PSI - Issue 79

D. Marhabi et al. / Procedia Structural Integrity 79 (2026) 34–52

38

(a)

(b)

Figure 1: Rotating bending and line Iso-energy W*

4. OVER-ENERGY IN ELLIPTICAL SECTION WITH THE AXIS , * * a b         This part is dedicated to analyze the over-energy expressed in the ellipse support is:   RB * , * Rb * W -W W 2 2 2 2 m Rb Rb A Rb Te ds where A y r E R E R             

(3)

Or



ds

W

2

2

2 , 1 



RB



( , ) W y r dydr Rb

( , ) W y r dydr Rb





*

, 1    Rb

Te

*

Ac

   * * Ae b a R R



(4.a)



WTe  



A

*

E

2 1 

 

A

RB





R



A c and A e are respectively the circular and the elliptical section (Fig. 1 (1b)) The Integral transformation in elliptical section (Appendix A.1) gives 3 3 2 * * * * 2 2 , 2 ( , ) Rb Rb Eq Rb R a b a b E R R R R A W y u udud e                                     3 3 2 * * * * 2 2 , 2 ( , ) Rb Rb Eq Rb R a b a b E R R R R W y u udud Ae                                            The Integral transformation in circular section (Appendix A.2) result is:   2 2 2 , 3 2 ( , ) Rb m Rb Rb R E A W y r rdrd c        The over-energy (4a) under dissymmetrical rotating bending when * 1 a R

(4.b)

(4.c)

 , gives:

3

3

    

   

    

*

*

* * R R          a b

a R R             b

2

2

2

2

, m Rb       3 ) Rb Rb





(

2

2

, Eq Rb



2

, 1    Te







(5)

, 1 

Rb

* * R R              b a

E

Rb

2 1 E

For us, this result is necessary to identify threshold stress in the mechanical component. An Asymptotic Method applied to Over-Energy [2] The basic concept in the influence area of no damage crack is expressed by (1.e). We allow writing two limits