# Wave Theory of Seismic Resistance of Underground Pipelines

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## Abstract

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## Featured Application

**Main gas and oil pipelines are critical elements of the world’s infrastructure. The integrity of pipelines during earthquakes is a subject of intensive research. At present, underground pipelines’ strength to the effect of seismic loads (seismic resistance) is calculated incorrectly. The existing methods do not consider seismic (dynamic) pressure, i.e., the stress normal to the pipeline’s outer surface arising from the propagation of a seismic wave in the soil. This leads to an incorrect determination of the pipeline’s stress; the longitudinal stress calculations leads to errors of 100% or more. A potential application of the results obtained is the calculation of seismic stability of underground pipelines.**

## Abstract

## 1. Introduction

- Under strong ground motion, the underground pipeline system is destroyed; these destructions are different: the destruction in the pipeline itself, in butt joints, the pipeline buckling from the ground; soil separation from the pipeline on the surfaces of their contact, etc.;
- Seismic forces are transmitted to underground pipelines through soil. Therefore, these forces’ values directly depend on the physical–mechanical and strain properties of the soil environment surrounding the underground pipeline.
- The assumption of an infinitely long straight horizontal pipeline in homogeneous soil simplifies the solution of the problem of seismic impact on the pipeline. This assumption is used in most theoretical studies. An exception is in [11]. The unsteady wave problem of a longitudinal monochromatic wave propagation and reflection from a rigid stationary barrier solved in a one-dimensional statement.
- During earthquakes, seismic forces are transmitted to underground pipelines through soil strain, i.e., the soil deforms under seismic wave effect and forces underground pipelines to deform. The pipelines inflict the most severe damage when the route is codirectional to the seismic strain vector [15].
- In the case of strong ground motions, the amplitudes of absolute displacements of soil particles reach 0.1–0.4 m at an oscillation period T = 0.05–20 s; under soil mass vibrations together with the underground pipeline, a difference between the vibrational motion of the pipeline and soil is formed (for various reasons) in vibration amplitude and phase; a displacement of the pipeline relative to the soil is observed, i.e., a relative displacement.
- There is a critical value of the relative displacement ${u}_{*}$ of the pipeline cross-section, below which there is an elastic bond between the soil particles and the outer surface of the pipeline in contact with the soil. At values of the relative displacement exceeding the critical value, the elastic bond between the pipeline and the ground is ruptured [14].
- After pipeline construction, due to the cohesion phenomena, a relative shear displacement occurs in the soil layer ${\delta}_{g}$ thick, called the contact soil layer, between the underground pipeline and soil on their contact surfaces.
- At an increase in relative displacement value greater than ${u}_{*}$, the contact soil layer undergoes significant shear strain and may collapse; the soil outside the contact layer may remain intact. In the case of strain and destruction of the contact soil layer under shear stresses, its limiting value is determined based on the Coulomb law.
- The processes of relative displacements $u$ formation on the contact layer, under its elastic, elastic–plastic strain ($u<{u}_{*}$) and destruction ($u\ge {u}_{*}$), are the two stages of one process—the process of deformation of the contact soil layer before and after destruction.

- To conduct a critical analysis of the dynamic theory grounds, to determine its disadvantages and advantages;
- To determine the ways of further development of the dynamic theory of seismic resistance of underground pipelines;
- To determine the grounds of an alternative wave theory of seismic resistance of underground pipelines and its advantages and disadvantages.

## 2. Materials and Methods

#### 2.1. Interaction Laws

#### 2.2. Formulation of the Problem

## 3. Results and Discussion

#### 3.1. Solution Methods

- (i).
- It is possible to obtain numerical solutions by the finite difference method directly. In this case, these partial differential equations are directly solved.
- (ii).
- However, there is another method. Partial differential equations of hyperbolic type have real characteristics and relations on them. In this case, the characteristic relations are already ordinary differential equations. In the case of boundary conditions (30)–(33) and laws of interaction (26), the wave front lines in soil and the pipeline remain linear. The characteristic lines also remain linear on the characteristic planes $xt$ for soil and the pipeline. The numerical finite difference method’s application to these ordinary differential equations significantly increases the accuracy of the solutions obtained. Previously the authors have used this algorithm in solving many wave problems and showed the high accuracy of the method. Examples of the application of the method are publications [35,38,39,40].

