Integration schemes in stiffness- and flexibility-based frame elements Francesco Marmo, Luciano Rosati Dipartimento di Ingegneria Strutturale, Universit` a di Napoli Federico II, Italy E-mail: f.marmo@unina.it, rosati@unina.it Keywords: Reinforced concrete, displacement formulation, flexibility formulation, nonlinear frame element, fiber-free, fiber method, gaussian quadrature, lobatto quadrature. We present the results of a numerical investigation on the performances of the most common integration schemes, both at the sectional and element level, used in nonlinear beam elements based either upon a displacement or a force formulation. Specifically the sectional response at each control point of the beam element has been addressed by comparing the fiber method and the fiber-free approach. At the element level, the Gauss and the Lobatto quadrature schemes have been investigated since both of them are widely used in beam elements for nonlinear analysis of framed structures. 1 INTEGRATION SCHEMES AT THE SECTION LEVEL The section internal forces and stiffness matrix of a beam element are expressed as 2D integrals extended to the beam cross section. In particular two integration schemes have been considered here: – the fiber method which is pervasively adopted in commercial and scientific softwares [3] and is based on the partition of the section in a number of sub-sections called fibers. The stress resultants 0 2 4 6 8 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 y x z 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 Top displacement [m] Base shear [MN] Push-over curve Figure 1: Framed structure and relevant push-over curve 1