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Theoretical model and solution for the rheological problem of anchor-grouting a soft rock tunnel
International Journal of Pressure Vessels and Piping 81 (2004) 739–748 www.elsevier.com/locate/ijpvp
Theoretical model and solution for the rheological problem of anchor-grouting a soft rock tunnel H.L. Daia,*, X. Wanga, G.X. Xieb, X.Y. Wanga a
Department of Engineering Mechanics, The School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Resources Exploration and Management Engineering, Anhui University of Science and Technology, Anhui 232001, China Received 8 August 2003; revised 12 May 2004; accepted 12 May 2004
Abstract The paper presents an analytical method for analysis of the soft rock tunnel, based on a model dividing soft rock tunnels into a region of anchor-grouting and a region of non-anchor-grouting surrounding rock. The Poynting – Thomson model and the Kelvin – Hooke model are applied to the region of non-anchor-grouting surrounding rock and the region of anchor-grouting, respectively. Stress expressions in the region of non-anchor-grouting surrounding rock and the region of anchor-grouting are obtained. Expanding the expression of displacements in the region of anchor-grouting into a Maclaurin series, and utilizing the cumulative displacement curve of the surrounding rock through observation, a theoretical model is set up. This model and its solution for the rheology problem of anchor-grouting a soft tunnel have been proved to be effective in practical engineering; according to the Mohr – Coulomb yield condition, a safe criterion for an anchor-grouting soft rock of a tunnel can be found. q 2004 Elsevier Ltd. All rights reserved. Keywords: Soft rock tunnel; Viscoelastic; Rheology; Elastoplastic
1. Introduction Recently, anchor-grouting support of soft rock is extensively utilized in railway tunnels [1 –3], mine laneways [4 – 6] and tunnel engineering of national defences [7 – 10]. Anchor-grouting support of a soft rock tunnel can increase cohesion and the internal friction angle of surrounding rock, so that this improves the strength of surrounding rock masses, the mechanical property of surrounding rock and stabilization of the tunnel. The strength of soft rock is lower and its uniaxial pressure strength is less than 30 MPa. Because soft rock mass shows strong ability to flow, soft rock mass is thought of as a nonlinear elastoplastic and viscous compressible medium. There are many studies on anchor-grouting support of soft rock tunnels, based on various methods. From its constitutive model beginning, solving the stress and deformation of a soft rock tunnel by numerical methods was presented in Ref. [11]. Ref. [12] applied the theory * Corresponding author. Tel.: þ86-215-474-5816; fax: þ 86-215-4745821. E-mail addresses: [email protected] (H.L. Dai), [email protected]. edu.cn (X. Wang). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.05.004
of chaos mechanics to explain the rheology rule of soft rock. Ref. [13] gave the fitting equation for soft rock deformation by means of measured observations. Ref. [14] presented a method of probabilistic statistics to solve the distribution of stress of soft rock. It was especially evident that the rheology of soft rock affected greatly its security, so that including time effects on deformation is important. At present, few investigations on the problem are done, so it is essential to conduct further research. This paper is based on the theory of rheology, and considers the flow of soft rock. Using the viscoelastic Poynting –Thomson model in the region of non-anchor-grouting surrounding rock and the viscoelastic Kelvin –Hooke model in the region of anchorgrouting, and applying Laplace transforms and inverse Laplace transforms, the stress expressions in the region of anchor-grouting and the region of non-anchor-grouting surrounding rock are, respectively, obtained. Expanding the expression of displacement in the region of anchor-grouting into a Maclaurin series, and comparing measured cumulative displacement of the surrounding rock, the theoretical analysis and the solution for the rheology problem of anchor-grouting soft rock tunnel are proved to be effective in practical engineering. According to the Mohr –Coulomb
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1r ; 1u ; gru
Nomenclature a b2a E; m Ec ; m c r; u
l G; Gc
K; Kc
P; lP ur ; uu u0r ucr ; ucu u r ; u cr sr ; s u ; t r u
sm s0r ; s0u ; t0ru sr ; su ; tru s_ r ; s_ u ; t_ru
internal radius of tunnel thickness of anchor-grouting layer elastic modulus and Poisson’s ratio in the region of non-anchor-grouting surrounding rock elastic modulus and Poisson’s ratio in the region of anchor-grouting radial direction and hoop direction in the soft tunnel Lame´’s constant shear moduli in the region of nonanchor-grouting surrounding rock and anchor-grouting volume strain moduli in the region of non-anchor-grouting surrounding rock and anchor-grouting loads along the vertical and horizontal directions displacements in the region of nonanchor-grouting surrounding rock initial displacements in the region of non-anchor-grouting surrounding rock displacements in the region of anchorgrouting Laplace transformation of ur and ucr stresses in the region of non-anchorgrouting surrounding rock average stress in the region of nonanchor-grouting surrounding rock initial stresses in the region of nonanchor-grouting surrounding rock Laplace transformation of sr ; su and tru derivative of sr ; su and tru
strains in the region of non-anchorgrouting surrounding rock 1m average strain in the region of nonanchor-grouting surrounding rock 10r ; 10u ; g0ru initial strains in the region of nonanchor-grouting surrounding rock 10m initial average strain in the region of non-anchor-grouting surrounding rock 1r ; 1u ; gru ; 1m Laplace transformation of 1r ; 1u ; gru and 1m 1_r ; 1_u ; g_ ru ; 1_m derivative of 1r ; 1u ; gru and 1m scr ; scu ; tcru stresses in the region of anchor-grouting s0cr ; s0cu ; t0cru initial stress in the region of anchorgrouting scr ; scu ; tcru Laplace transformation of scr ; scu and tcru s_ cr ; s_ cu ; t_cru derivative of scr ; scu ; tcru 1cr ; 1cu ; gcru ; 1cm strains in the region of anchor-grouting 10cr ; 10cu ; g0cru ; 10cm initial strains in the region of anchorgrouting s; 1 stress and strain of the soft tunnel s_ ; 1_ derivative of s and 1 E1 ; E2 elastic moduli in the P –T model E3 ; E4 elastic moduli in the K – H model h1 ; h2 viscosity coefficients in the P– T model and the K – H model Sij ; sii deviatoric stress tensor and spherical stress tensor eij ; 1ii deviatoric strain tensor and spherical strain tensor S_ ij ; e_ ij derivative of Sij and eij P1 ; Q1 ; P2 ; Q2 differential operators t time c; w cohesive force and internal friction angle
yield condition and stress expressions obtained, a safe criterion of anchor-grouting soft rock tunnel can be found.
2. Governing equations and solution A soft rock tunnel with an anchor-grouting supported layer is considered as an infinite long, cavity structure with a radius, a; as shown in Fig. 1. The anchor-grouting layer distributes along the longitudinal and radial directions of the tunnel. The thickness of the anchor-grouting layer is ðb 2 aÞ: The material property of the anchor-grouting supported layer is isotropic, and it’s elastic modulus and Poisson’s ratio are, respectively, Ec ; mc : The elastic modulus and Poisson’s ratio of the region of non-anchor-grouting surrounding rock are E and m; respectively. The four sides of the soft rock tunnel are subjected to loads P and lP as shown in Fig. 1.
Fig. 1. The theoretical model of the anchor-grouting soft rock tunnel.
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
The cavity structure with a supporting layer on the surface can be simplified as a problem of plane strain. The stress and displacement in the region of non-anchorgrouting surrounding rock are written as " # ! ! P g b2 2bb2 3 d b4 sr ¼ ð1þlÞ 12 2 þð12lÞ 12 2 2 4 cos2u 2 r r r
741
Fig. 2. P –T rheological model.
ð1Þ
" # ! ! P g b2 3db4 ð2Þ su ¼ ð1þlÞ 1þ 2 2ð12lÞ 12 4 cos2u 2 r r ! P bb 2 3 d b 4 tru ¼2 ð12lÞ 1þ 2 þ 4 sin2u ð3Þ 2 r r " # ( ) Pb2 2db2 2ð1 þ lÞg þ ð1 2 lÞ bð4 2 4mÞ þ 2 cos 2u ur ¼ 8Gr r " # Pb2 2d b2 ð1 2 lÞ bð2 2 4mÞ 2 2 sin 2u uu ¼ 2 8Gr r
ð4Þ ð5Þ
where the undetermined constants b; g; d are, respectively, given by 2
b¼2 g¼
2 3
Gm þ Gc ðb 2 a Þ Gm þ Gc ð10 2 12mc Þðb2 2 a2 Þ3
G½ð2 2 4mc Þb2 þ 2a2 2Gc ðb2 2 a2 Þ þ G½ð2 2 4mc Þb2 þ 2a2
d¼2
Gm þ Gc ð4 2 4mÞðb2 2 a2 Þ3 Gm þ Gc ð10 2 12mÞðb2 2 a2 Þ3
ð6Þ
ð9Þ
ð11Þ
1 {½ð2 2 4mÞa1 r 2 a2 r 21 þ ½24mc a4 r 3 2 a5 r ucr ¼ 2Gc ð12Þ
1 ½ð6 2 4mcc Þa4 r 3 þ a5 r 2 ð2 2 4mc Þa3 r 21 2Gc ð13Þ
The undetermined constants a1 –a6 in the above formulae are given by a1 ¼
P b2 ð1 þ lÞð1 2 gÞ 2 4 b 2 a2
P b2 a2 a2 ¼ 2 ð1 þ lÞð1 2 gÞ 2 2 b 2 a2
P b2 ðb2 2 3a2 Þ ð1 2 lÞð1 þ dÞ 2 4 ðb 2 a2 Þ3
ð17Þ
a5 ¼ 2 a6 ¼
3P b2 ðb4 þ b2 a2 þ 2a2 Þ ð1 2 lÞð1 þ dÞ 2 ðb2 2 a2 Þ3
P b2 a4 ð3b4 þ b2 a2 Þ ð1 2 lÞð1 þ dÞ 2 ðb2 2 a2 Þ3
ð18Þ ð19Þ
Utilizing the Poynting – Thomson model [15] in the region of non-anchor-grouting surrounding rock, as shown in Fig. 2, the constitutive equation is given as
s_ þ
E1 E E s ¼ ðE1 þ E2 Þ1_ þ 1 2 1 h1 h1
ð20Þ
h2 E þ E4 h s_ þ 3 s ¼ 2 1_ þ 1 E3 E4 E 3 E4 E4
tcru ¼ ða5 þ 6a4 r2 2 2a3 r22 þ 3a6 r24 Þsin 2u
2 a6 r 23 sin 2u
a4 ¼
ð8Þ
ð10Þ
u cu ¼
ð16Þ
ð7Þ
scu ¼ 2a1 2 a2 r 22 þ ða5 þ 12a4 r2 2 3a6 r24 Þcos 2u
þ ð4 2 4mc Þa3 r 21 2 a6 r 23 cos 2u}
3P b2 a2 ð2b4 þ a2 b2 þ a4 Þ ð1 2 lÞð1 þ dÞ 4 ðb2 2 a2 Þ3
For the region of anchor-grouting, the anchor-grouting layer shows greater elastic property, so the Kelvin –Hooke model as shown in Fig. 3 is adopted in the region of anchorgrouting, and the corresponding constitutive equation is
Stresses and displacements in the region of anchorgrouting are given by
scr ¼ ð2a1 þ a2 r22 Þ 2 ða5 þ 4a3 r22 2 3a6 r24 Þcos 2u
a3 ¼
ð21Þ
For the linear viscoelastic body under a complex stress state, the stress at a point can be divided into a deviatoric stress tensor Sij and a spherical stress tensor sii : Thus the general form of the constitutive equation of a linear viscoelastic body is P1 Sij ¼ Q1 eij ;
P2 sii ¼ Q2 1ii
ð22Þ
where P1 ;Q1 ; P2 ; Q2 are differential operators, and eij ;1ii are deviatoric strain tensor and spherical strain tensor, respectively. For the region of anchor-grouting under low stress, the volume deformation caused by the spherical stress tensor is considered as elasticity. The deviatoric stress tensor is satisfied with the K – H model, shown as follows Ac S_ ij þ Bc Sij ¼ 2Dc e_ ij þ 2eij
ð23Þ
sii ¼ 3Kc 1ii
ð24Þ
ð14Þ ð15Þ Fig. 3. K–H rheological model.
