Effects of central column aspect ratio on seismic performances of subway station structures

Abstract

The subway station structures have a contradiction in its design of the cross section of the central columns: larger aspect ratio increases the longitudinal traffic capacity, yet decreases the lateral seismic capacity. However, both capacities are important. Thus, finding a reasonable aspect ratio has become a significant problem to be studied in depth. In this article, a series of time history analyses are carried out on a typical two-story three-span subway station structure, and ultimate bearing capacity analysis of isolated columns of various sizes is performed. The static axial compression ratio (ranging from 0.13 to 0.68) and the aspect ratio (ranging from 0.04 to 26.67) are taken as parameters, and their influence on internal force and damage to the structure are explored. It is found that the increase in axial compression ratio of the columns increases both the compression and tension damage and reduces the seismic performance of the station structure. However, the influence of increasing the aspect ratio of columns on the seismic performance of structures is more complicated. The compression and tension damage rises first (ranging from 0.04 to 1.67) and then decreases (greater than 6.67), while the tension damage of the structure increases monotonously. Finally, (0.15, 0.6) is given as the recommended aspect ratio range.

Keywords

aspect ratio axial compression ratio central column seismic performance subway station structure

Introduction

As a convenient and efficient transportation, underground subway is principally used all over the world (Chen et al., 2016), and the construction of subway has gained a rapid development in China (Ying, 2011). Because of the serious damage to the Daikai station caused by the 1995 Hanshin earthquake in Japan (Nakamura et al., 1996), there is increasing research into the seismic performance of subway stations (Chen et al., 2014b, 2016; Khani and Homami 2014; Yamato et al., 1996). Iida et al. (1996), An et al. (1997), and Huo et al. (2005) studied the damage to the Daikai station caused by the earthquake and concluded that it was caused by the failure of central columns that led to the collapse of the top plate. Other studies have also concluded that the central columns were the weakest components of the subway station structure (Chen et al., 2013; Zhao et al., 2009).

The cross-sectional design of the central columns in subway station structures generally includes the following steps. First, the cross-sectional area (AC) is determined according to the axial compression ratio parameters, which is defined as follows

α_{F} = \frac{N}{AC \times fC}

(1)

where $α_{F}$ is the axial compression ratio, N refers to the axial force, AC is the cross-sectional area of column, and fC is the axial compressive strength of concrete. Then, the aspect ratio (H/B, ratio of column dimension along the longitudinal direction, H, to column dimension along the transverse direction of station, B) of the cross section is selected. After the axial compression ratio and H/B have been determined, the shear span ratio, reinforcement ratio, stirrup ratio, and details of the seismic design are calculated. The selection of an appropriate H/B should be considered from two perspectives: the lateral seismic capacity and the longitudinal traffic capacity of the subway station structures. Because the longitudinal size of subway station structure is far greater than the dimension of the cross section, the longitudinal stiffness of the subway station is typically much larger than the lateral stiffness. To increase the lateral stiffness of the station structure, H/B should be as small as possible. However, because of the increased requirement for the longitudinal traffic capacity of the station, B needs to be sufficiently small, which means that H/B needs to be sufficiently large, at least meeting the requirements of the limitation of platform width and clearance. Hence, there exists a contradiction in the design of the central columns with regard to optimal H/B.

At present, the cross-sectional design of the central columns of station structures is mainly based on the Code for Seismic Design of Buildings (GB50011, 2010) and the Code for Design of Metro (GB50157, 2013). The Code for Seismic Design of Buildings (GB50011, 2010) regulates the requirements of axial compression ratio and H/B for ground reinforced concrete (RC) frame structures. In the design of central columns of subway station structures, H/B should be more than 1/3 but less than 3. The Code for Design of Metro (GB50157, 2013) considers the higher safety requirements of the central columns of subway station structures but only provides a rough recommendation for the central columns to be replaced by central walls. For the surface RC structure, it has become the consensus that the H/B of the column needs to be close to 1 in order to meet the stiffness requirements in each direction. Compared with typical surface RC frame structures, subway station structures are very different in form. Subway stations are far longer than the width of the cross section (Hashash et al., 2001). Therefore, most subway station structures only have beams along the longitudinal direction; there are no beams along the lateral direction. Given the notable differences between RC frame structures on the ground and subway station structures, it is doubtful that the design of the central columns of subway station structures has the same requirements as surface RC frames.

Research into the parameters of RC columns is abundant (Atalay and Penzien, 1975; Iwasaki et al., 1985; Lam et al., 2003; Sheikh and Khoury, 1993; Wong et al., 1993; Zhang et al., 2005). Studies are mainly aimed at columns in surface RC structures and there is limited research into aspect ratio. The H/B parameters have a significant influence on the bearing and deformation capacity of the column, especially for subway station structures, because their design should consider both the seismic capacity of the station and also the longitudinal traffic capacity.

In this article, a series of time history analyses are performed of a typical two-story three-span subway station structure, using the general purposed finite element code ABAQUS/CAE (2009). Ultimate bearing capacity analysis of isolated columns which have various sizes is also performed. The static axial compression ratio (which means that the static axial force is used to calculate the axial compression ratio in equation (1)) and H/B are taken as parameters.

