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

601

2

whose procedure is precisely based on the fundamental analytical solution. In the field of pressure vessels there are analytical solutions for the cylindrical shape loaded with internal pressure. Solutions are known in both the elastic and elastic-plastic region. In this paper, the analytical solution is presented in an abbreviated version of a pressured cylindrical vessel in the elastic-plastic region up to the destruction level, allowing to calculation the burst pressure for a used bilinear material model. Effect of local wall thickness loss due to the corrosion e.g. it is then possible to solve by means of numerical simulation verified by analytical solution of pressure vessel with ideal geometry. 2. Solution of the elastic-plastic stress- strain state of pressured vessel In this derivation an incompressible material (ν = 0.5) is considered taking into account the fact that the plastic strain occurs at a constant volume and the collateral volume change is always elastic. As shown in Fig. 1, the z- axis is oriented in the direction of the axis of the vessel and r 2 will be marked the radius at the boundary of the elastic and plastic region. According to the theory of small elastic-plastic deformations (extended Hook's law for homogeneous isotropic material) we can use equations in the form that:

Fig. 1 Stresses on the element and deformation of the element in the wall of the cylindrical vessel loaded by internal pressure p 1 .

1 2

1 2

i 

i 

  

  

  

  

1 2

i 

  

  

(

)

(

)

,

,

(

)

(1)

r 

   − +

z 

   − +

=

=

t 

   − +

=

r

t

z

z

r

t

t

z

r

i 

i 

i 

Intensity of normal stresses (effective stress) is related to the second invariant of the stress deviator as follows: ( ) ( ) ( ) ( ) 2 2 2 2 1 2 2 3 3 1 2 3 3 2 i i I D          = = = − + − + − (2) The intensity of the normal stresses is chosen so that for the pure tension 1 i   = The strain intensity (effective strain) is given by equation:

2

2

( )

) ( 2 1 2   − + − + − 2 3   3 1   ) ( 2

)

(

2

(3)

2 I D 

i 

=

=

3

3

1 2

(

)

0 z  = ,

For the condition of incompressibility:

z t    = + and the axial force in closed vessel is equal r