PSI - Issue 65

D.S. Ivanov et al. / Procedia Structural Integrity 65 (2024) 102–108 D.S. Ivanov, G.S. Ammosov, Z.G. Kornilova, A.A. Antonov / Structural Integrity Procedia 00 (2024) 000–000

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2. Research methods

For obtaining the place function, the underground pipeline is considered as a beam lying on an elastic foundation by Ainbinder A.B.and Kamershtein A.G. (1982), Yasin E.M. and Chernikin V.I. (1967), described by the equation

4 d v d v 2

(1)

EI

2 dx    P cv

0

4

dx

where E is the modulus of elasticity, I is the inertia moment of the section, P is the longitudinal force on the pipeline, c is the coefficient of soil bedding, and v is the vertical displacement of the pipe. In the work by A.M. Shammazov et al (2005), an equation for a random underground pipeline in general is derived from the variational principle. In the case of a straight pipeline, the equation looks like this:

4 2 d v d v 2 4

(2)

EI

P r q

dx   

dx

The notations are the same as in equation (1), with r for the soil response, q for the vertical distributed load, and x is the longitudinal coordinate, m. Equation (1) assumes that the pipe depth is constant and the soil properties are homogeneous or piecewise homogeneous. Therefore, it cannot be used to model a pipeline subjected to uneven loading. In equations (1) and (2), the determination of the compressive forces involves difficulties. It depends on conditions when a pipe is attached by the edges, i.e., it requires measuring from one expansion pipe to another, which is kilometers. Measuring long sections increases the labor intensity of fieldwork. Numerical calculation becomes complicated as well since the number of mesh nodes increases. As a result, the computational costs and accuracy rise.

3. Underground pipeline equation

The choice of the underground pipeline model and the derivation of an equation that is not explicitly dependent on longitudinal forces and takes into account the inconsistency of the burial depth and the unevenness of soil reactions are presented in by Ivanov D.S. et al (2022). The moment of forces of a curved beam is expressed by the formula by Ivanov D.S. et al (2019)

2 d z

EI

(3)

M EI p  

2

dx

where M is the bending moment, E is the modulus of elasticity, I is the moment of inertia of the section, and ρ is the bending radius (it is inversely proportional to the second derivative at minor deformations). The equation of the vertical position function of the pipeline obtained in Ivanov D.S. et al (2022) coincides with the equations given in Ainbinder A.B.and Kamershtein A.G. (1982) . However, it does not require explicit specification of longitudinal forces and simplifies the solution. The distributed load from above p is calculated as the weight of the soil above the pipe, depending on the depth of the pipe. (4) where E is the modulus of elasticity; I is the moment of inertia of the section; D is the pipeline diameter; is the lineal weight of the pipe; g is the acceleration of gravity; q is the soil reaction; p is the vertical distributed load; Z is the vertical position of the pipe axis. 4 d Z D g d Z  2 4 2 * 0 2 l l EI q q     dx dx   