PSI - Issue 18

Vladimír Chmelko et al. / Procedia Structural Integrity 18 (2019) 600–607 Chmelko, V., Berta, I / Structural Integrity Procedia 00 (2019) 000 – 000

602

3

r

r

3

3

(

)

2 rdr rdr      = = +   z r t

(4)

P

0

r

r

1

1

The equation of equilibrium or forces acting on the element of Fig. 1 has a form ( ) ( ) 2 sin 0 2 r r r t d rd d r dr d dr        − + + + = ) Using the eq. (4) and (5) for the normal force is possible to derive equation ( ) ( )  − = = + = 3 1 2 3 3 2 1 1 2 0 2 r r r r p r p r ....... r d rdr P    

(5

(6)

Then for stress intensity from eq. 2

2

2 3

2 3

3

(

) r   ,

2

2

t  ... + = =

3

i 

=



−

 

−

(7)

r

t r

t

2

2

and the strain intensity from eq. 3 ( )     2 =

( ) r 

2

... 2 2 2 − + + − = =

t 

(8)

i

r

t

t

3

3

From the Fig.1 is clear that

r u

dr du

t = 

r = 

and

(9)

By eliminating u we get the differential equation of the strains in form ( ) 1 t r t d dr r    = −

(10)

By solving this equation, we arrive at a basic equation for solving closed cylindrical vessels in which r 2 is the radius at the elastic and plastic boundary

 

  

2

d

r

2 1 3 3 f r  

r 

(11)

=

K

2

2

dr

G r

( )

To calculate the course of tangential and radial stresses, you need to know the function f in the form of i f   = . For the analytically explicit bilinear material model, as an approximation of the true tensile diagram of Fig. 2 are valid the relations: i

i    ;

 

e K i i tg tg       = + − i

e

=

;

(12)

h tg tg =

i

tg