PSI - Issue 41

Christos F. Markides et al. / Procedia Structural Integrity 41 (2022) 351–360 Christos F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

354 4

1000 1200 1400 1600

y 

0 200 400 600 800

xy 

[MPa]

Contact pressure & friction

L

0

2

4

6

8 10

0 ≤ x ≤ L [mm]

Fig. 4. Distributions of contact stresses on the half flat edge of the FBD.

2(1 )  

2(1 ) (2 1)   

P

P

log 4 3 2 n   

  

 

j

frame

frame

cos

log

,

P

P





j

m

2

2 1 j



 

n t

 

2 8( 1) 4 4 3 n j j n      

t  





(2)

2(1 )  

P

log 4 3 2 n   

  

 

j

frame

sin

log

,

0

F

F





j

m

2

2 1 j

 



2 8( 1) 4 4 3 n j j n      

t  





The application scheme of P j , F j and P m is shown schematically in Fig.6. Letting now the number 2 n +1 of the infinitesimal holes-points Z j on the upper chord – LL (and similarly on the lower one) tend to infinity (Figs.5, 6), the influence of the shaded cyclic sectors to the rest of the complete disc may be ignored (in a first approximation), thus transforming the complete disc into the (truncated) FBD in question. In this context, – LL is from now on considered as the flat edge of the FBD and the point forces provided by Eqs.(2) represent the boundary conditions of the 1 st fundamental problem in question, the analytic solution of which will be given in next section.

y

P frame

y

P

P frame

y

j P

j P

P

m P

j P

L

L 

F L F

j F

j F

m Z

j Z

j Z 

j P

– L

2 1 ... ... n j Z Z Z Z

i e z r  

m Z

r

– L

L

R

j F



α R

2 1 ... ... n j Z Z Z Z ... ... 2 1 n j

x

Ο

m Z

j F

α R n

n

1

... ... 2 1 n j

j Z 

m Z

j Z

j F

j F

n

n

1

x

Ο

P

j P

j P

x

Ο

m

Fig. 5. Partition of – LL chord into 2 n +1 parts, and application of point forces to the 2 n +1 holes.

Fig. 6. Application scheme of P j , F j and P m .