- (iii).
- For underground pipeline$$\begin{array}{c}\frac{d{\sigma}_{c}}{dt}-{C}_{c}{\rho}_{0c}\frac{d{v}_{c}}{dt}=-{C}_{c}^{2}{\rho}_{0c}g\left({\sigma}_{c},{\epsilon}_{c}\right)+\chi {C}_{c}{\sigma}_{\tau c}\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{c}\mathit{h}\mathit{a}\mathit{r}\mathit{a}\mathit{c}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\text{}\frac{dx}{dt}={C}_{c}\\ \frac{d{\sigma}_{c}}{dt}+{C}_{c}{\rho}_{0c}\frac{d{v}_{c}}{dt}=-{C}_{c}^{2}{\rho}_{0c}g\left({\sigma}_{c},{\epsilon}_{c}\right)-\chi {C}_{c}{\sigma}_{\tau c}\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{c}\mathit{h}\mathit{a}\mathit{r}\mathit{a}\mathit{c}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\text{}\frac{dx}{dt}=-{C}_{c}\\ \frac{d{\sigma}_{c}}{dt}-{C}_{c}^{2}{\rho}_{0c}\frac{d{\epsilon}_{c}}{dt}=-{C}_{c}^{2}{\rho}_{0c}g\left({\sigma}_{c},{\epsilon}_{c}\right)\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\text{}\frac{dx}{dt}=0\end{array}\}$$
- (iv).
- For soil half-space$$\begin{array}{c}\frac{d{\sigma}_{g}}{dt}-{C}_{g}{\rho}_{0g}\frac{d{v}_{gc}}{dt}=-{C}_{g}^{2}{\rho}_{0g}g\left({\sigma}_{g},{\epsilon}_{g}\right)+\chi {C}_{g}{\sigma}_{\tau g}\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{c}\mathit{h}\mathit{a}\mathit{r}\mathit{a}\mathit{c}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\frac{dx}{dt}=+{C}_{g}\\ \frac{d{\sigma}_{g}}{dt}+{C}_{g}{\rho}_{0g}\frac{d{v}_{g}}{dt}=-{C}_{g}^{2}{\rho}_{0g}g\left({\sigma}_{g},{\epsilon}_{g}\right)-\chi {C}_{g}{\sigma}_{\tau g}\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{c}\mathit{h}\mathit{a}\mathit{r}\mathit{a}\mathit{c}\mathit{t}\mathit{e}\mathit{r}\mathit{i}\mathit{s}\mathit{t}\mathit{i}\mathit{c}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\frac{dx}{dt}=-{C}_{g}\\ \frac{d{\sigma}_{g}}{dt}-{C}_{g}^{2}{\rho}_{0g}\frac{d{\epsilon}_{g}}{dt}=-{C}_{g}^{2}{\rho}_{0g}g\left({\sigma}_{g},{\epsilon}_{g}\right)\\ \mathit{a}\mathit{l}\mathit{o}\mathit{n}\mathit{g}\text{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\frac{dx}{dt}=0\end{array}\}$$

#### 3.2. Numerical Calculations

#### 3.2.1. Initial Data for Calculations

- (i)
- For the underground pipeline:

^{3}; ${\mu}_{c}={10}^{4}$ s

^{−1}; ${C}_{c}=5000$ m/s; ${\gamma}_{c}=1.02$; ${E}_{Dc}={C}_{c}^{2}{\rho}_{0c}$; ${E}_{Sc}=\frac{{E}_{Ds}}{{\gamma}_{c}}$;

- (ii)
- For soil:

^{3}; ${C}_{g}=1000$ m/s; ${K}_{\sigma}=0.3$; ${\mu}_{g}=1000$ s

^{−1}; ${\gamma}_{g}=1.1$; ${E}_{Dg}={C}_{c}^{2}{\rho}_{0g}$; ${E}_{Sg}=\frac{{E}_{Dg}}{{\gamma}_{g}}$;

- (iii)
- For seismic load parameters:

- (iv)
- For interaction parameters:

^{−1}; $\beta =2$; $f=0.5$; ${u}_{*}=0.005$ m.

