“Computational Civil Engineering - "Intersections" International Journal

“Computational Civil Engineering 2005”, International Symposium 69can be computed [2]. A particularly important feature of the present method isbased on Green’s integration formula according to which the domain integralsappearing in the evaluation of internal resultant efforts and tangent stiffness matrixcoefficients of the section can be evaluated in terms of boundary integral [2]. Thenthe boundary integrals can be evaluated numerically using the Gauss-Lobattointegration rule.This approach is extremly rapid because stress integrals need only be evaluated at asmall number of points on the section boundary. In addition rapid convergence isassured by the inclusion of exactly determined tangent stiffnesses and, of greatimportance, it is assure convergence for any load case [2]. In this way, the states ofstrain, stress and yield stress are monitored explicitly during each step of theanalysis and the arbitrary cross-sectional shape and reinforcement layout, the effectof concrete tensile cracking, the nonlinear compressive response and strainsoftening of concrete in compression with different levels of confinement, areaccurately included in the analysis.2.2 Elasto-plastic tangent stiffness matrixFlexibility-based method is used to formulate the distributed plasticity model of a3D frame element (12 DOF) under the above assumptions. An element isrepresented by several cross sections (i.e. stations) that are located at the numericalintegration scheme points (Fig.2). The bending moments in these stations arecomputed through a second order transfer matrix as functions of the nodal elementforces following a detailed procedure given in [1].SteelConcretei≡PI1yIP2IP3IP4j=IP5xzL/2L/2Fig.2. 3-D fiber beam-column elementThe spread of inelastic zones within an element is captured considering the variablesection flexural EI y and EI z and axial EA rigidity along the member length,depending on the bending moments and axial force level, concrete cracking andnonlinear mechanical properties of the concrete in compresion and variation ofreinforcement bars in size and location along the member length. The tangent