“Computational Civil Engineering - "Intersections" International Journal

70 C. Chioreanflexural rigidities for major and minor axis bending under conditions of constantaxial load are evaluated by inverting the cross-section tangent stiffness matrix (1),imposing the condition of constant axial load to obtain a flexural flexibility matrixand inverting this to find the tangent flexural rigidities at constant axial load [2]. Asaforementioned, the cross-section stiffness is modeled by explicit integration ofstresses and strains over the cross-section area and then tangent stiffness propertiesof the cross sections are integrated along the member length to yield memberstiffness coefficients [1].2.3 The second-order effects on tangent stiffness matrixThe geometrical nonlinear effects for each element are taken into account in thepresent analysis, in a beam column approach, by the use of the inelastic stabilitystiffness functions and updating at each load increment the length, axial force andthe flexural rigidity about of each principal axes of the element. This wayminimizes modelling and solution time, generally only one or two elements areneeded per member.2.4 Member lateral loadsTo perform the nonlinear analysis of frame structures, in the majority of previouspublications, the loads are assumed to apply only at the nodes. In the presentinvestigation, the loading due to the member lateral loads and transferred to thenodes are allowed and included automatically in the analysis. This leads to asignificant saving in imputing the member loads, without the need to divide amember into several elements for simulation of these loads. The elasto-plasticequivalent nodal forces transferred to the nodes, from the member loads, will notbe constant during the analysis, and will be dependent on the variable flexuralrigidity along the member according with the process of gradual formation ofplastic zones [1].2.5 Diaphragm actionFor normal building frameworks, a floor slab may be modelled as a rigiddiaphragm, which is assumed to provide infinite in-plane stiffness and without anyout of plane stiffness. The lateral response of the floor slab is characterized by twotranslational and one rotational degrees of freedom located at the floor masternode. The multi-freedom constraints, required by the rigid body floor model, areimposed by augmenting the finite element model just described, with the penaltyelements. The main advantages of this technique with respect to the traditionalmaster-slave elimination technique is its lack of sensitivity with respect to whetherconstraints are linearly dependent and its straightforward computerimplementation. Considering the homogenous constraints Au=0 the penaltyaugmented system can be written compactly as: