PSI - Issue 8

E. Marotta et al. / Procedia Structural Integrity 8 (2018) 43–55

46 4

Author name / Structural Integrity Procedia 00 (2017) 000 – 000

 

M s

 



1

 



M s

ds



i 

(2)

i P     

EI

The expression of ds , in the Cartesian reference, results

(3)

2 2 ds dx dy  

This term introduces complexity in the analytical integration. An idea to circumvent these difficulties emerges by the change of variable, from the curvilinear abscissa to the angle given by the local tangent of the beam. This substitution yields   ds d     (4) where ρ(θ) is the radius of curvature and dθ is the elementary variation of attitude. This change of variable implicates that the new variable θ increases or decreases monotonically, for analytical integration needs. With the aim to include a generality of curved beams, but keeping the analytical integration possible, the choice fells on a polynomial expression, limited to the cubic interpolation of the curvature radius   3 2 a b c d             (5) Thanks to this substitution, the Cartesian coordinates follow by simple integration of product of a polynomial and cosine (or sine) functions

             cos d

x

(6)

0

             sin d

y

(7)

0

The solution of the above integrals, at a generic angle θ , are   3 2 2 sin sin 3 cos 6 sin sin x a b a a c                  

2 cos b

2 sin b

 

 

 

 





(8)

d

sin 6 cos a  

c

a c

cos 6 







 

  

3   

2   

2   

y

a

3 sin a

b

2 sin b

6 cos a

c

6 sin a

cos

cos

cos

 







 

 

 

 

 

 





(9)

c

sin 2 cos b  

d

b d

cos 2 







 

The knowledge of the Cartesian coordinates of the curve is crucial to carry out the bending moment acting along the curvilinear beam. Fig. 2 is an example of a curve that is manageable with the polynomial approach in the local coordinate system. It is clear that no matter if closed loops are considered.