PSI - Issue 58

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 150–156 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

152

3

1

n C



,

m

(2)

  t m 1 





C H D D t D e 3 2 1 1      

.

(3)

In formula (3), t is the time, 3 D ,  and m are parameters. The frame under consideration is made of a material that is functionally graded along the thickness. Therefore, the material parameters involved in formulas (1), (2) and (3) are smoothly distributed along the thickness. We use the following functions for treating the distributions of material parameters: 1 D , 2 D ,

1 

1  h E E h m m Q G  Q G 2  

2 h

  

  

E E

z

 



,

1

G

(4)

2 

2 h

  

  

m m

z

 



,

(5)

1

G

3 

3  h D D Q 1 

2 h

  

  

1

G

D D

z

 



,

(6)

1

1

1

G

4 

4  h D D Q h D D Q 2  5  3 

2 h h 2

     

     

2

G

D D

z

 



,

(7)

2

2

1

G

5 

3

G

D D

z

 



,

(8)

3

3

1

G

6 



 

2 h

  

  

6  h Q G

z

 



  G

,

(9)

1

where

z h    .

2 h

(10)

1

2

In formulas (4) – (9), 1 z is the centric axis of the cross-section, h is the thickness, 1  , 2  , 3  , 4  , 5  and 6  are constants regulating the distributions, the subscripts, G and Q , refer to outer and inner surface of the frame members, respectively. Investigation of the damping energy in the frame structure requires analysis of the stressed and strained state. Formula (11) presents the distribution of a  along the thickness of the frame member, 1 2 BB .   nn a z z 1 1     , (11) where  is the curvature, nn z 1 is the neutral axis coordinate. We determine  and nn z 1 by using equilibrium equations (12) and (13).