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Stress and deformation analysis on deep surrounding rock at different time stages and its application
International Journal of Mining Science and Technology 22 (2012) 301–306
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Stress and deformation analysis on deep surrounding rock at different time stages and its application Li Ming a,b,⇑, Mao Xianbiao a,b, Yu Yuanlin c, Li Kai a,b, Ma Chao a,b, Peng Yan a,b a
State Key Laboratory of Geomechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou 221008, China School of Mechanics & Civil Engineering, China University of Mining & Technology, Xuzhou 221008, China c Wanbei Coal-Electricity Group North Co. Production Technology Department, Taiyuan 030000, China b
a r t i c l e
i n f o
Article history: Received 12 August 2011 Received in revised form 15 September 2011 Accepted 17 November 2011 Available online 22 May 2012 Keywords: Surrounding rock Stress displacement Viscoelasticity Broken rock zone
a b s t r a c t Based on the characteristics of the deep circular tunnel, the surrounding rock was divided into three regions: the cracked region, the plastic region and the viscoelastic region. The process of rock stress deformation and change was divided into three stages after the roadway excavation. By using the elastic–plastic mechanics theory, the analytical solutions of the surrounding stress and displacement at different stages and the radii of cracked and plastic regions were formulated. We additionally explained the surrounding rock stress and displacement which appeared in practical project. Simultaneously, based on the problem which emerged from a mine in Xuzhou during the excavating process of rock roadway’s transport, we got the theoretical solutions for the stress and displacement in the process of rock roadway’s excavation and considered that the broken area of rock roadway was largely loosing circle. The results indicate that according to the rheological characteristics of surrounding rock, in the primeval excavation of rock roadway, we should increase the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame. In addition, when the deformation rate of the surrounding rock is descending after the 15 days’ excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note and anchor rope) and set aside about 20 cm as the reserved deformation layer. Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction With the increase of mining depth, the roadway fracture become more and more serious and the stability of the surrounding rock get weaker and weaker, which have great influence on safety of coal mine production. Therefore, it has important meaning to study the stress deformation of roadway for the coal mine production safety protection. Since 1980, the analysis of stress deformation in circular tunnels is increasing with the increase of underground construction [1–8]. In 1994, the surrounding rock loose circle theory was proposed, it provided important theoretical support for the analysis of stress and deformation in the surrounding rock [9]. Based on this theory, many scholars analyzed the influences of stress and deformation on stability of the surrounding rock, and some problems about the stability of surrounding rock roadway were solved [10–12]. The theoretical analysis of the surrounding rock stress and deformation in abroad is earlier than at home. Before the 1970s, Fenner and Kastner used the ideal elastoplastic model to calculate the roadway stress of the elastic and plastic zones [13]. ⇑ Corresponding author. Tel.: +86 13585477939. E-mail address: [email protected] (M. Li).
Based on this, many scholars used the elastic–brittle–plastic model to analyze the surrounding rock stress and deformation and obtained a lot of useful conclusions [14–16]. But these researches did not consider the time factors, so they had some limitations. In this paper, we divide the process of surrounding rock stress and deformation after roadway excavation into three stages. Based on the rheological features of the elastic zone and elastic mechanics theory, the analytical solutions of the surrounding stress and displacement in different stages were obtained. And according to the stress and displacement continuity condition, the computational formulae to determine the radii of cracked and plastic regions were obtained. For the analysis of practical problem, the maintenance measures of the stability in roadway were given, which provided the theoretical value for the safety production of coal mine. 2. Surrounding rock mechanics model Since the stress redistributed after excavation, the surrounding rock has the obvious rheological properties and was at last divided into three regions including: cracked region, plastic region, and viscoelastic region. The viscoelastic region was elastic at first before entering into the rheology, but the rheological form of cracked region and plastic region is too difficult to discuss. The mechanics
2095-2686/$ - see front matter Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2012.04.003
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M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
@ rr rr rh þ ¼0 @r r
p0
Choosing the Mohr–Coulomb yield criterion as the limit equilibrium condition, while the surrounding rock comes into plastic and break phase,
Viscoelastic region Plastic region Cracked region p0
r0
Rp
ð1Þ
p0
Rc
rh ¼ Nrr þ S
ð2Þ
where the material parameters N and S are represented by cohesion C and angle of internal friction u, respectively, but are different in the cracked and the plastic regions. Cracked region (r0 6 r 6 Rc)
8 < N ¼ Nc ¼ 1þsin uc 1sin u
p0
c
Fig. 1. Surrounding rock mechanics model.
ð3aÞ
: S ¼ Sc ¼ 2C c cos uc 1sin u c
Plastic region (Rc < r 6 Rp)
8 1þsin u < N ¼ Np ¼ 1sin up p
Deformation u u1 u2
p
According to the symmetry, the geometric equations of the problem are as follows:
(
III
u0
II I
0
ð3bÞ
: S ¼ S ¼ 2C p cos up p 1sin u
t0
t1
Time t
t2
Fig. 2. u–t curve of surrounding rock.
