PSI - Issue 12

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

545

8

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

Figure 8. Track segment under self-equilibrating bending moments decaying from left to right.

bending moments on the track constrained section and unloaded free end, for the rotations ψ i , the displacement v i and the nodal rotation ϕ i of the i section one obtains:

∆ ψ ( + ) ˜ λ i − 1 ˜ λ 2 Λ − i − 1 ˜ λ − 1 ˜ λ 2 Λ − 1 1 2 2 ˜ λ 2 Λ − ˜ λ + 1 ˜ λ Λ − ˜ λ + 1 ˜ λ − 1 ˜ λ 2 Λ − 1 ∆ ψ ( + ) l r ,

ψ i =

1 2 2 ˜ λ i − 1 + ˜ λ ˜ λ Λ + 1 + 2 ˜ λ 2 Λ − i ˜ λ − 1 ˜ λ 2 Λ − 1

˜ λ Λ + 1 ˜ λ 2 Λ − 1

∆ ψ ( + ) −

(5)

γ ψ ( + ) ,

ϕ i =

∆ ϕ ( + ) +

˜ λ i − 1 ˜ λ 2 Λ − i + 1 ˜ λ − 1 ˜ λ 2 Λ − 1 .

ψ ( + ) l r

i ˜ λ 2 Λ + 1 ˜ λ − 1 + − 1 / 2 ˜ λ i − 1 1 + ˜ λ ˜ λ 2 Λ − i + 1 + γ

v i =

λ 2 Λ − 1

˜ λ − 1

2 ˜

1 . For reason of brevity, the procedure adopted to derive previous results is not

being Λ = N = L / l r and ˜ λ = λ −

reported. A rigorous proof of eq.(5) will be given in a forthcoming paper. From previous results, a continuous approximation of the track response may be achieved by considering the limit case of unit-cell having infinitesimal length (with respect to the track length). This continuous track solution can be achieved observing that ˜ λ i = 1 − ˜ λ − 1 i =    1 + ˜ λ − 1 l r x i i    i , (6) with x i = i l r abscissa of the nodal section i in the reference frame of Fig. 8. When the cell sizes are very smaller than the track length L or equivalently the track is composed of a great number of cells per unit length, it will result in Under these conditions, to the last quantity in Eq. (6) we may substitute the corresponding limit as i approaches + ∞ :    1 + ˜ λ − 1 l r x i i    i e ( ˜ λ − 1 ) ζ i (8) with ζ i = x i / l r . When this limit value is inserted in Eq. (5) and the variable ζ i is substituted by the non-dimensional real variable ζ ranging in [0 , Λ ], three continuous functions ψ , ϕ and v are obtained. These latter define the deformed shape of a continuum equivalent medium, that may be thought as an asymptotic approximations of the track behaviour, exact only in the limit when i → + ∞ . x i >> l r and i >> 1 . (7)

5. Track analysis

By means of the equivalent continuum of Sec. 3 and the approximating solution of Sec. 4, the track behaviour for quite simple constraint and loading cases can be readily analysed.