Effect of soft layer on seismic response of subway station in layered stratum

2.1. Calculation method of internal force and safety factor of tunnel lining

Generally, the three-dimensional solid zones are usually chosen to simulate tunnel lining during numerical simulation, for the reason that solid zones can better simulate the relative interaction between tunnel lining and surrounding soil than structural elements. But it has the disadvantage that the internal force of secondary lining cannot be directly obtained. The internal force especially the axial force and bending moment of secondary lining are important for evaluating stability of secondary lining, and are the main indicators to calculate reinforcement. A fish command is written to conveniently calculate the axial force and bending moment of secondary lining through the obtained stress of solid zones. The obtained internal force can be then used to calculate safety factor of secondary lining. Based on the finite element theory, the calculated stress of solid zones reflects the stress state of the center points in solid zones. In other words, the stress in the center point of zones is accurate, and the stress in the boundary of solid zones is extrapolated through the stress in the center point. In the following parts, the computational method of stress in the boundary of solid zones, and the determination of safety factor is described in detail.

In order to calculate the internal force of secondary lining, the secondary lining is divided into two layers, with the thickness ranges from 20 to 40 cm, which is dependent on the thickness of secondary lining. In numerical model, the secondary lining is divided into two layers to calculate the internal force of secondary lining. Note that the secondary lining had better to be meshed using hexahedral element to improve the calculation accuracy. The stress and coordinates of the adjacent solid zones in secondary lining is shown in Fig. 1. The $x$-$y$ represents the global coordinate system, while the $n$-$s$ represents the local coordinate system. The angle between the $s$ direction and $x$ direction is defined as $\alpha$. It can be calculated according to the coordinates of solid zones through Eq. (1). Then the normal stress in the n direction of the discussed zone can be calculated by using Eq. (2):

where, ($x_1$,$\mathrm{}{y}_1$), ($x_2$,$\mathrm{}{y}_2$) are the coordinate of center point:

where, ${\sigma }_x$, ${\sigma }_y$,${\sigma }_{xy}$ is the stress components of the discussed zones.

Fig. 1Stress in the global and local system

The normal stress in the boundary of discussed zone is defined as ${\sigma }_{n1}$ and ${\sigma }_{n2}$, respectively (as shown in Fig. 1). Based on the theory of structural mechanics, the normal stress changes linearly along the boundary of solid zones. Thus, the stress in the outer boundary of these two discussed zones can be extrapolated by using Eq. (3) and Eq. (4), respectively:

where, ${\sigma }_1$ is the stress in the outer boundary of zone 1, ${\sigma }_2$ is the stress in the outer boundary of zone 2, ${\sigma }_{s1}$,${\sigma }_{s2}$ is the stress of zone 1 and 2 in the center point, $\epsilon$ represents the layers of solid zones. It is 0.5 when the secondary lining is divided into two layers.

Based on above calculated results, the bending moment and axial force can be calculated by using Eq. (5) and Eq. (6), respectively:

where, $M$ and $N$ are the bending moment and the axial force of the discussed section, respectively. $b$ and $h$ are the width and thickness of the discussed section, respectively.

According to the calculated internal force of tunnel lining, the safety factor of tunnel lining can be calculated to assess the safety of tunnel lining. According to Code for Design of Road tunnel [17], the safety factor of secondary lining is defined as the ratio of ultimate compressive strength to the actual internal force. The ultimate compressive strength is dependent on the failure mode of secondary lining. And the failure mode can be determined according to the eccentric distance e, which can be calculated through Eq. (7):

If $e$ is less than 0.2$h$, then the ultimate compressive strength of tunnel lining can be calculated by using Eq. (8):

where, $N_u$ is the ultimate compressive strength, $R_a$ is the compressive strength of concrete, $\phi$ is the longitudinal coefficient of tunnel lining, it can be taken as 1.0 for tunnel, $\alpha$ is the influence coefficient of axial force, and can be calculated through Eq. (9) which is an empirical formula:

When the eccentric distance e is larger than 0.2$h$, tunnel lining may be tensioned and damaged, and the ultimate compressive strength can be calculated through using Eq. (10):

where, $R_1$ is the ultimate tensile strength of concrete.

