Approximate Analysis of Post‐Limit Response of Frames (original) (raw)
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Nonlinear dynamic analysis of frame structures
Computers & Structures, 1987
A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida er af., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.
Postbuckling Analysis of Flexible Elastic Frames
Journal of Applied and Industrial Mathematics, 2018
A completely geometrically nonlinear beam model based on the hypothesis of plane sections and expressed in terms of engineering strains and apparent stresses is applied to the structural analysis of frames. The numerical results are obtained by the Raley-Ritz method with a representation of solutions as a sum of analytical basis functions which were previously proposed by the authors. The convergence of approximate solutions is investigated. High degree of accuracy is demonstrated for both determination of the solution components and the fulfillment of equilibrium equations. It is shown that the limit values of external loads can substantially differ from those predicted by the Euler buckling analysis, which may lead to catastrophic consequences in designing thin-walled structures.
Finite element post-buckling analysis for frames
International Journal for Numerical Methods in Engineering, 1977
Presented in this paper is a direct method of analysis of the elastic non-linear behaviour of frames. Emphasis is given to the method's capability of tracing the post-buckling path from a bifurcation point although the method can also trace the non-linear behaviour of frames with eccentricities. The method is proposed as an alternative to the main methods currently in use, the perturbation method and the incremental method. Conditions for equilibrium and stability are developed from a variational approach to the total potential. A finite element approximation is made and an efficient solution technique for the resulting non-linear equations is developed. Results for three frames are given demonstrating good agreement with solutions generated from other approaches.
Natural approach for geometric non-linear analysis of thin-walled frames
International Journal for Numerical Methods in Engineering, 1990
A new stiffness matrix for geometrically non-linear analysis of three-dimensional beam-columns with bisymmetrical, thin-walled, I-type cross-sections is presented. The displacement field is described in terms of natural and rigid-body displacements and both Cartesian and natural matrices are derived. A novelty of the resulting method is the use of this dual description in all aspects of the analysis including the computer implementation. A companion paper' that introduces the plastic behaviour of the material also justifies the need for this dual approach as a means to separate nodal forces from stress resultants in the cross-section of the 3D beam-column element. The current development uses the updated Lagrangian formulation and corrections in the element matrix are made to properly consider the behaviour under finite rotation. It is shown that accuracy and convergence of the solution process can be improved if the natural matrices derived here are used in a new force recovery process that is insensitive to rigid-body displacements.
International journal for numerical …, 1994
Adopting an updated Lagrangian approach, the general framework for the fully non-linear analysis of thin-walled framed structures is developed using a simple, two-node, Co-model (HMB2). The governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified Hellinger/Reissner type with independent assumptions for displacement and strain fields. All coupled significant modes of deformations, i.e. stretching, bending, shear, torsion and warping, are accounted for in the generalized-beam theory employed. Emphasis is placed on devising effective solution procedures to deal withjnite rotations in space, particularly with regard to their effect on the derivations of load-correction matrices corresponding to conjguration-dependent externally applied forces/moments. The effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space.
Post-buckling behavior and imperfection sensitivity of L-frames
2005
Beginning with the work of Koiter in 1945, valuable insights into the postbuckling behavior of structures have been gained by Lyapunov-Schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point. Here this methodology is generalized to include nonlinear prebuckling behavior, as well as multiple, not necessarily coincident buckling modes. The expansion of the reduced equilibrium equations is performed about a reference point (which need not coincide with any of the bifurcation points), and applies no matter whether the modes are coincident, closely spaced, or well separated. From a variety of possible decompositions of the admissible space of displacements, two are incorporated into a finite element program. Theoretical considerations, and numerical examples in which asymptotic results are compared to 'exact' results, indicate that one of the decompositions has some important advantages over the other. Examples include a shallow arch, and a beam on elastic foundation problem exhibiting symmetry-breaking modal interaction.