Exact closed‐form finite element solution for the bending static analysis of transversely cracked slender elastic beams on Winkler foundation

Abstract

Summary In this paper, bending problem of a transversely cracked beam resting on elastic foundations is addressed by means of the finite element method. The paper covers the derivation of a new finite element, where the soil is modelled by classical Winkler soil model, and the cracked beam is represented by the simplified computational model, as already widely used for various analyses of such transversely cracked slender beams. The derivation of transverse displacements' interpolation functions for the transversely cracked slender beam and the stiffness matrix, as well as the corresponding load vector for the linearly distributed continuous transverse load are presented. All expressions are derived at in closed symbolic forms, and they allow easy implementation in the existing finite element software. The presented approach is ideal for the effortless modelling of cracked beams in conditions where neither information about the crack's growth nor the stresses at the crack's tip is required. Numerical examples covering several load situations are presented to support the discussed approach. The results obtained are further compared with the values from corresponding coupled differential equations as well as from a large 2D plane finite elements model, where a detailed description of the crack was accomplished. It is evident that the drastic difference in the computational effort is not reflected in any significant difference in the results between the models.

Transverse displacements of cracked beams on Pasternak soil

“…All constants A ij and B ij ( i , j = 1, 2, 3, 4) appearing in the interpolation functions in equations (8) and (9) were recently derived [10].…”

“…The nodal load vector f q due to the action of a linearly distributed continuous load over the entire length of a finite element was already derived by using Winkler's soil model interpolation functions [10]. The vector's coefficients are defined as: …”

“…The Winkler's stiffness matrix K w has already been presented in closed form [10]. The complementary Pasternak's part K p of the complete stiffness matrix K was newly derived from the corresponding deformation energy U p .…”

“…Equation ( 18) is also applicable for the case of a uniform continuous load by considering q 1 = q 2 in the coefficients C j [10].…”

“…Subsequently, a more efficient 3‐node finite element using quadratic polynomial interpolation functions was presented [9]. In a recent study, the lack of polynomial interpolation functions was eliminated with interpolation functions obtained from exact solutions of the differential equations of a cracked beam on an elastic Winkler foundation [10]. Such a formulation leads to a computationally more efficient finite element, where all the transverse displacements’ values (either in the nodes or between them) are calculated accurately for any uniform continuous load.…”