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A model to assess the response of an arched roof of a lined tunnel
Tunnelling and Underground Space Technology 56 (2016) 211–225
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
A model to assess the response of an arched roof of a lined tunnel A.N. Dancygier ⇑, Y.S. Karinski, A. Chacha Department of Civil Engineering, National Building Research Institute, Technion – Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 2 September 2015 Received in revised form 11 January 2016 Accepted 23 March 2016
Keywords: Arched roof Arching coefficient Lined tunnel Soil-structure interaction
a b s t r a c t A model to analyze the response of an arched roof of a tunnel lining under a surface static loading is presented. It enhances a previous model by the authors, which is based on a discrete-continuous concept and is suitable for depths of burial at which ‘arching’ can develop. The current enhanced model takes into consideration the curvature of an arched roof of a lined tunnel. The proposed 2DOF system’s stiffness includes the influences of the soil side pressure as well as the arched geometry of the roof. For the case of zero curvature the analytical solution for the mid-roof deflection and average contact pressure that has been derived converges to the solution of a flat roof. The case of a relatively shallow buried structure has been calibrated and then verified against published experimental results. A case study shows that there is a certain opening angle of the roof at which the contact pressure has a maximum value. This angle coincides with the angle at which there is also a maximum value of the roof stiffness. However, it is also shown that approximately at this opening angle, the internal forces are minimal. It is therefore concluded that the average contact pressure is not necessarily the most important criterion for a design of an optimal shape of the roof. Furthermore, the angle that yields maximum contact pressure should be preferred for an optimal roof design. It is further shown that as the roof slenderness increases this optimal angle decreases. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction An important parameter in the design of lined tunnels is the load acting on its roof. A common loading case is the self-weight of the covering or backfill soil above the structure and an external surface pressure. For deeply buried structures the weight of the soil upper layers may be regarded as a uniformly distributed load acting on the layers below them. The covering soil adjacent to the lining roof interacts with the structure in a way known for static loads as ‘arching’. This phenomenon refers to a mechanism related to relative displacements in the soil media above the structure and far from it (e.g., Terzaghi, 1959; Newmark, 1964). Due to this phenomenon, the average contact pressure may be lower or higher than the undisturbed ‘free-field’ pressure. The first case is known as ‘positive’ or ‘active’ arching and the latter case is known as ‘negative’ or ‘passive’ arching. A common way to evaluate the response of a buried structure is by numerical analysis. Finite element simulation of a specific problem is most frequently used (e.g., Brachman et al., 2000; Papanikolaou and Kappos, 2014; Mai et al., 2014,). These methods require proper representation of the soil and structure properties, ⇑ Corresponding author. E-mail address: [email protected] (A.N. Dancygier). http://dx.doi.org/10.1016/j.tust.2016.03.009 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
as well as of the contact layer between them. They provide detailed analysis, but only for the specific problem that was analyzed. Thus, at the earlier stages of the design, when various alternatives need to be examined, such as structure geometry or element thicknesses, a theoretical model is preferred. Relatively simplistic models that are considered ‘‘classical” are known in the literature (e.g., Terzaghi, 1959; Newmark, 1964; Marston, 1930; Spangler, 1957). Simplistic models for structures with a rectangular crosssection have also been proposed (Weidlinger and Hinman, 1988; Higgins and Drake, 1995). One of the frequently used shapes of tunnels is the horseshoe cross-section (Szechy, 1973). A model based on a discretecontinuous concept for the prediction of the average dynamic and static pressure acting on a flat roof tunnel has been developed by the authors (Dancygier and Karinski, 1999a,b; Karinski et al., 2003). Unlike the models mentioned above, which refer only to buried structures with a flat roof, this paper proposes a model to assess the response of a lined tunnel with a non-flat, circular-arched roof under a surface static loading. It allows calculating the average contact pressure that acts on structures, such as horseshoeroofed tunnel’s lining. Although derivation of the model is relatively complicated, the resulted solution can be straightforwardly implemented. The paper starts with a description of the model,
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Nomenclature Aa b D Es Esoil E⁄ Effsoil ; Effb;soil Eff ; Effb h H I k(x) k0 ks K, K0 Kmsd KL L Mb, mb M, M0 N, n P Pext Pe0 Pv
cross sectional area of a unit width strip of the arch roof width of the structure depth of burial (from soil surface to the top of the arched roof) Young’s modulus of the structure roof Young’s modulus of the soil-column above the structure soil-column’s equivalent modulus of elasticity free field’s moduli of elasticity above and below the level of the structure floor free-field equivalent moduli of elasticity for a plane strain problem thickness of the roof cross-section total height of the structure’s cross-section moment of inertia of a unit width strip of the arch roof arching coefficient constant of the arching coefficient function side pressure coefficient roof and floor stiffnesses measured roof stiffness equivalent structure transformation factor span of the arched roof and flat floor bending moments caused by the contact pressure and by a vertical unit force acting at the top of the arch 2DOF masses and total masses of the structure roof and floor axial forces caused by the contact pressure and by a vertical unit force acting at the top of the arch roof perimeter external surface load (positive in compression) uniformly distributed load in the model (Pe0 = -Pext) total vertical contact force
followed by derivation of its equations and their solution. Finally, the solution is verified against experimental data and a case study is presented.
2. Description of the model
q qb ~b q R s S Ss U(x)
uniformly distributed load acting on the roof (positive for pressure) uniform pressure without taking into account free-field, ~b þ Eff U ffx ðD þ HÞ qb ¼ q average pressure that acts on the soil under the structure radius of the roof arch coordinate along the roof arch horizontal projection of the area of the roof, which is also equal to the floor area shear force vertical soil-column displacement above the structure
vertical free-field soil displacement Uff (x) Ub absolute mid-floor displacement Ux ðxÞ; Uffx ðxÞ derivatives with respect to ‘‘x” w, w0 mid-roof and mid-floor deflections relative to their supports (walls) W modulus of a unit width strip of the arch roof x depth coordinate y, y0 roof and floor vertical deflection surfaces, relative to their supports (walls) m; mb soil Poisson’s ratios above and below the level of the structure floor. a coefficient in the settlement expression b factor of the arching coefficient exponent csoil soil weight density l 2DOF spring stiffness per unit area /s soil internal friction angle h half of the arch opening angle q soil mass density r(x) axial stress in the soil column s(x) friction traction on the perimeter of the soil column
zero at the free surface). The coefficient of this relation k (‘arching coefficient’) may be either constant (Dancygier and Karinski, 1999a,b; Karinski et al., 2003) or some given function of the depth x (Chacha, 2014). The free field displacement for the case of a uniformly distributed load Pe0 acting on the surface of an infinite half plane is given by Karinski et al. (2003):
2.1. Basic discrete-continuous model The work described herein is an extension of a model that was developed previously by the authors for flat roofs (Karinski et al., 2003). For the clarity of presentation, key parts of the original model are presented in the following text. The model is suitable for depths of burial D, at which ‘arching’ (as explained above) can develop (at least about 15% of the roof span, e.g., see Dallriva and Hall, 1998). It comprises an equivalent structure subjected to a uniformly distributed equivalent load. The buried structure is represented here by an equivalent twodegree-of-freedom (2DOF) system interacting with an equivalent soil-column above it, where its Young’s modulus is denoted here Esoil. At its bottom, the 2DOF system is supported by a semiinfinite elastic medium, Fig. 1a and b. It is assumed that there is a full (perfect) contact between the structure and surrounding soil. The soil at the sides of the structure and far from it is represented in the model by the ‘free-field’ stress and displacement expressions (as shown in Karinski et al., 2003). The shear soil resistance (s) is represented in the model by a vertical friction traction that acts on the soil column perimeter and depends on the soil properties and on the relative displacement between the soil-column U(x) and that of the free field Uff (x) (where x is the depth, which is equal
U ff ðxÞ ¼ U 0 þ
8P qg 2 e0 > < Eff x 2Eff x ; > : PEe0ff x
0 6 x6 DþH 2 qg 1 1 1 1 ff x þ 2 ðD þ HÞ Eff ff ; D þ H 6 x ff x qg ðD þ H Þ Eff
qg
2
2Eb
Eb
Eb
ð1Þ
where H is the total height of the structure’s cross-section, q is the soil mass density, g is the gravitational acceleration and U0 is a constant, which is discussed in the following text. In Eq. (1) mb Þ mÞ Eff ¼ Effsoil ð1þðm1 ; Effb ¼ Effb;soil ð1þmð1 Þð12mÞ Þð12m b
bÞ
are the equivalent free-
field moduli of elasticity of the above and below the level of the structure floor, for a plane strain problem, where Effsoil ; Effb;soil are the corresponding ‘‘real” free field’s moduli of elasticity, and m and mb are the corresponding Poisson’s ratios. Note that Eq. (1) has been derived with the classical sign agreement that a tension force is positive. In the current problem external pressure, acting on the soil surface, Pext is positive in compression (pressure), and therefore, Pe0 = Pext in Eq. (1) and in Fig. 1b. It should also be noted that the proposed model is linear-elastic and is suitable for relatively small deflections and deformations that correspond to a ‘service-state’ condition (as opposed to an ‘ultimate state’ analysis).
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213
Fig. 1. Schemes of a buried structure and the model: (a) scheme of the problem, (b) equivalent discrete-continuous system, (c) loads and deflections, (d) free body diagrams of the 2DOF’s masses.
~b , which acts to the following expression for the average pressure q on the soil under the structure (positive for pressure, Fig. 2a):
2.2. Equivalent structure Consider a lined-tunnel with an arched roof and flat floor having equal spans (L). The top point of the structure’s roof is buried at depth D from the surface, which is loaded by uniformly distributed pressure Pext (positive for pressure). The total height of the tunnel’s cross-section is denoted H (Fig. 1a and c) and the roof radius is denoted R. The model assumes that the structure walls are rigid. Therefore, the following geometrical relations exist:
w þ w0 ¼ UðDÞ Ub ;
Ub ¼ UðDÞ ðw þ w0 Þ
ð2Þ
where w and w0 are the mid-roof and mid-floor deflections relative to their supports (walls), U(D) is the absolute mid-roof displacement (which is equal to the displacement of the bottom of the equivalent soil column), and Ub is the absolute mid-floor displacement, Fig. 1c. The equivalent 2DOF system (Fig. 1b and d) that represents the lining has two (total) masses, M (roof) and M0 (floor), and a spring, whose (equivalent) stiffness l (per unit area) is given in the following text (see Eq. (5)). The displacements of these masses are represented by the absolute displacements U(D) and Ub, described above (Fig. 1b and c). Equilibrium of the bottom mass M0 leads
~b ¼ q
M0 lðw þ w0 Þ gþ KL S
ð3aÞ
where KL is an equivalent structure transformation factor from the real structure to the 2DOF system (see Eq. (7)) and S is the horizontal projection of the area of the roof, which is also equal to the floor area. The settlement Ub is evaluated from a superposition of two vertical displacements (Fig. 2b and c): the free-field displacement at the structure’s base Uff(D + H) and an average displacement of the base area loaded with a uniform pressure qb (without taking ~b þ Eff U ffx ðD þ HÞ, see Fig. 2c). into account free-field, qb ¼ q Hereafter, the index ‘‘x” denotes derivative with respect to ‘‘x”. Note that negative derivative Ux(x) is associated with compression and therefore a negative sign appears in Fig. 2a. It can thus be seen from Fig. 2 that Ub is given by:
Ub ¼ Uff ðD þ HÞ þ a
qb Eb
ð3bÞ
where Eb is an equivalent Young’s modulus of the soil below the structure and Eb =a is its equivalent stiffness. The parameters a and Eb are given in Appendix A.
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Fig. 2. Superposition of displacements and pressures under the structure: (a) total pressure, (b) free-field pressure, (c) additional pressure caused by the structure.
Fig. 3. Schemes of an arch shaped element: (a) free body diagram of the roof, (b) fixed and (c) hinged support connections.
2.3. Equivalent stiffness of an arched roof The equivalent stiffness l has a dimension of force per unit area per displacement (w + w0). It is calculated according to the roof and floor stiffnesses K and K0, defined as the ratios between the total load acting on the roof (or floor) and its mid-span deflection. For a load and mass that are uniformly distributed per unit horizontal area, and considering equilibrium of the 2DOF system e b S M0 g ¼ q S þ Mg, refer to Fig. 1d), the stiffnesses are given (q by:
ðq S þ MgÞ w e b S M0 gÞ ðq S þ MgÞ ðq K0 ¼ w0 w0
K
ð4Þ
where q is a uniformly distributed load acting on the roof (see Fig. 1c). The definition of the 2DOF structure’s stiffness and Eq. (4) yield:
ðq S þ MgÞ ðq S þ MgÞ ðq S þ MgÞ KL ¼ lðw þ w0 Þ S ¼ l S þ K K0 1 KL )l¼1 1 ð5Þ þK S K 0
and once the sum of the roof and floor relative deflections (w + w0) has been calculated, the roof or floor deflection can be separately calculated according to their relative stiffnesses (Eq. (4)):
w¼
ðq S þ MgÞ ðw þ w0 Þ ¼ K 1 þ KK 0
ðw þ w0 Þ w0 ¼ 1 þ KK0
ð6Þ
The above mentioned transformation factor KL (for both load and stiffness) is obtained from equating the works of external forces in the real and equivalent structures, assuming that the maximum deflections of the two structures are equal, and for a mass that is uniformly distributed per unit horizontal area, it is given by (e.g., Szilard, 1974; Biggs, 1964):
RR KL ¼
S
ydS þ
RR
y0 dS
S
Sðw þ w0 Þ
ð7Þ
where y and y0 are respectively the roof and floor vertical deflection surfaces, relative to their supports (walls) in the real structure. Note that w and w0 are the maximum values of the deflection functions y and y0 (see Fig. 1c).
