PSI - Issue 40

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

217

4

2. Problem Statement Let us assume that an elastic pipeline is "fastened" on clamped sliding "supports" located at a distance L from each other, and in this case the "supports" do not hinder the fluid flow with density  i and velocity U i inside the pipeline along its axis (Fig. 1). The distance between the "supports" will be considered the arched ejection length of the pipeline. The deflection and the angle of rotation on the sliding supports are equal to zero. The acceleration G is oriented perpendicularly to the pipeline axis. It is assumed that fluids are ideal and incompressible. The pipeline is compressed by the longitudinal force Р . The force Р , pressures inside and outside the pipeline p i , p e and the velocity U i change independently of each other. The intensity of their increase from zero will be considered such that inertial forces affecting the system are small. At Р = 0, U i = 0, p i = 0, p e = 0 the pipeline has small x -axis misalignment in the form

2 0 sin β , β π , =

1, 2,...

0 n w W n x =

L n

=

(1)

where W 0 n is the amplitude of small initial misalignment.

Fig. 1. Pipeline layout.

In this case there are no residual stresses, for example, as a result of pipeline annealing [1]. The sum of z components affecting the element of dx length (Fig. 2) is equal to (Ilgamov, 2018)

Fig. 2. Scheme of forces effect on thin elastic pipeline under its bending in the zx -plane.

(

)

cos – Q Q dQ  +

cos(

) sin – sin( p F p F F  +  +   +  +  +   ) sin – – sin( d P P d

(

)

(

)

– p F p F F + i i e i

)  +  + d

+

 

 

(2)





i i

e

i

(

)

2 ( G F F dx G F F dx qdx U kdx +  +   +  ) – – –

0,

=

i i

e

i

i i

where Q is the cutting force, q is the intensity of distributed buoyancy force, k is the curvature of the pipeline axial line,  , h , F are the density, the wall thickness and the pipeline cross-sectional area, F i is the clear cross-sectional area of the pipeline. Pressures inside and outside the pipeline are determined according to the formulas

0 e e p p G w w p p G w w p p G H = +  + = +  + = +  0 0 0 0 0 ( ), ( ), i i i e e e

.

(3)