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Full stress and displacement fields for steel-lined deep pressure tunnels in transversely anisotropic rock
Tunnelling and Underground Space Technology 56 (2016) 125–135
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Full stress and displacement fields for steel-lined deep pressure tunnels in transversely anisotropic rock Antonio Bobet a, Haitao Yu b,⇑ a b
School of Civil Engineering, Purdue University, West Lafayette, IN, USA Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai, China
a r t i c l e
i n f o
Article history: Received 9 September 2015 Received in revised form 17 February 2016 Accepted 9 March 2016 Available online 24 March 2016 Keywords: Pressure tunnel Circular tunnel Steel liner Anisotropy Transversely anisotropic rock Water pressure
a b s t r a c t An analytical solution is derived that provides closed-form formulations for stresses and displacements for a deep pressure tunnel in a transversely anisotropic rock, with a steel liner, and subjected to a uniform internal pressure. For the derivation, it is assumed that the tunnel support includes a thin steel liner, concrete backfill and that there is an annulus of damaged rock around the concrete. It is also assumed that all materials remain elastic and that the concrete and the damaged rock cannot transmit shear or tangential stresses. The solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS. For thin steel liners, it can be assumed that the contact pressure between the different materials is uniform, and thus the bending moments in the liner are negligible. This is due to the low bending stiffness of the steel liner. The paper is inspired by and expands the work by Pachoud and Schleiss (2015) who conducted an extensive numerical parametric analysis to obtain correction factors that, when used with an analytical solution for isotropic materials, approximate the maximum principal stress in the liner and intact rock. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The purpose of pressure tunnels is to convey water under pressure with acceptable losses and without causing instabilities. Pressure tunnels are mostly built in hydroelectric power plant projects, but they are also used, albeit with smaller pressures, for water supply and wastewater conveyance. Pressure tunnels usually have a circular cross section because of its structural advantages due to the large internal pressures, and also for hydraulic reasons. The cross section area and the surface roughness (i.e. unlined tunnel or concrete or steel lining) depend on the head losses accepted over the length of the tunnel, provided that the opening is stable. The location of the ground water table, together with the hydraulic head in the tunnel, determines whether water will flow from the tunnel to the rock or vice versa (Merritt, 1999). If the hydraulic head inside the tunnel is larger than that from the water table, water will flow out from the tunnel, which requires that rock stresses around the opening are larger than the water pressures to prevent hydrofracturing or hydrojacking (Benson, 1989; Seeber, 1985a, b). If the tunnel is supported, the stiffness of the rock determines the contribution of the rock mass ⇑ Corresponding author at: Department of Geotechnical Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China. E-mail address: [email protected] (H. Yu). http://dx.doi.org/10.1016/j.tust.2016.03.005 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
to support the internal pressure, as the load transfer from the liner to the rock depends largely on the stiffness of the liner relative to the surrounding rock (e.g. Einstein and Schwartz, 1979; Bobet, 2001, 2003, 2011; Bobet and Nam, 2007). Finally, the permeability of the rock mass is required to estimate the leakage flow and thus the pore pressures behind the support. The layout of the pressure tunnel can be decided based on empirical methods and the information obtained during field exploration, if available. The empirical methods can provide initial estimates as to where a watertight liner is needed (Dann et al., 1964; Benson, 1989; Hartmaier et al., 1998; Bergh-Christensen, 1982; Broch, 1984a, b; Selmer-Olsen, 1985; Brekke and Ripley, 1989, 1993; Alvarez et al. 1999). Following this, calculations regarding flow rate, stresses in the liner and in the ground should be made to assess whether the leakage is acceptable and that the stress state in the liner and ground is adequate, and to provide an estimate of the factor of safety (see e.g. Schleiss, 1986, 1997; Hendron et al., 1989; Fernandez, 1994; Fernandez and Alvarez, 1994; Bobet and Nam, 2007). A steel liner is the industry standard for the sections of the tunnel where the minimum principal stress of the rock around the excavation is smaller than the internal tunnel pressure with a suitable factor of safety (Fernandez, 1994). Steel liners are also needed when the internal pressure is large and the surrounding rock has low modulus, so leakage control with reinforced concrete is not
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Nomenclature Notation As, Is cross-section and moment of inertia of the liner Young’s modulus and Poisson’s ratio of the backfill conEc, mc crete Ecrm, mcrm Young’s modulus and Poisson’s ratio of the damaged isotropic rock Erm, mrm Young’s modulus and Poisson’s ratio of the intact isotropic rock rm Erm Young’s modulus of intact rock in x- and y-axis (Ex = E x , Ey and Ey = E’ in Pachoud and Schleiss, 2015) crm Young’s modulus of damaged rock in x- and y-axis Ecrm x , Ey Es, ms Young’s modulus and Poisson’s ratio of the liner Grm shear modulus of the intact rock xy Gcrm shear modulus of the damaged rock xy radial stress at the steel-concrete contact pc pi internal water pressure r, h polar coordinates ri radius of the tunnel rcrm limit of cracked concrete liner rrm limit of damaged rock ts liner thickness axial force and moment of the liner Ts, Ms
possible. A concrete-encased steel cylinder is the most common type of liner (Eskilsson, 1999). There are two criteria that need to be satisfied for the design of the steel liner (Schleiss, 1988): (1) working stress and deformation of the steel liner; and (2) loadbearing capacity of the rock mass. The first criterion requires a design of the steel liner to adequately support the internal pressures during operation, the external hydraulic pressures during grouting and dewatering, the handling loads during transportation and erection, and that limits local deformations, e.g. crack bridging in the backfill concrete (Schleiss, 1988; Brekke and Ripley, 1993). The second criterion is needed to guarantee sufficient safety against rock failure, and thus ensure the load sharing between liner, backfill concrete and rock assumed for the first criterion. The design of a steel-lined pressure tunnel due to the internal pressure is generally based on an elastic analysis assuming that the support and rock mass are isotropic and elastic and that the concrete and an annulus of rock surrounding the concrete are damaged such that only radial pressures can be transmitted. Beyond the damaged rock, intact or undamaged rock is assumed (e.g. Schleiss, 1988; Moore, 1989; USACE, 1997; Hachem and Schleiss, 2009; ASCE 2012). Analytical solutions are available that account for these assumptions and are included in the next section for completeness. Pachoud and Schleiss (2015) provided a number of correction factors to the analytical solution to expand it to cases where the intact rock is transversely anisotropic. The factors were obtained after an extensive numerical analysis using a Finite Element Method and a genetic algorithm to minimize errors between the corrected analytical solution and the numerical results. The corrected factors were obtained to estimate the major principal stress in the liner and in the intact rock. This paper builds on the work from Pachoud and Schleiss (2015) and provides a full analytical solution for stresses and displacements for the steel liner, concrete, damaged rock and intact rock. It also expands the range of cases, as the solution accounts for the anisotropy of the damaged rock and does not have the limitation that the existing solution has of assuming that the shear modulus of the rock is equivalent to the empirical relation of Saint–Venant. Clearly, analytical formulations are limited due to the assumptions that need to be made to reach the solution. In most cases, the
Ux, Uy
displacements in Cartesian coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock Ur, Uh displacements in polar coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock w gap between steel liner and backfill concrete x, y Cartesian coordinates of axes of elastic symmetry zk complex number, zk = x + lky, k = 1, 2 ex, ey, cxy axial and shear strains in x- and y-axis er, eh, crh radial, tangential and shear stresses in polar coordinates fk complex number that depends on zk through conformal mapping /(zk), /0 (zk) stress function and its derivative l1, l2 roots of compatibility equation mxy, mxz, myz Poisson’s ratios in x–y axes rx, ry, sxy normal and shear stresses in Cartesian coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock rr, rh, srh radial, tangential, and shear stresses in polar coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock
design will require a numerical method that does not have the shortcomings of the analytical solutions, as it can consider the construction process, non-linear behavior, etc. Closed-form solutions however are invaluable to obtain a better understanding of the interplay that exists between loads, rock and support, to identify what are the most critical parameters for the problem, and to provide first estimates or even a preliminary design. An added advantage is that they can be used with very little cost to conduct sensitivity analysis and, most importantly, to provide benchmark values to check the results of the more complex numerical models.
2. Steel-lined pressure tunnel in isotropic rock Fig. 1 shows how the internal pressure is distributed between the steel, concrete and rock mass. As the internal pressure increases, the steel liner may initially take all the stress (this may occur if there is a gap between the steel and the concrete, e.g. due to differential thermal contraction between the steel and the concrete, creep of the rock, cycles of loading and unloading as the tunnel is pressurized and emptied, or the skin grouting of the steel-concrete interface is not done or is not effective). With further increase of internal pressure, as the steel liner expands, part of the loading is transferred to the concrete, to the rock damaged during excavation and to the undamaged/intact rock. It is generally assumed that the concrete and the damaged rock can only transmit radial deformations as the damage prevents transfer of tangential and shear stresses. This results in radial stresses that decrease with 1/r within the concrete and damaged rock zones, and 1/r2 in the undamaged rock zone. The magnitude of the radial stresses and corresponding radial displacements can be obtained by imposing equilibrium along the radial direction and compatibility of radial displacements at the liner-concrete, concrete-damaged rock, and damaged-undamaged rock boundaries (Moore, 1989; USACE, 1997; Hachem and Schleiss, 2009). Thus: For the liner:
T s ¼ ðpi þ pc Þr i r2
U sr ¼ ð1 m2s Þ Esits ðpi þ pc Þ
ð1Þ
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Fig. 1. Distribution of radial stresses in a concrete-encased steel lining.
For the cracked backfill concrete:
rcr ¼ pc ri r
ð2Þ
r2
U cr ¼ w þ ð1 m2s Þ Esits ðpi þ pc Þ þ ð1 m2c Þ Erci pc ln rr
i
For the damaged rock:
r
crm r
¼ rri pc r2
U crm ¼ w þ ð1 m2s Þ Esits ðpi þ pc Þ þ ð1 m2c Þ Erci pc ln rcrm r r þð1 m
i
ri 2 crm Þ Ecrm
ð3Þ
r
pc ln rcrm
For the intact rock: r i r rm rrm r ¼ r 2 pc rm rh ¼ rirr2rm pc
ð1þmrm Þ U rm r ¼ Erm
ð4Þ ri r rm r
pc
with
pc ¼ 1m2 s
Es t s
1m2s Es t s
ri þ
1m2c Ec
pi r i wri
ln rcrm þ ri
1m2crm Ecrm
mrm rm ln rrcrm þ 1þErm
ð5Þ
where Ur are the radial displacements (subscripts s, c, crm and rm denote steel, backfill concrete, damaged rock and intact rock, respectively); rr and rh are the radial and tangential stresses (subscripts s, c, crm and rm denote steel, backfill concrete, damaged rock and intact rock, respectively); w is the gap between the steel and the concrete; r is the radial distance from the center of the tunnel; ri, rcrm and rm are the internal radius of the tunnel, the radial extent of the concrete and of the damaged rock, respectively; Es, Ec, Ecrm and Erm are the Young’s modulus of the steel, concrete, damaged
rock and intact rock; ms, mc, mcrm, mrm are the Poisson’s ratios of steel, concrete, damaged rock and intact rock; ts is the thickness of the steel liner; pi is the water pressure and pc the radial stress at the liner-concrete interface. Note that because the problem has axial symmetry, shear stresses and tangential displacements are zero and, because of the assumption that a cracked material can only transmit radial stresses, the tangential stresses in the concrete and damaged rock are zero. Note also that the liner has been considered as a shell, i.e. the thickness is much smaller than the radius; in other words, ri ri + ts or ts/ri 1, which is typically the case for steel-lined pressure tunnels. For the above equations, it is assumed that all materials are elastic, and that tension is positive and compression is negative (the internal pressure pi is taken as positive); thus, displacements are positive in the positive direction of the axes of the coordinate system. Inspection of Eqs. (1-4) indicates that stresses and displacements strongly depend on the radial stress at the liner-concrete contact, pc (because of elasticity, all results are proportional, as they should, to the applied pressure pi). Eq. (5) provides the magnitude of pc, which, neglecting the gap parameter w, can be written as:
pc ¼
1m2s Eeq
pi
Eeq ¼ ð1 m2s Þ þ ð1 m2c Þ EEcs
ts ri
s ln rcrm þ ð1 m2crm Þ EEcrm r i
ts ri
rm ln rrcrm
ð6Þ
þð1 þ m
Es t s rm Þ Erm r i
The equation shows that the result is inversely proportional to a weighted average of the relative stiffness between the steel liner and the different materials. The solution strongly depends on the ratio of the thickness of the steel liner to the internal tunnel radius, ts/ri.