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Eq. (4) is valid for a planar roof (or floor), however when the shape of the roof (or floor) is arched it is also necessary to take into account the side pressure that acts on the roof (Fig. 3a). Consider the circular-arched roof of radius R and span L, related by the opening angle 2h (see Fig. 3a). The side pressure acting on the roof is commonly related to the vertical pressure by a ‘side pressure coefficient’ ks. Thus, the side pressure that acts on the arch is equal here to ks q, where q is the vertical contact pressure. The top point (mid-roof) displacement w can be calculated by the following expression:
Z
w¼
h
h
M b ð/Þ mb ð/Þ Rd/ þ Es I
Z
h
h
Nð/Þ nð/Þ Rd/ Es Aa
ð8Þ
where the bending moments Mb ð/Þ; mb ð/Þ and the axial forces N (/), n(/) are given in Appendix B. Es ; Aa and I are the Young’s modulus, cross sectional area and moment of inertia of a unit width strip of the arch (respectively); the variable angle / is defined in Fig. 3a. The arch stiffness per unit length of the tunnel is equal to the total vertical load (per horizontal unit length) divided by the mid-roof displacement and therefore:
K¼
q 2R sinðhÞ w
ð9Þ
Substitution of the moments and axial forces from Appendix B (Eqs. (B.1)–(B.8)) into Eq. (8), evaluation of the integrals and substitution of w into Eq. (9), yield the following expression for the stiffness (per unit width) K for both fixed and hinged support connections (Fig. 3b and c):
h F t ðh; h=LÞ K ¼ 48Es L F b ðh; h=L; ks Þ
ð10Þ
where h is the thickness of the roof cross-section. For the case of a fixed connection:
( 2 h 2 2 F t ¼ sin ðhÞ½1 þ cosðhÞ h sin ðhÞ½sinðhÞ cosðhÞ þ h L ) 2 2 þ3 2 cos ðhÞ þ h sinðhÞ cosðhÞ þ h 2
(
)
2
F b ¼ 3 sin ðhÞ ð1 ks ÞU1 ðhÞ 12 sinðhÞ½sinðhÞ þ ð1 2 cosðhÞÞ h ( 2 2 L h 4 9ð1 ks Þ U2 ðhÞ þ 4 sin ðhÞ ð1 ks ÞU3 ðhÞ h L ) 3 sinðhÞ ½h sinðhÞ where
þ h sinðhÞ 6 13 cosðhÞ 11 cos2 ðhÞ 3h2 1 2 cos2 ðhÞ ½1 þ cosðhÞ
U2 ðhÞ ¼ 6 cos3 ðhÞ 1 h2 4 6 cosðhÞ þ 3h2 cosðhÞ
þ 8 6h2 cos2 ðhÞ 4 cos4 ðhÞ 2h sinðhÞ þ h sinðhÞ cosðhÞ½3 þ 11 cosðhÞ þ 3h2 U3 ðhÞ ¼ h sinðhÞ 1 þ cosðhÞ þ cos2 ðhÞ sin2 ðhÞ ½2 þ cosðhÞ The stiffness per unit length of the tunnel’s flat floor is presented in the following form (see Karinski et al., 2003):
K ¼ 32Es
3 h 1 L 5 4aM
ð13Þ
where aM is equal to 1 and 0 for a fixed and hinged roof (respectively). It can be shown that Eq. (10) converges to Eq. (13) when h ! 0 while h R ! L. 3. Model’s equations and solution
ð11aÞ
As noted above, the vertical friction traction s(U(x), x) may be represented by the relative displacement between the soilcolumn U(x) and that of the free field Uff(x) in the following form (see Fig. 1):
h
i
ð14Þ
3
F b ¼ 3ð1 ks ÞF 1 ðhÞ þ 9h cos ðhÞðks 5Þ 9h sin ðhÞð1 þ 3ks Þ 2 L ð1 ks ÞF 2 ðhÞ þ 9h2 ð3 þ ks Þ þ 9 h 2 ( h þ4 ð1 ks ÞF 3 ðhÞ þ ð1 þ 2ks Þh sinðhÞ cos ð2hÞ cos4 ðhÞ L ) 2
U1 ðhÞ ¼ 2 sin2 ðhÞ 2 cos2 ðhÞ þ 5 cosðhÞ
sðUðxÞ; xÞ ¼ kðxÞ UðxÞ U ff ðxÞ 2
ð12bÞ
þh2 ð2 þ ks Þ þ ð4 ks Þh2 cos4 ðhÞ h2 cos2 ðhÞð5 þ ks Þ
ð11bÞ
In Karinski et al. (2003) the coefficient k(x) was taken as constant while in Chacha (2014) it was introduced as follows: – for a relatively shallow depth of burial (yet, sufficient for the development of arching):
kðxÞ ¼ k0 ebx
ð15Þ
– otherwise: where
F 1 ðhÞ ¼ ½5 cosðhÞ 8 1 þ cos4 ðhÞ þ cosðhÞ h i 3 16 cosðhÞ 10 cos2 ðhÞ þ 9h sin ðhÞ h i 2 þ 6h2 cosðhÞ cos3 ðhÞ sin ðhÞ
kðxÞ ¼
0 6 x < D nL k0 ; k0 ebðxDþnLÞ ; D nL 6 x 6 D
ð16Þ
where k0 and b are constants, L is the roof span and n is a parameter that must be obtained experimentally and depends on the soil properties. Note that Eq. (15) with b ¼ 0 converges to the particular case of k(x) = k0 = constant. The subsequent model verification and numerical analysis (given in Sections 4 and 5) refer only to the case of Eq. (15). Thus, the investigation of the parameter n is out of the scope of this paper.
2
F 2 ðhÞ ¼ ½cosðhÞ 4 sin ðhÞ 2h2 cosðhÞ½1 þ cosðhÞ þ h sinðhÞ½1 þ 5 cosðhÞ þ h2 F 3 ðhÞ ¼ h cosðhÞ cos ð2hÞ cos4 ðhÞ ½h sinðhÞ h cos5 ðhÞ For the case of a hinged connection:
(
3 F t ¼ sin ðhÞ 3 cosðhÞ 2 cos2 ðhÞ þ 2 cosðhÞ þ 1 h 3 sinðhÞð1 þ cosðhÞÞ þ h ) 2 h 2 þ sin ðhÞ½1 þ cosðhÞ½hþsinðhÞ cosðhÞ L
3.1. Equilibrium equations and boundary conditions The current problem is uniaxial with eyy ¼ ezz ¼ 0 and exx = dU (x)/dx (one-dimensional confined problem). It is well-known that in this case the axial stress is given by:
ð12aÞ
rxx rðxÞ ¼ E exx ¼ E Ux ðxÞ;
ð17Þ
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– For k = constant:
UðxÞ ¼ U ff ðxÞ þ A cosh ðkxÞ where A ¼ qgD E
2
al=E 1þal=Eb KL
ð24Þ
2 l MþEE la M0 þP KL EE la la EE þaP=D ðH 1 a H DÞ 2a=DðDÞ b b b , l K kD sinh ðkDÞþ a cosh ðkDÞ L
a=D
e0 ; M ¼ qM KL ; M0 ¼ qMDS0 ; P ¼ 1 qPgD P 1 andUff ðxÞ DS
la ¼ is
given by Eq. (5). – For k modeled by Eq. (15), Eq. (20) becomes:
U xx ðxÞ k2 ebx UðxÞ ¼
qg E
k2 ebx U ff ðxÞ
ð25Þ
The general solution of the homogeneous equation corresponding to Eq. (25) is as follows:
Fig. 4. Free body diagram of soil-column differential element.
where Ux ðxÞ ¼ dUðxÞ and E⁄ is the soil-column’s equivalent modulus dx of elasticity (‘confined modulus’), which is given by
UðxÞ ¼ AI0 ðzðxÞÞ þ BK 0 ðzðxÞÞ
ð26Þ
mÞ . E ¼ Esoil ð1þðm1 Þð12mÞ
where A and B are constants, I0(z), K0(z) are modified Bessel functions of the first and second kind (respectively) of order 0,
The equilibrium equation of a differential soil column element is given by (refer to Fig. 4):
and zðxÞ ¼ 2keb 2 .
S dr P dx s ¼ qg S dx
ð18Þ
where P and S are the roof perimeter and the horizontal projection of the area of the roof, which is also equal to the floor area (see above). Substitution of Eq. (17) into Eq. (18) yields the following equation of equilibrium in terms of the soil column displacement, U(x):
i P kðxÞ h qg ff U xx ðxÞ UðxÞ U ðxÞ ¼ S E E
ð19Þ
bx
The particular solution of Eq. (25) for homogeneous soil above the R zðnÞÞhðnÞ R ÞhðnÞ e dn I0 ðzðxÞÞ K 0 ðWðnÞ dn, structure is given by UðxÞ ¼ I0 ðzðnÞ WðnÞ bzðxÞ ff bzðxÞ qg qg 2 P e0 where hðxÞ ¼ k 2 U ðxÞ E ¼ k 2 U 0 þ Eff x 2Eff x qEg, WðxÞ ¼ bzðxÞ ½I0 ðzðxÞÞK 1 ðzðxÞÞ þ K 0 ðzðxÞÞI1 ðzðxÞÞ. 2 general solution of the equation is as follows:
Z Z
where Uxx ðxÞ ¼ d dxUðxÞ 2 . Eq. (19) can be re-written as follows: ff
2
U xx ðxÞ k f ðxÞ½UðxÞ U ðxÞ ¼
qg E
ð20Þ
qffiffiffiffiffiffiffi where k ¼ PS kE0 . For relatively deep structures Eq. (16) yields 1; 0 6 x < D nL f ðxÞ ¼ and for relatively shallow buried ebðxDþnLÞ ; D nL 6 x 6 D structures Eq. (15) yields f ðxÞ ¼ ebx , and when the arching coefficient is modeled as a constant, which is equal to k0 ; f ðxÞ ¼ 1. The boundary condition at the soil surface corresponds to x = 0 in Fig. 1b and is given by the following equation (refer to Eq. (17)):
E Ux ð0Þ ¼ Pe0
ð21Þ
Assuming full contact between the soil and the structure implies that the uniformly distributed load acting on the roof is given by q ¼ E U x ðDÞ. Therefore, the boundary condition at bottom of the soil column (x = D in Fig. 1b), in view of Eq. (5), is :
lðw þ w0 Þ ¼
K 0 ðzðnÞÞhðnÞ dn WðnÞ
I0 ðzðxÞÞ
Mg E Ux ðDÞ KL S
ð22Þ
Eqs. (2) and (3b) yield: lðw þ w0 Þ ¼ l½UðDÞ Ub ¼ h i l UðDÞ Uff ðD þ HÞ a qEb . Substitution of qb (see Fig. 2c), in view b
of Eq. (3a), yields:
0
1 UðDÞ U ff ðD þ HÞ Ea MS0 g þ Pe0 qg ðD þ HÞ b A l@ 1 þ El KaL b Mg E U x ðDÞ K L ¼ S
ð23Þ
The solution of Eq. (20) with the boundary conditions given in Eqs. (21) and (23) depends on the modeling of the arching coefficient k(x):
the
I0 ðzðnÞÞhðnÞ dn WðnÞ
UðxÞ ¼ AI0 ðzÞ þ BK 0 ðzÞK 0 ðzðxÞÞ þ
2
Therefore,
ð27Þ
Thus, the stress at the soil column is obtained by taking the first derivative of U(x) (multiplied by E ) and substituting into the boundary conditions (21) and (23), which yields the following expressions for the constants:
A¼
B¼
X
h i; Þ I0 ðzðDÞÞ þ E bzðDÞ I1 ðzðDÞÞj þ KI11ððzð0Þ K 0 ðzðDÞÞ E bzðDÞ K 1 ðzðDÞÞj zð0ÞÞ 2 2 Pe0 I1 ðzð0ÞÞðA Rð0ÞÞ K 1 ðzð0ÞÞTð0Þ kE ; K 1 ðzð0ÞÞ
ð28Þ where I1(z), K1(z) are modified Bessel functions of the first and second kind (respectively) of order 1, j ¼ K L þ lEa , X ¼ b nh i Mg þ E bzðDÞ ½I1 ðzðDÞÞRðDÞ þ K 1 ðzðDÞÞTðDÞ j þ I0 ðzðDÞÞRðDÞ K 0 ðzðDÞÞ 2 S h i h i P kI ðzð0ÞÞRð0ÞþkK 1 ðzð0ÞÞTð0Þþ Ee0 Þ Þ qg ðD þ HÞ Ea þ ðDþH þ 1 TðDÞ þ U 0 þ Ea MS0 g þ P e0 Ea þ ðDþH kK 1 ðzð0ÞÞ Eff 2Eff b b b h i R R bzðDÞ K 0 ðzðDÞÞ þ E 2 K 1 ðzðDÞÞj g; TðxÞ ¼ I0 ðzðxÞÞYðxÞdx; RðxÞ ¼ K 0 ðzðxÞÞYðxÞdx,
and YðxÞ ¼ bzðxÞ 2
bzðxÞ 2
k
P e0 qg x ff x2 Eff 2E
U0 þ
qg
þE
½I0 ðzðxÞÞK 1 ðzðxÞÞþK 0 ðzðxÞÞI1 ðzðxÞÞ
.
With the above expressions for the constants A and B, the solution of Eq. (20) is given by Eq. (27). As stated above, this solution, as well as the subsequent model verification and numerical analysis, deal with the case of an arching factor that varies exponentially with the soil depth (x), Eq. (15). One can see that for k = constant Eq. (24) is the closed form solution while for k that varies with depth the analytical solution that is given in Eq. (27) includes some integrals of Bessel functions that are simple for calculation. It should be noted that these solutions allow calculation of a relative displacement U(x) U0. When there is a depth, at which the displacement is known (commonly, the case of a bedrock at a known depth or a rigid floor
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of an experimental setup) then U0 can be explicitly determined. Even so, for the calculation of the average contact pressure one needs the derivative of these solutions, which is does not depend on the constant U0, i.e.:
q ¼ E U x ðDÞ
ð29Þ
Similarly, these solutions together with Eqs. (2) and (6) yield the roof deflection, relative to its supports (because the term U0 cancels out). 4. Verification The solution for the equivalent discrete-continuous system, given in Section 3 (Eqs. (24) or (27)) with the application of an equivalent stiffness given in Section 2.2 (Eqs. 4, 10-13) constitute a model for the assessment of the contact pressure on an arched roof of a lined tunnel. In order to verify the proposed model, its predictions have been compared with laboratory experimental results conducted by Dallriva and Hall (1998) of a semi-circular concrete arch buried in sand, where a uniform external pressure was applied on the backfill surface. The depth of burial (D) was equal to 19 cm and the arch span (L), width (b), and thickness (h), were 111.7, 91.4 and 5.1 cm, respectively (Fig. 5). According to the report, the arch specimen was assumed to be a fixed-fixed one. The specimen in these experiments was supported by two massive blocks that acted as a rigid ‘‘floor” (w0 = 0). The test included three cycles of loadingunloading-reloading, where the third one was applied after a significant structural damage (i.e., decreasing capacity) was observed. Therefore, the following calibration and verification correspond to the results reported from the first two cycles. Furthermore, the
217
reported load-displacement curve corresponding to the second cycle starts from a residual mid-span deflection. Consequently, for the purpose of the comparison with the current linear model, this cycle has been shifted to the origin. Normal contact pressures at nine discrete points along the arch, as well as the mid-point deflection were measured in the experiment. The pressure gages were located along the arch as shown in Fig. 5. Two types of soil-structure contact conditions were studied in this work: with and without a Teflon interface layer. The following comparison was performed with the experiment that did not include this layer because the current model refers to a state of full soil-structure contact. 4.1. Calibration of the side pressure coefficient ks Calibration of the side pressure coefficient ks has been obtained from comparison between the measured total vertical contact load-to-mid-span deflection ratio, Kmsd, and the stiffness (K) of the equivalent fixed-fixed structure (Eqs. 10,11) with h ¼ 90 ), i.e.:
h F t ðp=2; h=LÞ L F b ðp=2; h=L; ks Þ
K msd ¼ 48Es
ð30Þ
Solving this equation for ks yields:
ks ¼
Ak þ Bk Ck
ð31Þ
h2 i
2 2 where Ak ¼ 16 KEs b hL hL2 p4 þ 3 p4 2 ; Bk ¼ 8 þ 32p 32p 1 þ msd
2 2 2 2 2 3 hL 2 p2 þ p4 4 þ 23p hL2 ðp 1Þ; C k ¼ 8 þ 32p p2 3 3 hL2 p2 þ p4 4 þ p h2 ðp 4Þ. 3 L2
In these expressions, Es is Young’s modulus of the struc-
ture (taken by Dallriva and Hall, 1998, as equal to 30,000 MPa). Note that for a thin-wall arch (i.e., h/L 1) the above equations yield:
Es b h p2 8 K msd L3 p2 þ 4p 16 3
ks ¼ 1 16
ð32Þ
The measured stiffness Kmsd has been evaluated as follows:
Fig. 5. Schematic description of the setup and measurements in Dallriva and Hall (1998) experiment.
