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Redistribution and magnitude of stresses around horse shoe and circular excavations opened in anisotropic rock
International Journal of Mining Science and Technology 25 (2015) 615–621
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Redistribution and magnitude of stresses around horse shoe and circular excavations opened in anisotropic rock Mambou Ngueyep Luc Leroy a,b, Ndop Joseph a,c,⇑, Ndjaka Jean-Marie Bienvenu a a
Laboratory of Material Sciences, Department of Physics, Faculty of Sciences, University of Yaoundé 1, P.O. Box 812, Yaoundé, Cameroon Department of Mine Mineral Processing and Environment, School of Geology and Mining Engineering, University of Ngaoundéré, P.O. Box 115, Meiganga, Cameroon c Douala Institute of Technology, P.O. Box 1623, Douala, Cameroon b
a r t i c l e
i n f o
Article history: Received 26 November 2014 Received in revised form 21 January 2015 Accepted 5 February 2015 Available online 11 June 2015 Keywords: Horse shoe excavation Stress Strain Transverse isotropy Finite element
a b s t r a c t In this paper numerical analysis of underground structures, taking account the transverse isotropy system of rocks, was done using CAST 3M code by varying the shape of excavation and the coefficient of earth pressure k. Numerical results reveal that the anisotropy behavior, the shape of hole and the coefficient of earth pressure k have significant influence to the mining induced stress field and rock deformations which directly control the stability of underground excavation design. The magnitude of horizontal stress obtained for the horse shoe shape excavation (25.2 MPa for k = 1; 52.7 MPa for k = 2) is lower than the magnitude obtained for circular hole (26.4 MPa for k = 1; 59.5 MPa for k = 2). Therefore, we have concluded that the horse shoe shape offers the best stability and the best design for engineer. The anisotropy system presented by rock mass can also influence the redistribution of stresses around hole opened. Numerical results have revealed that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more the rock has the anisotropic behavior, the more the mass of rock around the tunnel is stable. Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Sustainable development of modern society requires the building of underground holes such as subways and railways, highways, material storage, sewage, water transport, hydropower structures [1–5]. All these underground excavations are very necessary, especially to ensure a greater transportation demand and preserve environmental quality [3–5]. Rock at depth is subjected to stresses resulting from the weight of the overlying strata and from horizontal stresses of tectonic origin. When an opening is excavated in this rock, the stress field is locally disrupted and a new set of stresses are induced in the rock surrounding the opening. Knowledge of the magnitudes and directions of these in situ and induced stresses is an essential component of underground excavation design since, in many cases, the strength of the rock is exceeded and the resulting instability can have serious consequences on the behavior of the excavations. ⇑ Corresponding author. Tel.: +237 77977902. E-mail address: [email protected] (J. Ndop).
The redistribution of near-field stresses following tunnel excavation has been studied extensively using a number of analytical, physical and numerical modeling techniques [1–5]. An analytical solution for the stress distribution in a stressed elastic plate containing a circular hole was published and this formed the basis for many early studies of rock behavior around tunnels. These studies concern in many cases, the circular shape of hole and the model most used are based on homogenous and isotropic rock with linear elastic behavior. However to the best of our knowledge, the numerical analysis of redistribution of stresses around the circular and the horse shoe shapes of excavation, in the case of transverse isotropic system mass of rock, remains unaddressed. The transverse isotropy system is a system typically found in the majority of rock and it has a great importance in rock mechanics [6]. In order to quantify state of stress and deformation for the designs of underground structures with different shapes, a two-dimensional boundary element numerical modeling will be performed by using CAST 3M code based on the finite element method. The aim of the study reported here is to examine the
http://dx.doi.org/10.1016/j.ijmst.2015.05.015 2095-2686/Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
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distribution of stresses around the opening after excavation and to decide how it varies with the shape and in situ stress state. This paper is organized as follows. The system under study is presented and modeled in Section 2 using Raleigh Ritz’s method and the constitutive law of rock mechanics. Section 3 is devoted to the numerical simulations of state of stress, deformation around the hole with different shapes (circular, horse shoe). Finally, the conclusion is given in section 4. 2. Theory In this section, we consider an underground excavation opened at 500 m of depth in the granitic mass of rock, and a bloc of rock around the hole opened as shown in Fig. 1. 2.1. In situ stresses Rock at depth is subjected to stresses resulting from the weight of the overlying strata and from horizontal stresses of tectonic origin. When an opening is excavated in this rock, the stress field is locally disrupted and a new set of stress is induced in the rock surrounding the opening. This is because rock, which previously contained stresses, has been removed and the load must be redistributed. The magnitudes of pre-existing in situ stresses have been found to vary widely, depending upon the geological history of the rock mass in which they are measured [7]. The weight of the overlying rock mass was recognized for more than a century as a primary source of stress around underground opening. The vertical stress can be estimated by the weight of the overburden.
rv ¼ cgH
ð1Þ
where c = 2600 kg/m3 is a density of granitic rock specimen, H = 500 m is the depth of the overlying rock mass, and g the constant of gravity. However it is very difficult to estimate the horizontal stresses. In general the horizontal stresses are estimated by assuming a suitable lateral earth pressure ratio [8].
m 1m
rv
ð2Þ
where m is the Poisson’s modulus and k is the stress ratio. High horizontal stresses are caused by factor relating to erosion, tectonics, rock anisotropy, local effects near discontinuities and scale effect.
Many underground excavations are irregular in shape and are frequently grouped close to other excavation. In addition, because of the presence of geological features such as faults and intrusions, the rock properties are seldom uniform within the rock volume of interest. Consequently, the closed form solutions described earlier are of limited value in calculating the stresses, displacements and failure of the rock mass surrounding underground excavations. These problems require using numerical techniques. In this work, CAST 3M code, based on finite element method, is applied to simulate the stress and strain distribution around different holes. The finite element method uses Raleigh– Ritz’s method based on principle of potential energy minimum to establish the equation of displacement as function of external forces. Recent field and laboratory studies have shown that, even at very small strains, many soils exhibit non-linear stress–strain behavior. Nevertheless, because of its convenience, linear elasticity will continue to play an important role in the analysis of such problems as settlement, deformation and soil-structure interaction. Let consider an elastic and isothermal bloc of rock around excavation, and submit to external force as shown in Fig. 1, the potential energy is given by:
P¼
Z 1 2 v
rt edv
Z v
ut fdv
Z X uti pi ut Tds
1 2
P ¼ qt Kq qt F
ð4Þ
where K is the matrix of rock rigidity which contains the elastic constancies of the rock (elastic modulus, Poisson’s modulus), q the global displacement vector of the knot, F the global charges vector. By minimizing the potential energy of the rock, we obtain the general equation of global displacement of the knot as a function of applied forces, given by:
½Kq ¼ F
ð5Þ
The algorithm of the resolution of this equation in the case of linear elastic and anisotropic medium is given by Fig. 2.
Vertical in situ stress σ y
Horizontal in situ stress σ h 2
σ1
σ h1
Horizontal tunnel
σ3
σ2
σz
τ xx
τ zy τ yx
τ yz
y
x z
τ xz
τxy
σx
σy
Induced principal stresses
(a)
ð3Þ
s
i
where r is the internal stress of the rock, e the strain of the rock, u the displacement of the rock, p the punctual external force, T the external stress applied, s the surface of the bloc of rock and V the volume of the bloc of rock. By applying the Raleigh Ritz‘s method, the potential energy becomes:
Horizontal in situ stress
rh ¼ krv ¼
2.2. Finite element method
(b)
Fig. 1. Geometry of tunnel opened (a) and the enlargement of the bloc of rock around excavation submitted to stresses (b).
