Flexural - Torsional Buckling Analysis of Beams by BEM (original) (raw)
Related papers
Elastic flexural buckling analysis of composite beams of variable cross-section by BEM
Engineering Structures, 2007
In this paper a boundary element method is developed for the elastic flexural buckling analysis of composite Euler-Bernoulli beams of arbitrary variable cross-section. The composite beam consists of materials in contact. Each of these materials can surround a finite number of inclusions or openings. All of the cross-section's materials are firmly bonded together. Since the cross-sectional properties of the beam vary along its axis, the coefficients of the governing differential equation are variable. The beam is subjected to a compressive centrally applied load together with arbitrarily axial and transverse distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problems are solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The influence of the boundary conditions on the buckling load is demonstrated through examples with great practical interest. The flexural buckling analysis of a homogeneous beam is treated as a special case.
International Journal of Mechanical …, 2012
The problem of dynamic elastic buckling of Euler-Bernoulli beams subjected to axial loads is studied analytically. The dynamic axial loading is accomplished by a constant displacement rate of one end of the beam with respect to the other. The axial loading rates are considered slow enough to obviate the need to account for axial wave propagation effects. The dynamics of the beam is formulated in the modal domain considering the lowest static buckling mode. The dynamic stability of the beam is investigated as a response problem assuming an initial deformed geometry in terms of an eccentricity favouring the mode shape. The governing partial differential equation is condensed into a unified ordinary differential equation for various boundary conditions using a single dimensionless parameter in terms of beam geometry, material properties, loading rate, and appropriate coefficients corresponding to different boundary conditions. The results obtained for the dynamic response of beams for various values of the dimensionless parameter and initial eccentricity suggest that the solutions can be combined into a single unified analytical expression for the dynamic buckling load. It is shown that the corresponding dynamic response curves can also be collapsed into a single curve using the dimensionless peak load and the associated time parameter. The accuracy of the unified analytical expression for dynamic buckling is verified against exact solution of the ordinary differential equation considering three problems in terms of dimensional quantities for different boundary conditions and loading rates. Further the validity of the single mode dynamic buckling formulation is examined by comparing the results obtained for various boundary conditions with the numerical results from the dynamic response of a large number of degree of freedom finite element model of the beam without any restriction on the deformation mode shape. This unified solution has potential application in optimum design for dynamic collapse of truss type structures subjected to dynamic loads.
Latin American Journal of Solids and Structures
In this research, two stress-based finite element methods including the curvature-based finite element method (CFE) and the curvature-derivative-based finite element method (CDFE) are developed for dynamics analysis of Euler-Bernoulli beams with different boundary conditions. In CFE, the curvature distribution of the Euler-Bernoulli beams is approximated by its nodal curvatures then the displacement distribution is obtained by its integration. In CDFE, the displacement distribution is approximated in terms of nodal curvature derivatives by integration of the curvature derivative distribution. In the introduced methods, compared with displacement-based finite element method (DFE), not only the required number of degrees of freedom is reduced, but also the continuity of stress at nodal points is satisfied. In this paper, the natural frequencies of beams with different type of boundary conditions are obtained using both CFE and CDFE methods. Furthermore, some numerical examples for the static and dynamic response of some beams are solved and compared with those obtained by DFE method.
Buckling of thin-walled beams by a refined theory
Journal of Zhejiang University SCIENCE A, 2012
The buckling of thin-walled structures is presented using the 1D finite element based refined beam theory formulation that permits us to obtain N -order expansions for the three displacement fields over the section domain. These higher-order models are obtained in the framework of the Carrera unified formulation (CUF). CUF is a hierarchical formulation in which the refined models are obtained with no need for ad hoc formulations. Beam theories are obtained on the basis of Taylor-type and Lagrange polynomial expansions. Assessments of these theories have been carried out by their applications to studies related to the buckling of various beam structures, like the beams with square cross section, I-section, thin rectangular cross section, and annular beams. The results obtained match very well with those from commercial finite element softwares with a significantly less computational cost. Further, various types of modes like the bending modes, axial modes, torsional modes, and circumferential shell-type modes are observed.