The arch was divided into 8 segments between the pressure gauges (see Fig. 5).
The normal contact force acting at each segment (Pi) was calculated, assuming a linear distribution of the contact pressure between the measuring points. The force at each segment is considered to be applied at the centroid of the pressure distribution, as shown in Fig. 6a.
The total vertical force (Pv) is calculated as the sum of the vertical components (Pvi) of Pi (see Fig. 6b).
Kmsd has been evaluated by dividing Pv by the measured midspan deflection (w) as explained in the following text.
Fig. 6. Schematic description of the calculation of the total vertical force acting on the arch.
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(a) Young's Modulus
Fig. 7. Force-deflection curve according to the experimental results (Dallriva and Hall, 1998).
Fig. 7 shows the measured Pv-w curve of the first experimental cycle. Because the proposed model is linear, calibration of the side pressure coefficient (ks) was performed according to a stiffness Kmsd = 805,593 kN/m, which was determined by a linear trendline of the initial segment of this curve (O-A in Fig. 7). Calculating ks from Eq. (31) with this Kmsd yields a value of 0.428. Note that using the simplified solution (Eq.(32)) yields a ks value of 0.366 (14% difference). It is also interesting to note that both these values are close to the ‘‘at-rest” value of the side pressure coefficient ks ¼ 1 sin ð/s Þ ¼ 0:377 for an internal friction angle /s ¼ 38:5 that was reported by Dallriva and Hall (1998). 4.2. Other soil parameters Dallriva and Hall (1998) reported a soil weight density csoil and Poisson’s ratio m of 16.6 kN/m3 and 0.3, respectively. The soil Young’s modulus above and below the structure (Esoil, Eb) and the arching coefficient parameters k0 and b (see Eq. (15)) were calibrated. The calibration was performed according to the average 2.015-mm mid-span deflection of two load cycles (first and shifted second; see above) under an external pressure of 1.96 MPa. Fig. 8 shows the variation of the calculated mid-roof deflection (solid lines) with varying values of these parameters, as well as the measured deflection (dotted line). It is important to note that the values shown at the X-axis in Fig. 8a are those of the sand Young’s moduli (and not of their equivalent values). It is evident from Fig. 8 that the calibrated values of the soil parameters are: Esoil = Eb = 200 MPa, k0 = 1175 MN/m3 and bD = 0.5 (where D is the depth of burial). 4.3. Comparison with the test results Fig. 9 shows comparisons of the model with the reported external load-deflection (at mid-span). The model’s prediction was calculated with the above calibrated soil parameters for the whole reported range of the external pressure, for both loading cycles mentioned above, where the calibration point is marked in the figure. Fig. 9a shows a comparison based on the initial stiffness corresponding to the line O-A in Fig. 7 (K = 805,593 kN/m). The figure shows good agreement of the model with the measured results up to an external pressure somewhat larger than 2 MPa. Above this pressure the increase in the predicted deflections is lower than the measured ones. This difference is related to the non-linear Pv-w curve in Fig. 7, based on which, the stiffness (K) has been calculated. This non-linearity is caused by the damage development in the concrete arch. Therefore, under relatively high external pressure, a lower stiffness is more realistic. Furthermore, the 2-mm deflection under an external pressure of 2 MPa corresponds to point ‘‘A” in Fig. 7, where a decrease of the stiffness is observed. Therefore, we used an enhanced model with an initial stiffness K = 805,593 kN/m up to point ‘‘A” in Fig. 7 and with a lower stiffness beyond this point. This stiffness has been evaluated
Fig. 8. Calibration of soil parameters based on Dallriva and Hall (1998).
in two ways: tangent stiffness (line A-B in Fig. 7, K = 345,363 kN/m) and secant stiffness (line A-C, K = 246,480 kN/m). Fig. 10 shows a comparisons of the enhanced model prediction of the average contact pressure (see Eq. (29)) with that obtained from the measured data (Pv/L, refer to Fig. 6b). Because the researchers reported malfunction of the pressure gauges during the second cycle, the comparison was carried out only with results from the first cycle. It is evident from the figure that there is a good agreement of the model with the test results, where the model with the secant stiffness shows a somewhat better agreement. It is also interesting to note that the model’s prediction of both contact pressure and deflection is conservative with respect to the experimental data. Additionally, one can see that both experimental and predicted results indicates contact pressure that is lower than the external pressure, i.e., positive arching. 5. Case study The solution for the equivalent discrete-continuous system, given in Section 3 (Eqs. (24) or (27)) has been applied to investigate the following problem. Consider a lined reinforced concrete tunnel, buried at a depth of D = 10 m under a surface external pressure Pext = 100 kPa. The tunnel has a flat rigid floor and walls, and an arched roof with an angle h (Fig. 3a) that varies from zero (limiting case of a flat roof) to 90° (semi-circular), Fig. 11. Note that for a given span (L) the radius of the roof arch varies with h as follows: L R ¼ 2 sin . It is further assumed that the roof has a fixed-fixed h
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Fig. 9. Comparison of mid-span deflections.
Fig. 10. Comparison of average contact pressure.
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Fig. 11. Structures of the case study.
Fig. 12. Normalized contact pressure for various spans (L) —— h = 250 mm – – – h = 150 mm.
scheme (refer to Fig. 3b). Young’s modulus and Poisson’s ratio of the structure were taken as 30,000 MPa and 0.15. The total height of the structure is 4 m (Fig. 11). Three spans (L), 2.5, 5 and 8 m, and two roof thicknesses (h), 150 and 250 mm, were studied. The parameters of the soil and arching coefficient were chosen as follows: csoil ¼ 16:87 kN/m3, Esoil = Eb = 75 MPa, m ¼ mb ¼ 0:3, k0 = 175 MN/m3 and bD = 0.4. The side pressure coefficient was taken as ks = 0.5 (corresponding to its ‘‘at-rest” value for an internal friction angle of 30°). 5.1. Contact pressure Fig. 12 shows the variation of the calculated average contact pressure (Eq. (29)) with the opening angle. The pressure has been
normalized with respect to the ‘free field’ pressure at the roof top point level, Pff ¼ Pext þ csoil D. One can see that there is a certain angle at which the contact pressure has a maximum value. This maximum is more pronounced at the longer span and smaller roof thickness. Note that the angle at which there is a maximum contact pressure is different for each roof span and stiffness. Fig. 12 also shows the development of positive arching, where for a given angle theta, longer spans L (i.e., lower roof stiffness) lead to lower contact pressure q. Noting that the arch stiffness K is also a function of h (Eqs. 10,11), the analysis shows that the angles at which the contact pressure has a maximum, coincide with the angles at which there is a maximum value of K. An example that illustrates this correspondence for L = 5 m and h = 150 mm is presented in Fig. 13.
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Fig. 13. Normalized contact pressure (– – –) and stiffness K (——) for L = 5 m and h = 150 mm.
Fig. 14. Normalized contact pressure vs. stiffness K for L = 5 m and h = 150 mm.
Fig. 14 shows peak values of the contact pressure versus peak values of the roof stiffness. Angles at which these maxima occur are also marked in the figure. It is evident from Fig. 14 that this relation is monotonic.
Presumably (see Fig. 12), the best angle for the design of the roof is the one that yields minimum contact pressure, i.e., flat roof (h ¼ 0). Evidently, however, this is not the case. The explanation for this outcome is the above shown relation between the roof
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Fig. 15. Bending moment (a) and shear force (b) for various opening angles (Theta) of the arch for L = 5 m and h = 150 mm (s is the ordinate along the arch neutral axis).
stiffness and the contact pressure (Fig. 13). Hence, a minimum contact pressure does not necessarily leads to minimum internal forces and moments, which control the design of the arch. Thus, in order to obtain the optimal opening angle of the arch the internal forces and moments should also be studied.
5.2. Internal forces Consider the bending moment (Mb), axial and shear forces (N and Ss), as well as the axial stresses at the extreme fibers of the cross-section, along the axis of the roof arch. The extreme stresses are given by MWb þ NA, where W and Aa are the crosssection modulus and area of a unit width strip of the arch roof,
‘‘+” and ‘‘” correspond to bottom and top fibers, respectively. The calculations by the present model were carried out for the loading shown in Fig. 3a and static schemes shown in Fig. 3b. Fig. 15 demonstrates examples of the bending moment and shear force diagrams for various opening angles of the arch for L = 5 m and h = 150 mm. The ordinate ‘‘s” in this figure denotes the coordinate along the arch (s = 0 at the top point). It is evident from the figure that although the contact pressure that acts on the flat roof is lower than the one that acts on the arched roof (see Fig. 13), the internal shear force and bending moment in the flat roof are significantly larger than those in the arched roof. For example, refer to Theta = 0 and 30–40° in Fig. 15. A similar finding has been obtained for the other cases (L = 2.5, 5 and 8 m, h = 150 and 250 mm).
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223
Fig. 16. Maximal values of the bending moment, shear force and axial stresses at the extreme fibers of the cross-section for L = 5 m and h = 150 mm.
Table 1 Optimal opening angles. L (m)
h (mm)
Slenderness L/h
h optimal [°] Shear force
Moment
Stress
2.5
150 250
16.7 10.0
49 61
58 70
63 73
5
150 250
33.3 20.0
36 45
42 53
48 59
8
150 250
53.3 32.0
28 36
33 43
39 48
5.3. Optimal opening angle According to this analysis one can obtain an optimal value for the opening angle (hÞ that will yield minimal internal forces and stresses. An example for evaluation of such optimal angel is shown in Fig. 16 for the case analyzed in Fig. 15 (for L = 5 m and
h = 150 mm). The figure shows maximal absolute values of the bending moment, shear force and axial stresses at the extreme fibers of the cross-section, for each opening angle. It can be seen in Fig. 16 that for the present case the optimal opening angle ranges between 35 and 45°: the shear force is minimal at 35° while the bending moment and normal stresses are minimal at 45°. Calculated optimal opening angles for all studied cases are given in Table 1. The table shows that as the roof slenderness increases, the optimal opening angle decreases. This finding is illustrated in Fig. 17. It is interesting to note that the range of optimal angles corresponds to the same range at which the contact pressure and the stiffness are maximal (refer to Figs. 16 and 13). 6. Conclusions A model to analyze the response of an arched roof of a tunnel lining under a surface static loading is presented. The model
Fig. 17. Optimal opening angles (filled markers denote the case shown in Fig. 16).
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simulates the soil and structure behavior under ‘service-state’ conditions and it is suitable for depths of burial at which ‘arching’ can develop. It enhances a previous model by the authors, which is based on a discrete-continuous concept: the buried structure is represented in the model by an equivalent 2DOF system interacting with an equivalent, continuous soil-column above it. The current enhanced model takes into consideration the roof curvature. The proposed 2DOF system’s stiffness includes the influences of the soil side pressure as well as the arched geometry of the roof (i.e., the span and opening angle). At the bottom, the 2DOF system is supported by either a semi-infinite elastic medium or by a rigid base. Opposite to numerical approaches (such as finite elements), the current model presents an analytical solution, which is a closedform for a constant arching coefficient (Eq. (24)). For a varying arching coefficient, it requires rather simple calculations of integrals of Bessel functions (Eqs. (27), (28)). Thus, the solution can be straightforwardly implemented by using only these (1 or 2) equations together with Eq. (29) for the evaluation of the average contact pressure. For the case of zero curvature the analytical solution for the mid-roof deflection and average contact pressure that has been derived converges to the solution of a flat roof. The case of a relatively shallow buried structure has been calibrated and then verified against published experimental results from a test of a semi-circular buried arch. A case study shows that there is a certain opening angle of the roof at which the contact pressure has a maximum value, which is more pronounced at a longer span and smaller roof thickness. This angle coincides with the angle at which there is also a maximum value of the roof stiffness. However, it is also shown that approximately at this opening angle, the internal moment, shear force and normal stresses are minimal. It is therefore concluded that the average contact pressure is not necessarily the most important criterion for a design of an optimal shape of the roof, which should be determined according to this angle. It is further shown that as the roof slenderness increases this optimal angle decreases. Acknowledgments This work was supported by a joint grant from the Centre for Absorption in Science of the Ministry of Immigrant Absorption and the Committee for Planning and Budgeting of the Council for Higher Education under the framework of the KAMEA Program and by the Ministry of Construction and Housing. The research grants are greatly appreciated.
