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

105

4

Equation (4) is specified in the section [a, b], where the complex deformation is observed. Outside the section, deformations are not evident – there are no bends. Therefore, the boundary conditions are set as the equality of the second derivative to zero.

2 d Z b dx

2 d Z a dx

2 ( )

2 ( )

0  ;

(5)

0



4. Soil response

Numerical solutions of equation (4) are well developed. The difficulty in this case lies in determining the soil response q. Generally, the soil reaction is considered according to the Winkler hypothesis proportional to the soil settlement Ainbinder A.B.and Kamershtein A.G. (1982). The Winkler hypothesis does not accurately describe the soil response, where the soil compression is directly proportional to the applied pressure. In practice, this dependence deviates from direct proportionality. The properties of frozen soil can change over a distance of several meters, which is the cause of uneven frost heaves. We decided to set up soil responses using the trial and error method. The soil reaction is specified at each node of the solution grid. Then, we solve the equation. If the resulting solution for the pipe position differs from those measured during the planned-high-altitude measurements, we correct the soil response at the nodes and perform further calculations. We repeat the iterations until the solutions to the equation fit the measured values. The soil responses were adjusted using the gradient descent method. The error function was introduced: 1 2 ( , , ,..., ,..., ) i l K Z x q q q q is solution to the equation (4) at for given soil responses at the grid nodes of the solution. Then we calculate the error function for the perturbed reactions ( 1, 2,..., ) l K  : 2 1 1 1 2 (() (, , ,..., ,..., )) N i i i l K Err V x Z x q q q q q      (7) 2 0 where ( ) i V x is measured vertical position of the pipeline at point, 1 1 2 (() (, , ,..., i i V x Z x q q q q q      ,..., )) N i l K Err (6)

and calculate the gradient:

Err Err 

(8)

grad



l

0

l

q



New values of soil responses are obtained using the formula:

(9)

q q

* grad   

l

l

l

where  is gradient descent step.

5. Calculation

The following values are used in the solution. Modulus of elasticity for 13G1S-U (13Mn1C-C) steel is E=206 hPa according to the design of the underground pipeline. Moment of inertia for a solid circular beam is by Iktin V.A (2004)