Exact Solutions for the Elastic Buckling Problem of Moderately Thick Beams
Revue des composites et des matériaux avancés, 2020
We present the elastic buckling problem of moderately thick and thick beams as a boundary value problem of the classical mathematical theory of elasticity. The study considered homogeneous, isotropic, linear elastic beams. Small deformation assumptions were used together with kinematic, constitutive relations and the differential equations of equilibrium to obtain the governing field equations as a fourth order non-homogeneous ordinary differential equation (ODE) when both axial, compressive and transverse loads are considered, and a fourth order homogeneous ODE when only axial compressive force is considered. Using trial function method, the homogeneous ODE is solved in general for any end support conditions to obtain a general solution for the buckled beam in terms of four unknown constants of integration. The boundary conditions corresponding to the four cases of end support conditions considered were used to obtain the characteristic buckling equations, which were expanded to obtain transcendental equations with an infinite number of roots in each case, thus yielding an infinite number of buckling loads. The least root of the transcendental equations was used to obtain the critical buckling load, which was found to depend on the ratio h/l and the Poisson's ratio, . Critical buckling loads for each end support condition was calculated and tabulated. The results show that for each end support condition, as h/l < 0.02, the critical buckling load coefficient obtained was approximately equal to the critical buckling load coefficient of Euler-Bernoulli beam. As h/l > 0.02, which is the threshold for thin beams, the critical buckling load is found to be much smaller than the critical buckling load obtained from Euler-Bernoulli theory. It is thus concluded that the shear deformable theory is necessary for a more realistic analysis of the critical load buckling capacities of moderately thick, and thick beams for safety in their design.
Some notes on finite element buckling formulations for beams
Computers & Structures, 1994
Based on a formulation for the elastic distortional buckling of tapered I-beams, some observations have been developed that are germane to the finite element modelling of the lateral buckling of beams. In particular, it is shown that a commonly cited formulation omits a boundary term, and can in some cases lead to erroneous results. A new set of degrees of freedom for distortional buckling are proposed, and these may be modified easily to account for lateral buckling in the simplified case.
A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams
Journal of Mechanics of Materials and Structures, 2006
In this paper, the direct one-dimensional beam model introduced by one of the authors is refined to take into account nonsymmetrical beam cross-sections. Two different beam axes are considered, and the strain is described with respect to both. Two inner constraints are assumed: a vanishing shearing strain between the cross-section and one of the two axes, and a linear relationship between the warping and twisting of the cross-section. Considering a grade one mechanical theory and nonlinear hyperelastic constitutive relations, the balance of power, and standard localization and static perturbation procedures lead to field equations suitable to describe the flexural-torsional buckling. Some examples are given to determine the critical load for initially compressed beams and to evaluate their post-buckling behavior.
Computers & Structures, 2011
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.
2019
The finite Fourier sine transform method was used in this work to solve the elastic buckling problem of thinwalled beams for the case of pinned ends, and uniform moments applied at the ends. The problem is a boundary value problem given by a fourth order ordinary differential equation and Dirichlet boundary conditions at the pinned ends. The Dirichlet boundary conditions at the pinned ends make the finite Fourier sine transform method ideally suited for the solution. The transformation of the governing domain equation converted the problem to an algebraic eigenvalue problem. The condition for nontrivial solution was used to obtain the characteristic buckling equation as a fourth degree polynomial. The eigenvalues of the characteristic buckling equation were used to obtain the n buckling moments. The critical buckling moment was found to correspond to the first buckling mode. The expressions obtained for the n buckling modes and the critical buckling moment were identical to those by...
Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections
Thin-Walled Structures, 2013
In this paper, a theoretical and numerical model based on the power series method is investigated for the lateral buckling stability of tapered thin-walled beams with arbitrary cross-sections and boundary conditions. Total potential energy is derived for an elastic behavior from strain energy and work of the applied loads. The effects of the initial stresses and load eccentricities are also considered in the study. The lateral-torsional equilibrium equations and the associated boundary conditions are obtained from the stationary condition. In presence of tapering, all stiffness coefficients are not constant. The power series approximation is then used to solve the fourth-order differential equations of tapered thinwalled beam with variable geometric parameters having generalized end conditions. Displacement components and cross-section properties are expanded in terms of power series of a known degree. The lateral buckling loads are determined by solving the eigenvalue problem of the obtained algebraic system. Several numerical examples of tapered thin-walled beams are presented to investigate the accuracy and the efficiency of the method. The obtained results are compared with finite element solutions using Ansys software and other available numerical or analytical approaches. It is observed that suggested method can be applied to stability of beams with constant cross-sections as well as tapered beams.