The average settlement under the floor, d (Fig. 2c), is obtained by:
d¼
1 L
Z
L=2
L=2
For the case of a long conduit with a flat floor of span L the relative vertical settlement, V(x,y) (absolute settlement minus the free field displacement) is obtained from the plane strain solution of a half plane loaded by a uniformly distributed load qb over a span L (see Fig. 2c). It is given by (e.g., Nowacki, 1976):
Eb V ðx; yÞ 1 þ mb ( !) qb r2 L r1 r2 ¼ ð1 2mb Þxðh2 h1 Þ þ 2ð1 þ mb Þ y ln ln 2 p r1 2 R0 ðA:1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x where r 1 ¼ x2 þ ðy þ L=2Þ2 ; r2 ¼ x2 þ ðy L=2Þ2 ; h1 ¼ arctan yþL=2 ; h2 ¼ x arctan L=2y , R0 is a constant obtained from boundary conditions
(and L ln(R0) is a rigid body term) and Eb ; mb are the modulus of elasticity and Poisson’s ratio of the soil under the structure.
1 m2b ½1 þ 2Lnðv=2Þ Eb p
ðA:2Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12mb d0 arctan L 1mb 2L d0 d0 and d0 has to be evaluated where v ¼ 1 þ L e as the depth at which the free field settlement and the settlement caused by the external load are similar. Usually a depth equal to about 5L may be suitable for the value of d0. The expression (A.2) may be re-written as follows:
da
q Eb
ðA:3Þ
where Eb ¼ Eb =ð1 m2b Þ and a ¼ 1 þ 2Ln v2 pL . Appendix B. Axial force and bending moment in the arched roof The following notations are used in this appendix: Es, Aa and I – Young’s modulus, cross sectional area and moment of inertia of a unit width strip of the arch (respectively); R, 2h – radius and opening angle of the arch (see Fig. 3a); q - vertical load per horizontal unit length; the variable angle / is defined in Fig. 3a, where h 6 / 6 h. For a hinged support connection (see Fig. 3c): The bending moment caused by the load q is as follows:
Mb ð/Þ ¼
qR2 M t1 ðh; ks ; /Þ M b1 ðhÞ 6
ðB:1Þ
where, M t1 ðh; ks ; /Þ ¼ R2 Es Aa ð1 ks Þu1 ðh; /Þ þ Es Iu2 ðh; ks ; /Þ; Mb1 ðhÞ ¼
EIðsinðhÞcosðhÞ þ hÞ þ R2 Es Aa h þ 2hcos2 ðhÞ 3 sinðhÞcosðhÞ ; u1 ðh; /Þ ¼ 2 2 ð3 sinðhÞ cosðhÞ hÞ þ 6hcos ðhÞ cos ðhÞ cosð/Þð1 þ 6h cosð/Þþ 7 sinðhÞÞ þ 4 sinðhÞ cosð/Þ þ 2cos3 ðhÞð3h cosð/Þ sinðhÞÞ cosðhÞ ð4 sinðhÞ þ 3h cosð/ÞÞ, u2 ðh; ks ; /Þ ¼ ð1 ks Þ cos2 ðhÞð4 sinðhÞ cosð/Þþ 3hÞ 3cos2 ð/ÞðsinðhÞ cosðhÞ hÞ sinðhÞcos3 ðhÞ þ 4 sinðhÞð1 þ 2ks Þ ðcosðhÞ cosð/ÞÞ. For the particular case of half a circle (h ¼ p/2):
Mb1 ¼ Es I þ R2 Es Aa p=2;
u1 ¼ ð3 p=2Þ þ 4 cosð/Þ;
u2 ¼ ð1 ks Þ3 cos ð/Þp=2 4 cosð/Þð1 þ 2ks Þ: 2
The bending moment caused by a vertical unit force acting at the top of the arch mb ð/Þ is given by:
mb ð/Þ ¼
Appendix A. Calculation of the parameters a and Eb
Vð0; yÞdy ¼ qb L
R mt1 ðh; /Þ 2 M b1 ðhÞ
ðB:2Þ
where, for 0 6 / 6 h : mt1 ðh; /Þ ¼ R2 Es Aa u3 ðh; /Þ þ Es Iu4 ðh; /Þ; u3
ðh; /Þ ¼ sinð/Þ h 1 þ 2cos2 ðhÞ 3 sinðhÞ cosðhÞ þ cosð/Þð2h cosðhÞ
2 sinðhÞ þ 3cos ðhÞ 2 cosðhÞ 1 Þ þ h sinðhÞ þ 2ðcosðhÞ 1Þ cosðhÞ, u4 ðh;/Þ ¼ ðcosðhÞsinðhÞ þ hÞsinð/Þ þ sinðhÞcosð/ÞsinðhÞ þ hsinðhÞ, and
for h 6 / 6 0 : mt1 ðh;/Þ ¼ sinðhÞ R2 Es Aa u5 ðh; /Þ þ Es Iu6 ðh; /Þ ; u5 ðh; /Þ ¼
cosð/Þ 2h sinðhÞ þ 3cos2 ðhÞ 2 cosðhÞ 1 sinð/Þ 2hcos2 ðhÞ þ h þ 3 sinðhÞÞ þ sinðhÞ þ ðcosðhÞ 1Þ2 cosðhÞ; u6 ðh; /Þ ¼ cosð/Þsin ðhÞ sinð/ÞðsinðhÞ cosðhÞ þ hÞ þ h sinðhÞ. The axial force caused by the load q is as follows: 2
Nð/Þ ¼
qR Nt1 ðh; ks ; /Þ 6 M b1 ðhÞ
ðB:3Þ
where Nt1 ðh; ks ; /Þ ¼ R2 Es Aa u7 ðh; ks ; /Þ þ Es Iu8 ðh; ks ; /Þ; u7 ðh; ks ; /Þ ¼
ð1 ks Þcosð/Þ 6cosð/Þ h þ 3sinðhÞcosðhÞ 2hcos2 ðhÞ þ þð7sinðhÞ cos2 ðhÞ 4 sinðhÞ 3h cosðhÞ cos ð2hÞÞ þ þ6ð3 sinðhÞ cosðhÞ h
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225
2h cos2 ðhÞÞ; u8 ðh; ks ; /Þ ¼ ð1 ks Þ ðsinðhÞ cosðhÞ þ hÞ6 cos2 ð/Þ 4 sinðhÞ
h i 2 2 2 sin ðhÞ h sinðhÞ cosðhÞ h2 þ 6 h2 þ 2 sin ðhÞ þ h sinðhÞ cosðhÞ ;
cos2 ðhÞ cosð/Þ þ 4 sinðhÞ cosð/Þð1 þ 2ks Þ 6ðsinðhÞ cosðhÞ þ hÞ. For the particular case of half a circle (h ¼ p/2):
u18 ðh; ks ; /Þ ¼ ð1 ks Þ cosð/Þ½sinðhÞ cosðhÞð4 cosðhÞ 6 cosð/ÞÞ 6h cosð/Þ 4 sinðhÞ cosð/Þð1 þ 2ks Þ þ 6ðsinðhÞ cosðhÞ þ hÞ.
For h ¼ p/2: u17 ¼ ð1 ks Þ cosð/Þ p=2 þ 6 cosð/Þ 2 p2 =4 þ 2 6 p =4 þ 2 , u18 ¼ 3pð1 ks Þ cos2 ð/Þ 4 cosð/Þð1 þ 2ks Þ þ 3p.
u7 ¼ ð1 ks Þ cosð/Þ½3 cosð/Þp 4 3p=2 þ 6 p=2; u8 ¼ 3ð1 ks Þp cos2 ð/Þ þ 4 cosð/Þð1 þ 2ks Þ p: The axial force caused by a vertical unit force acting at the top of the arch n(/Þ is given by:
nð/Þ ¼
1 nt1 ðh; /Þ 2 Mb1 ðhÞ
ðB:4Þ
6 / 6 h : nt1 ðh; /Þ ¼ Es Iu9 þ R2 Es Aa u10 ; u9 ðh; /Þ ¼
2 ðcosðhÞ sinðhÞ þ hÞ sinð/Þ þ cosð/Þ sin ðhÞ, u10 ðh; /Þ ¼ sinð/Þ h 1 þ 2cos2 ðhÞÞ 3 sinðhÞ cosðhÞÞ þ cosð/ÞðcosðhÞð2h sinðhÞ 2 þ 3 cosðhÞÞ 1Þ, and where,
for
0
for h 6 / 6 0 : u9 ðh; /Þ ¼ sin ðhÞ cosð/Þ sinð/ÞðcosðhÞ sinðhÞ þ hÞ; u10 ðh; /Þ ¼ cosð/Þ½cosðhÞð3 cosðhÞ 2 þ 2h sinðhÞÞ 1 þ sinð/Þð3 sinðhÞ cosðhÞ h 2hcos2 ðhÞÞ. For a fixed support connection (see Fig. 3b): 2
M b ð/Þ ¼
qR2 M t2 ðh; ks ; /Þ M b2 ðhÞ 12
ðB:5Þ
where, M t2 ðh; ks ; /Þ ¼ R2 Es Aa ð1 ks Þu11 ðh; /Þ þ Es Iu12 ðh; ks ; /Þ; Mb2 ðhÞ ¼ h i 2 R2 Es Aa hðsinðhÞ cosðhÞ þ hÞ þ 2sin ðhÞ þ Es IhðsinðhÞ cosðhÞ þ hÞ;
u11 ðh; /Þ ¼ 2 cosð/Þf2 cosðhÞ þ cos ð2hÞðh sinðhÞ cosðhÞÞ þ 3 cosð/Þ
h
i
2 hðsinðhÞ cosðhÞ þ hÞ 2sin ðhÞ g cos2 ðhÞ cos2 ðhÞ þ 7 3hðh þ sin ð2hÞÞ þ 8, u12 ðh; ks ; /Þ ¼ ð1 ks Þ 6hcos2 ð/Þðh þ sinðhÞ cosðhÞÞ 3h2 6h sinðhÞ cosðhÞ 8h sinðhÞcos2 ðhÞ cosð/Þ þ 8ð1 þ 2ks Þ 5cos4 ðhÞ ðh sinðhÞ cosð/Þ 1Þ þ cos2 ðhÞð13 þ 11ks Þ.
For h ¼ p/2: M b2 ¼ R2 Es Aa p2 =4 þ 2 þ Es Ip2 =4; u11 ¼ 2 cosð/Þ
2 3 cosð/Þ p =4 2 p=2 3p2 =4 þ 8, u12 ¼ ð1 ks Þ 6h cos2 ð/Þ
p=2 3p2 =4 þ 8ð1 þ 2ks Þðp=2 cosð/Þ 1Þ. mb ð/Þ ¼
R mt2 ðh; /Þ 2 Mb2 ðhÞ
ðB:6Þ
0 6 / 6 h : mt2 ðh; /Þ ¼ R2 Es Aa u13 ðh; /Þ þ Es Iu14 ðh; /Þ; u13 ðh; /Þ ¼ cosð/Þ sinðhÞ½2ðcosðhÞ 1Þ þ h sinðhÞ þ sinð/Þ h2 þ sinðhÞ ðh cosðhÞ 2 sinðhÞÞ þ ð1 cosðhÞÞðh sinðhÞÞ; u14 ðh; /Þ ¼ sinð/Þ where,
for
2
hðcosðhÞ sinðhÞ þ hÞ þ cosð/Þh sin ðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ,
and
for h 6 / 6 0 : mt2 ðh; /Þ ¼ R Es Aa u15 ðh; /Þ þ Es Iu16 ðh; /Þ; u15 ðh; /Þ ¼ 2 cosð/Þ sinðhÞðh sinðhÞ þ 2 cosðhÞ 2Þ þ sinð/Þ 2sin ðhÞ h2 h cosðhÞ u16 ðh; /Þ ¼ cosð/Þhsin2 ðhÞ sin sinðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ, 2
ð/ÞhðsinðhÞ cosðhÞ þ hÞ þ h sinðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ.
Nð/Þ ¼
qR Nt2 ðh; ks ; /Þ 6 M b2 ðhÞ
ðB:7Þ
where, Nt2 ðh; ks ; /Þ ¼ R2 Es Aa u17 ðh; ks ; /Þ þ Es Ihu18 ðh; ks ; /Þ; u17 ðh; ks ;
/Þ ¼ ð1 ks Þ cosð/Þ sinðhÞ 2h cos2 ðhÞ 3 sinðhÞ cosðhÞ þ h þ 6 cosð/Þ
nð/Þ ¼
1 nt1 ðh; /Þ 2 M b2 ðhÞ
ðB:8Þ
where, for 0 6 / 6 h : nt1 ðh; /Þ ¼ R2 Es Aa u19 ðh; /Þ þ Es Iu20 ðh; /Þ; u19 2 ðh; /Þ ¼ cosð/Þ sinðhÞðh sinðhÞ 2 þ 2 cosðhÞÞ þ sinð/Þ h2 2 sin ðhÞþ h cosðhÞ sinðhÞÞ; u20 ðh; /Þ ¼ h sinð/ÞððsinðhÞ cosðhÞ þ hÞÞ þ h sin ðhÞ cosð/Þ, and for h 6 / 6 0 : u19 ðh; /Þ ¼ cosð/Þ sinðhÞðh sinðhÞ 2 2 þ 2 cosðhÞÞ þ sinð/Þ 2sin ðhÞ h2 h cosðhÞ sinðhÞ ; u20 ðh; /Þ ¼ 2 h sin ðhÞ cosð/Þ sinð/ÞðsinðhÞ cosðhÞ þ hÞ . 2
References Biggs, J.M., 1964. Introduction to Structural Dynamics. McGraw-Hill, pp. 205–214. Brachman, R.W.I., Moore, I.D., Rowe, R.K., 2000. The design of a laboratory facility for evaluating the structural response of small-diameter buried pipes. Can. Geotech. J. 37 (2), 281–295. Chacha, A., 2014. A model to assess the load acting on a buried structure (M.Sc. Research Thesis). Technion – Israel Institute of Technologyr, Haifa (in Hebrew). Dallriva, F.D., Hall, R.L., 1998. Static tests of shallow-buried reinforced concrete arches, Final Report. US Army corps of Eng.. Dancygier, A.N., Karinski, Y.S., 1999a. Response of a buried structure to surface repetitive loading. Eng. Struct. 21, 416–424. Dancygier, A.N., Karinski, Y.S., 1999b. A simple model to assess the effect of soil shear resistance on the response of soil-buried structures under dynamic loads. Eng. Struct. 21 (12), 1055–1065. Higgins, C.J., Drake, J.L., 1995. An engineering approach to soil-structure interaction. In: Proc. 7th Int. Symposium on Interaction of the Effects of Munitions with Structures, pp. 510–516. Karinski, Y.S., Dancygier, A.N., Leviathan, I., 2003. An analytical model to evaluate the static soil pressure on a buried structure. Eng. Struct. 25, 91–101. Mai, V.T., Moore, I.D., Hoult, N.A., 2014. Performance of two-dimensional analysis: deteriorated metal culverts under surface live load. Tunn. Undergr. Space Technol. 42, 152–160. Marston, A., 1930. The theory of external loads on closed conduits in the light of the latest experiments. Iwoa Eng. Exp. Station, Bull. 96. Newmark, N.M., 1964. The basis of current criteria for the design of underground protective construction. In: Proc. of the Symp. Soil-Struct. Interaction. Tucson (Arizona), pp. 1–24. Nowacki, W., 1976. Theory of Elasticity. Springer-Verlag. Papanikolaou, V.K., Kappos, A.J., 2014. Practical nonlinear analysis of unreinforced concrete tunnel linings. Tunn. Undergr. Space Technol. 40, 127–140. Spangler, M.G., 1957. Stresses in pressure pipelines and protective pipe supporting earth loads and live loads. ASTM, 57. Szechy, K., 1973. The Art of Tunneling, second English ed. AkademiaiKiado, Budapest. Szilard, R., 1974. Theory and Analysis of Plates – Classical and Numerical Methods. Prentice-Hall, Englewood Cliffs, New Jersey. Terzaghi, K., 1959. Theoretical Soil Mechanics. Chapman and Hall, London. Weidlinger, P., Hinman, E., 1988. Analysis of underground protective structures. ASCE J. Struct. Eng. 114 (7), 1658–1672.