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The theoretical framework has been incorporated into recommendations for design of steel-lined pressure tunnels (e.g. USACE, 1997; ASCE 2012) and has been successfully used to estimate the load transfer between the steel liner and the backfill concrete and rock on a large number of pressure tunnels (Brekke and Ripley, 1989) and to provide estimates of the stress on the liner (e.g. Marulanda and Gutierrez, 1999). 3. Steel-lined pressure tunnel in transversely anisotropic rock The problem solved is defined in Fig. 2 that, as mentioned earlier, is identical to that considered by Pachoud and Schleiss (2015), with few exceptions that expand the scope of the investigation. It consists of a deep tunnel with circular cross section with internal radius ri, subjected to an internal water pressure pi, excavated in a transversely anisotropic rock. It is assumed that the steel-lined portion of the tunnel is very long, so plane-strain conditions apply. The tunnel support includes a circular steel liner with thickness ts and annular concrete backfill with thickness rcrm (ri + ts); with the assumption of ts/ri 1, the concrete thickness is rcrm ri. It is assumed that there is an annular volume of rock that is damaged during excavation, with constant thickness rrm – rcrm. As explained by Pachoud and Schleiss (2015), a circular shape of the damaged rock is a reasonable approximation of actual, more complex, conditions and may be sufficiently accurate in the case of grouted near-rock zone. All materials are assumed to remain elastic, with the following properties: Steel liner: Isotropic material with Young’s modulus Es and Poisson’s ratio ms; concrete backfill: Isotropic material with Ec and mc; damaged rock: Transversely crm crm crm crm (note that anisotropic with properties Ecrm x ; Ey ; Gxy ; mxz ; myx this is different from Pachoud and Schleiss, 2015 who considered
the damaged rock as isotropic); and intact rock: Transversely anirm rm rm rm sotropic with properties Erm x ; Ey ; Gxy ; mxz ; myx . The x- and y-axes in Fig. 2 denote the axes of elastic symmetry; the x axis, for convenience, is taken as horizontal. The concrete backfill and the damaged rock are assumed to be sufficiently cracked or damaged such that no tangential or shear stresses can be transmitted through them, which is a common assumption (e.g. Schleiss, 1988; Pachoud and Schleiss, 2015). The problem is solved by imposing the condition of equilibrium on each of the four different materials and by satisfying the boundary conditions. The latter is accomplished by just ensuring compatibility of radial stresses and radial displacements at the contact between materials. This is the only condition needed since the shear stress and the tangential stress in the concrete and damaged rock are zero, and thus at the interfaces the shear stress is also zero. For the liner, given that it is treated as a shell, equilibrium is given by the following equations (e.g. Flügge, 1966): s s ri dT dM ¼ r2i ss dh dh
ð7Þ
ri T s þ ddhM2 s ¼ r 2i rsr 2
where r and h are the polar coordinates, with origin at the center of the tunnel and h measured counterclockwise from the x-axis; Ts and Ms are the axial force and moment acting on the liner; and rsr and ss are the radial and shear stress at the contact with the concrete backfill. Deformations are expressed as (e.g. Flügge, 1966): d2 U sh dh2
dU sh dh
s
r þ dU ¼ dh
þ U sr þ r2IsA i
s
ð1m2s Þ 2 s ri Es As
s
4
U sr 4
d dh
2 s ð1m2 Þ þ 2 ddhU2r þ U sr ¼ Es Ass r 2i rsr
Fig. 2. Steel-lined pressure tunnel in transversely anistropic rock.
ð8Þ
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where U sr and U sh are the radial and tangential displacements, in polar coordinates, and As and Is are the area and moment of inertia, respectively, of the liner cross section (As = ts and Is = 1/12 t 3s since the liner is assumed to have a constant thickness). As the pressure inside the tunnel pi increases, the liner deforms outwards. If there is a gap, w, between the liner and the concrete backfill, a contact pressure pc between the liner and the concrete will not develop until the gap is closed. Thus, a distinction needs to be made between the total liner deformations and deformations at the liner-concrete interface, which are those needed to impose the boundary conditions. The following applies when the internal pressure is large enough to close the gap. In other words, when
pi >
Es t s w 1 m2s r 2i
ð9Þ
which is the pressure required to close the gap w. Eqs. (7) and (8) can be solved when the stresses at the steelconcrete contact are known. As discussed, the shear stress at the contact is zero because the concrete cannot transmit shear or tangential stresses. For the radial stresses, the following expression is attempted, given the symmetry of the problem (about the x and y axes):
pc ¼ po þ
1 X
ð10Þ
pn cos nh
n¼2;4;6
pc is the radial stress at the liner-concrete contact, and po and pn are constants that are obtained by imposing compatibility of radial displacements at each interface. Thus, 1 X
T s ¼ ðpo þ pi Þr i
n¼2;4;6
Ms ¼
1 X n¼2;4;6
U sr
¼ w þ
pn r 2i
n2 1
1m2s Es t s
pn r i n2 1
cos nh ð11Þ
cos nh
1m2s 4 r Es Is i
ðpi þ
1 X n¼2;4;6
po Þr2i
þ
1m2s 4 r Es Is i
1 X
pn n2 1
pn n2 1
cos nh ð12Þ
sin nh
Note that the radial displacements in (12) are produced after the gap is closed. Thus, the total radial displacements of the liner are those given in (12) plus the gap w. For the concrete backfill, with the assumption that only radial stresses can occur, the equation of equilibrium in polar coordinates takes the form:
drr rr þ ¼0 dr r
ð13Þ
which, after integration and compatibility of radial stresses at the contact with the liner, results in the following expression for the radial stresses in the concrete, rcr :
rcr ¼
1 X
ri pn cos nh po þ r n¼2;4;6
!
ð14Þ
1 X
1m 1m pn ðpi þ po Þr 2i þ cos nh Es ts Es I s n2 1 n¼2;4;6 ! 1 X 1 m2c r ri po þ pn cos nh ln þ r Ec i n¼2;4;6
U cr ¼ w þ
! 1 X ri ¼ pn cos nh po þ r n¼2;4;6
ð16Þ
In plane strain, for a transversely anisotropic material, strains are expressed as (e.g. Detournay and Cheng, 1993; Cheng, 1998; Wang 2000):
ex ¼ a1 rx a2 ry ey ¼ a2 rx þ a3 ry c ¼ Gsxyxy rz ¼ mxz rx þ EEyx myx ry
ð17Þ
a1 ¼ 1Emx xz 2
a2 ¼ ð1þmExzy Þmyx
a3 ¼ 1 EEyx m2yx
1 Ey
where rx, ry are the stresses in the x and y directions, Ex and Ey are the Young’s modulus in the x and y directions, mxz and myx are the Poisson’s ratios in the xz and yx directions, respectively, and Gxy is the shear modulus. Note that because of the symmetry of the strain tensor, mxy = myx Ex/Ey. Note also that the properties in the z and x directions are the same. It is convenient to express the strains in polar coordinates and as a function of radial, tangential and shear stresses. After manipulation of Eq. (17), one gets:
er ¼ a11 rr þ a12 rh þ a13 srh eh ¼ a21 rr þ a22 rh þ a23 rrh crh ¼ a31 rr þ a32 rh þ a33 rrh h i a11 ¼ 18 3a1 2a2 þ 3a3 þ G1xy þ 4ða1 a3 Þ cos 2h þ a1 þ 2a2 þ a3 G1xy cos 4h h
i
a1 6a2 þ a3 G1xy a1 þ 2a2 þ a3 G1xy cos 4h
a13 ¼ a31 ¼ 14 ½2ða1 a3 Þ sin2h þ a1 þ 2a2 þ a3 G1xy sin 4h h i a22 ¼ 18 3a1 2a2 þ 3a3 þ G1xy 4ða1 a3 Þ cos 2h þ a1 þ 2a2 þ a3 G1xy cos 4h h i a23 ¼ a32 ¼ 14 2ða1 a3 Þsin 2h a1 þ 2a2 þ a3 G1xy sin4h h i a33 ¼ 12 a1 þ 2a2 þ a3 þ G1xy a1 þ 2a2 þ a3 G1xy cos 4h
ð18Þ Given the assumption of no shear and tangential stresses and after integration of the radial strains, one gets the radial displaceas: ments U crm r
U crm ¼ w þ r
1m2s Es t s
ðpi þ po Þr 2i þ
1 X
1m2s 4 r Es I s i
n¼2;4;6
þ
1m2c r i ðpo Ec
þ
1 X n¼2;4;6
pn n2 1
cos nh
pn cos nhÞ ln rcrm þ ri 1 X
ðb1 þ b2 cos 2h þ b3 cos 4hÞðpo þ
r pn cos nhÞr i ln rcrm
ð19Þ
n¼2;4;6
Integration of the radial strains associated with (14) gives the radial displacements U cr : 2 s
r
crm r
a12 ¼ a21 ¼ 18
n¼2;4;6
U sh ¼
For the damaged rock, equilibrium is given by (13). The radial stresses rcrm r , after imposing compatibility of radial stresses at the contact with the concrete are:
2 s 4 ri
ð15Þ
1 b1 ¼ 18 ð3acrm 2acrm þ 3acrm þ Gcrm Þ 1 2 3 xy
acrm b2 ¼ 12 ðacrm 3 Þ 1 1 crm 1 þ acrm Gcrm b3 ¼ 8 a1 þ 2acrm 2 3 xy
Note that Eq. (19) satisfies compatibility of radial displacements at the concrete-damaged rock interface. Note also that Eq. (19) can be used when the damaged rock is isotropic (which was the assumption by Pachoud and Schleiss, 2015) by making Ex = Ey = Ecrm, mxz = myx = mcrm and Gxy = Ecrm/2(1 + mcrm), which results in b1 =(1 m2crm )/Ecrm and b2 = b3 = 0.
