J. Cent. South Univ. (2013) 20: 528–535

DOI: 10.1007/s11771-013-1515-x

Analytical solution to rock pressure acting on three shallow tunnels subjected to unsymmetrical loads

YANG Xiao-li(杨小礼), ZHANG Jia-hua(张佳华), JIN Qi-yun(金启云), MA Jun-qiu(马军秋)

School of Civil Engineering, Central South University, Changsha 410075, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: According to the interaction of three shallow tunnels with large section, the analytical solution to rock pressure has been derived and discussed. The load model is given when the bilateral tunnels are excavated. According to the model, the stresses of three tunnels and single tunnel are calculated and compared to analyze the distribution characteristics, where the stresses are influenced by controlling factors of clear distance, covering depth and inclination angle of ground surface. The results show that, in general, the bias distribution is more serious. Therefore, it is significant to settle down the load model of three shallow tunnels so as to determine the measure of reinforcement and design the structure of support. The model and results can be used as a theoretical basis in designation and further research of the three shallow tunnels.

Key words: three shallow tunnels; unsymmetrical loads; clear distance; rock pressure

1 Introduction

The problem of three closely built tunnels with large section is a novel design, the usage of which in the highway construction optimizes the overall linearity, and it is superior to other tunnels in cost and water handling. So, this new tunnel has been favored by the designers recently. However, the theoretical research about its stability and load model falls behind the needs of construction.

Rock pressure is adopted here to mean stresses caused by the perturbed rock, which is a feedback from supporting structure to the deformation of the rock. The stresses are directly affected in the tunnel design. The structure-load model is the commonly used calculation method to date in the tunnel structure design, which includes the wedge structure model and the equilibrium arch model. This solution is more precise than that of empirical equation, but the calculation equation is too complicated and contains too many parameters. Rock pressure of tunnels is often investigated using different methods, such as theoretical calculations [1–4], on-site measurements [5–8], model tests [9–11], and backward predication parsing methods [12–14]. However, those works focused on a single tunnel or two tunnels. There is little work referring to rock pressure of three tunnels.

In the practical engineering recently, the load model of single tunnel is often adopted. However, the load model of multi-tunnel has not been given in the codes [15–16]. While the load model of the multi-tunnel is far more intractable than the single one due to the interaction between tunnels, and the result of multi-tunnel obtained by the load model of single tunnel is neither accurate nor stable with the variation of rock-level [17]. So, it is necessary to study the rock pressure on the multi-tunnels.

Based on the Hanfu Mount Tunnel and the codes [15], the load model, including the stress distribution and the calculation equations, of three shallow tunnels with large section and clear distance will be proposed in this work. The model will take the interaction between tunnels into account, and the equation will be theoretically deduced.

2 Pressure calculations using limit equilibrium method

Based on the Hanfu Mount Tunnels, three shallow tunnels with clear distance, are analyzed in this section. The excavation order is that the side tunnels are prior to the middle tunnel. The value and distribution of stresses are calculated in the shallow and inclined rock.

2.1 Failure model

As shown in Fig. 1, the load model of three tunnels is established according to the model of single tunnel from the code [15].

Fig. 1 Calculation model of three tunnels subjected to unsymmetrical loads

From Fig. 1, it is found that the three tunnels have bottoms on the same level, a ground surface inclining to the left with angle a, excavation widths of B and clear distances of D between them. The height from the top of left and right sides of tunnel to the ground is H_i′ and H_i (H_i′< H_i) respectively. W_i ,W_i' and W_i" denote the masses of soil mass on the top, left and right sides of tunnel, respectively. F_i and F_i′ refer to the resistance forces on the failure surface beside tunnels, respectively. T_i and T_i′ are the resistance forces from the soil mass beside the one on top of tunnel to keep it from moving down, respectively. b_i and b_i′ are the angles of the failure surface A_iC_i and L_iM_i which occur when the values of T_i and T_i′ reach their limits, where the subscript i=1, 2, 3 identify the left, middle and right tunnels, respectively. Besides, j_c is the internal friction angle in the calculation, and q is the friction angle beside the top mass.

Some assumptions have been made for the calculation as below:

1) The rock is supposed to be a simple and continuous medium, in which the three shallow tunnels are buried with equidistance in between. The ground surface is an oblique line, as shown in Fig. 1. As mentioned above, the side tunnels are excavated first, then the mid-tunnel. Moreover, there is no interaction between side-tunnels.

