Stress concentration in parallel fluid-filled underground cylindrical pipes under harmonic and recorded seismic excitation

2.1. Geometry and material model of the pipe

Each of the two pipes is geometrically defined by its inner radius $R_1$, outer radius $R_2$, and wall thickness $h=R_2-R_1$. The two pipes are identical, and their axes are parallel and located at the same depth. The distance between the centres of the pipes is denoted by $d$, and the normalised spacing ratio $d$/$D$, with $D=2R_2$, is used as the main geometric parameter. Both pipes are assumed to be much longer than their diameters, so the problem is reduced to plane strain in the cross-sectional plane $\left(x,y\right)$.

The pipe wall is treated as a homogeneous, isotropic, linearly elastic continuum characterised by Young’s modulus $E_p$, Poisson’s ratio ${\nu }_p$, and mass density ${\rho }_p$, so that the corresponding Lamé constants are ${\lambda }_p=\frac{E_p{\nu }_p}{\left[\left(1+{\nu }_p\right)\left(1-2{\nu }_p\right)\right]}$ and ${\mu }_p=\frac{E_p}{\left[2\left(1+{\nu }_p\right)\right]}$. Although the wall thickness is small compared with the radius, the pipe is here modelled as a two-dimensional elastic continuum rather than as a thin shell, in order to retain a uniform elastodynamic formulation across the soil-pipe-fluid system and to allow direct evaluation of the radial and circumferential stress components in the wall, which control the dynamic stress concentration.

The internal fluid is assumed inviscid and barotropic, with bulk modulus $K_f$, density ${\rho }_f$, and acoustic sound speed $c_f={\left(K_f/{\rho }_f\right)}^{1/2}$. Its motion is described by an acoustic potential ${\phi }_f$ satisfying the Helmholtz equation ${\nabla }^2{\phi }_f+k_f^2{\phi }_f=0$, with $k_f=\omega /c_f$.

2.2. Surrounding soil and embedment conditions

The surrounding soil is treated as an unbounded, homogeneous, isotropic, linearly elastic medium of mass density ${\rho }_s$, Lamé constants ${\lambda }_s$ and ${\mu }_s$, P-wave velocity $c_p={\left[\frac{{\lambda }_s+2{\mu }_s}{{\rho }_s}\right]}^{1/2}$, and S-wave velocity $c_s={\left({\mu }_s/{\rho }_s\right)}^{1/2}$. The corresponding wave numbers are $k_p=\omega /c_p$ and $k_s=\omega /c_s$. Material damping is neglected in the frequency band of engineering interest; however, it can be reintroduced through complex Lamé constants without modifying the structure of the solution.

The pipes are buried at a depth $H$ sufficient for the incident seismic wave field to be considered approximately planar in the vicinity of the structures and for reflections from the free surface to be neglected over the time window of interest. This assumption is standard in the analytical study of stress concentration around buried inclusions [2, 4, 8, 9] and is consistent with pipes laid below the seismic active layer; if necessary, free-surface effects can be incorporated by image-source superposition without altering the analytical structure of the present solution.

The outer surface of each pipe is assumed to be in perfect bond with the surrounding soil, so that no slip or separation occurs along the pipe-soil interface. This represents a conservative upper bound on the dynamic load transmitted to the pipe wall and is consistent with field observations for steel pipelines installed in well-compacted backfill [2-4].

2.3. Wave-potential representation in bipolar coordinates

To describe the wave field around two parallel cylinders, bipolar cylindrical coordinates $\left(\xi ,\eta ,z\right)$ are introduced. Their relation to the Cartesian coordinates is:

where $a$ is the geometric parameter that defines the distance between the coordinate poles. The two pipe surfaces correspond to the bipolar coordinate lines $\eta =\pm {\eta }_0$, with ${\eta }_0$ chosen so that $R_2=a/\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}{\eta }_0$.

