This article is based on the planar beam theories presented in Eugster and Harsch (2020) and deals with the finite element analysis of their presented beam models. A Bubnov-Galerkin method, where B-splines are chosen for both ansatz and test functions, is applied for discretizing the variational formulation of the beam theories. Five different planar beam finite element formulations are presented: The Timoshenko beam, the Euler–Bernoulli beam obtained by enforcing the cross-section’s orthogonality constraint as well as the inextensible Euler–Bernoulli beam by additionally blocking the beam’s extension. Furthermore, the Euler–Bernoulli beam is formulated with a minimal set of kinematical descriptors together with a constrained version that satisfies inextensibility. Whenever possible, the numerical results of the different formulations are compared with analytical and semi-analytical solutions. Additionally, numerical results reported in classical beam finite element literature are collected and reproduced.