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

218

5

Here р 0 is the pressure affecting the fluid surface at a height Н from the pipeline, p i 0 , p e 0 are the fluid pressures inside and outside the pipeline at a design depth,  e is the fluid density outside the pipeline. As it follows from the equation of moments in the same approximation, Qdx – dM = 0, where w 0 does not enter the expression of bending moment M = D d 2 w / dx 2 because of the aforesaid assumption on the absence of stresses prior to external effects. Assuming in the linear problem that cos  = 1, sin(  + d  ) =  + d  and taking into account that  = d ( w 0 + w )/ dx , d  = ( d 2 ( w 0 + w )/ dx 2 ) dx , we obtain the equation of the pipeline bending relative to the actual deflection w ( x ) (Khakimov, 2018), in which the linearized equation is written in the form

( d w w D d w P p F p F F FU dx dx D EJ F R F R h R J +   + + − + +       = = = + − ) 2 0 0 0 4 2 2 2 π , 2 ( , π , π = + − i i e i i i i R h R = 2 4 4 ) 0,

(4)

(

)

(

)

 

4

4.







i

i

i

i

i

i

where E , J, R i are the modulus of elasticity, the moment of inertia and the inner radius of the pipeline cross section. 3. Pipeline Bending Let us take the partial solution of equation (4) in the form 2 sin β , 1, 2,... n w W n x n = = (5) Substituting (1), (5) in equation (4), we obtain the relationship between the actual deflection amplitude and the initial deflection amplitude in the form (Khakimov, 2018)

n W R W P n =

,

n

2 2

2 ( ) 4

R EJ

2  − 

0

n

E

n

(6)

(

)

2 2 ( ) . R P p F p F F FU n   = + − + +  

4 P D

,

=  =





0

0

E

n

i

i

e

i

i i i

2

L

4. Axisymmetric Pipeline Deformation

To determine the value  , we should consider axisymmetric deformation under the effect of pressure gradient p *= p i 0 – p e 0 . If inertia is ignored, circumferential force is equal to * * N p R  = . As it is follows from Hooke ’ s law (Kovalenko, 1970), ( ) ( ) * * * * * * 2 2 1 , 1 , 1 1 Eh Eh N T N T    =  +  − +   =  +  − +           − − (7) where * N  and longitudinal force * N are assumed to fall per unit length,  is the Poisson ’ s ratio, α 1 is the coefficient of linear thermal expansion, T is the change in the pipeline temperature, * * ,    is the deformation. In (Khakimov, 2019), we give the critical value of the compression force P 0 cr affecting the pipeline with account for axisymmetric deformation under the effect of pressure gradient p *= p i 0 - p e 0