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
A model to assess the response of an arched roof of a lined tunnel A.N. Dancygier ⇑, Y.S. Karinski, A. Chacha Department of Civil Engineering, National Building Research Institute, Technion – Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 2 September 2015 Received in revised form 11 January 2016 Accepted 23 March 2016
Keywords: Arched roof Arching coefficient Lined tunnel Soil-structure interaction
a b s t r a c t A model to analyze the response of an arched roof of a tunnel lining under a surface static loading is presented. It enhances a previous model by the authors, which is based on a discrete-continuous concept and is suitable for depths of burial at which ‘arching’ can develop. The current enhanced model takes into consideration the curvature of an arched roof of a lined tunnel. The proposed 2DOF system’s stiffness includes the influences of the soil side pressure as well as the arched geometry of the roof. For the case of zero curvature the analytical solution for the mid-roof deflection and average contact pressure that has been derived converges to the solution of a flat roof. The case of a relatively shallow buried structure has been calibrated and then verified against published experimental results. A case study shows that there is a certain opening angle of the roof at which the contact pressure has a maximum value. This angle coincides with the angle at which there is also a maximum value of the roof stiffness. However, it is also shown that approximately at this opening angle, the internal forces are minimal. It is therefore concluded that the average contact pressure is not necessarily the most important criterion for a design of an optimal shape of the roof. Furthermore, the angle that yields maximum contact pressure should be preferred for an optimal roof design. It is further shown that as the roof slenderness increases this optimal angle decreases. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction An important parameter in the design of lined tunnels is the load acting on its roof. A common loading case is the self-weight of the covering or backfill soil above the structure and an external surface pressure. For deeply buried structures the weight of the soil upper layers may be regarded as a uniformly distributed load acting on the layers below them. The covering soil adjacent to the lining roof interacts with the structure in a way known for static loads as ‘arching’. This phenomenon refers to a mechanism related to relative displacements in the soil media above the structure and far from it (e.g., Terzaghi, 1959; Newmark, 1964). Due to this phenomenon, the average contact pressure may be lower or higher than the undisturbed ‘free-field’ pressure. The first case is known as ‘positive’ or ‘active’ arching and the latter case is known as ‘negative’ or ‘passive’ arching. A common way to evaluate the response of a buried structure is by numerical analysis. Finite element simulation of a specific problem is most frequently used (e.g., Brachman et al., 2000; Papanikolaou and Kappos, 2014; Mai et al., 2014,). These methods require proper representation of the soil and structure properties, ⇑ Corresponding author. E-mail address: [email protected] (A.N. Dancygier). http://dx.doi.org/10.1016/j.tust.2016.03.009 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
as well as of the contact layer between them. They provide detailed analysis, but only for the specific problem that was analyzed. Thus, at the earlier stages of the design, when various alternatives need to be examined, such as structure geometry or element thicknesses, a theoretical model is preferred. Relatively simplistic models that are considered ‘‘classical” are known in the literature (e.g., Terzaghi, 1959; Newmark, 1964; Marston, 1930; Spangler, 1957). Simplistic models for structures with a rectangular crosssection have also been proposed (Weidlinger and Hinman, 1988; Higgins and Drake, 1995). One of the frequently used shapes of tunnels is the horseshoe cross-section (Szechy, 1973). A model based on a discretecontinuous concept for the prediction of the average dynamic and static pressure acting on a flat roof tunnel has been developed by the authors (Dancygier and Karinski, 1999a,b; Karinski et al., 2003). Unlike the models mentioned above, which refer only to buried structures with a flat roof, this paper proposes a model to assess the response of a lined tunnel with a non-flat, circular-arched roof under a surface static loading. It allows calculating the average contact pressure that acts on structures, such as horseshoeroofed tunnel’s lining. Although derivation of the model is relatively complicated, the resulted solution can be straightforwardly implemented. The paper starts with a description of the model,
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Nomenclature Aa b D Es Esoil E⁄ Effsoil ; Effb;soil Eff ; Effb h H I k(x) k0 ks K, K0 Kmsd KL L Mb, mb M, M0 N, n P Pext Pe0 Pv
cross sectional area of a unit width strip of the arch roof width of the structure depth of burial (from soil surface to the top of the arched roof) Young’s modulus of the structure roof Young’s modulus of the soil-column above the structure soil-column’s equivalent modulus of elasticity free field’s moduli of elasticity above and below the level of the structure floor free-field equivalent moduli of elasticity for a plane strain problem thickness of the roof cross-section total height of the structure’s cross-section moment of inertia of a unit width strip of the arch roof arching coefficient constant of the arching coefficient function side pressure coefficient roof and floor stiffnesses measured roof stiffness equivalent structure transformation factor span of the arched roof and flat floor bending moments caused by the contact pressure and by a vertical unit force acting at the top of the arch 2DOF masses and total masses of the structure roof and floor axial forces caused by the contact pressure and by a vertical unit force acting at the top of the arch roof perimeter external surface load (positive in compression) uniformly distributed load in the model (Pe0 = -Pext) total vertical contact force
followed by derivation of its equations and their solution. Finally, the solution is verified against experimental data and a case study is presented.
2. Description of the model
q qb ~b q R s S Ss U(x)
uniformly distributed load acting on the roof (positive for pressure) uniform pressure without taking into account free-field, ~b þ Eff U ffx ðD þ HÞ qb ¼ q average pressure that acts on the soil under the structure radius of the roof arch coordinate along the roof arch horizontal projection of the area of the roof, which is also equal to the floor area shear force vertical soil-column displacement above the structure
vertical free-field soil displacement Uff (x) Ub absolute mid-floor displacement Ux ðxÞ; Uffx ðxÞ derivatives with respect to ‘‘x” w, w0 mid-roof and mid-floor deflections relative to their supports (walls) W modulus of a unit width strip of the arch roof x depth coordinate y, y0 roof and floor vertical deflection surfaces, relative to their supports (walls) m; mb soil Poisson’s ratios above and below the level of the structure floor. a coefficient in the settlement expression b factor of the arching coefficient exponent csoil soil weight density l 2DOF spring stiffness per unit area /s soil internal friction angle h half of the arch opening angle q soil mass density r(x) axial stress in the soil column s(x) friction traction on the perimeter of the soil column
zero at the free surface). The coefficient of this relation k (‘arching coefficient’) may be either constant (Dancygier and Karinski, 1999a,b; Karinski et al., 2003) or some given function of the depth x (Chacha, 2014). The free field displacement for the case of a uniformly distributed load Pe0 acting on the surface of an infinite half plane is given by Karinski et al. (2003):
2.1. Basic discrete-continuous model The work described herein is an extension of a model that was developed previously by the authors for flat roofs (Karinski et al., 2003). For the clarity of presentation, key parts of the original model are presented in the following text. The model is suitable for depths of burial D, at which ‘arching’ (as explained above) can develop (at least about 15% of the roof span, e.g., see Dallriva and Hall, 1998). It comprises an equivalent structure subjected to a uniformly distributed equivalent load. The buried structure is represented here by an equivalent twodegree-of-freedom (2DOF) system interacting with an equivalent soil-column above it, where its Young’s modulus is denoted here Esoil. At its bottom, the 2DOF system is supported by a semiinfinite elastic medium, Fig. 1a and b. It is assumed that there is a full (perfect) contact between the structure and surrounding soil. The soil at the sides of the structure and far from it is represented in the model by the ‘free-field’ stress and displacement expressions (as shown in Karinski et al., 2003). The shear soil resistance (s) is represented in the model by a vertical friction traction that acts on the soil column perimeter and depends on the soil properties and on the relative displacement between the soil-column U(x) and that of the free field Uff (x) (where x is the depth, which is equal
U ff ðxÞ ¼ U 0 þ
8P qg 2 e0 > < Eff x 2Eff x ; > : PEe0ff x
0 6 x6 DþH 2 qg 1 1 1 1 ff x þ 2 ðD þ HÞ Eff ff ; D þ H 6 x ff x qg ðD þ H Þ Eff
qg
2
2Eb
Eb
Eb
ð1Þ
where H is the total height of the structure’s cross-section, q is the soil mass density, g is the gravitational acceleration and U0 is a constant, which is discussed in the following text. In Eq. (1) mb Þ mÞ Eff ¼ Effsoil ð1þðm1 ; Effb ¼ Effb;soil ð1þmð1 Þð12mÞ Þð12m b
bÞ
are the equivalent free-
field moduli of elasticity of the above and below the level of the structure floor, for a plane strain problem, where Effsoil ; Effb;soil are the corresponding ‘‘real” free field’s moduli of elasticity, and m and mb are the corresponding Poisson’s ratios. Note that Eq. (1) has been derived with the classical sign agreement that a tension force is positive. In the current problem external pressure, acting on the soil surface, Pext is positive in compression (pressure), and therefore, Pe0 = Pext in Eq. (1) and in Fig. 1b. It should also be noted that the proposed model is linear-elastic and is suitable for relatively small deflections and deformations that correspond to a ‘service-state’ condition (as opposed to an ‘ultimate state’ analysis).
A.N. Dancygier et al. / Tunnelling and Underground Space Technology 56 (2016) 211–225
213
Fig. 1. Schemes of a buried structure and the model: (a) scheme of the problem, (b) equivalent discrete-continuous system, (c) loads and deflections, (d) free body diagrams of the 2DOF’s masses.
~b , which acts to the following expression for the average pressure q on the soil under the structure (positive for pressure, Fig. 2a):
2.2. Equivalent structure Consider a lined-tunnel with an arched roof and flat floor having equal spans (L). The top point of the structure’s roof is buried at depth D from the surface, which is loaded by uniformly distributed pressure Pext (positive for pressure). The total height of the tunnel’s cross-section is denoted H (Fig. 1a and c) and the roof radius is denoted R. The model assumes that the structure walls are rigid. Therefore, the following geometrical relations exist:
w þ w0 ¼ UðDÞ Ub ;
Ub ¼ UðDÞ ðw þ w0 Þ
ð2Þ
where w and w0 are the mid-roof and mid-floor deflections relative to their supports (walls), U(D) is the absolute mid-roof displacement (which is equal to the displacement of the bottom of the equivalent soil column), and Ub is the absolute mid-floor displacement, Fig. 1c. The equivalent 2DOF system (Fig. 1b and d) that represents the lining has two (total) masses, M (roof) and M0 (floor), and a spring, whose (equivalent) stiffness l (per unit area) is given in the following text (see Eq. (5)). The displacements of these masses are represented by the absolute displacements U(D) and Ub, described above (Fig. 1b and c). Equilibrium of the bottom mass M0 leads
~b ¼ q
M0 lðw þ w0 Þ gþ KL S
ð3aÞ
where KL is an equivalent structure transformation factor from the real structure to the 2DOF system (see Eq. (7)) and S is the horizontal projection of the area of the roof, which is also equal to the floor area. The settlement Ub is evaluated from a superposition of two vertical displacements (Fig. 2b and c): the free-field displacement at the structure’s base Uff(D + H) and an average displacement of the base area loaded with a uniform pressure qb (without taking ~b þ Eff U ffx ðD þ HÞ, see Fig. 2c). into account free-field, qb ¼ q Hereafter, the index ‘‘x” denotes derivative with respect to ‘‘x”. Note that negative derivative Ux(x) is associated with compression and therefore a negative sign appears in Fig. 2a. It can thus be seen from Fig. 2 that Ub is given by:
Ub ¼ Uff ðD þ HÞ þ a
qb Eb
ð3bÞ
where Eb is an equivalent Young’s modulus of the soil below the structure and Eb =a is its equivalent stiffness. The parameters a and Eb are given in Appendix A.
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Fig. 2. Superposition of displacements and pressures under the structure: (a) total pressure, (b) free-field pressure, (c) additional pressure caused by the structure.
Fig. 3. Schemes of an arch shaped element: (a) free body diagram of the roof, (b) fixed and (c) hinged support connections.
2.3. Equivalent stiffness of an arched roof The equivalent stiffness l has a dimension of force per unit area per displacement (w + w0). It is calculated according to the roof and floor stiffnesses K and K0, defined as the ratios between the total load acting on the roof (or floor) and its mid-span deflection. For a load and mass that are uniformly distributed per unit horizontal area, and considering equilibrium of the 2DOF system e b S M0 g ¼ q S þ Mg, refer to Fig. 1d), the stiffnesses are given (q by:
ðq S þ MgÞ w e b S M0 gÞ ðq S þ MgÞ ðq K0 ¼ w0 w0
K
ð4Þ
where q is a uniformly distributed load acting on the roof (see Fig. 1c). The definition of the 2DOF structure’s stiffness and Eq. (4) yield:
ðq S þ MgÞ ðq S þ MgÞ ðq S þ MgÞ KL ¼ lðw þ w0 Þ S ¼ l S þ K K0 1 KL )l¼1 1 ð5Þ þK S K 0
and once the sum of the roof and floor relative deflections (w + w0) has been calculated, the roof or floor deflection can be separately calculated according to their relative stiffnesses (Eq. (4)):
w¼
ðq S þ MgÞ ðw þ w0 Þ ¼ K 1 þ KK 0
ðw þ w0 Þ w0 ¼ 1 þ KK0
ð6Þ
The above mentioned transformation factor KL (for both load and stiffness) is obtained from equating the works of external forces in the real and equivalent structures, assuming that the maximum deflections of the two structures are equal, and for a mass that is uniformly distributed per unit horizontal area, it is given by (e.g., Szilard, 1974; Biggs, 1964):
RR KL ¼
S
ydS þ
RR
y0 dS
S
Sðw þ w0 Þ
ð7Þ
where y and y0 are respectively the roof and floor vertical deflection surfaces, relative to their supports (walls) in the real structure. Note that w and w0 are the maximum values of the deflection functions y and y0 (see Fig. 1c).