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The final problem is that of a circular opening of radius rrm in a transversely anisotropic rock, with principal axes of elastic symmetry x and y (Fig. 2). The solution can be obtained following Lekhnitskii (1963). Equilibrium is satisfied by finding a stress function F(x,y) such that stresses are:
rx ¼ @@y2F 2
ry ¼ @@x2F @2 F sxy ¼ @x@y 2
ð20Þ
The compatibility equation is, in terms of the function F(x,y):
a1
4
4
ð21Þ
where a1, a2, a3 are obtained from Eq. (17) using the properties of the intact rock. The solution of Eq. (21) gives F(x,y). Lekhnitskii (1963) found a solution by introducing the complex variable zk = x + lky, where lk is a complex number. Expressing (21) as a function of the complex variable z, one gets:
a1 l4k þ
1 2a2 Gxy
l2k þ a3
@ F ¼0 @z4
ð22Þ
Defining /(zk) = F (zk) = oF/ozk, stresses can be obtained from:
l
l
2 0 2 0 x ¼ 2Re½ 1 /1 ðz1 Þ þ 2 /2 ðz2 Þ 0 0 y ¼ 2Re½/1 ðz1 Þ þ /2 ðz2 Þ ¼ 2Re½ 1 /01 ðz1 Þ þ 2 /02 ðz2 Þ
l
ð23Þ
l
where l1 and l2 are the roots of the equation:
a1 l4k þ
1 2a2 Gxy
l2k þ a3 ¼ 0
ð24Þ
The stress functions are, after Lekhnitskii (1963):
/2 ðz2 Þ ¼
1 X
l
1
1 ðbn 2 an Þ n 1 2 n¼1 1 X 1 l 1 l ðbn 1 an Þ n 2 2 1 2 n¼1
/1 ðz1 Þ ¼ l 1 l 1 2
l
ð25Þ
1
n are constants and f and f are complex variables, n and b where a 1 2 related to the variables zk = x + lky using the conformal mapping relation:
zk ¼
1 1 ð1 lk iÞr rm 1k þ ð1 þ lk iÞr rm 11 k 2 2
where i is the imaginary number i ¼ obtained from: 1 X R an enhi ¼ n¼1 1 X
bn enhi ¼
R
pffiffiffiffiffiffiffi 1. The constants are
rr sin hrrm dh
rr cos hrrm dh
ð27Þ
! 1 X ri po þ pn cos nh r rm n¼2;4;6
1 ¼ ðpo 12 p2 Þr i a 1 n ¼ 2n a ðpn1 pnþ1 Þri n ¼ 3; 5; 7; . . . n ¼ 0 n ¼ 2; 4; 6; . . . a 1 ¼ iðp þ 1 p Þr b 2
n¼1 1 X
1 ð1þil2 Þð1il2 Þ122
nðbn l1 an Þ1nþ1 2
ð28Þ
i
¼ i ðp b n ¼ 3; 5; 7; . . . n n1 þ pnþ1 Þr i 2n n ¼ 0 n ¼ 2; 4; 6; . . . b
ð30Þ
n¼1
Displacements are the result of the integration of the strains in (17) given the definition of stresses in (23). They are:
U x ¼ 2Re½ða1 l21 a2 Þ/1 ðz1 Þ þ ða1 l22 a2 Þ/2 ðz2 Þ
U y ¼ 2Re½ða2 l1 þ la3 Þ/1 ðz1 Þ þ ða2 l2 þ la3 Þ/2 ðz2 Þ
ð31Þ
2
It is informative to expand the equations for displacements using the stress functions given in (25) with (29). The result is: ( ) 1 X n ðarm þ l l arm Þa n ½cos nh i sin nh U x ¼ Re ½ðl þ l Þarm b 1
2
1
2
1
2
1
n¼1;3;5
(
U y ¼ Re l 1l 1 2
)
1 X
rm rm ½ðl1 þ l2 Þarm 3 an ða3 þ l1 l2 a2 Þbn ½cos nh i sin nh
n¼1;3;5
ð32Þ
Given the form or the characteristic Eq. (24), the roots must satisfy the following:
l21 þ l22 ¼ a11
1 Gxy
2a2
ð33Þ
l21 l22 ¼ aa31
which results in real numbers. The roots of the characteristic equation (24) are either complex numbers or pure imaginary numbers, but cannot be real numbers (Lekhnitskii, 1963). The roots l1 = a + ib and l2 = c + id, given (33), can only have one of the two forms: l1 = ib and l2 = id, or l1 = a + ib and l2 = a + ib. This is interesting because l1 + l2 = i(b + d) or l1 + l2 = 2ib, and l1l2 = bd or l1l2 = (a2 + b2). In other words, the sum of the roots is a pure imaginary number and the product a negative real number. Using this property, the radial displacements, at the interface with the damage rock, can be expressed as: 1 X
rm rm ðl1 þ l2 Þarm 1 bn ða2 þ l1 l2 a1 Þan
1 U rm r ¼ 2
n¼1;3;5
½cosðn þ 1Þh þ cosðn 1Þh 1 X
1 i rm rm ðl1 þ l2 Þarm þ2 l l 3 an ða3 þ l1 l2 a2 Þbn
ð34Þ
n¼1;3;5
½cosðn þ 1Þh cosðn 1Þh Finally, coefficients po, pn are obtained by making the radial displacements in (19) equal to the radial displacements in (34), at r = rrm. Doing this term by term, i.e. for the constant term, cos 2h, cos 4h, etc., results in the following system of linear equations:
po
1m2s r Es t s i
þ
1m2c Ec
rm ln rcrm þ b1 ln rrcrm þ 12 A3 þ ri i
1m2s Es t s
pi r i
rm po ðb2 ln rrcrm þ 12 A1 Þ 1m2 1m2 1 rm þ b1 ln rrcrm þ 14 A2 þ 12 A3 þp2 13 Es Iss r 3i þ Ec c ln rcrm r i
rm rm þ 16 A4 Þ þ p6 12 b3 ln rrcrm ¼0 þp4 12 ðb2 ln rrcrm rm rm po b3 ln rrcrm þ p2 12 ðb2 ln rrcrm þ 16 A1 Þ 2 2 1m 1 1ms 3 1 1 rm þp4 15 r þ Ec c ln rcrm þ b1 ln rrcrm þ 12 A2 þ 20 A3 Es I s i ri
The solution of (27) is:
2
1 X nðbn l2 an Þ1nþ1 1
rm rm þ 12 A4 Þ þ p4 12 b3 ln rrcrm ¼ wr p2 12 ðb2 ln rrcrm
where rr is the radial stress at r = rrm; that is, from (16):
o
¼ ðl l Þrrm 1 2 1
1 2
ð26Þ
n¼1
rr ¼
/02 ðz2 Þ
1 ð1þil1 Þð1il1 Þ121
4
0
r r s
/01 ðz1 Þ ¼ ðl l1 Þrrm 1 2
1
4
@ F @ F 1 @ F þ a3 4 þ 2a2 ¼0 @y4 @x Gxy @x2 @y2
Stresses are obtained from the derivatives of (25), which are:
1 rm rm þp6 12 ðb2 ln rrcrm þ 10 A4 Þ þ p8 12 b3 ln rrcrm ¼0
ð29Þ
1 rm rm pn4 12 b3 ln rrcrm þ pn2 12 ðb2 ln rrcrm þ 2ðn1Þ A1 Þ 2 2 1mc A2 A3 r crm r rm 1 1ms 3 þpn n2 1 Es Is r i þ Ec ln r þ b1 ln rcrm þ 4ðn1Þ þ 4ðnþ1Þ i
1 rm rm þ 2ðnþ1Þ A4 Þ þ pnþ4 12 b3 ln rrcrm ¼ 0 for n > 4 þpnþ2 12 ðb2 ln rrcrm
ð35Þ
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135 Table 1 Comparison between analytical solution and ABAQUS. Input parameters. Case 1 Isotropic rock 1 Anisotropic rock 1 Anisotropic rock & G0 2 Isotropic rock 2 Anisotropic rock 2 Anisotropic rock & G0
pi (MPa)
ri (m)
Es (GPa)
ms
ts (m)
Ec (GPa)
mc
10 10 10
1.5 1.5 1.5
210 210 210
0.3 0.3 0.3
0.03 0.03 0.03
20 20 20
0 0 0
2.0 2.0 2.0
2 2 2
0 0 0
15 15 15
2.5 2.5 2.5
210 210 210
0.3 0.3 0.3
0.02 0.02 0.02
20 20 20
0 0 0
3.0 3.0 3.0
20 20 20
0 0 0
with
A1 ¼ ðl1 þ l2 Þa1 þ ða2 A2 ¼ ðl1 þ l2 Þa1 ða2 A3 ¼ ðl1 þ l2 Þa1 þ ða2 A4 ¼ ðl1 þ l2 Þa1 ða2
l1 l2 a1 Þ ll1 þll2 a3 1 2 l1 l2 a1 Þ þ ll1 þll2 a3 1 2 l1 l2 a1 Þ þ ll1 þll2 a3 1 2 l1 l2 a1 Þ ll1 þll2 a3 1 2
þ a3 ll1ll2 a2 1 2 þ a3 ll1ll2 a2 1 2 a3 ll1ll2 a2 1 2 a3 ll1ll2 a2 1 2
ð36Þ
Note that in (36), given the properties of the roots of the characteristic equation previously discussed, i.e. l1 + l2 = b + d or 2b, and l1l2 = bd or (a2 + b2), the expressions l1 + l2 and l1l2 are real numbers. From a practical point of view, only a finite number of equations is needed, given that their contribution to the solution diminishes as n increases. It is important however to truncate the system of equations at a number that minimizes errors. Further comments are provided later in the next section. The solution obtained is verified by comparing its results from those obtained using the numerical program ABAQUS (ABAQUS, 2015). ABAQUS is a Finite Element code that has been extensively used and validated in geotechnical engineering. All the different cases are discretized with eight-node isoparametric elements, with only one quarter of the tunnel modeled due to the symmetry along the vertical and horizontal axes. A tied contact between the different materials is used, which effectively acts as frictionless since the shear stresses in the concrete and damaged rock are zero. No horizontal displacements are allowed along the vertical axis through the center of the tunnel and no vertical movements along the horizontal axis. A constant pressure equal to pi is imposed to the internal perimeter of the tunnel. The external boundary is placed far from the center of the tunnel such that it does not have an effect on the solution. The two same cases used by Pachoud and Schleiss (2015) are chosen for the comparisons, which is done to facilitate a discussion on the advantages and limitations of the two approaches. The first case corresponds to a circular tunnel with internal radius ri = 1.5 m, internal pressure pi = 10 MPa, steel liner with a thickness of 0.03 m, concrete backfill 0.5 m thick, and damaged rock with a thickness of 0.5 m. The second case is analogous, with ri = 2.5 m, pi = 15 MPa, and thickness of steel, concrete and damage rock of 0.02 m, 0.5 m, and 0.7 m, respectively. Three scenarios are considered in each case: one, where all materials are isotropic, and the other two with transversely anisotropic intact rock. All scenarios assume that the damaged rock is isotropic and that there is no gap between the liner and the backfill concrete. The first scenario in each case considers, as mentioned, isotropic intact rock; case one, scenario two, includes properties of a transversely anisotropic rock taken from Tonon and Amadei (2003), and case two, scenario two, from Amadei (1996). Scenarios three from cases one and two are analogous to those with the anisotropic rock, except the shear modulus is estimated based on the following Saint–Venant empirical relation:
G0rm xy ¼
Erm y 1þ
Erm y Erm x
þ 2myx
ð37Þ
rcrm (m)
Ecrm (GPa)
mcrm
Erm y
mxz
myx
Grm xy
2.5 2.5 2.5
Erm x (GPa) 2.4 7.8 7.8
(GPa) 2.4 2.4 2.4
0.22 0.22 0.22
0.22 0.07 0.07
(GPa) 983 830 1658
3.7 3.7 3.7
23.9 29.3 29.3
23.9 23.9 23.9
0.18 0.18 0.18
0.18 0.13 0.13
10,127 6200 11,514
rrm (m)
Table 1 lists the input properties used for both the analytical solution and for ABAQUS. In the numerical method, the condition of no shear and tangential stresses in the concrete and damaged rock is approximated by assuming that these materials have cylindrical anisotropy, with the radial Young’s modulus equal to the value shown in Table 1, tangential Young’s modulus equal to 1012 MPa, the smallest number that the code accepts, and the Poisson’s ratio in the radial and tangential directions equal to zero (this is necessary to have in the numerical model a positive definite stiffness matrix, given the small tangential Young’s modulus used as input). Figs. 3 and 4 provide a comparison between the results obtained from ABAQUS and from the analytical solution for cases 1 and 2, respectively. More specifically, Figs. 3(a) and 4(a) are plots of the normalized tangential stress at the interior fiber of the liner; Figs. 3(b) and 4(b), of the normalized tangential stresses in the intact rock at its contact with the damaged rock; Figs. 3 (c) and 4(c), of the normalized radial displacements of the liner and Figs. 3(d) and 4(d) of the normalized radial displacements of the intact rock at the contact with the damaged rock. In the figures, white symbols represent results from ABAQUS and black symbols from the analytical solutions. The comparison is excellent, with differences smaller than 1–2% for all the values plotted. Pachoud and Schleiss (2015) only reported results for the maximum principal stress for the liner and the intact rock. For case 1, for the liner, they reported the values 24.87, 24.27, 22.27 for scenarios 1, 2, and 3, respectively; for the intact rock, 0.1, 0.156, and 0.133. For case 2, 5.44, 6.19, 4.99 for the liner and 0.237, 0.265 and 0.239 for the intact rock. The values given are normalized as in Figs. 3 and 4. The differences between their results and the results obtained with the new simulations are all below 1%4%, except for the maximum principal stress for the intact rock for the scenarios with anisotropic rock and G0 from Eq. (37), where the errors are of the order of 20–25%. These larger errors could be due to small differences in the models, and/or how the condition of no-shear and no-tangential stresses in the concrete and damaged rock are imposed. There is no limitation in using Eq. (35) for isotropic problems. When both the damaged rock and the intact rock are isotropic, b1 = (1 – m2crm )/Ecrm and b2 = b3 = 0 for the damaged rock; and rm 2 l1 = l2 = 1, arm arm 3 ¼ a1 ¼ ð1 mrm Þ/Erm, 2 ¼ ð1 þ mrm Þ mrm =Erm , A1 = A4 = 0, A2 = 2(1 + mrm)(3–5mrm)/Erm and A3 = 2(1 + mrm)/Erm for the intact rock. With these values, the system of Eq. (35) has a solution pn = 0 for n > 0 and pc = po, which has the same result as that given in (6).