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where P1 ¼ A c
› þ Bc ; ›t
Q1 ¼ 2Dc
› þ 2; ›t
P2 ¼ 1;
ð25Þ
Q2 ¼ 3Kc where Ac ; Bc and Dc are constants in the K – H model, and Kc represents the volume strain modulus of the region of anchor-grouting. For the region of non-anchor-grouting surrounding rock, the deviatoric strain tensor caused by deviatoric stress satisfies the P –T model. Assuming the material of the surrounding rock is incompressible gives m ¼ 0:5; K ! 1 and 1v ¼ 1ii ¼ 0: The constitutive equation of the P – T model in the non-anchor-grouting region is S_ ij þ ASij ¼ 2D_eij þ 2Beij
ð26Þ
where P1 ¼
› þ A; ›t
Q1 ¼ 2D
› þ 2B; ›t
P2 ¼ 1;
ð27Þ
Q2 ¼ 3K ! 1 where A; B and D are constants in the P– T model, and K denotes the volume strain modulus of the region of nonanchor-grouting. From Eq. (20), stress and displacement in the nonanchor-grouting region are expressed as
E1 E E 3K sr þ s_ r ¼ 1 2 1r þ 2 1 1m E h1 h1
2 3K 2 1 1_m þ ðE1 þ E2 Þ 1_r þ ð28Þ E2
E1 E E 3K su þ s_ u ¼ 1 2 1u þ 2 1 1m E h1 h1
2 3K 2 1 1_m þ ðE1 þ E2 Þ 1_u þ E2 E1 EE tru þ t_ru ¼ ðE1 þ E2 Þg_ru þ 1 2 gru h1 h1
ð29Þ
ð30Þ
In Eqs. (28)–(30), sm ¼ 3K1m ; where sm ¼ 13 ðsr þ su þ sz Þ represents average stress and 1m ¼ 13 ð1r þ 1u þ 1z Þ represents average strain. To solve Eqs. (28) – (30), we apply Laplace transforms E1 þ s sr 2 s0r h1
E1 E2 3K ¼ þ ðE1 þ E2 Þs 1r þ 2 1 1m E2 h1
3K 2 ðE1 þ E2 Þ 10r þ 2 1 10m ð31Þ E2
E1 þ s su 2 s0u h1
E1 E2 3K ¼ þ ðE1 þ E2 Þs 1u þ 2 1 1m E2 h1
3K 2 ðE1 þ E2 Þ 10u þ 2 1 10m E2 E1 þ s tru 2 t0ru h1
E1 E2 ¼ þ ðE1 þ E2 Þs gru 2 ðE1 þ E2 Þg0ru h1
ð32Þ
ð33Þ
where sr denotes the Laplace transformation of sr and s0r denotes the initial stress value at t ¼ 0: Eqs. (31) – (33) can be rewritten as E E =h þ ðE1 þ E2 Þs ½1r þ ð3K=E2 2 1Þ1m L r ¼ 1 2 1 E 1 = h1 þ s
ð34Þ
E E =h þ ðE1 þ E2 Þs ½1u þ ð3K=E2 2 1Þ1m L u ¼ 1 2 1 E1 = h1 þ s
ð35Þ
E E =h þ ðE1 þ E2 Þs L ru ¼ 1 2 1 gru E1 =h1 þ s
ð36Þ
where
s0 2 ðE1 þ E2 Þ½10r þ ð3K=E2 2 1Þ10m L r ¼ sr 2 r E 1 = h1 þ s
ð37Þ
s0 2 ðE1 þ E2 Þ½10u þ ð3K=E2 2 1Þ10m L u ¼ su 2 u E1 =h1 þ s
ð38Þ
t 2 ðE1 þ E2 Þg0ru L ru ¼ tru 2 ru E1 =h1 þ s
ð39Þ
Comparing Eqs. (34) –(36) with the generalized Hooke’s law of the plane strain problem [15], it is seen that the elastic solution of a plane strain problem is the same as Laplace transforms of the viscoelastic solution for the same problem. Thus by applying inverse Laplace transforms to the elastic solution of a plane strain problem, the corresponding viscoelastic solution can be obtained. Applying Laplace transforms to the expressions ((1) – (5)) of stresses and displacements in the region of non-anchor-grouting around rock gives " ! ! P gb2 2bb2 3db4 ð1 þ lÞ 1 2 2 þ ð1 2 lÞ 1 2 2 2 4 sr ¼ 2 r r r # ð40Þ
cos 2u
" # ! ! P gb2 3db4 ð1 þ lÞ 1 þ 2 2 ð1 2 lÞ 1 2 4 cos 2u su ¼ 2 r r !
P bb2 3db4 tru ¼ 2 ð1 2 lÞ 1 þ 2 þ 4 sin 2u 2 r r
ð41Þ ð42Þ
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
" # ( ) Pb2 2db2 u r ¼ 2ð1 þ lÞg þ ð1 2 lÞ bð4 2 4mÞ þ 2 cos 2u 8Gr r "
Pb 2db ð1 2 lÞ bð2 2 4mÞ 2 2 8Gr r 2
u r ¼ 2
2
ð43Þ
# sin 2u
ð44Þ
where b; g; d are Laplace transformations of the undetermined constants shown in Eqs. (6) – (8). Substituting Eqs. (40) – (42) into Eqs. (34) –(36) yield
s0 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m L r ¼ sr 2 r E1 =h1 þs " # ! ! P gb2 2bb2 3db4 ¼ ð1þlÞ 12 2 þð12lÞ 12 2 2 4 cos2u 2 r r r 2
s0r 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m E1 =h1 þs
ð45Þ
s0 2ðE1 þE2 Þ½10u þð3K=E2 21Þ10m L u ¼ su 2 u E1 =h1 þs " # ! ! P gb2 3db4 ¼ ð1þlÞ 1þ 2 2ð12lÞ 12 4 cos2u 2 r r 2
s0u 2ðE1 þE2 Þ½10u þð3K=E2 21Þ10m E1 =h1 þs 0
tcru ¼ða5 þ6a4 r2 22a3 r22 þ3a6 r24 Þsin2u u cr ¼
743
ð53Þ
1 {½ð224mc Þa1 r2a 2 r 21 þ½24mc a 4 r 3 2a 5 r 2Gc þð424mc Þa3 r 21 2a 6 r 23 cos2u}
u cu ¼
ð54Þ
1 ½ð624mc Þa4 r 3 þ a 5 r 2Gc 2ð224mc Þa3 r 21 2 a 6 r 23 sin 2u
L cr ¼ 2a1 þ a 2 r 22 2ða5 þ4a3 r 22 23a6 r 24 Þcos 2u h2 0 h2 0 3K h2 h2 0 s 2 1 2 2 1 E E cr E4 cr E3 E4 E4 cm 2 3 4 E3 þE4 h þ 2 s E3 E 4 E3 E4 L cu ¼ 2a1 2 a 2 r 22 þða5 þ12a4 r 22 23a6 r 24 Þcos 2u h 2 0 h2 0 3K h2 h2 0 s cu 2 1 cu 2 2 1 E E E4 E3 E4 E4 cm 2 3 4 E3 þE4 h þ 2 s E3 E4 E3 E4
ð55Þ
ð56Þ
ð57Þ
L cru ¼ ða5 þ6a4 r 2 22a3 r 22 þ3a6 r 24 Þsin 2u ð46Þ
0
t 2ðE1 þE2 Þgru L ru ¼tru 2 ru E1 =h1 þs
! P bb2 3db4 t0 2ðE1 þE2 Þg0ru ¼2 ð12lÞ 1þ 2 þ 4 sin2u2 ru 2 E1 =h1 þs r r
2
h2 0 h t þ 2 g0 E3 E4 cru E4 cru
ð58Þ
Because radial displacement and radial stress are continuous at the interface between the region of non-anchor-grouting surrounding rock and the region of anchor-grouting, and there is no friction at the interface, the corresponding boundary conditions are
ð47Þ
ur ¼ ucr ; at r ¼ b
ð59Þ
The constitutive Eq. (21) of the K – H model can be rewritten as E3 þE4 h E þE h scr þ 2 s_ cr ¼1cr þ 3K 3 4 21 1cm þ 2 1_ cr E 3 E4 E3 E4 E3 E4 E4 3K h2 h2 þ 2 ð48Þ 1_ E3 E4 E4 cm E3 þE4 h E þE h s þ 2 s_ ¼1 þ 3K 3 4 21 1cm þ 2 1_ cu E 3 E 4 cu E 3 E 4 cu cu E3 E4 E4 3K h2 h2 þ 2 ð49Þ 1_ E3 E4 E4 cm
Lr ¼ Lcr ; at r ¼ b
ð60Þ
Lru ¼ Lcru ¼ 0; at r ¼ b
ð61Þ
E3 þE4 h h t þ 2 t_ ¼ g þ 2 g_ E3 E4 cru E3 E4 cru cru E4 cru
ð50Þ
Utilizing a similar solving method as used for the region of non-anchor-grouting, the stress and the displacement in the region of anchor-grouting are represented as
scr ¼2a1 þa 2 r
22
2ða5 þ4a3 r
22
23a6 r
24
Þcos2u
scu ¼2a1 2a 2 r22 þða5 þ12a4 r22 23a6 r24 Þcos2u
ð51Þ ð52Þ
Since there is no radial stress in the inner boundary region of anchor-grouting, the corresponding boundary conditions are Lcr ¼ 0; at r ¼ a
ð62Þ
Lcru ¼ 0; at r ¼ a
ð63Þ