#### 3.2.2. The Case of a Large Critical Relative Displacement

#### 3.2.3. Influence of Shear Stresses on Longitudinal Stresses in Pipeline Cross-Sections

#### 3.2.4. The Case of Variable Stress Normal to the Outer Surface of the Pipeline

## 4. Conclusions

- The main disadvantage of the dynamic theory of seismic resistance of underground pipelines is the neglect of dynamic stress state in soil under seismic wave propagation. The next drawback of the dynamic theory is an inaccurate, approximate accounting for the displacement of the soil medium to which the underground pipeline is embedded.
- The complete interaction process includes the stages of nonlinear changes in the interaction force (the friction force) by manifesting its peak value and the Coulomb friction. The contact layer of soil undergoes shear deformations until complete structural destruction of the soil contact layer. The interaction force is the friction force, and its peak value does not appear.
- The problems of seismic resistance of underground pipelines should be considered based on the theory of propagating seismic waves in a soil medium and the interaction of seismic waves with underground pipelines, i.e., based on the wave theory of seismic resistance of underground pipelines.
- A one-dimensional coupled problem of seismic resistance of underground pipelines under seismic impacts was posed based on the wave theory. An algorithm and a program for the numerical solution of the stated wave problems were developed using the method of characteristics and the method of finite differences.
- An analysis of the laws of interaction of underground pipelines with soil under seismic influences shows that it is necessary to use in the calculations the laws of interaction that account for the complete interaction processes observed in experiments.
- The analysis of the obtained numerical solutions and the posed coupled problems of the wave theory of seismic resistance of underground pipelines show the occurrence mechanisms of longitudinal stresses in underground pipelines under seismic influences.
- The results of calculations stated that an account for the dynamic stress normal to the underground pipeline’s outer surface leads to multiple increases in longitudinal stress in the underground pipeline. This multiple increase is due to the transformation of the interaction force into an active frictional force, resulting from a greater strain in soil than the one in the underground pipeline.
- The wave theory’s efficiency and reliability are shown in comparison with the dynamic theory of seismic resistance of underground pipelines.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic dependencies of the shear stress $\tau \left(u\right)$ arising on the contact surface of the pipeline with the soil on relative displacement $u$ at various normal stresses.

**Figure 2.**Changes in longitudinal stress in time in the sections of an underground pipeline at ${\sigma}_{N}={\sigma}_{N}^{S}=const$ for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 3.**Dependence of shear stress on relative displacement at ${\sigma}_{N}={\sigma}_{N}{}^{s}=const$.

**Figure 4.**Changes in relative velocity of pipe sections in time for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 5.**Changes in relative displacements in time for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 6.**Time dependence of shear stresses for x = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).

**Figure 7.**Dependence of shear stress on relative displacement without considering ${\sigma}_{N}^{D}$ for x = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).

**Figure 8.**Change in stresses in pipeline cross-sections for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 9.**Stress change in time in the sections of soil medium for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 10.**The pattern of stress change in time in pipeline cross-sections at ${\sigma}_{N}\ne const$ for x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).

**Figure 11.**Calculated dependences of shear stress in time at ${\sigma}_{N}\ne const$ for x = 5 m, 10 m, and 15 m (curves 4, 5, and 6, respectively).

**Figure 12.**Stress change in time in the pipeline cross-sections with accounting of ${\sigma}_{N}^{D}$ for x = 30 m, 60 m, and 90 m (curves 4, 5, and 6, respectively).

**Figure 13.**Relationship between shear stress and relative displacement, accounting for ${\sigma}_{N}^{D}$ at x = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).

**Figure 14.**The change pattern in the pipeline cross-sections velocity in time at x = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, 3, respectively).

**Figure 16.**Relative displacement depending on time for x = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).

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**MDPI and ACS Style**

Sultanov, K.S.; Vatin, N.I.
Wave Theory of Seismic Resistance of Underground Pipelines. *Appl. Sci.* **2021**, *11*, 1797.
https://doi.org/10.3390/app11041797

**AMA Style**

Sultanov KS, Vatin NI.
Wave Theory of Seismic Resistance of Underground Pipelines. *Applied Sciences*. 2021; 11(4):1797.
https://doi.org/10.3390/app11041797

**Chicago/Turabian Style**

Sultanov, Karim Sultanovich, and Nikolai Ivanovich Vatin.
2021. "Wave Theory of Seismic Resistance of Underground Pipelines" *Applied Sciences* 11, no. 4: 1797.
https://doi.org/10.3390/app11041797