4. Discussion Figs. 3 and 4 show that, when the rock is isotropic, stresses and displacements do not depend on the angular coordinate h. This is expected because the problem is axisymmetric. What is interesting is the observation that the liner tangential stresses are insensitive to the tangential coordinate, even when the rock is anisotropic. The
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
50.0
/pi
1.0 rm
40.0
Liner Tangential Stresses
35.0 30.0 25.0
Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
20.0
white symbols: ABAQUS black symbols: analytical solution
15.0 10.0 5.0
Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
0.8
Intact Rock Tangential Stresses
s
int
/pi
45.0
white symbols: ABAQUS black symbols: analytical solution
0.6
0.4
0.2
0.0
0.0 0
15
30
45
60
75
0
90
15
30
45
i i
rm
30.0 25.0 Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
5.0 0.0 0
15
30
45
60
75
90
2.0
E /p r
rm
35.0
10.0
90
Isotropic Rock Anisotropic Rock
r
40.0
Intact Rock Radial Displacements U
s
45.0
r
s
Liner Radial Displacements U E /p r
i i
50.0
15.0
75
(b)
(a)
20.0
60
60
75
90
Anisotropic Rock & G'
1.5
white symbols: ABAQUS black symbols: analytical solution
1.0
0.5
0.0 0
15
30
45
(d)
(c) Fig. 3. Comparison between ABAQUS and analytical solution. Case 1.
reason for this is the low stiffness of the steel liner, which then can deform with small or negligible moments. This notion is supported by the normalized plots of the radial stress between the liner and the backfill concrete in Fig. 5. The figure shows a fairly constant pressure distribution on the liner. As a result, the liner experiences a uniform tangential stress (i.e. small bending moments). The observation is in agreement with the concept of flexible liner by Peck (1969), who suggested that a flexible liner would have a uniform pressure distribution and a deflected shape such that the bending moments at all points in the liner would be negligible; which is the case at hand due to the small thickness of the steel liner. Indeed, in the calculations performed, the values of the constants pn in Eq. (10) are several orders of magnitude smaller than the value of po, which supports the notion that the radial stresses are dominated by the constant term po. Given that the concrete and damaged rock, as hypothesized, cannot tansmit shear or tangential stresses, the radial stresses at the contact between the damaged and the intact rock are also uniform. Because of the anisotropy of the intact rock, the radial deformations are not uniform even when the radial stress applied is uniform. Thus, larger radial displacements occur at the crown than at the springline given that in all cases considered the Young’s modulus of the intact rock is larger in the horizontal direction than in the vertical direction.
The observation that the radial stress at the liner-concrete contact can be approximated as a uniform pressure can be used to simplify greatly the equations derived in the previous section, using for the radial stress at the liner-concrete interface:
pc ¼ po
ð38Þ
Following the same procedure as that used before, the following results are obtained: For the steel liner:
T s ¼ ðpo þ pi Þr i Ms ¼ 0
1m2 1m2 rm U sr ¼ U crm Ec c ri po ln rcrm Ecrmcrm r i po ln rrcrm þw r r r¼rm
ð39Þ
i
where U crm is given by (42). r r¼rm For the backfill concrete:
rcr ¼ rri po U cr ¼ U sr þ
1m2c r i po Ec
ð40Þ
ln rr
i
For the damaged rock, assuming isotropic properties:
rcrm ¼ rri po r U crm ¼ U sr þ r
1m2c ri po Ec
ln rcrm þ r i
1m2crm r i po Ecrm
r ln rcrm
ð41Þ
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
1.0
/pi rm
45.0 40.0
Isotropic Rock
35.0
Anisotropic Rock Anisotropic Rock & G'
0.8
Intact Rock Tangential Stresses
Liner Tangential Stresses
s
int
/pi
50.0
white symbols: ABAQUS black symbols: analytical solution
30.0 25.0 20.0 15.0 10.0 5.0 0.0
0.6
0.4 Isotropic Rock Anisotropic Rock
0.2
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
0.0 0
15
30
45
60
75
0
90
15
30
(a)
45
60
75
90
(b)
50.0
E /p r
rm
rm
Isotropic Rock Anisotropic Rock
35.0
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
30.0 25.0 20.0 15.0 10.0 5.0 0.0 0
15
30
45
60
r
40.0
Intact Rock Radial Displacements U
s
45.0
r
s
Liner Radial Displacements U E /p r
i i
i i
2.0
75
90
1.5
1.0 Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
0.5
white symbols: ABAQUS black symbols: analytical solution
0.0 0
15
30
(c)
45
60
75
90
(d)
Fig. 4. Comparison between ABAQUS and analytical solution. Case 2.
and for the intact rock, at the interface with the damaged rock:
rm r r¼rm U rm r r¼rm
r
¼
po rm ¼ 12 lpolri ðl1 þ l2 l1 l2 Þl1 l2 arm 1 þ 2l 1 l 2 a 2 1 2 þðl1 þ l2 1Þarm 3
rm rm A3 ¼ ðl1 þ l2 Þarm 1 þ ða2 l1 l2 a1 Þ þ
rm þ½ðl1 þ l2 l1 l2 Þl1 l2 arm 1 ðl1 þ l2 1Þa3 cos 2h ð42Þ
And the tangential stresses, after Lekhnitskii (1963):
nkþnðk1Þ cos hþ½ðkþ1Þ n sin h cos rrm h r¼rm ¼ sin4 hþ½n2 2k sin2 h cos2 hþk2 cos4 h
k¼
2
qffiffiffiffiffi ffi rm Ex Erm y
Erm x
>
2
2
2
2
h ri rrm
po
Erm y
Erm x Grm xy
m¼ 2mxy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 2k þ m
ð43Þ
The value of po is given by the following expression:
1 m2s 1 m2c r crm 1 m2crm r rm 1 po ri þ ln þ ln þ A3 Es t s Ec ri Ecrm r crm 2 ¼
with
ri r rm
w 1 m2s p ri ri Es t s i
ð44Þ
l1 þ l2 rm arm l1 l2 arm 2 a3 3 l1 l2 l1 l2 ð45Þ
The differences between the results obtained from the simplified solution and the complete solution, for the cases and scenarios investigated, is smaller than 0.3%. Eq. (44) provides a good estimate of the pressures behind the liner. In practice, the bending moments, even if they are small, should be evaluated, e.g. using the general formulation or numerical methods, due to the potential risk of failure in the case of high strength steel and flaws in weldings. One of the advantages of an analytical solution is that it provides insight into the variables that are most relevant. Inspection of Eq. (44) shows that the contact pressure between the liner and the concrete, neglecting the gap, is proportional to the applied pressure (expected for an elastic analysis), is inversely proportional to the weighed equivalent relative stiffness, Eeq, of the different materials, as defined in Eq. (46), and it strongly depends on the ratio ts/ri. Note that these observations are exactly the same as
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
conditions, circular cross section, tunnel support composed of a thin shell, concrete backfill, and an annulus of damaged rock around the concrete. It is assumed that all the materials remain elastic and that the damaged rock and intact rock are transversely anisotropic, with one of the axes of elastic symmetry parallel to the tunnel axis. It is also assumed that the concrete and damaged rock cannot develop shear or tangential stresses. The paper is inspired by the work by Pachoud and Schleiss (2015), who conducted an extensive numerical parametric analysis for steel-lined pressure tunnels in transversely anisotropic rock to obtain correction factors that could be used with the analytical solution for isotropic rock. Only correction factors for the maximum stress in the liner and intact rock were provided. They showed that the maximum stresses obtained after the corrections compared well with those obtained with the numerical method, except for the intact rock and for those cases when the shear modulus of the intact rock G was larger than the empirical relation of Saint–Venant G0 . The solutions obtained in the present work do not have these limitations. They provide the complete stress and displacement fields for all the materials and they do not require that the damaged rock is isotropic. The analytical solution presented has been verified by providing comparisons between its results and the results obtained from the Finite Element program ABAQUS, for a number of scenarios. The differences obtained are smaller than 1–2%, which provides confidence in the formulation. The equations provided may be of limited practical use because of the strong assumptions made for their derivation. There are clear advantages however of closed-form solutions over numerical methods. One of the advantages is that they can be used to readily identify the most important factors that determine the results. The paper shows that there are two parameters that strongly influence the results. The first one is the relative stiffness between the liner and the different materials, weighted with factors related to the geometry of the problem, and on the relative thickness of the steel liner with respect to the internal radius of the tunnel. Another important conclusion, given the results obtained from the cases analyzed, is that the tensile stress of flexible steel liners can be approximated assuming a uniform pressure. A simplified formulation is derived and presented that incorporates this observation. Other advantages over numerical methods are that they can be used for preliminary design, for sensitivity analysis, or to evaluate the results of complex numerical computations. This is supported in the technical literature by the widespread use of existing formulations for the analysis of steel-lined pressure tunnels.
-1.0
Isotropic Rock Anisotropic Rock
-0.8
white symbols: ABAQUS black symbols: analytical solution
c
Liner Radial Stresses p /p
i
Anisotropic Rock & G'
-0.6
-0.4
-0.2
0.0 0
15
30
45
60
75
90
(a) Case 1 -1.0
c
Liner Radial Stresses p /p
i
-0.8
-0.6 Isotropic Rock Anisotropic Rock
-0.4
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
-0.2
0.0 0
15
30
45
60
75
90
(b) Case 2 Fig. 5. Radial stress, pc, at liner-concrete contact.
