Shock and Vibration

Volume 2017 (2017), Article ID 9152632, 7 pages

https://doi.org/10.1155/2017/9152632

## Vibration of a Cylindrical Tunnel under a Centric Point-Source Explosion

^{1}College of Defense Engineering, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China^{2}State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China^{3}Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki, Thessaloniki, Greece

Correspondence should be addressed to Cheng Chu; moc.361@iccmai

Received 15 March 2017; Accepted 30 April 2017; Published 18 June 2017

Academic Editor: Abdul Qadir Bhatti

Copyright © 2017 Yuetang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Underground tunnels are vulnerable to terrorists’ bombing attacks, which calls for studies on tunnel’s response to internal explosive loading. In this paper, the dynamic response of a cylindrical tunnel to an ideal centric point explosion was treated as an axisymmetric 2-dimensional problem, in which the tunnel was modeled with a continuous anisotropic shell, while the ground medium’s effect was accounted for with linear elastic Winkler springs and the explosive loading described by a temporal and spatial function. The governing equation of the motion is a fourth-order partial differential equation, for which a numerical method combining finite difference with the implicit Newmark-*β* method was adopted. This method avoided complicated integral transform and numerical inverse transformation, thus allowing efficient parameter study. The maximum radial displacement was found on the cricle of the center of explosive, where hoop stress is the maximum principal stress. The anisotropy showed little influence on maximum hoop stress. Within the range of ground medium’s modulus, minor influence on maximum hoop stress was incurred. This research may be helpful to hazard assessment and protective design for some critical subway tunnels.

#### 1. Introduction

In recent years, increased terrorist bombing attacks on subway system have been observed [1–3]; the most recent one was the notorious 2017 Saint Petersburg subway bombing [4]. Besides threats from terrorism, accidental explosion may be another potential hazard. The aftermath of an internal blast may include casualties and structural damage [5]. A deformation beyond threshold may cause the collapse of the structure. Also many of these tunnels may be built across beneath water bodies, such as these in New York, London, Shanghai, and Paris. A blast may breach or tear apart the wall of the tunnel lining and cause a flooding, for which the aftermath may be intractable [6, 7]. Thus the resistance of a tunnel structure to an internal blast should be studied.

In the designing of tunnels, the effects from some natural hazards and man-made disturbances such as seismic events and fires are generally considered, whereas blast loads are rarely considered for civilian tunnels [8]. Therefore, under the background of increasing probability of terrorist bombing attacks, safety of municipal tunnels under explosions needs more attention.

So far experimental study on tunnel’s response to internal explosion can be scarcely found in literature. Preece et al. [9] and Liu and Nezili [10] conducted a series of scale model tests using centrifuge facility, where the tunnel was simulated by monolithic aluminum tube. Their results may not be applicable to reinforced concrete tunnel since mechanical behaviors of metals like aluminum are quite different from that of reinforced concrete. Bonalumi et al. [11] conducted internal blast tests on a concrete pipe embedded in soft soil purposed to study internal blast loading in cylindrical enclosure, whereas the response of structure is not accounted for. Zhao et al. [12] carried out a few full-scale tests on staggered segmental tunnel’s response to internal point-source explosions, concluding that damage mainly concentrated in joint areas; however, no theoretical study was made on the tests.

Tunnels’ response to blast loading was more studied with finite element modeling. Based on the results of numerical simulation with ABAQUS, Colombo et al. [13, 14] plotted pressure-impulse (P-I) diagrams, which is also called iso-damage diagram for segmental tunnels in case of an internal explosion. This provided a useful tool for damage assessment and antiexplosion design of segmental tunnel lining. However, their results were obtained on the presumption of an axisymmetric plane problem, that is, the blast loading keeps constant along the tunnel axis, which is unlikely the true case. Choi et al. [15, 16] did nonlinear finite element analyses on internal explosion of tunnels and evaluated the vulnerability of lining structure, only concluding with some descriptive remarks. Yu et al. [17] also simulated internal explosion effects on tunnel in soil, and the influences from the soil and charge location were analyzed; however, there were no clear conclusions. These FEM analyses allowed for reproduction of the process involving blast wave propagation and its interaction with the tunnel and the structure’s response. However, these researches rely much on commercial finite element code, and the validity of the results depends on the material models’ parameters they used. Until now, no validation against any test results can be found in literature. Also it is not efficient to do parameter studies when the model scale is big.

As for theoretical investigation, a tunnel’s response to internal blast loads is generally treated as an axisymmetric plane strain problem, where the blast loading is simplified as uniformly distributed loading which keeps constant along the axial direction, and the tunnel is modeled with continuous isotropic thin shell [18–20], but this can hardly be the real case since the explosion could not be a centric infinite line charge detonation. Among those theoretical studies, Gao et al. [21] studied a 3D problem of point-source explosion’s effects on a long straight cylindrical tunnel in soil. However, the method includes complicated integral transform and numerical inverse transforms, which makes their research inconvenient for engineering use. Another problem in their research is that the orthogonal anisotropy of tunnel structure, that is, the difference in mechanic behavior between the circumferential and axial directions, which might be important in its overall dynamic response, was seldom accounted for.

In this paper a 3-dimensional problem of a long cylindrical thin-walled tunnel under a point-source explosion was studied, where both the lining structure and the blast loading were described more realistically. To begin with, a description of the blast loading on the inner wall of the structure with simplified spatial and temporal function was elaborated. Then the tunnel structure was modeled as a long straight anisotropic cylindrical thin shell, of which the governing equation of its motion was established on the basis of classical theory of shell. Thereafter a solving method combining finite difference method and Newmark-*β* implicit integration method was developed and some numerical results of shell’s dynamic response and parameter study were displayed and discussed. At last some concluding remarks were given on the basis of the analyses and discussions. The method developed in this paper may aid in both blast hazard assessment and protective design for some critical part of subway tunnel systems.

#### 2. The Description of Blast Loads due to a Centric Point-Source Explosion

Considering a point charge of explosive locating on the centre axis. After detonation, the blast wave’s propagation and its loading effects can be categorized into two phases. First is a period of spherical pressure wave propagation and its interaction with the structure in the near-field. After a distance of 4~6 times’ radii from the explosive centre, it evolves into second phase; that is, after complicated interactions in first phase the blast wave propagates along tunnel axis like a 1-dimensional plane wave [22, 23]. Therefore, the loading on the tunnel from inner explosion is quite complicated.

Blast wave with pressure of propagates at a velocity of along the inner surface of a cylindrical shell with a certain length; the overpressure of incident blast wave could be described by a temporal and spatial function [21] as follows:where , are attenuation parameters of blast waves with respect to time and space, respectively; the centric point source of explosion is chosen as the origin of the coordinate system; is overpressure of incident blast wave, which could be estimated with empirical equation given by M. A. Sadovskii [24]:where is scaled distance, with and denoting stand-off distance (m) to the centre of explosive charge and weight of the charge (kg).

Knowing incident blast wave loading in confined space of tunnel's enclosure, the effective loading acts on inner wall of the lining due to the wave's reflection, that is, reflected overpressure can be calculated with the following empirical formula for normal relfection [25]:where is incident overpressure and is ambient atmospheric pressure.

Pokrovsky [24] proved by experimental data that when incidence pressure is smaller than MPa, reflected pressure of an oblique reflection can be calculated by the formula for normal reflection.

#### 3. Governing Motion Equation and Its Solution

Taking account of the stiffness difference between circumferential and axial directions, the circular tunnel lining structure could be treated as an anisotropic cylindrical shell, as graphically shown in Figure 1, and the ground resistance effects are modeled with radial linear elastic Winkler springs.