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617
Let us consider that the massif has transverse isotropic behavior and the influence of heterogeneity is minimized. It means that each point within the massif is traversed by a plane where the elastic properties are the same regardless of the direction considered. From Lekhnitskii [10], we have:
Definition of the geometry and meshing creation of a medium
Enter the mechanical parameters of the medium and boundary conditions
10 exx 1 0 c11 c12 c13 0 0 0 rxx 1 C B eyy C B c B 0 0 0 C C B 21 c11 c23 CB ryy C B C B CB C B B ezz C B c31 c23 c33 0 0 0 CB rzz C C¼B CB C B C Bc C B 0 B 0 0 2ðc11 c12 Þ 0 0 C CB sxy C B xy C B C B CB C B @ czy A @ 0 0 0 0 c44 0 A@ syz A czx sxz 0 0 0 0 0 c44 0
Definition of a model behaviour, symmetry system and calculation of rigidity matrix Calculation of the displacements of nodel points Calculation of stresses and strains of a medium
ð6Þ
where No
Convergence test
c11 ¼ E11 ; c33 ¼ E12 ; E1 ¼ Ey ¼ Ex ;
Yes
c12 ¼ m1 =E1 ; c13 ¼ m2 =E2 ; c44 ¼ 1=G2 ; 2ðc11 c12 Þ ¼ 1=G1
Print results Fig. 2. Algorithm of CAST 3M code for elastic problem.
2.3. Shape of underground excavations The stability of underground excavations is affected by its shape, size of opening, in situ stress, and soil conditions [4,5]. Even though the shape of the opening mainly depends on the purpose for which it is to be used, the safe design and construction of an underground opening requires the knowledge of the stress distribution and the displacements which occur in and around the openings. In this study, circular and horse shoe geometries are considered for the design of underground excavations of tunnels. These different shapes of excavations are the shape mostly found in literature and the underground structures. We have considered that all these excavations have the same surface S = 24 m2 but different shapes. We use triangular element for mesh creation as shown in Fig. 3.
The parameters E1 and m1 are, respectively, Young’s modulus and Poisson’s ratio in the plane of isotropy, E2 is Young’s modulus perpendicular to the plane of isotropy, m2 is Poisson’s ratio giving the transversal strain in the plane of isotropy resulting from an axial strain perpendicular to the same plane, and G2 the shear modulus in the planes perpendicular to the plane of isotropy. As in Ref. [11], the mechanic parameters used for modeling are
E1 ¼ 50 GPa;E2 ¼ 1 GPa;V 1 ¼ 0:1; G1 ¼ 8:065 GPa;V 2 ¼ 0:09; G2 ¼ 7 GPa In the next section the analyses were done by varying the in situ stress ratio (lateral earth pressure ratio) as 0.25, 0.5, 1, 1.5, 2 and 2.5 to see how the value affects the redistribution of stresses. The analyses were done for circular and horse shoe shape of tunnels. The redistribution of stresses around the excavations was observed by plotting principal stresses occurring in x (horizontal) and y (vertical) directions. 3. Numerical analysis
2.4. Mechanical properties of granitic mass of rock
15 m
In this work we choose to deal with the granitic rock due to the fact that it is largely used in civil engineering, building of rail way, road and tunnels. Many tunnels are built in the granitic rock massifs [9]. Rock-mass quality (i.e. adopted mechanical parameters) has an important influence on the development of mining induced stress field and characteristics of rock deformation, which directly controls the stability and safety of the underground excavation of the tunnel. Let assume that all these underground excavations are opened in the granitic mass of rock. In this case of hard rocks (brittle material), it will not yield and no plastic behavior in the material before failure will be observed. Thus, the material is considered to behave in a linear elastic mode.
15 m
(a) Circular hole
(b) Horse shoe hole
Fig. 3. Meshing of holes.
During underground excavation and mining, vertical loading and horizontal displacement on the rock strata can change noticeably. In the following subsections we plan to analyze the stresses and strains occurring around the different shapes of excavations as a function of stress ratio k. We will consider the massif with transverse isotropic system and then compare to the result obtained from isotropic system. The in situ vertical stress is considered as constant and equal to rv=13 MPa. Then only the lateral in situ stress is variable. In addition, we suppose that the excavations are only submitted to in situ stresses. In Figs. 4–11, as a principle in rock mechanics, the stress value indicated in minus sign mean compressive stress and the positive sign mean tensional stress. 3.1. Redistribution and magnitudes of stresses and strains around circular excavation Fig. 4 presents the magnitude and redistribution of principal stresses and strains around tunnel which is confined by a vertical stress greater than the horizontal stress: k = 0.5. In Fig. 4a and c, the parts of the excavation colored in blue represent the zone of compressive stress. The strain and the stress value at this zone are indicated in minus sign. The maximum value of the horizontal stress is 9.93 MPa and the maximum value of horizontal principal strain is 9.9 104 m. The maximum of the tensional stress is negligible (1.03 103 Pa) compared to compressive stresses. The parts of the hole colored in red represent the zone of tensional stress. This zone is smaller than the zone of compressive stress.
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7.98E 1.03E 1.97E 2.92E 3.86E 4.81E 5.75E 6.70E 7.64E 8.59E 9.53E
04 06 06 06 06 06 06 06 06 06 06
(a)
2.85E 4.41E 8.54E 1.27E 1.68E 2.09E 2.51E 2.92E 3.33E 3.74E 4.16E
05 06 06 07 07 07 07 07 07 07 07
1.40E 7.04E 1.55E 2.39E 3.24E 4.08E 4.92E 5.77E 6.61E 7.45E 8.30E
05 05 04 04 04 04 04 04 04 04 04
(b) 7.67E 2.60E 1.29E 2.31E 3.34E 4.37E 5.39E 6.42E 7.45E 8.47E 9.50E
05 05 04 04 04 04 04 04 04 04 04
(c)
(d)
In Fig. 4b and d, the parts of the tunnel colored in red represent the zone of tensional stress and the part colored in blue represents the zone of compressive stresses. This red zone is smaller than the zone of compressive stress. The maximum value of vertical principal stress is 43.3 MPa and the maximum value of vertical principal strain is 5.65 104 m. The maximum value of vertical stress (43.3 MPa) is more than the maximum value of horizontal stress (9.93 MPa). Globally this excavation is confined by the compressive stresses. This hole is more affected by vertical principal stress than the horizontal principal stress. The magnitude and redistribution of principal stresses and strains around excavation which is confined to a hydrostatic stress (k = 1) are present in Fig. 5. In Fig. 5, the compressive stress is concentrated in the part of the tunnel colored in red and the zone colored in blue represents the zone of tensional stress. The maximum value of horizontal stress is 26.4 MPa and the maximum value of vertical stress is 26.7 MPa. Fig. 6 presents the redistribution and the magnitude of principal stresses and strains around a tunnel which is submitted to a vertical stress greater than the horizontal stress: k = 2.
05 06 06 06 07 07 07 07 07 07 07
(a)
(a)
1.71E 2.72E 5.27E 7.82E 1.04E 1.29E 1.55E 1.80E 2.06E 2.31E 2.57E
05 06 06 06 07 07 07 07 07 07 07
4.89E 7.09E 6.31E 1.19E 1.75E 2.31E 2.87E 3.43E 3.99E 4.55E 5.11E
05 05 05 04 04 04 04 04 04 04 04
6.68E 3.37E 5.77E 3.25E 6.57E 9.88E 1.32E 1.65E 1.98E 2.31E 2.64E
06 06 04 06 06 06 07 07 07 07 07
1.33E 6.94E 5.87E 5.76E 1.21E 1.85E 2.48E 3.12E 3.75E 4.39E 5.02E
04 05 06 05 04 04 04 04 04 04 04
(b) 4.05E 5.67E 1.14E 1.71E 2.28E 2.85E 3.42E 4.00E 4.57E 5.14E 5.71E
06 04 03 03 03 03 03 03 03 03 03
(d)
Fig. 6. Magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy developed around the circular hole opened with stress ratio k = 2.
