PSI - Issue 79

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

51

Appendix A.1: The Integral transformation in ellipse section 2 2 2 2 1 * * , 2 (1.1) 0 * sin *cos * ( *sin *cos ) * * * cos *sin 2 2 2 * * * 2 2 2 2 sin cos y r m Rb Rb udud E R E R a ua r u a i b j and DetJ Det ua b b ub r a b u u                                                                     

R    

R    

R    

2

2



   





1 cos2 

* 1 cos2 

b       R

2 * a

u      

  



R

2

2



2            *

   

2

1

 

Rb r E R



A a

udud

.1.1. )

2

  

 

0





2

2

2 * 



*         b R

* * R R             a b

 



1

a        



1 Cos2 2   

1 cos2 

   

3

Rb







R

u dud

2

 

  

E R

2

2





0

3

3

2

2

    

2 * *                             * * R b a  a b  2

2 * *                                  * * R a b  a b R R R R 2

2

2





Rb

Rb



  

  

E

R R R R E

2

2

2

*         y R

*         b R

* 1 cos2

  



b

2

2 2

R      







A b

u

u

.1.1. )

cos

2

2

3 2 

2

    

    



2



1

*         b

* * R R a b

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







       

   



R

1 cos2 



, m Rb

2

3 * * * u du d

, m Rb







R

2

 

E

2

E R

2

0





The first result (1.1) is: 3 3 2 * * * * R b a a b                            2

3 2 R R E    

   



2

2 * * R a b  

           



, m Rb

Rb R R R R E  



  

  

2



 A.2: The Integral transformation in circular section



 

2

*        y R

2

    

2

2



R



2

1 cos2 

1 cos2 





, R m Rb 

2

, m Rb













and

rdrd

cos

2



 

E

2

2

2

E



0

 



2



2

2

2 2    R

   

R

*        

 





r

*

Rb r E R

Rb

  





(sin cos ) i j   



rdrd

where

2

3

 

R

2

E



0



2

The Integral circular section result is:   2 2 2 ( , ) , 3 2 R A WRb y r rdrd m Rb Rb c E       

A.3: Multiaxial energy Criterion The multiaxial energy of the material under rotating bending combined to torsion is: