Theoretical and experimental research on flexural-torsional buckling of perforated cold-formed steel lipped channel beam-columns

Two applications of the principle of stationary potential energy to the finite straining of a neo-Hookean (rubberlike) material are given in this paper. The major purpose of the work presented is to illustrate the suitability of energy methods for the solution of problems in finite strain theory since the literature of the subject does not contain mention of such solutions. One problem not amenable to the usual inverse methods of finite elasticity is studied approximately. The other problem, involving a stability question of an unusual sort, is handled with ease by means of the energy principle.

*† This study presents a semi-analytical solution method to perform both pre-buckling and bucking analyses of laminates having a cutout by employing a simple {3,0}-order plate theory. In this theory, the in-plane and out-of-plane displacement fields are respectively assumed in the form of cubic and uniform through-the-thickness expansions. The cubic expansion ensures the correct behavior of transverse shear deformations while satisfying the condition of zero transverse shear stresses at the laminate faces, thus avoiding the computation of the shear correction factor. This semi-analytical method utilizes the principle of stationary potential energy in order to derive the equations of equilibrium for the prebuckling and buckling states of the laminate with a cutout. Buckling load comparisons against the classical laminate and higher-order theories prove this semi-analytical method credible. I. Introduction AMINATED panels with cutouts are common structural components of the aircraft industry. The cutouts in these panels are unavoidable because they provide access to various aircraft sections. These panels are made of advanced composite materials whose in-plane tensile moduli are several times higher than their transverse shear moduli, thus making transverse shear deformations a crucial factor. Furthermore, these panels are prone to buckling behavior under compressive loading. Hence, the accurate design of these composite panels requires an analysis tool that is capable of predicting the pre-buckling and buckling responses of these panels while accounting for the presence of cutout geometry and the effect of transverse shear deformations in order to accurately predict the complete stress field. An analytical prediction of buckling response of laminates in the presence of cutouts is rather difficult to obtain. This is mainly because of the difficulties arising from the cutout geometry and the governing stability equation that render the problem analytically intractable. For this reason, the majority of the investigations resorted to semianalytical procedures. 1-3 In these semi-analytical approaches, the complex potential theory was utilized to predict the pre-buckling response of plates while energy-based series approximations were adopted to achieve the buckling response of the plates with cutouts. However, their buckling analysis did not, mathematically, take into account the effect of cutout geometry in their solution forms. Later, Barut and Madenci 4 included the presence of cutouts in their buckling analysis formulations by assuming the out-of-plane displacement field near the cutout in the form of a complex Laurent series. With this approach, they studied the buckling response of laminated plates with eccentrically located cutouts under compression. None of the aforementioned semi-analytical solution techniques, however, include the effect of transverse shear deformations in their formulations. As for the buckling analysis of laminates with cutouts in the presence of transverse shear deformation, Hadi and Matthews 5 performed the prebuckling analysis using the complex potential theory in conjunction with the boundary collocation method, and utilized Mindlin’s theory along with the principle of stationary potential energy for the buckling of a sandwich panel with a cutout. In view of the scarcity of semi-analytical methods concerning the stress and buckling analysis of moderately thick laminates with a cutout, the objective of this study is to develop a semi-analytical solution method for stress and buckling analysis of shear deformable composite laminates with central elliptical cutouts based on the {3,0}order laminated plate theory introduced by Ray.

Note on the principle of stationary complementary energy applied to free vibration of an elastic body

A Note on the Principle of Stationary Complementary Energy in Nonlinear Elasticity

The present study develops a theory for the elastic analysis of a preloaded wide flange steel beam, strengthened with two glass fiber-reinforced polymer (GFRP) plates bonded to both flanges, then subjected to additional loads. Starting with the principle of stationary potential energy, the governing equilibrium equations and corresponding boundary conditions are formulated prior to and after GFRP strengthening. The resulting theory involves four coupled equilibrium equations and 10 boundary conditions. A general closed form solution is then provided for general loading and boundary conditions. Detailed comparisons with three-dimensional finite-element solutions show that the theory provides reliable predictions for the displacements and stresses. A parametric study is then developed to quantify the effects of strengthening, GFRP plate thicknesses, and preexisting loads on the capacity of the strengthened beam.

Cylindrical storage bins with flat bottom and conical roof have been analyzed for grain and wind loads using the equations of shell bending theory. Both the cases of static and flow grain pressures are considered. The cylindrical container and the conical roof are made of corrugated and plane steel sheets respectively. The boundary conditions at the top of the cone with a ring beam and the cone cylinder junction are derived from the principle of stationary potential energy. The governing equations for the isotropic cone and the orthotropic cylinder subjected to the appropriate boundary conditions are solved for the stresses and displacements using a numerical procedure of piecewise polynomials in conjunction with the partition method. For a considerable part of the cylinder the circumferential moment must be taken into account for an accurate calculation of the hoop stress. Also the stresses produced are much higher during the flow of grain.

A novel approach to thick plate theory suggested by studies in foundation theory

Static and dynamic nonlinear behavior of a multistable structural system consisting of two coupled von Mises trusses

Starting with the principle of stationary potential energy, this paper develops the governing differential equations of equilibrium and boundary conditions for shear deformable thin-walled beams with open cross-section. Unlike conventional solutions, the formulation is based on a non-orthogonal coordinate system, in which the selected origin is generally offset from the section centroid. The exact solution of the resulting coupled differential equations of equilibrium is derived and used to develop exact shape functions. A finite element based on the exact shape functions is then formulated. Through a series of examples, the adoption of non-orthogonal coordinates is shown to enable the seamless modelling of structural members with eccentric boundary conditions and (or) stepwise cross-sectional variations.

Closed form solutions for shear deformable thin-walled beams including global and through-thickness warping effects

Large deformations under axial force and moment loads of initially flat annular membranes

Using the third-order shear deflection beam theory (TBT), nonlinear bending of functionally graded (FG) beams composed with various amounts of ceramic and metal is analyzed utilizing the differential quadrature method (DQM). The properties of beam material are supposed to accord with the power law index along to thickness. First, according to the principle of stationary potential energy, the partial differential control formulae of the FG beams subjected to a distributed lateral force are derived. To obtain numerical results of the nonlinear bending, non-dimensional boundary conditions and control formulae are dispersed by applying the DQM. To verify the present solution, several examples are analyzed for nonlinear bending of homogeneous beams with various edges. A minute parametric research is in progress about the effect of the law index, transverse shear deformation, distributed lateral force and boundary conditions.

Shell analysis of thin-walled pipes. Part II – Finite element formulation

A shear deformable theory for the analysis of steel beams reinforced with GFRP plates

A numerical method is developed for the analysis of general quadrilateral, moderately thick orthotropic plates having arbitrary boundary conditions. The procedure is based on the application of the Rayleigh–Ritz method in conjunction with the Reissner–Mindlin thick plate theory. A set of complete polynomials in terms of natural co‐ordinates comprising of boundary and domain terms, which satisfy the boundary conditions, is deployed as the basic functions to approximate the real displacement field. The generalized displacements and the eigenvalues are determined by imposing the principle of stationary potential energy. Although the procedure has the ability to solve problems involving thick quadrilateral plates, the numerical examples presented are mostly for skew plates. The results herein are compared with their counterparts determined by other investigators. Copyright © 2002 John Wiley & Sons, Ltd.