Upper bound analysis of collapse failure of deep tunnel under karst cave considering seismic force

Soil Dynamics and Earthquake Engineering 132 (2020) 106003

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Upper bound analysis of collapse failure of deep tunnel under karst cave considering seismic force Cheng Lyu a, b, Li Yu a, b, Mingnian Wang a, b, *, Pengxi Xia a, b, Yuan Sun a, b a b

School of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China Key Laboratory of Traffic Tunnel Engineering, Ministry of Education, Southwest Jiaotong University, Chengdu, 610031, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Upper bound theorem Karst tunnel Nonlinear failure criterion Karst cave Variational principle

When tunnels are excavated in karst areas, the existence of karst caves in the surrounding rock usually results in the collapse failure of tunnels. Tunnel collapse has always been a challenging issue because it seriously threatens the safety of tunnel builders. To investigate the collapse of a deep tunnel under a karst cave, we constructed the collapse mechanisms of the rectangular and circular tunnels under a karst cave, and obtained the expressions of the collapse failure in tunnel considering seismic force and seepage force by means of the variational principle and limit analysis theory. Compared with previous results and numerical solutions, it is found that the results in this paper are consistent with those results and numerical solutions, which indicates that the proposed method in this work is rational. According to the analytical solutions, the shape of the collapsing blocks of the deep rect­ angular and circular tunnels under a karst cave are drawn, and the effects of different parameters, especially the vertical seismic coefficient kv, on the collapsing blocks are analyzed. The findings in this paper indicate that the range of collapsing blocks decreases with increased kv. The two formulas for calculating the critical height be­ tween the tunnel and the karst cave are given by derivation.

1. Introduction With the rapid development of transportation facilities, traffic tun­ nels inevitably pass through karst areas because of their wide distribu­ tion in the world, especially in the southwest of China [1,2]. Buried karst, which is a serious threat to the stability of traffic tunnels, is characterized by complex distribution, high fissure water content, var­ iable shapes, and high permeability [1,3]. When tunnel excavation is carried out near a cave, it is easy to cause the failure of the surrounding rock between tunnel and cave due to the disturbance. The locations of concealed caves in actual projects are difficult to detect, and tunnel collapse is sudden, which seriously affects the safety of tunnel builders. Therefore, the stability of tunnels in karst areas has attracted wide attention, and encouraging progress has been made [1–7]. Cui [2] described the characteristics of seven karst areas in China in detail and proposed a method to control geological hazards caused by karst caves near tunnel excavation. Wang [3] calculated the horizontal critical distance between karst tunnels and karst caves by means of numerical simulation and dichotomy, and by analyzing the influence of diverse parameters on the critical horizontal distance, suggesting that the karst cave span has the most significant effect. Huang [7] combined the

analytic hierarchy process and statistical information of related engi­ neering to propose a model for evaluating the influence of karst caves on the stability of surrounding rock of tunnels in karst areas, which has been applied in some projects. Yang [8] obtained an expression of minimum thickness of rock plugs in tunnels by combining the varia­ tional principle and limit analysis, and analyzed the influence of diverse parameters on the minimum thickness of rock plugs. However, tunnel collapse in karst areas has been rarely studied out until recently. To evaluate tunnel face stability, Mollon [9] constructed a three-dimensional model of a circular tunnel face using the spatial dis­ cretization technique, and this method showed improved solutions compared to previous methods but is complex. Lippmann [10] analyzed the upper and lower bound method to investigate collapse in circular and rectangular coal tunnels and the supporting force needed to prevent collapse. Fraldi [11,12] presented a theoretical method for predicting the collapsing range of deep tunnels by combining limit analysis and the variational principle, and the validity of this method was verified by comparing with the numerical results. Then the expression of collapse surface was given and the influence of different parameters on the collapse surface was analyzed. Referring to the findings of Fraldi [11, 12], Yang [13] constructed a new collapse mechanism of shallow

* Corresponding author. School of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail address: [email protected] (M. Wang). https://doi.org/10.1016/j.soildyn.2019.106003 Received 2 July 2019; Received in revised form 29 October 2019; Accepted 8 December 2019 Available online 14 February 2020 0267-7261/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. The collapse mechanism of deep tunnel under karst cave: (a) rectangular tunnel; (b) circular tunnel.

