Arbitrary Transverse Loads Research Articles

Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions

A new elegant, powerful and accurate superposition method is presented for isotropic/specially orthotropic rectangular plates with arbitrary transverse load and arbitrary combination of free/simply-supported/clamped/guided/elastically supported edges. All the component solutions used here are infinite series equivalents of the complicated closed-form Levy-type solutions employed in the conventional superposition method; it is shown that these series equivalents are easily derived. The mathematical equations pertaining to the various component solutions required for the application of this new method to any plate problem are clearly presented. A number of validation studies are carried out to verify the accuracy of the method. The method can be directly extended to the analysis of more complicated plates made of multifunctional or functionally graded materials.

State space formulation for composite beam-columns with partial interaction

A state space formulation is established for analyzing static responses of composite members with partial shear interaction under the combined action of an arbitrary transverse load and a constant axial force. Three generalized displacements (deflection, rotation angle, and interface slip) and three generalized forces (bending moment, shear force, and axial force) are combined into a state vector, which satisfies a state equation whose solution is easily obtained using matrix theory. The interfacial normal contact stress between the two subelements of a composite member is derived in order to check the validity of the basic assumption of identical deflection (or curvature) possessed by the two subelements. We find that, when a concentrated load alone acts on clamped–clamped or clamped–free beam-columns, tensile normal contact stress does appear at certain part of the interface. The formulation is then readily extended to analyze continuous composite beam-columns and inhomogeneous composite beam-columns. In particular, a non-continuous model of slip stiffness along the interface with discrete rectangular pulses is incorporated into the analysis, and the numerical results indicate a significant effect of slip stiffness ununiformity on the critical axial load as well as internal actions of the composite members.

Exact shape functions of imperfect beam element for stability analysis

In this paper, a versatile and efficient solution for the displacement function of a geometrical imperfect beam element under high axial load is derived. The initial imperfection is included by a sine curve approximation. The aim of this work is to propose a natural extension of the element by stability functions. For this purpose, exact displacement shape functions are solved from the differential equilibrium equation, by using arbitrary boundary conditions. These shape functions offer the possibility to express the equivalent nodal forces for any arbitrary transverse loads, and are further extended to obtain the equivalent nodal forces relating to the effect of the initial curvature without affecting the boundary conditions of the element. The accuracy of this element makes the convergence rate better than the conventional cubic element or other used elements. Euler’s critical load is computed readily with no more than two iterations for each step loading with a single element per member. By using these exact shape functions in computer software, the second-order analysis and the nonlinear integrated design become reliable and easy to use for practical purposes.

An exact finite element for a beam on a two-parameter elastic foundation: a revisit

An analytical solution for the shape functions of a beam segment supported on a generalized two-parameter elastic foundation is derived. The solution is general, and is not restricted to a particular range of magnitudes of the foundation parameters. The exact shape functions can be utilized to derive exact analytic expressions for the coefficients of the element stiffness matrix, work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. As illustration, each distinct coefficient of the element stiffness matrix is compared with its conventional counterpart for a beam segment supported by no foundation at all for the entire range of foundation parameters.

Static Analysis of Infinite Laminated Cylindrical Shell Panel

This paper presents an approximate three-dimensional elasticity solution for an infinitely long, cross-ply laminated circular cylindrical shell panel with simply supported boundary conditions, subjected to an arbitrary discontinuous transverse loading. The solution is based on the principal assumption that the ratio of the thickness of the lamina to its middle surface radius is negligible compared to unity. The validity of this assumption and the range of application of this approximate solution have been established through a comparison with an exact solution. Results of classical and first-order shear deformation shell theories have been compared with the results of the present solution to bring out the accuracy of these theories. It is also shown that for very shallow shell panels the definition of a thin shell should be based on the ratio of thickness to chord width rather than the ratio of thickness to mean radius.

Beams on generalized foundations: supplementary element matrices

A general analytical solution for the shape functions of a linear beam segment supported on a two-parameter elastic foundation is derived. The solution is not restricted to a particular range of magnitudes of the foundation parameters. The closed form shape functions are utilized to derive analytic expressions for work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. Each work equivalent nodal force and each coefficient of the element matrices is compared with its conventional counterpart for an ordinary beam segment supported by no foundation.

Improved engineering theory for uniform beams

A small static symmetric bending deformation of isotropic linear elastic beams under arbitrary transverse loading varying slowly with the axial coordinate is considered. An asymptotic analysis of the three-dimensional variational equation — in which the small parameter is the ratio of maximum cross-sectional dimension to beam length — gives Timoshenko type governing equations, corresponding boundary conditions, improved formulae for the displacements, and, unlike known beam theories, for all stresses a plane problem in the cross-sectional domain to be solved. Predictions of the theory for beams of narrow rectangular and circular cross-sections are compared with explicit elasticity solutions.

Improved engineering theory for uniform beams

A small static symmetric bending deformation of isotropic linear elastic beams under arbitrary transverse loading varying slowly with the axial coordinate is considered. An asymptotic analysis of the three-dimensional variational equation — in which the small parameter is the ratio of maximum cross-sectional dimension to beam length — gives Timoshenko type governing equations, corresponding boundary conditions, improved formulae for the displacements, and, unlike known beam theories, for all stresses a plane problem in the cross-sectional domain to be solved. Predictions of the theory for beams of narrow rectangular and circular cross-sections are compared with explicit elasticity solutions.