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Eq. (4) is valid for a planar roof (or floor), however when the shape of the roof (or floor) is arched it is also necessary to take into account the side pressure that acts on the roof (Fig. 3a). Consider the circular-arched roof of radius R and span L, related by the opening angle 2h (see Fig. 3a). The side pressure acting on the roof is commonly related to the vertical pressure by a ‘side pressure coefficient’ ks. Thus, the side pressure that acts on the arch is equal here to ks q, where q is the vertical contact pressure. The top point (mid-roof) displacement w can be calculated by the following expression:
Z
w¼
h
h
M b ð/Þ mb ð/Þ Rd/ þ Es I
Z
h
h
Nð/Þ nð/Þ Rd/ Es Aa
ð8Þ
where the bending moments Mb ð/Þ; mb ð/Þ and the axial forces N (/), n(/) are given in Appendix B. Es ; Aa and I are the Young’s modulus, cross sectional area and moment of inertia of a unit width strip of the arch (respectively); the variable angle / is defined in Fig. 3a. The arch stiffness per unit length of the tunnel is equal to the total vertical load (per horizontal unit length) divided by the mid-roof displacement and therefore:
K¼
q 2R sinðhÞ w
ð9Þ
Substitution of the moments and axial forces from Appendix B (Eqs. (B.1)–(B.8)) into Eq. (8), evaluation of the integrals and substitution of w into Eq. (9), yield the following expression for the stiffness (per unit width) K for both fixed and hinged support connections (Fig. 3b and c):
h F t ðh; h=LÞ K ¼ 48Es L F b ðh; h=L; ks Þ
ð10Þ
where h is the thickness of the roof cross-section. For the case of a fixed connection:
( 2 h 2 2 F t ¼ sin ðhÞ½1 þ cosðhÞ h sin ðhÞ½sinðhÞ cosðhÞ þ h L ) 2 2 þ3 2 cos ðhÞ þ h sinðhÞ cosðhÞ þ h 2
(
)
2
F b ¼ 3 sin ðhÞ ð1 ks ÞU1 ðhÞ 12 sinðhÞ½sinðhÞ þ ð1 2 cosðhÞÞ h ( 2 2 L h 4 9ð1 ks Þ U2 ðhÞ þ 4 sin ðhÞ ð1 ks ÞU3 ðhÞ h L ) 3 sinðhÞ ½h sinðhÞ where
þ h sinðhÞ 6 13 cosðhÞ 11 cos2 ðhÞ 3h2 1 2 cos2 ðhÞ ½1 þ cosðhÞ
U2 ðhÞ ¼ 6 cos3 ðhÞ 1 h2 4 6 cosðhÞ þ 3h2 cosðhÞ
þ 8 6h2 cos2 ðhÞ 4 cos4 ðhÞ 2h sinðhÞ þ h sinðhÞ cosðhÞ½3 þ 11 cosðhÞ þ 3h2 U3 ðhÞ ¼ h sinðhÞ 1 þ cosðhÞ þ cos2 ðhÞ sin2 ðhÞ ½2 þ cosðhÞ The stiffness per unit length of the tunnel’s flat floor is presented in the following form (see Karinski et al., 2003):
K ¼ 32Es
3 h 1 L 5 4aM
ð13Þ
where aM is equal to 1 and 0 for a fixed and hinged roof (respectively). It can be shown that Eq. (10) converges to Eq. (13) when h ! 0 while h R ! L. 3. Model’s equations and solution
ð11aÞ
As noted above, the vertical friction traction s(U(x), x) may be represented by the relative displacement between the soilcolumn U(x) and that of the free field Uff(x) in the following form (see Fig. 1):
h
i
ð14Þ
3
F b ¼ 3ð1 ks ÞF 1 ðhÞ þ 9h cos ðhÞðks 5Þ 9h sin ðhÞð1 þ 3ks Þ 2 L ð1 ks ÞF 2 ðhÞ þ 9h2 ð3 þ ks Þ þ 9 h 2 ( h þ4 ð1 ks ÞF 3 ðhÞ þ ð1 þ 2ks Þh sinðhÞ cos ð2hÞ cos4 ðhÞ L ) 2
U1 ðhÞ ¼ 2 sin2 ðhÞ 2 cos2 ðhÞ þ 5 cosðhÞ
sðUðxÞ; xÞ ¼ kðxÞ UðxÞ U ff ðxÞ 2
ð12bÞ
þh2 ð2 þ ks Þ þ ð4 ks Þh2 cos4 ðhÞ h2 cos2 ðhÞð5 þ ks Þ
ð11bÞ
In Karinski et al. (2003) the coefficient k(x) was taken as constant while in Chacha (2014) it was introduced as follows: – for a relatively shallow depth of burial (yet, sufficient for the development of arching):
kðxÞ ¼ k0 ebx
ð15Þ
– otherwise: where
F 1 ðhÞ ¼ ½5 cosðhÞ 8 1 þ cos4 ðhÞ þ cosðhÞ h i 3 16 cosðhÞ 10 cos2 ðhÞ þ 9h sin ðhÞ h i 2 þ 6h2 cosðhÞ cos3 ðhÞ sin ðhÞ
kðxÞ ¼
0 6 x < D nL k0 ; k0 ebðxDþnLÞ ; D nL 6 x 6 D
ð16Þ
where k0 and b are constants, L is the roof span and n is a parameter that must be obtained experimentally and depends on the soil properties. Note that Eq. (15) with b ¼ 0 converges to the particular case of k(x) = k0 = constant. The subsequent model verification and numerical analysis (given in Sections 4 and 5) refer only to the case of Eq. (15). Thus, the investigation of the parameter n is out of the scope of this paper.
2
F 2 ðhÞ ¼ ½cosðhÞ 4 sin ðhÞ 2h2 cosðhÞ½1 þ cosðhÞ þ h sinðhÞ½1 þ 5 cosðhÞ þ h2 F 3 ðhÞ ¼ h cosðhÞ cos ð2hÞ cos4 ðhÞ ½h sinðhÞ h cos5 ðhÞ For the case of a hinged connection:
(
3 F t ¼ sin ðhÞ 3 cosðhÞ 2 cos2 ðhÞ þ 2 cosðhÞ þ 1 h 3 sinðhÞð1 þ cosðhÞÞ þ h ) 2 h 2 þ sin ðhÞ½1 þ cosðhÞ½hþsinðhÞ cosðhÞ L
3.1. Equilibrium equations and boundary conditions The current problem is uniaxial with eyy ¼ ezz ¼ 0 and exx = dU (x)/dx (one-dimensional confined problem). It is well-known that in this case the axial stress is given by:
ð12aÞ
rxx rðxÞ ¼ E exx ¼ E Ux ðxÞ;
ð17Þ
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– For k = constant:
UðxÞ ¼ U ff ðxÞ þ A cosh ðkxÞ where A ¼ qgD E
2
al=E 1þal=Eb KL
ð24Þ
2 l MþEE la M0 þP KL EE la la EE þaP=D ðH 1 a H DÞ 2a=DðDÞ b b b , l K kD sinh ðkDÞþ a cosh ðkDÞ L
a=D
e0 ; M ¼ qM KL ; M0 ¼ qMDS0 ; P ¼ 1 qPgD P 1 andUff ðxÞ DS
la ¼ is
given by Eq. (5). – For k modeled by Eq. (15), Eq. (20) becomes:
U xx ðxÞ k2 ebx UðxÞ ¼
qg E
k2 ebx U ff ðxÞ
ð25Þ
The general solution of the homogeneous equation corresponding to Eq. (25) is as follows:
Fig. 4. Free body diagram of soil-column differential element.
where Ux ðxÞ ¼ dUðxÞ and E⁄ is the soil-column’s equivalent modulus dx of elasticity (‘confined modulus’), which is given by
UðxÞ ¼ AI0 ðzðxÞÞ þ BK 0 ðzðxÞÞ
ð26Þ
mÞ . E ¼ Esoil ð1þðm1 Þð12mÞ
where A and B are constants, I0(z), K0(z) are modified Bessel functions of the first and second kind (respectively) of order 0,
The equilibrium equation of a differential soil column element is given by (refer to Fig. 4):
and zðxÞ ¼ 2keb 2 .
S dr P dx s ¼ qg S dx
ð18Þ
where P and S are the roof perimeter and the horizontal projection of the area of the roof, which is also equal to the floor area (see above). Substitution of Eq. (17) into Eq. (18) yields the following equation of equilibrium in terms of the soil column displacement, U(x):
i P kðxÞ h qg ff U xx ðxÞ UðxÞ U ðxÞ ¼ S E E
ð19Þ
bx
The particular solution of Eq. (25) for homogeneous soil above the R zðnÞÞhðnÞ R ÞhðnÞ e dn I0 ðzðxÞÞ K 0 ðWðnÞ dn, structure is given by UðxÞ ¼ I0 ðzðnÞ WðnÞ bzðxÞ ff bzðxÞ qg qg 2 P e0 where hðxÞ ¼ k 2 U ðxÞ E ¼ k 2 U 0 þ Eff x 2Eff x qEg, WðxÞ ¼ bzðxÞ ½I0 ðzðxÞÞK 1 ðzðxÞÞ þ K 0 ðzðxÞÞI1 ðzðxÞÞ. 2 general solution of the equation is as follows:
Z Z
where Uxx ðxÞ ¼ d dxUðxÞ 2 . Eq. (19) can be re-written as follows: ff
2
U xx ðxÞ k f ðxÞ½UðxÞ U ðxÞ ¼
qg E
ð20Þ
qffiffiffiffiffiffiffi where k ¼ PS kE0 . For relatively deep structures Eq. (16) yields 1; 0 6 x < D nL f ðxÞ ¼ and for relatively shallow buried ebðxDþnLÞ ; D nL 6 x 6 D structures Eq. (15) yields f ðxÞ ¼ ebx , and when the arching coefficient is modeled as a constant, which is equal to k0 ; f ðxÞ ¼ 1. The boundary condition at the soil surface corresponds to x = 0 in Fig. 1b and is given by the following equation (refer to Eq. (17)):
E Ux ð0Þ ¼ Pe0
ð21Þ
Assuming full contact between the soil and the structure implies that the uniformly distributed load acting on the roof is given by q ¼ E U x ðDÞ. Therefore, the boundary condition at bottom of the soil column (x = D in Fig. 1b), in view of Eq. (5), is :
lðw þ w0 Þ ¼
K 0 ðzðnÞÞhðnÞ dn WðnÞ
I0 ðzðxÞÞ
Mg E Ux ðDÞ KL S
ð22Þ
Eqs. (2) and (3b) yield: lðw þ w0 Þ ¼ l½UðDÞ Ub ¼ h i l UðDÞ Uff ðD þ HÞ a qEb . Substitution of qb (see Fig. 2c), in view b
of Eq. (3a), yields:
0
1 UðDÞ U ff ðD þ HÞ Ea MS0 g þ Pe0 qg ðD þ HÞ b A l@ 1 þ El KaL b Mg E U x ðDÞ K L ¼ S
ð23Þ
The solution of Eq. (20) with the boundary conditions given in Eqs. (21) and (23) depends on the modeling of the arching coefficient k(x):
the
I0 ðzðnÞÞhðnÞ dn WðnÞ
UðxÞ ¼ AI0 ðzÞ þ BK 0 ðzÞK 0 ðzðxÞÞ þ
2
Therefore,
ð27Þ
Thus, the stress at the soil column is obtained by taking the first derivative of U(x) (multiplied by E ) and substituting into the boundary conditions (21) and (23), which yields the following expressions for the constants:
A¼
B¼
X
h i; Þ I0 ðzðDÞÞ þ E bzðDÞ I1 ðzðDÞÞj þ KI11ððzð0Þ K 0 ðzðDÞÞ E bzðDÞ K 1 ðzðDÞÞj zð0ÞÞ 2 2 Pe0 I1 ðzð0ÞÞðA Rð0ÞÞ K 1 ðzð0ÞÞTð0Þ kE ; K 1 ðzð0ÞÞ
ð28Þ where I1(z), K1(z) are modified Bessel functions of the first and second kind (respectively) of order 1, j ¼ K L þ lEa , X ¼ b nh i Mg þ E bzðDÞ ½I1 ðzðDÞÞRðDÞ þ K 1 ðzðDÞÞTðDÞ j þ I0 ðzðDÞÞRðDÞ K 0 ðzðDÞÞ 2 S h i h i P kI ðzð0ÞÞRð0ÞþkK 1 ðzð0ÞÞTð0Þþ Ee0 Þ Þ qg ðD þ HÞ Ea þ ðDþH þ 1 TðDÞ þ U 0 þ Ea MS0 g þ P e0 Ea þ ðDþH kK 1 ðzð0ÞÞ Eff 2Eff b b b h i R R bzðDÞ K 0 ðzðDÞÞ þ E 2 K 1 ðzðDÞÞj g; TðxÞ ¼ I0 ðzðxÞÞYðxÞdx; RðxÞ ¼ K 0 ðzðxÞÞYðxÞdx,
and YðxÞ ¼ bzðxÞ 2
bzðxÞ 2
k
P e0 qg x ff x2 Eff 2E
U0 þ
qg
þE
½I0 ðzðxÞÞK 1 ðzðxÞÞþK 0 ðzðxÞÞI1 ðzðxÞÞ
.