2. Theoretical basis

tunnels. Similarly, the expression of collapse was obtained by combining limit analysis and the variational principle, and the collapsing range could be further obtained. To assess the effects of changing water tables on the potential progressive collapse of tunnel roofs, Qin [14] con­ structed 2D and 3D collapse failure modes of tunnel roofs and obtained upper bound solutions of collapsing block with the help of the upper bound theorem of limit analysis and the variational method. Yu [15] analyzed the three-dimensional collapse of deep tunnels considering pore water pressure based on the nonlinear Mohr–Coulomb criterion and the nonassociative flow rule. Afterwards, Yang [6] put forward a collapse mechanism of the tunnel floor of an underlying karst cave, and then derived the expression of the detaching curve of the collapsing block and the formula for calculating the collapsing range based on the variational principle. Huang [16] proposed a passive failure mode, which was different from Yang’s [6]. With the same theoretical deri­ vation, the expression of the detaching curve describing the shape of the collapse and the formula for calculating the collapsing range was ob­ tained, which are different from Yang’s [6] results. Although rigid analytical solutions of tunnel roof and tunnel floor collapse in karst areas were obtained from the above-mentioned studies, the collapse of a karst tunnel roof under akarst cave has not been sufficiently resolved, and requires further investigation. Moreover, there are now many studies on the stability of slopes and retaining walls considering seismic force, but the influence of seismic force on tunnel collapse has not been reported by limit analysis [17–22]. The quasi-seismic force method is employed to investigate the effect of seismic force on the collapse of a tunnel under a karst cave in this work. In this paper, we construct a collapse mechanism of the roof of the rectangular and circular tunnels under akarst cave, which extends to the upper karst cave. By incorporating seismic force into the limit analysis theory and combining the nonlinear Hoek-Brown failure criterion with the variational principle, we derive analytical expressions of the detaching curves of collapsing blocks and theoretical formulas for calculating the collapsing range and weight. The results of this paper are simply transformed and compared with those of Fraldi [11,12] and numerical solutions, verifying the rationality of this method. Addition­ ally, the influence of variational parameters on the collapsing range can be obtained by plotting the collapse shape of rectangular and circular tunnels under a karst cave with different parameters.

2.1. Upper bound limit analysis subjected to seepage force Groundwater resources are abundant in karst areas, and the seepage force of groundwater affects the stability of geotechnical engineering [23–30]. Thus, the seepage force should be incorporated into the upper limit theorem of limit analysis to analyze the collapse of karst tunnels. The upper bound theorem of limit analysis insists that the work done by all external loads is no more than the dissipated energy in any kine­ matically admissible velocity field [15,31]. The seepage force is considered to be the sum of external loads acting on the skeleton of rock and soil, and then the upper limit theorem of the limit analysis consid­ ering seepage force can be expressed as [23–25]. Z Z Z Z grad u⋅vi dΩ þ Ti vi ds þ Xi vi dV � σij ε_ ij dV (1) Ω

s

Ω

Ω

where -grad u represents excess pore pressure, v signifies velocity along the detaching surface, Ω means the volume of the collapsing block, Ti denotes the load acting on boundary s, Xi represents a body force, and σij and ε_ ij stand for the stress tensor and plastic strain rate, respectively, in the kinematically admissible velocity field. It should be assumed that the geotechnical material is completely plastic and obeys the associated flow rule, and the deformation of the rock mass is negligibly small. 2.2. Nonlinear Hoek-Brown failure criterion Since Hoek and Brown established the Hoek-Brown failure criterion, it has been extensively studied and applied in geotechnical engineering [32–34]. The failure criterion is well known in two classical expressions: one for major and minor principal stress, and the other for normal and shear stress, as shown in Eq. (2): � � σ σt B τ ¼ Aσ c n (2) fA; B 2 ð0; 1Þ; σt � 0; σc � 0g

σc

where τ is shear stress; A and B are mechanical constants of the rock mass, which need to be determined by experiments; σn represents normal stress; and σt and σc indicate tensile strength and uniaxial compressive strength, respectively.

2

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Fig. 2. The numerical result: (a) numerical model of a rectangular tunnel under a karst cave; (b) the vertical displacement contour plot.

Fig. 3. Comparisons between the numerical result and upper bound solution: (a) numerical model of a circular tunnel under a karst cave; (b) results of numerical method and the method presented in this paper.

3. Limit analysis of the collapse of tunnels under a karst cave

the expressions of the lower and upper semicircular contours in the karst cave are shown by g1(x) and g2(x), respectively. Similarly, in the collapse mechanism of a circular tunnel, as illustrated in Fig. 1(b), the expres­ sions of the lower and upper semicircular contours of the karst cave can be denoted by g3(x) and g4(x), and the contour of the roof in a circular tunnel can be expressed by g5 (x). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) g1 ðxÞ ¼ ðH þ R1 Þ þ R21 x2

3.1. Collapse failure mechanism of tunnels under a karst cave Rectangular and circular tunnels are widely adopted in tunnel en­ gineering to simplify karst caves into circular ones in the twodimensional plane to simplify mathematical calculation. The karst cave lies directly above the tunnel, and the collapse mechanism of the rectangular and circular tunnels is constructed, which extends to the upper karst cave. For tunnels with the homogeneous surrounding rock, bilateral symmetry of the collapse has been demonstrated [11–13]. The outlines of the collapsing blocks of rectangular and circular tunnels can be expressed by detaching curves f1(x) and f2(x), respectively. In the collapse failure mechanism, h1 and h2 show the height of the collapsing block in the rectangular and circular tunnels, respectively. 3.2. Energy analysis In the collapse mechanism of a rectangular tunnel shown in Fig. 1(a), 3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 x2