Acknowledgments
those obtained for the isotropic scenario (see Eq. (6)), and that the structure of Eqs. (6) and (46) is the same.
pc ¼
1m2s Eeq
pi
Eeq ¼ ð1 m2s Þ þ ð1 m2c Þ EEcs
ts ri
s ln rcrm þ ð1 m2crm Þ EEcrm r i
ts ri
This work was sponsored by the National Science Foundation, Geomechanics and Geotechnical Systems Program, under Grant CMMI-1162082 to Purdue University. The authors are grateful to the sponsor for the support.
rm ln rrcrm þ 12 Es A3 trs i
ð46Þ When the intact rock is isotropic, A3 = 2(1 + mrm)/Erm, as already discussed, which makes Eq. (46) identical to Eq. (6), as it should. 5. Summary and conclusions An analytical solution is presented that provides close-form solutions for stresses and displacements for deep steel-lined pressure tunnels for a uniform internal pressure. The following assumptions are made in deriving the solution: plane strain
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Full stress and displacement fields for steel-lined deep pressure tunnels in transversely anisotropic rock Antonio Bobet a, Haitao Yu b,⇑ a b
School of Civil Engineering, Purdue University, West Lafayette, IN, USA Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai, China
a r t i c l e
i n f o
Article history: Received 9 September 2015 Received in revised form 17 February 2016 Accepted 9 March 2016 Available online 24 March 2016 Keywords: Pressure tunnel Circular tunnel Steel liner Anisotropy Transversely anisotropic rock Water pressure
a b s t r a c t An analytical solution is derived that provides closed-form formulations for stresses and displacements for a deep pressure tunnel in a transversely anisotropic rock, with a steel liner, and subjected to a uniform internal pressure. For the derivation, it is assumed that the tunnel support includes a thin steel liner, concrete backfill and that there is an annulus of damaged rock around the concrete. It is also assumed that all materials remain elastic and that the concrete and the damaged rock cannot transmit shear or tangential stresses. The solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS. For thin steel liners, it can be assumed that the contact pressure between the different materials is uniform, and thus the bending moments in the liner are negligible. This is due to the low bending stiffness of the steel liner. The paper is inspired by and expands the work by Pachoud and Schleiss (2015) who conducted an extensive numerical parametric analysis to obtain correction factors that, when used with an analytical solution for isotropic materials, approximate the maximum principal stress in the liner and intact rock. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The purpose of pressure tunnels is to convey water under pressure with acceptable losses and without causing instabilities. Pressure tunnels are mostly built in hydroelectric power plant projects, but they are also used, albeit with smaller pressures, for water supply and wastewater conveyance. Pressure tunnels usually have a circular cross section because of its structural advantages due to the large internal pressures, and also for hydraulic reasons. The cross section area and the surface roughness (i.e. unlined tunnel or concrete or steel lining) depend on the head losses accepted over the length of the tunnel, provided that the opening is stable. The location of the ground water table, together with the hydraulic head in the tunnel, determines whether water will flow from the tunnel to the rock or vice versa (Merritt, 1999). If the hydraulic head inside the tunnel is larger than that from the water table, water will flow out from the tunnel, which requires that rock stresses around the opening are larger than the water pressures to prevent hydrofracturing or hydrojacking (Benson, 1989; Seeber, 1985a, b). If the tunnel is supported, the stiffness of the rock determines the contribution of the rock mass ⇑ Corresponding author at: Department of Geotechnical Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China. E-mail address: [email protected] (H. Yu). http://dx.doi.org/10.1016/j.tust.2016.03.005 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
to support the internal pressure, as the load transfer from the liner to the rock depends largely on the stiffness of the liner relative to the surrounding rock (e.g. Einstein and Schwartz, 1979; Bobet, 2001, 2003, 2011; Bobet and Nam, 2007). Finally, the permeability of the rock mass is required to estimate the leakage flow and thus the pore pressures behind the support. The layout of the pressure tunnel can be decided based on empirical methods and the information obtained during field exploration, if available. The empirical methods can provide initial estimates as to where a watertight liner is needed (Dann et al., 1964; Benson, 1989; Hartmaier et al., 1998; Bergh-Christensen, 1982; Broch, 1984a, b; Selmer-Olsen, 1985; Brekke and Ripley, 1989, 1993; Alvarez et al. 1999). Following this, calculations regarding flow rate, stresses in the liner and in the ground should be made to assess whether the leakage is acceptable and that the stress state in the liner and ground is adequate, and to provide an estimate of the factor of safety (see e.g. Schleiss, 1986, 1997; Hendron et al., 1989; Fernandez, 1994; Fernandez and Alvarez, 1994; Bobet and Nam, 2007). A steel liner is the industry standard for the sections of the tunnel where the minimum principal stress of the rock around the excavation is smaller than the internal tunnel pressure with a suitable factor of safety (Fernandez, 1994). Steel liners are also needed when the internal pressure is large and the surrounding rock has low modulus, so leakage control with reinforced concrete is not
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Nomenclature Notation As, Is cross-section and moment of inertia of the liner Young’s modulus and Poisson’s ratio of the backfill conEc, mc crete Ecrm, mcrm Young’s modulus and Poisson’s ratio of the damaged isotropic rock Erm, mrm Young’s modulus and Poisson’s ratio of the intact isotropic rock rm Erm Young’s modulus of intact rock in x- and y-axis (Ex = E x , Ey and Ey = E’ in Pachoud and Schleiss, 2015) crm Young’s modulus of damaged rock in x- and y-axis Ecrm x , Ey Es, ms Young’s modulus and Poisson’s ratio of the liner Grm shear modulus of the intact rock xy Gcrm shear modulus of the damaged rock xy radial stress at the steel-concrete contact pc pi internal water pressure r, h polar coordinates ri radius of the tunnel rcrm limit of cracked concrete liner rrm limit of damaged rock ts liner thickness axial force and moment of the liner Ts, Ms
possible. A concrete-encased steel cylinder is the most common type of liner (Eskilsson, 1999). There are two criteria that need to be satisfied for the design of the steel liner (Schleiss, 1988): (1) working stress and deformation of the steel liner; and (2) loadbearing capacity of the rock mass. The first criterion requires a design of the steel liner to adequately support the internal pressures during operation, the external hydraulic pressures during grouting and dewatering, the handling loads during transportation and erection, and that limits local deformations, e.g. crack bridging in the backfill concrete (Schleiss, 1988; Brekke and Ripley, 1993). The second criterion is needed to guarantee sufficient safety against rock failure, and thus ensure the load sharing between liner, backfill concrete and rock assumed for the first criterion. The design of a steel-lined pressure tunnel due to the internal pressure is generally based on an elastic analysis assuming that the support and rock mass are isotropic and elastic and that the concrete and an annulus of rock surrounding the concrete are damaged such that only radial pressures can be transmitted. Beyond the damaged rock, intact or undamaged rock is assumed (e.g. Schleiss, 1988; Moore, 1989; USACE, 1997; Hachem and Schleiss, 2009; ASCE 2012). Analytical solutions are available that account for these assumptions and are included in the next section for completeness. Pachoud and Schleiss (2015) provided a number of correction factors to the analytical solution to expand it to cases where the intact rock is transversely anisotropic. The factors were obtained after an extensive numerical analysis using a Finite Element Method and a genetic algorithm to minimize errors between the corrected analytical solution and the numerical results. The corrected factors were obtained to estimate the major principal stress in the liner and in the intact rock. This paper builds on the work from Pachoud and Schleiss (2015) and provides a full analytical solution for stresses and displacements for the steel liner, concrete, damaged rock and intact rock. It also expands the range of cases, as the solution accounts for the anisotropy of the damaged rock and does not have the limitation that the existing solution has of assuming that the shear modulus of the rock is equivalent to the empirical relation of Saint–Venant. Clearly, analytical formulations are limited due to the assumptions that need to be made to reach the solution. In most cases, the
Ux, Uy
displacements in Cartesian coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock Ur, Uh displacements in polar coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock w gap between steel liner and backfill concrete x, y Cartesian coordinates of axes of elastic symmetry zk complex number, zk = x + lky, k = 1, 2 ex, ey, cxy axial and shear strains in x- and y-axis er, eh, crh radial, tangential and shear stresses in polar coordinates fk complex number that depends on zk through conformal mapping /(zk), /0 (zk) stress function and its derivative l1, l2 roots of compatibility equation mxy, mxz, myz Poisson’s ratios in x–y axes rx, ry, sxy normal and shear stresses in Cartesian coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock rr, rh, srh radial, tangential, and shear stresses in polar coordinates; superscripts s, c, crm or rm denote steel, concrete, damaged rock or intact rock
design will require a numerical method that does not have the shortcomings of the analytical solutions, as it can consider the construction process, non-linear behavior, etc. Closed-form solutions however are invaluable to obtain a better understanding of the interplay that exists between loads, rock and support, to identify what are the most critical parameters for the problem, and to provide first estimates or even a preliminary design. An added advantage is that they can be used with very little cost to conduct sensitivity analysis and, most importantly, to provide benchmark values to check the results of the more complex numerical models.
2. Steel-lined pressure tunnel in isotropic rock Fig. 1 shows how the internal pressure is distributed between the steel, concrete and rock mass. As the internal pressure increases, the steel liner may initially take all the stress (this may occur if there is a gap between the steel and the concrete, e.g. due to differential thermal contraction between the steel and the concrete, creep of the rock, cycles of loading and unloading as the tunnel is pressurized and emptied, or the skin grouting of the steel-concrete interface is not done or is not effective). With further increase of internal pressure, as the steel liner expands, part of the loading is transferred to the concrete, to the rock damaged during excavation and to the undamaged/intact rock. It is generally assumed that the concrete and the damaged rock can only transmit radial deformations as the damage prevents transfer of tangential and shear stresses. This results in radial stresses that decrease with 1/r within the concrete and damaged rock zones, and 1/r2 in the undamaged rock zone. The magnitude of the radial stresses and corresponding radial displacements can be obtained by imposing equilibrium along the radial direction and compatibility of radial displacements at the liner-concrete, concrete-damaged rock, and damaged-undamaged rock boundaries (Moore, 1989; USACE, 1997; Hachem and Schleiss, 2009). Thus: For the liner:
T s ¼ ðpi þ pc Þr i r2
U sr ¼ ð1 m2s Þ Esits ðpi þ pc Þ
ð1Þ
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Fig. 1. Distribution of radial stresses in a concrete-encased steel lining.
For the cracked backfill concrete:
rcr ¼ pc ri r
ð2Þ
r2
U cr ¼ w þ ð1 m2s Þ Esits ðpi þ pc Þ þ ð1 m2c Þ Erci pc ln rr
i
For the damaged rock:
r
crm r
¼ rri pc r2
U crm ¼ w þ ð1 m2s Þ Esits ðpi þ pc Þ þ ð1 m2c Þ Erci pc ln rcrm r r þð1 m
i
ri 2 crm Þ Ecrm
ð3Þ
r
pc ln rcrm
For the intact rock: r i r rm rrm r ¼ r 2 pc rm rh ¼ rirr2rm pc
ð1þmrm Þ U rm r ¼ Erm
ð4Þ ri r rm r
pc
with
pc ¼ 1m2 s
Es t s
1m2s Es t s
ri þ
1m2c Ec
pi r i wri
ln rcrm þ ri
1m2crm Ecrm
mrm rm ln rrcrm þ 1þErm
ð5Þ
where Ur are the radial displacements (subscripts s, c, crm and rm denote steel, backfill concrete, damaged rock and intact rock, respectively); rr and rh are the radial and tangential stresses (subscripts s, c, crm and rm denote steel, backfill concrete, damaged rock and intact rock, respectively); w is the gap between the steel and the concrete; r is the radial distance from the center of the tunnel; ri, rcrm and rm are the internal radius of the tunnel, the radial extent of the concrete and of the damaged rock, respectively; Es, Ec, Ecrm and Erm are the Young’s modulus of the steel, concrete, damaged
rock and intact rock; ms, mc, mcrm, mrm are the Poisson’s ratios of steel, concrete, damaged rock and intact rock; ts is the thickness of the steel liner; pi is the water pressure and pc the radial stress at the liner-concrete interface. Note that because the problem has axial symmetry, shear stresses and tangential displacements are zero and, because of the assumption that a cracked material can only transmit radial stresses, the tangential stresses in the concrete and damaged rock are zero. Note also that the liner has been considered as a shell, i.e. the thickness is much smaller than the radius; in other words, ri ri + ts or ts/ri 1, which is typically the case for steel-lined pressure tunnels. For the above equations, it is assumed that all materials are elastic, and that tension is positive and compression is negative (the internal pressure pi is taken as positive); thus, displacements are positive in the positive direction of the axes of the coordinate system. Inspection of Eqs. (1-4) indicates that stresses and displacements strongly depend on the radial stress at the liner-concrete contact, pc (because of elasticity, all results are proportional, as they should, to the applied pressure pi). Eq. (5) provides the magnitude of pc, which, neglecting the gap parameter w, can be written as:
pc ¼
1m2s Eeq
pi
Eeq ¼ ð1 m2s Þ þ ð1 m2c Þ EEcs
ts ri
s ln rcrm þ ð1 m2crm Þ EEcrm r i
ts ri
rm ln rrcrm
ð6Þ
þð1 þ m
Es t s rm Þ Erm r i
The equation shows that the result is inversely proportional to a weighted average of the relative stiffness between the steel liner and the different materials. The solution strongly depends on the ratio of the thickness of the steel liner to the internal tunnel radius, ts/ri.