2) Caused by the excavation of mid-tunnel, the failure surface L₂M₂ will cross with the existing one A₁C₁ which caused by the formerly excavated left tunnel, thereby forming a triangular mass A₁K₁L₂. The mass will move slightly along the failure surface A₁C₁ when the left tunnel excavated, which will weaken the cohesion on A₁K₁. When the mid-tunnel is excavated, the mass will slide down the failure surface L₂M₂. In order to simplify the analysis, a vertical collapse J₁K₁ is supposed to appear in the triangular mass which is separated into two parts and the force on J₁K₁ is zero. The same assumption is made for triangle mass A₂K₂L₃.

3) The distances D₁ and D₂, from the cross point of failure surface to the sidewall of tunnel, can be approximated to D/2, because the clear distance of the tunnel model herein is quite small.

2.2 Calculation process

According to the calculation model and the assumptions above, the calculation process can be done in two steps:

1) Before the excavation of mid-tunnel, the side tunnels are excavated independently as a single tunnel, therefore the stresses can refer to the load model of the codes [15–16].

2) After the excavation of mid-tunnel, the stresses on the left wall of left tunnel and the right wall of right tunnel remain the same, and the other parts affected by the interaction will be recalculated.

As the deduction of the right tunnel is quite similar to the left tunnel, in this work, the deducing process of the left tunnel is given only.

2.3 Load analysis before excavation of mid-tunnel

The left and right tunnels can be treated as independent ones in this stage, and the load model of them is shown in Fig. 2. The solution is presented below.

2.3.1 Horizontal stresses on sidewall of left tunnel

The horizontal stresses can be calculated by Eq. (1).

e_i =lgh_i                                    (1)

where h_i is the depth of underground point i, and l is the characteristic coefficient which can be determined by Eq. (2) and Eq. (3).

Fig. 2 Load model of side tunnels before excavation of mid-tunnel

         (2)

          (3)

The angle of failure surface can be obtained by Eq. (4) and Eq. (5).

(4)

(5)

2.3.2 Vertical stresses on top of left tunnel

Assuming that the distribution shape of vertical stress is parallel to the ground surface, the calculation equations can be easily obtained as bellow.

The total pressure on top of left tunnel is:

    (6)

Then,

 (7)

 (8)

2.4 Load calculation after excavation of mid-tunnel

Figure 3 and 4 show the computing model and the stress distribution model of mid-tunnel and left tunnel after the excavation of mid-tunnel.

2.4.1 Horizontal stresses on sidewall of mid-tunnel

For the left wall of mid-tunnel, the equilibrium equation is:

            (9)

 (10)

where is taken as the characteristic coefficient of stresses on left side of mid-tunnel. When reaches the maximum, and can then be determined.

 (11)

 (12)

Substituting  into Eq. (1), the stresses on left wall of mid-tunnel can then be calculated. For the right wall of mid-tunnel, the deduction is the same.

 (13)

 (14)

where h₂ is the distance from bottom of tunnel to ground surface.

Fig. 3 Load model and force distribution of mid-tunnel after excavation

Fig. 4 Load model and force distribution of left tunnel after excavation of mid-tunnel

2.4.2 Vertical stresses on top of mid-tunnel

   (15)

 (16)

 (17)

2.4.3 Horizontal stresses on sidewall of left tunnel

The stresses on the left wall of left tunnel are not affected by the excavation of mid-tunnel, and the characteristic coefficient is still as Eq. (2), while the stresses on the right wall should be revised.

  (18)

where h₁ is the distance from bottom of right wall to ground surface.

2.4.4 Vertical stresses on top of left tunnel

       (19)

(20)

It should be mentioned here that the distribution angle of the stresses is no longer the same as the ground surface.

2.5 Applicable conditions

From the load model, it is found that, only keeping the tunnel within a certain distance, the failure surfaces intersect under the ground surface and the side tunnels are affected by the excavation of mid-tunnel. This means

              (21)

Moreover, the left and right tunnels should keep a spacing to make sure that the two tunnels will not interact with each other. So, it is obtained:

   (22)

That is,

       (23)

Putting tanb₁ and tanb' into Eq. (21) and Eq. (23), the threshold of the clear distance is set.

3 Results and discussion

In order to analyze the distribution characteristic of rock pressure on the three tunnels, the model selected is a bi-directional tunnel built in V-level rock. The calculation parameters are as below: the width and height of each tunnel B=14 m and H_g=12 m; the unit weight of rock g=20 kN/m³; the internal friction angle ; calculation friction angles and . The covering depth, the inclination angle of the ground and the clear distance between tunnels are calculated accordingly.

