PSI - Issue 12

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

542

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

5

Figure 4. Unit cell deformations generating the transversal (a) and the longitudinal (b) shears.

Finally, the components d a and f a of the axial force transmission mode s a are: d a = ¿ , 0 , ¿ , 0 , 0 T , f a = ˜ u β a , 0 , 0 , 0 , 0 T where β a = (2 β r ) / ( l r l 2

s ) = 2 E r A r / l r , ˜ u denotes a rigid unit cell translation in the axial direction and the symbol ¿ is

adopted for indeterminate quantities.

3. Track continuum model

In order to write the expression of the cell strain energy associated with the inner forces transmission modes, we adopt the following re-parametrization of the sectional and nodal rotations: ψ = ¯ ψ + ξ, ϕ = ¯ ψ, where ¯ ψ is the common part of ψ and ϕ due to both bending and transversal shear while ξ is the di ff erence ψ − ϕ caused by the longitudinal shear. Furthermore, we decompose the unit cell response to the shear and bending moment in the three following parts: pure bending, symmetric bending due to shear and antisymmetric bending and shear (Fig. 5). Then, denoting by ∆ ( . ) the change of a generic quantity ( . ) over the cell range, the cell strain energy due to the transmission modes is written as:

s + 12 η r ¯ ψ −

l r

1 2

2 η r ∆ ¯ ψ

2 + 12 η t ξ

1 2

∆ v

β r

2 + E

2 +

2 .

E u =

β a ∆ u

2 +

where E s is the strain energy pertaining to the symmetric bending due to shear. A 1-D equivalent continuum for the track can be built by the following two-steps procedure. Firstly, the elastic strain energy ¯ E per (track) unit length is evaluated by dividing the unit cell energy E u by the cell length l r . Then, considering that l r is usually very smaller than the length L of the track to be analysed or equivalently that ε = l r / L 1, the ratios ∆ ¯ ψ/ l r , ∆ v / l r and ∆ u / l r may be interpreted as the incremental ratios of the di ff erentiable functions ¯ ψ ( x ) , v ( x ) and u ( x ) that define together with ξ ( x ) the deformed shape of the substitute medium. Consequently, they may be approximated

Figure 5. Decomposition of shear and bending moment change in symmetric and antisymmetric parts.