2.2. Project description

Fengtianlu subway station of metro line 1 in Chongqing is selected to study the effect of soft layer on seismic behavior of subway station. Based on geological surveying report, a soft layer is located in the lower part of subway station, with the thickness of approximate 2 m. The altitude of it is approximately level, and the relative distance between subway station and it is about 6 m. The designed width and height of Fengtainlu subway station are 22 m and 24 m respectively. The thickness of secondary lining is 0.6 m. An island platform is designed and the thickness of the plate is 0.5 m. The concrete girder is transversely constructed, with the size of 1.0 m×1.2 m. The secondary beam is designed along the longitudinal direction of subway station, with the size of 0.8 m×0.5 m. Column is designed to bear load transferred from concrete girder. The size of central columns is 0.8 m×0.8 m, with the center to center spacing of 8.4 m.

2.3. Numerical modeling

Considering the length of subway station is much larger than the size of cross section. The study is simplified as a plain strain problem. The established numerical model is shown in Fig. 2. To reduce the effect of boundary on calculation results, the lateral distance between horizontal boundary and subway station is selected as 64 m (about 3 times of the diameter of subway station). The length and height of subway station are 150 m and 70 m, respectively. Note that, the blue zones represent soft layer, while the red zones represent normal strata. Fig. 2(b) described the mesh of secondary lining and internal structures in subway station. The secondary lining is divided into two layers to facilitate the calculation of internal force. Surrounding rock masses and secondary lining are both simulated using solid elements, while internal structures in subway station including beams and columns are simulated using structural beam element. The structural elements are rigidly connected with solid elements. Totally, the established model is composed of 4385 zones and 65 structural elements. The central column in the island platform is simplified to a longitudinal wall in numerical model. Thus the elastic modulus of central columns should be reduced based on the equivalent stiffness principle. The local damp is selected to simulate the damping of rock masses. Based on site geological conditions and local engineering experiences, the damp of solid zones and structural zones are set as 0.05 and 0.02, respectively. The free field boundary combined with displacement boundary is applied around numerical model. The earthquake excitation is applied in the bottom of numerical model. In numerical simulation, the normal stratum and soft layer bot obey the Mohr Coulomb criterion. The structures in subway station including plate, beams and columns are simulated conform to linear elastic constitutive model. According to the geological survey report, the physical and mechanical parameters of materials in numerical simulation are listed in Table 1, which are chosen according to geological survey report.

Fig. 2Numerical model

a) Numerical model of model A

b) Detail size of Fengtianlu subway station

2.4. Earthquake excitation

The ground acceleration of Kobe earthquake happened in 1995 is completely recorded, and it is widely used in seismic study. The duration of this seismic record is approximate 50 s. This paper focuses on studying the peak seismic response of subway station, thus only 8.5 s of the record containing the peak velocity was selected as input excitation. For briefly, the initially 8.5 s of seismic record is adopted (as shown in Fig. 3). Generally, seismic wave contains a wide range of spectrum, but most of the energy is concentrated in the part with a frequency of less than10 Hz. And the high-frequency seismic wave does not play an important role on geotechnical engineering [18]. A seismic wave is usually a series of discrete acceleration points, which should be filtered and rectified before use in numerical simulations [19]. A higher frequency requires a smaller mesh dimension, a larger grid number, and a longer calculating time under the same calculation accuracy condition. It is suggested that the high-frequency of input history can be removed without significantly affecting the results [20]. In this paper, the filtering procedure is accomplished with a low-pass filter routine in SeismoSignal (http://www.seismosoft.com). The selected acceleration record was filtered with a 20 Hz low-pass filter. The input earthquake excitation and its corresponding Fourier amplitude after filtering are plotted in Fig. 3. The input acceleration is scaled down to a magnitude of 0.15 g according to local engineering experiences.

Fig. 3The selected earthquake excitation

a) The acceleration versus time data of input earthquake

b) The Fourier frequency of input earthquake

3.1. The effect of soil layer on displacement of subway station

Fig. 4 shows the time-history curve of relative lateral displacement between tunnel crown a bottom. It can be seen that the curve of the case with soft layer is always within the range of the case without soft layer. The relative displacement when soft layer is included is much less than the case without soft layer. In detail, the maximum relative displacement without considering soft layer is approximate 29.86 mm, at the moment of 1.89 s, while the maximum relative displacement of the case considering soft layer is 14.66 mm, at the moment of 5.15 s. The decrease rate of maximum relative displacement in the case considering soft layer is about 50.9 % when compared with the case without considering soft layer. The existence of soft layer is beneficial for decreasing relative displacement of subway station.

