PSI - Issue 40

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

219

6

(

)

(

)

2

2

2

,

0 P P n p p F p F p F F FU T W EhR EhR Eh CL L = + −  − + + − − +      =  =  =  = +  +  +  0 0 0 0 1 3 1 1 2 2 2 , , , , 1 (1 ) 2(1 ) cr E i e i i i e i i i i i i

n

(8)

where С is the longitudinal stiffness of the support with a single arch. From (8) we obtain the critical value of compression force affecting the pipeline ( ) ( ) 2 2 2 0 0 0 0 1 1 1 . cr E i i e e i i i i n P P n p F p F p F FU T W = − −  + + −  − − +  (9) which represents a generalized classical critical value in Euler ’ s problem with account being taken for pressures inside and outside the pipeline p i 0 , p e 0 , velocity head inside the pipeline 2 i i U  , temperature of the pipe wall T , and maximum deflection of the pipeline W n during the formation of arched ejection. As it is seen from (9), the critical value of the compression force P 0 affecting the pipeline is greater, if the critical value of Euler’s static longitudinal compression force P E affecting the pipeline and pressure outside the pipeline p e 0 are higher while pressure inside the pipeline p i 0 and velocity head inside the pipeline 2 i i U  and temperature of the pipeline wall T are lower. It should also be noted that an increase in the maximum pipeline deflection W n is accompanied by an increase in the critical value of compression force P 0 affecting the pipeline. The first term in the right-hand side of expression (8) represents the Euler ’ s critical axial compression force; the second term takes place during the axisymmetric expansion of the pipe and its longitudinal contraction under the effect of temperature gradient p *= p i 0 - p e 0 , which depending on the Poisson’s ratio of the pipe material and conditions of its fastening on the supports results in the formation of tensile longitudinal force; the third term multiplied by curvature k represents longitudinal distribution force oriented in the direction of axial line convexity; the forth term multiplied by the curvature k represents lateral distribution force oriented in the direction of the axial line during the deflection of the pipe under the effect of external pressure p e 0 ; the fifth term multiplied by the curvature k represents lateral distribution force oriented in the direction of axial line convexity during the deflection of the pipe under the effect of velocity head 2 i i U  ; the sixth term is the compression force that occurs during an increase in the pipeline wall temperature T and takes account for the conditions of the pipe fastening on the supports; the seventh term is the tensile force during the formation of arched ejection that takes account for the conditions of the pipe fastening on the supports. 5. Loss of Stability If arched ejection occurs at maximum stresses less than the yield point of the pipeline material, the critical axial compressive load in the effective pipeline is determined by formula (8), which will be rewritten in the form ( ) ( ) 2 2 1 2 1 0 0 1 1 2 2 0 0 , χ β γ , ρ , cr cr cr cr E i e i n cr i i e i i i i P P P P P n p p F T W P p F p F F FU = + = + − − + = − + + − (10) where P cr 1 is the effective axial compressive load in the pipeline, P cr 2 is the change in the critical axial compressive load due to the effect of lateral distribution forces. From the standpoint of the arched ejection formation, the most unfavourable case is λ = 0 ( C → ∞), i.e. at both ends of arched ejection there is stiff clamping (for example, frozen ground), while the segment of the pipeline L long floats in water or melted boggy environment. Therefore,

3

EhR

0, 2 ,   =  =   =   = 2 , EhR

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i

1

i

2

2

L