Dynamic analysis of buried steel pipeline subjected to blast seismic waves

3.1. Incident wave potential function

The incident plane P wave with the frequency $\omega$ has an angle ${\theta }_{\alpha }$ pendicular to the axis of the pipeline. In the rectangular coordinates system, the potential function can be expressed as:

where ${\alpha }_1$ denotes the incident wave number, ${\alpha }_1=\omega /C_{P1}$; $C_{P1}$ denotes the P wave velocity in half infinite space; the time factor $\mathrm{e}\mathrm{x}\mathrm{p}(-i\omega t)$ is ignored.

Fig. 1Analysis model and the coordinate system

3.2. Reflected wave potential function

The incident plane P wave is reflected from the free surface, generating both P wave ${\phi }^{\left(r\right)}$ and SV wave ${\mathrm{\Psi }}^{\left(r\right)}$. The incident and reflected waves in the polar coordinate systems are written in the form of Fourier-Bessel series:

where:

and $J_n$ denotes the first kind of Hankel function; ${\epsilon }_n$ denotes Neumann factor: when $n=$ 0, ${\epsilon }_n=\text{1}$; when $n>$ 0, ${\epsilon }_n=\text{2}$.

3.3. Scattering wave potential function

In the infinite half space, the incident P wave generates both scattering P wave and SV wave at the interface and the convex surface. There are four scattering waves in the pipeline: two outward propagating waves and two inward propagating waves excited by the incident plane wave. Their potential function can be expressed as:

where ${\alpha }_2$, ${\beta }_2$ denotes the wave number of P wave and SV wave respectively.

3.4. Solution of the model

At the convex free surface $r_2=L$, the continuity conditions are:

At the outer surface of the pipeline $r_1=b$, the stress and displacement continuity conditions are:

At the inner surface of the pipeline $r_1=a$, the continuity conditions are:

For the plane strain problem of a poroelastic medium, the displacements and the stresses have the following expressions:

where ${\nabla }^2\phi =\frac{{\partial }^2\phi }{\partial r^2}+\frac{1}{r}\frac{\partial \phi }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2\phi }{\partial {\theta }^2}$; $u_r$ and $u_{\theta }$ denote the radial and circumferential displacement in the polar coordinate system; ${\sigma }_{rr}$, ${\sigma }_{\theta \theta }$, ${\sigma }_{r\theta }$ denote radial stress, the hoop stress and shear stress respectively; $\lambda$, $\mu$ denote Lame constant. Through the above equations we can acquire the stress and displacement of the buried pipeline.

4.1. Influence of incident wave number

Figs. 2(a) and 2(b) show the distribution of $K_{\theta \theta }$ on the inner surface of the pipeline for $\eta =$ 1.05, $\nu =$ 2.0, ${\theta }_{\alpha }=\mathrm{}$30°. The figures indicate that the hoop stress of both Q235 and X70 pipelines are more sensitive to higher space frequency $\delta$. And Q235 are more sensitive compared with X70.

Fig. 2Distribution of Kθθ on the inner surface of the pipeline

a) Q235

b) X70

Fig. 3Relationship between Kθθ and the incident P wave number

a) Q235

b) X70

Fig. 3 shows the relationships between $K_{\theta \theta }$ and the incident P wave number. From Fig. 3(a) and 3(b) we can see $K_{\theta \theta }$ decrease exponentially on the intervals $\delta \in \left(\text{0, 0.5}\right)$ with the increase of the incident wave number. This means that the hoop stress of shallow buried pipeline is more sensitive to low frequency blast seismic waves. The value of $K_{\theta \theta }$ keeps oscillating on the intervals $\delta \in \left(\text{0.5, 3}\right)$, with some peak values. For $\theta =$ 0º, the peak value increase with the increase of $\delta$, and reaches to the maximum value when $\delta =$ 2.65. For $\theta =$ 60º, the peak value decrease with the increase of $\delta$, and the peak value reaches to the maximum value when $\delta =$ 0.85.

4.2. Influence of the diameter-thick ratio

Figs. 4(a) and 4(b) show the relationship between $K_{\theta \theta }$ and diameter-thick ratio. In a certain thickness range, the hoop stress of Q235 has three different peak values. The first two values are small and the last one is large. For ${\theta }_{\alpha }=$5º, the hoop stress reaches to the maximum when the diameter-thick ratio is 1.075. The hoop stress of X70 also has three peak values. But the values are not so quite different compare with Q235. For $\theta =$ 60º, the peak value of the hoop stress reaches to the maximum when the incident angle is 15º.

Fig. 4Relationship between Kθθ and diameter-thick ratio

a) Q235

b) X70

4.3. Influence of the buried depth

Figs. 5(a) and 5(b) show relationship between $K_{\theta \theta }$ and buried depth of Q235. From the figures we can see that $K_{\theta \theta }$ presented a periodic change with the increase of the buried depth. For $\eta =$ 1.05, the periodic number reaches to maximum when $\theta =$ 0º, $\delta =$ 2.0. For $\delta =$ 2.0, the periodic number reaches to maximum when $\theta =$ 60º, $\eta =$ 1.05.

Fig. 5Relationship between Kθθ and buried depth of Q235

a)$\eta =$ 1.05

b)$\delta =$ 2.0

Figs. 6(a) and 6(b) show relationship between $K_{\theta \theta }$ and buried depth of X70. From the figures we can see that $K_{\theta \theta }$ also presented a periodic change with the increasing of the buried depth. For $\eta =$ 1.05, $\theta =$ 60º, the periodic number reaches to maximum when $\theta =$ 60º, $\delta =$ 3.0. For $\delta =$ 2.0, the periodic number reaches to maximum when $\eta =$ 1.02, $\theta =$ 0º.

Fig. 6Relationship between Kθθ and buried depth of X70

a)$\eta =$ 1.05

b)$\delta =$ 2.0