PSI - Issue 12

M. Catena et al. / Procedia Structural Integrity 12 (2018) 538–552

548

M. Catena et Al. / Structural Integrity Procedia 00 (2018) 000–000

11

Figure 12. E ff ects of changes in fasteners sti ff ness on the response of track segments with N = 20 ( upper row ) and N = 100 ( lower row ) cells .

takes place. To eliminate it, we introduce in the loaded section two hinges to constrain the nodes of the contiguous elementary cells. Then on the right and left side parts of the section j we apply self-equilibrated bending moments of intensity m 1 such that the jump ∆ ϕ 1 = − ∆ ϕ for the nodal rotations occurs. By the compliance given in Eq. (9) and taking into account the signs of the nodal rotations caused by a positive self-equilibrated moment at the left and right hand side of the loaded section, for the jump ∆ ϕ 1 we obtain:

g x f ,

2 Γ b ∆ η

(11)

∆ ϕ 1 = m 1

where

g x f = exp [2 ( λ − 1) L / l r ] − 1 exp 2 ( λ − 1) ( L − x f ) / l r − 1 exp 2 ( λ − 1) x f / l r − 1 . Hence, the self-equilibrated bending moment active in the loaded section is

g x f −

1 48

α H ∆ η κ H Γ b

1

(12)

m 1 =

.

Both equilibrated and self-equilibrated solutions to be superimposed are obtained by combining the results achieved for the cantilever of Case 1 by a procedure analogous to the one usually adopted for simply supported Euler-Bernoulli beams.

6. Homogenization Error and Validation Study

To examine in detail the homogenization errors of the interpolating and asymptotic approximations, a sensitivity study has been carried out. For this purpose, a track with the following geometrical and mechanical properties, has been considered: l s = 1500 mm , l r = 600 mm , k θ = 3 . 5 × 10 7 Nmm A r = 7670 mm 2 , I r = 5 . 123 × 10 6 mm 4 , I s = 2 . 50 × 10 8 mm 4 (13)