Project background

The two-story three-span subway station structure chosen for our model is located in Shanghai, China. The ground elevation is 3.70–4.70 m. Landforms are coastal plain type and strata are quaternary sedimentary. The underground water depth of the site is 0.50–2.60 m. According to the Code for Seismic Design of Buildings (GB50011, 2010), the site soil belongs to a weak soil type and a building site of Grade IV. The designed peak ground acceleration (PGA) is 0.1 g. The station is an island-style station, which has a total of three entrances and three shafts. The upper layer is the station hall, while the lower layer is the island platform. The station is constructed by the cut and cover method, and a diaphragm wall is used for retaining structure. The maximum length of the station is 470 m. Figure 1(a) shows the standard cross section of the station structure. The excavation depth of the center position of the standard cross section is 15.27 m, and the width and height are 20.9 and 12.37 m, respectively. The roof depth of the structure is 2.9 m. The station structure is provided with a lining wall within the diaphragm wall, and the lining wall and diaphragm wall are connected by embedded parts which ensure that the diaphragm wall bears the load in conjunction with the lining wall. Central columns of both floors are of the same size: 1.0-m long and 0.6-m wide. The spacing between central columns along the longitudinal direction of the station structure is 8 m, while the spacing between central columns along the cross-sectional direction of the station structure is 5.3 m.

Figure 1.

Standard cross section of subway station structure and reinforcement details of central column cross section (unit: mm): (a) cross section of the station and (b) reinforcement details of central column cross section.

Setup of the numerical model of the station

Numerical model

A soil–structural dynamic interaction analysis model was established in the finite element code ABAQUS. The size of the structure is shown by the standard cross section in Figure 1(a). According to the Code for Seismic Design of Buildings (GB50011, 2010), the width of each side of soil around the structure should be at least three times larger than the width of the structure. In the model, the width of each side of soil was 250 m.

An infinite element boundary was used for the soil lateral boundary. The infinite elements provided by ABAQUS are based on the static analysis of Zienkiewicz et al. (1991) and the dynamic response analysis of Lysmer and Kuhlemeyer (1969). On the traditional artificial truncation boundary, the wave will be reflected on the boundary surface, which will return the energy to the analysis grid. In fact, the waves travel to infinity, and we do not care too much about the propagation of waves in the far field, but it is necessary to ensure that it has little impact on the analysis area. The infinite element in ABAQUS does this well by setting damping on the boundary, which means that it can simulate no reflection. Thus, it can be used as the lateral boundary of the model.

Because of the lack of experimental data concerning deep drilling soil samples, the soil was modeled down to a 60.55 m depth from ground surface. The depth of soil from the bottom plate was 45.28 m, which was more than three times the height of the station. The X and Y direction degrees of freedom of the soil element nodes at the bottom boundary were imposed before the ground motion was input. The established analytical model is shown in Figure 2.

Figure 2.

Numerical analysis model.

Constitutive models and properties of materials

Mohr–Coulomb elastic–plastic constitutive model which adopts kinematic hardening rule was applied for soil mass, and the specific properties are shown in Table 1. According to the engineering geological exploration, the soil mass in the depth range was roughly divided into 10 layers. The effective shear strength indexes (effective internal friction angle, $φ'_{CU}$ , and effective cohesion,) which were according to Mohr–Coulomb strength theory and measured by consolidated undrained (CU) tri-axial tests were selected as the model properties $C'_{CU}$ . The groundwater level was located 0.5m below the ground. Submerged unit weight is used for the unit weight of soil mass below the groundwater level.

Table 1.

Soil parameters of station site.

Soil texture	Depth (m)	Unit weight (kN/m³)	Elastic modulus (MPa)	Poisson’s ratio υ	Internal friction angle, $φ'_{CU}$ (^o)	Cohesion (kPa)
Artificial fill	0–0.5	19			15	20
Artificial fill	0.5–1.32	9	20.34	0.32	15	20
Yellowish dark brown silty clay	1.32–2.41	9.2	20.34	0.32	31.3	9.5
Yellow gray silty clay	2.41–3.3	8	14.00	0.34	33.8	15.1
Gray mucky silty clay	3.3–6.92	7.4	10.85	0.38	28.3	5.3
Gray mucky clay	6.92–14.77	6.7	7.39	0.4	24.9	7.2
Gray clay	14.77–16.73	7.4	11.55	0.35	29.7	10
Dark green silty clay	16.73–21.13	9.5	24.85	0.29	29.1	31.3
Olive drab-gray sandy silt	21.13–28.01	8.2	32.20	0.29	31.1	2
Gray clay	28.01–43.02	7.7	15.09	0.33	32.5	8.1
Gray silty clay and silty sand interbedded soil	43.02–60.55	8.4	28.70	0.32	28.1	8

A beam element (B21) was used to simulate the station structure. As the central columns are the main structural components that are studied in this article, the computational elements of the central columns were relatively dense to other components; each central column was divided into 90 elements.

To better simulate the dynamic response of the structure in the elastic–plastic stage, the concrete damaged plasticity model was adopted. According to the model (established on the basis of the theories of Lubliner et al. (1989) and Lee and Fenves (1998)), a damage factor D is used to describe the degree of damage and stiffness degradation of the concrete

\bar{σ} = (1 - D) E_{0} : (ε - ε^{pl})

(2)

where $\bar{σ}$ is the effective stress, $E_{0}$ is the initial elastic stiffness, $ε$ is the strain tensor, and $ε^{pl}$ is the plastic strain tensor.

D is divided into two types, D_t and D_c, which are used to describe the tension and compression damage of concrete materials, respectively

D = 1 - (1 - D_{t}) (1 - D_{c}) \bar{σ}

(3)

Their value range lies between 0 and 1. The larger the value of the structure, the more serious the damage is; when the value is 0, the structure is in the elastic stage.

Based on this theory, the degradation of the elastic stiffness is characterized by two damage variables, dt and dc (ABAQUS/CAE, 2009). dt and dc are both a function of plastic strain, temperature, and field variables.