(4)

g2 ðxÞ ¼

ðH þ R1 Þ

g3 ðxÞ ¼

� R2

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23 þ H þ R1 þ R21 x2

(5)

g4 ðxÞ ¼

� R2

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23 þ H þ R1

(6)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 x2

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the following formula: 0

γ ¼γ

(13)

γw

where γ w means the water weight per unit volume and is generally 10 kN/m3. In addition, the geometric relationships of the height of the collapsing block can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ¼ H þ R1 (14) R21 L22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 L24 þ R2

h2 ¼ H þ R1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23

(15)

In the same way, when the collapsing block extends to the top of the karst cave, the work done by gravity on the collapsing block can be expressed as Z L1 Z L2 1 0 0 0 Pγ’1 ¼ γ f1 ðxÞvdx þ γ g2 ðxÞvdx þ γ πR21 v (16) 2 L2 0 Z

L3

Pγ’2 ¼ L4

Fig. 4. Collapsing range of tunnels under karst caves with vertical seismic coefficient.

g5 ðxÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 x2

8

L3 <

PD2 ¼ L4

:

σt þ

B

(7)

9 �1 1 B = 1 � 0 vdx σ c AB⋅f 2 ðxÞ ; B

Z

L3

Z

L4

0

L4

Z

L3

0

γ g3 ðxÞvdx 0

0

γ g5 ðxÞvdx 0

L3

0

γ g5 ðxÞvdx 0

(17)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 L24 þ R2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23

(19)

Ps1 ¼ kv Pγ’1

(20)

Ps2 ¼ kv Pγ’2

(21)

where kv is the vertical seismic coefficient, which normally ranges from 0.3 to 0.3. The work done by the supporting force in rectangular and circular tunnels can be shown as [15].

(9)

(22)

Pq1 ¼ qv⋅L1 ⋅cos π Pq2 ¼ R1 qv⋅arc sin

L3 ⋅cos π R2

(23)

where q indicate the supporting force. The excess pore pressure can be obtained by Eq. (24) [23,24]: grad u ¼ γ w

(24)

rp γ

where γp denotes the pore pressure coefficient. Afterwards, the work of the seepage forces in the collapse mechanism can be expressed by Z Z L1 � γ w rp γ f1 ðxÞvdx Pu1 ¼ grad u⋅vdΩ ¼ (25) Ω

L2

Z

Z

Pu2 ¼

L3

γw

grad u⋅vdΩ ¼ Ω

L4

� rp γ f2 ðxÞvdx

(26)

Based on the virtual work principle, the dissipated energy of the collapsing block is equal to the work done by all external loads [11,12]. To obtain the analytical solutions of the collapse of tunnels under a karst cave, it is necessary to construct the objective functions of rectangular and circular tunnels, which are composed of dissipated energy and external loads [15]. Thus, the expressions of two objective functions are

0

γ f2 ðxÞvdx þ

Z

The quasi-static method is employed to consider the effect of vertical seismic force, while horizontal seismic force is perpendicular to the di­ rection of the falling velocity of the collapsing block, so it can be ignored. According to the pseudo-static method, the vertical seismic force acting on the collapsing block in the rectangular and circular tunnels can be written as [22,35–37].

When the collapsing block extends to the bottom of the upper karst cave, the expressions of the work done by gravity on the collapsing block are Z L1 Z L2 0 0 Pγ’1 ¼ γ f1 ðxÞvdx þ γ g1 ðxÞvdx (11)

Pγ’2 ¼

1 0 0 γ g4 ðxÞvdx þ γ πR21 2

0

h2 ¼ H þ R1 þ

where L1 and L2 denote the half width of the bottom and top of the collapsing block in the rectangular tunnel, respectively. L3 and L4 stand for the half width of the bottom and top of the collapsing block in the circular tunnel, separately. f1ʹ(x) and f2ʹ(x) are the first derivative of f1(x) and f2(x). The collapse of the tunnel may extend to the bottom or top of the upper karst cave, so the work done by gravity on the collapsing block needs to be analyzed in two ways. When the collapsing block extends to the bottom of the upper karst cave, and those geometric constraints can be expressed as follows: � 0 � h 1 � H þ R1 (10) H þ R1 � h1 � H þ 2R1

L2

L4

The geometric relationships of the height of the collapsing block are denoted as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (18) h1 ¼ H þ R1 þ R21 L22

where L3 is the half width of the bottom of the collapsing block in a circular tunnel; R1 and R2 represent the radius of the karst cave and the circular tunnel, respectively; and H is the vertical height between the karst cave and the tunnel. The dissipated energy along any point on the detaching curve is integrated to obtain the total dissipated energy, with reference to Ref. [11]. The expressions of the dissipated energy of the collapse in the rectangular and circular tunnels can be written as follows: 8 9 Z L1 < �1 1 B = B 1 � 0 PD1 ¼ vdx (8) σt þ σ c AB⋅f 1 ðxÞ ; B L2 : Z

Z 0

γ f2 ðxÞvdx þ

(12)

where γʹ is the buoyant weight per unit volume, which can be gotten by 4

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Fig. 5. Collapsing range of tunnels under karst caves with different parameters: (a) parameter A, (b) pore pressure coefficient rp, (c) uniaxial compressive strength σ c, (d) tensile strength σ t, (e) parameter B, (f) unit weight γ, (g) supporting force q, (h) vertical height H. 5

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functionals, and f1(x), f2(x), f1ʹ(x), and f2ʹ(x) are the functionals to be assured. The Euler equations corresponding to the functionals can be derived with the help of the variation principle. � � � � ∂ϕ1 ∂ ∂ϕ1 0 δϕ1 f1 ðxÞ; f 1 ðxÞ; x ¼ 0⇒ ¼0 (31) ∂f1 ðxÞ ∂x ∂f 01 ðxÞ � � ∂ϕ2 0 δϕ2 f2 ðxÞ; f 2 ðxÞ; x ¼ 0⇒ ∂f2 ðxÞ

�

�

∂ ∂ϕ2 ¼0 ∂x ∂f 02 ðxÞ

(32)

Substituting Eqs. (29) and (30) into Eqs. (31) and (32), the following Euler equations can be figured out: 1 1

B 1

1

B

1

0

2B 1 1 B

1

0

2B 1 1 B

σc ðABÞ1 B f 1 ðxÞ σc ðABÞ1 B f 2 ðxÞ

0

� γ 1

� 0� rp þ k v γ ¼ 0

(33)

0

� γ 1

� 0� rp þ k v γ ¼ 0

(34)

⋅ f 10 ðxÞ ⋅ f 20 ðxÞ

where f1ʹʹ(x) and f2ʹʹ(x) are the second derivative of f1(x) and f2(x), respectively. According to Eqs. (33) and (34), the preliminary expres­ sions of the first derivative of the detaching curves can be obtained: � � �1 BB 0 �1 B � 1 1 γ 1 rp þ kv γ B a 0 � f 1 ðxÞ ¼ A B ⋅ x (35) 0 B σc γ 1 rp þ kv γ

Fig. 6. Weight of collapsing block in a circular tunnel under different unit weight γ.

1 0 f 2 ðxÞ ¼ A B

1 B

� 0 �1 B � rp þ kv γ B ⋅ x

� γ 1

σc

γ 1

b �

rp þ kv γ 0

�1 BB

(36)

where a and b are constants to be defined. There is no shear stress dis­ tribution on the inner surface of the karst cave, that is to say, τ is equal to 0 on the inner surface of the karst cave.

τxz ðx ¼ L2 ; z ¼

h1 Þ ¼ τxz ðx ¼ L4 ; z ¼

h2 Þ ¼ 0

(37)

On the basis of previous studies, only if f1ʹ(L2) and f2ʹ(L4) are equal to 0 can make Eq. (37) be satisfied [13]. 0

(38)

0

f 1 ðL2 Þ ¼ f 2 ðL4 Þ ¼ 0

a and b are separately worked out by plugging Eqs. (35) and (36) into Eq. (38): � � 0� a ¼ γ 1 rp þ kv γ L2 (39)

Fig 7. Weight of collapsing block in a circular tunnel under a karst cave with different values of H.

�

0

�

ς1 f1 ðxÞ; f 1 ðxÞ; x ¼ PD1 Z

L1

¼ L2

Pγ 0 1

� � 0 ϕ1 f1 ðxÞ; f 1 ðxÞ; x vdx

� � 0 ς2 f2 ðxÞ; f 2 ðxÞ; x ¼ PD2 Z

L3

¼ L4

Pq1 Pu1 Z L2 0 ð1 þ kv Þ γ g1 ðxÞvdx

� b¼ γ 1

Ps1

qv⋅L1 ⋅cos π

(27)

Ps2

Pq2

Pu2 Z

� � 0 ϕ2 f2 ðxÞ; f 2 ðxÞ; x vdx Z

þð1 þ kv Þ

L3

0

γ g5 ðxÞvdx 0

L4

ð1 þ kv Þ

0

γ g3 ðxÞvdx 0

L3 R1 qv⋅arc sin ⋅cos π R1

1 0 f 2 ðxÞ ¼ A B

� � 0 ϕ2 f2 ðxÞ; f 2 ðxÞ; x ¼

σt þ

σt þ

B B

B B

�1 1 � 0 σ c AB⋅f 1 ðxÞ 1 B

� γ 1

� γ 1

�

� γ 1

� 0 �1 B rp þ kv γ B

σc

ðx

1 B B

L4 Þ

(42)

σc

� 0� rp þ kv γ f1 ðxÞ (29)

�1 1 � 0 σ c AB⋅f 2 ðxÞ 1 B

1 B

The expressions of the detaching curves of collapsing block in rect­ angular and circular tunnels are acquired by integrating Eqs. (41) and (42): � � 0 �1 B rp þ k v γ B 1 1 γ 1 f1 ðxÞ ¼ A B ðx L2 ÞB þ n1 (43)

(28)

where ϕ1[f1(x), f1ʹ(x),x] and ϕ2[f2(x), f2ʹ(x),x] are functionals, which are respectively represented as: � � 0 ϕ1 f1 ðxÞ; f 1 ðxÞ; x ¼

(40)

Putting Eqs. (39) and (40) into Eqs. (35) and (36), the explicit ex­ pressions of the first derivative of the detaching curves can be ascertained: � � 0 �1 B 1 B 1 B1 γ 1 rp þ kv γ B 0 ðx L2 Þ B f 1 ðxÞ ¼ A (41) B σc

0

Pγ’2

� 0� rp þ kv γ L4

f2 ðxÞ ¼ A

� rp þ kv γ f2 ðxÞ

1 B

� γ 1

� 0 �1 B rp þ k v γ B

σc

ðx

1

L4 ÞB þ n2

(44)

0

where n1 and n2 are the integral constants to be solved. From the collapse mechanism shown in Fig. 1, some geometric relationships that contribute to receiving the analytical solutions of the collapse in tunnels

(30)

The extrema of the objective functions are determined by the 6

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Fig. 8. Critical height and half width of collapsing block with different parameters: (a) vertical seismic coefficient kv, (b) parameter A, (c) parameter B, (d) unit weight γ, (e) pore pressure coefficient rp, (f) uniaxial compressive strength σ c, (g) tensile strength σ t, (h) supporting force q. 7

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should be taken seriously.

Furthermore, the expressions of the weight of the collapsing block in rectangular and circular tunnels can be gained separately, according to the combination of Eqs. (11) and (12) and Eqs. (50) and (51). � � � 0 �1 B 1þB 1 0 B1 γ 1 rp þ kv γ B 0 γA ðL1 L2 Þ B γ G1 ¼ 1þB σc � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 R2 L2 ðH þ R1 ÞL2 þ (56) R21 L22 þ 1 arc sin 2 2 R1

(45)

f1 ðL1 Þ ¼ f2 ðL3 Þ ¼ 0 f ðL2 Þ ¼

h1 ¼

� R1

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 L22 þ H

f2 ðL4 Þ ¼

h2 ¼

� R1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R21 L24 þ R2

(46) � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23 þ H

(47)

The exact expressions of integral constants n1 and n2 are figured out by bringing Eqs. (43) and (44) into Eq. (45). � � 0 �1 B 1 rp þ kv γ B 1 γ 1 B ðL1 L2 ÞB n1 ¼ h1 ¼ A (48)

� � 0 �1 B 1þB 1 0 B1 γ 1 rp þ kv γ B ðL3 L4 Þ B γA 1þB σc � � � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L4 R2 L4 0 R2 γ R22 L23 þ H þ R1 L4 þ R21 L24 þ 1 arc sin 2 2 R1 � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2 L3 R2 L3 0 þγ R22 L23 arc sin 2 2 R2 (57)

G2 ¼

σc

n2 ¼

h2 ¼

A

1 B

� 0 �1 B rp þ kv γ B

� γ 1

(49)

1

L4 ÞB

ðL3

σc

Therefore, the final expressions of the detaching curves are derived. 3 2 � � 0 �1 B 1 1 rp þ kv γ B 4 1 γ 1 B B f1 ðxÞ ¼ A B (50) ⋅ ðx L2 Þ ðL1 L2 Þ 5

Likewise, plugging Eqs. (4), (6) and (7), and Eqs. (52) and (53) into Eqs. (27) and (28) and setting the objective function to 0, we can get the equations for working out the analytic solutions when the collapsing block extends to the top of the karst cave. 8 9 > > < 1 �γ 1 r � þ k γ 0 �B1 = 1 p v ðL1 L2 ÞB σt ðL1 L2 Þ A B σc > > σ c : ;

σc

f2 ðxÞ ¼ A

1 B

2 � 0 �1 B rp þ kv γ B 4 ⋅ ðx

� γ 1

3 1 B

L4 Þ

σc

ðL3

1 B

L4 Þ 5

(51)

Eqs. (50) and (51) are taken into Eq. (45), and then the analytic expressions of the functionals can be revealed as: � � 0 �1 1 γ 1 rp þ kv γ B 1 ϕ1 ¼ A B σ c ðL1 L2 ÞB σt

� � 0 �1 1þB γ 1 rp þ kv γ B 1 1 A B σc ðL1 L2 Þ B Bþ1 σc � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 R2 L2 0 ð1 þ kv Þ⋅γ R21 L22 þ 1 arc sin ðH þ R1 ÞL2 þ 2 2 R1

σc

� 0 �1 rp þ kv γ B

� 1 B1 γ 1 A σc B 1

ϕ2 ¼ A B σ c

σc �

� γ 1

1 B1 A σc B

rp þ kv γ

σc � γ 1

0

�

ðx

1 0 ð1 þ kv Þγ πR21 2

�B1 ðL3

rp þ kv γ

(52)

1

L2 ÞB

1 B

L4 Þ

0 �1 B

σc

ðx

(58)

qL1 cos π ¼ 0

σt The expressions of the weight of the collapsing block in rectangular and circular tunnels are respectively:

(53)

1

L4 ÞB

When the collapsing block extends to the bottom of the karst cave, the equations for solving the analytic solutions can be obtained by inducing Eqs. (3), (5) and (7), and Eqs. (52) and (53) into Eqs. (27) and (28) and setting the objective function to 0.

8 > < 1 �γ 1 A B σc > : � ð1

kv Þ⋅γ

0

8 > < 1 �γ 1 A B σc > : �

�

rp þ kv γ

0

�B1

σc

ðL1

1

L2 ÞB

9 > = σ t ðL1 > ;

L2 Þ

� γ 1 1 1 A B σc Bþ1

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 R2 L2 ðH þ R1 ÞL2 þ R21 L22 þ 1 arc sin 2 2 R1 � 0 �1 rp þ kv γ B

σc

ðL3

1 B

L4 Þ

9 > = σ t ðL3 > ;

L4 Þ

� 0 �1 rp þ kv γ B

σc

kv Þ⋅γ

0

1þB B

L2 Þ

(54)

qL1 cos π ¼ 0

� γ 1 1 1 A B σc Bþ1

� 0 �1 rp þ kv γ B

σc

� � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L4 R2 L4 R22 L23 þ H þ R1 L4 þ R21 L24 þ 1 arc sin 2 2 R1 � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � L3 L3 R22 L3 0 R2 q⋅arc sin ⋅cos π þ ð1 kv Þγ ¼0 R22 L23 arc sin R2 2 2 R2

ð1

ðL1

�

R2

8

ðL3

1þB B

L4 Þ

(55)

C. Lyu et al.

Soil Dynamics and Earthquake Engineering 132 (2020) 106003

8 > < 1 �γ 1 A B σc > :

�

rp þ kv γ

0

�B1 ðL3

σc

�

1

L4 ÞB

9 > = σ t ðL3 > ;

L4 Þ

� γ 1 1 1 A B σc Bþ1

� 0 �1 rp þ kv γ B

σc

1þB B

L4 Þ

ðL3

(59)

� � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L4 R2 L4 R22 L23 þ H þ R1 L4 þ R21 L24 þ 1 arc sin 2 2 R1 � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2 L3 1 L3 R2 L3 0 0 R2 q⋅arc sin ⋅cos π ¼0 ð1 kv Þγ πR21 þ ð1 þ kv Þγ R22 L23 arc sin 2 R2 2 2 R2

ð1 þ kv Þ⋅γ

G1 ¼

�

0

1 0 γA 1þB

R2

1 B

� 0 �1 B rp þ kv γ B

� γ 1

σc

ðH þ R1 ÞL2 þ

L2 2

1þB B

L2 Þ

� qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 L2 R21 L22 þ 1 arc sin 2 R1

� 1 γ 1

�

0

Therefore, the method in this paper can be used as a convenient sup­ plementary method to investigate the collapse of rectangular tunnel under a karst cave.

� ðL1

γ

0

1 0 2 γ π R1 2

(60)

4.2. Comparison of circular tunnels

�1 BB

1þB rp þ kv γ 1 0 B γA ðL3 L4 Þ B 1þB σc � � � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L4 R2 L4 0 γ R22 L23 þ H þ R1 L4 þ R2 R21 L24 þ 1 arc sin 2 2 R1 � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � L3 R22 L3 1 0 2 0 γ π R1 þγ R22 L23 arc sin 2 2 2 R2 (61)

G2 ¼

After the same transformation as in Section 4.1, the collapse mech­ anism of the circular tunnel consistent with Fraldi [12] can be obtained without considering the seepage forces, and the same formulas (Eqs. (64) and (65)) for figuring out the height and half width of the collapsing block as Fraldi [12] are as follows: 3 2 � �1 BB γ 1 1 1 1 B 1 1 1þB 4γA B A B σ cB γB L B LB σ t 5L Bþ1 σc (64) � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � L R21 L 2 2 R1 L γ arc sin ¼0 2 R1 2

4. Comparison To validate the rationality of the method proposed in this paper, which is conducive to investigating the collapse of the rectangular and circular tunnels under a karst cave, it is necessary to compare this research with previous results and numerical solutions.

h¼A

Since there is no research on the collapse in tunnels under a karst cave, the equations in this paper need to be transformed and compared with the previous research of Fraldi [11] mentioned in Section 1. When L1 ¼ kv ¼ γ p ¼ γ w ¼ 0, the same formulas (Eqs. (62) and (63)) for calcu­ lating the height and half width of the collapsing block as Fraldi [11] can be obtained. In addition, the collapse mechanism in this paper can be turned into the collapse mechanism put forward by Yang [24] when seepage forces are taken into account; that is, let L1 ¼ kv ¼ 0. It is found that our results are completely consistent with previous research, and the studies of Fraldi [11] and Yang [24] are only special cases of this work, indicating that the proposed method in this paper for investi­ gating the collapse of the rectangular tunnel under a karst cave is rational. B