model shown in Fig. 1, regard the surrounding rock as plane strain problem. Where the p0 is the virgin stress, r0, Rc and Rp expresses the radius of the roadway, the cracked region and the plastic region respectively. After excavation, the process of the surrounding rock deformation can be divided into three stages. (I) The instantaneous elastic stage. Here, the surrounding rock deformation is elastic deformation at this stage, which is pretty instantaneous. And because of the stress redistributed, the deformation of the surrounding rock is large. (II) The cracked region, in which the plastic and viscoelastic regions are formed. The leading cause of the surrounding rock deformation is the influence of the ambient stress in this stage, of which the time is very short. Since the plastic strains firstly appear inside the surrounding rock, then rock fracture and the cracked region are formed. So the cracked region is formed from inside to plastic zone. (III) Here the elastic region enters into rheology and the rock of the cracked region is broken. The deformation of surrounding rock includes the additional deformations in two ways, of which the time is long. The whole process of deformation of surrounding rock deformation is shown in Fig. 2. After excavation, the changing process of the surrounding rock stress and deformation is continuous. In order to ensure easy and correct analysis, it is divided into three processes. 3. Stress and deformation of surrounding rock 3.1. Basic theory In the process of computation and analysis, the subscripted variables c, p and e express the component of the cracked region, the plastic region and the elastic region, respectively. In addition, rr,rh are regarded as the radial and tangential stresses respectively and u is the displacement. In the calculation, by ignoring the volume force, the stress in every region accords to the equilibrium equation:
er ¼ @u @r eh ¼ ur
ð4Þ
And the stress and displacement continuity conditions of the surrounding rock are as follows:
8 rrc ¼ 0 > > > r ¼ r ; u ¼ u rp re p e > > : rre ¼ p0
ðr ¼ r 0 Þ ðr ¼ Rc Þ ðr ¼ Rp Þ
ð5Þ
ðr ! 1Þ
3.2. Stress of the surrounding rock (1) The stress of the surrounding rock at stage I Before mining, the state of the surrounding rock stress is virgin, and the stress components are as follows:
rr0e ¼ p0 rh0e ¼ p0
ð6Þ
(2) The stress of the surrounding rock at stage II Cracked region (r0 6 r 6 Rc) From the balance Eq. (1), failure criteria (2) and (3a) and the 1st formula in boundary condition (5), it is easy to get the expression of the stress component in the cracked region
8 " # 2 sin uc > > r 1sin uc > > r ¼ C cot u 1 c > c r0 < rc " # 2 sin uc > > 1þsin uc > r 1sin uc > r ¼ C cot u 1 > c c 1sin uc r0 : hc
ð7Þ
Plastic region (Rc < r 6 Rp) In the same way, from the balance Eq. (1), failure criteria (2) and (3b) and the 2nd formula in boundary condition (5), the expression of the stress component in plastic region is
8 " # 2 sin up 2 sin up > > r 1sin up r 1sin up > > r ¼ rRC c Rc þ C p cot up Rc 1 > < rp " # 2 sin up 2 sin up > > > > rhp ¼ 1þsin up rRC c r 1sin up þ C p cot up 1þsin up r 1sin up 1 > Rc 1sin up 1sin up Rc : ð8Þ
M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
In the expression,
2
rRC c
rRC c is given as
3.3. Deformation of the surrounding rock
3 2 sin uc 1sin uc R C ¼ C c cot uc 4 15 r0
It represents the radial stress value on the interface between cracked region and plastic region. Elastic region (Rp < r < 1) According to the elasticity, combining the 2nd and 3rd formulae and boundary condition (5), the expression of the stress component in elastic region is
8 2 > < rre ¼ ðrRp p p0 Þ Rrp þ p0 2 > : r ¼ ðr p Þ Rp þ p he Rp p 0 0 r
ð9Þ
(1) The deformation of the surrounding rock at stage I According to the physical and the geometric equations of the elastic problem, at the excavating moment, the deformation of the surrounding rock is as follows:
ue0 ¼
" # R2p 1 ue ¼ ðrRp p p0 Þ þ ð1 2lÞp0 r 2G r
ð10Þ
r ¼ 3ke n
where D is the derivative operator to t, Dn ¼ @t@ n ; srs = rrs r, ers = ers e the deviatoric stress and strain tensors; r ¼ 13 ðrx þ ry þ rz Þ; e ¼ 13 ðex þ ey þ ez Þ the average stress and strain respectively; and k is the volumetric deformation modulus. The Laplace variation coefficients of constitutive equation are as follows:
f ðsÞ ¼ s þ g1
re1
2pRp ue ¼ 2pRc up
ð14Þ
ð15Þ
Substituting Eq. (14) into Eq. (15), the deformation of the plastic region is as follows:
up ¼
Rp ue Rc
ð16Þ
where ueis the deformation on the interface of the cracked and plastic regions. (3) The deformation of the surrounding rock at stage III Viscoelastic region (Rp < r < 1) By substituting the Laplace coefficient into Eq. (14), the viscoelastic solution is obtained by inverse Laplace transform.
R2 1 1 1 G1 G1 t rRp p p þ e gre1 G0 1 2 r G0 G0 gre1 G0 " 2 # R p 1 1 1 G1 t p e gre1 G0 þ 0 þ þ ð1 2lÞr G1 G0 G1 2 r
u0e ¼
ð11Þ
gðsÞ ¼ G0 s þ gG1
ð13Þ
Plastic region (Rc < r 6 Rp) The deformation of the plastic region is caused by volume deformation, therefore, the volume deformation of the outer and inner boundaries of the plastic region are equal to Eq. (15):
After the excavation and formation of the regions, the surrounding rock in the elastic region shows a stability creep property. And the rock shows the obvious characteristics of stress relaxation because of the rheological properties. Based on the deformation characteristics of the rock, the Poyting-Thomson model was chose to analyze the viscoelastic region. The model is shown in Fig. 3. The derivative operators of the constitutive equation are as follows:
h i ( D þ kg1 srs ¼ ðk1 þ k2 ÞD þ k1gk2 ers
1 2l p0 r 0 2G
E where l is the Poisson’s ratio, G ¼ 2ð1þ lÞ the rigidity modulus, and E is the elastic modulus. (2) The deformation of the surrounding rock at stage II Elastic region (Rp < r < 1) According to the physical equations, the geometric equations and (9), the expression of ue is as follows:
(3) The stress of the surrounding rock at stage III In this stage, the expression of stress in cracked region and same as (7). Viscoelastic region (Rp < r < 1)
(
303
ð17Þ
re1
In the practical engineering, the deformation of the viscoelastic region is shown at r = Rp, that is
where G1 = k1 represent the long-term shear modulus of the rock; G0 = k1 + k2 represent the instantaneous shear modulus of the rock; gre1 ¼ kg1 represent the relaxation time of the rock. In the elastic solution of stress and displacement, substituting G with
gðsÞ ; f ðsÞ
p0 with
p0 , s
1 1 1 G1 G1 t rRp p Rp þ e gre1 G0 1 2 G0 G0 gre1 G0 p0 1 1 1 G1 t e gre1 G0 þ þ ½Rp þ ð1 2lÞRp G1 G0 G1 2
u0Rp e ¼
then the viscoelastic solution is obtained by
inverse Laplace transform. So the expressions of the stress in the viscoelastic region are as follow:
8 2 > < r0re ¼ ðrRp p p0 Þ Rrp þ p0 2 > : r0 ¼ ðr p Þ Rp þ p Rp p 0 0 he r
k1
ð12Þ
σ k2 Fig. 3. Poyting–Thomson viscoelastic model.
Cracked region (r0 6 r 6 Rc) The deformation of cracked region is mainly caused by the rock volume changes. Comparing with the elasticity, there is a relation between the ecr and ech at one point of the cracked region:
ecr ¼ l0 ech
η
σ
ð18Þ
ð19Þ
In this expression, l0 is defined as the break Poisson’s ratio. At the inner boundary of the roadway r ¼ r0 ; l0r0 ¼ 1 and is the maximum. Since the volume of the plastic region is invariable, the position of the minimum l0 is at r = Rp. Substituting (9) into (19),
l0Rp ¼
rRp p 2p0 rRp p
ð20Þ
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M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
To facilitate the calculations, the inside cracked region can be approximately regarded as:
1 2
l0 ¼ ðl0r0 þ l0Rp Þ ¼
p0 2p0 rRp p
Using the Eqs. (13, 18, 25 and 29), we get:
1 1 1 G1 G1 t rRp p Rp þ e gre1 G0 1 2 G0 G0 gre1 G0 p0 1 1 1 G1 t þ ½Rp þ ð1 2lÞRp e gre1 G0 þ G1 G0 G1 2 0 Rp 1 2 l p0 r 0 þ 1þl0 ue r l0 2G Rc
u¼ ð21Þ
According to the geometric Eq. (4), we get:
duc uc ¼ l0 dr r
ð22Þ
So
ð30Þ
4. Engineering application
uc ¼ br
l0
ð23Þ
Combining the 2nd formula of Eqs. (5,16 and 23), the deformation of the cracked region is as follows:
uc ¼
Rp
0
l R1þ c
0
ue r l
ð24Þ
Inside the roadway, the dilatancy deformation is as follows:
ur0 c ¼
Rp l R1þ c
l0
0
ue r0
ð25Þ
3.4. Solving Rp and Rc The breaking of the surrounding rock appear from the inside, the plastic region stabilize before the formation of the cracked region. The expression of Rp is as follows:
" Rp ¼ r 0
#1sin up ðp0 þ C p cot up Þð1 sin up Þ 2 sin up C p cot up
4.1. Surrounding rock stress
ð26Þ
At r = Rp, the regions stabilizes and it accords with Eq. (27):
rrp þ rhp ¼ 2p0
During the process of excavating the main transport haulage roadway at a mine in Xuzhou, the completed roadways were destroyed in tandem and the surrounding rock took severe deformations with the obvious rheological characteristic. Through geological exploration, the black, mudstone and deep grey argillaceous sandstone which are the main components of the surrounding rock, had no obvious geology structure and faults. Because the vault of the roadway excavating equipment is semicircle and bottom plate is an inverted arch, so the section of the roadway is regarded as a circle and r0 = 2.5 m. At 984 m depth of roadway, the virgin stress of several points is measured and the horizontal and vertical stresses are 22.86 and 22.04 MPa respectively. Therefore the initial rock stress is considered isobaric at the two directions, p0 = 22.5 MPa. The basic mechanical parameters of the rock are shown in Table 1. And during the period of measurement, we got the parameters related to rheological property of the rock, which are shown in Table 2.
ð27Þ
Using the parameters and the expressions that we got, the stress expressions of the three regions are obtained (unit of stress: MPa). Viscoelastic region (Rp < r < 1)
(
Substituting (8) into (27),
2 3 2 sin up 2 sin up Rp 1sin up Rp 1sin up 4 rRC c þ C p cot up 15 Rc Rc 2 sin up 1sin up 1 þ sin up R rRC c p þ 1 sin up Rc 2 3 2 sin up 1 þ sin up Rp 1sin up 4 þ C p cot up 15 ¼ 2p0 1 sin up Rc
r0re ¼ 264:05 þ 22:5 r2 r0he ¼ 264:05 þ 22:5 2 r
Plastic region (Rc < r 6 Rp)
(
rrp ¼ 0:82r2 9:94 rhp ¼ 2:46r2 9:94
Cracked region (r0 6 r 6 Rc)
ð28Þ
(
rrc ¼ 4:17r0:7 7:912 rhc ¼ 7:08r0:7 7:912
Substituting the variant expression into Eq. (28), the radius expression Rc of cracked region is obtained.
In the surrounding rock, the position showing stress is inside of roadway, that is r = r0, and the stress of the cracked region is as follows:
3.5. Appearance of the stress and deformation of the surrounding rock
(
After the stress and deformation stability of the surrounding rock, the components of stress include the viscoelastic, plastic and cracked regions. However, in practice we can only test for the stress of the cracked region inside the roadway by using equipment. The deformation of the surrounding rock includes three parts: the instantaneous elastic deformation, the elastoplastic deformation and the deformation caused by rock breaking. But the deformation, which has influences on the supports, is the elastoplastic deformation and the deformation caused by the rock breaking and removing the instantaneous elastic deformation.
u¼
u0Rp e
þ ur0 c ue0
ð29Þ
rrc =r¼r0 ¼ 0 rhc =r¼r0 ¼ 5:53
From the stress expression, inside the roadway, the radial and tangential stresses are 0 and 5.53 MPa respectively, but the value is not large. This is because most of the energy from the rock breaking is obviously between the broken massive rock and the surrounding
Table 1 Basic mechanical parameters of rock. E (GPa)
l
cp (MPa)
up (o)
cc (MPa)
uc(o)
30.0
0.25
5.74
30.0
2.15
15.2
M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306 Table 2 Rock rheological parameters. G0 (MPa)
G1 (MPa)
gre1 (d)
860
610
10
weakened rock, leading to the weak action of the virgin stress on the cracked region. 4.2. Solving RP and RC Using the parameters and Eqs. (25 and 27), the radius of plastic and cracked regions and the thickness d of the rock loose zone are obtained as follows:
8 > < Rp ¼ 5:3 m Rc ¼ 3:98 m > : d ¼ 1:48 m Through computation, the thickness of the rock loose zone is nearly 1.5 m, which is the largest rock loose zone. A large scale of the loosened rock undermines the stability control of the roadway and badly hurt the safety of coal mine production.
Using the parameters shown in Tables 1 and 2 and the Eq. (30), the deformation expression of the surrounding rock was obtained. The u–t curve of the surrounding rock is shown in Fig. 3: 27
u ¼ 87:488 42:9e320t According to the three stages of the surrounding rock deformation, from Fig. 4, it can be seen that at stage I, t = 0 d, u = 44.79 cm; at the stage II, t = 0.2 d, u = 45.31 cm; about 40 d after the roadway excavation, the deformation of the surrounding rock nearly does not change and eventually stables at u = 87.488 cm. In order to validate the theatrical conclusion of the main haulage roadway deformation, we chose the deformation of Qujiang main haulage roadway for our measurements, because the geological and mining conditions of Qujiang are similar to those of Xuzhou. We tested the relative movement of the roof and floor, the two sides at instantaneous, after the exact 0.2 d and the final deformation. After testing, when t = 0, u = 48.21 cm and when t = 0.2 d, u = 50.24 cm. Because the roadway was supported after a short time, so we did not get the final deformation of the surrounding roadway. About after 40 d, the deformation remained stable at about 60 cm. And as time went on, the surrounding rock losing and the loose circle is disappeared. Basing on the stress and deformation characters of the transport roadway, when supporting the surrounding rock, we should in88 80
u (cm)
72 64 56 48 0
20
40
60
crease the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame. And when the rate of deformation of the surrounding rock is descending after the 15 days of excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note and anchor rope) and set aside about 20 cm as the reserved deformation layer. 5. Conclusions (1) Based on the deformation characteristics of the surrounding rock after excavation, the whole stress and deformation process from the excavation to the formation of the cracked region was divided into three stages, and the viscoelastic– plastic–burst mechanics model of the surrounding rock was established. In addition, the deformation curve of the whole excavating-burst process was established. (2) The stress and deformation of the surrounding rock and the respective radius of the plastic and cracked regions in three stages were obtained from the analysis. The deformation which really has influence on the supports is the elastoplastic deformation and is the deformation caused by the rock breaking and removing of the instantaneous elastic deformation:
u ¼ u0Rp e þ ur0 c ue0
4.3. Deformation of the surrounding rock
40
305
80
t (d) Fig. 4. Roadway u–t curve.
100
120
(3) Aiming at the problem which emerged in a mine in Xuzhou during the process of excavating the transport roadway, we need increase the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame when supporting the surrounding rock. And when the rate of deformation of the surrounding rock is decreasing after the 15 days of excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note, anchor rope) and set aside about 20 cm as the reserved deformation layer.
Acknowledgments Financial support for this work, provided by the National Basic Research Program of China (No. 2007CB209408), the National Natural Science Foundation of China (No. 51074166) and the Research Innovation Program for College Graduates of Jiangsu (No.CX098B_ 108Z). References [1] Yao GS, Li JP, Gu SC. Analytic solution to deformation of soft rock tunnel considering dilatancy and plastic softening of rock mass. Rock Soil Mech 2009;30(2):463–7. [2] Jiao CM, Lv AZ. Analytical solution of loads on supporting structure for circular tunnel and stresses in viscoelastic surrounding rock. Rock Soil Mech 2004;25(Supp):103–6. [3] Zhang GL, Ma LW, Pan SH. The elastic displacement analyzing of he surrounding rock of four layer in roundness tunnel. Shan Xi Archit 2007;33(9):107–9. [4] Yu XF, Deng Y, Liu HY, Fang ZC. The stability of country rock in underground projects. Beijing: The Publishing House of Coal industry; 1983. pp. 204–236. [5] Chen YG, Lu SL, Xu YQ. Control technology of surrounding rock of roadway in China. Xuzhou: China University of Mining and Technology Press; 1994. [6] Bai JB, Wang XY, Jia MK, Hou CJ. Theory and application of supporting in deep soft roadway. Chinese J Geotech Eng 2008;30(5):632–4. [7] Li DZ, Li BB, Tan YY. Analysis and control of distortion of surrounding rock of roadway at deep ground of mine. Rock Soil Mech 2009;30(1):109–12. [8] He MC, Yuan HS. Chinese coal mines bolting support theory and practice. Beijing: Science Press; 2004. [9] Dong FT, Song HW, Guo ZH, et al. Roadway support theory on broken rock zone. J China Coal Soc 1994;19(1):21–32. [10] Jing HW, Li YH, Xu GA. Analysis of surrounding rock stability of deeply buried roadways and study on its control techniques. Rock Soil Mech 1994;19(1):21–32. [11] Jiang BS, Zhang Q, He YN, Han LJ. Elastoplastic analysis of cracked surrounding rocks in deep circular openings. J Rock Mech Eng 2007;26(5):982–6.
