Abstract
This paper derives an original finite element for the static bending analysis of a transversely cracked uniform beam resting on a two-parametric elastic foundation. In the simplified computational model based on the Euler–Bernoulli theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic members. The derivations of approximate transverse displacement functions, stiffness matrix coefficients, and the load vector for a linearly distributed load along the entire beam element are based on novel cubic polynomial interpolation functions, including the second soil parameter. Moreover, all derived expressions are obtained in closed forms, which allow easy implementation in existing finite element software. Two numerical examples are presented in order to substantiate the discussed approach. They cover both possible analytical solution forms that may occur (depending on the problem parameters) from the same governing differential equation of the considered problem. Therefore, several response parameters are studied for each example (with additional emphasis on their convergence) and compared with the corresponding analytical solution, thus proving the quality of the obtained finite element.
Highlights
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
FEM bending analysis of slender cracked uniform beams resting on a two-parametric soil was considered
The second soil parameter was directly implemented in the new transverse displacement interpolation functions
Summary
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. 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 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].
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