Tunnel portal sections often suffer serious damage in strong earthquake events. Earthquake waves may propagate in different directions, producing various dynamic responses in the tunnel portal. Based on the Galongla tunnel, which is located in a seismic region of China, three-dimensional seismic analysis is conducted to investigate the dynamic response of a tunnel portal subjected to earthquake waves with different vibration directions. In order to simulate the mechanic behavior of slope rock effectively, an elastoplastic damage model is adopted and applied to ABAQUS software by a self-compiled user material (UMAT) subroutine. Moreover, the seismic wave input method for tunnel portal is established to realize the seismic input under vertically incident earthquake waves with different vibration directions, e.g., S waves with a vibration direction perpendicular or parallel to the tunnel axis and P waves with a vibration direction perpendicular to the tunnel axis. The numerical results indicate that the seismic response and damage mechanisms of the tunnel portal section are related to the vibration direction of the earthquake waves. For vertically incident S waves running perpendicular to the tunnel axis, the hoop tensile strain at the spandrel and arch foot and the hoop shear strain at the vault and arch bottom are the main contributors to the plastic damage of the tunnel. The strain is initially concentrated around the tunnel foot and spandrel, before shifting to the tunnel vault and bottom farther away from the tunnel entrance. For vertically incident S waves running parallel to the tunnel axis, very large hoop shear strain and plastic damage appear at the tunnel haunches. This strain first increases and then decreases with distance from the tunnel entrance. For vertically incident P waves running perpendicular to the tunnel axis, the maximum damage factor of the slope rock and the maximum plastic strain of the tunnel are significantly lower than for S waves. Moreover, with increasing distance from the tunnel entrance, the plastic damage to the tunnel lining rapidly decreases.

The thousands of mountain tunnels that have recently been built in China play an important role in local development. Such tunnels may pass through highly active seismic areas, especially in Southwest China. Several strong earthquakes in recent decades, some mountain tunnel portals suffered particularly serious damage to some mountain tunnel portals, which is just less critical than that of the fault [

There has been considerable research on the seismic damage suffered by tunnel portals, mainly through model tests, numerical simulations, and in-situ investigations. Sun et al. [

Because of the smaller overburden depth, the tunnel portal has a seismic response that is strongly affected by the slope around the structure. The slope rock is often severely weathered, reducing its ability to restrain the portal structure compared with other tunnel parts that are buried deeper underground [

In this study, an elastoplastic damage model is integrated into the ABAQUS software through a self-compiled user material (UMAT) subroutine, allowing the mechanical behavior of the slope rock to be simulated. First, the accuracy of the model is verified. Based on the Galongla tunnel project, a dynamic time-history model is then constructed to simulate the seismic response. A viscous-spring boundary is used to absorb the wave radiation. In addition, a wave input method for the slope rock is established, and the applicability and accuracy of this method are verified. Vertical incident seismic waves in different vibration directions are simulated by the proposed method, and the response of the portal structure of Galongla tunnel is numerically studied. Finally, the seismic response characteristics and damage mechanism of the tunnel portal are discussed.

Galongla tunnel is a critical part of the Zhamo highway in Tibet, the only highway leading to Motuo County in China. the local geography is shown in

Galongla tunnel is located in the Himalayan fault zone, which is subjected to frequent geological activity, i.e., an earthquake-prone area. According to the seismic data issued by the China Earthquake Administration, the basic intensity degree is VIII. Therefore, the tunnel, especially the tunnel portal, faces a serious earthquake risk. To investigate the seismic damage mechanics of the tunnel portal part, the results of dynamic numerical analysis are described in the following sections.

Based on the nonlinear unified strength criterion, Du et al. [

The following nonlinear unified strength criterion developed by Lu et al. [_{β}_{r}_{c}

To describe the rock dilatancy effectively, a non-associated flow rule is adopted. The expression for the plastic potential function is similar to the yield function, with the parameter _{β}_{g}

The hardening variable ^{p}_{o}_{c} _{o}_{γ}_{c}^{p}

The damage factor _{1} is a parameter controlling the slope of the softening curve; _{d}_{pv}_{1} controls the rate of increase of _{d}_{2} is a constant that affects the value of χ_{1}.

For the elastoplastic damage model, the increment in the Cauchy stress

The relationship between increments in the effective stress

The plastic strain increment

Substituting

The consistency condition then gives

Substituting

Combining

Combining

The expressions for

Based on the derived relationship between the stress and strain, the elastoplastic damage model is integrated into the ABAQUS software through a self-developed UMAT subroutine. To verify the accuracy of the implemented model, a three-dimensional (3-D) solid unit cube model measuring 1 m × 1 m × 1 m is built and discretized into a single grid using reduction integral elements (C3D8R). The numerical results are compared with the results of conventional triaxial experiments by Kawamoto et al. [

_{o} |
_{c}/^{o} |
^{o} |
A | _{o} |
_{1} |
_{2} |
||||
---|---|---|---|---|---|---|---|---|---|---|

Lin et al. [ |
25.3 | 47.4 | 0.25 | 0.276 | 26.1 | 13.1 | 0.0015 | 0.8 | 7600 | 1.5 |

Kawamoto et al. [ |
51.6 | 24.7 | 0.25 | 0.243 | 46.9 | 23.5 | 0.002 | 0.9 | 1500 | 1.3 |

To investigate the seismic response of the Galongla tunnel portal, dynamic history simulations were performed using the ABAQUS software. The 3-D finite element model and its dimensions are shown in ^{3}, Young's modulus ^{3}, _{o}_{c }=_{ }46.9, _{o}_{1} = 501, _{2} =1.07.

