A discrete differential geometry-based approach to numerical simulation of Timoshenko beam

Abstract We report a discrete differential geometry-based numerical method for the simulation of geometrically nonlinear dynamics of thick beam — known as Timoshenko beam. Our numerical framework discretizes the beam into a number nodes and uses the degrees of freedom of each node – position and rotation angle – to construct discrete elastic energies. Equations of motion resulting from balance of forces are formulated at each degree of freedom. These equations are integrated using a second order, implicit Newmark-beta time marching scheme. We find that the structural rigidity and natural frequency computed in Timoshenko beam framework are always lower than the one obtained using Euler–Bernoulli beam method for both naturally straight and curved beams. For quantitative comparison, we analytically solve the Euler–Lagrange equations using both Euler–Bernoulli and Timoshenko beam theories for a number of examples. A good match between the analytical solution and the numerical results in the geometrically linear regime indicates the correctness of our discrete model. The simulation can seamlessly handle geometrically nonlinear deformation that is often not amenable to an analytical approach.