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[12] Zhang Q, Wang SL, Ge XR. Elastoplastic analysis of circular openings instrainsoftening rock masses. J Rock Mech Eng 2010;29(5):1031–5. [13] Yuan WB, Chen J. Analysis of plastic zone and loose zone around opening in softening rock mass. J China Coal Soc 1986;11(3):77–86. [14] Carranza-torres C, Fairhurst C. The elastoplastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 1999;36(6):777–809.
[15] Diering DH. Mining at ultra depths in the 21st century. CIM Bull 2000;93:141–5. [16] Cichowicz A, Miller AM, Durrheim RJ. Rock mass behavior under seismic loading in a deep mine environment: implications for stope support. J South Afr Inst Min Metall 2000;100(2):121–8.
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Stress and deformation analysis on deep surrounding rock at different time stages and its application Li Ming a,b,⇑, Mao Xianbiao a,b, Yu Yuanlin c, Li Kai a,b, Ma Chao a,b, Peng Yan a,b a
State Key Laboratory of Geomechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou 221008, China School of Mechanics & Civil Engineering, China University of Mining & Technology, Xuzhou 221008, China c Wanbei Coal-Electricity Group North Co. Production Technology Department, Taiyuan 030000, China b
a r t i c l e
i n f o
Article history: Received 12 August 2011 Received in revised form 15 September 2011 Accepted 17 November 2011 Available online 22 May 2012 Keywords: Surrounding rock Stress displacement Viscoelasticity Broken rock zone
a b s t r a c t Based on the characteristics of the deep circular tunnel, the surrounding rock was divided into three regions: the cracked region, the plastic region and the viscoelastic region. The process of rock stress deformation and change was divided into three stages after the roadway excavation. By using the elastic–plastic mechanics theory, the analytical solutions of the surrounding stress and displacement at different stages and the radii of cracked and plastic regions were formulated. We additionally explained the surrounding rock stress and displacement which appeared in practical project. Simultaneously, based on the problem which emerged from a mine in Xuzhou during the excavating process of rock roadway’s transport, we got the theoretical solutions for the stress and displacement in the process of rock roadway’s excavation and considered that the broken area of rock roadway was largely loosing circle. The results indicate that according to the rheological characteristics of surrounding rock, in the primeval excavation of rock roadway, we should increase the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame. In addition, when the deformation rate of the surrounding rock is descending after the 15 days’ excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note and anchor rope) and set aside about 20 cm as the reserved deformation layer. Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction With the increase of mining depth, the roadway fracture become more and more serious and the stability of the surrounding rock get weaker and weaker, which have great influence on safety of coal mine production. Therefore, it has important meaning to study the stress deformation of roadway for the coal mine production safety protection. Since 1980, the analysis of stress deformation in circular tunnels is increasing with the increase of underground construction [1–8]. In 1994, the surrounding rock loose circle theory was proposed, it provided important theoretical support for the analysis of stress and deformation in the surrounding rock [9]. Based on this theory, many scholars analyzed the influences of stress and deformation on stability of the surrounding rock, and some problems about the stability of surrounding rock roadway were solved [10–12]. The theoretical analysis of the surrounding rock stress and deformation in abroad is earlier than at home. Before the 1970s, Fenner and Kastner used the ideal elastoplastic model to calculate the roadway stress of the elastic and plastic zones [13]. ⇑ Corresponding author. Tel.: +86 13585477939. E-mail address: [email protected] (M. Li).
Based on this, many scholars used the elastic–brittle–plastic model to analyze the surrounding rock stress and deformation and obtained a lot of useful conclusions [14–16]. But these researches did not consider the time factors, so they had some limitations. In this paper, we divide the process of surrounding rock stress and deformation after roadway excavation into three stages. Based on the rheological features of the elastic zone and elastic mechanics theory, the analytical solutions of the surrounding stress and displacement in different stages were obtained. And according to the stress and displacement continuity condition, the computational formulae to determine the radii of cracked and plastic regions were obtained. For the analysis of practical problem, the maintenance measures of the stability in roadway were given, which provided the theoretical value for the safety production of coal mine. 2. Surrounding rock mechanics model Since the stress redistributed after excavation, the surrounding rock has the obvious rheological properties and was at last divided into three regions including: cracked region, plastic region, and viscoelastic region. The viscoelastic region was elastic at first before entering into the rheology, but the rheological form of cracked region and plastic region is too difficult to discuss. The mechanics
2095-2686/$ - see front matter Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2012.04.003
302
M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
@ rr rr rh þ ¼0 @r r
p0
Choosing the Mohr–Coulomb yield criterion as the limit equilibrium condition, while the surrounding rock comes into plastic and break phase,
Viscoelastic region Plastic region Cracked region p0
r0
Rp
ð1Þ
p0
Rc
rh ¼ Nrr þ S
ð2Þ
where the material parameters N and S are represented by cohesion C and angle of internal friction u, respectively, but are different in the cracked and the plastic regions. Cracked region (r0 6 r 6 Rc)
8 < N ¼ Nc ¼ 1þsin uc 1sin u
p0
c
Fig. 1. Surrounding rock mechanics model.