With the above expressions for the constants A and B, the solution of Eq. (20) is given by Eq. (27). As stated above, this solution, as well as the subsequent model verification and numerical analysis, deal with the case of an arching factor that varies exponentially with the soil depth (x), Eq. (15). One can see that for k = constant Eq. (24) is the closed form solution while for k that varies with depth the analytical solution that is given in Eq. (27) includes some integrals of Bessel functions that are simple for calculation. It should be noted that these solutions allow calculation of a relative displacement U(x) U0. When there is a depth, at which the displacement is known (commonly, the case of a bedrock at a known depth or a rigid floor
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of an experimental setup) then U0 can be explicitly determined. Even so, for the calculation of the average contact pressure one needs the derivative of these solutions, which is does not depend on the constant U0, i.e.:
q ¼ E U x ðDÞ
ð29Þ
Similarly, these solutions together with Eqs. (2) and (6) yield the roof deflection, relative to its supports (because the term U0 cancels out). 4. Verification The solution for the equivalent discrete-continuous system, given in Section 3 (Eqs. (24) or (27)) with the application of an equivalent stiffness given in Section 2.2 (Eqs. 4, 10-13) constitute a model for the assessment of the contact pressure on an arched roof of a lined tunnel. In order to verify the proposed model, its predictions have been compared with laboratory experimental results conducted by Dallriva and Hall (1998) of a semi-circular concrete arch buried in sand, where a uniform external pressure was applied on the backfill surface. The depth of burial (D) was equal to 19 cm and the arch span (L), width (b), and thickness (h), were 111.7, 91.4 and 5.1 cm, respectively (Fig. 5). According to the report, the arch specimen was assumed to be a fixed-fixed one. The specimen in these experiments was supported by two massive blocks that acted as a rigid ‘‘floor” (w0 = 0). The test included three cycles of loadingunloading-reloading, where the third one was applied after a significant structural damage (i.e., decreasing capacity) was observed. Therefore, the following calibration and verification correspond to the results reported from the first two cycles. Furthermore, the
217
reported load-displacement curve corresponding to the second cycle starts from a residual mid-span deflection. Consequently, for the purpose of the comparison with the current linear model, this cycle has been shifted to the origin. Normal contact pressures at nine discrete points along the arch, as well as the mid-point deflection were measured in the experiment. The pressure gages were located along the arch as shown in Fig. 5. Two types of soil-structure contact conditions were studied in this work: with and without a Teflon interface layer. The following comparison was performed with the experiment that did not include this layer because the current model refers to a state of full soil-structure contact. 4.1. Calibration of the side pressure coefficient ks Calibration of the side pressure coefficient ks has been obtained from comparison between the measured total vertical contact load-to-mid-span deflection ratio, Kmsd, and the stiffness (K) of the equivalent fixed-fixed structure (Eqs. 10,11) with h ¼ 90 ), i.e.:
h F t ðp=2; h=LÞ L F b ðp=2; h=L; ks Þ
K msd ¼ 48Es
ð30Þ
Solving this equation for ks yields:
ks ¼
Ak þ Bk Ck
ð31Þ
h2 i
2 2 where Ak ¼ 16 KEs b hL hL2 p4 þ 3 p4 2 ; Bk ¼ 8 þ 32p 32p 1 þ msd
2 2 2 2 2 3 hL 2 p2 þ p4 4 þ 23p hL2 ðp 1Þ; C k ¼ 8 þ 32p p2 3 3 hL2 p2 þ p4 4 þ p h2 ðp 4Þ. 3 L2
In these expressions, Es is Young’s modulus of the struc-
ture (taken by Dallriva and Hall, 1998, as equal to 30,000 MPa). Note that for a thin-wall arch (i.e., h/L 1) the above equations yield:
Es b h p2 8 K msd L3 p2 þ 4p 16 3
ks ¼ 1 16
ð32Þ
The measured stiffness Kmsd has been evaluated as follows:
Fig. 5. Schematic description of the setup and measurements in Dallriva and Hall (1998) experiment.
The arch was divided into 8 segments between the pressure gauges (see Fig. 5).
The normal contact force acting at each segment (Pi) was calculated, assuming a linear distribution of the contact pressure between the measuring points. The force at each segment is considered to be applied at the centroid of the pressure distribution, as shown in Fig. 6a.
The total vertical force (Pv) is calculated as the sum of the vertical components (Pvi) of Pi (see Fig. 6b).
Kmsd has been evaluated by dividing Pv by the measured midspan deflection (w) as explained in the following text.
Fig. 6. Schematic description of the calculation of the total vertical force acting on the arch.
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(a) Young's Modulus
Fig. 7. Force-deflection curve according to the experimental results (Dallriva and Hall, 1998).
Fig. 7 shows the measured Pv-w curve of the first experimental cycle. Because the proposed model is linear, calibration of the side pressure coefficient (ks) was performed according to a stiffness Kmsd = 805,593 kN/m, which was determined by a linear trendline of the initial segment of this curve (O-A in Fig. 7). Calculating ks from Eq. (31) with this Kmsd yields a value of 0.428. Note that using the simplified solution (Eq.(32)) yields a ks value of 0.366 (14% difference). It is also interesting to note that both these values are close to the ‘‘at-rest” value of the side pressure coefficient ks ¼ 1 sin ð/s Þ ¼ 0:377 for an internal friction angle /s ¼ 38:5 that was reported by Dallriva and Hall (1998). 4.2. Other soil parameters Dallriva and Hall (1998) reported a soil weight density csoil and Poisson’s ratio m of 16.6 kN/m3 and 0.3, respectively. The soil Young’s modulus above and below the structure (Esoil, Eb) and the arching coefficient parameters k0 and b (see Eq. (15)) were calibrated. The calibration was performed according to the average 2.015-mm mid-span deflection of two load cycles (first and shifted second; see above) under an external pressure of 1.96 MPa. Fig. 8 shows the variation of the calculated mid-roof deflection (solid lines) with varying values of these parameters, as well as the measured deflection (dotted line). It is important to note that the values shown at the X-axis in Fig. 8a are those of the sand Young’s moduli (and not of their equivalent values). It is evident from Fig. 8 that the calibrated values of the soil parameters are: Esoil = Eb = 200 MPa, k0 = 1175 MN/m3 and bD = 0.5 (where D is the depth of burial). 4.3. Comparison with the test results Fig. 9 shows comparisons of the model with the reported external load-deflection (at mid-span). The model’s prediction was calculated with the above calibrated soil parameters for the whole reported range of the external pressure, for both loading cycles mentioned above, where the calibration point is marked in the figure. Fig. 9a shows a comparison based on the initial stiffness corresponding to the line O-A in Fig. 7 (K = 805,593 kN/m). The figure shows good agreement of the model with the measured results up to an external pressure somewhat larger than 2 MPa. Above this pressure the increase in the predicted deflections is lower than the measured ones. This difference is related to the non-linear Pv-w curve in Fig. 7, based on which, the stiffness (K) has been calculated. This non-linearity is caused by the damage development in the concrete arch. Therefore, under relatively high external pressure, a lower stiffness is more realistic. Furthermore, the 2-mm deflection under an external pressure of 2 MPa corresponds to point ‘‘A” in Fig. 7, where a decrease of the stiffness is observed. Therefore, we used an enhanced model with an initial stiffness K = 805,593 kN/m up to point ‘‘A” in Fig. 7 and with a lower stiffness beyond this point. This stiffness has been evaluated
Fig. 8. Calibration of soil parameters based on Dallriva and Hall (1998).
in two ways: tangent stiffness (line A-B in Fig. 7, K = 345,363 kN/m) and secant stiffness (line A-C, K = 246,480 kN/m). Fig. 10 shows a comparisons of the enhanced model prediction of the average contact pressure (see Eq. (29)) with that obtained from the measured data (Pv/L, refer to Fig. 6b). Because the researchers reported malfunction of the pressure gauges during the second cycle, the comparison was carried out only with results from the first cycle. It is evident from the figure that there is a good agreement of the model with the test results, where the model with the secant stiffness shows a somewhat better agreement. It is also interesting to note that the model’s prediction of both contact pressure and deflection is conservative with respect to the experimental data. Additionally, one can see that both experimental and predicted results indicates contact pressure that is lower than the external pressure, i.e., positive arching. 5. Case study The solution for the equivalent discrete-continuous system, given in Section 3 (Eqs. (24) or (27)) has been applied to investigate the following problem. Consider a lined reinforced concrete tunnel, buried at a depth of D = 10 m under a surface external pressure Pext = 100 kPa. The tunnel has a flat rigid floor and walls, and an arched roof with an angle h (Fig. 3a) that varies from zero (limiting case of a flat roof) to 90° (semi-circular), Fig. 11. Note that for a given span (L) the radius of the roof arch varies with h as follows: L R ¼ 2 sin . It is further assumed that the roof has a fixed-fixed h
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Fig. 9. Comparison of mid-span deflections.
Fig. 10. Comparison of average contact pressure.
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Fig. 11. Structures of the case study.
Fig. 12. Normalized contact pressure for various spans (L) —— h = 250 mm – – – h = 150 mm.
scheme (refer to Fig. 3b). Young’s modulus and Poisson’s ratio of the structure were taken as 30,000 MPa and 0.15. The total height of the structure is 4 m (Fig. 11). Three spans (L), 2.5, 5 and 8 m, and two roof thicknesses (h), 150 and 250 mm, were studied. The parameters of the soil and arching coefficient were chosen as follows: csoil ¼ 16:87 kN/m3, Esoil = Eb = 75 MPa, m ¼ mb ¼ 0:3, k0 = 175 MN/m3 and bD = 0.4. The side pressure coefficient was taken as ks = 0.5 (corresponding to its ‘‘at-rest” value for an internal friction angle of 30°). 5.1. Contact pressure Fig. 12 shows the variation of the calculated average contact pressure (Eq. (29)) with the opening angle. The pressure has been
normalized with respect to the ‘free field’ pressure at the roof top point level, Pff ¼ Pext þ csoil D. One can see that there is a certain angle at which the contact pressure has a maximum value. This maximum is more pronounced at the longer span and smaller roof thickness. Note that the angle at which there is a maximum contact pressure is different for each roof span and stiffness. Fig. 12 also shows the development of positive arching, where for a given angle theta, longer spans L (i.e., lower roof stiffness) lead to lower contact pressure q. Noting that the arch stiffness K is also a function of h (Eqs. 10,11), the analysis shows that the angles at which the contact pressure has a maximum, coincide with the angles at which there is a maximum value of K. An example that illustrates this correspondence for L = 5 m and h = 150 mm is presented in Fig. 13.
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221
Fig. 13. Normalized contact pressure (– – –) and stiffness K (——) for L = 5 m and h = 150 mm.
Fig. 14. Normalized contact pressure vs. stiffness K for L = 5 m and h = 150 mm.
Fig. 14 shows peak values of the contact pressure versus peak values of the roof stiffness. Angles at which these maxima occur are also marked in the figure. It is evident from Fig. 14 that this relation is monotonic.
Presumably (see Fig. 12), the best angle for the design of the roof is the one that yields minimum contact pressure, i.e., flat roof (h ¼ 0). Evidently, however, this is not the case. The explanation for this outcome is the above shown relation between the roof
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Fig. 15. Bending moment (a) and shear force (b) for various opening angles (Theta) of the arch for L = 5 m and h = 150 mm (s is the ordinate along the arch neutral axis).
stiffness and the contact pressure (Fig. 13). Hence, a minimum contact pressure does not necessarily leads to minimum internal forces and moments, which control the design of the arch. Thus, in order to obtain the optimal opening angle of the arch the internal forces and moments should also be studied.
5.2. Internal forces Consider the bending moment (Mb), axial and shear forces (N and Ss), as well as the axial stresses at the extreme fibers of the cross-section, along the axis of the roof arch. The extreme stresses are given by MWb þ NA, where W and Aa are the crosssection modulus and area of a unit width strip of the arch roof,
‘‘+” and ‘‘” correspond to bottom and top fibers, respectively. The calculations by the present model were carried out for the loading shown in Fig. 3a and static schemes shown in Fig. 3b. Fig. 15 demonstrates examples of the bending moment and shear force diagrams for various opening angles of the arch for L = 5 m and h = 150 mm. The ordinate ‘‘s” in this figure denotes the coordinate along the arch (s = 0 at the top point). It is evident from the figure that although the contact pressure that acts on the flat roof is lower than the one that acts on the arched roof (see Fig. 13), the internal shear force and bending moment in the flat roof are significantly larger than those in the arched roof. For example, refer to Theta = 0 and 30–40° in Fig. 15. A similar finding has been obtained for the other cases (L = 2.5, 5 and 8 m, h = 150 and 250 mm).
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223
Fig. 16. Maximal values of the bending moment, shear force and axial stresses at the extreme fibers of the cross-section for L = 5 m and h = 150 mm.
Table 1 Optimal opening angles. L (m)
h (mm)
Slenderness L/h
h optimal [°] Shear force
Moment
Stress
2.5
150 250
16.7 10.0
49 61
58 70
63 73
5
150 250
33.3 20.0
36 45
42 53
48 59
8
150 250
53.3 32.0
28 36
33 43
39 48
5.3. Optimal opening angle According to this analysis one can obtain an optimal value for the opening angle (hÞ that will yield minimal internal forces and stresses. An example for evaluation of such optimal angel is shown in Fig. 16 for the case analyzed in Fig. 15 (for L = 5 m and
h = 150 mm). The figure shows maximal absolute values of the bending moment, shear force and axial stresses at the extreme fibers of the cross-section, for each opening angle. It can be seen in Fig. 16 that for the present case the optimal opening angle ranges between 35 and 45°: the shear force is minimal at 35° while the bending moment and normal stresses are minimal at 45°. Calculated optimal opening angles for all studied cases are given in Table 1. The table shows that as the roof slenderness increases, the optimal opening angle decreases. This finding is illustrated in Fig. 17. It is interesting to note that the range of optimal angles corresponds to the same range at which the contact pressure and the stiffness are maximal (refer to Figs. 16 and 13). 6. Conclusions A model to analyze the response of an arched roof of a tunnel lining under a surface static loading is presented. The model
Fig. 17. Optimal opening angles (filled markers denote the case shown in Fig. 16).
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simulates the soil and structure behavior under ‘service-state’ conditions and it is suitable for depths of burial at which ‘arching’ can develop. It enhances a previous model by the authors, which is based on a discrete-continuous concept: the buried structure is represented in the model by an equivalent 2DOF system interacting with an equivalent, continuous soil-column above it. The current enhanced model takes into consideration the roof curvature. The proposed 2DOF system’s stiffness includes the influences of the soil side pressure as well as the arched geometry of the roof (i.e., the span and opening angle). At the bottom, the 2DOF system is supported by either a semi-infinite elastic medium or by a rigid base. Opposite to numerical approaches (such as finite elements), the current model presents an analytical solution, which is a closedform for a constant arching coefficient (Eq. (24)). For a varying arching coefficient, it requires rather simple calculations of integrals of Bessel functions (Eqs. (27), (28)). Thus, the solution can be straightforwardly implemented by using only these (1 or 2) equations together with Eq. (29) for the evaluation of the average contact pressure. For the case of zero curvature the analytical solution for the mid-roof deflection and average contact pressure that has been derived converges to the solution of a flat roof. The case of a relatively shallow buried structure has been calibrated and then verified against published experimental results from a test of a semi-circular buried arch. A case study shows that there is a certain opening angle of the roof at which the contact pressure has a maximum value, which is more pronounced at a longer span and smaller roof thickness. This angle coincides with the angle at which there is also a maximum value of the roof stiffness. However, it is also shown that approximately at this opening angle, the internal moment, shear force and normal stresses are minimal. It is therefore concluded that the average contact pressure is not necessarily the most important criterion for a design of an optimal shape of the roof, which should be determined according to this angle. It is further shown that as the roof slenderness increases this optimal angle decreases. Acknowledgments This work was supported by a joint grant from the Centre for Absorption in Science of the Ministry of Immigrant Absorption and the Committee for Planning and Budgeting of the Council for Higher Education under the framework of the KAMEA Program and by the Ministry of Construction and Housing. The research grants are greatly appreciated.
