The study of varying weights in rational basis functions in isogeometric large-displacement analysis of beams

This study presents novel planar curved Euler-Bernoulli and Timoshenko-Ehrenfest beam elements for analyzing beam structures subjected to large displacements but small strains. The total Lagrangian description is employed, using the initial configuration as the reference for the kinematics. By assuming small strains and thin beams, higher-order terms of the Green-Lagrange strains are discarded, simplifying the formulations. Equilibrium is stated through the principle of virtual work. Subsequently, the proposed elements are formulated using the isogeometric analysis approach. Cubic non-uniform rational B-spline (NURBS) representations are employed for the Euler-Bernoulli beam element, while quadratic and cubic NURBS representations are used for the Timoshenko-Ehrenfest beam elements. In contrast to the traditional isogeometric analysis concept, the isoparameterization between the reference geometry and kinematic unknowns is relaxed by treating selected weights in the NURBS basis functions as degrees of freedom (DOFs). The obtained equation systems are inherently nonlinear with respect to the DOFs. To solve these nonlinear systems, this study employs the Newton-Raphson method, with linearization performed using MATLAB’s symbolic computing capability.The efficiency of the proposed elements and their superiority over constant-weight elements are demonstrated through several beam problems. The obtained solutions underscore the proposed elements’ effectiveness in analyzing beam structures that experience large displacements with complex behaviors, such as buckling and snap responses. Additionally, the varying-weight elements consistently outperform their constant-weight counterparts in providing highly accurate solutions. In this regard, the constant-weight elements always require significantly more DOFs than the varying-weight counterparts to achieve comparable accuracy. The advantage of the varying-weight elements becomes even more apparent in extreme regions such as buckling points, limit points, or substantial displacements.For Euler-Bernoulli beams, the superiority of the varying-weight element is also evident in its ability to accurately recover strain measures. While the cubic varying-weight element exhibits no locking effects, the cubic constant-weight element shows slight locking effects in some problems. For Timoshenko-Ehrenfest beams, the varying-weight elements effectively mitigate locking effects, confirming their reliability for solving thin-beam problems. Furthermore, while maintaining their locking-free characteristics, the varying-weight elements yield accuracy comparable to that of the locking-free B-bar elements but with significantly lower computational costs. This effectively addresses the trade-off challenge faced by B-bar elements between mitigating locking effects and maintaining computational efficiency.