In this study, the tunnel portal responses under three different earthquake incident waves are investigated. The first case considers the incident waves to be plane P waves with a vibration direction perpendicular to the tunnel axis. The second and third cases consider the earthquake waves to be vertically incident S waves with vibration directions perpendicular and parallel to the tunnel axis, respectively. The Kobe University earthquake records for rock strata were obtained in similar geological conditions to those of the Galongla tunnel project site. The Kobe waves are real seismic records with rich spectra, as shown in

During the numerical analysis, a viscous-spring artificial boundary is employed to simulate the radiation damping effect at the truncated boundary, as introduced in

In recent decades, many artificial boundaries have been developed which can be found in the works of Lysmer et al. [_{s}_{p}_{r}_{r}

For node _{x}_{y}_{z}_{x}_{y}_{z}_{yy}_{yx}_{3l} is a quarter of the total area of all boundary elements containing boundary node _{3l} = (_{1} + _{2} + _{3} + _{4})/4.

For node _{x}_{xx}_{xy}_{xz}

For node _{z}_{zz}

As shown in _{x}_{y}_{z}_{xx}_{yy}_{zz}_{xy}_{zy}_{zx}

In the 2-D model for slope seismic analysis, the viscous-spring boundary developed by Du et al. [

In addition, the incident earthquake waves are converted into equivalent nodal forces that apply at each boundary node. For node _{2l} is half of the total length of all boundary elements containing boundary node _{2l} = (_{1} + _{2})/2.

For node _{x}

In general, the seismic input method for the 3-D model is as illustrated in

The proposed seismic input method is now verified. The 3-D slope site model illustrated in ^{3},

The displacement contours of the 2-D model and the 3-D model are compared in

The maximum principal plastic strain contours of the tunnel lining under S waves with a vibration direction perpendicular to the tunnel axis is presented in

The strain can be decomposed into the hoop axial strain _{h}, hoop shear strain _{h}, and axial tensile strain _{a}, as shown in _{h} appears at the tunnel foot and spandrel, and the hoop shear strain _{h} around the tunnel bottom and vault is considerable. Though the rock-tunnel model is bilaterally symmetric, the shaking produced by an earthquake is not symmetric, and neither are the vertically incident S waves with a vibration direction perpendicular to the tunnel axis. Thus, _{h} and _{h} are not symmetric at the cross-sections. Compared with _{h} and _{h}, the axial tensile strain _{a} is very small, which indicates that the tunnel suffers only slight tensile deformation in the tunnel axial direction.

The distributions of hoop axial strain _{h} at the left spandrel and right arch foot along the tunnel axis are given in _{h} at the tunnel vault and bottom are shown in _{a} is very small, the distribution of this component along the tunnel axis is not presented. From _{h} at the spandrel and arch foot are dominant at the tunnel entrance, and then quickly decrease to a stable value with distance from the tunnel entrance. Therefore, the plastic strain at the spandrel and arch foot are higher around the tunnel entrance than at other positions, as shown in

The distributions of the strain components at two tunnel cross-sections for S waves with a vibration direction parallel to the tunnel axis are shown in

The seismic response of the tunnel portal under vertically incident P waves with a vibration direction perpendicular to the tunnel axis is now analyzed. The damage factor contours of the rock near the tunnel portal are shown in

_{h} and hoop shear strain _{h} are larger than the axial tensile strain _{a}. Moreover, _{h} and _{h} are dominant around the tunnel haunches, and change at the same pace as the plastic strain concentrated around the tunnel haunches. _{h} and _{h} at the tunnel haunches along the tunnel axis. With increasing distance from the tunnel entrance, the hoop axial strain gradually decreases, whereas the hoop shear strain increases at first and then decreases.

During past earthquake events, tunnel portals have suffered more serious damage than other tunnel sections. Earthquake waves may propagate in different vibration directions, producing a complex dynamic response in the tunnel portal. Based on the Galongla tunnel project, which is located in a seismic region of China, a 3-D numerical model of the tunnel portal was established to study such effects. The seismic mechanical behavior of the slope rock was simulated using an elastoplastic damage model implemented in ABAQUS through a self-compiled UMAT subroutine. The viscous-spring artificial boundary was used to simulate the radiation damping effect at the truncated boundary. Moreover, a seismic input method for the slope site was established for vertically incident earthquake waves with different vibration directions. The damage characteristics of the Galongla tunnel portal under different vibration directions of earthquake waves were then numerically studied. The conclusions are as follows:

For vertically incident S waves with a vibration direction perpendicular to the tunnel axis, the hoop tensile strain at the spandrel and arch foot and the hoop shear strain at the vault and arch bottom are the main contributors to the plastic damage of the tunnel. For vertically incident S waves with a vibration direction parallel to the tunnel axis, a very large hoop shear strain occurs at the tunnel haunches. For vertically incident P waves with a vibration direction perpendicular to the tunnel axis, the tunnel is mainly subjected to hoop tensile strain and hoop shear strain.

The vibration direction of the earthquake waves significantly influences the dynamic response of the tunnel portal. For vertically incident S waves with a vibration direction perpendicular to the tunnel axis, the hoop axial strain at the spandrel and arch foot are dominant at the tunnel entrance, and then rapidly decrease to a stable value away from the tunnel entrance. The hoop shear strain increases at first and then decreases with increasing distance from the tunnel entrance. For vertically incident S waves with a vibration direction parallel to the tunnel axis, the hoop shear strain increases and then decreases with distance from the tunnel entrance. For vertically incident P waves with a vibration direction perpendicular to the tunnel axis, the hoop axial strain decreases gradually, and the hoop shear strain first increases and then decreases with increasing distance from the tunnel entrance.

The authors wish to acknowledge financial support from the Beijing Natural Science Foundation Program (JQ19029) and the National Natural Science Foundation of China (41672289; U1839201; 51421005).

where

The expression of _{β}_{g}