The average settlement under the floor, d (Fig. 2c), is obtained by:
d¼
1 L
Z
L=2
L=2
For the case of a long conduit with a flat floor of span L the relative vertical settlement, V(x,y) (absolute settlement minus the free field displacement) is obtained from the plane strain solution of a half plane loaded by a uniformly distributed load qb over a span L (see Fig. 2c). It is given by (e.g., Nowacki, 1976):
Eb V ðx; yÞ 1 þ mb ( !) qb r2 L r1 r2 ¼ ð1 2mb Þxðh2 h1 Þ þ 2ð1 þ mb Þ y ln ln 2 p r1 2 R0 ðA:1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x where r 1 ¼ x2 þ ðy þ L=2Þ2 ; r2 ¼ x2 þ ðy L=2Þ2 ; h1 ¼ arctan yþL=2 ; h2 ¼ x arctan L=2y , R0 is a constant obtained from boundary conditions
(and L ln(R0) is a rigid body term) and Eb ; mb are the modulus of elasticity and Poisson’s ratio of the soil under the structure.
1 m2b ½1 þ 2Lnðv=2Þ Eb p
ðA:2Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12mb d0 arctan L 1mb 2L d0 d0 and d0 has to be evaluated where v ¼ 1 þ L e as the depth at which the free field settlement and the settlement caused by the external load are similar. Usually a depth equal to about 5L may be suitable for the value of d0. The expression (A.2) may be re-written as follows:
da
q Eb
ðA:3Þ
where Eb ¼ Eb =ð1 m2b Þ and a ¼ 1 þ 2Ln v2 pL . Appendix B. Axial force and bending moment in the arched roof The following notations are used in this appendix: Es, Aa and I – Young’s modulus, cross sectional area and moment of inertia of a unit width strip of the arch (respectively); R, 2h – radius and opening angle of the arch (see Fig. 3a); q - vertical load per horizontal unit length; the variable angle / is defined in Fig. 3a, where h 6 / 6 h. For a hinged support connection (see Fig. 3c): The bending moment caused by the load q is as follows:
Mb ð/Þ ¼
qR2 M t1 ðh; ks ; /Þ M b1 ðhÞ 6
ðB:1Þ
where, M t1 ðh; ks ; /Þ ¼ R2 Es Aa ð1 ks Þu1 ðh; /Þ þ Es Iu2 ðh; ks ; /Þ; Mb1 ðhÞ ¼
EIðsinðhÞcosðhÞ þ hÞ þ R2 Es Aa h þ 2hcos2 ðhÞ 3 sinðhÞcosðhÞ ; u1 ðh; /Þ ¼ 2 2 ð3 sinðhÞ cosðhÞ hÞ þ 6hcos ðhÞ cos ðhÞ cosð/Þð1 þ 6h cosð/Þþ 7 sinðhÞÞ þ 4 sinðhÞ cosð/Þ þ 2cos3 ðhÞð3h cosð/Þ sinðhÞÞ cosðhÞ ð4 sinðhÞ þ 3h cosð/ÞÞ, u2 ðh; ks ; /Þ ¼ ð1 ks Þ cos2 ðhÞð4 sinðhÞ cosð/Þþ 3hÞ 3cos2 ð/ÞðsinðhÞ cosðhÞ hÞ sinðhÞcos3 ðhÞ þ 4 sinðhÞð1 þ 2ks Þ ðcosðhÞ cosð/ÞÞ. For the particular case of half a circle (h ¼ p/2):
Mb1 ¼ Es I þ R2 Es Aa p=2;
u1 ¼ ð3 p=2Þ þ 4 cosð/Þ;
u2 ¼ ð1 ks Þ3 cos ð/Þp=2 4 cosð/Þð1 þ 2ks Þ: 2
The bending moment caused by a vertical unit force acting at the top of the arch mb ð/Þ is given by:
mb ð/Þ ¼
Appendix A. Calculation of the parameters a and Eb
Vð0; yÞdy ¼ qb L
R mt1 ðh; /Þ 2 M b1 ðhÞ
ðB:2Þ
where, for 0 6 / 6 h : mt1 ðh; /Þ ¼ R2 Es Aa u3 ðh; /Þ þ Es Iu4 ðh; /Þ; u3
ðh; /Þ ¼ sinð/Þ h 1 þ 2cos2 ðhÞ 3 sinðhÞ cosðhÞ þ cosð/Þð2h cosðhÞ
2 sinðhÞ þ 3cos ðhÞ 2 cosðhÞ 1 Þ þ h sinðhÞ þ 2ðcosðhÞ 1Þ cosðhÞ, u4 ðh;/Þ ¼ ðcosðhÞsinðhÞ þ hÞsinð/Þ þ sinðhÞcosð/ÞsinðhÞ þ hsinðhÞ, and
for h 6 / 6 0 : mt1 ðh;/Þ ¼ sinðhÞ R2 Es Aa u5 ðh; /Þ þ Es Iu6 ðh; /Þ ; u5 ðh; /Þ ¼
cosð/Þ 2h sinðhÞ þ 3cos2 ðhÞ 2 cosðhÞ 1 sinð/Þ 2hcos2 ðhÞ þ h þ 3 sinðhÞÞ þ sinðhÞ þ ðcosðhÞ 1Þ2 cosðhÞ; u6 ðh; /Þ ¼ cosð/Þsin ðhÞ sinð/ÞðsinðhÞ cosðhÞ þ hÞ þ h sinðhÞ. The axial force caused by the load q is as follows: 2
Nð/Þ ¼
qR Nt1 ðh; ks ; /Þ 6 M b1 ðhÞ
ðB:3Þ
where Nt1 ðh; ks ; /Þ ¼ R2 Es Aa u7 ðh; ks ; /Þ þ Es Iu8 ðh; ks ; /Þ; u7 ðh; ks ; /Þ ¼
ð1 ks Þcosð/Þ 6cosð/Þ h þ 3sinðhÞcosðhÞ 2hcos2 ðhÞ þ þð7sinðhÞ cos2 ðhÞ 4 sinðhÞ 3h cosðhÞ cos ð2hÞÞ þ þ6ð3 sinðhÞ cosðhÞ h
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225
2h cos2 ðhÞÞ; u8 ðh; ks ; /Þ ¼ ð1 ks Þ ðsinðhÞ cosðhÞ þ hÞ6 cos2 ð/Þ 4 sinðhÞ
h i 2 2 2 sin ðhÞ h sinðhÞ cosðhÞ h2 þ 6 h2 þ 2 sin ðhÞ þ h sinðhÞ cosðhÞ ;
cos2 ðhÞ cosð/Þ þ 4 sinðhÞ cosð/Þð1 þ 2ks Þ 6ðsinðhÞ cosðhÞ þ hÞ. For the particular case of half a circle (h ¼ p/2):
u18 ðh; ks ; /Þ ¼ ð1 ks Þ cosð/Þ½sinðhÞ cosðhÞð4 cosðhÞ 6 cosð/ÞÞ 6h cosð/Þ 4 sinðhÞ cosð/Þð1 þ 2ks Þ þ 6ðsinðhÞ cosðhÞ þ hÞ.
For h ¼ p/2: u17 ¼ ð1 ks Þ cosð/Þ p=2 þ 6 cosð/Þ 2 p2 =4 þ 2 6 p =4 þ 2 , u18 ¼ 3pð1 ks Þ cos2 ð/Þ 4 cosð/Þð1 þ 2ks Þ þ 3p.
u7 ¼ ð1 ks Þ cosð/Þ½3 cosð/Þp 4 3p=2 þ 6 p=2; u8 ¼ 3ð1 ks Þp cos2 ð/Þ þ 4 cosð/Þð1 þ 2ks Þ p: The axial force caused by a vertical unit force acting at the top of the arch n(/Þ is given by:
nð/Þ ¼
1 nt1 ðh; /Þ 2 Mb1 ðhÞ
ðB:4Þ
6 / 6 h : nt1 ðh; /Þ ¼ Es Iu9 þ R2 Es Aa u10 ; u9 ðh; /Þ ¼
2 ðcosðhÞ sinðhÞ þ hÞ sinð/Þ þ cosð/Þ sin ðhÞ, u10 ðh; /Þ ¼ sinð/Þ h 1 þ 2cos2 ðhÞÞ 3 sinðhÞ cosðhÞÞ þ cosð/ÞðcosðhÞð2h sinðhÞ 2 þ 3 cosðhÞÞ 1Þ, and where,
for
0
for h 6 / 6 0 : u9 ðh; /Þ ¼ sin ðhÞ cosð/Þ sinð/ÞðcosðhÞ sinðhÞ þ hÞ; u10 ðh; /Þ ¼ cosð/Þ½cosðhÞð3 cosðhÞ 2 þ 2h sinðhÞÞ 1 þ sinð/Þð3 sinðhÞ cosðhÞ h 2hcos2 ðhÞÞ. For a fixed support connection (see Fig. 3b): 2
M b ð/Þ ¼
qR2 M t2 ðh; ks ; /Þ M b2 ðhÞ 12
ðB:5Þ
where, M t2 ðh; ks ; /Þ ¼ R2 Es Aa ð1 ks Þu11 ðh; /Þ þ Es Iu12 ðh; ks ; /Þ; Mb2 ðhÞ ¼ h i 2 R2 Es Aa hðsinðhÞ cosðhÞ þ hÞ þ 2sin ðhÞ þ Es IhðsinðhÞ cosðhÞ þ hÞ;
u11 ðh; /Þ ¼ 2 cosð/Þf2 cosðhÞ þ cos ð2hÞðh sinðhÞ cosðhÞÞ þ 3 cosð/Þ
h
i
2 hðsinðhÞ cosðhÞ þ hÞ 2sin ðhÞ g cos2 ðhÞ cos2 ðhÞ þ 7 3hðh þ sin ð2hÞÞ þ 8, u12 ðh; ks ; /Þ ¼ ð1 ks Þ 6hcos2 ð/Þðh þ sinðhÞ cosðhÞÞ 3h2 6h sinðhÞ cosðhÞ 8h sinðhÞcos2 ðhÞ cosð/Þ þ 8ð1 þ 2ks Þ 5cos4 ðhÞ ðh sinðhÞ cosð/Þ 1Þ þ cos2 ðhÞð13 þ 11ks Þ.
For h ¼ p/2: M b2 ¼ R2 Es Aa p2 =4 þ 2 þ Es Ip2 =4; u11 ¼ 2 cosð/Þ
2 3 cosð/Þ p =4 2 p=2 3p2 =4 þ 8, u12 ¼ ð1 ks Þ 6h cos2 ð/Þ
p=2 3p2 =4 þ 8ð1 þ 2ks Þðp=2 cosð/Þ 1Þ. mb ð/Þ ¼
R mt2 ðh; /Þ 2 Mb2 ðhÞ
ðB:6Þ
0 6 / 6 h : mt2 ðh; /Þ ¼ R2 Es Aa u13 ðh; /Þ þ Es Iu14 ðh; /Þ; u13 ðh; /Þ ¼ cosð/Þ sinðhÞ½2ðcosðhÞ 1Þ þ h sinðhÞ þ sinð/Þ h2 þ sinðhÞ ðh cosðhÞ 2 sinðhÞÞ þ ð1 cosðhÞÞðh sinðhÞÞ; u14 ðh; /Þ ¼ sinð/Þ where,
for
2
hðcosðhÞ sinðhÞ þ hÞ þ cosð/Þh sin ðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ,
and
for h 6 / 6 0 : mt2 ðh; /Þ ¼ R Es Aa u15 ðh; /Þ þ Es Iu16 ðh; /Þ; u15 ðh; /Þ ¼ 2 cosð/Þ sinðhÞðh sinðhÞ þ 2 cosðhÞ 2Þ þ sinð/Þ 2sin ðhÞ h2 h cosðhÞ u16 ðh; /Þ ¼ cosð/Þhsin2 ðhÞ sin sinðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ, 2
ð/ÞhðsinðhÞ cosðhÞ þ hÞ þ h sinðhÞ þ ð1 cosðhÞÞðh sinðhÞÞ.
Nð/Þ ¼
qR Nt2 ðh; ks ; /Þ 6 M b2 ðhÞ
ðB:7Þ
where, Nt2 ðh; ks ; /Þ ¼ R2 Es Aa u17 ðh; ks ; /Þ þ Es Ihu18 ðh; ks ; /Þ; u17 ðh; ks ;
/Þ ¼ ð1 ks Þ cosð/Þ sinðhÞ 2h cos2 ðhÞ 3 sinðhÞ cosðhÞ þ h þ 6 cosð/Þ
nð/Þ ¼
1 nt1 ðh; /Þ 2 M b2 ðhÞ
ðB:8Þ
where, for 0 6 / 6 h : nt1 ðh; /Þ ¼ R2 Es Aa u19 ðh; /Þ þ Es Iu20 ðh; /Þ; u19 2 ðh; /Þ ¼ cosð/Þ sinðhÞðh sinðhÞ 2 þ 2 cosðhÞÞ þ sinð/Þ h2 2 sin ðhÞþ h cosðhÞ sinðhÞÞ; u20 ðh; /Þ ¼ h sinð/ÞððsinðhÞ cosðhÞ þ hÞÞ þ h sin ðhÞ cosð/Þ, and for h 6 / 6 0 : u19 ðh; /Þ ¼ cosð/Þ sinðhÞðh sinðhÞ 2 2 þ 2 cosðhÞÞ þ sinð/Þ 2sin ðhÞ h2 h cosðhÞ sinðhÞ ; u20 ðh; /Þ ¼ 2 h sin ðhÞ cosð/Þ sinð/ÞðsinðhÞ cosðhÞ þ hÞ . 2
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