In Fig. 6a and c, the zone of the excavation colored in blue represents the zone of compressive stress. The strain and the stress value at this zone are indicated in minus sign. The maximum value of the horizontal stress is 59.5 MPa and the maximum value of horizontal principal strain is 5.95 103 m. The parts of the hole colored in red represent the zone of tensional stress. In Fig. 6b and d, the redistribution of stresses is less concentrated around the excavation. The maximum value of vertical principal stress is 27.8 MPa and the maximum value of vertical principal strain is 5.29 104 m. The maximum value of vertical stress (59.5 MPa) is more than the maximum value of horizontal stress (27.8 MPa). Globally the redistribution of stresses around the circular hole increases with the stress ration k.
3.2. Redistribution and magnitudes of stresses and strains around horse shoe excavation Fig. 7 presents horse shoe tunnel which is confined to a vertical stress greater than the horizontal stress k = 0.5.
5.95E 1.70E 2.80E 3.90E 5.00E 6.11E 7.21E 8.31E 9.41E 1.05E 1.16E
05 06 06 06 06 06 06 06 06 07 07
(a)
1.67E 3.40E 8.47E 1.35E 1.86E 2.37E 2.87E 3.38E 3.89E 4.39E 4.90E
06 06 06 07 07 07 07 07 07 07 07
5.21E 5.08E 1.54E 2.57E 3.59E 4.62E 5.65E 6.68E 7.71E 8.74E 9.77E
05 05 04 04 04 04 04 04 04 04 04
(b)
(b) 3.06E 2.25E 4.81E 7.38E 9.94E 1.25E 1.51E 1.76E 2.02E 2.27E 2.53E
(c)
04 06 07 07 07 07 07 07 07 07 07
(c)
Fig. 4. Plot of magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the circular hole opened with stress ratio k = 0.5.
2.22E 2.73E 5.24E 7.75E 1.03E 1.28E 1.53E 1.78E 2.03E 2.28E 2.53E
6.20E 5.65E 1.14E 1.71E 2.28E 2.85E 3.42E 3.99E 4.57E 5.14E 5.71E
05 04 04 04 04 03 03 03 03 03 03
4.90E 1.18E 2.31E 3.44E 4.57E 5.71E 6.84E 7.97E 9.10E 1.02E 1.14E
(c)
06 04 04 04 04 04 04 04 04 03 03
(d)
(d)
Fig. 5. Magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy developed around a circular hole opened with stress ratio k = 1.
Fig. 7. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 0.5.
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In Fig. 7a and c, the zone of the excavation colored in blue color represents the zone of compressive stress. The maximum value of the horizontal stress is 12.1 MPa and the maximum value the horizontal principal strain is 1.18 103 m. The parts of the hole with a red color represent the zone of tensional stress. The roof and the two corners of excavations are confined to compressive stresses and the two sides are solicited by tensional stresses. We observe that the roof and the corners of the tunnel are more solicited than the sides of the tunnel. The maximum value of the horizontal strain is observed at the roof and the corners of hole. In Fig. 7b and d, the maximum value of vertical principal stress is 51.1 MPa and the maximum value of vertical principal strain is 1.02 104 m. Only the corners of hole are more solicited by the vertical principal stress (51.1 MPa). Fig. 8 presents the redistribution of stresses and strains around horse shoe tunnel submitted to hydrostatic stress k = 1. In Fig. 8, the maximum value of horizontal stress is 25.2 MPa and the maximum value of horizontal strain is 2.5 103 m. The Maximum value of vertical stress is 54.42 MPa and maximum value of vertical strain is 1.08 103 m. Fig. 9 presents the redistribution of stresses and strains around horse shoe tunnel submitted to vertical stress less than the horizontal stress (k = 2). The maximum value of horizontal stress is 52.7 MPa and the maximum value of horizontal strain is 5.26 103 m. The maximum value of vertical stress is 60.1 MPa and the maximum value of vertical strain is 1.19 103 m. The hole is more affected by the vertical. Figs. 8 and 9 show the similar results obtained in Fig. 7. The maximum value of horizontal stress is 52.7 MPa and the maximum value of horizontal strain is 5.26 103 m. The maximum value of vertical stress is 60.1 MPa and the maximum value of vertical strain is 1.19 103 m. The roof and the two corners are more solicited than the case of k < 1. The increase in lateral pressure affects more the roof and the corners than the vertical in situ stress. 3.3. Influence on the shape of a tunnel opened in an isotropic transverse rock mass In Fig. 10, we study the influence on the shape of an excavation opened in an isotropic transverse rock.
8.11E 3.15E 5.50E 7.84E 1.02E 1.25E 1.49E 1.72E 1.95E 2.19E 2.42E
05 06 06 06 07 07 07 07 07 07 07
(a)
05 06 06 07 07 07 07 07 07 07 07
5.13E 5.72E 1.66E 2.74E 3.83E 4.91E 6.00E 7.08E 8.17E 9.25E 1.03E
05 05 04 04 04 04 04 04 04 04 03
(b) 3.36E 2.70E 5.07E 7.43E 9.80E 1.22E 1.45E 1.69E 1.93E 2.16E 2.40E
(c)
6.55E 4.61E 9.88E 1.51E 2.04E 2.57E 3.09E 3.62E 4.15E 4.68E 5.20E
05 04 04 04 04 03 03 03 03 03 03
(d)
Fig. 8. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 1.
1.04E 6.00E 1.10E 1.59E 2.09E 2.58E 3.08E 3.57E 4.07E 4.57E 5.06E
06 06 07 07 07 07 07 07 07 07 07
(a)
06 06 06 07 07 07 07 07 07 07 07
1.94E 6.06E 7.23E 2.05E 3.38E 4.71E 6.04E 7.37E 8.70E 1.00E 1.14E
04 05 05 04 04 04 04 04 04 03 03
(b) 7.61E 5.74E 1.07E 1.57E 2.07E 2.56E 3.06E 3.56E 4.06E 4.55E 5.05E
(c)
9.49E 2.81E 3.87E 1.06E 1.72E 2.39E 3.06E 3.73E 4.40E 5.06E 5.73E
05 04 03 03 03 03 03 03 03 03 03
(d)
Fig. 9. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 2.
Fig. 10a shows that, for the shapes considered, the maximum horizontal stresses increase when lateral earth pressure ratio increases. This means that an increase in the horizontal in situ stresses causes the high stress around hole and can lead to rapid failure of the rock around the opened excavation. This figure also shows that the circular shape of an excavation makes the appearance of the highest horizontal stresses around the excavation and the horse shoe excavation makes the appearance of the lowest horizontal stresses around the excavation. In poor quality soil masses or in tunnels at great depth, the circular shape is not a good choice because of the high stress concentrations at the corners. In some cases, failures initiated at these corners can lead to severe floor heave and even to failure of the entire tunnel perimeter. Similar observations as shown in Fig. 10a can be found as for the horizontal strains in Fig. 10c. Meanwhile, in Fig. 10b and d, according to vertical stress and strain, the observations are contrary. In Fig. 10b and d, it is clear that the magnitude of stress distribution and strain distribution are more for a horse shoe tunnels as compared to the circular shape. Thus, they are not preferable for the best design of excavations. The lowest magnitude of stress and strain distribution is obtained for circular shapes. For the circular shape, vertical stress begins to decrease with increase in the stress ratio and then starts to increase above k = 2. For the horse shoe shape, the vertical stress increases when stress ratio increases. Globally in Fig. 10a and b, we observe that the horizontal stress mostly affects mass rock around excavations than vertical stress. It is clear that the shape of excavation has a great influence on the redistribution of stresses around opened excavation. In order to verify if all these observations are the same in the case of linear elastic isotropic body, let us plot the stress and strain distribution around opened excavations in an isotropic mass of rock. 3.4. Influence on the shape of an opened tunnel in an isotropic mass of rock The influence on the shape of an opened excavation in an isotropic mass of rock is shown in Fig. 11. Contrary to the results obtained in the Fig. 10, Fig. 11 shows that the stress and the strain distributions (horizontal and vertical) are lower for circular shape than the horse shoe shape. It means that for isotropic rock, the circular shape offers the best stability of
40
Circular hole
20
Horseshoe hole
0 1 2 3 Lateral earth pressure ratio (k)
60 40 20
6 4 2 0
0 1 2 3 Lateral earth pressure ratio (k)
(a)
8
Maximum vertical strain ε xx (mm)
60
Maximum horizontal strain ε xx (mm)
80
Maximum vertical stress σ yy (MPa)
N.L.L. Mambou et al. / International Journal of Mining Science and Technology 25 (2015) 615–621
Maximum horizontal stress σ xx (MPa)
620
2
4
0.8 0.4
1 2 3 0 Lateral earth pressure ratio (k)
Lateral earth pressure ratio (k)
(b)
1.2
(c)
(d)
Circular hole Horseshoe hole
40
0
1 2 3 Lateral earth pressure ratio (k)
120 80 40
0 1 2 3 Lateral earth pressure ratio (k)
(a)
2.5
2.0
Maximum vertical strain ε xx (mm)
80
Maximum horizontal strain ε xx (mm)
120
Maximum vertical stress σ yy (MPa)
Maximum horizontal stress σ xx (MPa)
Fig. 10. Influence on the shape of a tunnel in the case of isotropic transverse rock of mass (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) horizontal principal strains exx versus lateral earth pressure ratio and (d) vertical principal strains eyy versus lateral earth pressure ratio.
