engleski

Updated Lagrangian formulation for nonlinear stability analysis of flexibly connected thin-walled frames

Frames composed of thin-walled sections are extensively used as load-carrying structures in engineering applications. Unfortunately, such weight-optimized structural components, especially those with open profiles, are commonly susceptive to instability failure. Stability limit state evaluation is frequently carried out under the assumption of idealised connection behaviours, i.e. structural connections are assumed to be either fully restrained or frictionless pinned. Stability analysis of such structures can be performed using two different approaches, both of them furnish two different kinds of valid and important information. In the first case, the stability analysis, known as the linear one, is performed in an eigenvalue manner and it attempts to determine the instability load in a direct manner without calculating the deformations. The lowest eigenvalue is recognised as a critical or buckling load and the corresponding eigenvector represents a corresponding critical or buckling mode. In this, ideal or perfect structures and loading conditions are considered and the pre-buckling deformations prior to the attainment of buckling load are neglected. In the second case, stability problems are approached by load-deflection manner, which attempts to solve a stability problem by predicting the structural behaviour through the entire range of loading, including pre-buckling as well as post-buckling phase. This approach, known as the nonlinear stability analysis, when compared with the linear one, is procedurally more complex and computationally more expensive, but provides information more reliably for imperfect or real structures and loading conditions with or without material nonlinearity, for which the eigenvalue approach generally gives overestimated results. To perform such an analysis for a beam-type structure, a nonlinear beam model should be available and the corresponding load-displacement behaviour of a structure can be obtained by one of the incremental descriptions such as the total Lagrangian (TL), updated Lagrangian (UL) or the Eulerian description. This paper presents a two-nodded (14 d.o.f.) beam element for the numerical simulation of nonlinear stability behaviour of flexibly connected space frames composed of the straight and prismatic thin-walled beam members. Within the framework of UL formulation, the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping as well as second-order displacement terms due to large rotations, the equations of equilibrium are firstly derived of a straight beam element. Internal moments are represented as the stress resultants calculated by the engineering theories, i.e. the Euler-Bernoulli-Navier theory for bending and the Timoshenko-Vlasov theory for torsion. Due to the nonlinear displacement field, the joint moment equilibrium conditions of adjacent non-collinear elements are ensured. The influence of connection behaviour is introduced in the numerical model by transforming stiffness matrices of a conventional beam element. For this purpose a special transformation matrix is derived. The effects of partially restrained warping are also considered.

Thin-walled framed structures; Nonlinear stability; Beam element; Updated Lagrangian formulation; Flexible connections

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