Elastic-plastic analysis with pre-integrated beam finite element based on state diagrams: Elastic-perfectly plastic flow

The behaviour of a structure beyond its elastic limit involves a non-linear analysis, commonly conducted in engineering design with the benefit of Finite Element Analysis. Many structures are an assembly of one-dimensional components (beams) to form a frame. There are two main FE approaches to studying axial plastic propagation and the distribution within the cross-section of a beam: solid elements or beam elements with cross-section sampling (fiber section). The first requires a considerable number of d.o.f.’s to obtain reliable results; therefore, it is not generally used for extensive size modelling; the second, although requiring less computational efforts, necessitates an extensive sampling of cross-sections to evaluate the plastic distribution at every fiber. Furthermore, a high number of elements are needed to assess the plastic propagation. This paper provides a new approach that allows to considerably reduce the numerical heaviness by providing two ideas: i) elastic-plastic State Diagrams that allow the pre-integration of the cross-section response (that overcame the sampling within the section) ii) section-slices subdivision, that enable to build-up the tangent stiffness matrix of an entire partially plasticized beam (whatever is the plastic distribution and the section shape), with the minimum number of d.o.f.’s. The pre-integrated approach here developed assume planar bending and an elastic-perfectly plastic flow. The proposed element is validated by three test cases, comparing with the solutions obtained through FEA using solid elements and fiber-section beams. A significant enhancement in the overall efforts emerges, obtaining very similar results among the three approaches but with a considerable reduction of the computational time when using the proposed element.