Figures 5 and 6 show the stresses on the sidewall varying with the covering depth and clear distance, when the inclination angle of the ground a =20°.

It is indicated from Fig. 5 that, broadly, under a certain covering depth, the vertical stresses of the three tunnels increase when the clear distance becomes narrower and are higher than that of single tunnel. When D≥1.2B , these stresses nearly overlap with each other.

Fig. 5 Wall vertical stresses curves of three tunnels with different clear distances:

Fig. 6 Wall horizontal stresses curves of three tunnels depending on clear distance:

On the other hand, the vertical stresses are very susceptible to clear distance. The stresses on the right of left tunnel are the most susceptible, followed by that of mid-tunnel and right tunnel. Comparing Figs. 5 (a) and (b), (c) and (d) , the covering depth influence is obvious. That is, the difference between the stresses of three tunnels and single tunnel and the value of point on D/B-axis, where coincidence appears, is larger with the deeper overlying. Besides, the unusual V-shaped rebound in the pictures can be explained by error of approximation made for D₁, D₂, D₃ and D₄.

The horizontal stresses, as shown in Fig. 6, are on top of the sidewall. When the clear distance is growing under a certain overlying, the stresses of three tunnels increase till equivalent to that of single tunnel. Comparing Figs. 6 (a) and (c), (b) and (d), it can be manifested that, with the increase of covering depth, the influence of clear distance on the value of stresses is more obvious. And the value of clear distance is greater, when the difference of stresses between three tunnels and single tunnel can be ignored. Again, a fall after the peak in the conditions is explained by error of approximation made for D₁, D₂, D₃ and D₄.

Figures 7 and 8 show how the ratio of right stresses to left stresses is affected by clear distance in different combinations of covering depth and inclination angle of the ground.

The distribution pattern of the vertical stresses on top of the three tunnels differs from each other, as shown in Fig. 7. That is, with the decrease of clear distance, the ratio of left tunnel increases. This means a distinct asymmetrical distribution, and the pattern of mid-tunnel is almost the same as single tunnel. For the right tunnel, the difference between left and right stresses is similar to the single tunnel at first, and is dwindling till equally distributed, then the left stresses gradually exceed the right, a reverse bias shows and becomes larger. What’s more, the bias is more serious on the left tunnel with the growth. For the middle and right tunnel, it depends contrary on the latter. The bias degree of the former is alleviated with a deeper covering and flatter ground.

In Fig. 8, the ratio of the stresses on top of the sidewall is illustrated. For the left tunnel, when the clear distance becomes narrower, the left stresses are catching up with the right little by little and finally exceed it. For the mid-tunnel, the stresses are symmetrical, and they gradually approach the pattern of single tunnel when the distance becomes broader. For the right tunnel, the ratio is always beyond that of single tunnel, especially when D ≤0.4B. The disparity is so big that it has to be underlined.

Fig. 7 Ratio of vertical stresses of three tunnels varying with clear distance:

Fig. 8 Ratio of horizontal stresses of three tunnels varying with clear distance:

From the comparison, it is found that the biased phenomenon for all the three tunnels is more obvious with a shallower and steeper overlying.

4 Conclusions

1) Three shallow tunnels subjected to unsymmetrical loads are excavated. For the left tunnel, the bias distribution of stress is evident and even more serious with a deeper covering, closer neighbor and steeper ground. For the mid-tunnel, the excavation stresses magnify on top and minus on sidewall. The variation tendency is basically the same as a single tunnel. For the right tunnel, the stress distribution depends on the clear distance between tunnels. The load condition of three tunnels is different. Therefore, the parameters of support should be decided according to the worst situation in the structure design.

2) The error of biased load, caused by the approximation, increases with the value of distance D. Especially when D ≥1.6B, the error is more than 5%. Due to the specification of the model, the results should be modified before use in some practical engineering, such as when the ground is right inclined or composed of several inclination angles rather than a left-inclined slope in the chosen model. Besides, if considering the soil layers and underground water, a further study could be done on the basis of the proposed model.

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(Edited by HE Yun-bin)

Foundation item: Projects(2013CB036004, 2011CB013800) supported by the National Basic Research Program of China; Project(51178468, 50908234) supported by the National Natural Science Foundation of China; Project(2011G103–B) supported by the Science and Technology Development of Railway in China

Received date: 2011–11–07; Accepted date: 2012–04–03

Corresponding author: YANG Xiao-li, Professor; Tel: +86–14789933669; E-mail: yxnc@yahoo.com.cn