Post-buckling analysis of composite delaminated beams

A general one-dimensional geometrical non-linear model of a composite delaminated beam under arbitrary axial and transverse loading for any boundary conditions is derived for predicting the post-buckling behavior. The model comprises nonlinear equations based on the Von Karman kinematic approach, suitable for checking the effect of delamination and initial imperfection on the overall non-linear behavior. The differential equations, which incorporate the bendingstretching coupling effect, are solved by Newton's method using a finite-difference scheme. The delamination ratio parameter, bending stretching coupling and the initial imperfection are examined by means of parameteric analysis.

Transverse vibration analysis of plane trusses by analytical method

The type of plane truss considered in this paper is the Warren truss, which is built up with basic units consisting of four sets of bars pin-jointed at the nodes and subjected to transverse nodal loads. The natural vibration and dynamic response are studied for the simply supported truss with lumped masses at the nodes. Closed form solutions are deduced by using the U-transformation technique which leads to the uncoupling of the governing equations. In order to explain the procedure of the present method, the response functions for the nodal displacements of a truss subjected to the action of a suddenly applied load at a node are derived. It is obvious that the present method is also applicable to the forced vibration analysis of trusses subjected to arbitrary transverse loads applied at the nodes.

Exact solution for finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads

This paper deals with finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads. By the use of a method of variable replacement, the nonlinear differential equation with varied coefficient for the problem can be transformed into an equation with variable separable. The exact solution can be obtained by the integration method. Some examples are given in the paper, and the results of these examples show that this exact solution includes the existing solutions in references as special cases.

Effect of delamination on stability of laminated composite strip

A general one-dimensional model of a composite delaminated elongated strip (wide beam) under arbitrary axial and transverse loading and boundary conditions is used for predicting the classical buckling load, with the bending-stretching coupling effect as well as the prebuckling geometrical nonlinearity taken into account. The differential equations are solved by Newton's method, using a finite-difference scheme. The contribution of the coupling effect is examined by means of parametric analysis.

Fourier series solution for a rectangular thick plate with free edges on an elastic foundation

AbstractA Fourier series solution is presented for a system of first‐order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.

Finite difference solution of a system of first‐order partial differential equations

AbstractA finite difference solution of a system of first‐order partial differential equations, using a central difference scheme, is presented. The equations describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. The main deflections and stresses predicted by the method for a particular case are given, together for purposes of comparison, with results from a finite element analysis.

The general solution of bending of a spherical thin shallow shell with a circular hole at the center under arbitrary transverse loads

Basing several suggestions appeared in [1], we find out the general solution of the bending of a spherical thin shallow shell with a circular hole at the center. As we know, very few papers had touched upon these problems.

Transverse vibrations of viscoelastic shallow shells

A method for the time history of motion of shallow shells of viscoelastic material under arbitrary time-dependent transverse load is presented. The method is based upon the concept of iso-amplitude contour lines on the surface of the shell. It is shown that the time behavior can be found by using the frequency of free vibration of the associated elastic shallow shell. As an illustration of the technique, the problem of a shallow dome upon an elliptic base is discussed, all details of which are explained by graphs.

Nonlinear Analysis of Laminated Cross-Ply Plates

Nonlinear bending of unsymmetrically laminated cross-ply plates has been the subject of some investigations. Utilizing the multiple Fourier Series method the writers have examined the post-buckling of loosely clamped cross-ply plates. Nonlinear bending of rigidly clamped cross-ply plates has been considered by one of the writers using the perturbation method. Zaghloul and Kennedy have used the finite-difference technique to investigate the large deflections of unsymmetric cross-ply plates with all-clamped and all-simply supported edges. They have modified the boundary conditions to render the analysis simple but have not described how the boundary condition for zero normal bending moment at the edges of a simply supported plate. In this note the method is extended to study the large deflection behavior of a simply supported unsymmetric cross-ply rectangular plate under arbitrary transverse loading. The notations used are adopted in the present analysis.

ON THE BENDING OF SIMPLY-SUPPORTED COMPOSITE PLATES

Most previous treatments of problems of bonded materials in linear elastostatics do not establish the possible functional relationship between the fields in the single and composite material. In this paper, the interior of one of two orthotropic semi-infinite plate strips joined end-on and simply-supported along the longitudinal edges is subjected to an arbitrary transverse load. Using a real-function approach, it is shown that the fields for the composite plate are directly computable from the corresponding fields for the homogeneous infinite plate, and that the connecting relations remain unchanged irrespective of the shape of the loaded area. A knowledge of this fact gives important advantages in economy and simplicity.

A comparison of approximate methods for the solution of plate bending problems.

It is well known that classical, exact methods of analytical solution cannot be applied to the plate bending problem in cases of irregular boundary shape and arbitrary transverse loading and constraint. The paper summarizes and compares results obtained from using nine approximate analytical methods: point matching, boundary point least squares, TrefftzMorley, interior collocation, interior least squares, subdomain, Galerkin, Ritz, and Kantorovich. For purposes of concrete comparison, each of the methods is applied to two problems for which exact, although intricate, solutions are known: 1) a uniformly loaded, simply supported elliptical plate; 2) a square plate having free edges, supported at four asymmetrically located interior points, and loaded by its own weight. Comparative results are presented. Each technique is rated good, fair, or poor according to 11 important technical criteria and the underlying rationale is explained.

Simultaneous bending and compression (or tension) of frames composed of members with linearly varying depth of cross sections

The paper deals with the derivation of the formulae for the initial end bending moments and shear forces, occuring in the case of bars having a constant width of cross sections while their depth varies linearly. The bars are subjected to the axial forces and an arbitrary transverse load. These formulae together with the slope deflection equations for the displacement method deduced in a recent paper make it possible to solve the problems of simultaneous bending and compression or tension of framed structures composed of bars with varying cross sections in a way specified in the paper. An illustrative example is given.