PSI - Issue 39

Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000

13

5

2.2. Stresses into a semi-infinite elastic plate subjected to line loading For the aim of calculating the stresses in a body due to a patch loading, an elastic body subjected to a normal surface traction p(s) and a tangential surface traction q(s) acting on a portion of its surface from –c ≤ x ≤ c , see Figure 3, can be approximated as an elastic half-space if its dimensions are large compared with the extension of the contact, i.e. if the stresses due to contact tend to zero within the body. The validity of this hypothesis will be checked in the following sections for the considered geometry, by comparing the results with those of a finite element model. Under the normal and tangential patch loads p(s) and q(s) , the normal and tangential stresses at the point ( x,z ) can be calculated by making use of the solution for a tangential and normal force, which are available from theory of elasticity. This is done considering the superposition of the stress states caused by the summation of single forces p ds and q ds acting on an infinitesimal area ds , considering the solutions for single forces, see Johnson (1987):

p ( s )( x-s ) 2 [( x-s ) 2 +z 2 ] 2 c p ( s ) [( x-s ) 2 + 2 ] 2 c p ( s ) ( x-s ) [( x-s ) 2 + 2 ] 2 c

q ( s )( x-s ) 3 [( x-s ) 2 +z 2 ] 2 c q ( s )( x-s ) [( x-s ) 2 + 2 ] 2 c q ( s )( x-s ) 2 [( x-s ) 2 + 2 ] 2 c - c - c - c

π � π � 2z 1 2 π �

2 π � 2 π �

2z 1

- c - c

(6)

σ x =-

ds -

ds

2z 1 3

(7)

σ z 1 =-

ds -

ds

2 π �

τ x 1 z 1 =- (8) where p(s) and q(s) are the normal and tangential pressure distributions, and s is depicted in Figure 3. A closed form solution of the integrals of Equations 6-8 might exist, depending on the functions p(s) and q(s) . Irrespective of this, for the goal of this paper, the solution is obtained by numerical integration. - c ds - ds

s

p(s)

q(s)

x

c

c

z

Elastic half-space

Fig. 3. Mechanical scheme of the cracked beam supported by the elastic foundation under a moving contact load.

2.3. Beam supported by an elastic foundation The scheme of a beam supported by an elastic foundation is used to determine the internal forces and moments in the rail. In this model, some real conditions of railway tracks are neglected, with the advantage that closed form solutions exist for some basic load cases, see Hetényi (1946).This formulation assumes that the elastic foundation is a bilateral constraint, therefore reacting also with a tensile reaction. This approximation reflects an ideal condition which reasonably approximates the case of a ballasted track for realistic values of section properties and stiffness of the track foundation, see Esveld (2001) and Lichtberger (2011). The direct solution of the differential equation in the case of space varying distributed load is non-trivial, depending on the function p(s) . The analytical solution of the displacements for beam supported by an elastic foundation under a concentrated force P exists in a closed form: