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Mechanical model of control of key strata in deep mining
Mining Science and Technology (China) 21 (2011) 267e272
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Mining Science and Technology (China) journal homepage: www.elsevier.com/locate/mstc
Mechanical model of control of key strata in deep mining Pu Hai a, b, *, Zhang Jian a, b a b
State Key Laboratory of Geomechanics and Deep Underground Engineering, Xuzhou 221008, China School of Mechanics and Civil Engineering, China University of Mining & Technology, Xuzhou 221116, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 August 2010 Received in revised form 10 October 2010 Accepted 15 November 2010
Based on the characteristic of deep rock layers and the theory of key strata, we analysed elastic mechanical characteristics of key strata by using elastic plate theory. The results show that the deformation and distribution of internal forces of key strata vary with different mine boundary conditions. The boundary values of key strata with three point boundaries and one fixed boundary is greater than that with four fixed boundaries. Considering the rheology of key strata under low stress conditions, we selected a generalized Kelvin model to analyse the rheology characteristics of the key strata and discovered their instantaneous elastic phases. The rate of deformation decreased over time to the point where the key strata reached stability. But over this time, the effect on deformation became very clear. For high stress conditions, we chose a Burgers model and found deformation of key strata in the form of attenuation and steady-state creep and although the rate of deformation remained constant, secondary creep was obvious, causing instability in the system. As well, we analysed the effect of creep buckling and derived a relation between buckling force and time. Copyright Ó 2011, China University of Mining & Technology. All rights reserved.
Keywords: Deep mining Key strata Rheology Creep buckling
1. Introduction With the development of our economy, the demand for energy keeps increasing. However, shallow resources are dwindling and coal mines in both China and abroad have to get used to deep mining. In China, the depth of mines increases annually at a rate of 8e12 m [1,2]. The number of disasters in coal mines continues to grow because of their increasing mining depth. Disasters such as rock bursts, gas explosions, aggravated strata behaviour and the deformation of rock surrounding roadways severely threaten the efficiency and security of exploitation. Therefore, it is necessary to study the mechanical problems caused by deep mining [3]. We refer to deep areas as those areas where nonlinear mechanical phenomena of engineering rock mass begin their downward effect. Projects at this depth fall in the realm of deep engineering [4]. Deep rock masses show some particular mechanical properties, such as brittleeductile transition or strong rheology, because of their complex geophysical environment and stress fields. For deep mining, it is urgent to solve problems of controlling deep strata, because damage to the rocks surrounding roadways, is compounded when the depth increases and rock rheology becomes more obvious. Currently, many investigators are engaged in research
* Corresponding author. State Key Laboratory of Geomechanics and Deep Underground Engineering, Xuzhou 221008, China. Tel.: þ86 516 83885205. E-mail address: [email protected] (P. Hai).
related to these problems. Ju applied a DDA (Discontinuous Deformation Analysis) method to simulate and analysed the stress distribution and deformation of rock strata during excavation of coal mines [5]. Wang developed a creep equation of a soft rock tunnel surrounded by rock, subject to high ground stresses in a deep mine, the data for which was obtained from creep experiments [6]. Dai analysed the effect of the size of face-separated pillars on ground movements in geological conditions by material similarity tests in a Jiaoping Mine. The results achieved so far, showing that faceseparated pillars with a definite width can effectively control ground subsidence [7]. Xu et al. studied the rules and characteristics of subsidence caused by deep coal mining. Their results show that the subsidence movement of overburden layers is characterized by consistent and uniform compression, displacement and deformation. The behaviour of surface subsidence is such that surface deformation is continuous and slows down in a long period of movement. In the end, they proposed a prediction method with amendable probability to be integrated in deep coal mining projects [8]. Yin et al. studied the deformation behaviour of covered rock strata in deep mines or inclined seams and developed a mechanical model for the deformation of covered rock strata. Their results can be used to provide scientific basis in the calculations carried out for covered rock strata in deep mining of steep or inclined seams [9]. Bai et al. studied the stability of rocks surrounding deep roadways. They pointed to methods to control rocks surrounding deep roadways which improve rock strength, transfer the high stresses of surrounding rocks and adopt rational supporting technology [10].
1674-5264/$ e see front matter Copyright Ó 2011, China University of Mining & Technology. All rights reserved. doi:10.1016/j.mstc.2011.02.014
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The research team led by Qian and Miao has come up with key strata theory in ground control [11e15]. In this theory, the processes of rock movement from the bottom-up are to be studied as a complete entity, such as rock pressure, fracture development, ground subsidence etc. This can provide an important theoretical basis for mining engineering, especially in strata control. However, this powerful tool is generally limited to elastic analysis and not much useful to study time-dependent effects. Based on key strata theory, we developed a key mechanical stratum model for deep mining combined with rheological constitutive equations, taking into account the special situation of deep mining. 2. Rheological constitutive relations and the corresponding elasticerheological principle With the increase in mining depth, the load on key strata gradually changes. The effect of the same load varies with depth. At different depths and affected by stress conditions, the rules of rock rheology vary. Rock rheology is attenuation creep when the load on key strata becomes low. After some time, it reaches stability. At this stage, we can select a generalized Kelvin model to describe the constitutive relations of the key strata. The following equations are 1: a differential constitutive equation
s þ p1 s_ ¼ q0 3 þ q1 3_
3. Mechanical rheological model of key strata We have simplified the key strata by assuming them to be plate structures without the impact of a support medium. We have taken four fixed boundaries in consideration of the boundary conditions of the plates. 3.1. Temporal correlation of key strata deflection 3.1.1. Temporal correlation of key strata deflection under conditions of low stress Applying a generalized Kelvin model and the corresponding elasticerheological principle, a Laplace transformation of the key strata is given as:
wðs; x; yÞ ¼
s0 E2
þ
s0 E1
wðx; y; tÞ ¼
1 þ cos
b
4qa4 b4
p4 3a4 þ 3b4 þ 2a2 b2
12 1 3 þ h3 2q0 2ð6K þ q0 Þ
q0 t p1 1 p1 e q1 þ 2q1 2q0 6p1 K þ q1 6K þ q0 t px 3 1 þ cos e 6p1 K þ q1 2ð6K þ q0 Þ a py 1 þ cos b
(6)
If we let t ¼ 0, the instantaneous elastic deflection is obtained as:
E1
w0 ¼
Rock rheology is constant in velocity when the load on the key strata is high. We can choose a Burgers model to describe the rheological properties, i.e.: A differential constitutive equation:
s þ p1 s_ þ p2 s€ ¼ q1 3_ þ q2€3
s0 E1
þ
t
h2
þ
s0 E3
1 et=s1
(4)
4qa4 b4
12 p1 p1 þ h3 2q1 6p1 K þ q1
p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
a
1 þ cos
b
wN ¼
4qa4 b4
12 1 3 þ h3 2q0 2ð6K þ q0 Þ
p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
a
1 þ cos
b
where
h3 E3
Except for these two constitutive relations, there are no differences in boundary value problems between rheological and elastic bodies. The elastic stressestrain relation is just a special form of the rheology constitutive equation. The linear boundary value problem of a rheological body corresponds to an elastic problem in a Laplace transformation space, and is referred to as the corresponding elasticerheological principle. The relation of the rheological boundary value problem with time can be developed from the inversion of the Laplace transformation after obtaining the Laplace space solution.
(7)
If we let t ¼ N, the steady-state elastic deflection is obtained as:
(3)
where p1 ¼ (h2/E1) þ (h2 þ h3/E3), p2 ¼ h2h3/E1E3, q1 ¼ h2, q2 ¼ h2h3/ E3 and a creep equation:
s1 ¼
(5)
(2)
h1
3ðtÞ ¼
a
þ
where
s1 ¼
Substituting qðsÞ ¼ q0 =s and EðsÞ, mðsÞ into Eq. (5) and inverting the Laplace transformation, we have:
(1)
1 et=s1
12 1 mðsÞ2
EðsÞh3 p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
where p1 ¼ h1/E1 þ E2, q0 ¼ E1E2/E1 þ E2, q1 ¼ E1h1/E1 þ E2 and 2: a creep equation:
3ðtÞ ¼
4qðsÞa4 b4
Fig. 1. Key strata deflection as a function of time.
(8)
P. Hai, Z. Jian / Mining Science and Technology (China) 21 (2011) 267e272
It can be seen in Fig. 1 that the key strata deflection has an instantaneous elastic phase at the start. The rate of deformation decreases as a function of time; after time tc, the deformation reaches stability. As shown in Fig. 2, curves of key strata deflect at midpoint over time, when the viscosity coefficient h1 is 200, 300, 400 or 500 GPa d. Fig. 2 shows that the flow strain rate decreases as the viscosity coefficient increases. It is seen that the deflection of rocks with higher rheological coefficients requires more time to reach steady state. But the viscosity coefficient has little effect once rheology has reached this steady state. In the end, all deflections will have the same steady value. As well, the modulus of elasticity, E2, of an elastic body has a great effect on the instantaneous elastic deflection, but little effect on the changes of the deflection as a function of time in this constitutive model. Instead, the modulus of elasticity, E1, of a Kelvin body only affects the changes in deflection. At the start, deflections are almost the same: the smaller E1, the more obvious the rheology and the longer the steady time, the larger the steady-state rheology value.
269
Fig. 2. Key strata deflections at midpoint as a function of time.
3.1.2. Temporal correlation of key strata deflection under conditions of high stress The solution of a fixed key stratum deflection in a Laplace space, applying a Burgers model, describing a constant velocity rheology, is:
wðs;x;yÞ ¼
4q0 a4 b4
p4 3a4 þ3b4 þ2a2 b2
12 1 1þp1 sþp2 s2 h3 2s q1 sþq2 s2 s2
Fig. 3. Variation of key stratum deflection as a function of time in high stress.
px
3 1þp1 sþp2 1þcos 2s ð6p2 K þq2 Þs2 þð6Kp1 þq1 Þsþ6K a py 1þcos b þ
(9)
Based on the inversion of the Laplace transformation, Eq. (9) can be written as:
wðx; y; tÞ ¼
4qa4 b4
p4 3a4 þ 3b4 þ 2a2 b2 py
1 þ cos where
AðtÞ ¼
b
px 12 AðtÞ 1 þ cos 3 a h
We see that the key stratum deflection has an instantaneous elastic phase. The rate of deformation decreases over time; after tc, the speed in rheology remains constant and deflection increases linearly, as shown in Fig. 3. 3.2. Temporal correlation of internal key strata forces
(10)
3.2.1. Temporal correlation of internal key strata forces under low stress conditions Based on its corresponding elasticerheological principle, the Laplace transformation of My, which is a bending moment of the fixed key stratum, is written as:
! q1 t p1 q t p1 q p 1 22 þ 22 2 e q2 þ 2q1 2q1 2q1 2q1 2q1 2q2 4K 2 3 144q2 p1 p2 K 3 þ24p1 q22 þ24q1 q22 K 2 þ4q1 q22 K 5 412q1 p2 K þ2q1 q2 2 24p2 K 2 þ4Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þq1 K sin 2 12p2 K þ2q2 ffi e 12K p2 þ2Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 144p2 K þ24Kq2 36p21 K 2 12p1 q1 K q21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þq1 K cos 2 12p2 K þ2q2 q2 e 12K p2 þ2Kq2 24p2 K 2 þ4Kq2
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P. Hai, Z. Jian / Mining Science and Technology (China) 21 (2011) 267e272
Fig. 4. Key stratum moment at midpoint over time under low stress conditions.
M y ðs; x; yÞ ¼
p4
p2 px py cos 1 þ cos a b a2 3a4 þ 3b4 þ 2a2 b2 4qðsÞa4 b4
Fig. 5. Key strata moment at midpoint over time under high stress condition.
My ðs; x; yÞ ¼
px py 1 þ cos cos a b
b2
4q0 a4 b4
p4 3a4 þ 3b4 þ 2a2 b2
p2 a2
(11) From the inversion of the Laplace transformation, Eq. (11), it follows that:
px py 4q0 1þcos My ðx;y;tÞ ¼ cos 2 a b p4 3a4 þ3b4 þ2a2 b2 b 1 3q0 3q0 3q1 þ 2 2ð6K þq0 Þ 2ð6K þq0 Þ 2ð6Kp1 þq0 Þ 6K þq0 t p2 px py cos ð12Þ 1þcos e 6Kp1 þq1 a b a2 a4 b4
px
cos
a
1 þ cos
py b
p2 px py mðsÞ 2 1 þ cos cos a b b
p2
mðsÞ
p2
For example, let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m and the uniform load on the key stratum q ¼ 10 MPa then, according to the creep test data, Fig. 4 shows that the curve of the key strata deflects at the midpoint over time when E1 ¼ 60 GPa, h1 ¼ 600 GPa d and E2 ¼ 20 GPa.
(13) where
mðsÞ ¼
1 3K 1 þ p1 s þ p2 s2 q1 s þ q2 s2 s 6K 1 þ p1 s þ p2 s2 þ q1 s þ q2 s2
Based on the inversion of the Laplace transformation, Eq. (13) can be written as:
p2 px py cos 1 þ cos 2 a b a p4 3a4 þ 3b4 þ 2a2 b2 p2 px py cos BðtÞ 2 1 þ cos a b b
My ðx; y; tÞ ¼
4q0 a4 b4
(14) where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ 24Kq2 36p21 K 2 12p1 q1 K q21
6K 2 þ q1 K cos 1 2 p þ 2Kq 12K 2 2 BðtÞ ¼ 3q2 e 2
12p2 K þ 2q2 12p2 K þ 2q2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ 24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þ q1 K sin 12q2 p12 K þ 3q2 q1 2 12p2 K þ 2q2 ffi e 12K p2 þ 2Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3q1 12p2 K þ 2q2 2 144p2 K þ 24Kq2 36p21 K 2 12p1 q1 K q21
From Fig. 4, we see that the moment exhibits an exponential attenuation distribution over time with an instantaneous elastic phase. The instantaneous value of the moment is 17.2 MN. It reaches a steady state after 20 days at a stable value of 22.3 MN, 24 per cent higher than its instantaneous value. Hence, the bending moment also has a time-dependent effect. 3.2.2. Temporal correlation of internal key strata forces under conditions of high stress The Laplace transformation of My which is the bending moment of the fixed key strata, is written as:
Let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m and the uniform load on the key strata q ¼ 5 MPa. According to the creep test, Fig. 5 shows that the key stratum deflection at the midpoint over time, when E1 ¼ 20 GPa is h2 ¼ 600 GPa d and when E3 ¼ 60 GPa, h3 ¼ 300 GPa d. We see that, in Fig. 5, the distribution of the bending moment over time is very different from the deflection. The moment exhibits an exponential decay function over time and becomes steady after 150 days. The bending moment varies over a large range and doubles from the start to its steady state.
P. Hai, Z. Jian / Mining Science and Technology (China) 21 (2011) 267e272
271
N x
a b
N
Fig. 7. Model of creep buckling of key strata.
Fig. 6. Fracture distance of first break over time under high stress conditions.
4
DV w
3.3. Temporal correlation of first break of key strata Assuming that the key strata are four fixed boundaries, we have:
Mmax
Assuming that a key stratum is a simple four-edged supported rectangle, without the effect of shear stress on its boundaries, the model shown in Fig. 7 is created as follows: The key buckling stratum equation is:
2 p 1 3q0 2 þ2 ¼ 2 2ð6K þ q0 Þ b2 p4 3a4 þ 3b4 þ 2a2 b2 4qa4 b4
Setting the long-term strength of key strata rocks as sf, we obtain:
2 sf h2 p p2 2 þ 2BðtÞ ¼ 2 2 6 b a 3a4 þ 3b4 þ 2a2 b2
The fracture distance from the first break is written as:
L0 ðtÞ ¼
3
16qBðtÞa2 sf h2 p2
¼ 0
(17)
Amn sin
mpx mpy sin a b
(18)
Combining the boundary conditions and Nx ¼ Ny ¼ N in this problem, we have:
Nc ¼
p2 D a2
m2 þ n2
a2 b2
(19)
According to its corresponding elasticerheological principle and assuming m(t) ¼ m0 (where m0 is a constant), the Laplace transformation of a critical buckling load is:
4qa4 b4
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi u 4 2 2 2 u 2t 72qa 3sf h p a 6 CðtÞ
v2 y
m¼1 n¼1
(15)
p4
þ Ny
N X N X
wðx; yÞ ¼
6K þ q0 t 2 p 3q0 3q1 e 6Kp1 þ q1 þ 2ð6K þ q0 Þ 2ð6Kp1 þ q0 Þ a2
v2 x
! v2 w
where Nx ¼ sxh, Ny ¼ syh Assuming the surface is unstable, the deflection expression is:
Nx
v2 w
N ðsÞ ¼ c
(16)
where C(t) ¼ 144q2a8 12qsfh2p2a6 2s2p4h4a4 þ 36B(t)qsfh2p2a6 Let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m, the uniform load on key strata q ¼ 5 MPa and the compressive strength 15 MPa. Then, according to the creep test, Fig. 6 shows that the fracture distance of the first break over time, when E1 ¼ 20 GPa, is h2 ¼ 600 GPa d and when E3 ¼ 60 GPa, h3 ¼ 300 GPa d. Fig. 6 shows that the fracture distance of the first break decays over time and reaches a steady state after 150 days. The rock strength decreases and is just 70 per cent of its elastic strength. Although in general the change is very small, we cannot ignore that the fracture distance is affected by rheology. For deep mining, we must consider the rheological effect on fracture distance because it has a strong time-dependent effect. 4. Creep buckling of key strata In deep mining, ground stress is quite high and transverse loads above the goaf are much lower than ground stress. However, the coal wall surrounding the goaf is a high stress concentration zone. Under the effect of high stress, the coal walls of these key strata are in a state of buckling, causing instability and bending of the goaf. It is also an important reason for breaks in key strata.
p2 2G ðsÞ a2 m2 þ n2 2 2 m a 1 0 b
(20)
Based on the inversion of the Laplace transformation, Eq. (20) can be written as:
Nc ðtÞ ¼
p2 a2
3 a2 2 t q1 7 b2 6 q0 e p1 5 4q0 þ 1 m0 p1
m 2 þ n2
(21)
Setting E1 ¼ 60 GPa, h1 ¼ 300 GPa d, E2 ¼ 20 GPa, Eq. (21) yields:
a2 2
3
2 2 4t p2 m þ n b2 4 Nc ðtÞ ¼ 2 15 þ 5e 15 5 1 m0 a
(22)
Fig. 8 shows the buckling load of the plate as a function of time: From Fig. 8, we can derive three cases about plate rheological buckling caused by different loads. 1) When the horizontal load Nc is higher than the instantaneous critical elastic load Nc0, the plate will immediately become unstable because of buckling. This is the same as an elastic situation. 2) When the horizontal load Nc is below the instantaneous critical elastic load Nc0 but higher than the long-term steady load NcN, the plate will lose stability after tc by reason of its material rheological properties. This is a case of delayed instability. 3) When the horizontal load Nc is below the long-term steady load NcN, the plate will not lose stability owing to damage.
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P. Hai, Z. Jian / Mining Science and Technology (China) 21 (2011) 267e272
instability due to buckling sets in immediately; second, when the horizontal load Nc is below the instantaneous critical elastic load Nc0 but higher than the long-term steady load NcN, instability of the plate will be delayed, caused by the rheological properties of the material and third, when the horizontal load Nc is below the long-term steady load NcN, the plate will not lose stability due to damage. These three forms of instability need to be considered, especially delayed instability, when the key strata are in a precarious mechanical condition. Acknowledgements
Fig. 8. Buckling load as a function of time.
5. Conclusions For deep mining, the behaviour of strata surrounding a stope appears severe. Reasonable explanations are provided by the key stratum theory of stratum behaviour and movement. We introduced rheological properties of deep rock masses. According to the principle of correspondence, we analysed these characteristics, selected a generalized Kelvin model under conditions of low stress and a Burgers model under high stress conditions. Key strata are in a state of buckling instability because of the squeezing action of the rock surrounding them. In considering of this condition, we carried out rheological mechanical analyses. Based on our results, we draw the following conclusions: 1) Considering the rheology of a deep rock mass under low stress conditions, we selected a generalized Kelvin model to analyse the rheology characteristics of key strata and found that the rate of rheological strains of key internal strata forces and deflection shows exponential decay as a function of time, which became steady after 20 days. Both deflection and bending moment are time-dependent. The stability value is 20e30 per cent higher than the instantaneous elastic value. 2) When the load on key strata is high, the deformation of key strata has two phases: attenuation rheology and constant velocity rheology. Attenuation rheology lasts only a short time, but the constant velocity rheology endures. In the constant velocity rheology phase, rheology causes considerable damage to key strata. However, the corresponding bending moment shows an attenuated trend. The period of stability is relatively long and the extent of the damage quite large. 3) We analysed the instability of key strata caused by buckling and obtained the critical buckling load, given the squeezing action of the surrounding rock to the key strata. The results show that the critical load is basically unchanged when the length of the face is much longer than the advance distance; the critical load is largest when the goaf is square. 4) Given that rheology is a fact of life in deep mines, we considered three forms of creep buckling: first, once a plate is loaded,
This project was supported by the National Natural Science Foundation of China (No. 50904065) and the Program for New Century Excellent Talents in University (No. NCET-09-0728). As well, this project was sponsored by the Qinglan Project and the Fundamental Research Funds for the Central Universities (China University of Mining and Technology). References [1] He MC, Xie HP, Peng SP, Jiang YD. Study on rock mechanics in deep mining engineering. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2803e13 [in Chinese]. [2] Sun XM, He MC. Numerical simulation research on coupling support theory of roadway within soft rock at depth. Journal of China University of Mining & Technology 2005;34(2):166e9 [in Chinese]. [3] He YN, Han LJ, Shao P, Jiang BS. Some problems of rock mechanics for roadways stability in depth. Journal of China University of Mining & Technology 2006;35(3):288e95 [in Chinese]. [4] He MC. Conception system and evaluation indexes for deep engineering. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2854e8 [in Chinese]. [5] Ju Y, Zuo JP, Song ZD, Tian LL, Zhou HW. Numerical simulation of stress distribution and displacement of rock strata of coal mines by means of DDA method. Chinese Journal of Geotechnical Engineering 2007;29(2):268e73 [in Chinese]. [6] Wang YY, Wei J, Qi J, Yang CH, Li JG. Study on prediction for nonlinear creep deformation of deep rocks. Journal of China Coal Society 2005;30(4):409e13 [in Chinese]. [7] Dai HY, Wang SB, Yi SH, Kong LY, Gao YB. Influential laws on strata and ground movement of face-separated pillars at a great depth. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2929e33 [in Chinese]. [8] Xu NZ, Wang B, Qi YC. Prediction of surface subsidence in the deep coal mining. Journal of Mining & Safety Engineering 2006;23(1):66e9 [in Chinese]. [9] Yin GZ, Wang DK, Zhang WZ. Mechanics model to deformation of covered rock strata and its application in deep mining of steep or inclined seam. Journal of Chongqing University 2006;29(2):79e82 [in Chinese]. [10] Bai JB, Hou CJ. Control principle of surrounding rocks in deep roadway and its application. Journal of China University of Mining & Technology 2006;35(2):145e8 [in Chinese]. [11] Qian MG, Miao XX, Xu JL, Mao XB. Key strata theory in ground control. Xuzhou: China University of Mining and Technology Press; 2003 [in Chinese]. [12] Qian MG, Miao XX, Xu JL. Theoretical study of key stratum in ground control. Journal of China Coal Society 1996;21(3):225e30 [in Chinese]. [13] Qian MG, Mao XB, Miao XX. Variation of loads on the key layer of the overlying strata above the workings. Journal of China Coal Society 1998;23(2):135e9 [in Chinese]. [14] Mao XB, Miao XX, Qian MG. Study on broken laws of key strata in mining overlying strata. Journal of China University of Mining & Technology 1998;27(1):39e42 [in Chinese]. [15] Miao XX, Qian MG. Advance in the key strata theory of mining rockmass. Journal of China University of Mining & Technology 2000;29(1):25e9 [in Chinese].
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Mining Science and Technology (China) journal homepage: www.elsevier.com/locate/mstc
Mechanical model of control of key strata in deep mining Pu Hai a, b, *, Zhang Jian a, b a b
State Key Laboratory of Geomechanics and Deep Underground Engineering, Xuzhou 221008, China School of Mechanics and Civil Engineering, China University of Mining & Technology, Xuzhou 221116, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 August 2010 Received in revised form 10 October 2010 Accepted 15 November 2010
Based on the characteristic of deep rock layers and the theory of key strata, we analysed elastic mechanical characteristics of key strata by using elastic plate theory. The results show that the deformation and distribution of internal forces of key strata vary with different mine boundary conditions. The boundary values of key strata with three point boundaries and one fixed boundary is greater than that with four fixed boundaries. Considering the rheology of key strata under low stress conditions, we selected a generalized Kelvin model to analyse the rheology characteristics of the key strata and discovered their instantaneous elastic phases. The rate of deformation decreased over time to the point where the key strata reached stability. But over this time, the effect on deformation became very clear. For high stress conditions, we chose a Burgers model and found deformation of key strata in the form of attenuation and steady-state creep and although the rate of deformation remained constant, secondary creep was obvious, causing instability in the system. As well, we analysed the effect of creep buckling and derived a relation between buckling force and time. Copyright Ó 2011, China University of Mining & Technology. All rights reserved.
Keywords: Deep mining Key strata Rheology Creep buckling
1. Introduction With the development of our economy, the demand for energy keeps increasing. However, shallow resources are dwindling and coal mines in both China and abroad have to get used to deep mining. In China, the depth of mines increases annually at a rate of 8e12 m [1,2]. The number of disasters in coal mines continues to grow because of their increasing mining depth. Disasters such as rock bursts, gas explosions, aggravated strata behaviour and the deformation of rock surrounding roadways severely threaten the efficiency and security of exploitation. Therefore, it is necessary to study the mechanical problems caused by deep mining [3]. We refer to deep areas as those areas where nonlinear mechanical phenomena of engineering rock mass begin their downward effect. Projects at this depth fall in the realm of deep engineering [4]. Deep rock masses show some particular mechanical properties, such as brittleeductile transition or strong rheology, because of their complex geophysical environment and stress fields. For deep mining, it is urgent to solve problems of controlling deep strata, because damage to the rocks surrounding roadways, is compounded when the depth increases and rock rheology becomes more obvious. Currently, many investigators are engaged in research
* Corresponding author. State Key Laboratory of Geomechanics and Deep Underground Engineering, Xuzhou 221008, China. Tel.: þ86 516 83885205. E-mail address: [email protected] (P. Hai).
related to these problems. Ju applied a DDA (Discontinuous Deformation Analysis) method to simulate and analysed the stress distribution and deformation of rock strata during excavation of coal mines [5]. Wang developed a creep equation of a soft rock tunnel surrounded by rock, subject to high ground stresses in a deep mine, the data for which was obtained from creep experiments [6]. Dai analysed the effect of the size of face-separated pillars on ground movements in geological conditions by material similarity tests in a Jiaoping Mine. The results achieved so far, showing that faceseparated pillars with a definite width can effectively control ground subsidence [7]. Xu et al. studied the rules and characteristics of subsidence caused by deep coal mining. Their results show that the subsidence movement of overburden layers is characterized by consistent and uniform compression, displacement and deformation. The behaviour of surface subsidence is such that surface deformation is continuous and slows down in a long period of movement. In the end, they proposed a prediction method with amendable probability to be integrated in deep coal mining projects [8]. Yin et al. studied the deformation behaviour of covered rock strata in deep mines or inclined seams and developed a mechanical model for the deformation of covered rock strata. Their results can be used to provide scientific basis in the calculations carried out for covered rock strata in deep mining of steep or inclined seams [9]. Bai et al. studied the stability of rocks surrounding deep roadways. They pointed to methods to control rocks surrounding deep roadways which improve rock strength, transfer the high stresses of surrounding rocks and adopt rational supporting technology [10].
1674-5264/$ e see front matter Copyright Ó 2011, China University of Mining & Technology. All rights reserved. doi:10.1016/j.mstc.2011.02.014
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The research team led by Qian and Miao has come up with key strata theory in ground control [11e15]. In this theory, the processes of rock movement from the bottom-up are to be studied as a complete entity, such as rock pressure, fracture development, ground subsidence etc. This can provide an important theoretical basis for mining engineering, especially in strata control. However, this powerful tool is generally limited to elastic analysis and not much useful to study time-dependent effects. Based on key strata theory, we developed a key mechanical stratum model for deep mining combined with rheological constitutive equations, taking into account the special situation of deep mining. 2. Rheological constitutive relations and the corresponding elasticerheological principle With the increase in mining depth, the load on key strata gradually changes. The effect of the same load varies with depth. At different depths and affected by stress conditions, the rules of rock rheology vary. Rock rheology is attenuation creep when the load on key strata becomes low. After some time, it reaches stability. At this stage, we can select a generalized Kelvin model to describe the constitutive relations of the key strata. The following equations are 1: a differential constitutive equation
s þ p1 s_ ¼ q0 3 þ q1 3_
3. Mechanical rheological model of key strata We have simplified the key strata by assuming them to be plate structures without the impact of a support medium. We have taken four fixed boundaries in consideration of the boundary conditions of the plates. 3.1. Temporal correlation of key strata deflection 3.1.1. Temporal correlation of key strata deflection under conditions of low stress Applying a generalized Kelvin model and the corresponding elasticerheological principle, a Laplace transformation of the key strata is given as:
wðs; x; yÞ ¼
s0 E2
þ
s0 E1
wðx; y; tÞ ¼
1 þ cos
b
4qa4 b4
p4 3a4 þ 3b4 þ 2a2 b2
12 1 3 þ h3 2q0 2ð6K þ q0 Þ
q0 t p1 1 p1 e q1 þ 2q1 2q0 6p1 K þ q1 6K þ q0 t px 3 1 þ cos e 6p1 K þ q1 2ð6K þ q0 Þ a py 1 þ cos b
(6)
If we let t ¼ 0, the instantaneous elastic deflection is obtained as:
E1
w0 ¼
Rock rheology is constant in velocity when the load on the key strata is high. We can choose a Burgers model to describe the rheological properties, i.e.: A differential constitutive equation:
s þ p1 s_ þ p2 s€ ¼ q1 3_ þ q2€3
s0 E1
þ
t
h2
þ
s0 E3
1 et=s1
(4)
4qa4 b4
12 p1 p1 þ h3 2q1 6p1 K þ q1
p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
a
1 þ cos
b
wN ¼
4qa4 b4
12 1 3 þ h3 2q0 2ð6K þ q0 Þ
p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
a
1 þ cos
b
where
h3 E3
Except for these two constitutive relations, there are no differences in boundary value problems between rheological and elastic bodies. The elastic stressestrain relation is just a special form of the rheology constitutive equation. The linear boundary value problem of a rheological body corresponds to an elastic problem in a Laplace transformation space, and is referred to as the corresponding elasticerheological principle. The relation of the rheological boundary value problem with time can be developed from the inversion of the Laplace transformation after obtaining the Laplace space solution.
(7)
If we let t ¼ N, the steady-state elastic deflection is obtained as:
(3)
where p1 ¼ (h2/E1) þ (h2 þ h3/E3), p2 ¼ h2h3/E1E3, q1 ¼ h2, q2 ¼ h2h3/ E3 and a creep equation:
s1 ¼
(5)
(2)
h1
3ðtÞ ¼
a
þ
where
s1 ¼
Substituting qðsÞ ¼ q0 =s and EðsÞ, mðsÞ into Eq. (5) and inverting the Laplace transformation, we have:
(1)
1 et=s1
12 1 mðsÞ2
EðsÞh3 p4 3a4 þ 3b4 þ 2a2 b2 px py 1 þ cos
where p1 ¼ h1/E1 þ E2, q0 ¼ E1E2/E1 þ E2, q1 ¼ E1h1/E1 þ E2 and 2: a creep equation:
3ðtÞ ¼
4qðsÞa4 b4
Fig. 1. Key strata deflection as a function of time.
(8)
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It can be seen in Fig. 1 that the key strata deflection has an instantaneous elastic phase at the start. The rate of deformation decreases as a function of time; after time tc, the deformation reaches stability. As shown in Fig. 2, curves of key strata deflect at midpoint over time, when the viscosity coefficient h1 is 200, 300, 400 or 500 GPa d. Fig. 2 shows that the flow strain rate decreases as the viscosity coefficient increases. It is seen that the deflection of rocks with higher rheological coefficients requires more time to reach steady state. But the viscosity coefficient has little effect once rheology has reached this steady state. In the end, all deflections will have the same steady value. As well, the modulus of elasticity, E2, of an elastic body has a great effect on the instantaneous elastic deflection, but little effect on the changes of the deflection as a function of time in this constitutive model. Instead, the modulus of elasticity, E1, of a Kelvin body only affects the changes in deflection. At the start, deflections are almost the same: the smaller E1, the more obvious the rheology and the longer the steady time, the larger the steady-state rheology value.
269
Fig. 2. Key strata deflections at midpoint as a function of time.
3.1.2. Temporal correlation of key strata deflection under conditions of high stress The solution of a fixed key stratum deflection in a Laplace space, applying a Burgers model, describing a constant velocity rheology, is:
wðs;x;yÞ ¼
4q0 a4 b4
p4 3a4 þ3b4 þ2a2 b2
12 1 1þp1 sþp2 s2 h3 2s q1 sþq2 s2 s2
Fig. 3. Variation of key stratum deflection as a function of time in high stress.
px
3 1þp1 sþp2 1þcos 2s ð6p2 K þq2 Þs2 þð6Kp1 þq1 Þsþ6K a py 1þcos b þ
(9)
Based on the inversion of the Laplace transformation, Eq. (9) can be written as:
wðx; y; tÞ ¼
4qa4 b4
p4 3a4 þ 3b4 þ 2a2 b2 py
1 þ cos where
AðtÞ ¼
b
px 12 AðtÞ 1 þ cos 3 a h
We see that the key stratum deflection has an instantaneous elastic phase. The rate of deformation decreases over time; after tc, the speed in rheology remains constant and deflection increases linearly, as shown in Fig. 3. 3.2. Temporal correlation of internal key strata forces
(10)
3.2.1. Temporal correlation of internal key strata forces under low stress conditions Based on its corresponding elasticerheological principle, the Laplace transformation of My, which is a bending moment of the fixed key stratum, is written as:
! q1 t p1 q t p1 q p 1 22 þ 22 2 e q2 þ 2q1 2q1 2q1 2q1 2q1 2q2 4K 2 3 144q2 p1 p2 K 3 þ24p1 q22 þ24q1 q22 K 2 þ4q1 q22 K 5 412q1 p2 K þ2q1 q2 2 24p2 K 2 þ4Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þq1 K sin 2 12p2 K þ2q2 ffi e 12K p2 þ2Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 144p2 K þ24Kq2 36p21 K 2 12p1 q1 K q21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þq1 K cos 2 12p2 K þ2q2 q2 e 12K p2 þ2Kq2 24p2 K 2 þ4Kq2
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Fig. 4. Key stratum moment at midpoint over time under low stress conditions.
M y ðs; x; yÞ ¼
p4
p2 px py cos 1 þ cos a b a2 3a4 þ 3b4 þ 2a2 b2 4qðsÞa4 b4
Fig. 5. Key strata moment at midpoint over time under high stress condition.
My ðs; x; yÞ ¼
px py 1 þ cos cos a b
b2
4q0 a4 b4
p4 3a4 þ 3b4 þ 2a2 b2
p2 a2
(11) From the inversion of the Laplace transformation, Eq. (11), it follows that:
px py 4q0 1þcos My ðx;y;tÞ ¼ cos 2 a b p4 3a4 þ3b4 þ2a2 b2 b 1 3q0 3q0 3q1 þ 2 2ð6K þq0 Þ 2ð6K þq0 Þ 2ð6Kp1 þq0 Þ 6K þq0 t p2 px py cos ð12Þ 1þcos e 6Kp1 þq1 a b a2 a4 b4
px
cos
a
1 þ cos
py b
p2 px py mðsÞ 2 1 þ cos cos a b b
p2
mðsÞ
p2
For example, let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m and the uniform load on the key stratum q ¼ 10 MPa then, according to the creep test data, Fig. 4 shows that the curve of the key strata deflects at the midpoint over time when E1 ¼ 60 GPa, h1 ¼ 600 GPa d and E2 ¼ 20 GPa.