ð3aÞ
: S ¼ Sc ¼ 2C c cos uc 1sin u c
Plastic region (Rc < r 6 Rp)
8 1þsin u < N ¼ Np ¼ 1sin up p
Deformation u u1 u2
p
According to the symmetry, the geometric equations of the problem are as follows:
(
III
u0
II I
0
ð3bÞ
: S ¼ S ¼ 2C p cos up p 1sin u
t0
t1
Time t
t2
Fig. 2. u–t curve of surrounding rock.
model shown in Fig. 1, regard the surrounding rock as plane strain problem. Where the p0 is the virgin stress, r0, Rc and Rp expresses the radius of the roadway, the cracked region and the plastic region respectively. After excavation, the process of the surrounding rock deformation can be divided into three stages. (I) The instantaneous elastic stage. Here, the surrounding rock deformation is elastic deformation at this stage, which is pretty instantaneous. And because of the stress redistributed, the deformation of the surrounding rock is large. (II) The cracked region, in which the plastic and viscoelastic regions are formed. The leading cause of the surrounding rock deformation is the influence of the ambient stress in this stage, of which the time is very short. Since the plastic strains firstly appear inside the surrounding rock, then rock fracture and the cracked region are formed. So the cracked region is formed from inside to plastic zone. (III) Here the elastic region enters into rheology and the rock of the cracked region is broken. The deformation of surrounding rock includes the additional deformations in two ways, of which the time is long. The whole process of deformation of surrounding rock deformation is shown in Fig. 2. After excavation, the changing process of the surrounding rock stress and deformation is continuous. In order to ensure easy and correct analysis, it is divided into three processes. 3. Stress and deformation of surrounding rock 3.1. Basic theory In the process of computation and analysis, the subscripted variables c, p and e express the component of the cracked region, the plastic region and the elastic region, respectively. In addition, rr,rh are regarded as the radial and tangential stresses respectively and u is the displacement. In the calculation, by ignoring the volume force, the stress in every region accords to the equilibrium equation:
er ¼ @u @r eh ¼ ur
ð4Þ
And the stress and displacement continuity conditions of the surrounding rock are as follows:
8 rrc ¼ 0 > > >
ðr ¼ r 0 Þ ðr ¼ Rc Þ ðr ¼ Rp Þ
ð5Þ
ðr ! 1Þ
3.2. Stress of the surrounding rock (1) The stress of the surrounding rock at stage I Before mining, the state of the surrounding rock stress is virgin, and the stress components are as follows:
rr0e ¼ p0 rh0e ¼ p0
ð6Þ
(2) The stress of the surrounding rock at stage II Cracked region (r0 6 r 6 Rc) From the balance Eq. (1), failure criteria (2) and (3a) and the 1st formula in boundary condition (5), it is easy to get the expression of the stress component in the cracked region
8 " # 2 sin uc > > r 1sin uc > > r ¼ C cot u 1 c > c r0 < rc " # 2 sin uc > > 1þsin uc > r 1sin uc > r ¼ C cot u 1 > c c 1sin uc r0 : hc
ð7Þ
Plastic region (Rc < r 6 Rp) In the same way, from the balance Eq. (1), failure criteria (2) and (3b) and the 2nd formula in boundary condition (5), the expression of the stress component in plastic region is
8 " # 2 sin up 2 sin up > > r 1sin up r 1sin up > > r ¼ rRC c Rc þ C p cot up Rc 1 > < rp " # 2 sin up 2 sin up > > > > rhp ¼ 1þsin up rRC c r 1sin up þ C p cot up 1þsin up r 1sin up 1 > Rc 1sin up 1sin up Rc : ð8Þ
M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
In the expression,
2
rRC c
rRC c is given as
3.3. Deformation of the surrounding rock
3 2 sin uc 1sin uc R C ¼ C c cot uc 4 15 r0
It represents the radial stress value on the interface between cracked region and plastic region. Elastic region (Rp < r < 1) According to the elasticity, combining the 2nd and 3rd formulae and boundary condition (5), the expression of the stress component in elastic region is
8 2 > < rre ¼ ðrRp p p0 Þ Rrp þ p0 2 > : r ¼ ðr p Þ Rp þ p he Rp p 0 0 r
ð9Þ
(1) The deformation of the surrounding rock at stage I According to the physical and the geometric equations of the elastic problem, at the excavating moment, the deformation of the surrounding rock is as follows:
ue0 ¼
" # R2p 1 ue ¼ ðrRp p p0 Þ þ ð1 2lÞp0 r 2G r
ð10Þ
r ¼ 3ke n
where D is the derivative operator to t, Dn ¼ @t@ n ; srs = rrs r, ers = ers e the deviatoric stress and strain tensors; r ¼ 13 ðrx þ ry þ rz Þ; e ¼ 13 ðex þ ey þ ez Þ the average stress and strain respectively; and k is the volumetric deformation modulus. The Laplace variation coefficients of constitutive equation are as follows:
f ðsÞ ¼ s þ g1
re1
2pRp ue ¼ 2pRc up
ð14Þ
ð15Þ
Substituting Eq. (14) into Eq. (15), the deformation of the plastic region is as follows:
up ¼
Rp ue Rc
ð16Þ
where ueis the deformation on the interface of the cracked and plastic regions. (3) The deformation of the surrounding rock at stage III Viscoelastic region (Rp < r < 1) By substituting the Laplace coefficient into Eq. (14), the viscoelastic solution is obtained by inverse Laplace transform.
