A three-dimensional (3D) detailed numerical model of an immersed tunnel in a horizontally layered site is established in this study. The 3D seismic response of the immersed tunnel in a horizontally layered site subjected to obliquely incident waves is analyzed based on the precise dynamic stiffness matrix of the soil layer and half-space via combined viscous-spring boundary and equivalent node stress methods. The nonlinear effects of external and internal site conditions on the whole model were determined by equivalent linearization algorithm and Mohr–Coulomb model, respectively. The proposed model was then applied to investigate the nonlinear seismic response of an immersed tunnel in the Haihe River subjected to seismic waves of oblique incidence. The dislocation (opening) of pipe joints in the immersed tunnel were analyzed to determine the response characteristics of the shear keys and overall displacement of the tunnel; the dynamic responses of the immersed tunnel subjected to obliquely incident seismic waves markedly differ from those of vertically incident seismic SV waves. The maximum stress value of shear keys and the maximum dislocation of the pipe joint appear as upon critical angle. The overall displacement of the tunnel increases as incident angle increases. Under severe earthquake conditions, both the pipe corners and midspan section of the roof and floor are likely to produce crack. These areas need careful consideration in the seismic design of immersed tunnel structures.
Immersed tunnel structures have grown increasingly common in cross-sea and cross-river traffic engineering projects. Many of these structures are located in highly seismic regions. The unique characteristics of the immersed tunnel make any structural damage caused by an earthquake particularly difficult to be repaired, so antiseismic design is particularly important. The complexity of the immersed tunnel system makes it difficult to test by shaking table. Numerical analysis methods are an important means to study the seismic performance of immersed tunnels. Immersed tunnels tend to be lengthy and are usually built on weak ground [
Recent researchers have conducted many shaking table tests [
It is generally assumed that seismic waves in a deep-focus earthquake strike perpendicularly to the bedrock and transfer it to the structure. For shallow-focus earthquakes with small epicentral distance, the influence of seismic wave incident angle on the seismic response of surrounding structures is an important consideration [
Previous researchers have mainly focused on 2D cross sections. However, it is difficult to accurately describe the nonlinear response state by simply decomposing the immersed tunnel into in-plane and out-of-plane components under the oblique incidence of seismic waves. Unfortunately, it is very challenging to secure an accurate 3D model due to its complexity. Huang et al. [
In the present study, we established a 3D refined model of horizontal layered site immersed tunnel. We used the dynamic stiffness matrix of the soil layer and half-space combined with viscous-spring boundary and equivalent node stress methods to conduct accurate simulations of 3D seismic responses in a horizontally layered immersed tunnel structure. We applied the equivalent linearization algorithm (self-programmed) to determine the nonlinear effects of the external site of the overall model. We also used the Mohr–Coulomb model to determine the nonlinear effects of soil inside the overall model. The nonlinear effects of concrete were determined by a plasticity damage model of concrete in ABAQUS. The nonlinear seismic response of an immersed tunnel in the Haihe River and the seismic behavior of shear keys and pipe joints under obliquely incident SV waves were then determined based on the proposed model.
We used the 3D viscous-spring boundary set provided by Huang et al. [
Here, we combine the finite-element method with the time-domain method of viscous-spring boundaries. The displacement and stress of the artificial boundary caused by equivalent load input on the artificial boundary must be consistent with that of the original free field. In this way, the earthquake input is translated into a free-field motion problem on artificial boundary nodes, whereas free-field motion can be transformed into an equivalent nodal load on artificial boundary nodes [
The schematic diagram of the input wave field when the SV wave is obliquely incident is shown in Figure
Layered site model of SV wave under oblique incidence.
Wolf [
The stiffness matrix of each soil layer can be set and adjusted to obtain the overall space stiffness matrix
The horizontal displacement and vertical displacement at any point within each soil layer can be expressed as equations (
In this study, we investigated the nonlinear seismic response of the site by the equivalent linearization method. The soil characteristics in the matrix are replaced by the equivalent shear modulus
The coordinate system in the direct stiffness method differs from that in ABAQUS. After coordinate transformation, the solution is plugged into equation (
We established a 3D horizontally layered finite-element model (Figure
3D model of the horizontally layered site. (a) Finite-element model; (b) angle of incidence.
Soil layer parameters.
Category | Thickness (m) | Density (kg·m−3) | Poisson’s ratio | S wave velocity (m·s−1) | P wave velocity (m·s−1) |
---|---|---|---|---|---|
Soil layer 1 | 40 | 2000 | 0.3 | 200 | 374.1657 |
Soil layer 2 | 40 | 2100 | 0.3 | 400 | 748.3315 |
Soil layer 3 | 40 | 2200 | 0.3 | 800 | 1496.663 |
Displacement diagram and velocity diagram of pulse wave. (a) Displacement diagram of pulse wave; (b) velocity diagram of pulse wave.