Applying Laplace transforms to Eqs. (59) – (63) yields u r ¼ u cr ; at r ¼ b
ð64Þ
L r ¼ L cr ; at r ¼ b
ð65Þ
L ru ¼ L cru ¼ 0; at r ¼ b
ð66Þ
L cr ¼ 0; at r ¼ a
ð67Þ
L cru ¼ 0; at r ¼ a
ð68Þ
Substituting Eqs. (45) – (47) and Eqs. (56) –(58) into Eqs. (63) – (68), and utilizing Eqs. (40) – (46) and Eqs. (51) – (55), where the detailed expressions are given in Appendix A, finally, the expressions of stress
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and displacement in the region of non-anchor-grouting surrounding rock are expressed as 1þ l b2 P2 2 sr ¼ 2 r " # 2E2 ðPþu0r Kc Þ Kc ð2E2 u0r 2PbÞ b 1 h1 2 exp 2 t 2E2 þbKc 2E2 þbKc E1 b2 KðE3 u0r 2PbÞ b h exp 2 2 2 t cos 2u 2 2 ð69Þ 2E3 þbK E3 þE4 r
1þ l b2 Pþ 2 su ¼ 2 r " # 2E2 ðPþu0r Kc Þ Kc ð2E2 u0r 2PbÞ b 1 h1 2 exp 2 t 2E2 þbKc 2E2 þbKc E1 b2 KðE3 u0r 2PbÞ b 2 h2 exp 2 þ 2 t cos 2u ð70Þ 2E3 þbK E3 þE4 r " # b2 Pþu0r Kc 2E2 u0r 2b h1 b 1 exp 2 þ t ur ¼ 2 E1 r 2E2 þbKc bð2E2 þbKc Þ b2 E3 u0r Kc b2 h 2 b2 þ 2 t cos 2u exp 2 E3 þE4 r Gc ð2E4 þbKc Þðb2 2a2 Þr ð71Þ The expressions of stress and displacement in the region of anchor-grouting are represented as ! Pb22E2 u0r a2 scr ¼ 12 2 ð2E2 þbKc Þðb2 2a2 Þ r
hb a2 K ½E u0 þð12 lÞPb 12exp 2 1 1 t 2 2 c 3 r 2E3 þbK E1 r h 2 b2 exp 2 t cos 2u ð72Þ E3 þE4 !
hb 12exp 2 1 1 t E1 2 0 a K ½E u þð12 lÞPb hb exp 2 2 2 t cos 2u þ 2 c 3 r 2E3 þbK E3 þE4 r ð73Þ
Pb22E2 u0r a2 sc u ¼ 1þ 2 2 2 ð2E2 þbKc Þðb 2a Þ r
ucr ¼
ðPb22E2 u0r ÞKc b2 ½ðKc 21Þr 2 þ2a2 4Gc ð2E2 þbKc Þðb2 2a2 Þr
hb a2 lPbKc u0r 12exp 2 1 1 t þ 2 E1 r Gc ð2E3 þbKc Þðb2 2a2 Þr hb exp 2 2 2 t ð74Þ E3 þE4
where b1 ¼ ð2E2 þbKc Þ=ð2bþbKc Þ; b2 ¼ ðE3 E4 Þ=h2 ; Kc ¼ 324mc ; K ¼ 324m:
3. Safe criterion for anchor-grouting support The expression of displacement in the region of anchorgrouting can be written in the form ucr ¼ A 2 A e2Bt þ C e2Dt
ð75Þ
where A¼
ðPb 2 2E2 u0r ÞKc b2 ½ðKc 2 1Þr 2 þ 2a2 ; 4Gc ð2E2 þ bKc Þðb2 2 a2 Þr
B¼
h1 b 1 E1
C¼
a2 lPbKc u0r ; 2 r Gc ð2E3 þ bKc Þðb2 2 a2 Þr
ð76Þ D¼
h 2 b2 E 3 þ E4
Expanding Eq. (75) into a Maclaurin series at t ¼ 0; gives ucr ¼ C þ ðAB 2 CDÞt 2 ðAB2 2 CD2 Þ þ ðAB3 2 CD3 Þ
t2 2
t3 t4 2 ðAB4 2 CD4 Þ þ ··· 6 24
ð77Þ
A measured curve of displacement of the tunnel’s two walls can be described as f ðtÞ ¼ A0 þ A1 t þ A2 t2 þ A3 t3 þ · · ·
ð78Þ
Comparing the coefficients of Eq. (77) with Eq. (78), we can get coupled equations as follows C ¼ A0 ;
AB 2 CD ¼ A1 ;
20:5ðAB2 2 CD2 Þ ¼ A2 ; ðAB2 2 CD2 Þ=6 ¼ A3
ð79Þ
Solving the coupled Eq. (79) yields A¼
A0 D þ A1 ; B
B¼
A0 D 2 2A1 ; A0 D þ A1
C ¼ A0
ð80Þ
where D should satisfy the following equation A0 A1 D3 þ 4A0 A2 D2 þ 6A0 A3 D þ 6A1 A2 2 4A22 ¼ 0
ð81Þ
The tested tunnel lies at the 2 340 m level at the XinJi mine in China. The cross-section of the tunnel is a semicircle, net width 3.4 m, net height 3.0 m. The original support form is U-sectioned steel and whitewashing support, as Fig. 4a shows. The surrounding rock is mainly made up of mudstone, metalstone and calleystone, which are extremely soft and fragile. The block diagram of the rock stratum is shown in Fig. 5. The tested tunnel lies in the sixth floor. Fig. 6 is a measuring picture at the location. The measured displacement curve of the tunnel’s two sides is shown in Fig. 7. Fig. 4b shows the location pictures after the test.
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
745
For the region of anchor-grouting, the yield criterion of Mohr – Coulomb is applied scu þ scr ð84Þ sin w R # c £ cot w þ 2 where R ¼ ðscu 2 scr Þ=2; c is cohesive force, and w is internal friction angle. Substituting Eqs. (72) and (73) into Eq. (84), yield
a2 Pb22E2 u0r h 1 a1 t 12exp 2 E1 r 2 ð2E2 þbKc Þðb2 2a2 Þ a2 K ½E u0 þð12 lÞPb ha þ 2 c 3 r exp 2 2 2 t cos 2u 2E3 þbK E3 þE4 r # cð1þsin wÞcos w
ð85Þ
For a given tunnel, c and w are certain values. To make Eq. (85) true for any value of time t; the following condition must be met a2 Pb22E2 u0r 2 r ð2E2 þbKc Þðb2 2a2 Þ þ
Fig. 4. (a) The status of tunnel before the test of anchor-grouting. (b) The status of tunnel after the test of anchor-grouting.
The coefficients of the equation are shown in Table 1. From Table 1 and Eqs. (80) and (81), we have A ¼ 40:0787514; C ¼ 13:044026;
B ¼ 0:058362381;
ð82Þ
D ¼ 0:10062504
Substituting Eq. (82) into Eq. (75), we can get the cumulative displacement creep function at the measured points in the tunnel’s two walls 20:0584t
ucr ¼ 40:08 2 40:08 e
þ 13:044 e
20:101t
ð83Þ
where the unit of the cumulative displacement ucr is mm, and the unit of time t is day.
a2 Kc ½E3 u0r þð12 lÞPb cos 2u # const 2E3 þbK r2
ð86Þ
where const ¼ cð1þsin wÞcos w: When r ¼ a and cos 2u ¼ 1; the condition to make the left of Eq. (86) reach the maximum value is Pb 2 2E2 u0r Kc ½E3 u0r þ ð1 2 lÞPb # const þ 2E3 þ bK ð2E2 þ bKc Þðb2 2 a2 Þ ð87Þ Eq. (87) can be simplified into ðP 2 Kc u0r Þb 2ð1 2 lÞP 2 Ku0r b þ 2 2 2ð2E3 þ bKÞ ð2E2 þ bKc Þðb 2 a Þ Kc 1 þ 2 2 u0r # const 2 b 2 a2
ð88Þ
It is seen from inequality (88) that increasing the elastic coefficients E2 ; E3 makes the inequality more easily satisfied, thus helping the surrounding rock of a tunnel to
Fig. 5. The tunnel terrane column.
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the supported region can be determined, which is related to the property of the surrounding rock, the material of the anchor-grouting layer and the method of anchor-grouting, etc. In practical engineering, the thickness ðb 2 aÞ of the supported region is taken as 2.5 m.
4. Conclusions
Fig. 6. The disposition of observation points on the surface of tunnel.