To better represent the two different behaviors in the tensile region and compressive region, a yield function (Lubliner et al., 1989) is given as equations (4) to (7)

F = \frac{1}{1 - α} (\bar{q} - 3 α \bar{p} + β ({\tilde{ε}}^{pl}) 〈 {\hat{\bar{σ}}}_{\max} 〉) - {\bar{σ}}_{c} ({\tilde{ε}}_{c}^{pl}) = 0

(4)

α = \frac{(σ_{b 0} / σ_{c 0}) - 1}{2 (σ_{b 0} / σ_{c 0}) - 1}; 0 \leq α \leq 0.5

(5)

β = \frac{{\bar{σ}}_{c} ({\tilde{ε}}_{c}^{pl})}{{\bar{σ}}_{t} ({\tilde{ε}}_{c}^{pl})} (1 - α) - (1 + α)

(6)

γ = \frac{3 (1 - K_{c})}{2 K_{c} - 1}

(7)

where ${\tilde{ε}}_{t}^{pl}, {\tilde{ε}}_{c}^{pl}$ are the equivalent plastic strain, $K_{c}$ is the ratio of the second stress invariant on the tensile meridian, ${\hat{\bar{σ}}}_{\max}$ is the maximum principal effective stress, $σ_{b 0} / σ_{c 0}$ is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, ${\bar{σ}}_{t} ({\tilde{ε}}_{t}^{pl})$ is the effective tensile cohesion stress, and ${\bar{σ}}_{c} ({\tilde{ε}}_{c}^{pl})$ is the effective compressive cohesion stress. Calculation of damage parameters has been performed in the previous studies (Chen et al., 2016).

The central columns of the structure were concrete of grade C45 (GB50010, 2010). Its elastic modulus, Poisson’s ratio, tensile, and compression strength were 33.5 GPa, 0.2, 2.51 MPa, and 29.6 MPa, respectively. Other parts of the structure were concrete of C35 grade, whose parameters were 31.5 GPa, 0.2, 2.20 MPa, and 23.4 MPa, respectively.

Idealized elastic–plastic model (bilinear isotropic model) which adopts kinematic hardening rule was selected for rebar. Reinforcement ratios of central column, sidewall, top plate, middle plate, and bottom plate were 2.7%, 1.9%, 1.1%, 1.2%, and 1.4%, respectively. Particularly, the reinforcement details of the central column in the station are illustrated in Figure 1(b). The column was reinforced with 26 longitudinal bars of 28 mm diameter. Stirrups with 90° hooks of transverse reinforcement were used in the central columns of the standard cross section, and transverse reinforcement consisted of 14-mm bars spaced at 150 mm. Rebar used in structures was HRB400, of which the elastic modulus and yield strength were 200 GPa and 400 MPa, respectively.

The soil–structure interaction was defined by the Coulomb friction law. The coefficient of friction was assumed to be 0.4, which is equivalent to a friction angle of 22° (Chen et al., 2014a; Huo et al., 2005).

Input ground motion

The input ground motion at the bottom of the numerical analysis model was based upon the El Centro earthquake. The time history data of the wave was from the Pacific Earthquake Engineering Research Center (2000) in the United States. The N-S and vertical acceleration time histories are shown in Figure 3.

Figure 3.

Acceleration time histories of El Centro: (a) south–north component and (b) vertical component.

It should be noted that the peak values of the input ground motion in the horizontal direction at the bottom of the numerical analysis model were adjusted to two levels of 0.033 and 0.073 g. Then, the PGA responses in the horizontal direction were about 0.1 and 0.22 g, respectively, which were in accordance with the intensity of a moderate earthquake (seismic intensity with a 10% exceedance probability in 50 years) and rare earthquake (seismic intensity with a 2% exceedance probability in 50 years). Peak accelerations of input ground motion in the vertical direction were set as 65% of the horizontal ones.

Design of the calculation conditions and choice of components and analysis of cross sections

The length (L) and width (W) of the column in this typical metro station is 1.0 and 0.6 m, respectively. The cross-sectional area (A_C) of the central columns is 0.6 m² and H/B is 1.67. To study the influences of the axial compression ratio and H/B of the central columns on the seismic performance of columns and structures, a single variable method is adopted. This means that $α_{FS}$ and H/B of the central columns are kept constant, respectively, and the influences of the other variables are studied. The specific calculation conditions are shown in Tables 2 and 3. $α_{FS}$ of 1-1 cross section for all calculation conditions are within the range of (0.13, 0.68), while H/B is within the range of (0.04, 26.67). When the cross-sectional size of the central columns is changed, the reinforcement ratio of the central columns in all calculations remained unchanged, and longitudinal rebar is uniformly distributed along the cross section.

Table 2.

Calculation conditions of central columns with different $α_{FS}$ .

Calculation conditions	Size of central columns		A_C (m²)	$α_{FS}$ (1-1 cross section)	H/B	M/Vh₀ (1-1 cross section)
Calculation conditions	Length (m)	Width (m)	A_C (m²)	$α_{FS}$ (1-1 cross section)	H/B	M/Vh₀ (1-1 cross section)
1	1.25	0.75	0.94	0.13	1.67	3.84
2	1.1	0.66	0.73	0.16	1.67	4.09
3	1	0.6	0.60	0.20	1.67	4.36
4	0.9	0.54	0.49	0.24	1.67	4.71
5	0.75	0.45	0.34	0.34	1.67	5.54
6	0.5	0.3	0.15	0.68	1.67	8.70

Table 3.

Calculation conditions of central columns with different H/B.

