Comparison of Penzien and Wang analytical methods with the finite difference numerical method in seismic analyses of the Qazvin-Rasht (Kuhin) tunnel

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Underground facilities are an integral part of the infrastructure of modern society and are used for a wide range of applications, including subways and railways, highways, material storage, and sewage and water transport. Underground facilities built in areas subject to earthquake activity must withstand both seismic and static loading. Historically, underground facilities have experienced a lower rate of damage than surface structures. Nevertheless, some underground structures have experienced significant damage in recent large earthquakes: 1990 in Manjil, Iran; 1995 in Kobe, Japan; 1999 in Chi-Chi, Taiwan; and 1999 in Kocaeli, Turkey. Several studies have documented earthquake damage to underground facilities. The American Society of Civil Engineers describes the damage in the Los Angeles area as a result of the 1971 San Fernando earthquake; the Japanese Society of Civil Engineers describes the performance of several underground structures, including an immersed tube tunnel during shaking in Japan.
Earthquake effects on underground structures can be grouped into two categories: ground shaking and ground failure such as liquefaction, fault displacement and slope instability. The component that has the most important influence on the tunnel lining under seismic loading, except for the case of the tunnel being directly sheared by a fault, is the ovaling or racking deformations. Studies propose that while ovaling may be caused by waves propagating horizontally or obliquely, vertically propagating shear waves are the predominant form of earthquake loading that causes these types of deformations. The ovaling deformation is commonly simulated as a two-dimensional, plane-strain condition. Since the inertia effect can be relatively small, the ovaling deformation is further simplified as a quasistatic case, and hence without the dynamic interaction. Wang and Penzien present closed form solutions to compute displacements and forces in the lining due to equivalent static ovaling deformations. The analytical solutions are frequently used in estimation moment and forces in tunnels. Hashash et al. identifies a significant discrepancy in the computed lining thrust between the Wang and Penzien solutions. This discrepancy has important implications as far as lining design and is of concern to many design engineers.
This paper discusses the discrepancy between the two analytical and finite difference methods. Numerical analysis is performed using the explicit finite difference model by FLAC2D software to evaluate the analytical solutions for ovaling deformation. In this method, shear loading is applied at the lower ends of the boundaries to simulate pure shear condition. In FLAC2D, no-slip condition between the tunnel lining and ground is simulated.
A comparison of the two analytical methods shows that the calculated forces and displacements are identical for the condition of full-slip between the tunnel lining and ground. However, the calculated lining thrusts differ by an order of magnitude when considering no-slip between the tunnel lining and the ground. The analytical solutions are compared to numerical analyses of the no-slip condition, using the finite difference method to validate which of the two methods provide the correct method. To achieve the goal: numerical analysis is performed in a two-dimensional elasticity-plasticity domain, using the explicit finite difference model by FLAC2D software on the field data of the Kuhin Railway tunnel and characterize Manjil earthquake. Numerical analysis results agree with one of the analytical solutions that provides a higher estimate of the thrust on the tunnel lining, thus highlighting the limitation of the other analytical solution. The maximum axial thrusts from numerical analyses result in an almost perfect match with Wang’s solutions, whereby the differences are within 1.8% for this case. However, the difference between the numerical and Penzien’s solutions are significant. The difference is higher than 590%, in which Penzien’s solutions highly underestimate the thrust for this case. The comparisons clearly demonstrate that Wang’s solution provides a realistic estimate of the thrust in the tunnel linings for the no-slip condition. It is recommended that the Penzien’s solution not be used for no-slip condition.
Keywords: Earthquake design, Seismic design, Kuhin railway tunnel, Manjil earthquake
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