(13) where
mðsÞ ¼
1 3K 1 þ p1 s þ p2 s2 q1 s þ q2 s2 s 6K 1 þ p1 s þ p2 s2 þ q1 s þ q2 s2
Based on the inversion of the Laplace transformation, Eq. (13) can be written as:
p2 px py cos 1 þ cos 2 a b a p4 3a4 þ 3b4 þ 2a2 b2 p2 px py cos BðtÞ 2 1 þ cos a b b
My ðx; y; tÞ ¼
4q0 a4 b4
(14) where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ 24Kq2 36p21 K 2 12p1 q1 K q21
6K 2 þ q1 K cos 1 2 p þ 2Kq 12K 2 2 BðtÞ ¼ 3q2 e 2
12p2 K þ 2q2 12p2 K þ 2q2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144p2 K 2 þ 24Kq2 36p21 K 2 12p1 q1 K q21 2 6K þ q1 K sin 12q2 p12 K þ 3q2 q1 2 12p2 K þ 2q2 ffi e 12K p2 þ 2Kq2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3q1 12p2 K þ 2q2 2 144p2 K þ 24Kq2 36p21 K 2 12p1 q1 K q21
From Fig. 4, we see that the moment exhibits an exponential attenuation distribution over time with an instantaneous elastic phase. The instantaneous value of the moment is 17.2 MN. It reaches a steady state after 20 days at a stable value of 22.3 MN, 24 per cent higher than its instantaneous value. Hence, the bending moment also has a time-dependent effect. 3.2.2. Temporal correlation of internal key strata forces under conditions of high stress The Laplace transformation of My which is the bending moment of the fixed key strata, is written as:
Let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m and the uniform load on the key strata q ¼ 5 MPa. According to the creep test, Fig. 5 shows that the key stratum deflection at the midpoint over time, when E1 ¼ 20 GPa is h2 ¼ 600 GPa d and when E3 ¼ 60 GPa, h3 ¼ 300 GPa d. We see that, in Fig. 5, the distribution of the bending moment over time is very different from the deflection. The moment exhibits an exponential decay function over time and becomes steady after 150 days. The bending moment varies over a large range and doubles from the start to its steady state.
P. Hai, Z. Jian / Mining Science and Technology (China) 21 (2011) 267e272
271
N x
a b
N
Fig. 7. Model of creep buckling of key strata.
Fig. 6. Fracture distance of first break over time under high stress conditions.
4
DV w
3.3. Temporal correlation of first break of key strata Assuming that the key strata are four fixed boundaries, we have:
Mmax
Assuming that a key stratum is a simple four-edged supported rectangle, without the effect of shear stress on its boundaries, the model shown in Fig. 7 is created as follows: The key buckling stratum equation is:
2 p 1 3q0 2 þ2 ¼ 2 2ð6K þ q0 Þ b2 p4 3a4 þ 3b4 þ 2a2 b2 4qa4 b4
Setting the long-term strength of key strata rocks as sf, we obtain:
2 sf h2 p p2 2 þ 2BðtÞ ¼ 2 2 6 b a 3a4 þ 3b4 þ 2a2 b2
The fracture distance from the first break is written as:
L0 ðtÞ ¼
3
16qBðtÞa2 sf h2 p2
¼ 0
(17)
Amn sin
mpx mpy sin a b
(18)
Combining the boundary conditions and Nx ¼ Ny ¼ N in this problem, we have:
Nc ¼
p2 D a2
m2 þ n2
a2 b2
(19)
According to its corresponding elasticerheological principle and assuming m(t) ¼ m0 (where m0 is a constant), the Laplace transformation of a critical buckling load is:
4qa4 b4
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi u 4 2 2 2 u 2t 72qa 3sf h p a 6 CðtÞ
v2 y
m¼1 n¼1
(15)
p4
þ Ny
N X N X
wðx; yÞ ¼
6K þ q0 t 2 p 3q0 3q1 e 6Kp1 þ q1 þ 2ð6K þ q0 Þ 2ð6Kp1 þ q0 Þ a2
v2 x
! v2 w
where Nx ¼ sxh, Ny ¼ syh Assuming the surface is unstable, the deflection expression is:
Nx
v2 w
N ðsÞ ¼ c
(16)
where C(t) ¼ 144q2a8 12qsfh2p2a6 2s2p4h4a4 þ 36B(t)qsfh2p2a6 Let the length of the face a ¼ 100 m, the advance distance b ¼ 30 m, the uniform load on key strata q ¼ 5 MPa and the compressive strength 15 MPa. Then, according to the creep test, Fig. 6 shows that the fracture distance of the first break over time, when E1 ¼ 20 GPa, is h2 ¼ 600 GPa d and when E3 ¼ 60 GPa, h3 ¼ 300 GPa d. Fig. 6 shows that the fracture distance of the first break decays over time and reaches a steady state after 150 days. The rock strength decreases and is just 70 per cent of its elastic strength. Although in general the change is very small, we cannot ignore that the fracture distance is affected by rheology. For deep mining, we must consider the rheological effect on fracture distance because it has a strong time-dependent effect. 4. Creep buckling of key strata In deep mining, ground stress is quite high and transverse loads above the goaf are much lower than ground stress. However, the coal wall surrounding the goaf is a high stress concentration zone. Under the effect of high stress, the coal walls of these key strata are in a state of buckling, causing instability and bending of the goaf. It is also an important reason for breaks in key strata.