The reflection and refraction waveforms of the SV wave are shown in Figure
Displacement and velocity time curve of each node under incident angle (60°, 30°). (a) Displacement of point A; (b) velocity of point A; (c) displacement of point B; (d) velocity of point B.
The tunnel immersed below the Tianjin Haihe River is a two-hole, three-pipe gallery structure (Figure
Pipe model and overall model. (a) A single pipe model; (b) overall model.
Structural material parameters.
Material | Density (kg·m−3) | Modulus of elasticity (MPa) | Poisson’s ratio |
---|---|---|---|
Concrete | 2500 | 32500 | 0.16 |
Steel cable | 7000 | 200000 | 0.3 |
Crushed-stone soil (superimposed soil) | 1500 | 150 | 0.15 |
Soil parameters.
Type | Depth (m) | Density (kg·m−3) | Shear velocity (m) | Poisson’s ratio | Cohesion (kPa) | Internal friction angle (°) |
---|---|---|---|---|---|---|
Soil layer 1 | 0–25 | 2000 | 200 | 0.27 | 14.6 | 24.3 |
Soil layer 2 | 25–40 | 2100 | 300 | 0.30 | 22.0 | 18.0 |
Soil layer 3 | 40–55 | 2200 | 400 | 0.30 | 24.0 | 20.0 |
We established a 3D finite-element model (300 m × 255 m × 55 m) of the Haihe River immersed tunnel using the parameters discussed above. The overall model includes three pipes (
A Tianjin wave was input at the bottom of the model with a 20 s duration, 0.02 s interval, and peak acceleration of 3.1 m/s2 (Figure
Tianjin wave acceleration time history.
The pipe joint and shear key are significant components in the immersed tunnel—their inherent weakness may result in serious, even irreversible damage to the whole tunnel in the case of an earthquake. We first calculated and analyzed the mechanical properties of the pipe joint and the shear key, then the deformation of the tunnel (including three pipes) under typical earthquake conditions.
The three shear keys on the left side of the
Shear key sketches. (a) Shear key distribution at joint interface; (b) horizontal shear key A; (c) vertical shear key B and C.
Maximum principal stress (MPa).
Points | Angle | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0°, 0° | 0°, 15° | 0°, 30° | 30°, 0° | 30°, 15° | 30°, 30° | 60°, 0° | 60°, 15° | 60°, 30° | ||
A | 1 | 0.611 | 0.407 | 1.481 | 0.619 | 0.458 | 2.647 | 0.418 | 0.290 | 1.580 |
2 | 0.270 | 0.299 | 0.808 | 0.334 | 0.343 | 2.769 | 0.245 | 0.264 | 1.513 | |
3 | 2.285 | 2.095 | 4.394 | 2.014 | 1.842 | 3.548 | 1.537 | 1.270 | 3.003 | |
4 | 1.039 | 1.128 | 3.659 | 1.207 | 1.076 | 6.312 | 1.047 | 0.944 | 4.629 | |
|
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B | 1 | 0.0079 | 0.0086 | 0.0536 | 0.0114 | 0.0094 | 0.823 | 0.0147 | 0.0135 | 0.853 |
2 | 0.163 | 0.159 | 0.703 | 0.723 | 0.701 | 1.146 | 0.710 | 0.794 | 0.729 | |
3 | 0.251 | 0.234 | 0.546 | 0.222 | 0.213 | 0.958 | 0.153 | 0.161 | 1.213 | |
4 | 0.288 | 0.212 | 0.755 | 0.724 | 0.709 | 2.063 | 0.768 | 0.839 | 0.762 | |
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C | 1 | 0.0061 | 0.0046 | 0.0258 | 0.0098 | 0.0101 | 0.0688 | 0.0163 | 0.0045 | 0.257 |
2 | 0.0628 | 0.0024 | 0.379 | 0.211 | 0.182 | 0.473 | 0.283 | 0.352 | 0.431 | |
3 | 0.0620 | 0.0629 | 0.219 | 0.0590 | 0.0143 | 0.165 | 0.0862 | 0.0501 | 0.723 | |
4 | 0.124 | 0.119 | 0.410 | 0.260 | 0.236 | 0.681 | 0.402 | 0.443 | 0.517 |
Minimum principal stress (MPa).