Calculation conditions	Size of central columns		A_C (m²)	$α_{FS}$ (1-1 cross section)	H/B	M/Vh₀ (1-1 cross section)
Calculation conditions	Length (m)	Width (m)	A_C (m²)	$α_{FS}$ (1-1 cross section)	H/B	M/Vh₀ (1-1 cross section)
7	4	0.15	0.6	0.20	26.67	23.94
8	3	0.2	0.6	0.20	15.00	15.10
9	2	0.3	0.6	0.20	6.67	9.09
3	1	0.6	0.6	0.20	1.67	4.36
10	0.6	1	0.6	0.20	0.6	2.82
11	0.3	2	0.6	0.20	0.15	1.62
12	0.2	3	0.6	0.20	0.07	1.17
13	0.15	4	0.6	0.20	0.04	0.92

According to the symmetry of the structure, 1-1 cross section in central columns is chosen because it has the maximum axial compression ratio, and 2-2 cross section is chosen because it has the maximum shear force and sidewall bending moment when analyzing internal forces; No. 1 and No. 2 central column drifts and left sidewall drift to represent the structure drift are chosen when analyzing the relative displacement, as shown in Figure 4.

Figure 4.

Analysis components and cross sections.

Numerical results and discussion

Story drifts

Figure 5 shows the variation curves of No. 1 and No. 2 central column drifts and the structural drift when $α_{FS}$ and H/B of central columns are changed.

Figure 5.

Variation curves of central column drifts and the structure drift: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

As shown in Figure 5(a), for PGA = 0.1 g, the No. 1 and No. 2 central column drifts and the structural drift increase by 63%, 86%, and 36%, respectively; for PGA = 0.22 g, these three drifts increase by 82%, 99%, and 42%, respectively. As shown in Figure 5(b), for PGA = 0.1 g, the No. 1 and No. 2 central column drifts and the structural drift increase by 99%, 117%, and 50%, respectively; for PGA = 0.22 g, these three drifts increase by 106%, 130%, and 56%, respectively. Overall, the variation regularities of the drifts of the columns and the structure with the change of $α_{FS}$ are almost the same as those with the change of H/B. Increases in $α_{FS}$ and H/B of the central columns both increase the drifts of the central columns and the structure. The data also show a distinct characteristic: the increase in central column drifts is much larger than that of the structural drift with the increase in $α_{FS}$ and H/B. Studies show that drift is related to the stiffness ratio of soil to structure (Chen et al., 2014b; Parra-Montesinos et al., 2006). As the soil stiffness has not changed, these varying degrees of drift increase reflect the changing degrees of lateral stiffness of column and structure. With the increase in $α_{FS}$ and H/B, both the lateral stiffness of the columns and the lateral stiffness of the structure are decreasing, and the lateral stiffness of the column decreases more rapidly than the lateral stiffness of the structure. This is because that the change of the cross-sectional size of the column affects the lateral stiffness of the column itself first and greatly, and as the central column is one of the structure components, the change of the column size has influence on the structure lateral stiffness but is smaller than the influence on column lateral stiffness.

Internal forces of columns and sidewalls

Axial force

Figure 6 shows the variation regularities of the axial force of the central columns (1-1 cross section) and the sidewalls (2-2 cross section) in a static state (static axial force) and with underground motions (dynamic axial force) when $α_{FS}$ and H/B of the central columns are changed, respectively. The total axial forces of 1-1 section and 2-2 section are also shown in Figure 6.

Figure 6.

Variation curves of axial force: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

As can be seen from Figure 6(a), when H/B = 1.67, as $α_{FS}$ increases from 0.13 to 0.68, the axial forces of the central column in static state, for PGA = 0.1 and 0.22 g, are reduced by 632 kN (15%), 672 kN (16%), and 580 kN (13%), respectively; conversely, the axial forces of the sidewall are increased by 470 kN (13%), 561 kN (15%), and 402 kN (10%), respectively. Thus, after the increase in $α_{FS}$ , the main reason for the decrease in the axial force on the column is not the reduction in its weight, it is the increase in the pressure and strain of the column, which distributes a part of the vertical load which is carried by central columns to the sidewalls. The increase in $α_{FS}$ of the central columns changes the load-bearing ratio of the column and the sidewall. The change of PGA has an effect on this trend; the variation of axial force is slightly enlarged under small PGA, while the variation reduces under large PGA. It is related to the large plastic damage in central columns and the structure under rare earthquake.

As mentioned in the “Introduction” section, AC is considered to be the most important parameter affecting the axial compression ratio in the design, and the influence of H/B on the axial compression ratio of central columns is not considered. In this article, the design of calculation conditions is also based on the control of axial compression ratio by AC. However, according to equation (1), although H/B does not affect the AC and fC, we cannot exclude the possibility that the axial force of components’ changes to some extent. While $α_{FS} = 0.2$ , as shown in Figure 6(b), when H/B increases from 0.04 to 26.67, the total axial forces both in the static state and underground motions are reduced by about 0.4%, which reflects that the H/B parameter has little effect on the total axial forces. The static and dynamic axial forces of the column are both reduced by about 120 kN (3%), while the static and dynamic axial force of the sidewalls both are increased by about 90 kN (2%). It indicates that the axial forces of the central columns and the sidewalls have been changed after the increase in H/B. But in contrast to the influence of the change of $α_{FS}$ on the axial force, it is found that the degree of influence exerted by the change of H/B is relatively small, and it is not sensitive to changes of PGA. From the above analysis, a conclusion can be made that the influence of H/B on the axial force is not more than 5%, which is indeed a negligible figure in the design.

In short, although both the increase in $α_{FS}$ and H/B can increase the axial force of central columns and decrease that of sidewalls, $α_{FS}$ is the main factor that affects the axial force of central columns and sidewalls. In addition, although the axial forces of the central columns are changed, the load-bearing capacity of the central columns changed simultaneously.