3.2. The effect of soft layer on acceleration of secondary lining

In order to investigate the effect of soft layer on acceleration of secondary lining, five key points in secondary lining (as shown in Fig. 2(b)) are selected to record the variation of acceleration in secondary lining during earthquake. Fig. 5 shows the time-history curve of acceleration except point b, for the record command of point b is missing in numerical simulation. It can be seen that the curve of acceleration with dynamic time when considering soft layer is within the range of the case without conspiring soft layer. The acceleration with soft layer is much less than that without soft layer. The corresponding moment of the maximum acceleration without considering soft layer is earlier than the case considering soft layer. In detail, the maximum acceleration without considering soft layer appears at the moment of 2.34 s, which is 2.81 s earlier than the case considering soft layer. The maximum acceleration of these four monitoring points is extracted to further investigate the influence degree of soft layer on acceleration of secondary lining. Table 2 lists the maximum acceleration and calculation errors of these four monitoring points. Note that the calculation error is defined as the change rate of numerical results considering soft layer with the case without soft layer. It can be seen that acceleration when the soft layer is much less than the case considering soft layer, which means the presence of soft layer is beneficial for reducing acceleration in secondary lining. The decrease rate from point a to point e are –49.05 %, –49.70 %, –35.56 %, and –39.79 %, respectively. The maximum decrease rate is in tunnel crown, and the minimum value occurs in arch springing.

Fig. 4Time-history curve of the relative lateral displacement between tunnel crown and bottom

3.3. The effect of soft layer on safety factor of secondary lining

Safety factor plays a key role in evaluating stability of secondary lining. Safety factor of five key points in secondary lining is calculated through the calculation method discussed in Section 2.1. It should be noted that since the safety factor is not an intrinsic indicator in Flac3D, it can’t be directly and continuously recorded during numerical simulation. To derive the variation of safety factor during simulation, 425 calculation steps were divided, with the calculation time of each steps 0.02 s. Fig. 6 shows the time-history curve of safety factor. It can be seen the curve of the case considering soft layer is within the range of the case without softy layer. The safety factor of secondary lining fluctuates around a certain value except for point a and e. For point a and e, the safety factor decreases rapidly and then tends to keep stable. Besides, the appearance moment of rapid decrease in point a and e without considering soft layer is earlier than the case with soft layer. Note that the dashed line in Fig. 6 represents the minimum safety factor during earthquake. To further investigate the influence degree of soft layer on safety factor of secondary lining, the minimum safety factor of five monitoring points is extracted and listed in Table 3. It can be seen the distribution of safety factor in two cases are similar. The maximum safety factor occurs in point a (tunnel crown), and the minimum value appears in Point d (tunnel arch feet). To ensure the stability of subway station, the thickness and strength of secondary lining in tunnel arch feet should be improved. The safety factor of secondary lining in the case that soft layer is included shows great increase compared with the case without soft layer. The increase rate from Points a to e are 72.57 %, 43.25 %, 13.42 %, 50.72 %, and 54.94 %, respectively. The presence of soft layer can greatly enhance the stability of subway station.

Fig. 5Time-history curve of acceleration in X direction in secondary lining

Fig. 7 shows the distribution of safety factor in secondary lining at the end of earthquake. It can be seen that the distribution of safety factor is not symmetric along the central axes of tunnel, for the reason that the stress condition of tunnel lining in two sides are opposite. Generally, the moving trend of tunnel lining is always identical with the direction of input earthquake. When the direction of input earthquake is along the positive half of the $X$ axis, tunnel lining in the right part will be extruded by surrounding soil, but that in the left part will detach from surrounding soil, which will introduce the differences in stress condition. It can be seen in Fig. 7 that secondary lining in tunnel crown is the safest area, and tunnel arch feet is the most unfavorable part. Safety factor of the case with soft layer is much greater than the case without soft layer in any part of tunnel lining.

Fig. 6Time-history curve of safety factor in secondary lining

Fig. 7Safety factor of secondary lining at the end of earthquake

a) Without soft layer

b) With soft layer

3.4. The effect of soil layer on internal force of internal structures

The Kobe earthquake happened in Japan in 1995 caused serious seismic damage in subway station. The most attracting phenomenon is that the columns in subway station suffered serious shear failure. In addition, many existing studies have pointed that the bottom of columns in subway station is vulnerable to damages in earthquake. To make clear the effect of soft layer on internal force of columns in subway station, the internal force, including axial force, shear force and bending moment in the bottom of columns are highlighted. Fig. 8 shows the time-history curve of internal force. Note that positive and negative values of axial force refer to tension and compression, respectively. If the shear force tends to make the column rotate clockwise, it is regarded as positive. Otherwise, it is negative. If the bending moment tends to put the column in the right part into tension, it is regarded as positive. Otherwise, it is negative. Generally, the value of internal force when the soft layer is included are less than the case when the soft layer is not included except for axial force. Table 4 lists the maximum value of internal force in central column. The variation rates are 7.19 %, –48.24 %, and –56.33 %, respectively. The axial force is less influenced by soft layer compared with shear force and bending moment. The decrease of internal force in central column will greatly increase the stability of subway station.