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The theoretical framework has been incorporated into recommendations for design of steel-lined pressure tunnels (e.g. USACE, 1997; ASCE 2012) and has been successfully used to estimate the load transfer between the steel liner and the backfill concrete and rock on a large number of pressure tunnels (Brekke and Ripley, 1989) and to provide estimates of the stress on the liner (e.g. Marulanda and Gutierrez, 1999). 3. Steel-lined pressure tunnel in transversely anisotropic rock The problem solved is defined in Fig. 2 that, as mentioned earlier, is identical to that considered by Pachoud and Schleiss (2015), with few exceptions that expand the scope of the investigation. It consists of a deep tunnel with circular cross section with internal radius ri, subjected to an internal water pressure pi, excavated in a transversely anisotropic rock. It is assumed that the steel-lined portion of the tunnel is very long, so plane-strain conditions apply. The tunnel support includes a circular steel liner with thickness ts and annular concrete backfill with thickness rcrm (ri + ts); with the assumption of ts/ri 1, the concrete thickness is rcrm ri. It is assumed that there is an annular volume of rock that is damaged during excavation, with constant thickness rrm – rcrm. As explained by Pachoud and Schleiss (2015), a circular shape of the damaged rock is a reasonable approximation of actual, more complex, conditions and may be sufficiently accurate in the case of grouted near-rock zone. All materials are assumed to remain elastic, with the following properties: Steel liner: Isotropic material with Young’s modulus Es and Poisson’s ratio ms; concrete backfill: Isotropic material with Ec and mc; damaged rock: Transversely crm crm crm crm (note that anisotropic with properties Ecrm x ; Ey ; Gxy ; mxz ; myx this is different from Pachoud and Schleiss, 2015 who considered
the damaged rock as isotropic); and intact rock: Transversely anirm rm rm rm sotropic with properties Erm x ; Ey ; Gxy ; mxz ; myx . The x- and y-axes in Fig. 2 denote the axes of elastic symmetry; the x axis, for convenience, is taken as horizontal. The concrete backfill and the damaged rock are assumed to be sufficiently cracked or damaged such that no tangential or shear stresses can be transmitted through them, which is a common assumption (e.g. Schleiss, 1988; Pachoud and Schleiss, 2015). The problem is solved by imposing the condition of equilibrium on each of the four different materials and by satisfying the boundary conditions. The latter is accomplished by just ensuring compatibility of radial stresses and radial displacements at the contact between materials. This is the only condition needed since the shear stress and the tangential stress in the concrete and damaged rock are zero, and thus at the interfaces the shear stress is also zero. For the liner, given that it is treated as a shell, equilibrium is given by the following equations (e.g. Flügge, 1966): s s ri dT dM ¼ r2i ss dh dh
ð7Þ
ri T s þ ddhM2 s ¼ r 2i rsr 2
where r and h are the polar coordinates, with origin at the center of the tunnel and h measured counterclockwise from the x-axis; Ts and Ms are the axial force and moment acting on the liner; and rsr and ss are the radial and shear stress at the contact with the concrete backfill. Deformations are expressed as (e.g. Flügge, 1966): d2 U sh dh2
dU sh dh
s
r þ dU ¼ dh
þ U sr þ r2IsA i
s
ð1m2s Þ 2 s ri Es As
s
4
U sr 4
d dh
2 s ð1m2 Þ þ 2 ddhU2r þ U sr ¼ Es Ass r 2i rsr
Fig. 2. Steel-lined pressure tunnel in transversely anistropic rock.
ð8Þ
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where U sr and U sh are the radial and tangential displacements, in polar coordinates, and As and Is are the area and moment of inertia, respectively, of the liner cross section (As = ts and Is = 1/12 t 3s since the liner is assumed to have a constant thickness). As the pressure inside the tunnel pi increases, the liner deforms outwards. If there is a gap, w, between the liner and the concrete backfill, a contact pressure pc between the liner and the concrete will not develop until the gap is closed. Thus, a distinction needs to be made between the total liner deformations and deformations at the liner-concrete interface, which are those needed to impose the boundary conditions. The following applies when the internal pressure is large enough to close the gap. In other words, when
pi >
Es t s w 1 m2s r 2i
ð9Þ
which is the pressure required to close the gap w. Eqs. (7) and (8) can be solved when the stresses at the steelconcrete contact are known. As discussed, the shear stress at the contact is zero because the concrete cannot transmit shear or tangential stresses. For the radial stresses, the following expression is attempted, given the symmetry of the problem (about the x and y axes):
pc ¼ po þ
1 X
ð10Þ
pn cos nh
n¼2;4;6
pc is the radial stress at the liner-concrete contact, and po and pn are constants that are obtained by imposing compatibility of radial displacements at each interface. Thus, 1 X
T s ¼ ðpo þ pi Þr i
n¼2;4;6
Ms ¼
1 X n¼2;4;6
U sr
¼ w þ
pn r 2i
n2 1
1m2s Es t s
pn r i n2 1
cos nh ð11Þ
cos nh
1m2s 4 r Es Is i
ðpi þ
1 X n¼2;4;6
po Þr2i
þ
1m2s 4 r Es Is i
1 X
pn n2 1
pn n2 1
cos nh ð12Þ
sin nh
Note that the radial displacements in (12) are produced after the gap is closed. Thus, the total radial displacements of the liner are those given in (12) plus the gap w. For the concrete backfill, with the assumption that only radial stresses can occur, the equation of equilibrium in polar coordinates takes the form:
drr rr þ ¼0 dr r
ð13Þ
which, after integration and compatibility of radial stresses at the contact with the liner, results in the following expression for the radial stresses in the concrete, rcr :
rcr ¼
1 X
ri pn cos nh po þ r n¼2;4;6
!
ð14Þ
1 X
1m 1m pn ðpi þ po Þr 2i þ cos nh Es ts Es I s n2 1 n¼2;4;6 ! 1 X 1 m2c r ri po þ pn cos nh ln þ r Ec i n¼2;4;6
U cr ¼ w þ
! 1 X ri ¼ pn cos nh po þ r n¼2;4;6
ð16Þ
In plane strain, for a transversely anisotropic material, strains are expressed as (e.g. Detournay and Cheng, 1993; Cheng, 1998; Wang 2000):
ex ¼ a1 rx a2 ry ey ¼ a2 rx þ a3 ry c ¼ Gsxyxy rz ¼ mxz rx þ EEyx myx ry
ð17Þ
a1 ¼ 1Emx xz 2
a2 ¼ ð1þmExzy Þmyx
a3 ¼ 1 EEyx m2yx
1 Ey
where rx, ry are the stresses in the x and y directions, Ex and Ey are the Young’s modulus in the x and y directions, mxz and myx are the Poisson’s ratios in the xz and yx directions, respectively, and Gxy is the shear modulus. Note that because of the symmetry of the strain tensor, mxy = myx Ex/Ey. Note also that the properties in the z and x directions are the same. It is convenient to express the strains in polar coordinates and as a function of radial, tangential and shear stresses. After manipulation of Eq. (17), one gets:
er ¼ a11 rr þ a12 rh þ a13 srh eh ¼ a21 rr þ a22 rh þ a23 rrh crh ¼ a31 rr þ a32 rh þ a33 rrh h i a11 ¼ 18 3a1 2a2 þ 3a3 þ G1xy þ 4ða1 a3 Þ cos 2h þ a1 þ 2a2 þ a3 G1xy cos 4h h
i
a1 6a2 þ a3 G1xy a1 þ 2a2 þ a3 G1xy cos 4h
a13 ¼ a31 ¼ 14 ½2ða1 a3 Þ sin2h þ a1 þ 2a2 þ a3 G1xy sin 4h h i a22 ¼ 18 3a1 2a2 þ 3a3 þ G1xy 4ða1 a3 Þ cos 2h þ a1 þ 2a2 þ a3 G1xy cos 4h h i a23 ¼ a32 ¼ 14 2ða1 a3 Þsin 2h a1 þ 2a2 þ a3 G1xy sin4h h i a33 ¼ 12 a1 þ 2a2 þ a3 þ G1xy a1 þ 2a2 þ a3 G1xy cos 4h
ð18Þ Given the assumption of no shear and tangential stresses and after integration of the radial strains, one gets the radial displaceas: ments U crm r
U crm ¼ w þ r
1m2s Es t s
ðpi þ po Þr 2i þ
1 X
1m2s 4 r Es I s i
n¼2;4;6
þ
1m2c r i ðpo Ec
þ
1 X n¼2;4;6
pn n2 1
cos nh
pn cos nhÞ ln rcrm þ ri 1 X
ðb1 þ b2 cos 2h þ b3 cos 4hÞðpo þ
r pn cos nhÞr i ln rcrm
ð19Þ
n¼2;4;6
Integration of the radial strains associated with (14) gives the radial displacements U cr : 2 s
r
crm r
a12 ¼ a21 ¼ 18
n¼2;4;6
U sh ¼
For the damaged rock, equilibrium is given by (13). The radial stresses rcrm r , after imposing compatibility of radial stresses at the contact with the concrete are:
2 s 4 ri
ð15Þ
1 b1 ¼ 18 ð3acrm 2acrm þ 3acrm þ Gcrm Þ 1 2 3 xy
acrm b2 ¼ 12 ðacrm 3 Þ 1 1 crm 1 þ acrm Gcrm b3 ¼ 8 a1 þ 2acrm 2 3 xy
Note that Eq. (19) satisfies compatibility of radial displacements at the concrete-damaged rock interface. Note also that Eq. (19) can be used when the damaged rock is isotropic (which was the assumption by Pachoud and Schleiss, 2015) by making Ex = Ey = Ecrm, mxz = myx = mcrm and Gxy = Ecrm/2(1 + mcrm), which results in b1 =(1 m2crm )/Ecrm and b2 = b3 = 0.