2.0 1.5 1.0 0.5
1.6 1.2 0.8 0.4
1 2 3 0 Lateral earth pressure ratio (k)
(b)
1 2 3 0 Lateral earth pressure ratio (k)
(c)
(d)
Fig. 11. Influence on the shape of a tunnel in the case of isotropic rock of mass (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) horizontal principal strains exx versus lateral earth pressure ratio and (d) vertical principal strains eyy versus lateral earth pressure ratio.
60 40 20
0 1 2 3 Lateral earth pressure ratio (k)
(a)
2.0
100 80 60 40 20 0 1 2 3 Lateral earth pressure ratio (k)
8
Maximum vertical strain ε yy (mm)
80
120
Maximum horizontal strain ε xx (mm)
Circular hole (isotropic mass of rock) Circular hole (anisotropic mass of rock) Horseshoe hole (isotropic mass of rock) Horseshoe hole (anisotropic mass of rock)
100
Maximum vertical stress σ yy (MPa)
Maximum horizontal stress σ xx (MPa)
120
6 4 2 1 2 3 0 Lateral earth pressure ratio (k)
(b)
(c)
1.6 1.2 0.8 0.4 1 2 3 0 Lateral earth pressure ratio (k)
(d)
Fig. 12. Influence of the anisotropy of the mass of rock tunnel: (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) maximum horizontal principal strains exx versus lateral earth pressure ratio and (d) maximum vertical principal strains eyy versus lateral earth pressure ratio.
underground excavation. For all the shapes, the redistribution of horizontal stresses and strains increases gradually when lateral pressure increases. The variation of the redistribution of vertical stress and vertical strain is more complex. They increase slowly for horse shoe hole when stress ratio increases; but decrease first and slowly increase above k = 1.5 for circular shape. In order to better understand the influence of anisotropy of rock, let us compare stress and strain distribution in the case of an isotropic mass of rock to transverse isotropic mass of rock.
isotropic rock is less than those obtained in the case of isotropic rock. Numerical result reveals that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more the rock has the anisotropic behavior, the more the mass of rock around the tunnel is stable.
3.5. Influence of the transverse isotropy system on the redistribution of stresses around an opened excavation
4. Conclusions
In Fig. 12, we study the influence on the transverse isotropy system on the redistribution of stresses around the circular and horse shoe shapes of opened excavations. Fig. 12 clearly shows that the anisotropic behavior affects more the redistribution stresses and strains in the case of horse shoe shape than the circular shape. We observed that for all the shapes, the magnitude of redistribution of stresses obtained for transverse
This work dealt with the numerical analysis of underground structures, taking accounts the transverse isotropy system of rocks. The analysis was done using CAST 3M code by varying the shape of excavation and the coefficient of earth pressure k. Numerical results have shown that the many factors can influence the behavior of rocks around underground openings: the in-situ stress condition; the rock mass strength; the anisotropy behavior of the rock mass; and the geometry of excavation.
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The in situ horizontal stress (lateral earth pressure) causes more instability of rock around the hole than the vertical in situ stress. For all the shapes considered, the redistribution of horizontal stresses and strains increases gradually when lateral pressure increases. For circular shape, the maximum horizontal stress is 9.93, 26.4 and 59.5 MPa respectively for k = 0.5; 1; 2. For horse shoe shape, the maximum horizontal stress is 12.1, 25.2 and 52.7 MPa respectively for k = 0.5; 1; 2. The high horizontal and vertical strains observed around the tunnels indicate maximum volume change toward those regions. These results can help to design a rock bolts installed systematically around tunnels (regularly spaced around the tunnel) excavated in rock masses and which permit to stabilize instable rock mass. The design of geometry of tunnel influences the redistribution of stress and strain. The magnitude of horizontal stress obtained for the horse shoe shape excavation (25.2 MPa for k = 1; 52.7 MPa for k = 2) is lower than the magnitude obtained for circular hole (26.4 MPa for k = 1; 59.5 MPa for k = 2). Therefore, we conclude that the horse shoe shape offers the best stability and the best design for engineer. However according to vertical stress and strain, the circular shape offers the best design. These results help the engineers, geoscientists and tunnel designers to understand mining induced stress and possible deformation that developed for the advancement of circular and horse shoe tunnel and its possible influences on the rock strata. The anisotropy system presented by rock mass can also influence the redistribution of stresses around hole opened. Numerical result reveals that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more
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the rock has the anisotropic behavior the more the mass of rock around the tunnel is stable. An interesting work under investigation is the analysis of the redistribution and magnitude of stresses and strains around particular shapes of hole, as trapezium, rectangular and vault shapes; where the rock presents the anisotropic behavior. References [1] Lu Y, Yang W. Analytical solutions of stress and displacement in strain softening rock mass around a newly formed cavity. J Cent South Univ 2013; 20:1397–404. [2] Duddeck H. Application of numerical analysis for tunnelling. Int J Numer Anal Methods Geomech 1991;15(4):223–39. [3] Raji M, Sitharam TG. Stress distribution around the tunnel: influence of in situ stress and shape of tunnel. In: Proceedings of Indian geotechnical conference, Kochi; 2011. [4] Alejano LR, Rodriguez-Dono A, Alonso E, Fdez-Manin G. Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behavior. Tunnel Undergr Space Technol 2009;24(6): 689–705. [5] Carranza-Torres C. Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 2004;41(1):1–11. [6] Berry P, Crea G, Martino D, Ribacchi R. The influence of fabric on the deformability of anisotropic rocks. In: Proc 3rd Int Cong Isrm, Denver; 1974. p. 105–10. [7] Brown ET, Bray JW, Ladanyi B, Hoek E. Ground response curve for rock tunnels. J Geotech Eng 1983;109(1):15–39. [8] Eberhardt E, Stead D, Reeves MJ, Connors C. Design of tabular excavations in foliated rock: an integrated numerical modeling approach. Geotech Geol Eng 1997;15(1):47–85. [9] Takarli M, Prince-Agbodjan W. Temperature effects on physical properties and mechanical behavior of granite: experimental investigation of material damage. J ASTM Int 2008;5(3):13. [10] Gonzaga Gomez G, Leite MH, Corthesy R. Determination of anisotropic deformability parameters from a single standard rock specimen. Int J Rock Mech Min Sci 2008;45:1420–38. [11] Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body. San Francisco: Holden Day; 1963.
Contents lists available at ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Redistribution and magnitude of stresses around horse shoe and circular excavations opened in anisotropic rock Mambou Ngueyep Luc Leroy a,b, Ndop Joseph a,c,⇑, Ndjaka Jean-Marie Bienvenu a a
Laboratory of Material Sciences, Department of Physics, Faculty of Sciences, University of Yaoundé 1, P.O. Box 812, Yaoundé, Cameroon Department of Mine Mineral Processing and Environment, School of Geology and Mining Engineering, University of Ngaoundéré, P.O. Box 115, Meiganga, Cameroon c Douala Institute of Technology, P.O. Box 1623, Douala, Cameroon b
a r t i c l e
i n f o
Article history: Received 26 November 2014 Received in revised form 21 January 2015 Accepted 5 February 2015 Available online 11 June 2015 Keywords: Horse shoe excavation Stress Strain Transverse isotropy Finite element
a b s t r a c t In this paper numerical analysis of underground structures, taking account the transverse isotropy system of rocks, was done using CAST 3M code by varying the shape of excavation and the coefficient of earth pressure k. Numerical results reveal that the anisotropy behavior, the shape of hole and the coefficient of earth pressure k have significant influence to the mining induced stress field and rock deformations which directly control the stability of underground excavation design. The magnitude of horizontal stress obtained for the horse shoe shape excavation (25.2 MPa for k = 1; 52.7 MPa for k = 2) is lower than the magnitude obtained for circular hole (26.4 MPa for k = 1; 59.5 MPa for k = 2). Therefore, we have concluded that the horse shoe shape offers the best stability and the best design for engineer. The anisotropy system presented by rock mass can also influence the redistribution of stresses around hole opened. Numerical results have revealed that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more the rock has the anisotropic behavior, the more the mass of rock around the tunnel is stable. Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Sustainable development of modern society requires the building of underground holes such as subways and railways, highways, material storage, sewage, water transport, hydropower structures [1–5]. All these underground excavations are very necessary, especially to ensure a greater transportation demand and preserve environmental quality [3–5]. Rock at depth is subjected to stresses resulting from the weight of the overlying strata and from horizontal stresses of tectonic origin. When an opening is excavated in this rock, the stress field is locally disrupted and a new set of stresses are induced in the rock surrounding the opening. Knowledge of the magnitudes and directions of these in situ and induced stresses is an essential component of underground excavation design since, in many cases, the strength of the rock is exceeded and the resulting instability can have serious consequences on the behavior of the excavations. ⇑ Corresponding author. Tel.: +237 77977902. E-mail address: [email protected] (J. Ndop).
The redistribution of near-field stresses following tunnel excavation has been studied extensively using a number of analytical, physical and numerical modeling techniques [1–5]. An analytical solution for the stress distribution in a stressed elastic plate containing a circular hole was published and this formed the basis for many early studies of rock behavior around tunnels. These studies concern in many cases, the circular shape of hole and the model most used are based on homogenous and isotropic rock with linear elastic behavior. However to the best of our knowledge, the numerical analysis of redistribution of stresses around the circular and the horse shoe shapes of excavation, in the case of transverse isotropic system mass of rock, remains unaddressed. The transverse isotropy system is a system typically found in the majority of rock and it has a great importance in rock mechanics [6]. In order to quantify state of stress and deformation for the designs of underground structures with different shapes, a two-dimensional boundary element numerical modeling will be performed by using CAST 3M code based on the finite element method. The aim of the study reported here is to examine the
http://dx.doi.org/10.1016/j.ijmst.2015.05.015 2095-2686/Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
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distribution of stresses around the opening after excavation and to decide how it varies with the shape and in situ stress state. This paper is organized as follows. The system under study is presented and modeled in Section 2 using Raleigh Ritz’s method and the constitutive law of rock mechanics. Section 3 is devoted to the numerical simulations of state of stress, deformation around the hole with different shapes (circular, horse shoe). Finally, the conclusion is given in section 4. 2. Theory In this section, we consider an underground excavation opened at 500 m of depth in the granitic mass of rock, and a bloc of rock around the hole opened as shown in Fig. 1. 2.1. In situ stresses Rock at depth is subjected to stresses resulting from the weight of the overlying strata and from horizontal stresses of tectonic origin. When an opening is excavated in this rock, the stress field is locally disrupted and a new set of stress is induced in the rock surrounding the opening. This is because rock, which previously contained stresses, has been removed and the load must be redistributed. The magnitudes of pre-existing in situ stresses have been found to vary widely, depending upon the geological history of the rock mass in which they are measured [7]. The weight of the overlying rock mass was recognized for more than a century as a primary source of stress around underground opening. The vertical stress can be estimated by the weight of the overburden.
rv ¼ cgH
ð1Þ
where c = 2600 kg/m3 is a density of granitic rock specimen, H = 500 m is the depth of the overlying rock mass, and g the constant of gravity. However it is very difficult to estimate the horizontal stresses. In general the horizontal stresses are estimated by assuming a suitable lateral earth pressure ratio [8].
m 1m
rv
ð2Þ
where m is the Poisson’s modulus and k is the stress ratio. High horizontal stresses are caused by factor relating to erosion, tectonics, rock anisotropy, local effects near discontinuities and scale effect.
Many underground excavations are irregular in shape and are frequently grouped close to other excavation. In addition, because of the presence of geological features such as faults and intrusions, the rock properties are seldom uniform within the rock volume of interest. Consequently, the closed form solutions described earlier are of limited value in calculating the stresses, displacements and failure of the rock mass surrounding underground excavations. These problems require using numerical techniques. In this work, CAST 3M code, based on finite element method, is applied to simulate the stress and strain distribution around different holes. The finite element method uses Raleigh– Ritz’s method based on principle of potential energy minimum to establish the equation of displacement as function of external forces. Recent field and laboratory studies have shown that, even at very small strains, many soils exhibit non-linear stress–strain behavior. Nevertheless, because of its convenience, linear elasticity will continue to play an important role in the analysis of such problems as settlement, deformation and soil-structure interaction. Let consider an elastic and isothermal bloc of rock around excavation, and submit to external force as shown in Fig. 1, the potential energy is given by:
P¼
Z 1 2 v
rt edv
Z v
ut fdv
Z X uti pi ut Tds
1 2
P ¼ qt Kq qt F
ð4Þ
where K is the matrix of rock rigidity which contains the elastic constancies of the rock (elastic modulus, Poisson’s modulus), q the global displacement vector of the knot, F the global charges vector. By minimizing the potential energy of the rock, we obtain the general equation of global displacement of the knot as a function of applied forces, given by:
½Kq ¼ F
ð5Þ
The algorithm of the resolution of this equation in the case of linear elastic and anisotropic medium is given by Fig. 2.
Vertical in situ stress σ y
Horizontal in situ stress σ h 2
σ1
σ h1
Horizontal tunnel
σ3
σ2
σz
τ xx
τ zy τ yx
τ yz
y
x z
τ xz
τxy
σx
σy
Induced principal stresses
(a)
ð3Þ
s
i
where r is the internal stress of the rock, e the strain of the rock, u the displacement of the rock, p the punctual external force, T the external stress applied, s the surface of the bloc of rock and V the volume of the bloc of rock. By applying the Raleigh Ritz‘s method, the potential energy becomes:
Horizontal in situ stress
rh ¼ krv ¼
2.2. Finite element method
(b)
Fig. 1. Geometry of tunnel opened (a) and the enlargement of the bloc of rock around excavation submitted to stresses (b).
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617
Let us consider that the massif has transverse isotropic behavior and the influence of heterogeneity is minimized. It means that each point within the massif is traversed by a plane where the elastic properties are the same regardless of the direction considered. From Lekhnitskii [10], we have:
Definition of the geometry and meshing creation of a medium
Enter the mechanical parameters of the medium and boundary conditions
10 exx 1 0 c11 c12 c13 0 0 0 rxx 1 C B eyy C B c B 0 0 0 C C B 21 c11 c23 CB ryy C B C B CB C B B ezz C B c31 c23 c33 0 0 0 CB rzz C C¼B CB C B C Bc C B 0 B 0 0 2ðc11 c12 Þ 0 0 C CB sxy C B xy C B C B CB C B @ czy A @ 0 0 0 0 c44 0 A@ syz A czx sxz 0 0 0 0 0 c44 0
Definition of a model behaviour, symmetry system and calculation of rigidity matrix Calculation of the displacements of nodel points Calculation of stresses and strains of a medium
ð6Þ
where No
Convergence test
c11 ¼ E11 ; c33 ¼ E12 ; E1 ¼ Ey ¼ Ex ;
Yes
c12 ¼ m1 =E1 ; c13 ¼ m2 =E2 ; c44 ¼ 1=G2 ; 2ðc11 c12 Þ ¼ 1=G1
Print results Fig. 2. Algorithm of CAST 3M code for elastic problem.
2.3. Shape of underground excavations The stability of underground excavations is affected by its shape, size of opening, in situ stress, and soil conditions [4,5]. Even though the shape of the opening mainly depends on the purpose for which it is to be used, the safe design and construction of an underground opening requires the knowledge of the stress distribution and the displacements which occur in and around the openings. In this study, circular and horse shoe geometries are considered for the design of underground excavations of tunnels. These different shapes of excavations are the shape mostly found in literature and the underground structures. We have considered that all these excavations have the same surface S = 24 m2 but different shapes. We use triangular element for mesh creation as shown in Fig. 3.
The parameters E1 and m1 are, respectively, Young’s modulus and Poisson’s ratio in the plane of isotropy, E2 is Young’s modulus perpendicular to the plane of isotropy, m2 is Poisson’s ratio giving the transversal strain in the plane of isotropy resulting from an axial strain perpendicular to the same plane, and G2 the shear modulus in the planes perpendicular to the plane of isotropy. As in Ref. [11], the mechanic parameters used for modeling are
E1 ¼ 50 GPa;E2 ¼ 1 GPa;V 1 ¼ 0:1; G1 ¼ 8:065 GPa;V 2 ¼ 0:09; G2 ¼ 7 GPa In the next section the analyses were done by varying the in situ stress ratio (lateral earth pressure ratio) as 0.25, 0.5, 1, 1.5, 2 and 2.5 to see how the value affects the redistribution of stresses. The analyses were done for circular and horse shoe shape of tunnels. The redistribution of stresses around the excavations was observed by plotting principal stresses occurring in x (horizontal) and y (vertical) directions. 3. Numerical analysis
2.4. Mechanical properties of granitic mass of rock
15 m
In this work we choose to deal with the granitic rock due to the fact that it is largely used in civil engineering, building of rail way, road and tunnels. Many tunnels are built in the granitic rock massifs [9]. Rock-mass quality (i.e. adopted mechanical parameters) has an important influence on the development of mining induced stress field and characteristics of rock deformation, which directly controls the stability and safety of the underground excavation of the tunnel. Let assume that all these underground excavations are opened in the granitic mass of rock. In this case of hard rocks (brittle material), it will not yield and no plastic behavior in the material before failure will be observed. Thus, the material is considered to behave in a linear elastic mode.
15 m
(a) Circular hole
(b) Horse shoe hole
Fig. 3. Meshing of holes.
During underground excavation and mining, vertical loading and horizontal displacement on the rock strata can change noticeably. In the following subsections we plan to analyze the stresses and strains occurring around the different shapes of excavations as a function of stress ratio k. We will consider the massif with transverse isotropic system and then compare to the result obtained from isotropic system. The in situ vertical stress is considered as constant and equal to rv=13 MPa. Then only the lateral in situ stress is variable. In addition, we suppose that the excavations are only submitted to in situ stresses. In Figs. 4–11, as a principle in rock mechanics, the stress value indicated in minus sign mean compressive stress and the positive sign mean tensional stress. 3.1. Redistribution and magnitudes of stresses and strains around circular excavation Fig. 4 presents the magnitude and redistribution of principal stresses and strains around tunnel which is confined by a vertical stress greater than the horizontal stress: k = 0.5. In Fig. 4a and c, the parts of the excavation colored in blue represent the zone of compressive stress. The strain and the stress value at this zone are indicated in minus sign. The maximum value of the horizontal stress is 9.93 MPa and the maximum value of horizontal principal strain is 9.9 104 m. The maximum of the tensional stress is negligible (1.03 103 Pa) compared to compressive stresses. The parts of the hole colored in red represent the zone of tensional stress. This zone is smaller than the zone of compressive stress.
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7.98E 1.03E 1.97E 2.92E 3.86E 4.81E 5.75E 6.70E 7.64E 8.59E 9.53E
04 06 06 06 06 06 06 06 06 06 06
(a)
2.85E 4.41E 8.54E 1.27E 1.68E 2.09E 2.51E 2.92E 3.33E 3.74E 4.16E
05 06 06 07 07 07 07 07 07 07 07
1.40E 7.04E 1.55E 2.39E 3.24E 4.08E 4.92E 5.77E 6.61E 7.45E 8.30E
05 05 04 04 04 04 04 04 04 04 04
(b) 7.67E 2.60E 1.29E 2.31E 3.34E 4.37E 5.39E 6.42E 7.45E 8.47E 9.50E
05 05 04 04 04 04 04 04 04 04 04
(c)
(d)
In Fig. 4b and d, the parts of the tunnel colored in red represent the zone of tensional stress and the part colored in blue represents the zone of compressive stresses. This red zone is smaller than the zone of compressive stress. The maximum value of vertical principal stress is 43.3 MPa and the maximum value of vertical principal strain is 5.65 104 m. The maximum value of vertical stress (43.3 MPa) is more than the maximum value of horizontal stress (9.93 MPa). Globally this excavation is confined by the compressive stresses. This hole is more affected by vertical principal stress than the horizontal principal stress. The magnitude and redistribution of principal stresses and strains around excavation which is confined to a hydrostatic stress (k = 1) are present in Fig. 5. In Fig. 5, the compressive stress is concentrated in the part of the tunnel colored in red and the zone colored in blue represents the zone of tensional stress. The maximum value of horizontal stress is 26.4 MPa and the maximum value of vertical stress is 26.7 MPa. Fig. 6 presents the redistribution and the magnitude of principal stresses and strains around a tunnel which is submitted to a vertical stress greater than the horizontal stress: k = 2.
05 06 06 06 07 07 07 07 07 07 07
(a)
(a)
1.71E 2.72E 5.27E 7.82E 1.04E 1.29E 1.55E 1.80E 2.06E 2.31E 2.57E
05 06 06 06 07 07 07 07 07 07 07
4.89E 7.09E 6.31E 1.19E 1.75E 2.31E 2.87E 3.43E 3.99E 4.55E 5.11E
05 05 05 04 04 04 04 04 04 04 04
6.68E 3.37E 5.77E 3.25E 6.57E 9.88E 1.32E 1.65E 1.98E 2.31E 2.64E
06 06 04 06 06 06 07 07 07 07 07
1.33E 6.94E 5.87E 5.76E 1.21E 1.85E 2.48E 3.12E 3.75E 4.39E 5.02E
04 05 06 05 04 04 04 04 04 04 04
(b) 4.05E 5.67E 1.14E 1.71E 2.28E 2.85E 3.42E 4.00E 4.57E 5.14E 5.71E
06 04 03 03 03 03 03 03 03 03 03
(d)
Fig. 6. Magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy developed around the circular hole opened with stress ratio k = 2.
In Fig. 6a and c, the zone of the excavation colored in blue represents the zone of compressive stress. The strain and the stress value at this zone are indicated in minus sign. The maximum value of the horizontal stress is 59.5 MPa and the maximum value of horizontal principal strain is 5.95 103 m. The parts of the hole colored in red represent the zone of tensional stress. In Fig. 6b and d, the redistribution of stresses is less concentrated around the excavation. The maximum value of vertical principal stress is 27.8 MPa and the maximum value of vertical principal strain is 5.29 104 m. The maximum value of vertical stress (59.5 MPa) is more than the maximum value of horizontal stress (27.8 MPa). Globally the redistribution of stresses around the circular hole increases with the stress ration k.