Shear force and bending moment

The variation of the shear forces and bending moments is more significant than that of the axial forces. Theoretically, changes in $α_{FS}$ and H/B of the central columns can affect the relative stiffness between the central column and the sidewall; thus, the distribution of shear forces and the bending moments of the two kinds of components changes accordingly. However, the influence of the two parameters on the shear force and bending moment of the structure is different. Figure 7 shows the variation regularities of the shear forces of the central columns (1-1 cross section) and the sidewalls (2-2 cross section) when $α_{FS}$ and H/B of the central columns are changed under different PGA conditions, and the total shear forces of the 1-1 cross section and the 2-2 cross section are also shown in Figure 7. The variation curves of the bending moments of the central columns (1-1 cross section) and the sidewalls (2-2 cross section) when $α_{FS}$ and H/B of central columns are changed under different PGA conditions are shown in Figure 8. The total bending moments of the 1-1 cross section and the 2-2 cross section are also shown in Figure 8.

Figure 7.

Variation curves of shear force: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

Figure 8.

Variation curves of bending moment: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

When H/B = 1.67, with the increase in $α_{FS}$ from 0.13 to 0.68, as can be seen from Figures 7(a) and 8(a), the dynamic shear force and bending moment of the column decrease, even tending to 0. At the same time, the static and dynamic shear forces and bending moment of the sidewall are rising. These results indicate that with the increase in $α_{FS}$ (decrease in Ac), central columns are no longer considered to be one of the main components of the shear and bending moment, and sidewalls bear almost all the shear and bending moments. The reason is obvious, which is with the increase in $α_{FS}$ (decrease in Ac) of the column, the lateral stiffness ratio of the sidewall to the central column becomes larger (increased from 9 to 359 approximately). Notably, the total shear forces as well as the total bending moments of the two sections are increasing. The total shear forces in a static state, for PGA = 0.1 and 0.22 g, are increased by 347 kN (18%), 350 kN (14%), and 374 kN (11%), respectively, while the total bending moments are increased by 1560 kN m (55%), 2121 kN m (43%), and 1947 kN m (26%), respectively. The data indicate that the actual increase in shear forces and bending moments of the sidewalls is larger than the actual decrease in that of the central columns. This is probably because the change of the stiffness of the column affects the stiffness of the structure and then affects the interaction between the structure and the soil (Chen et al., 2014b). Meanwhile, the change of PGA has a certain influence on the variation of shear forces and bending moments, which is revealed in the phenomenon that the increase in PGA reduces the proportion of shear force and bending moment increments. The larger the PGA is, the higher is the nonlinearity degree of the structure, which means that the structure may be in the case of less internal forces increment but larger drifts increment, and the increase in drifts in Figure 5 confirms this to some extent.

As mentioned before, the change of H/B changes the relative stiffness between the sidewall and central column, which may affect the distribution of static and dynamic shear force and bending moment between them, and the distribution is significant for structural design. When $α_{FS} = 0.2$ , with the increase in H/B from 0.04 to 26.67, as can be seen from Figures 7(b) and 8(b), the variation of static shear forces and static bending moments of both the central column and the sidewall is nearly negligible. However, the dynamic shear forces and dynamic bending moments change significantly. For PGA = 0.1 and 0.22 g, the dynamic shear forces of the central column are both reduced by about 91%, and the dynamic bending moments are both reduced by about 95%; for PGA = 0.1 g, the dynamic shear force of the sidewall is increased by 449 kN (21%), and the dynamic bending moment is increased by 2007 kN m (54%); for PGA = 0.22 g, the dynamic shear force of the sidewall is increased by 976 kN (40%), and the dynamic bending moment is increased by 3503 kN m (72%). The influence that the change of H/B exerts on the total shear forces and total bending moments is different from that of $α_{FS}$ . The influence that the change of H/B exerts on both is not monotonic. It can be seen that there is a minimum value on the total bending moment curve in Figure 8(b) when H/B equals to 0.6. There are two aspects to explain. On one hand, just as mentioned above, the increase in H/B decreases the stiffness of the structure, which has a close relationship with the interaction between the structure and the soil. And, it affects the effect of the interaction, which is, with the increase in H/B, structure drift in Figure 5(b) increased, but there were no significant changes in total shear force curve in Figure 7(b). It indicates that the lateral interaction force between soil and structure is not changed after H/B has been changed. On the other hand, the increase in H/B decreases the column stiffness. It causes the inflection point of central column moving downward, and the downward trend became slow when H/B is greater than 0.6 (this can be confirmed in Table 4). Meanwhile, the position of inflection point of the sidewall is not changed. As shown in Figure 7(b), shear force of sidewalls increases, while that of columns decreases. The change of downward trend of the column inflection point caused the change of decline rate of bending moment at the column bottom, resulting in the trough of total bending moment curve in Figure 8(b).

Table 4.

Ultimate bearing capacity and ultimate drift of central columns with different sizes.

Calculation conditions	$α_{FS}$	H/B	Inflection point height, l (mm)	Maximum dynamic axial force, N (kN)	V_max (kN)	M_max (kNm)	Ultimate drift Δ_max (mm)
1	0.13	1.67	4133	4443	1456	3832	17.39
2	0.16	1.67	3990	4365	1100	2676	14.72
3	0.20	1.67	3920	4246	901	2102	14.80
4	0.24	1.67	3777	4252	729	1630	15.24
5	0.34	1.67	3706	4144	453	966	16.33
6	0.68	1.67	3635	4863	91	186	11.50
7	0.20	26.67	3635	4237	142	289	37.90
8	0.20	15.00	3635	4238	233	475	38.82
9	0.20	6.67	3706	4217	429	916	25.35
10	0.20	0.6	4204	4291	1350	3554	19.49
11	0.20	0.15	4703	4353	2186	6837	16.23
12	0.20	0.07	4987	4259	2861	9807	16.92
13	0.20	0.04	5201	4351	3619	13,125	16.11

In short, the increase in $α_{FS}$ and H/B of the central column leads to the decrease in the relative stiffness between the central column and sidewall, which reduces the shear forces and bending moments of columns and enhances that of sidewalls. The increase in $α_{FS}$ causes the shear forces and bending moments of the central columns and the sidewalls increasing monotonically, but the influence of H/B on the two kinds of internal forces is not monotonic.

