An improvement for the Euler-Bernoulli fiber force-based beam through Simpson Integration

Abstract

This work proposes an improvement to the formulation of the Euler-Bernoulli fiber beam force-based element by incorporating the Simpson integration scheme for 2D field, for the integration of the stresses field over the cross section. This integration is done for the calculation of the element’s general forces (N,M) as well as the element’s flexibility matrix in both the bending and the torsional terms prior to the inversion for obtaining the stiffness matrix. This improvement has been employed in the open source computational mechanics code MSolve and a comparison with the classic trapezoidal rule integration that is implemented in Ansys and Opensees is performed. The results indicate that the proposed model provides a more robust integration for the flexibility matrix and subsequently for the stiffness matrix. Moreover, the stability of the load displacement curve in cyclic nonlinear analysis is increased, and the percentage divergence of both methods is substantially low. Specifically, in all examples, the largest relative divergence is in the order of magnitude of 5%. This applicability holds for different one-dimensional material constitutive models such as the combined nonlinear hardening, the Ramberg-Osgood and the Kent-Park concrete model. The aforementioned problems are in both uniaxial and biaxial bending, indicating the efficiency of the 2D Simpson integration scheme coherence. Finally, the examples are imposed with cyclic and monotonic static loading, which in computational terms are the most detrimental occasions for presenting a numerical instability. It is depicted that the proposed framework can result to a more stable and accurate simulations in nonlinear loading of structures.