Fig. 7. The curve of cumulative displacements in the observation station of the tunnel.
be maintained in an elastic deformation state rather than suffer plastic failure. According to the Poynting – Thomson model, it is not easy for the region of non-anchor-grouting to go through plastic yield. From the yield criterion (88) of Mohr –Coulomb for the anchor-grouting supported region, the thickness of
1. Based on the theory of rheology, the paper presents an analytical method for the viscoplastic stress and displacement distribution around a soft rock tunnel. It is seen from the solution that the stresses and displacements of the surrounding rock are related to time, so that they increase with time, and gradually reach a stabilized value. 2. The soft rock tunnel is composed of a supported region and an unsupported region. Because the unsupported region around rock belongs to soft rock having less elasticity, the P –T model is applied to the unsupported region. The K – H model is applied to the supported region that belongs to the anchor-grouting layer having greater elastic behavior. 3. From Eq. (77), the best support time may be determined. The later the time of anchor-grouting support, the larger the displacement of the surrounding rock. The variation of the displacement is less after anchor-grouting support. 4. Expanding the expression of displacement in the supported region into a Maclaurin series, and comparing measured cumulative displacements around rock, the theoretical analysis and the solution for the rheology problem of the anchor-grouting soft rock tunnel are proved to be effective in the practical engineering. According to the Mohr – Coulomb yield criterion and stress expressions obtained, a safe criterion for anchor-grouting soft rock tunnel can be set up.
Acknowledgements The authors thank the referees for their valuable comments.
Appendix A Table 1 The fitting curve coefficients of displacements in the observation station of the tunnel Items
Coefficients A0
A1
A2
A3
Average 13.044026 1.0265357 20.0022175707 21.9372207 £ 1025 Squared 2.16559 0.25746 0.00822 0.0001 deviation
The initial displacements of surrounding rock are represented by u0r and u0u ; and 10r ; 10u ; g0ru and 10m represent initial strains of surrounding rock. Applying Laplace transforms on the elastic solution without support, and solving these equations, the corresponding solution of the viscoelastic equations are expressed as ur ¼
P ½ð1 þ lÞu1 þ ð1 2 lÞu11 cos 2u 2
ðA1Þ
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
P uu ¼ ð1 2 lÞv1 sin 2u 2
ðA2Þ
1r ¼
P ½ð1 þ lÞ11r þ ð1 2 lÞ111 r cos 2u 2
ðA3Þ
1u ¼
P ½ð1 þ lÞ11u þ ð1 2 lÞ111 u cos 2u 2
ðA4Þ
gru ¼ Pð1 2 lÞg11 ru cos 2u
ðA5Þ
P ð1 2 lÞe1 cos 2u 2
ðA6Þ
1m ¼
11r
11u
(
! 1 E1 ðE1 þ E2 Þ exp 2 t E1 þ E2 E 2 h1 " !#) 1 E1 ðE1 þ E2 Þ 1 2 exp 2 t þ E2 E 2 h1
b2 u ¼ 2r
ðA7Þ
! ( b2 1 E 1 E2 t ¼ 2 2 exp 2 ðE1 þ E2 Þ 2r E1 þ E2 " !#) 1 E 1 E2 1 2 exp 2 þ t E2 ðE1 þ E2 Þh1
ðA8Þ
(
ðA9Þ
!( ! b2 b2 1 E1 E 2 4 2 4m 2 2 exp 2 t u ¼ E1 þ E2 2r ðE1 þ E2 Þh1 r " !#) 1 E1 E2 1 2 exp 2 t ðA10Þ þ E2 ðE1 þ E2 Þh11 11
!( ! b2 b2 1 E1 E 2 2 2 4m þ 2 v ¼2 exp 2 t E 1 þ E2 2r ðE1 þ E2 Þh1 r " !#) 1 E1 E2 1 2 exp 2 t ðA11Þ þ E2 ðE1 þ E2 Þh11 11
!(
!
b2 3b2 1 E 1 E2 424m 2 2 exp 2 t 2 E1 þE2 ðE1 þE2 Þh1 2r r " !#) 1 E 1 E2 12exp 2 t ðA12Þ þ E2 ðE1 þE2 Þh11
111 r ¼2
!(
111 u ¼
Pb2 1 ð1þ lÞg ¼ ½ð224mc Þa1 b2 a 2 b21 Gc 2Gb
!
b2 3b2 1 E1 E2 4m 2 2 exp 2 t 2 E1 þE2 ðE1 þE2 Þh1 2r r " !#) 1 E 1 E2 12exp 2 t ðA13Þ þ E2 ðE1 þE2 Þh11
ðA16Þ
Pb2 ð12 lÞ½bð424mÞþ2d 4Gb ¼
! 1 E1 E 2 exp 2 t E 1 þ E2 ðE1 þ E2 Þh1 " !#) 1 E 1 E2 1 2 exp 2 t þ E2 ðE1 þ E2 Þh1
b2 ¼ 2 2r
!( ! b2 3b2 1 E1 E2 22 2 exp 2 t E1 þE2 ðE1 þE2 Þh1 2r 2 r " !#) 1 E1 E 2 12exp 2 t ðA14Þ þ E2 ðE1 þE2 Þh11 ( ! b2 1 E1 ðE1 þE2 Þ 11 exp 2 t 1m ¼ 2 2 ð224mÞ E1 þE2 E2 h1 3r " !#) 1 E1 ðE1 þE2 Þ 12exp 2 t ðA15Þ þ E2 E2 h1
g11 ru ¼ 2
Substituting Eqs. (45) – (47) and Eqs. (56) –(63) into Eqs. (64) – (68), and separating the uniform part which is independent of u and the non-uniform part which is dependent on u; we have
where 1
747
1 ½24mc a 4 b3 2 a 5 b2ð424mc Þa3 b21 2 a 6 b23 Gc
ðA17Þ
P s0 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m ð1þ lÞð12 gÞ2 r 2 E1 =h1 þs h2 0 h 2 0 3K h2 h2 0 s 2 1 2 2 1 E E cr E4 cr E3 E4 E4 cm ¼ 2a1 þ a 2 b22 2 3 4 E3 þE4 h þ 2 s E3 E4 E 3 E4 ðA18Þ P ð12 lÞð122b 23dÞ ¼ 2ða5 þ4a3 b22 23a6 b24 Þ 2
ðA19Þ
a 5 þ6a4 22a3 b22 þ3a6 b24 ¼ 0
ðA20Þ
h2 0 h2 0 3K h2 h2 0 scr 2 1cr 2 2 1 E E E E3 E4 E4 cm 4 ¼0 2a1 þ a 2 a22 2 3 4 E3 þE4 h þ 2 s E3 E4 E3 E4 ðA21Þ a 5 þ4a3 a22 23a6 a24 ¼ 0
ðA22Þ
a 5 þ6a4 a2 22a3 a22 þ3a6 a24 ¼ 0
ðA23Þ
Solving the above Eqs. (A16) – (A23) yield (" ) # 1 a2 G P 0 0 12 2 12 a 1 ¼ Lcr þLr 2ð1þ lÞ n Gc 2 b
a2 G a 2 ¼ 2ð1þ lÞPþ2 ð122mc ÞL0cr þ2L0r Gc n
ðA24Þ ðA25Þ
a 3 ¼ 2
3Pb ð12 lÞða4 þa2 b2 þ2b4 Þð3sþ424mÞ 8Gm2
ðA26Þ
a 4 ¼ 2
3Pb ð12 lÞð3a2 þb2 Þð3sþ424mÞ 4Gm4
ðA27Þ
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a 5 ¼
3Pb ð12 lÞð2a4 þa2 b2 þb4 Þð3sþ424mÞ 4Gm3
References ðA28Þ
Pb a 6 ¼ 2 ð12 lÞð3sþ424mÞ 4Gm1
ðA29Þ
3ð12 lÞð4a4 2a2 b2 þb4 Þð3sþ424mÞ b ¼ 2þ 2a4 bGm1 ða2 þ3b2 Þ
ðA30Þ
2G ½2ð122mc Þa1 b2 a 2 b21 PbGc ð1þ lÞ
ðA31Þ
g ¼
ð12 lÞð4a4 2a2 b2 þb4 Þð3sþ424mÞ d ¼ 212 2a4 bGm1 ða2 þ3b2 Þ
ðA32Þ
where
s0r 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m ðA33Þ E1 =h1 þs h2 0 h2 0 3K h2 h2 0 scr 2 1cr 2 2 1 E E E4 E3 E4 E4 cm L0cr ¼ 3 4 ðA34Þ E3 þE4 h2 þ s E 3 E4 E3 E 4 " 1 1 m1 ¼ 4 3 2 ð625mÞð5a6 23a4 b2 þ9a2 b4 23b6 Þ a b ða þ3b2 Þ G L0r ¼
2
1 2m ð5a6 23a4 b2 þ9a2 b4 23b6 Þþ c ð3a6 þ8a4 b2 Gc Gc #
þ5a2 b4 þb6 Þ
ðA35Þ
m2 ¼ a2 b2 ða2 þ3b2 Þm1
ðA36Þ
m3 ¼ a4 b2 ða2 þ3b2 Þm1
ðA37Þ
m4 ¼ a2 ða2 þ3b2 Þm1 G G a2 G n ¼ 2 1þ 22 mc 22 2 12 Gc Gc Gc b
ðA38Þ ðA39Þ
Substituting Eqs. (A14) – (A32) into Eqs. (40) – (44) and Eqs. (51) –(55), and utilizing Laplace transforms, we can get Eqs. (69) – (74).