Ultimate bearing capacity of the central columns

With the increase in $α_{FS}$ or H/B of the central column, the shear force and bending moment of the central column decrease, and the bearing capacity of the column also decreases. Hence, further study is required to identify whether the column is damaged more seriously or not. To solve this problem, a numerical model of the isolated No. 2 central column was developed.

When the column cross-sectional size is close to (or even larger than) longitudinal beam, the column stiffness is close to (or larger than) the longitudinal beam stiffness (such as calculation conditions 12 and 13). Thus, in order to account for these extreme situations, the longitudinal beam and the column should be isolated and analyzed together. Figure 9(a) shows the isolated No. 2 central column with the longitudinal beams in both ends of the column. The length of the central column is 4270 mm, while the cross-sectional heights of the longitudinal beam at the top and the bottom of the column are 850 and 1500 mm, respectively. Because the linear stiffness ratio of bottom plate to vertical member is usually bigger than 10 in this junction, which means that bottom plate can be considered as a rigid body. Thus, it is reasonable that the bottom of the isolated system could be regarded as a fixed end. The upper boundary condition depended on the relative stiffness of each member of the joint, between the fixed and free constraints. If the column section size is small enough, the position of the inflection point (M = 0) should be located at the half-height of the column (l = 3635 mm), as shown in Figure 9(b), and if the column section size is large enough, the top of the isolated system (l = 6620 mm) is the point that meets the condition of M = 0, as shown in Figure 9(c). All the analysis conditions are located between these two limits, coupling the two limit conditions to different degrees. Therefore, as shown in Figure 9(d), the position of the inflection point is between 3635 and 6620 mm. According to the seismic damage of Daikai Station (Nakamura et al., 1996), the bottom of the column can be regarded as the most vulnerable location. So, the isolated system contains the most vulnerable location. The upper end of the isolated system has no bending moment due to the position of the inflection point; thus, the upper end of the isolated system could be free, then the vertical load which equals to the maximum dynamic axial force under rare ground motion (PGA = 0.22 g) and lateral load were applied at the position of the inflection point. After that, the corresponding ultimate lateral bearing capacity of the isolated system under the vertical load was obtained, as shown in Figure 9(e). Aschheim et al. (1997) and Moehle and Lehman (2006) both argue that the total deformation, Δ, of the RC cantilever column is composed of three parts, which are the bending deformation, the shear deformation and the bond-slip deformation. In the numerical analysis, the bond-slip deformation was not considered. Thus, the ultimate drift of the inflection point to the bottom of the isolated system can be described as the total deformation of this part.

Figure 9.

(a) Isolated No. 2 central column and (b–e) ultimate bearing capacity analysis model of No. 2 central column.

Beam element (B21) was used to simulate the isolated system, and it was divided into 185 elements. The concrete damaged plasticity model was adopted in the constitutive model, while the steel bar was simulated by the command *rebar, and materials and parameters were described in section “Constitutive models and properties of materials.”Table 4 reflects the inflection point height of the column with different sizes under rare earthquake as well as the ultimate bearing capacity (V_max and M_max) and ultimate drift capacity (Δ_max) under the maximum dynamic axial force (N).

To enable a visual comparison, under rare ground motion (PGA = 0.22 g), the ratio of the maximum dynamic shear force of the 1-1 cross section to the ultimate shear force (V/V_max), and the ratio of the maximum dynamic bending moment of the same cross section to the ultimate bending moment (M/M_max) of each column size conditions are shown in Figure 10. The ratio of drift between the inflection point and the bottom of the beam to the ultimate drift (Δ/Δ_max) of these conditions is shown in Figure 10. It is clear that with the increase in $α_{FS}$ and H/B, the ratio of internal force to ultimate bearing capacity (V/V_max, M/M_max) increases continuously. When H/B is constant (H/B = 1.67) and $α_{FS}$ increases from 0.13 to 0.68, V/V_max and M/M_max both increase from 36% to 80%, as shown in Figure 10(a). And, when $α_{FS}$ is constant $(α_{FS} = 0.02)$ and H/B increases from 0.04 to 26.67, V/V_max and M/M_max both increase from 25% to 50%, as shown in Figure 10(b). Therefore, from the point of view of the ultimate bearing capacity, the increases in $α_{FS}$ and H/B of the central column both reduce the seismic capacity of the central column.

Figure 10.

Variation curves of V/V_max, M/M_max, and Δ/Δ_max: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

But if Δ/Δ_max is analyzed, the results are different. As shown in Figure 10(a), when H/B is 1.67, Δ/Δ_max increases from 34% to 60% and $α_{FS}$ increases from 0.13 to 0.68, and when $α_{FS}$ is constant $(α_{FS} = 0.2)$ , H/B increases from 0.04 to 26.67 and Δ/Δ_max decreases from 42% to 19%, as shown in Figure 10(b). Thus, from the point of view of the ultimate drift, the increase in $α_{FS}$ is not conducive to the column bearing. However, the increase in H/B improves the ductility of the column, which is beneficial to the bearing of the central column.

In addition, as can be seen from Figure 10(b), the rising curves (V/V_max, M/M_max) and the falling curve (Δ/Δ_max) have an intersection, roughly between 0.15 and 0.6. The intersection means that the column has the same seismic capacity in terms of the internal force and the lateral displacement at this point. Therefore, considering both the seismic capacity of internal force and displacement, 0.15–0.6 is a reasonable range for H/B.