R2 1 1 1 G1 G1 t rRp p p þ e gre1 G0 1 2 r G0 G0 gre1 G0 " 2 # R p 1 1 1 G1 t p e gre1 G0 þ 0 þ þ ð1 2lÞr G1 G0 G1 2 r
u0e ¼
ð11Þ
gðsÞ ¼ G0 s þ gG1
ð13Þ
Plastic region (Rc < r 6 Rp) The deformation of the plastic region is caused by volume deformation, therefore, the volume deformation of the outer and inner boundaries of the plastic region are equal to Eq. (15):
After the excavation and formation of the regions, the surrounding rock in the elastic region shows a stability creep property. And the rock shows the obvious characteristics of stress relaxation because of the rheological properties. Based on the deformation characteristics of the rock, the Poyting-Thomson model was chose to analyze the viscoelastic region. The model is shown in Fig. 3. The derivative operators of the constitutive equation are as follows:
h i ( D þ kg1 srs ¼ ðk1 þ k2 ÞD þ k1gk2 ers
1 2l p0 r 0 2G
E where l is the Poisson’s ratio, G ¼ 2ð1þ lÞ the rigidity modulus, and E is the elastic modulus. (2) The deformation of the surrounding rock at stage II Elastic region (Rp < r < 1) According to the physical equations, the geometric equations and (9), the expression of ue is as follows:
(3) The stress of the surrounding rock at stage III In this stage, the expression of stress in cracked region and same as (7). Viscoelastic region (Rp < r < 1)
(
303
ð17Þ
re1
In the practical engineering, the deformation of the viscoelastic region is shown at r = Rp, that is
where G1 = k1 represent the long-term shear modulus of the rock; G0 = k1 + k2 represent the instantaneous shear modulus of the rock; gre1 ¼ kg1 represent the relaxation time of the rock. In the elastic solution of stress and displacement, substituting G with
gðsÞ ; f ðsÞ
p0 with
p0 , s
1 1 1 G1 G1 t rRp p Rp þ e gre1 G0 1 2 G0 G0 gre1 G0 p0 1 1 1 G1 t e gre1 G0 þ þ ½Rp þ ð1 2lÞRp G1 G0 G1 2
u0Rp e ¼
then the viscoelastic solution is obtained by
inverse Laplace transform. So the expressions of the stress in the viscoelastic region are as follow:
8 2 > < r0re ¼ ðrRp p p0 Þ Rrp þ p0 2 > : r0 ¼ ðr p Þ Rp þ p Rp p 0 0 he r
k1
ð12Þ
σ k2 Fig. 3. Poyting–Thomson viscoelastic model.
Cracked region (r0 6 r 6 Rc) The deformation of cracked region is mainly caused by the rock volume changes. Comparing with the elasticity, there is a relation between the ecr and ech at one point of the cracked region:
ecr ¼ l0 ech
η
σ
ð18Þ
ð19Þ
In this expression, l0 is defined as the break Poisson’s ratio. At the inner boundary of the roadway r ¼ r0 ; l0r0 ¼ 1 and is the maximum. Since the volume of the plastic region is invariable, the position of the minimum l0 is at r = Rp. Substituting (9) into (19),
l0Rp ¼
rRp p 2p0 rRp p
ð20Þ
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M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306
To facilitate the calculations, the inside cracked region can be approximately regarded as:
1 2
l0 ¼ ðl0r0 þ l0Rp Þ ¼
p0 2p0 rRp p
Using the Eqs. (13, 18, 25 and 29), we get:
1 1 1 G1 G1 t rRp p Rp þ e gre1 G0 1 2 G0 G0 gre1 G0 p0 1 1 1 G1 t þ ½Rp þ ð1 2lÞRp e gre1 G0 þ G1 G0 G1 2 0 Rp 1 2 l p0 r 0 þ 1þl0 ue r l0 2G Rc
u¼ ð21Þ
According to the geometric Eq. (4), we get:
duc uc ¼ l0 dr r
ð22Þ
So
ð30Þ
4. Engineering application
uc ¼ br
l0
ð23Þ
Combining the 2nd formula of Eqs. (5,16 and 23), the deformation of the cracked region is as follows:
uc ¼
Rp
0
l R1þ c
0
ue r l
ð24Þ
Inside the roadway, the dilatancy deformation is as follows:
ur0 c ¼
Rp l R1þ c
l0
0
ue r0
ð25Þ
3.4. Solving Rp and Rc The breaking of the surrounding rock appear from the inside, the plastic region stabilize before the formation of the cracked region. The expression of Rp is as follows:
" Rp ¼ r 0
#1sin up ðp0 þ C p cot up Þð1 sin up Þ 2 sin up C p cot up
4.1. Surrounding rock stress
ð26Þ
At r = Rp, the regions stabilizes and it accords with Eq. (27):
rrp þ rhp ¼ 2p0
During the process of excavating the main transport haulage roadway at a mine in Xuzhou, the completed roadways were destroyed in tandem and the surrounding rock took severe deformations with the obvious rheological characteristic. Through geological exploration, the black, mudstone and deep grey argillaceous sandstone which are the main components of the surrounding rock, had no obvious geology structure and faults. Because the vault of the roadway excavating equipment is semicircle and bottom plate is an inverted arch, so the section of the roadway is regarded as a circle and r0 = 2.5 m. At 984 m depth of roadway, the virgin stress of several points is measured and the horizontal and vertical stresses are 22.86 and 22.04 MPa respectively. Therefore the initial rock stress is considered isobaric at the two directions, p0 = 22.5 MPa. The basic mechanical parameters of the rock are shown in Table 1. And during the period of measurement, we got the parameters related to rheological property of the rock, which are shown in Table 2.