For harmonic motion with time factor $e^{-i\omega t}$, the displacement field in each elastic medium (soil or pipe wall) is decomposed by the Helmholtz decomposition into a scalar potential $\phi$ and a vector potential $\psi$; under plane-strain conditions the latter reduces to a single scalar component. The potentials satisfy:

where $k_p$ and $k_s$ are the wave numbers of the compressional and shear waves in the relevant medium. The general solution in bipolar coordinates is sought in series form:

where $Z_n\left(·\right)$ denotes a cylindrical wave function: Bessel $J_n$ in the pipe annular region (regularity inside the wall), Hankel $H_n^{\left(1\right)}$ in the unbounded soil (radiation condition at infinity), and a linear combination of $J_n$ and $Y_n$ across the annular pipe wall. The constants $A_n$ and $B_n$ are unknown complex coefficients to be determined from the boundary conditions. A similar expansion is used for the acoustic potential ${\phi }_f$ of the internal fluid, with $Z_n=J_n$ to ensure regularity at the pipe axis.

2.4. Boundary conditions

Six sets of conditions are imposed simultaneously on the coupled problem:

(a) At each outer pipe-soil interface ($r=R_2$) – continuity of radial and tangential displacements and tractions:

(b) At each inner pipe–fluid interface ($r=R_1$) – continuity of normal velocity, equality of fluid pressure with the normal traction, and vanishing tangential traction in contact with the inviscid fluid:

(c) Regularity of the fluid acoustic potential at each pipe axis ($r\to$ 0).

(d) Sommerfeld radiation condition at infinity in the soil for the scattered wave field, which guarantees that no energy is reflected back from infinity.

(e) The incident seismic action is prescribed as a plane harmonic P-, SV-, or SH-wave of unit amplitude propagating at angle $\gamma$ with respect to the line connecting the pipe centres.

(f) Absence of body forces in the soil, in the pipe walls, and in the fluid.

2.5. Solution procedure

By substituting the wave-potential expansions Eq. (3) into the boundary conditions Eqs. (4-5) and projecting onto the bipolar harmonics $\mathrm{c}\mathrm{o}\mathrm{s}\left(n\xi \right)$ and $\mathrm{s}\mathrm{i}\mathrm{n}\left(n\xi \right)$, the problem is reduced to a complex linear algebraic system for the unknown coefficients $A_n$, $B_n$, and the corresponding fluid coefficients. The infinite system is truncated at a sufficiently large order $N$ and solved by Gaussian elimination with partial pivoting. Series convergence is verified by progressively increasing $N$ until the maximum dynamic stress concentration coefficient changes by less than 0.5 %; in the parameter range of practical interest ($k_p{R}_2\le$ 3, $d$/$D$ ∈ [0.5, 4]) this is achieved with $N=$ 12-18.

The obtained solution makes it possible to evaluate the dynamic stress concentration in the pipe system as a function of pipe spacing $d$/$D$, the type and incidence angle of the seismic wave, and the properties of the internal fluid. These parameters are used in the next sections to analyse the maximum dynamic concentration coefficient and the limiting non-resonant distance between the pipes.

4.1. Verification and computational parameters

The numerical results presented below were obtained for representative parameters of a buried steel pipeline embedded in stiff soil: pipe outer diameter $D=$ 1.0 m, wall thickness $h=$ 0.012 m, $E_p=$ 210 GPa, ${\nu }_p=$ 0.30, ${\rho }_p=$ 7850 kg/m³; soil $c_p=$ 600 m/s, $c_s=$ 300 m/s, ${\rho }_s=$ 1900 kg/m³; internal fluid (water) $c_f=$ 1480 m/s, ${\rho }_f=$ 1000 kg/m³. The truncation order $N=$ 16 was verified to provide three-digit convergence of ${\eta }_{max}$ for all spacings considered. In the limit $d$/$D$ → ∞ and in the absence of fluid, the obtained value of ${\eta }_{max}$ recovers the well-known single-pipe result reported in [2], which validates the numerical implementation.

4.2. Results under harmonic excitation

The analytical solution of Section 2 was first used to evaluate the dynamic stress concentration in the parallel underground fluid-filled pipes under harmonic seismic excitation. Special attention was paid to the influence of the normalised spacing ratio, the angle of wave incidence, and the presence of fluid inside the pipes. The results show that the interaction between adjacent pipes significantly modifies the stress state of the system in comparison with a single buried pipe.