Fig. 8Time-history curve of internal force in central column

4.1. Effect of soft layer location on relative displacement

Fig. 10 shows the time-history curve of relative displacement. It can be seen that the maximum relative displacement in model B and C are 70.17 mm and 30 mm, respectively. Through comparing numerical results with the results presented in Section 3.1, it can be seen that the maximum relative displacement in model B has greater increase than model C compared with the case without considering soft layer. The amplification factor of relative displacement is defined as the ratio of relative displacement with soft layer to that without soft layer. Thus, the amplification factor of model B and C are 2.81 and 1.01, respectively. The effect of soft layer on relative displacement almost can be neglected when it is above subway station. The most favorable and unfavorable case in reducing relative displacement are model A and B, respectively.

4.2. Effect of soft layer location on acceleration of secondary lining

The variation of maximum acceleration in secondary lining due to the change of soft layer location is shown in Table 5. The calculation results of three cases with soft layer are compared with the case without soft layer. Note that calculation error is defined as the change rate of results considering soft layer to the case without considering soft layer. It can be seen from Table 5 that although the presence of soft layer is generally beneficial for reducing acceleration, special attention should be paid to model B. Since the acceleration in point b has a significant increase, with the value and increase rate of 20.45 and 327.82 %, respectively. The induced great acceleration by soft layer in model B may cause failure of secondary lining. Besides, the location of point c is near to the location of soft layer. It can be concluded that when the soft layer is in the lower part or upper part of subway station, the presence of soft layer is effective in reducing acceleration. While the soft layer is within the range of subway station, acceleration of the range that is close to soft layer will have an increase, but the area that is far away from it will be reduced. The effect of soft layer in model A and C are similar, since they are both favorable for reducing acceleration. The soft location in model A is more preferable than that in model C.

Fig. 10Time-history curve of relative displacement in four cases

4.3. Effect of soft layer location on safety factor of secondary lining

Table 6 compares the safety factor of secondary lining in five key points of four cases. As can be seen from Table 6, the contribution of soft layer location to safety factor is opposite to that of acceleration, which reveals that the safety factor has a close relationship with acceleration. In detail, safety factor will greatly reduce while the acceleration increases significantly. The distribution of safety factor in four cases is almost identical. The most favorable and unfavorable point is point a and point d in each case. The effect of soft layer in model A and C are consistent. The presence of soft layer in these two cases will greatly increase the safety factor, with the maximum increase rate 72.57 % and 56.19 %, respectively. Soft layer in model A and C are helpful to improve the stability of subway station. The soft layer location in model A is more preferable on improving safety factor than that in model C. The calculated safety factor in model B is much less than the case without soft layer except point e, with the maximum decrease rate –22.37 %. This phenomenon reveals that the location of soft layer in model B is harmful to the stability of subway station.

4.4. Effect of soft layer location on internal force of central column

In order to study the location of soft layer on internal force of central column, the maximum induced internal force of central column of four cases during earthquake are listed in Table 7. As can be seen from Table 7, the calculated values in model A and C both decrease significantly compared with the case without soft layer. However, soft layer in model B caused an increase of shear force and bending moment except axial force. From the observations of past earthquakes, it seems that the bottom of columns in subway station is more vulnerable to shear failure. Shear force in model A and C has a significantly decrease, and the stability of columns will be greatly improved. While in model B, the shear force and bending moment both increased, which may lead to unstable of columns during earthquake. In addition, the decrease rate of bending moment is greater than that of axial force, which will lead to the decrease of eccentric distance and further improve the stress state of columns. However, the axial force decreased while the bending moment increased in model B, while will lead to a significant increase of eccentric distance and further decrease the bearing capacity of columns. Hence, the soft layer location in model A and C is beneficial for improving stress state of columns. Nevertheless, the soft layer location in model B is harmful for the stability of columns.