3.2. Redistribution and magnitudes of stresses and strains around horse shoe excavation Fig. 7 presents horse shoe tunnel which is confined to a vertical stress greater than the horizontal stress k = 0.5.
5.95E 1.70E 2.80E 3.90E 5.00E 6.11E 7.21E 8.31E 9.41E 1.05E 1.16E
05 06 06 06 06 06 06 06 06 07 07
(a)
1.67E 3.40E 8.47E 1.35E 1.86E 2.37E 2.87E 3.38E 3.89E 4.39E 4.90E
06 06 06 07 07 07 07 07 07 07 07
5.21E 5.08E 1.54E 2.57E 3.59E 4.62E 5.65E 6.68E 7.71E 8.74E 9.77E
05 05 04 04 04 04 04 04 04 04 04
(b)
(b) 3.06E 2.25E 4.81E 7.38E 9.94E 1.25E 1.51E 1.76E 2.02E 2.27E 2.53E
(c)
04 06 07 07 07 07 07 07 07 07 07
(c)
Fig. 4. Plot of magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the circular hole opened with stress ratio k = 0.5.
2.22E 2.73E 5.24E 7.75E 1.03E 1.28E 1.53E 1.78E 2.03E 2.28E 2.53E
6.20E 5.65E 1.14E 1.71E 2.28E 2.85E 3.42E 3.99E 4.57E 5.14E 5.71E
05 04 04 04 04 03 03 03 03 03 03
4.90E 1.18E 2.31E 3.44E 4.57E 5.71E 6.84E 7.97E 9.10E 1.02E 1.14E
(c)
06 04 04 04 04 04 04 04 04 03 03
(d)
(d)
Fig. 5. Magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy developed around a circular hole opened with stress ratio k = 1.
Fig. 7. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 0.5.
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In Fig. 7a and c, the zone of the excavation colored in blue color represents the zone of compressive stress. The maximum value of the horizontal stress is 12.1 MPa and the maximum value the horizontal principal strain is 1.18 103 m. The parts of the hole with a red color represent the zone of tensional stress. The roof and the two corners of excavations are confined to compressive stresses and the two sides are solicited by tensional stresses. We observe that the roof and the corners of the tunnel are more solicited than the sides of the tunnel. The maximum value of the horizontal strain is observed at the roof and the corners of hole. In Fig. 7b and d, the maximum value of vertical principal stress is 51.1 MPa and the maximum value of vertical principal strain is 1.02 104 m. Only the corners of hole are more solicited by the vertical principal stress (51.1 MPa). Fig. 8 presents the redistribution of stresses and strains around horse shoe tunnel submitted to hydrostatic stress k = 1. In Fig. 8, the maximum value of horizontal stress is 25.2 MPa and the maximum value of horizontal strain is 2.5 103 m. The Maximum value of vertical stress is 54.42 MPa and maximum value of vertical strain is 1.08 103 m. Fig. 9 presents the redistribution of stresses and strains around horse shoe tunnel submitted to vertical stress less than the horizontal stress (k = 2). The maximum value of horizontal stress is 52.7 MPa and the maximum value of horizontal strain is 5.26 103 m. The maximum value of vertical stress is 60.1 MPa and the maximum value of vertical strain is 1.19 103 m. The hole is more affected by the vertical. Figs. 8 and 9 show the similar results obtained in Fig. 7. The maximum value of horizontal stress is 52.7 MPa and the maximum value of horizontal strain is 5.26 103 m. The maximum value of vertical stress is 60.1 MPa and the maximum value of vertical strain is 1.19 103 m. The roof and the two corners are more solicited than the case of k < 1. The increase in lateral pressure affects more the roof and the corners than the vertical in situ stress. 3.3. Influence on the shape of a tunnel opened in an isotropic transverse rock mass In Fig. 10, we study the influence on the shape of an excavation opened in an isotropic transverse rock.
8.11E 3.15E 5.50E 7.84E 1.02E 1.25E 1.49E 1.72E 1.95E 2.19E 2.42E
05 06 06 06 07 07 07 07 07 07 07
(a)
05 06 06 07 07 07 07 07 07 07 07
5.13E 5.72E 1.66E 2.74E 3.83E 4.91E 6.00E 7.08E 8.17E 9.25E 1.03E
05 05 04 04 04 04 04 04 04 04 03
(b) 3.36E 2.70E 5.07E 7.43E 9.80E 1.22E 1.45E 1.69E 1.93E 2.16E 2.40E
(c)
6.55E 4.61E 9.88E 1.51E 2.04E 2.57E 3.09E 3.62E 4.15E 4.68E 5.20E
05 04 04 04 04 03 03 03 03 03 03
(d)
Fig. 8. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 1.
1.04E 6.00E 1.10E 1.59E 2.09E 2.58E 3.08E 3.57E 4.07E 4.57E 5.06E
06 06 07 07 07 07 07 07 07 07 07
(a)
06 06 06 07 07 07 07 07 07 07 07
1.94E 6.06E 7.23E 2.05E 3.38E 4.71E 6.04E 7.37E 8.70E 1.00E 1.14E
04 05 05 04 04 04 04 04 04 03 03
(b) 7.61E 5.74E 1.07E 1.57E 2.07E 2.56E 3.06E 3.56E 4.06E 4.55E 5.05E
(c)
9.49E 2.81E 3.87E 1.06E 1.72E 2.39E 3.06E 3.73E 4.40E 5.06E 5.73E
05 04 03 03 03 03 03 03 03 03 03
(d)
Fig. 9. Plot of the magnitude and distribution of (a) horizontal principal stresses rxx, (b) vertical principal stresses ryy, (c) horizontal principal strains exx and (d) vertical principal strains eyy, developed around the horse shoe tunnel opened with stress ratio k = 2.
Fig. 10a shows that, for the shapes considered, the maximum horizontal stresses increase when lateral earth pressure ratio increases. This means that an increase in the horizontal in situ stresses causes the high stress around hole and can lead to rapid failure of the rock around the opened excavation. This figure also shows that the circular shape of an excavation makes the appearance of the highest horizontal stresses around the excavation and the horse shoe excavation makes the appearance of the lowest horizontal stresses around the excavation. In poor quality soil masses or in tunnels at great depth, the circular shape is not a good choice because of the high stress concentrations at the corners. In some cases, failures initiated at these corners can lead to severe floor heave and even to failure of the entire tunnel perimeter. Similar observations as shown in Fig. 10a can be found as for the horizontal strains in Fig. 10c. Meanwhile, in Fig. 10b and d, according to vertical stress and strain, the observations are contrary. In Fig. 10b and d, it is clear that the magnitude of stress distribution and strain distribution are more for a horse shoe tunnels as compared to the circular shape. Thus, they are not preferable for the best design of excavations. The lowest magnitude of stress and strain distribution is obtained for circular shapes. For the circular shape, vertical stress begins to decrease with increase in the stress ratio and then starts to increase above k = 2. For the horse shoe shape, the vertical stress increases when stress ratio increases. Globally in Fig. 10a and b, we observe that the horizontal stress mostly affects mass rock around excavations than vertical stress. It is clear that the shape of excavation has a great influence on the redistribution of stresses around opened excavation. In order to verify if all these observations are the same in the case of linear elastic isotropic body, let us plot the stress and strain distribution around opened excavations in an isotropic mass of rock. 3.4. Influence on the shape of an opened tunnel in an isotropic mass of rock The influence on the shape of an opened excavation in an isotropic mass of rock is shown in Fig. 11. Contrary to the results obtained in the Fig. 10, Fig. 11 shows that the stress and the strain distributions (horizontal and vertical) are lower for circular shape than the horse shoe shape. It means that for isotropic rock, the circular shape offers the best stability of
40
Circular hole
20
Horseshoe hole
0 1 2 3 Lateral earth pressure ratio (k)
60 40 20
6 4 2 0
0 1 2 3 Lateral earth pressure ratio (k)
(a)
8
Maximum vertical strain ε xx (mm)
60
Maximum horizontal strain ε xx (mm)
80
Maximum vertical stress σ yy (MPa)
N.L.L. Mambou et al. / International Journal of Mining Science and Technology 25 (2015) 615–621
Maximum horizontal stress σ xx (MPa)
620
2
4
0.8 0.4
1 2 3 0 Lateral earth pressure ratio (k)
Lateral earth pressure ratio (k)
(b)
1.2
(c)
(d)
Circular hole Horseshoe hole
40
0
1 2 3 Lateral earth pressure ratio (k)
120 80 40
0 1 2 3 Lateral earth pressure ratio (k)
(a)
2.5
2.0
Maximum vertical strain ε xx (mm)
80
Maximum horizontal strain ε xx (mm)
120
Maximum vertical stress σ yy (MPa)
Maximum horizontal stress σ xx (MPa)
Fig. 10. Influence on the shape of a tunnel in the case of isotropic transverse rock of mass (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) horizontal principal strains exx versus lateral earth pressure ratio and (d) vertical principal strains eyy versus lateral earth pressure ratio.
