## 1 Introduction  
The occurrence of cracks in the structure is considered to be one of the most unfavorable effects, since their presence may lead to the collapse in extreme cases. It is well known that the stress and strain in structural elements such as beams or columns increase significantly at the location of cracking. This leads not only to a reduction in stiffness but also to a change in the response pattern, the magnitude of which depends mainly on the location and depth of the crack. In the static analysis of structures, cracked beams are usually considered as a one-dimensional continuum with local stiffness reduction, since such models allow simpler and faster but still reliable response calculations. When the stress distribution near the crack is not of interest, a simplified computational model in which two adjacent elastic non-cracked segments are connected at the crack location by a corresponding (rotational and/or translational) point spring is particularly suitable. In addition, such a model allows an effective representation of the depth and location of the crack. In the pure bending of slender Euler-Bernoulli beams, the presence of cracks mainly affects the bending stiffness, so the point spring can be justifiably represented by the rotational contribution only [1].  
This rotational spring can also be effectively implemented into the computational model of cracked beams on an elastic soil, which has already been confirmed by many studies in the field of structural mechanics [2; 3; 4; 5]. Most studies idealize soils by the classical one-parameter Winkler model [6], in which the real compressive strength of soils is represented by individual and unconnected vertical compression springs. Furthermore, a recent study introduces a new exact finite element for static analysis, which contains accurate interpolation functions obtained from the differential equation solutions of a transversely cracked beam on Winkler soil [7].  
In addition to Winkler's compressive contribution, the mechanical strength of the actual soil includes the shear contribution of deformations. Therefore, an additional parameter is included for a more detailed description of the soil behavior that also takes into account the shear interaction between Winkler springs. By implementing the shear interaction between the springs as a supplementary parameter, deformations and displacements of the soil can be calculated even outside the loaded area, allowing the behavior of the soil to be described much more realistically compared to single parameter formulation. In the two-parametric model, the actual shear interaction between Winkler springs is considered by introducing a second soil parameter. Several two-parametric models were presented by many authors based on different physical assumptions. The shear interaction between springs was first represented by an elastic membrane layer ([PERSON] model), then by an elastic beam (Heteny model), and finally by an elastic shear layer (Pasternak model), which is the most natural representation of the shear contribution of the soil. Generally, all these models mathematically share the same governing differential equation [8].  
Although the analytical approach to solving differential equations provides exact solutions, the finite element method is primarily used in modern static analysis of beams on elastic soil, which is mostly due to the mathematical complexity of the analytical solution of differential equations. The accuracy and speed of convergence of the results depend not only on the applied mesh of elements but also on the quality of the applied finite elements.  
Therefore, the novelty of this article is a closed-form original finite element for static bending analysis of the uniform Euler-Bernoulli transversely cracked beam on two-parametric soil. All expressions were derived from newly derived cubic polynomial interpolation functions in which the second soil parameter was included directly.