Table 1 presents the variation of the maximum dynamic concentration coefficient ${\eta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for different spacing ratios in the case of an incident P-wave. The results show that ${\eta }_{max}$ increases from 1.68 at $d$/$D=$ 0.5 to 1.76 at $d$/$D=$ 1.0, and then decreases to 1.61 and 1.60 at $d$/$D=$ 2.0 and $d$/$D=$ 4.0, respectively. Thus, the strongest interaction is observed at relatively small spacing, while at larger distances the effect gradually stabilises and approaches the response of a single pipe. This behaviour indicates that the mutual influence of parallel pipes is significant mainly when $d$/$D\le$ 4.0.

The increase in ${\eta }_{max}$ at small spacing is associated with superposition and interference of waves reflected between the surfaces of adjacent pipes. From an engineering point of view, this effect is important because it may lead to local amplification of the dynamic soil pressure transmitted to the buried pipeline system. According to Table 1, in the considered range the most favourable spacing is $d$/$D$ = 0.5, where the dynamic concentration coefficient is lower than at $d$/$D$ = 1.0.

An additional design aspect concerns resonance-like effects in multi-line pipelines. As shown in Table 2, the limiting non-resonant distance between the pipe centres increases with the angle of wave incidence $\gamma$: from 5.0 m at 0° to 10.0 m at 90°. Therefore, when the seismic wave approaches the pipeline at smaller angles, the pipes should be placed closer to each other in order to avoid resonance and to ensure seismic safety.

The numerical analysis also shows that the wave type affects the stress concentration level. The local resonance effect is more pronounced for SV-wave excitation than for P-wave excitation. The presence of fluid inside the pipes increases the seismic effect by approximately 10-20 %, while for two-line pipe systems the increase in the maximum dynamic pressure is generally about 5-10 %. For pipes arranged in two lines at $d<3.0D$, the maximum dynamic soil pressure may exceed that of a single pipe by up to 15 %.

4.3. Results under recorded earthquake excitation

The FFT-based procedure described in Section 3 was applied to the worst harmonic case under P-wave excitation, namely $d$/$D$ = 1.0. With the El Centro NS record as input, the peak stress concentration coefficient is ${\eta }_{peak}\approx$ 1.71. This value is approximately 3 % below the harmonic peak ${\eta }_{max}=$ 1.76 reported in Table 1, because the recorded record does not concentrate all of its energy at the resonance frequency of the two-pipe system. Conversely, ${\eta }_{peak}$ is approximately 8 % higher than the value that would be obtained from a single-pipe analysis under the same record, which confirms that the pipe-pipe interaction effect identified in the harmonic case persists under realistic broadband loading.

The 1966 Tashkent record produces ${\eta }_{peak}\approx$ 1.66 at the same spacing, with a similar relative increase over the corresponding single-pipe value. This confirms the practical relevance of the harmonic-domain results in the seismic context of the region considered. From a resilience point of view, the small reduction of ${\eta }_{peak}$ with respect to ${\eta }_{max}$ indicates that the harmonic analysis used in design provides a slightly conservative – but physically consistent – estimate of the dynamic stress concentration; the safety margin is, however, narrow at the resonance spacing $d$/$D$ ≈ 1.0, which reinforces the importance of avoiding this spacing in multi-line layouts.

4.4. Engineering implications

The obtained tendencies are of practical interest for the seismic design of underground multi-line pipeline systems. They show that an unfavourable spacing between adjacent pipes may intensify dynamic stresses, whereas a rational choice of spacing can reduce the stress concentration and prevent resonance-like amplification. The results also confirm that denser embankment soil reduces the seismic effect on underground pipes, and that at sufficiently large distances the dynamic problem tends towards a quasi-static response. The 10-20 % increase produced by the internal fluid implies that water-filled fire-protection lines and oil-filled pipelines should not be assessed on the basis of empty-pipe formulae.