2.0 1.5 1.0 0.5
1.6 1.2 0.8 0.4
1 2 3 0 Lateral earth pressure ratio (k)
(b)
1 2 3 0 Lateral earth pressure ratio (k)
(c)
(d)
Fig. 11. Influence on the shape of a tunnel in the case of isotropic rock of mass (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) horizontal principal strains exx versus lateral earth pressure ratio and (d) vertical principal strains eyy versus lateral earth pressure ratio.
60 40 20
0 1 2 3 Lateral earth pressure ratio (k)
(a)
2.0
100 80 60 40 20 0 1 2 3 Lateral earth pressure ratio (k)
8
Maximum vertical strain ε yy (mm)
80
120
Maximum horizontal strain ε xx (mm)
Circular hole (isotropic mass of rock) Circular hole (anisotropic mass of rock) Horseshoe hole (isotropic mass of rock) Horseshoe hole (anisotropic mass of rock)
100
Maximum vertical stress σ yy (MPa)
Maximum horizontal stress σ xx (MPa)
120
6 4 2 1 2 3 0 Lateral earth pressure ratio (k)
(b)
(c)
1.6 1.2 0.8 0.4 1 2 3 0 Lateral earth pressure ratio (k)
(d)
Fig. 12. Influence of the anisotropy of the mass of rock tunnel: (a) maximum horizontal principal stresses rxx versus lateral earth pressure ratio, (b) maximum vertical principal stresses ryy versus lateral earth pressure ratio, (c) maximum horizontal principal strains exx versus lateral earth pressure ratio and (d) maximum vertical principal strains eyy versus lateral earth pressure ratio.
underground excavation. For all the shapes, the redistribution of horizontal stresses and strains increases gradually when lateral pressure increases. The variation of the redistribution of vertical stress and vertical strain is more complex. They increase slowly for horse shoe hole when stress ratio increases; but decrease first and slowly increase above k = 1.5 for circular shape. In order to better understand the influence of anisotropy of rock, let us compare stress and strain distribution in the case of an isotropic mass of rock to transverse isotropic mass of rock.
isotropic rock is less than those obtained in the case of isotropic rock. Numerical result reveals that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more the rock has the anisotropic behavior, the more the mass of rock around the tunnel is stable.
3.5. Influence of the transverse isotropy system on the redistribution of stresses around an opened excavation
4. Conclusions
In Fig. 12, we study the influence on the transverse isotropy system on the redistribution of stresses around the circular and horse shoe shapes of opened excavations. Fig. 12 clearly shows that the anisotropic behavior affects more the redistribution stresses and strains in the case of horse shoe shape than the circular shape. We observed that for all the shapes, the magnitude of redistribution of stresses obtained for transverse
This work dealt with the numerical analysis of underground structures, taking accounts the transverse isotropy system of rocks. The analysis was done using CAST 3M code by varying the shape of excavation and the coefficient of earth pressure k. Numerical results have shown that the many factors can influence the behavior of rocks around underground openings: the in-situ stress condition; the rock mass strength; the anisotropy behavior of the rock mass; and the geometry of excavation.
N.L.L. Mambou et al. / International Journal of Mining Science and Technology 25 (2015) 615–621
The in situ horizontal stress (lateral earth pressure) causes more instability of rock around the hole than the vertical in situ stress. For all the shapes considered, the redistribution of horizontal stresses and strains increases gradually when lateral pressure increases. For circular shape, the maximum horizontal stress is 9.93, 26.4 and 59.5 MPa respectively for k = 0.5; 1; 2. For horse shoe shape, the maximum horizontal stress is 12.1, 25.2 and 52.7 MPa respectively for k = 0.5; 1; 2. The high horizontal and vertical strains observed around the tunnels indicate maximum volume change toward those regions. These results can help to design a rock bolts installed systematically around tunnels (regularly spaced around the tunnel) excavated in rock masses and which permit to stabilize instable rock mass. The design of geometry of tunnel influences the redistribution of stress and strain. The magnitude of horizontal stress obtained for the horse shoe shape excavation (25.2 MPa for k = 1; 52.7 MPa for k = 2) is lower than the magnitude obtained for circular hole (26.4 MPa for k = 1; 59.5 MPa for k = 2). Therefore, we conclude that the horse shoe shape offers the best stability and the best design for engineer. However according to vertical stress and strain, the circular shape offers the best design. These results help the engineers, geoscientists and tunnel designers to understand mining induced stress and possible deformation that developed for the advancement of circular and horse shoe tunnel and its possible influences on the rock strata. The anisotropy system presented by rock mass can also influence the redistribution of stresses around hole opened. Numerical result reveals that the magnitude of redistribution of horizontal stresses obtained for transverse isotropic rock (12.1 MPa for k = 0.5; 25.2 MPa for k = 1 and 52.7 MPa for k = 2) is less than those obtained in the case of isotropic rock (27.6 MPa for k = 1; 48.6 MPa for k = 2 and 90.81 MPa for k = 2). The more
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the rock has the anisotropic behavior the more the mass of rock around the tunnel is stable. An interesting work under investigation is the analysis of the redistribution and magnitude of stresses and strains around particular shapes of hole, as trapezium, rectangular and vault shapes; where the rock presents the anisotropic behavior. References [1] Lu Y, Yang W. Analytical solutions of stress and displacement in strain softening rock mass around a newly formed cavity. J Cent South Univ 2013; 20:1397–404. [2] Duddeck H. Application of numerical analysis for tunnelling. Int J Numer Anal Methods Geomech 1991;15(4):223–39. [3] Raji M, Sitharam TG. Stress distribution around the tunnel: influence of in situ stress and shape of tunnel. In: Proceedings of Indian geotechnical conference, Kochi; 2011. [4] Alejano LR, Rodriguez-Dono A, Alonso E, Fdez-Manin G. Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behavior. Tunnel Undergr Space Technol 2009;24(6): 689–705. [5] Carranza-Torres C. Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 2004;41(1):1–11. [6] Berry P, Crea G, Martino D, Ribacchi R. The influence of fabric on the deformability of anisotropic rocks. In: Proc 3rd Int Cong Isrm, Denver; 1974. p. 105–10. [7] Brown ET, Bray JW, Ladanyi B, Hoek E. Ground response curve for rock tunnels. J Geotech Eng 1983;109(1):15–39. [8] Eberhardt E, Stead D, Reeves MJ, Connors C. Design of tabular excavations in foliated rock: an integrated numerical modeling approach. Geotech Geol Eng 1997;15(1):47–85. [9] Takarli M, Prince-Agbodjan W. Temperature effects on physical properties and mechanical behavior of granite: experimental investigation of material damage. J ASTM Int 2008;5(3):13. [10] Gonzaga Gomez G, Leite MH, Corthesy R. Determination of anisotropic deformability parameters from a single standard rock specimen. Int J Rock Mech Min Sci 2008;45:1420–38. [11] Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body. San Francisco: Holden Day; 1963.