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The final problem is that of a circular opening of radius rrm in a transversely anisotropic rock, with principal axes of elastic symmetry x and y (Fig. 2). The solution can be obtained following Lekhnitskii (1963). Equilibrium is satisfied by finding a stress function F(x,y) such that stresses are:
rx ¼ @@y2F 2
ry ¼ @@x2F @2 F sxy ¼ @x@y 2
ð20Þ
The compatibility equation is, in terms of the function F(x,y):
a1
4
4
ð21Þ
where a1, a2, a3 are obtained from Eq. (17) using the properties of the intact rock. The solution of Eq. (21) gives F(x,y). Lekhnitskii (1963) found a solution by introducing the complex variable zk = x + lky, where lk is a complex number. Expressing (21) as a function of the complex variable z, one gets:
a1 l4k þ
1 2a2 Gxy
l2k þ a3
@ F ¼0 @z4
ð22Þ
Defining /(zk) = F (zk) = oF/ozk, stresses can be obtained from:
l
l
2 0 2 0 x ¼ 2Re½ 1 /1 ðz1 Þ þ 2 /2 ðz2 Þ 0 0 y ¼ 2Re½/1 ðz1 Þ þ /2 ðz2 Þ ¼ 2Re½ 1 /01 ðz1 Þ þ 2 /02 ðz2 Þ
l
ð23Þ
l
where l1 and l2 are the roots of the equation:
a1 l4k þ
1 2a2 Gxy
l2k þ a3 ¼ 0
ð24Þ
The stress functions are, after Lekhnitskii (1963):
/2 ðz2 Þ ¼
1 X
l
1
1 ðbn 2 an Þ n 1 2 n¼1 1 X 1 l 1 l ðbn 1 an Þ n 2 2 1 2 n¼1
/1 ðz1 Þ ¼ l 1 l 1 2
l
ð25Þ
1
n are constants and f and f are complex variables, n and b where a 1 2 related to the variables zk = x + lky using the conformal mapping relation:
zk ¼
1 1 ð1 lk iÞr rm 1k þ ð1 þ lk iÞr rm 11 k 2 2
where i is the imaginary number i ¼ obtained from: 1 X R an enhi ¼ n¼1 1 X
bn enhi ¼
R
pffiffiffiffiffiffiffi 1. The constants are
rr sin hrrm dh
rr cos hrrm dh
ð27Þ
! 1 X ri po þ pn cos nh r rm n¼2;4;6
1 ¼ ðpo 12 p2 Þr i a 1 n ¼ 2n a ðpn1 pnþ1 Þri n ¼ 3; 5; 7; . . . n ¼ 0 n ¼ 2; 4; 6; . . . a 1 ¼ iðp þ 1 p Þr b 2
n¼1 1 X
1 ð1þil2 Þð1il2 Þ122
nðbn l1 an Þ1nþ1 2
ð28Þ
i
¼ i ðp b n ¼ 3; 5; 7; . . . n n1 þ pnþ1 Þr i 2n n ¼ 0 n ¼ 2; 4; 6; . . . b
ð30Þ
n¼1
Displacements are the result of the integration of the strains in (17) given the definition of stresses in (23). They are:
U x ¼ 2Re½ða1 l21 a2 Þ/1 ðz1 Þ þ ða1 l22 a2 Þ/2 ðz2 Þ
U y ¼ 2Re½ða2 l1 þ la3 Þ/1 ðz1 Þ þ ða2 l2 þ la3 Þ/2 ðz2 Þ
ð31Þ
2
It is informative to expand the equations for displacements using the stress functions given in (25) with (29). The result is: ( ) 1 X n ðarm þ l l arm Þa n ½cos nh i sin nh U x ¼ Re ½ðl þ l Þarm b 1
2
1
2
1
2
1
n¼1;3;5
(
U y ¼ Re l 1l 1 2
)
1 X
rm rm ½ðl1 þ l2 Þarm 3 an ða3 þ l1 l2 a2 Þbn ½cos nh i sin nh
n¼1;3;5
ð32Þ
Given the form or the characteristic Eq. (24), the roots must satisfy the following:
l21 þ l22 ¼ a11
1 Gxy
2a2
ð33Þ
l21 l22 ¼ aa31
which results in real numbers. The roots of the characteristic equation (24) are either complex numbers or pure imaginary numbers, but cannot be real numbers (Lekhnitskii, 1963). The roots l1 = a + ib and l2 = c + id, given (33), can only have one of the two forms: l1 = ib and l2 = id, or l1 = a + ib and l2 = a + ib. This is interesting because l1 + l2 = i(b + d) or l1 + l2 = 2ib, and l1l2 = bd or l1l2 = (a2 + b2). In other words, the sum of the roots is a pure imaginary number and the product a negative real number. Using this property, the radial displacements, at the interface with the damage rock, can be expressed as: 1 X
rm rm ðl1 þ l2 Þarm 1 bn ða2 þ l1 l2 a1 Þan
1 U rm r ¼ 2
n¼1;3;5
½cosðn þ 1Þh þ cosðn 1Þh 1 X
1 i rm rm ðl1 þ l2 Þarm þ2 l l 3 an ða3 þ l1 l2 a2 Þbn
ð34Þ
n¼1;3;5
½cosðn þ 1Þh cosðn 1Þh Finally, coefficients po, pn are obtained by making the radial displacements in (19) equal to the radial displacements in (34), at r = rrm. Doing this term by term, i.e. for the constant term, cos 2h, cos 4h, etc., results in the following system of linear equations:
po
1m2s r Es t s i
þ
1m2c Ec
rm ln rcrm þ b1 ln rrcrm þ 12 A3 þ ri i
1m2s Es t s
pi r i
rm po ðb2 ln rrcrm þ 12 A1 Þ 1m2 1m2 1 rm þ b1 ln rrcrm þ 14 A2 þ 12 A3 þp2 13 Es Iss r 3i þ Ec c ln rcrm r i
rm rm þ 16 A4 Þ þ p6 12 b3 ln rrcrm ¼0 þp4 12 ðb2 ln rrcrm rm rm po b3 ln rrcrm þ p2 12 ðb2 ln rrcrm þ 16 A1 Þ 2 2 1m 1 1ms 3 1 1 rm þp4 15 r þ Ec c ln rcrm þ b1 ln rrcrm þ 12 A2 þ 20 A3 Es I s i ri
The solution of (27) is:
2
1 X nðbn l2 an Þ1nþ1 1
rm rm þ 12 A4 Þ þ p4 12 b3 ln rrcrm ¼ wr p2 12 ðb2 ln rrcrm
where rr is the radial stress at r = rrm; that is, from (16):
o
¼ ðl l Þrrm 1 2 1
1 2
ð26Þ
n¼1
rr ¼
/02 ðz2 Þ
1 ð1þil1 Þð1il1 Þ121
4
0
r r s
/01 ðz1 Þ ¼ ðl l1 Þrrm 1 2
1
4
@ F @ F 1 @ F þ a3 4 þ 2a2 ¼0 @y4 @x Gxy @x2 @y2
Stresses are obtained from the derivatives of (25), which are:
1 rm rm þp6 12 ðb2 ln rrcrm þ 10 A4 Þ þ p8 12 b3 ln rrcrm ¼0
ð29Þ
1 rm rm pn4 12 b3 ln rrcrm þ pn2 12 ðb2 ln rrcrm þ 2ðn1Þ A1 Þ 2 2 1mc A2 A3 r crm r rm 1 1ms 3 þpn n2 1 Es Is r i þ Ec ln r þ b1 ln rcrm þ 4ðn1Þ þ 4ðnþ1Þ i
1 rm rm þ 2ðnþ1Þ A4 Þ þ pnþ4 12 b3 ln rrcrm ¼ 0 for n > 4 þpnþ2 12 ðb2 ln rrcrm
ð35Þ
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135 Table 1 Comparison between analytical solution and ABAQUS. Input parameters. Case 1 Isotropic rock 1 Anisotropic rock 1 Anisotropic rock & G0 2 Isotropic rock 2 Anisotropic rock 2 Anisotropic rock & G0
pi (MPa)
ri (m)
Es (GPa)
ms
ts (m)
Ec (GPa)
mc
10 10 10
1.5 1.5 1.5
210 210 210
0.3 0.3 0.3
0.03 0.03 0.03
20 20 20
0 0 0
2.0 2.0 2.0
2 2 2
0 0 0
15 15 15
2.5 2.5 2.5
210 210 210
0.3 0.3 0.3
0.02 0.02 0.02
20 20 20
0 0 0
3.0 3.0 3.0
20 20 20
0 0 0
with
A1 ¼ ðl1 þ l2 Þa1 þ ða2 A2 ¼ ðl1 þ l2 Þa1 ða2 A3 ¼ ðl1 þ l2 Þa1 þ ða2 A4 ¼ ðl1 þ l2 Þa1 ða2
l1 l2 a1 Þ ll1 þll2 a3 1 2 l1 l2 a1 Þ þ ll1 þll2 a3 1 2 l1 l2 a1 Þ þ ll1 þll2 a3 1 2 l1 l2 a1 Þ ll1 þll2 a3 1 2
þ a3 ll1ll2 a2 1 2 þ a3 ll1ll2 a2 1 2 a3 ll1ll2 a2 1 2 a3 ll1ll2 a2 1 2
ð36Þ
Note that in (36), given the properties of the roots of the characteristic equation previously discussed, i.e. l1 + l2 = b + d or 2b, and l1l2 = bd or (a2 + b2), the expressions l1 + l2 and l1l2 are real numbers. From a practical point of view, only a finite number of equations is needed, given that their contribution to the solution diminishes as n increases. It is important however to truncate the system of equations at a number that minimizes errors. Further comments are provided later in the next section. The solution obtained is verified by comparing its results from those obtained using the numerical program ABAQUS (ABAQUS, 2015). ABAQUS is a Finite Element code that has been extensively used and validated in geotechnical engineering. All the different cases are discretized with eight-node isoparametric elements, with only one quarter of the tunnel modeled due to the symmetry along the vertical and horizontal axes. A tied contact between the different materials is used, which effectively acts as frictionless since the shear stresses in the concrete and damaged rock are zero. No horizontal displacements are allowed along the vertical axis through the center of the tunnel and no vertical movements along the horizontal axis. A constant pressure equal to pi is imposed to the internal perimeter of the tunnel. The external boundary is placed far from the center of the tunnel such that it does not have an effect on the solution. The two same cases used by Pachoud and Schleiss (2015) are chosen for the comparisons, which is done to facilitate a discussion on the advantages and limitations of the two approaches. The first case corresponds to a circular tunnel with internal radius ri = 1.5 m, internal pressure pi = 10 MPa, steel liner with a thickness of 0.03 m, concrete backfill 0.5 m thick, and damaged rock with a thickness of 0.5 m. The second case is analogous, with ri = 2.5 m, pi = 15 MPa, and thickness of steel, concrete and damage rock of 0.02 m, 0.5 m, and 0.7 m, respectively. Three scenarios are considered in each case: one, where all materials are isotropic, and the other two with transversely anisotropic intact rock. All scenarios assume that the damaged rock is isotropic and that there is no gap between the liner and the backfill concrete. The first scenario in each case considers, as mentioned, isotropic intact rock; case one, scenario two, includes properties of a transversely anisotropic rock taken from Tonon and Amadei (2003), and case two, scenario two, from Amadei (1996). Scenarios three from cases one and two are analogous to those with the anisotropic rock, except the shear modulus is estimated based on the following Saint–Venant empirical relation:
G0rm xy ¼
Erm y 1þ
Erm y Erm x
þ 2myx
ð37Þ
rcrm (m)
Ecrm (GPa)
mcrm
Erm y
mxz
myx
Grm xy
2.5 2.5 2.5
Erm x (GPa) 2.4 7.8 7.8
(GPa) 2.4 2.4 2.4
0.22 0.22 0.22
0.22 0.07 0.07
(GPa) 983 830 1658
3.7 3.7 3.7
23.9 29.3 29.3
23.9 23.9 23.9
0.18 0.18 0.18
0.18 0.13 0.13
10,127 6200 11,514
rrm (m)
Table 1 lists the input properties used for both the analytical solution and for ABAQUS. In the numerical method, the condition of no shear and tangential stresses in the concrete and damaged rock is approximated by assuming that these materials have cylindrical anisotropy, with the radial Young’s modulus equal to the value shown in Table 1, tangential Young’s modulus equal to 1012 MPa, the smallest number that the code accepts, and the Poisson’s ratio in the radial and tangential directions equal to zero (this is necessary to have in the numerical model a positive definite stiffness matrix, given the small tangential Young’s modulus used as input). Figs. 3 and 4 provide a comparison between the results obtained from ABAQUS and from the analytical solution for cases 1 and 2, respectively. More specifically, Figs. 3(a) and 4(a) are plots of the normalized tangential stress at the interior fiber of the liner; Figs. 3(b) and 4(b), of the normalized tangential stresses in the intact rock at its contact with the damaged rock; Figs. 3 (c) and 4(c), of the normalized radial displacements of the liner and Figs. 3(d) and 4(d) of the normalized radial displacements of the intact rock at the contact with the damaged rock. In the figures, white symbols represent results from ABAQUS and black symbols from the analytical solutions. The comparison is excellent, with differences smaller than 1–2% for all the values plotted. Pachoud and Schleiss (2015) only reported results for the maximum principal stress for the liner and the intact rock. For case 1, for the liner, they reported the values 24.87, 24.27, 22.27 for scenarios 1, 2, and 3, respectively; for the intact rock, 0.1, 0.156, and 0.133. For case 2, 5.44, 6.19, 4.99 for the liner and 0.237, 0.265 and 0.239 for the intact rock. The values given are normalized as in Figs. 3 and 4. The differences between their results and the results obtained with the new simulations are all below 1%4%, except for the maximum principal stress for the intact rock for the scenarios with anisotropic rock and G0 from Eq. (37), where the errors are of the order of 20–25%. These larger errors could be due to small differences in the models, and/or how the condition of no-shear and no-tangential stresses in the concrete and damaged rock are imposed. There is no limitation in using Eq. (35) for isotropic problems. When both the damaged rock and the intact rock are isotropic, b1 = (1 – m2crm )/Ecrm and b2 = b3 = 0 for the damaged rock; and rm 2 l1 = l2 = 1, arm arm 3 ¼ a1 ¼ ð1 mrm Þ/Erm, 2 ¼ ð1 þ mrm Þ mrm =Erm , A1 = A4 = 0, A2 = 2(1 + mrm)(3–5mrm)/Erm and A3 = 2(1 + mrm)/Erm for the intact rock. With these values, the system of Eq. (35) has a solution pn = 0 for n > 0 and pc = po, which has the same result as that given in (6).