Points | Angle | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0°, 0° | 0°, 15° | 0°, 30° | 30°, 0° | 30°, 15° | 30°, 30° | 60°, 0° | 60°, 15° | 60°, 30° | ||
A | 1 | −3.737 | −2.599 | −7.800 | −3.797 | −2.279 | −18.67 | −2.908 | −1.261 | −17.22 |
2 | −3.461 | −3.389 | −6.089 | −4.080 | −3.140 | −25.53 | −2.879 | −2.875 | −17.62 | |
3 | −3.737 | −2.599 | −7.800 | −3.797 | −2.279 | −18.67 | −2.908 | −1.261 | −17.22 | |
4 | −3.461 | −3.389 | −6.089 | −4.070 | −3.140 | −25.53 | −2.879 | −2.875 | −17.62 | |
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B | 1 | −1.281 | −0.995 | −2.160 | −1.495 | −1.414 | −6.879 | −2.653 | −2.029 | −7.228 |
2 | −1.787 | −1.943 | −6.621 | −4.193 | −3.727 | −13.04 | −6.094 | −6.270 | −9.048 | |
3 | −1.890 | −1.889 | −2.879 | −2.219 | −2.091 | −6.879 | −2.705 | −3.100 | −7.228 | |
4 | −1.787 | −1.943 | −6.621 | −4.193 | −3.927 | −13.04 | −6.094 | −6.270 | −9.048 | |
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C | 1 | −0.641 | −0.640 | −1.458 | −1.036 | −0.898 | −2.052 | −1.761 | −1.269 | −4.529 |
2 | −1.248 | −1.307 | −4.695 | −3.144 | −2.601 | −6.897 | −4.464 | −3.885 | −7.460 | |
3 | −1.266 | −1.329 | −2.138 | −1.517 | −1.373 | −2.755 | −1.950 | −1.754 | −4.529 | |
4 | −1.232 | −1.307 | −4.695 | −3.144 | −2.601 | −6.897 | −4.464 | −3.885 | −7.460 |
Shear stress (MPa).
Points | Angle | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0°, 0° | 0°, 15° | 0°, 30° | 30°, 0° | 30°, 15° | 30°, 30° | 60°, 0° | 60°, 15° | 60°, 30° | ||
A | 1 | 1.105 | 0.938 | 2.273 | 1.134 | 0.770 | 5.050 | 0.938 | 0.490 | 4.782 |
2 | 1.127 | 1.103 | 2.366 | 1.307 | 1.051 | 7.600 | 1.009 | 0.988 | 5.372 | |
3 | 1.105 | 0.938 | 2.273 | 1.134 | 0.770 | 5.050 | 0.938 | 0.490 | 4.782 | |
4 | 1.127 | 1.103 | 2.221 | 1.307 | 1.051 | 7.600 | 1.009 | 0.988 | 5.372 | |
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B | 1 | 0.326 | 0.343 | 1.055 | 0.731 | 0.690 | 1.900 | 0.986 | 0.991 | 1.226 |
2 | 0.359 | 0.419 | 1.723 | 0.945 | 0.885 | 3.678 | 1.505 | 1.517 | 2.691 | |
3 | 0.318 | 0.248 | 0.939 | 0.560 | 0.516 | 1.900 | 0.856 | 0.853 | 1.345 | |
4 | 0.359 | 0.419 | 1.723 | 0.945 | 0.885 | 3.549 | 1.505 | 1.517 | 2.691 | |
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C | 1 | 0.194 | 0.218 | 0.711 | 0.506 | 0.437 | 1.005 | 0.714 | 0.620 | 1.161 |
2 | 0.217 | 0.252 | 1.218 | 0.752 | 0.580 | 1.878 | 1.096 | 0.924 | 2.179 | |
3 | 0.194 | 0.202 | 0.653 | 0.392 | 0.315 | 1.061 | 0.633 | 0.520 | 1.078 | |
4 | 0.217 | 0.252 | 1.174 | 0.752 | 0.499 | 1.729 | 1.096 | 0.918 | 2.175 |
Shear key stress nephogram at incidence of (30°, 30°). (a) Maximum principal stress of shear key A; (b) maximum principal stress of shear key B; (c) minimum principal stress of shear key A; (d) minimum principal stress of shear key B; (e) shear stress of shear key A; and (f) shear stress of shear key B.
Tables The shear key produces significant compressive stress during an earthquake while the shear stress and tensile stress are relatively low. When the seismic wave incidence transforms from vertical to oblique along the pipe cross section (that is, When the incidence direction of the seismic wave transforms from the tunnel transverse direction to the axial (that is, The horizontal shear stress distribution of compressive stress and shear stress is very uniform, the top corner has stress concentration phenomenon, and the tensile stress is mainly concentrated at the root of the shear key. The stress of the vertical shear keys B and C assumes a trapezoidal distribution, that is, the stress at the lower part of the shear key is greater than the upper stress. Tables
As shown in Figure
Nodes at the pipe joint interface.