[1] Sveinn T, Matthias L, Jensson O. Monitoring of deformations, rock support and testing of rock anchors in the powerhouse cavern. Tunnel Underground Space Technol 1991;6(2):235–9 [in English]. [2] Ito T, Akagi T, Tuchiyama S. Separation of the convergence caused by progression of tunnel face into one and another due to creep characteristics of rock. Proc Jpn Soc Civil Engr 1990;6:279–82 [in Japanese]. [3] Li N, Chen F, Jiang F, Wei J. Stability analysis for the tunneling in swelling-creep clay-rock masses. Chin J Rock Mech Eng 2000; 19(11):731–5 [in Chinese]. [4] Scott JJ. Dynamic rock anchors a superior coal mine roof support system. Prepr—Soc Mining Engr AIME 1990;26(5):67 – 72 [in English]. [5] Vassiliev VV. Yielding glass-reinforced plastics anchors for the support of mine workings. Coal Int (Redhill, Engl) 1999;24(2):52–4 [in English]. [6] Selvadurai APS. Mechanics of a rock anchor with a penny-shaped basal crack Author. Int J Rock Mech Mining Sci Geomech Abstr 1993;30(11):1285–90 [in English]. [7] Baeumle R, Hoetzl H, Witthueser K. Flow pattern and transport behaviour of granitic rock intersected by a highly permeable fault zone. Int Assoc Hydrol Sci 2000;26(5):283–8 [in English]. [8] Mysore GV, Liapis SI, Plaut RH. Dynamic analysis of single-anchor inflatable dams. J Sound Vib 1998;215(2):251–72 [in English]. [9] Khun’ba KT. Effect of soil creep and the spatial factor on the distribution of vertical stresses s over tunnels constructed in the body of a high earth-and-rock dam. Gidrotekhnich Stroitel’stvo 1993;9: 40– 3 [in Russian]. [10] Zhang L, Zhou D. Cause of creep landslide formation and mechanism of deformation of tunnel. Chin J Rock Mech Eng 1999;18(2):217 –21 [in Chinese]. [11] Zhang XD, Zheng YT, Xiao YX. Creep research on the soft rock of tertiary formation. J Northeastern Univ 1997;18(1):31–5 [in Chinese, English]. [12] Lee SL, Parnploy U, Yong KY, Lee FH. Time-dependent deformation of anchored excavation in soft clay. Symp Underground Excavations Soils Rocks, Including Earth Press Theories, Buried Struct Tunnels 1991;(6):241. [13] Li Z, Wang XY, Liang WM, Zhu J. Experimental research and analysis of the friction forces of earth acting on the anchor bar and tension. Rock Soil Mech 1999;20(1):48 –52 [in Chinese]. [14] Lee IM, Kim DH, Lo KY. A statistical approach on geotechnical parameter estimation for underground structures. Int Proc Ninth Int Conf Comput Meth Arch Geomech, Wuhan 1997;775–80. [15] Slater RAC. Engineering plasticity. New York, USA: The Macmillan Press; 1977.
Theoretical model and solution for the rheological problem of anchor-grouting a soft rock tunnel H.L. Daia,*, X. Wanga, G.X. Xieb, X.Y. Wanga a
Department of Engineering Mechanics, The School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Resources Exploration and Management Engineering, Anhui University of Science and Technology, Anhui 232001, China Received 8 August 2003; revised 12 May 2004; accepted 12 May 2004
Abstract The paper presents an analytical method for analysis of the soft rock tunnel, based on a model dividing soft rock tunnels into a region of anchor-grouting and a region of non-anchor-grouting surrounding rock. The Poynting – Thomson model and the Kelvin – Hooke model are applied to the region of non-anchor-grouting surrounding rock and the region of anchor-grouting, respectively. Stress expressions in the region of non-anchor-grouting surrounding rock and the region of anchor-grouting are obtained. Expanding the expression of displacements in the region of anchor-grouting into a Maclaurin series, and utilizing the cumulative displacement curve of the surrounding rock through observation, a theoretical model is set up. This model and its solution for the rheology problem of anchor-grouting a soft tunnel have been proved to be effective in practical engineering; according to the Mohr – Coulomb yield condition, a safe criterion for an anchor-grouting soft rock of a tunnel can be found. q 2004 Elsevier Ltd. All rights reserved. Keywords: Soft rock tunnel; Viscoelastic; Rheology; Elastoplastic
1. Introduction Recently, anchor-grouting support of soft rock is extensively utilized in railway tunnels [1 –3], mine laneways [4 – 6] and tunnel engineering of national defences [7 – 10]. Anchor-grouting support of a soft rock tunnel can increase cohesion and the internal friction angle of surrounding rock, so that this improves the strength of surrounding rock masses, the mechanical property of surrounding rock and stabilization of the tunnel. The strength of soft rock is lower and its uniaxial pressure strength is less than 30 MPa. Because soft rock mass shows strong ability to flow, soft rock mass is thought of as a nonlinear elastoplastic and viscous compressible medium. There are many studies on anchor-grouting support of soft rock tunnels, based on various methods. From its constitutive model beginning, solving the stress and deformation of a soft rock tunnel by numerical methods was presented in Ref. [11]. Ref. [12] applied the theory * Corresponding author. Tel.: þ86-215-474-5816; fax: þ 86-215-4745821. E-mail addresses: [email protected] (H.L. Dai), [email protected]. edu.cn (X. Wang). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.05.004
of chaos mechanics to explain the rheology rule of soft rock. Ref. [13] gave the fitting equation for soft rock deformation by means of measured observations. Ref. [14] presented a method of probabilistic statistics to solve the distribution of stress of soft rock. It was especially evident that the rheology of soft rock affected greatly its security, so that including time effects on deformation is important. At present, few investigations on the problem are done, so it is essential to conduct further research. This paper is based on the theory of rheology, and considers the flow of soft rock. Using the viscoelastic Poynting –Thomson model in the region of non-anchor-grouting surrounding rock and the viscoelastic Kelvin –Hooke model in the region of anchorgrouting, and applying Laplace transforms and inverse Laplace transforms, the stress expressions in the region of anchor-grouting and the region of non-anchor-grouting surrounding rock are, respectively, obtained. Expanding the expression of displacement in the region of anchor-grouting into a Maclaurin series, and comparing measured cumulative displacement of the surrounding rock, the theoretical analysis and the solution for the rheology problem of anchor-grouting soft rock tunnel are proved to be effective in practical engineering. According to the Mohr –Coulomb
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H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
1r ; 1u ; gru
Nomenclature a b2a E; m Ec ; m c r; u
l G; Gc
K; Kc
P; lP ur ; uu u0r ucr ; ucu u r ; u cr sr ; s u ; t r u
sm s0r ; s0u ; t0ru sr ; su ; tru s_ r ; s_ u ; t_ru
internal radius of tunnel thickness of anchor-grouting layer elastic modulus and Poisson’s ratio in the region of non-anchor-grouting surrounding rock elastic modulus and Poisson’s ratio in the region of anchor-grouting radial direction and hoop direction in the soft tunnel Lame´’s constant shear moduli in the region of nonanchor-grouting surrounding rock and anchor-grouting volume strain moduli in the region of non-anchor-grouting surrounding rock and anchor-grouting loads along the vertical and horizontal directions displacements in the region of nonanchor-grouting surrounding rock initial displacements in the region of non-anchor-grouting surrounding rock displacements in the region of anchorgrouting Laplace transformation of ur and ucr stresses in the region of non-anchorgrouting surrounding rock average stress in the region of nonanchor-grouting surrounding rock initial stresses in the region of nonanchor-grouting surrounding rock Laplace transformation of sr ; su and tru derivative of sr ; su and tru
strains in the region of non-anchorgrouting surrounding rock 1m average strain in the region of nonanchor-grouting surrounding rock 10r ; 10u ; g0ru initial strains in the region of nonanchor-grouting surrounding rock 10m initial average strain in the region of non-anchor-grouting surrounding rock 1r ; 1u ; gru ; 1m Laplace transformation of 1r ; 1u ; gru and 1m 1_r ; 1_u ; g_ ru ; 1_m derivative of 1r ; 1u ; gru and 1m scr ; scu ; tcru stresses in the region of anchor-grouting s0cr ; s0cu ; t0cru initial stress in the region of anchorgrouting scr ; scu ; tcru Laplace transformation of scr ; scu and tcru s_ cr ; s_ cu ; t_cru derivative of scr ; scu ; tcru 1cr ; 1cu ; gcru ; 1cm strains in the region of anchor-grouting 10cr ; 10cu ; g0cru ; 10cm initial strains in the region of anchorgrouting s; 1 stress and strain of the soft tunnel s_ ; 1_ derivative of s and 1 E1 ; E2 elastic moduli in the P –T model E3 ; E4 elastic moduli in the K – H model h1 ; h2 viscosity coefficients in the P– T model and the K – H model Sij ; sii deviatoric stress tensor and spherical stress tensor eij ; 1ii deviatoric strain tensor and spherical strain tensor S_ ij ; e_ ij derivative of Sij and eij P1 ; Q1 ; P2 ; Q2 differential operators t time c; w cohesive force and internal friction angle
yield condition and stress expressions obtained, a safe criterion of anchor-grouting soft rock tunnel can be found.
2. Governing equations and solution A soft rock tunnel with an anchor-grouting supported layer is considered as an infinite long, cavity structure with a radius, a; as shown in Fig. 1. The anchor-grouting layer distributes along the longitudinal and radial directions of the tunnel. The thickness of the anchor-grouting layer is ðb 2 aÞ: The material property of the anchor-grouting supported layer is isotropic, and it’s elastic modulus and Poisson’s ratio are, respectively, Ec ; mc : The elastic modulus and Poisson’s ratio of the region of non-anchor-grouting surrounding rock are E and m; respectively. The four sides of the soft rock tunnel are subjected to loads P and lP as shown in Fig. 1.
Fig. 1. The theoretical model of the anchor-grouting soft rock tunnel.
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
The cavity structure with a supporting layer on the surface can be simplified as a problem of plane strain. The stress and displacement in the region of non-anchorgrouting surrounding rock are written as " # ! ! P g b2 2bb2 3 d b4 sr ¼ ð1þlÞ 12 2 þð12lÞ 12 2 2 4 cos2u 2 r r r
741
Fig. 2. P –T rheological model.