1 1 B B c t

L ¼ AB ð1 þ BÞ γ σ h¼

1þB 1 γ σt B

σ

� �1 BB γ

σc

1

LB

(65)

Similarly, a numerical model of a circular tunnel under a karst cave is established as shown in Fig. 3. Given the geometric parameters (R1 ¼ 4 m, R2 ¼ 7 m, and H ¼ 2 m) and rock mechanics parameters (A ¼ 0.22, B ¼ 0.78, σc ¼ 5.0 MPa, and σt ¼ 0.05 MPa), their corre­ sponding parameters (a ¼ 0.61, mb ¼ 0.22, s ¼ 0, σc ¼ 5.0 MPa, and GSI ¼ 8), which can be substituted into the numerical simulation and the method presented in this paper. The numerical result and the upper bound solution of this method are shown in Fig. 3(b), which shows that two methods have consistent results. On the whole, the method presented in this paper has been verified by comparing with previous studies and numerical solutions, which makes it available for analyzing the collapse of tunnels under a karst cave.

4.1. Comparison of rectangular tunnels

B

1 B

5. Results and discussion 5.1. Sensitivity analysis of parameters

(62)

The sensitivity analysis of parameters needs to be performed to predict the collapse of the rectangular and circular tunnels under a karst cave more accurately. To find out the influence of the vertical seismic coefficient on tunnel collapse, the shape of the collapsing blocks in rectangular and circular tunnels under different vertical seismic co­ efficients are drawn when given parameters. Fig. 4 indicates variations of the collapsing range in rectangular and circular tunnels with pa­ rameters corresponding to A ¼ 0.25, B ¼ 0.8, γ ¼ 25 kN/m3, rp ¼ 0.2, σ c ¼ 5 MPa, σt ¼ σc/100, D ¼ R2 ¼ 7 m, R1 ¼ 4 m, q ¼ 10 kPa, and H ¼ 2 m with the vertical seismic coefficient kv ranging from 0.3 to 0.3. As demonstrated in Fig. 4, the collapsing range of rectangular and circular tunnels decreases as the vertical seismic coefficient increases. When the most disadvantageous situation is considered, that is, kv ¼ 0.3, the collapsing range is larger than that for tunnels without

(63)

A numerical model as shown in Fig. 2 is established to simulate the collapse process of a rectangular tunnel under a karst cave by using FLAC3D software, and the geometric parameters (R1 ¼ 4 m, D ¼ 7 m, and H ¼ 2 m) are given. Moreover, the rock mechanics parameters (A ¼ 0.4, B ¼ 0.76, σc ¼ 5.0 MPa, and σt ¼ 0.05 MPa) can be converted into cor­ responding parameters (a ¼ 0.55, mb ¼ 0.69, s ¼ 0, σc ¼ 5.0 MPa, and GSI ¼ 20) in another form of the Hoek–Brown criterion, according to previous studies [34,38]. Fig. 2(b) shows the vertical displacement contour plot of the rectangular tunnel under a upper karst cave, revealing that it is consistent with the shape of the collapsing block [11]. 9

C. Lyu et al.

Soil Dynamics and Earthquake Engineering 132 (2020) 106003

with parameters corresponding to A ¼ 0.25, B ¼ 0.8, γ ¼ 25 kN/m3, rp ¼ 0.2, σc ¼ 5 MPa, σc ¼ σt/100, D ¼ R2 ¼ 7 m, and q ¼ 10 kPa. The in­ fluence of vertical seismic force on the collapsing range is again confirmed, which reveals that the greater the vertical seismic force, the worse the safety of tunnels. The half width of collapsing blocks in rectangular and circular tun­ nels increases with increased values of A, rp, σ c, and σt when other pa­ rameters are the same as those described in Fig. 8(a). Notice that the critical height of collapsing blocks in rectangular and circular tunnels increases with increased values of rp, σc, and σ t; however, with increased values of A, the critical height increases in circular tunnels but remains unchanged in rectangular tunnels, as shown in Fig. 8(b), (e), (g) and (h). Furthermore, increased values of B, γ, and q, when A ¼ 0.35 and other parameters are the same as those described in Fig. 6(a) can lead to a decrease in the collapsing range, including the half width and critical height. In brief, the existence of karst caves makes tunnel roofs more vulnerable to occured the collapse failure by reason of construction disturbance, and reducing the vertical height between the karst cave and the tunnel will be detrimental to the stability of tunnel roofs.