Joint dislocation (opening) at different angles of incidence. (a) Dislocation (opening) of roof and floor at pipe joints under incident angle (0°, 0°); (b) dislocation (opening) of roof and floor at pipe joints under incident angle (0°, 15°); (c) dislocation (opening) of roof and floor at pipe joints under incident angle (0°, 30°); (d) dislocation (opening) of roof and floor at pipe joints under incident angle (30°, 0°); (e) dislocation (opening) of roof and floor at pipe joints under incident angle (30°, 15°); (f) dislocation (opening) of roof and floor at pipe joints under incident angle (30°, 30°); (g) dislocation (opening) of roof and floor at pipe joints under incident angle (60°, 0°); (h) dislocation (opening) of roof and floor at pipe joints under incident angle (60°, 15°); and (i) dislocation (opening) of roof and floor at pipe joints under incident angle (60°, 30°).
The vertical dislocation value can be obtained without indicating the direction of dislocation, as shown in Figure
The dislocation volume at the joint is significantly affected by the seismic wave incidence angle. The horizontal and vertical dislocations of the roof and floor of the immersed tunnel joints increase as the local seismic wave incidence changes from vertical to oblique along the pipe cross section (
When the incidence direction of the seismic wave transforms from the tunnel transverse direction to the axial (
The horizontal dislocations at joints are similar at a certain angle of incidence. The end nodes are larger than the intermediate nodes in the vertical dislocation, and the direction of dislocation is opposite; in other words, the immersed tunnel shows torsional deformation. When the incident angle is (30°, 30°), the D-value of the vertical dislocation between the end node and the intermediate node of the floor joint is maximal (0.014°), which is in the scope of security.
The nodes from the immersed tunnel’s central axis including three pipes were used to draw horizontal and vertical dislocation curves at the maximum stress (Figure
Horizontal and vertical displacements of immersed tunnel at different incident angles. (a) Horizontal and vertical displacement of immersed tunnel under (0°, 0°); (b) horizontal and vertical displacement of immersed tunnel under (0°, 15°); (c) horizontal and vertical displacement of immersed tunnel under (0°, 30°); (d) horizontal and vertical displacement of immersed tunnel under (30°, 0°); (e) horizontal and vertical displacement of immersed tunnel under (30°, 15°); (f) horizontal and vertical displacement of immersed tunnel under (30°, 30°); (g) horizontal and vertical displacement of immersed tunnel under (60°, 0°); (h) horizontal and vertical displacement of immersed tunnel under (60°, 15°); (i) horizontal and vertical displacement of immersed tunnel under (60°, 30°).
Figure
Furthermore, when the seismic wave is transversely incident (
The distribution of tensile damage factors of the cross section of the immersed tunnel joint under incident angle (30°, 30°) is shown in Figure
Crack distribution of joint cross section under incident angle (30°, 30°). (a)
As shown in Figure
The 3D seismic response of an underwater immersed tunnel under obliquely incident seismic waves was investigated in this study based on simulations of the viscous-spring artificial boundary and seismic load. The influence of seismic SV wave incident angle on the 3D immersed tunnel joint and overall deformation of the tunnel is discussed by modelling the Tianjin Haihe River tunnel structure. Our main conclusions can be summarized as follows. The oblique incidence of seismic SV waves, unlike the vertical incidence, has a marked effect on the seismic response of the immersed tunnel. Among nine seismic wave combinations, the stress of the shear key is the largest under 30°, 30°; this is the worst-case angle to the site discussed in this paper. The tensile stress of the horizontal shear key is mainly concentrated where the shear key is connected to the main body of the immersed tube. The stress in the lower part of the vertical shear key is larger than that in the upper part, which makes it an important design point. The horizontal and vertical dislocation of the roof and the floor of the immersed tunnel joints increase as the seismic wave incidence angle increases; they are maximal at an incident angle of 30°, 30°. The maximum relative dislocation of tunnel joints observed in this study was about 10 mm, while the average residual compression of GINA water stops is 15 mm–25 mm. In other words, the dislocation is within a safe range. The vertical dislocation of the end section is greater than that of the intermediate node and the dislocation directions are opposite, which suggests that the deformation of the immersed tunnel is torsional. The overall horizontal and vertical displacement of the immersed tunnel increases as the seismic wave incidence angle increases. The displacement difference between the two ends of the same tunnel segment also increases, which can cause axial tensile deformation of the tunnel joints. Under the oblique incidence of SV waves, cracks appear in the ends of the side walls (the corners of the tunnel), the midspan of the roof and the floor, and the part of the horizontal shear key connected to the floor. In an actual engineering scenario, construction measures and anticracking treatment should be targeted to these key areas.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research described in this paper was financially supported by the National Natural Science Foundation of China under grant numbers 51678389 and 51678390.