Structural damage

The previous analysis shows that the variation of $α_{FS}$ and H/B of the central column changes V/V_max, M/M_max, and Δ/Δ_max. And the variation of the degree of damage to the central column and the structure is a comprehensive reflection of these changes. In all of the calculation conditions, the compression damage of the structure is concentrated in the central column. In order to depict the compression damage degree of the column, the maximum compressive damage (DC_max) is defined as the maximum value of the compression damage factor dc of the central column. Figure 11 shows the variation curves of DC_max with the variation of $α_{FS}$ and H/B of the central column.

Figure 11.

Variation curves of DC_max: (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

As shown in Figure 11(a), when the value of H/B is 1.67, with the increase in $α_{FS}$ , the DC_max increases monotonically. It is consistent with the previous analysis that the increase in $α_{FS}$ enlarges V/V_max, M/M_max, and Δ/Δ_max, which reduces the seismic capacity of the central column. However, with the increase in H/B, the curve shows a trend of increasing first and then decreasing, as shown in Figure 11(b). There are two aspects need to be focus on. For one thing, compared with the influence of $α_{FS}$ on DC_max, the influence of aspect ratio on DC_max is relatively weak, which can be seen from the range of changes of DC_max in Figure 11. This implies that it is reasonable to determine the cross-sectional area of central columns first according to the axial compression ratio and then select H/B in the design of metro station. For another, H/B of the central column of the studied subway station was designed to be 1.67, which is almost at the highest point of the DC_max curve. It indicates that the existing design of H/B is unreasonable from the point of view of seismic resistance.

Figure 12 shows the variation regularities of structural tension damage (dt) with the variation of $α_{FS}$ and H/B of the central column. It can be seen from Figure 12(a) and (b) that with the increase in $α_{FS}$ or H/B, not only does the tension damage of sidewalls increases because of the increase in internal forces, but the tension damage of the floor also increases in varying degrees. The variation of the two parameters exerts a similar effect to dt. Tension damage first appears at the foot of the sidewalls and at both ends of the column, then appears at the corner where the bottom plate is connected to the sidewall, followed by the junction point of the middle plates and central columns, and finally, the junction point of the sidewalls and the top plate as well as the junction point of the sidewalls and the middle plates.

Figure 12.

Variation regularities of structure tension damage (dt): (a) H/B = 1.67 and (b) $α_{FS} = 0.20$ .

The variation regularities of dc and dt show that the increase in $α_{FS}$ has a negative effect on all of the components of the structure, and the increase in H/B also had a negative influence on all the components except the central columns. In addition, according to the curve in Figure 11, when H/B is larger than 6.67, the increase in H/B leads to the decline of dc of the column, which means that it is conducive to reducing the seismic damage of the column.

Therefore, based on the above research and discussion, after determining the appropriate axial compression ratio parameters according to relevant design specification, the strategy of choosing H/B can be worked out. When H/B is between 1.67 and 6.67, the compressive damage of column is most serious; therefore, (1.67–6.67) is the design range to be avoided. When H/B is larger than 6.67, the compressive damage of column is declined; however, the tension damage of the structure increased significantly, which is not conducive to waterproofing of structure. In addition, central columns seem more like central wall under this condition, lowering the lateral traffic capacity of the station noticeably. Thus, (6.67–26.67) is not suitable for H/B. When H/B is smaller than 1.67, the decrease in H/B reduces the damage of the structure (both tension damage and compressive damage), but it does not mean that the smaller the H/B is, the better is the seismic performance of the structure. The intersection in Figure 10(b) shows that when H/B is less than the intersection point, column seismic capacity declines. Since the intersection point is roughly in the range of (0.15, 0.6), from the point of view of seismic resistance, the ideal H/B should be between 0.15 and 0.6. To select the specific point in the range, it is necessary to determine the actual demand for longitudinal traffic capacity.

Conclusion

In this article, time history analyses and ultimate bearing capacity analysis of central columns were carried out on the seismic behavior of a typical two-story three-span subway station structure. Emphasis was placed on the static axial compression ratio, $α_{FS}$ , and aspect ratio, H/B, of the central columns. The main conclusions are drawn as follows:

The increase in $α_{FS}$ of the central column reduces the structural seismic capacity. Given constant H/B, the dynamic shear force and bending moment of the central column decrease when $α_{FS}$ increases from 0.13 to 0.68. The ratio of internal force to ultimate bearing capacity (V/V_max, M/M_max) and the ratio of column drift to ultimate drift of the central column (Δ/Δ_max) both increase from 36% to 80% and 34% to 60%, respectively. And, as a result, the maximum compression damage (DC_max) of the central column increases from 0.02 to 0.24 under rare ground motion (PGA = 0.22 g); meanwhile, the tension damage of each component is increasing.

The influence of H/B on the seismic performance of structures is more complicated. Given constant $α_{FS}$ , V/V_max and M/M_max, both increase from 25% to 50% when H/B increases from 0.04 to 26.67. However, Δ/Δ_max declines from 42% to 19% meanwhile. Under the combined effect of these two aspects, DC_max of the column first rises from 0 to 0.07 then decreases to 0.02 under rare ground motion. And the tension damage of each component is increasing monotonously.

From the point of view of seismic resistance, the current design of H/B (1.67) in the subway station is unreasonable. Based on the research of this article, the ideal H/B should be between 0.15 and 0.6. To select the specific point in the range, it is necessary to determine the actual demand for longitudinal traffic capacity.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant no. 41472246) and the Key Laboratory of Transportation Tunnel Engineering (TTE2014-01). Both supports are gratefully acknowledged.

