PSI- Issue 9

Ch.F. Markides et al. / Procedia Structural Integrity 9 (2018) 108–115 Author name / Structural Integrity Procedia 00 (2018) 000–000

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5

2 2 2 R ogR R ogR    2

  

  

(         ) 2 ( 2 )

1 2

(z)   

ogz

z

(12)

1 2

1







2 2 R R 

1

2 2 1 2 )R R R 1 2

(    

(13)

(z)  

og





2 ( 2 )(R R )     

2 2

1 R z

1

  

  

2

(         ) 2 ( 2 )

z

(z)   

ogz

(14)





2 R R 

2 2

1



   

2 2 1 2 R R 1

 

(15)

(z)  

ogz











2

2 ( 2 )    

2 z 2 R R 

2 2

 

1

In turn, introducing the preceding formulae in the well-known expression (Muskhelishvili (1963)):

1

3   

 

  

,

(16)

,   u iv

(z) z (z)     

(z)

,    

,  

,  

,  

2

    

the displacement components are obtained as follows (over-bar indicates the complex conjugate expression):

2.1 The case of bending by couples

 



 

2 2 2 R ogR R ogR    2

( 4 ( 2 )                 ) 3

1 2 3



u

1    

ogr r cos 





 

1 2

1





2 2 R R 



 

1 2 2 1 2

(17)

  

2R R R

3   

cos

  

  

  



1 r sin    

r cos

og

 

2



2 2 R R 

2

R r

  

  

1

1

 



 

2 2 2 R ogR R ogR    2

( 4 ( 2 )                 ) 3

1 2



v

1    

ogr r sin 





 

1 2

1





2 2 R R 



 

1 2 2 1 2

(18)

  

2R R R

3   

3   

sin

  

  



1 r cos    

r sin

og

 

2



2 2 R R 

2

R r

  

  

1

1

2.2 The case of bending by transverse forces

   

2 2 1 2

  

  

R R

2

2

( 3 1                 4 ( 2 )   )

3 r

1

1

2r

  

u

ogr

1  

cos 2

(19)





 









2 R R 

2 2

2

2 R R 

2 2

r

  

  

2 2 R R 

2 2

1

1

1

    

   

( 3 1              )

2 2 1 2

R R

2

3 r

1

1

v

1  

sin 2

(20)



 



 

 







2 R R 

2 2

2

4 ( 2 )       

2 r 2 R R 

  

  

2 2

1

1

3. Validating the analytic solution by means of experimental data In order to validate the analytic solution an experimental protocol was implemented with specimens made of poly methyl-methacrylate-A (commercially known as PMMA or plexiglass). The material was chosen given that it is relati-