ð1Þ
" # ! ! P g b2 3db4 ð2Þ su ¼ ð1þlÞ 1þ 2 2ð12lÞ 12 4 cos2u 2 r r ! P bb 2 3 d b 4 tru ¼2 ð12lÞ 1þ 2 þ 4 sin2u ð3Þ 2 r r " # ( ) Pb2 2db2 2ð1 þ lÞg þ ð1 2 lÞ bð4 2 4mÞ þ 2 cos 2u ur ¼ 8Gr r " # Pb2 2d b2 ð1 2 lÞ bð2 2 4mÞ 2 2 sin 2u uu ¼ 2 8Gr r
ð4Þ ð5Þ
where the undetermined constants b; g; d are, respectively, given by 2
b¼2 g¼
2 3
Gm þ Gc ðb 2 a Þ Gm þ Gc ð10 2 12mc Þðb2 2 a2 Þ3
G½ð2 2 4mc Þb2 þ 2a2 2Gc ðb2 2 a2 Þ þ G½ð2 2 4mc Þb2 þ 2a2
d¼2
Gm þ Gc ð4 2 4mÞðb2 2 a2 Þ3 Gm þ Gc ð10 2 12mÞðb2 2 a2 Þ3
ð6Þ
ð9Þ
ð11Þ
1 {½ð2 2 4mÞa1 r 2 a2 r 21 þ ½24mc a4 r 3 2 a5 r ucr ¼ 2Gc ð12Þ
1 ½ð6 2 4mcc Þa4 r 3 þ a5 r 2 ð2 2 4mc Þa3 r 21 2Gc ð13Þ
The undetermined constants a1 –a6 in the above formulae are given by a1 ¼
P b2 ð1 þ lÞð1 2 gÞ 2 4 b 2 a2
P b2 a2 a2 ¼ 2 ð1 þ lÞð1 2 gÞ 2 2 b 2 a2
P b2 ðb2 2 3a2 Þ ð1 2 lÞð1 þ dÞ 2 4 ðb 2 a2 Þ3
ð17Þ
a5 ¼ 2 a6 ¼
3P b2 ðb4 þ b2 a2 þ 2a2 Þ ð1 2 lÞð1 þ dÞ 2 ðb2 2 a2 Þ3
P b2 a4 ð3b4 þ b2 a2 Þ ð1 2 lÞð1 þ dÞ 2 ðb2 2 a2 Þ3
ð18Þ ð19Þ
Utilizing the Poynting – Thomson model [15] in the region of non-anchor-grouting surrounding rock, as shown in Fig. 2, the constitutive equation is given as
s_ þ
E1 E E s ¼ ðE1 þ E2 Þ1_ þ 1 2 1 h1 h1
ð20Þ
h2 E þ E4 h s_ þ 3 s ¼ 2 1_ þ 1 E3 E4 E 3 E4 E4
tcru ¼ ða5 þ 6a4 r2 2 2a3 r22 þ 3a6 r24 Þsin 2u
2 a6 r 23 sin 2u
a4 ¼
ð8Þ
ð10Þ
u cu ¼
ð16Þ
ð7Þ
scu ¼ 2a1 2 a2 r 22 þ ða5 þ 12a4 r2 2 3a6 r24 Þcos 2u
þ ð4 2 4mc Þa3 r 21 2 a6 r 23 cos 2u}
3P b2 a2 ð2b4 þ a2 b2 þ a4 Þ ð1 2 lÞð1 þ dÞ 4 ðb2 2 a2 Þ3
For the region of anchor-grouting, the anchor-grouting layer shows greater elastic property, so the Kelvin –Hooke model as shown in Fig. 3 is adopted in the region of anchorgrouting, and the corresponding constitutive equation is
Stresses and displacements in the region of anchorgrouting are given by
scr ¼ ð2a1 þ a2 r22 Þ 2 ða5 þ 4a3 r22 2 3a6 r24 Þcos 2u
a3 ¼
ð21Þ
For the linear viscoelastic body under a complex stress state, the stress at a point can be divided into a deviatoric stress tensor Sij and a spherical stress tensor sii : Thus the general form of the constitutive equation of a linear viscoelastic body is P1 Sij ¼ Q1 eij ;
P2 sii ¼ Q2 1ii
ð22Þ
where P1 ;Q1 ; P2 ; Q2 are differential operators, and eij ;1ii are deviatoric strain tensor and spherical strain tensor, respectively. For the region of anchor-grouting under low stress, the volume deformation caused by the spherical stress tensor is considered as elasticity. The deviatoric stress tensor is satisfied with the K – H model, shown as follows Ac S_ ij þ Bc Sij ¼ 2Dc e_ ij þ 2eij
ð23Þ
sii ¼ 3Kc 1ii
ð24Þ
ð14Þ ð15Þ Fig. 3. K–H rheological model.
742
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where P1 ¼ A c
› þ Bc ; ›t
Q1 ¼ 2Dc
› þ 2; ›t
P2 ¼ 1;
ð25Þ
Q2 ¼ 3Kc where Ac ; Bc and Dc are constants in the K – H model, and Kc represents the volume strain modulus of the region of anchor-grouting. For the region of non-anchor-grouting surrounding rock, the deviatoric strain tensor caused by deviatoric stress satisfies the P –T model. Assuming the material of the surrounding rock is incompressible gives m ¼ 0:5; K ! 1 and 1v ¼ 1ii ¼ 0: The constitutive equation of the P – T model in the non-anchor-grouting region is S_ ij þ ASij ¼ 2D_eij þ 2Beij
ð26Þ
where P1 ¼
› þ A; ›t
Q1 ¼ 2D
› þ 2B; ›t
P2 ¼ 1;
ð27Þ
Q2 ¼ 3K ! 1 where A; B and D are constants in the P– T model, and K denotes the volume strain modulus of the region of nonanchor-grouting. From Eq. (20), stress and displacement in the nonanchor-grouting region are expressed as
E1 E E 3K sr þ s_ r ¼ 1 2 1r þ 2 1 1m E h1 h1
2 3K 2 1 1_m þ ðE1 þ E2 Þ 1_r þ ð28Þ E2
E1 E E 3K su þ s_ u ¼ 1 2 1u þ 2 1 1m E h1 h1
2 3K 2 1 1_m þ ðE1 þ E2 Þ 1_u þ E2 E1 EE tru þ t_ru ¼ ðE1 þ E2 Þg_ru þ 1 2 gru h1 h1
ð29Þ
ð30Þ
In Eqs. (28)–(30), sm ¼ 3K1m ; where sm ¼ 13 ðsr þ su þ sz Þ represents average stress and 1m ¼ 13 ð1r þ 1u þ 1z Þ represents average strain. To solve Eqs. (28) – (30), we apply Laplace transforms E1 þ s sr 2 s0r h1
E1 E2 3K ¼ þ ðE1 þ E2 Þs 1r þ 2 1 1m E2 h1
3K 2 ðE1 þ E2 Þ 10r þ 2 1 10m ð31Þ E2
E1 þ s su 2 s0u h1
E1 E2 3K ¼ þ ðE1 þ E2 Þs 1u þ 2 1 1m E2 h1
3K 2 ðE1 þ E2 Þ 10u þ 2 1 10m E2 E1 þ s tru 2 t0ru h1
E1 E2 ¼ þ ðE1 þ E2 Þs gru 2 ðE1 þ E2 Þg0ru h1
ð32Þ
ð33Þ
where sr denotes the Laplace transformation of sr and s0r denotes the initial stress value at t ¼ 0: Eqs. (31) – (33) can be rewritten as E E =h þ ðE1 þ E2 Þs ½1r þ ð3K=E2 2 1Þ1m L r ¼ 1 2 1 E 1 = h1 þ s
ð34Þ
E E =h þ ðE1 þ E2 Þs ½1u þ ð3K=E2 2 1Þ1m L u ¼ 1 2 1 E1 = h1 þ s
ð35Þ
E E =h þ ðE1 þ E2 Þs L ru ¼ 1 2 1 gru E1 =h1 þ s
ð36Þ
where
s0 2 ðE1 þ E2 Þ½10r þ ð3K=E2 2 1Þ10m L r ¼ sr 2 r E 1 = h1 þ s
ð37Þ
s0 2 ðE1 þ E2 Þ½10u þ ð3K=E2 2 1Þ10m L u ¼ su 2 u E1 =h1 þ s
ð38Þ
t 2 ðE1 þ E2 Þg0ru L ru ¼ tru 2 ru E1 =h1 þ s
ð39Þ
Comparing Eqs. (34) –(36) with the generalized Hooke’s law of the plane strain problem [15], it is seen that the elastic solution of a plane strain problem is the same as Laplace transforms of the viscoelastic solution for the same problem. Thus by applying inverse Laplace transforms to the elastic solution of a plane strain problem, the corresponding viscoelastic solution can be obtained. Applying Laplace transforms to the expressions ((1) – (5)) of stresses and displacements in the region of non-anchor-grouting around rock gives " ! ! P gb2 2bb2 3db4 ð1 þ lÞ 1 2 2 þ ð1 2 lÞ 1 2 2 2 4 sr ¼ 2 r r r # ð40Þ
cos 2u
" # ! ! P gb2 3db4 ð1 þ lÞ 1 þ 2 2 ð1 2 lÞ 1 2 4 cos 2u su ¼ 2 r r !
