<p>This work presents a new type of non-conventional finite elements (FE) that utilize trigonometric-based shape functions. The selection of the shape functions is inspired by the analytical expression of structural modeshapes. The proposed element consists of two “regular” nodes and a middle “hidden” node that basically enriches the local approximation and leads to the partition-of-unity property. Two different element types are constructed: a rod element with axial degrees of freedom and a Timoshenko beam element with two degrees of freedom: vertical displacement and rotation. Both the proposed elements are tested against conventional 3-node FE in free vibration and transient dynamic simulations of isotropic rod and beam structures. Numerical results show that the proposed trigonometric FE yield more accurate estimations of natural frequencies than the traditional 3-node FE. Also, the maximum natural frequency of each case is not only more accurate but also has smaller numerical value. This leads to the selection of larger time steps when employing explicit time integration, resulting in lower computing times. Finally, the presented elements evince higher convergence rates than the conventional 3-node FE in wave propagation simulations of rods and beams, further increasing the proposed method’s efficiency. This is explicitly quantified, since the proposed FE appears to be twice as fast as the conventional 3-node FE, in obtaining a transient wave response with the same level of accuracy.</p>