References

ABAQUS/CAE (2009) User’s Manual 6.9: Dassault Systèmes Simulia Corp. Providence, RI: ABAQUS/CAE.

Shawky

Maekawa

(1997) The collapse mechanism of a subway station during the Great Hanshin Earthquake. Cement and Concrete Composites 19(3): 241–257.

Aschheim

Moehle

Mahin

(1997) Design and evaluation of reinforced concrete bridges for seismic resistance. Report no. UCB/EERC-97/04. Berkeley, CA: EERC, University of California.

Atalay

Penzien

(1975) The seismic behavior of critical regions of reinforced concrete components as influenced by moment, shear and axial force. Report no. 75-19. Berkeley, CA: EERC, University of California.

Chen

Wang

Zuo

et al . (2013) Shaking table test on the seismic failure characteristics of a subway station structure on liquefiable ground. Earthquake Engineering & Structural Dynamics 42(10): 1489–1507.

Chen

Bian

(2014a) Seismic performance upgrading for underground structures by introducing shear panel dampers. Advances in Structural Engineering 17(9): 1343–1357.

Chen

Zhang

(2014b) Seismic performance evaluation of multi-story subway structure based on pushover analysis. In: Proceedings of the advances in soil dynamics and foundation engineering, Shanghai, China, 26–28 May 2014, pp. 444–454. New York: ASCE.

Chen

Zhang

et al . (2016) Effects of axial compression ratio of central columns on seismic performance of a multi-story underground structure. International Journal of Computational Methods 13: 1641014 (16 pp.).

GB50010 (2010) Code for Design of Concrete Structures. Beijing, China: China Architecture & Building Press.

10.

GB50011 (2010) Code for Seismic Design of Buildings. Beijing, China: China Architecture & Building Press.

11.

GB50157 (2013) Code for Design of Metro. Beijing, China: China Architecture & Building Press.

12.

Hashash

Hook

Schmidt

et al . (2001) Seismic design and analysis of underground structures. Tunnelling and Underground Space Technology 16(4): 247–293.

13.

Huo

Bobet

Fernández

et al . (2005) Load transfer mechanisms between underground structure and surrounding ground: evaluation of the failure of the Daikai Station. Journal of Geotechnical and Geoenvironmental Engineering 131: 1522–1533.

14.

Iida

Hiroto

Yoshida

et al . (1996) Damage to Daikai subway station. Soils and Foundations 36: 283–300.

15.

Iwasaki

Kawashima

Hagiwara

et al . (1985) Experimental investigation on hysteretic behavior of reinforced concrete bridge pier columns. In: Proceedings of the 17th Joint Panel Meeting of the U.S.-Japan Cooperative Program in Wind and Seismic Effects, May 21-24, 1985. Gaithersburg, MD: Center for Building Technology, U.S. National Bureau of Standards.

16.

Khani

Homami

(2014) Seismic performance of underground shallow subway stations in soft soil. Journal of Engineering Geology 8(1): 1983–2002.

17.

Lam

SSE

Wang

Liu

et al . (2003) Drift capacity of rectangular reinforced concrete columns with low lateral confinement and high-axial load. Journal of Structural Engineering 129(6): 733–742.

18.

Lee

Fenves

(1998) Plastic-damage model for cyclic loading of concrete structures. Journal of Engineering Mechanics 124: 892–900.

19.

Lubliner

Oliver

Oller

et al . (1989) A plastic-damage model for concrete. International Journal of Solids and Structures 25(3): 299–326.

20.

Lysmer

Kuhlemeyer

(1969) Finite dynamic model for infinite media. Journal of the Engineering Mechanics Division 95(4): 859–878.

21.

Moehle

Lehman

(2006) Seismic performance of confined concrete bridge columns. Special Publication International Symposium on Confined Concrete SP-238: 23–42.

22.

Nakamura

Yoshida

Iwatate

(1996) Damage to Daikai subway station during the 1995 Hyogoken-Nambu earthquake and its investigation. Journal of JSCE287–295.

23.

Pacific Earthquake Engineering Research Center (2000) PEER Strong Motion Database. Berkeley, CA: University of California Press.

24.

Parra-Montesinos

Bobet

Ramirez

(2006) Evaluation of soil-structure interaction and structural collapse in Daikai subway station during Kobe earthquake. ACI Structural Journal 103: 113–122.

25.

Sheikh

Khoury

(1993) Confined concrete columns with stubs. ACI Structural Journal 90(4): 414–431.

26.

Wong

Paulay

Priestley

MJN

(1993) Response of circular reinforced concrete columns to multi-directional seismic attack. ACI Structural Journal 90(2): 180–191.

27.

Yamato

Umehara

Aoki

et al . (1996) Damage to Daikai Subway Station of Kobe Rapid Transit System and Estimation of Its Reason During The 1995 Hyogoken-Nanbu Earthquake. Proceedings of the Japan Society of Civil Engineers 537: 303–320 (in Japanese).

28.

Ying

(2011) Focus on Shanghai metro’s 12th-plan period taking network operation security as a priority. Traffic and Transportation 27(4): 1–3 (in Chinese).

29.

Zhang

Bai

(2005) Experimental study on seismic behavior of frame columns with high axial load ratio subjected to low cyclic loads. Earthquake Engineering and Engineering Vibration 25(6): 70–75 (in Chinese).

30.

Zhao

Jiang

Chen

(2009) The response analysis of underground railway station structure in soft soil under strong earthquake. China Railway Science 3: 10 (in Chinese).

31.

Zienkiewicz

Bettess

et al . (1991) Mapped infinite elements for exterior wave problems. International Journal for Numerical Methods in Engineering 32(1): 207–209.