According to the elementary static bending theory, transverse and axial displacements for non-cracked straight structural elements are not interdependent. However, the occurrence of a transverse single-sided crack causes a redistribution of stresses within the non-cracked section and changes the deflection behavior mechanism along the element's axis due to the shift of the neutral axis deeper into the non-cracked cross-section. Consequently, both transverse and axial loading simultaneously induce transverse and axial (static and kinematic) quantities on transversely cracked structural elements. This paper presents a new analytical two-parameter model with an eccentric substitute rotational spring, focusing on more precise modeling of this phenomenon for slender structural elements. In this study, the crack is still modeled with a rotational spring, but displaced from the neutral axis of the non-cracked part of the element to its new position. This approach involves solving the differential equations for the transverse and axial displacements of non-cracked segments of the beam using the Euler–Bernoulli bending theory, where determining the integration constants is based on the continuity and boundary conditions. The model was initially validated using numerical examples of a cantilever beam with a crack, and the obtained results were compared to those of a more precise 2D model in the commercial software SAP2000. The results obtained show that the analytical model consistently predicts the transverse and axial displacements of the beam, achieving exceptionally low average relative errors. Additional analyses for different crack depths further confirmed that the simplified model's results align closely with the discrete results of the 2D model. The key finding of this research is that the new analytical model enables accurate prediction of the mechanical responses of beams with cracks, which is crucial for the safe design as well as maintenance of structures.