P bb2 3db4 tru ¼ 2 ð1 2 lÞ 1 þ 2 þ 4 sin 2u 2 r r
ð41Þ ð42Þ
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
" # ( ) Pb2 2db2 u r ¼ 2ð1 þ lÞg þ ð1 2 lÞ bð4 2 4mÞ þ 2 cos 2u 8Gr r "
Pb 2db ð1 2 lÞ bð2 2 4mÞ 2 2 8Gr r 2
u r ¼ 2
2
ð43Þ
# sin 2u
ð44Þ
where b; g; d are Laplace transformations of the undetermined constants shown in Eqs. (6) – (8). Substituting Eqs. (40) – (42) into Eqs. (34) –(36) yield
s0 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m L r ¼ sr 2 r E1 =h1 þs " # ! ! P gb2 2bb2 3db4 ¼ ð1þlÞ 12 2 þð12lÞ 12 2 2 4 cos2u 2 r r r 2
s0r 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m E1 =h1 þs
ð45Þ
s0 2ðE1 þE2 Þ½10u þð3K=E2 21Þ10m L u ¼ su 2 u E1 =h1 þs " # ! ! P gb2 3db4 ¼ ð1þlÞ 1þ 2 2ð12lÞ 12 4 cos2u 2 r r 2
s0u 2ðE1 þE2 Þ½10u þð3K=E2 21Þ10m E1 =h1 þs 0
tcru ¼ða5 þ6a4 r2 22a3 r22 þ3a6 r24 Þsin2u u cr ¼
743
ð53Þ
1 {½ð224mc Þa1 r2a 2 r 21 þ½24mc a 4 r 3 2a 5 r 2Gc þð424mc Þa3 r 21 2a 6 r 23 cos2u}
u cu ¼
ð54Þ
1 ½ð624mc Þa4 r 3 þ a 5 r 2Gc 2ð224mc Þa3 r 21 2 a 6 r 23 sin 2u
L cr ¼ 2a1 þ a 2 r 22 2ða5 þ4a3 r 22 23a6 r 24 Þcos 2u h2 0 h2 0 3K h2 h2 0 s 2 1 2 2 1 E E cr E4 cr E3 E4 E4 cm 2 3 4 E3 þE4 h þ 2 s E3 E 4 E3 E4 L cu ¼ 2a1 2 a 2 r 22 þða5 þ12a4 r 22 23a6 r 24 Þcos 2u h 2 0 h2 0 3K h2 h2 0 s cu 2 1 cu 2 2 1 E E E4 E3 E4 E4 cm 2 3 4 E3 þE4 h þ 2 s E3 E4 E3 E4
ð55Þ
ð56Þ
ð57Þ
L cru ¼ ða5 þ6a4 r 2 22a3 r 22 þ3a6 r 24 Þsin 2u ð46Þ
0
t 2ðE1 þE2 Þgru L ru ¼tru 2 ru E1 =h1 þs
! P bb2 3db4 t0 2ðE1 þE2 Þg0ru ¼2 ð12lÞ 1þ 2 þ 4 sin2u2 ru 2 E1 =h1 þs r r
2
h2 0 h t þ 2 g0 E3 E4 cru E4 cru
ð58Þ
Because radial displacement and radial stress are continuous at the interface between the region of non-anchor-grouting surrounding rock and the region of anchor-grouting, and there is no friction at the interface, the corresponding boundary conditions are
ð47Þ
ur ¼ ucr ; at r ¼ b
ð59Þ
The constitutive Eq. (21) of the K – H model can be rewritten as E3 þE4 h E þE h scr þ 2 s_ cr ¼1cr þ 3K 3 4 21 1cm þ 2 1_ cr E 3 E4 E3 E4 E3 E4 E4 3K h2 h2 þ 2 ð48Þ 1_ E3 E4 E4 cm E3 þE4 h E þE h s þ 2 s_ ¼1 þ 3K 3 4 21 1cm þ 2 1_ cu E 3 E 4 cu E 3 E 4 cu cu E3 E4 E4 3K h2 h2 þ 2 ð49Þ 1_ E3 E4 E4 cm
Lr ¼ Lcr ; at r ¼ b
ð60Þ
Lru ¼ Lcru ¼ 0; at r ¼ b
ð61Þ
E3 þE4 h h t þ 2 t_ ¼ g þ 2 g_ E3 E4 cru E3 E4 cru cru E4 cru
ð50Þ
Utilizing a similar solving method as used for the region of non-anchor-grouting, the stress and the displacement in the region of anchor-grouting are represented as
scr ¼2a1 þa 2 r
22
2ða5 þ4a3 r
22
23a6 r
24
Þcos2u
scu ¼2a1 2a 2 r22 þða5 þ12a4 r22 23a6 r24 Þcos2u
ð51Þ ð52Þ
Since there is no radial stress in the inner boundary region of anchor-grouting, the corresponding boundary conditions are Lcr ¼ 0; at r ¼ a
ð62Þ
Lcru ¼ 0; at r ¼ a
ð63Þ
Applying Laplace transforms to Eqs. (59) – (63) yields u r ¼ u cr ; at r ¼ b
ð64Þ
L r ¼ L cr ; at r ¼ b
ð65Þ
L ru ¼ L cru ¼ 0; at r ¼ b
ð66Þ
L cr ¼ 0; at r ¼ a
ð67Þ
L cru ¼ 0; at r ¼ a
ð68Þ
Substituting Eqs. (45) – (47) and Eqs. (56) –(58) into Eqs. (63) – (68), and utilizing Eqs. (40) – (46) and Eqs. (51) – (55), where the detailed expressions are given in Appendix A, finally, the expressions of stress
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and displacement in the region of non-anchor-grouting surrounding rock are expressed as 1þ l b2 P2 2 sr ¼ 2 r " # 2E2 ðPþu0r Kc Þ Kc ð2E2 u0r 2PbÞ b 1 h1 2 exp 2 t 2E2 þbKc 2E2 þbKc E1 b2 KðE3 u0r 2PbÞ b h exp 2 2 2 t cos 2u 2 2 ð69Þ 2E3 þbK E3 þE4 r
1þ l b2 Pþ 2 su ¼ 2 r " # 2E2 ðPþu0r Kc Þ Kc ð2E2 u0r 2PbÞ b 1 h1 2 exp 2 t 2E2 þbKc 2E2 þbKc E1 b2 KðE3 u0r 2PbÞ b 2 h2 exp 2 þ 2 t cos 2u ð70Þ 2E3 þbK E3 þE4 r " # b2 Pþu0r Kc 2E2 u0r 2b h1 b 1 exp 2 þ t ur ¼ 2 E1 r 2E2 þbKc bð2E2 þbKc Þ b2 E3 u0r Kc b2 h 2 b2 þ 2 t cos 2u exp 2 E3 þE4 r Gc ð2E4 þbKc Þðb2 2a2 Þr ð71Þ The expressions of stress and displacement in the region of anchor-grouting are represented as ! Pb22E2 u0r a2 scr ¼ 12 2 ð2E2 þbKc Þðb2 2a2 Þ r
hb a2 K ½E u0 þð12 lÞPb 12exp 2 1 1 t 2 2 c 3 r 2E3 þbK E1 r h 2 b2 exp 2 t cos 2u ð72Þ E3 þE4 !
hb 12exp 2 1 1 t E1 2 0 a K ½E u þð12 lÞPb hb exp 2 2 2 t cos 2u þ 2 c 3 r 2E3 þbK E3 þE4 r ð73Þ
Pb22E2 u0r a2 sc u ¼ 1þ 2 2 2 ð2E2 þbKc Þðb 2a Þ r
ucr ¼
ðPb22E2 u0r ÞKc b2 ½ðKc 21Þr 2 þ2a2 4Gc ð2E2 þbKc Þðb2 2a2 Þr
hb a2 lPbKc u0r 12exp 2 1 1 t þ 2 E1 r Gc ð2E3 þbKc Þðb2 2a2 Þr hb exp 2 2 2 t ð74Þ E3 þE4
where b1 ¼ ð2E2 þbKc Þ=ð2bþbKc Þ; b2 ¼ ðE3 E4 Þ=h2 ; Kc ¼ 324mc ; K ¼ 324m:
3. Safe criterion for anchor-grouting support The expression of displacement in the region of anchorgrouting can be written in the form ucr ¼ A 2 A e2Bt þ C e2Dt
ð75Þ
where A¼
ðPb 2 2E2 u0r ÞKc b2 ½ðKc 2 1Þr 2 þ 2a2 ; 4Gc ð2E2 þ bKc Þðb2 2 a2 Þr
B¼
h1 b 1 E1
C¼
a2 lPbKc u0r ; 2 r Gc ð2E3 þ bKc Þðb2 2 a2 Þr
ð76Þ D¼
h 2 b2 E 3 þ E4
Expanding Eq. (75) into a Maclaurin series at t ¼ 0; gives ucr ¼ C þ ðAB 2 CDÞt 2 ðAB2 2 CD2 Þ þ ðAB3 2 CD3 Þ
t2 2
t3 t4 2 ðAB4 2 CD4 Þ þ ··· 6 24
ð77Þ
A measured curve of displacement of the tunnel’s two walls can be described as f ðtÞ ¼ A0 þ A1 t þ A2 t2 þ A3 t3 þ · · ·
ð78Þ
Comparing the coefficients of Eq. (77) with Eq. (78), we can get coupled equations as follows C ¼ A0 ;
AB 2 CD ¼ A1 ;
20:5ðAB2 2 CD2 Þ ¼ A2 ; ðAB2 2 CD2 Þ=6 ¼ A3
ð79Þ
Solving the coupled Eq. (79) yields A¼
A0 D þ A1 ; B
B¼
A0 D 2 2A1 ; A0 D þ A1
C ¼ A0
ð80Þ
where D should satisfy the following equation A0 A1 D3 þ 4A0 A2 D2 þ 6A0 A3 D þ 6A1 A2 2 4A22 ¼ 0
ð81Þ
The tested tunnel lies at the 2 340 m level at the XinJi mine in China. The cross-section of the tunnel is a semicircle, net width 3.4 m, net height 3.0 m. The original support form is U-sectioned steel and whitewashing support, as Fig. 4a shows. The surrounding rock is mainly made up of mudstone, metalstone and calleystone, which are extremely soft and fragile. The block diagram of the rock stratum is shown in Fig. 5. The tested tunnel lies in the sixth floor. Fig. 6 is a measuring picture at the location. The measured displacement curve of the tunnel’s two sides is shown in Fig. 7. Fig. 4b shows the location pictures after the test.
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
745
For the region of anchor-grouting, the yield criterion of Mohr – Coulomb is applied scu þ scr ð84Þ sin w R # c £ cot w þ 2 where R ¼ ðscu 2 scr Þ=2; c is cohesive force, and w is internal friction angle. Substituting Eqs. (72) and (73) into Eq. (84), yield
a2 Pb22E2 u0r h 1 a1 t 12exp 2 E1 r 2 ð2E2 þbKc Þðb2 2a2 Þ a2 K ½E u0 þð12 lÞPb ha þ 2 c 3 r exp 2 2 2 t cos 2u 2E3 þbK E3 þE4 r # cð1þsin wÞcos w
ð85Þ
For a given tunnel, c and w are certain values. To make Eq. (85) true for any value of time t; the following condition must be met a2 Pb22E2 u0r 2 r ð2E2 þbKc Þðb2 2a2 Þ þ
Fig. 4. (a) The status of tunnel before the test of anchor-grouting. (b) The status of tunnel after the test of anchor-grouting.
The coefficients of the equation are shown in Table 1. From Table 1 and Eqs. (80) and (81), we have A ¼ 40:0787514; C ¼ 13:044026;
B ¼ 0:058362381;
ð82Þ
D ¼ 0:10062504
Substituting Eq. (82) into Eq. (75), we can get the cumulative displacement creep function at the measured points in the tunnel’s two walls 20:0584t
ucr ¼ 40:08 2 40:08 e
þ 13:044 e
20:101t
ð83Þ
where the unit of the cumulative displacement ucr is mm, and the unit of time t is day.
a2 Kc ½E3 u0r þð12 lÞPb cos 2u # const 2E3 þbK r2
ð86Þ
where const ¼ cð1þsin wÞcos w: When r ¼ a and cos 2u ¼ 1; the condition to make the left of Eq. (86) reach the maximum value is Pb 2 2E2 u0r Kc ½E3 u0r þ ð1 2 lÞPb # const þ 2E3 þ bK ð2E2 þ bKc Þðb2 2 a2 Þ ð87Þ Eq. (87) can be simplified into ðP 2 Kc u0r Þb 2ð1 2 lÞP 2 Ku0r b þ 2 2 2ð2E3 þ bKÞ ð2E2 þ bKc Þðb 2 a Þ Kc 1 þ 2 2 u0r # const 2 b 2 a2
ð88Þ
It is seen from inequality (88) that increasing the elastic coefficients E2 ; E3 makes the inequality more easily satisfied, thus helping the surrounding rock of a tunnel to
Fig. 5. The tunnel terrane column.