ð27Þ
Using the parameters and the expressions that we got, the stress expressions of the three regions are obtained (unit of stress: MPa). Viscoelastic region (Rp < r < 1)
(
Substituting (8) into (27),
2 3 2 sin up 2 sin up Rp 1sin up Rp 1sin up 4 rRC c þ C p cot up 15 Rc Rc 2 sin up 1sin up 1 þ sin up R rRC c p þ 1 sin up Rc 2 3 2 sin up 1 þ sin up Rp 1sin up 4 þ C p cot up 15 ¼ 2p0 1 sin up Rc
r0re ¼ 264:05 þ 22:5 r2 r0he ¼ 264:05 þ 22:5 2 r
Plastic region (Rc < r 6 Rp)
(
rrp ¼ 0:82r2 9:94 rhp ¼ 2:46r2 9:94
Cracked region (r0 6 r 6 Rc)
ð28Þ
(
rrc ¼ 4:17r0:7 7:912 rhc ¼ 7:08r0:7 7:912
Substituting the variant expression into Eq. (28), the radius expression Rc of cracked region is obtained.
In the surrounding rock, the position showing stress is inside of roadway, that is r = r0, and the stress of the cracked region is as follows:
3.5. Appearance of the stress and deformation of the surrounding rock
(
After the stress and deformation stability of the surrounding rock, the components of stress include the viscoelastic, plastic and cracked regions. However, in practice we can only test for the stress of the cracked region inside the roadway by using equipment. The deformation of the surrounding rock includes three parts: the instantaneous elastic deformation, the elastoplastic deformation and the deformation caused by rock breaking. But the deformation, which has influences on the supports, is the elastoplastic deformation and the deformation caused by the rock breaking and removing the instantaneous elastic deformation.
u¼
u0Rp e
þ ur0 c ue0
ð29Þ
rrc =r¼r0 ¼ 0 rhc =r¼r0 ¼ 5:53
From the stress expression, inside the roadway, the radial and tangential stresses are 0 and 5.53 MPa respectively, but the value is not large. This is because most of the energy from the rock breaking is obviously between the broken massive rock and the surrounding
Table 1 Basic mechanical parameters of rock. E (GPa)
l
cp (MPa)
up (o)
cc (MPa)
uc(o)
30.0
0.25
5.74
30.0
2.15
15.2
M. Li et al. / International Journal of Mining Science and Technology 22 (2012) 301–306 Table 2 Rock rheological parameters. G0 (MPa)
G1 (MPa)
gre1 (d)
860
610
10
weakened rock, leading to the weak action of the virgin stress on the cracked region. 4.2. Solving RP and RC Using the parameters and Eqs. (25 and 27), the radius of plastic and cracked regions and the thickness d of the rock loose zone are obtained as follows:
8 > < Rp ¼ 5:3 m Rc ¼ 3:98 m > : d ¼ 1:48 m Through computation, the thickness of the rock loose zone is nearly 1.5 m, which is the largest rock loose zone. A large scale of the loosened rock undermines the stability control of the roadway and badly hurt the safety of coal mine production.
Using the parameters shown in Tables 1 and 2 and the Eq. (30), the deformation expression of the surrounding rock was obtained. The u–t curve of the surrounding rock is shown in Fig. 3: 27
u ¼ 87:488 42:9e320t According to the three stages of the surrounding rock deformation, from Fig. 4, it can be seen that at stage I, t = 0 d, u = 44.79 cm; at the stage II, t = 0.2 d, u = 45.31 cm; about 40 d after the roadway excavation, the deformation of the surrounding rock nearly does not change and eventually stables at u = 87.488 cm. In order to validate the theatrical conclusion of the main haulage roadway deformation, we chose the deformation of Qujiang main haulage roadway for our measurements, because the geological and mining conditions of Qujiang are similar to those of Xuzhou. We tested the relative movement of the roof and floor, the two sides at instantaneous, after the exact 0.2 d and the final deformation. After testing, when t = 0, u = 48.21 cm and when t = 0.2 d, u = 50.24 cm. Because the roadway was supported after a short time, so we did not get the final deformation of the surrounding roadway. About after 40 d, the deformation remained stable at about 60 cm. And as time went on, the surrounding rock losing and the loose circle is disappeared. Basing on the stress and deformation characters of the transport roadway, when supporting the surrounding rock, we should in88 80
u (cm)
72 64 56 48 0
20
40
60
crease the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame. And when the rate of deformation of the surrounding rock is descending after the 15 days of excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note and anchor rope) and set aside about 20 cm as the reserved deformation layer. 5. Conclusions (1) Based on the deformation characteristics of the surrounding rock after excavation, the whole stress and deformation process from the excavation to the formation of the cracked region was divided into three stages, and the viscoelastic– plastic–burst mechanics model of the surrounding rock was established. In addition, the deformation curve of the whole excavating-burst process was established. (2) The stress and deformation of the surrounding rock and the respective radius of the plastic and cracked regions in three stages were obtained from the analysis. The deformation which really has influence on the supports is the elastoplastic deformation and is the deformation caused by the rock breaking and removing of the instantaneous elastic deformation:
u ¼ u0Rp e þ ur0 c ue0
4.3. Deformation of the surrounding rock
40
305
80
t (d) Fig. 4. Roadway u–t curve.
100
120
(3) Aiming at the problem which emerged in a mine in Xuzhou during the process of excavating the transport roadway, we need increase the length of anchor bolt and cooperate it with anchor nets cable-U steel supporting frame when supporting the surrounding rock. And when the rate of deformation of the surrounding rock is decreasing after the 15 days of excavation, we should use the ‘‘three anchor’’ supporting method (anchor bolt spray, anchor note, anchor rope) and set aside about 20 cm as the reserved deformation layer.
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