PSI - Issue 40

A.G. Khakimov et al. / Procedia Structural Integrity 40 (2022) 214–222 A.G. Khakimov / Structural Integrity Procedia 00 (2022) 000 – 000

220

7

The internal pressure p i 0 , velocity head inside the pipeline 2 i i U  and external pressure p e 0 for the effective pipeline are determined by the technological regime for exploitation. Thus, the change in the temperature of the pipeline wall T is found as an inequality between the temperature of its wall and the ambient temperature during the construction. Maximum compression stresses σ max in the dangerous section B taken with the positive sign should not exceed the yield point σ t

1 P M F W  = −   max cr B

,

(11)

t

where W = J / R i is the moment of resistance of the pipeline cross section, M B is the bending moment in the section B (Fig. 3) determined by relationships

Fig. 3. Pipeline with arched ejection.

2

L

( ) L M M M M M q x x dx   = + = + ,

0 

2

2 − = 

,

2 cr n EJW P W + n

 

 

1

2

1

B

B

B

B

A

2

(12)

2

( ) d w M EJW q x P P W x M P W dx =  = =   = − 2 2 2 2 2 1 cr n 2 2 , 2 cos 2 , A n cr cr n B

,

where M B 1 , M B 2 are the bending moments from lateral distribution forces in the section B and effective axial compressive load in the pipeline, M A is the bending moment in the section A , q ( x ) are the lateral distribution forces. Condition (11) with account for relationships (12) is written in the form ( ) ( ) 2 2 2 2 11 2 11 11 0 0 1 1 2 , , n cr n cr n n cr n t cr E i e i EJW P W P W W P W P P n p p F T F W  + − +  +  −   = + −  − from which the equation is derived for the maximum deflection, where the maximum compressive stress is equal to the yield point of the pipeline material

3 2 aW bW cW d a W b F c ER P P W d P F + + + = =  =  = −  + − = −  0, , 2 , n n n 2

,

(13)

(

)

.

1

2

11

i

cr

cr

cr

t

Calculations are performed for the following parameters of the pipe: modulus of elasticity of the pipeline material E = 2.0  10 11 N/m 2 , density  = 7800 kg/m 3 , Poisson’s ratio  = 0.3, coefficient of linear thermal expansion α = 11.3  10 – 6 1/°C , inner radius of the pipeline cross section R i = 0.259 m, wall thickness h = 5 mm, wall temperature T = 10 ° С , gas pressure inside the pipeline p i 0 =1 MPa, gas density ρ i = 913 kg/m 3 , flow velocity U i = 10 m/s, longitudinal stiffness of support with a single arch C = 10 7 N/ м 2 , pressure outside the pipeline p e 0 = 0 MPa. Fig. 4 ( а ) gives the dependences of the acceptable amplitude W n of arched ejection with the yield point σ t for different distances between the supports L = 100, 150, 200 m (solid, dashed and dotted lines, respectively) at pressure inside the pipeline of 1 MPa. As it is seen, with an increase in the yield point of the pipeline material and the distance