746
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
the supported region can be determined, which is related to the property of the surrounding rock, the material of the anchor-grouting layer and the method of anchor-grouting, etc. In practical engineering, the thickness ðb 2 aÞ of the supported region is taken as 2.5 m.
4. Conclusions
Fig. 6. The disposition of observation points on the surface of tunnel.
Fig. 7. The curve of cumulative displacements in the observation station of the tunnel.
be maintained in an elastic deformation state rather than suffer plastic failure. According to the Poynting – Thomson model, it is not easy for the region of non-anchor-grouting to go through plastic yield. From the yield criterion (88) of Mohr –Coulomb for the anchor-grouting supported region, the thickness of
1. Based on the theory of rheology, the paper presents an analytical method for the viscoplastic stress and displacement distribution around a soft rock tunnel. It is seen from the solution that the stresses and displacements of the surrounding rock are related to time, so that they increase with time, and gradually reach a stabilized value. 2. The soft rock tunnel is composed of a supported region and an unsupported region. Because the unsupported region around rock belongs to soft rock having less elasticity, the P –T model is applied to the unsupported region. The K – H model is applied to the supported region that belongs to the anchor-grouting layer having greater elastic behavior. 3. From Eq. (77), the best support time may be determined. The later the time of anchor-grouting support, the larger the displacement of the surrounding rock. The variation of the displacement is less after anchor-grouting support. 4. Expanding the expression of displacement in the supported region into a Maclaurin series, and comparing measured cumulative displacements around rock, the theoretical analysis and the solution for the rheology problem of the anchor-grouting soft rock tunnel are proved to be effective in the practical engineering. According to the Mohr – Coulomb yield criterion and stress expressions obtained, a safe criterion for anchor-grouting soft rock tunnel can be set up.
Acknowledgements The authors thank the referees for their valuable comments.
Appendix A Table 1 The fitting curve coefficients of displacements in the observation station of the tunnel Items
Coefficients A0
A1
A2
A3
Average 13.044026 1.0265357 20.0022175707 21.9372207 £ 1025 Squared 2.16559 0.25746 0.00822 0.0001 deviation
The initial displacements of surrounding rock are represented by u0r and u0u ; and 10r ; 10u ; g0ru and 10m represent initial strains of surrounding rock. Applying Laplace transforms on the elastic solution without support, and solving these equations, the corresponding solution of the viscoelastic equations are expressed as ur ¼
P ½ð1 þ lÞu1 þ ð1 2 lÞu11 cos 2u 2
ðA1Þ
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
P uu ¼ ð1 2 lÞv1 sin 2u 2
ðA2Þ
1r ¼
P ½ð1 þ lÞ11r þ ð1 2 lÞ111 r cos 2u 2
ðA3Þ
1u ¼
P ½ð1 þ lÞ11u þ ð1 2 lÞ111 u cos 2u 2
ðA4Þ
gru ¼ Pð1 2 lÞg11 ru cos 2u
ðA5Þ
P ð1 2 lÞe1 cos 2u 2
ðA6Þ
1m ¼
11r
11u
(
! 1 E1 ðE1 þ E2 Þ exp 2 t E1 þ E2 E 2 h1 " !#) 1 E1 ðE1 þ E2 Þ 1 2 exp 2 t þ E2 E 2 h1
b2 u ¼ 2r
ðA7Þ
! ( b2 1 E 1 E2 t ¼ 2 2 exp 2 ðE1 þ E2 Þ 2r E1 þ E2 " !#) 1 E 1 E2 1 2 exp 2 þ t E2 ðE1 þ E2 Þh1
ðA8Þ
(
ðA9Þ
!( ! b2 b2 1 E1 E 2 4 2 4m 2 2 exp 2 t u ¼ E1 þ E2 2r ðE1 þ E2 Þh1 r " !#) 1 E1 E2 1 2 exp 2 t ðA10Þ þ E2 ðE1 þ E2 Þh11 11
!( ! b2 b2 1 E1 E 2 2 2 4m þ 2 v ¼2 exp 2 t E 1 þ E2 2r ðE1 þ E2 Þh1 r " !#) 1 E1 E2 1 2 exp 2 t ðA11Þ þ E2 ðE1 þ E2 Þh11 11
!(
!
b2 3b2 1 E 1 E2 424m 2 2 exp 2 t 2 E1 þE2 ðE1 þE2 Þh1 2r r " !#) 1 E 1 E2 12exp 2 t ðA12Þ þ E2 ðE1 þE2 Þh11
111 r ¼2
!(
111 u ¼
Pb2 1 ð1þ lÞg ¼ ½ð224mc Þa1 b2 a 2 b21 Gc 2Gb
!
b2 3b2 1 E1 E2 4m 2 2 exp 2 t 2 E1 þE2 ðE1 þE2 Þh1 2r r " !#) 1 E 1 E2 12exp 2 t ðA13Þ þ E2 ðE1 þE2 Þh11
ðA16Þ
Pb2 ð12 lÞ½bð424mÞþ2d 4Gb ¼
! 1 E1 E 2 exp 2 t E 1 þ E2 ðE1 þ E2 Þh1 " !#) 1 E 1 E2 1 2 exp 2 t þ E2 ðE1 þ E2 Þh1
b2 ¼ 2 2r
!( ! b2 3b2 1 E1 E2 22 2 exp 2 t E1 þE2 ðE1 þE2 Þh1 2r 2 r " !#) 1 E1 E 2 12exp 2 t ðA14Þ þ E2 ðE1 þE2 Þh11 ( ! b2 1 E1 ðE1 þE2 Þ 11 exp 2 t 1m ¼ 2 2 ð224mÞ E1 þE2 E2 h1 3r " !#) 1 E1 ðE1 þE2 Þ 12exp 2 t ðA15Þ þ E2 E2 h1
g11 ru ¼ 2
Substituting Eqs. (45) – (47) and Eqs. (56) –(63) into Eqs. (64) – (68), and separating the uniform part which is independent of u and the non-uniform part which is dependent on u; we have
where 1
747
1 ½24mc a 4 b3 2 a 5 b2ð424mc Þa3 b21 2 a 6 b23 Gc
ðA17Þ
P s0 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m ð1þ lÞð12 gÞ2 r 2 E1 =h1 þs h2 0 h 2 0 3K h2 h2 0 s 2 1 2 2 1 E E cr E4 cr E3 E4 E4 cm ¼ 2a1 þ a 2 b22 2 3 4 E3 þE4 h þ 2 s E3 E4 E 3 E4 ðA18Þ P ð12 lÞð122b 23dÞ ¼ 2ða5 þ4a3 b22 23a6 b24 Þ 2
ðA19Þ
a 5 þ6a4 22a3 b22 þ3a6 b24 ¼ 0
ðA20Þ
h2 0 h2 0 3K h2 h2 0 scr 2 1cr 2 2 1 E E E E3 E4 E4 cm 4 ¼0 2a1 þ a 2 a22 2 3 4 E3 þE4 h þ 2 s E3 E4 E3 E4 ðA21Þ a 5 þ4a3 a22 23a6 a24 ¼ 0
ðA22Þ
a 5 þ6a4 a2 22a3 a22 þ3a6 a24 ¼ 0
ðA23Þ
Solving the above Eqs. (A16) – (A23) yield (" ) # 1 a2 G P 0 0 12 2 12 a 1 ¼ Lcr þLr 2ð1þ lÞ n Gc 2 b
a2 G a 2 ¼ 2ð1þ lÞPþ2 ð122mc ÞL0cr þ2L0r Gc n
ðA24Þ ðA25Þ
a 3 ¼ 2
3Pb ð12 lÞða4 þa2 b2 þ2b4 Þð3sþ424mÞ 8Gm2
ðA26Þ
a 4 ¼ 2
3Pb ð12 lÞð3a2 þb2 Þð3sþ424mÞ 4Gm4
ðA27Þ
748
H.L. Dai et al. / International Journal of Pressure Vessels and Piping 81 (2004) 739–748
a 5 ¼
3Pb ð12 lÞð2a4 þa2 b2 þb4 Þð3sþ424mÞ 4Gm3
References ðA28Þ
Pb a 6 ¼ 2 ð12 lÞð3sþ424mÞ 4Gm1
ðA29Þ
3ð12 lÞð4a4 2a2 b2 þb4 Þð3sþ424mÞ b ¼ 2þ 2a4 bGm1 ða2 þ3b2 Þ
ðA30Þ
2G ½2ð122mc Þa1 b2 a 2 b21 PbGc ð1þ lÞ
ðA31Þ
g ¼
ð12 lÞð4a4 2a2 b2 þb4 Þð3sþ424mÞ d ¼ 212 2a4 bGm1 ða2 þ3b2 Þ
ðA32Þ
where
s0r 2ðE1 þE2 Þ½10r þð3K=E2 21Þ10m ðA33Þ E1 =h1 þs h2 0 h2 0 3K h2 h2 0 scr 2 1cr 2 2 1 E E E4 E3 E4 E4 cm L0cr ¼ 3 4 ðA34Þ E3 þE4 h2 þ s E 3 E4 E3 E 4 " 1 1 m1 ¼ 4 3 2 ð625mÞð5a6 23a4 b2 þ9a2 b4 23b6 Þ a b ða þ3b2 Þ G L0r ¼
2
1 2m ð5a6 23a4 b2 þ9a2 b4 23b6 Þþ c ð3a6 þ8a4 b2 Gc Gc #
þ5a2 b4 þb6 Þ
ðA35Þ
m2 ¼ a2 b2 ða2 þ3b2 Þm1
ðA36Þ
m3 ¼ a4 b2 ða2 þ3b2 Þm1
ðA37Þ
m4 ¼ a2 ða2 þ3b2 Þm1 G G a2 G n ¼ 2 1þ 22 mc 22 2 12 Gc Gc Gc b
ðA38Þ ðA39Þ
Substituting Eqs. (A14) – (A32) into Eqs. (40) – (44) and Eqs. (51) –(55), and utilizing Laplace transforms, we can get Eqs. (69) – (74).
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