Improved numerical integration for locking treatment in isogeometric structural elements, Part I: Beams

A general mathematical framework is proposed, in this paper, to define new quadrature rules in the context of B-spline/NURBS-based isogeometric analysis. High order continuity across the elements within a patch turned out to have higher accuracy than C⁰ finite elements, as well as a better time efficiency. Unfortunately, a maximum regularity accentuates the shear and membrane locking in thick structural elements. The improved selective reduced integration schemes are given for uni-dimensional beam problems, with basis functions of order two and three, and can be easily extended to higher orders. The resulting B-spline/NURBS finite elements are free from membrane and transverse shear locking. Moreover, no zero energy modes are generated. The performance of the approach is evaluated on the classical test of a cantilever beam subjected to a distributed moment, and compared to Lagrange under-integrated finite elements.