Integration methods for fixed integration point at the end of force-based beam-column elements

The plastic hinge area generally appears at the ends because of the nonlinear material response of beam-column members, but there is no integration point fixed at the end of the element for Gauss-Legendre Integration that is commonly used in FE programs. So an integration method for the fixed integration point at the ends of the element is proposed and developed based on OpenSees. The study case shows that the result of Gauss-Radau integration (it places an integration point at only one end of the element) and Gauss-Lobatto integration (it places an integration point at each end of the element) is better than the result of Newton-Cotes integration (it also places an integration point at each end of the element), but they all perform better than the result of Gauss-Legendre integration. It is suggested to use Gauss-Radau integration or Gauss-Lobatto integration in the nonlinear analysis.