considering seismic force, illustrating that the role of seismic force should be paied attention to the study of tunnel collapse. Fig. 5(a), (b), (c) and (d) reveal the variation of collapsing range in the rectangular tunnel with parameter A varying from 0.1 to 0.8 and the circular tunnel with parameter A varying from 0.15 to 0.35, and other rock parameters corresponding to rp ¼ 0–0.35, σ c ¼ 3.5–5.5 MPa,σ t ¼ σ c/ 90 - σc/130, and kv ¼ 0.2, while other parameters remain the same as those in Fig. 4. When values of A, rp, σc, and σ t increase, the collapsing range of the rectangular and circular tunnels increases continuously. The increase of values of A, σ c, and σ t will result in an increase of the dissipated energy along the boundary of the potential collapsing block, which requires a larger collapsing range to make the work rate by external force and the dissipated energy equal. The increased values of rp also requires an increased collapsing range to make the dissipated en­ ergy equal to the total work rate by external force. Fig. 5(e), (f) and (g) show the changes of collapsing range in rect­ angular and circular tunnels with different values of B, γ, and q when kv ¼ 0.2 and A ¼ 0.35 and other parameters corresponding with those in Fig. 4. It can be found that the collapsing range of rectangular and cir­ cular tunnels decreases with increased values of B, γ, and q.Because increased values of B leads to increased the dissipated energy and increased values of γ and q bring about an increased in the work rate generated by external forces, requiring a larger collapsing range to ensure that the dissipated energy and the work rate produced by external forces always conform to Eqs. (54) and (55). What needs special attention is that although the collapsing range decreases with increased values of γ, the changing law of the weight of the collapsing block is uncertain due to that increase. Therefore, the variation of the collapsing weight in rectangular and circular tunnels with the per unit weight γ in Fig. 6 is drawn according to Eqs. (56) and (57). It is remarkable that the collapsing weight of rectangular tunnels decreases with increased values of γ, while that of circular tunnels increases. Fig. 5(h) illuminates the influence of the distance between the tunnel and the karst cave on the collapsing range in karst tunnels when H changes from 1 m to 3 m, A ¼ 0.35, and kv ¼ 0.2, and other parameters are consistent with those of Fig. 4. According to Fig. 5(h), the collapsing range of rectangular tunnels increases as a result of the increased values of H. The height of the collapsing block increases and the top half width decreases with increased values of H, while the bottom half width de­ creases first and then increases, which leads to a changing law of the real collapsing range that cannot be directly concluded. Thus, Fig. 7 shows the changing law of the weight of the collapsing block in a circular tunnel with increased values of H. Increasing the value of H changes the weight of the collapsing block, which clearly demonstrates the role of H in the collapse of circular tunnels.

6. Conclusion The upper bound theorem of limit analysis was performed to inves­ tigating the collapse failure mechanism of the rectangular and circular tunnels under a karst cave in this paper. As a result, the collapse failure mechanism that extends from the rectangular and circular tunnels to the upper karst cave is established. Then the analytical expressions of the detaching curves of the collapsing block and the equations for working out the collapsing range and weidht in rectangular and circular tunnels taking into account seismic forces are derived by adopting the nonlinear Hoek-Brown failure criterion and the variational principle. The results of this paper are transformed and compared with previous studies and numerical solutions, providing evidence for the rationality of the method for investigating the collapse of the tunnels under a karst cave. By analyzing the influence of variable parameters on the collapsing range in rectangular and circular tunnels under a karst cave, it is found that increased values of A, rp, σ c, σ t, and H result in an increased collapsing range, whereas increased values of B, γ, and q result in a decreased collapsing range. It is especially noted that the collapsing range in rectangular and circular tunnels under a karst cave decreases with the vertical seismic coefficient kv ranging from 0.3 to 0.3, which suggests that the effect of the vertical seismic force on the collapse must be attended to. The formulas for the critical height in rectangular and circular tunnels are separately given when L2 and L4 are reduced to zero, and the results show that an increased collapsing range results from increased values of A, rp, σ c, and σt and decreased values of kv, B, γ, and q. This work can provide guidance for the design and construction of tunnels under a karst cave.

5.2. Critical height We determined the critical height between the tunnel and the upper karst cave and its calculation formulas based on the upper bound the­ orem, which plays a dominant role in the design and construction of tunnels under a karst cave. According to Fig. 7, it is found that critical height Hcr1 and Hcr2, which consider vertical seismic force and seepage force, are the height when L2 and L4, respectively, are reduced to zero. � � 0 �1 B rp þ kv γ B B1 1 γ 1 Hcr1 ¼ A B L1 (66)

Declaration of competing interest None. Acknowledgments This study was supported by the China Railway Corporation Science and Technology Research and Development Program (2017G007-G-2). Their financial support is greatly appreciated.

σc

Hcr2 ¼ A

1 B

� γ 1

� 0 �1 B rp þ k v γ B

σc

1

L3B

R2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 L23

(67)

Appendix A. Supplementary data

We can solve the collapsing range of rectangular and circular tunnels according to Eqs. (54) and (55) separately, when L1 ¼ L3 ¼ 0, and then draw the collapsing range with different parameters. The critical height and half width of the collapsing block monotonically decrease with an increased vertical seismic coefficient kv, which is shown in Fig. 8(a),

Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.106003.

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Soil Dynamics and Earthquake Engineering 132 (2020) 106003

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