubic splines. The strength of singular forces is determined by solving a small system of equations via the GMRES method. The Stokes equations are discretized using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate Stokes flows on irregular domains.     

Buckling of eccentrically stringer-stiffened cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of eccentrically stringer-stiffened circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. As the panel width is increased, the results show that the complete cylinder buckling load is reached only for one set of boundary conditions (SS3, classical simple support conditions). However, for 180° and wider panels, the panel buckling loads are within ± 10% of the complete cylinder load for all cases except SS1 panels (free in-plane edge displacements) with outside stringers. Low buckling loads, as low as half the complete cylinder load, are found for some SS 1 panels. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction is more effective in increasing the buckling loads of panels with free circumferential edge displacement υ that it is for panels with υ = 0. From stringer-eccentricity studies, it is shown that buckling loads are generally at least 40–50% higher for the case of outside stringers, and that eccentricity effects are generally similar for clamped and simply supported panels with the same in-plane boundary conditions.     

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.     

Maximization of buckling load of laminated composite plates with central circular holes using MFD method

This paper addresses optimal design of simply supported symmetrically laminated composite plates with central circular holes. The design objective is the maximization of the buckling load, and the design variable is considered as the fiber orientation. The first-order shear deformation theory is used for the finite element analysis. The study is complicated because the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used to solve the optimization problems. Finally, the effect of different number of layers, boundary conditions, width-to-thickness ratio, plate aspect ratios, hole daimeter-to-width ratio, and load ratios on the results is investigated.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

A numerical and experimental investigation of the buckling behavior of composite panels

The experimental buckling behavior of axially compressed, fiber-reinforced, circular cylindrical panels is compared with numerical predictions obtained from an energy-based, finite difference, computer program. Test specimens were clamped along the curved edges and either unsupported or simply supported along the straight edges.Numerical predictions were obtained for a perfect cylindrical geometry and for small axially symmetric initial imperfections. Agreement between experimental and numerical buckling behaviors are in good agreement for both types of boundary conditions.     

A new formulation of the SADI method for the prediction of natural convection flows in cavities

A numerical method involving cubic splines is used to solve the classical problem of natural convection in cavities. The derivation of a new formulation of the spline alternating direction implicit method (SADI) is presented in detail. The technique developed allows the reduction of the usual block-tridiagonal matrix to three tridiagonal ones. The implementation of the method for the specific problem under consideration is also explained in detail. The advantages of such a formulation are that all the boundary conditions required by the spline method (function and its first two derivatives) may be obtained by using the basic cubic spline relations and that computational time is saved significantly. Solutions obtained for air enclosed in square cavities agree well with existing solutions. The results presented illustrate the capacity of the method to handle problems in natural convection at high Rayleigh numbers.     

Elasto-plastic buckling of curved webs

This paper presents an investigation on elastic buckling strength of curved girder webs subjected to uniform shears or bending stresses at the edges.An elastic 20 degrees of freedom finite element model was used to formulate the eigenvalue problem and a Gauss-Seidel iterative procedure was employed to yield the lowest critical edge loads.In case of pure bending, the investigation is extended into the plastic range. The deformation theory of plasticity in conjunction with a new formulation of the secant modulus is used to derive the elasto-plastic buckling equations. The same Gauss-Seidel iterative procedure was used to find the critical load for each assumed stress level. Further iterations with incremental stress were done to match the elasto-plastic buckling stresses. The material is assumed to be elastic-perfectly plastic and incompressible.In order to aid design professions, the dimensions of the web panel studied are within the practical ranges of curved plate girders. Four boundary conditions that represent various constrain conditions from flanges to stiffeners of plate girder designs, were considered.The results are presented in graphical forms. Interaction curves relating to various dimensionless parameters are constructed. Comparisons and convergence studies were made with existing available data. It is found that boundary conditions and aspect ratio influence the buckling stresses greatly. However, curvature effect is relatively insignificant over the range of practical application.     

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.     

Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.