4. Discussion Figs. 3 and 4 show that, when the rock is isotropic, stresses and displacements do not depend on the angular coordinate h. This is expected because the problem is axisymmetric. What is interesting is the observation that the liner tangential stresses are insensitive to the tangential coordinate, even when the rock is anisotropic. The
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
50.0
/pi
1.0 rm
40.0
Liner Tangential Stresses
35.0 30.0 25.0
Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
20.0
white symbols: ABAQUS black symbols: analytical solution
15.0 10.0 5.0
Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
0.8
Intact Rock Tangential Stresses
s
int
/pi
45.0
white symbols: ABAQUS black symbols: analytical solution
0.6
0.4
0.2
0.0
0.0 0
15
30
45
60
75
0
90
15
30
45
i i
rm
30.0 25.0 Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
5.0 0.0 0
15
30
45
60
75
90
2.0
E /p r
rm
35.0
10.0
90
Isotropic Rock Anisotropic Rock
r
40.0
Intact Rock Radial Displacements U
s
45.0
r
s
Liner Radial Displacements U E /p r
i i
50.0
15.0
75
(b)
(a)
20.0
60
60
75
90
Anisotropic Rock & G'
1.5
white symbols: ABAQUS black symbols: analytical solution
1.0
0.5
0.0 0
15
30
45
(d)
(c) Fig. 3. Comparison between ABAQUS and analytical solution. Case 1.
reason for this is the low stiffness of the steel liner, which then can deform with small or negligible moments. This notion is supported by the normalized plots of the radial stress between the liner and the backfill concrete in Fig. 5. The figure shows a fairly constant pressure distribution on the liner. As a result, the liner experiences a uniform tangential stress (i.e. small bending moments). The observation is in agreement with the concept of flexible liner by Peck (1969), who suggested that a flexible liner would have a uniform pressure distribution and a deflected shape such that the bending moments at all points in the liner would be negligible; which is the case at hand due to the small thickness of the steel liner. Indeed, in the calculations performed, the values of the constants pn in Eq. (10) are several orders of magnitude smaller than the value of po, which supports the notion that the radial stresses are dominated by the constant term po. Given that the concrete and damaged rock, as hypothesized, cannot tansmit shear or tangential stresses, the radial stresses at the contact between the damaged and the intact rock are also uniform. Because of the anisotropy of the intact rock, the radial deformations are not uniform even when the radial stress applied is uniform. Thus, larger radial displacements occur at the crown than at the springline given that in all cases considered the Young’s modulus of the intact rock is larger in the horizontal direction than in the vertical direction.
The observation that the radial stress at the liner-concrete contact can be approximated as a uniform pressure can be used to simplify greatly the equations derived in the previous section, using for the radial stress at the liner-concrete interface:
pc ¼ po
ð38Þ
Following the same procedure as that used before, the following results are obtained: For the steel liner:
T s ¼ ðpo þ pi Þr i Ms ¼ 0
1m2 1m2 rm U sr ¼ U crm Ec c ri po ln rcrm Ecrmcrm r i po ln rrcrm þw r r r¼rm
ð39Þ
i
where U crm is given by (42). r r¼rm For the backfill concrete:
rcr ¼ rri po U cr ¼ U sr þ
1m2c r i po Ec
ð40Þ
ln rr
i
For the damaged rock, assuming isotropic properties:
rcrm ¼ rri po r U crm ¼ U sr þ r
1m2c ri po Ec
ln rcrm þ r i
1m2crm r i po Ecrm
r ln rcrm
ð41Þ
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
1.0
/pi rm
45.0 40.0
Isotropic Rock
35.0
Anisotropic Rock Anisotropic Rock & G'
0.8
Intact Rock Tangential Stresses
Liner Tangential Stresses
s
int
/pi
50.0
white symbols: ABAQUS black symbols: analytical solution
30.0 25.0 20.0 15.0 10.0 5.0 0.0
0.6
0.4 Isotropic Rock Anisotropic Rock
0.2
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
0.0 0
15
30
45
60
75
0
90
15
30
(a)
45
60
75
90
(b)
50.0
E /p r
rm
rm
Isotropic Rock Anisotropic Rock
35.0
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
30.0 25.0 20.0 15.0 10.0 5.0 0.0 0
15
30
45
60
r
40.0
Intact Rock Radial Displacements U
s
45.0
r
s
Liner Radial Displacements U E /p r
i i
i i
2.0
75
90
1.5
1.0 Isotropic Rock Anisotropic Rock Anisotropic Rock & G'
0.5
white symbols: ABAQUS black symbols: analytical solution
0.0 0
15
30
(c)
45
60
75
90
(d)
Fig. 4. Comparison between ABAQUS and analytical solution. Case 2.
and for the intact rock, at the interface with the damaged rock:
rm r r¼rm U rm r r¼rm
r
¼
po rm ¼ 12 lpolri ðl1 þ l2 l1 l2 Þl1 l2 arm 1 þ 2l 1 l 2 a 2 1 2 þðl1 þ l2 1Þarm 3
rm rm A3 ¼ ðl1 þ l2 Þarm 1 þ ða2 l1 l2 a1 Þ þ
rm þ½ðl1 þ l2 l1 l2 Þl1 l2 arm 1 ðl1 þ l2 1Þa3 cos 2h ð42Þ
And the tangential stresses, after Lekhnitskii (1963):
nkþnðk1Þ cos hþ½ðkþ1Þ n sin h cos rrm h r¼rm ¼ sin4 hþ½n2 2k sin2 h cos2 hþk2 cos4 h
k¼
2
qffiffiffiffiffi ffi rm Ex Erm y
Erm x
>
2
2
2
2
h ri rrm
po
Erm y
Erm x Grm xy
m¼ 2mxy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 2k þ m
ð43Þ
The value of po is given by the following expression:
1 m2s 1 m2c r crm 1 m2crm r rm 1 po ri þ ln þ ln þ A3 Es t s Ec ri Ecrm r crm 2 ¼
with
ri r rm
w 1 m2s p ri ri Es t s i
ð44Þ
l1 þ l2 rm arm l1 l2 arm 2 a3 3 l1 l2 l1 l2 ð45Þ
The differences between the results obtained from the simplified solution and the complete solution, for the cases and scenarios investigated, is smaller than 0.3%. Eq. (44) provides a good estimate of the pressures behind the liner. In practice, the bending moments, even if they are small, should be evaluated, e.g. using the general formulation or numerical methods, due to the potential risk of failure in the case of high strength steel and flaws in weldings. One of the advantages of an analytical solution is that it provides insight into the variables that are most relevant. Inspection of Eq. (44) shows that the contact pressure between the liner and the concrete, neglecting the gap, is proportional to the applied pressure (expected for an elastic analysis), is inversely proportional to the weighed equivalent relative stiffness, Eeq, of the different materials, as defined in Eq. (46), and it strongly depends on the ratio ts/ri. Note that these observations are exactly the same as
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A. Bobet, H. Yu / Tunnelling and Underground Space Technology 56 (2016) 125–135
conditions, circular cross section, tunnel support composed of a thin shell, concrete backfill, and an annulus of damaged rock around the concrete. It is assumed that all the materials remain elastic and that the damaged rock and intact rock are transversely anisotropic, with one of the axes of elastic symmetry parallel to the tunnel axis. It is also assumed that the concrete and damaged rock cannot develop shear or tangential stresses. The paper is inspired by the work by Pachoud and Schleiss (2015), who conducted an extensive numerical parametric analysis for steel-lined pressure tunnels in transversely anisotropic rock to obtain correction factors that could be used with the analytical solution for isotropic rock. Only correction factors for the maximum stress in the liner and intact rock were provided. They showed that the maximum stresses obtained after the corrections compared well with those obtained with the numerical method, except for the intact rock and for those cases when the shear modulus of the intact rock G was larger than the empirical relation of Saint–Venant G0 . The solutions obtained in the present work do not have these limitations. They provide the complete stress and displacement fields for all the materials and they do not require that the damaged rock is isotropic. The analytical solution presented has been verified by providing comparisons between its results and the results obtained from the Finite Element program ABAQUS, for a number of scenarios. The differences obtained are smaller than 1–2%, which provides confidence in the formulation. The equations provided may be of limited practical use because of the strong assumptions made for their derivation. There are clear advantages however of closed-form solutions over numerical methods. One of the advantages is that they can be used to readily identify the most important factors that determine the results. The paper shows that there are two parameters that strongly influence the results. The first one is the relative stiffness between the liner and the different materials, weighted with factors related to the geometry of the problem, and on the relative thickness of the steel liner with respect to the internal radius of the tunnel. Another important conclusion, given the results obtained from the cases analyzed, is that the tensile stress of flexible steel liners can be approximated assuming a uniform pressure. A simplified formulation is derived and presented that incorporates this observation. Other advantages over numerical methods are that they can be used for preliminary design, for sensitivity analysis, or to evaluate the results of complex numerical computations. This is supported in the technical literature by the widespread use of existing formulations for the analysis of steel-lined pressure tunnels.
-1.0
Isotropic Rock Anisotropic Rock
-0.8
white symbols: ABAQUS black symbols: analytical solution
c
Liner Radial Stresses p /p
i
Anisotropic Rock & G'
-0.6
-0.4
-0.2
0.0 0
15
30
45
60
75
90
(a) Case 1 -1.0
c
Liner Radial Stresses p /p
i
-0.8
-0.6 Isotropic Rock Anisotropic Rock
-0.4
Anisotropic Rock & G'
white symbols: ABAQUS black symbols: analytical solution
-0.2
0.0 0
15
30
45
60
75
90
(b) Case 2 Fig. 5. Radial stress, pc, at liner-concrete contact.
Acknowledgments
those obtained for the isotropic scenario (see Eq. (6)), and that the structure of Eqs. (6) and (46) is the same.
pc ¼
1m2s Eeq
pi
Eeq ¼ ð1 m2s Þ þ ð1 m2c Þ EEcs
ts ri
s ln rcrm þ ð1 m2crm Þ EEcrm r i
ts ri
This work was sponsored by the National Science Foundation, Geomechanics and Geotechnical Systems Program, under Grant CMMI-1162082 to Purdue University. The authors are grateful to the sponsor for the support.
rm ln rrcrm þ 12 Es A3 trs i
ð46Þ When the intact rock is isotropic, A3 = 2(1 + mrm)/Erm, as already discussed, which makes Eq. (46) identical to Eq. (6), as it should. 5. Summary and conclusions An analytical solution is presented that provides close-form solutions for stresses and displacements for deep steel-lined pressure tunnels for a uniform internal pressure. The following assumptions are made in deriving the solution: plane strain
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