SUMMARYA three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined. Elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures. The continuum-type equations of plastic model of J, mixed hardening are transformed into the beam equations by satisfying beam hypotheses. An effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so-called multistep method which is a method of successive control of residuals on yield surfaces. It avoids severe divergence when the displacement increments become large which is usual for the continuation methods. Material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix. Different from other elements, the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion. A great number of numerical examples are analysed and compared with the literature. The proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre-buckling and large deflection collapse of spatial framed structures.
Summary
In this paper, a 3‐node C0 triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second‐order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first‐order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node‐based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.
In the context of kinematic models, a general procedure for the bending crush analysis of thin-walled members is proposed and applied to the trapezoidal and rectangular tubes in bending. The main emphasis of this study is given to the determination of collapse mode parameters and the prediction of post-collapse behaviors of the structural member. The mode shapes are determined by minimizing plastic energy rate of the members. Moment-rotation curves are obtained by differentiating the absorbed plastic energy with respect to the rotation angle. For small rotation angles the solution is modified by considering elastic buckling and plastic collapse moments. Through several examples a full range of post-collapse response of the thin-walled members is shown to be effectively calculated by the proposed method without making any recourse to experimental data.
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