p2 2G ðsÞ a2 m2 þ n2 2 2 m a 1 0 b
(20)
Based on the inversion of the Laplace transformation, Eq. (20) can be written as:
Nc ðtÞ ¼
p2 a2
3 a2 2 t q1 7 b2 6 q0 e p1 5 4q0 þ 1 m0 p1
m 2 þ n2
(21)
Setting E1 ¼ 60 GPa, h1 ¼ 300 GPa d, E2 ¼ 20 GPa, Eq. (21) yields:
a2 2
3
2 2 4t p2 m þ n b2 4 Nc ðtÞ ¼ 2 15 þ 5e 15 5 1 m0 a
(22)
Fig. 8 shows the buckling load of the plate as a function of time: From Fig. 8, we can derive three cases about plate rheological buckling caused by different loads. 1) When the horizontal load Nc is higher than the instantaneous critical elastic load Nc0, the plate will immediately become unstable because of buckling. This is the same as an elastic situation. 2) When the horizontal load Nc is below the instantaneous critical elastic load Nc0 but higher than the long-term steady load NcN, the plate will lose stability after tc by reason of its material rheological properties. This is a case of delayed instability. 3) When the horizontal load Nc is below the long-term steady load NcN, the plate will not lose stability owing to damage.
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instability due to buckling sets in immediately; second, when the horizontal load Nc is below the instantaneous critical elastic load Nc0 but higher than the long-term steady load NcN, instability of the plate will be delayed, caused by the rheological properties of the material and third, when the horizontal load Nc is below the long-term steady load NcN, the plate will not lose stability due to damage. These three forms of instability need to be considered, especially delayed instability, when the key strata are in a precarious mechanical condition. Acknowledgements
Fig. 8. Buckling load as a function of time.
5. Conclusions For deep mining, the behaviour of strata surrounding a stope appears severe. Reasonable explanations are provided by the key stratum theory of stratum behaviour and movement. We introduced rheological properties of deep rock masses. According to the principle of correspondence, we analysed these characteristics, selected a generalized Kelvin model under conditions of low stress and a Burgers model under high stress conditions. Key strata are in a state of buckling instability because of the squeezing action of the rock surrounding them. In considering of this condition, we carried out rheological mechanical analyses. Based on our results, we draw the following conclusions: 1) Considering the rheology of a deep rock mass under low stress conditions, we selected a generalized Kelvin model to analyse the rheology characteristics of key strata and found that the rate of rheological strains of key internal strata forces and deflection shows exponential decay as a function of time, which became steady after 20 days. Both deflection and bending moment are time-dependent. The stability value is 20e30 per cent higher than the instantaneous elastic value. 2) When the load on key strata is high, the deformation of key strata has two phases: attenuation rheology and constant velocity rheology. Attenuation rheology lasts only a short time, but the constant velocity rheology endures. In the constant velocity rheology phase, rheology causes considerable damage to key strata. However, the corresponding bending moment shows an attenuated trend. The period of stability is relatively long and the extent of the damage quite large. 3) We analysed the instability of key strata caused by buckling and obtained the critical buckling load, given the squeezing action of the surrounding rock to the key strata. The results show that the critical load is basically unchanged when the length of the face is much longer than the advance distance; the critical load is largest when the goaf is square. 4) Given that rheology is a fact of life in deep mines, we considered three forms of creep buckling: first, once a plate is loaded,
This project was supported by the National Natural Science Foundation of China (No. 50904065) and the Program for New Century Excellent Talents in University (No. NCET-09-0728). As well, this project was sponsored by the Qinglan Project and the Fundamental Research Funds for the Central Universities (China University of Mining and Technology). References [1] He MC, Xie HP, Peng SP, Jiang YD. Study on rock mechanics in deep mining engineering. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2803e13 [in Chinese]. [2] Sun XM, He MC. Numerical simulation research on coupling support theory of roadway within soft rock at depth. Journal of China University of Mining & Technology 2005;34(2):166e9 [in Chinese]. [3] He YN, Han LJ, Shao P, Jiang BS. Some problems of rock mechanics for roadways stability in depth. Journal of China University of Mining & Technology 2006;35(3):288e95 [in Chinese]. [4] He MC. Conception system and evaluation indexes for deep engineering. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2854e8 [in Chinese]. [5] Ju Y, Zuo JP, Song ZD, Tian LL, Zhou HW. Numerical simulation of stress distribution and displacement of rock strata of coal mines by means of DDA method. Chinese Journal of Geotechnical Engineering 2007;29(2):268e73 [in Chinese]. [6] Wang YY, Wei J, Qi J, Yang CH, Li JG. Study on prediction for nonlinear creep deformation of deep rocks. Journal of China Coal Society 2005;30(4):409e13 [in Chinese]. [7] Dai HY, Wang SB, Yi SH, Kong LY, Gao YB. Influential laws on strata and ground movement of face-separated pillars at a great depth. Chinese Journal of Rock Mechanics and Engineering 2005;24(16):2929e33 [in Chinese]. [8] Xu NZ, Wang B, Qi YC. Prediction of surface subsidence in the deep coal mining. Journal of Mining & Safety Engineering 2006;23(1):66e9 [in Chinese]. [9] Yin GZ, Wang DK, Zhang WZ. Mechanics model to deformation of covered rock strata and its application in deep mining of steep or inclined seam. Journal of Chongqing University 2006;29(2):79e82 [in Chinese]. [10] Bai JB, Hou CJ. Control principle of surrounding rocks in deep roadway and its application. Journal of China University of Mining & Technology 2006;35(2):145e8 [in Chinese]. [11] Qian MG, Miao XX, Xu JL, Mao XB. Key strata theory in ground control. Xuzhou: China University of Mining and Technology Press; 2003 [in Chinese]. [12] Qian MG, Miao XX, Xu JL. Theoretical study of key stratum in ground control. Journal of China Coal Society 1996;21(3):225e30 [in Chinese]. [13] Qian MG, Mao XB, Miao XX. Variation of loads on the key layer of the overlying strata above the workings. Journal of China Coal Society 1998;23(2):135e9 [in Chinese]. [14] Mao XB, Miao XX, Qian MG. Study on broken laws of key strata in mining overlying strata. Journal of China University of Mining & Technology 1998;27(1):39e42 [in Chinese]. [15] Miao XX, Qian MG. Advance in the key strata theory of mining rockmass. Journal of China University of Mining & Technology 2000;29(1):25e9 [in Chinese].