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Contact interface in seismic analysis of circular tunnels
Tunnelling and Underground Space Technology 24 (2009) 482–490
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Technical Note
Contact interface in seismic analysis of circular tunnels Hassan Sedarat a, Alexander Kozak a, Youssef M.A. Hashash b,*, Anoosh Shamsabadi c, Alex Krimotat a a b c
SC Solutions, 1261 Oakmead Parkway, Sunnyvale, CA 94085, USA University of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana, IL 61801, USA Caltrans, Department of Transportation, State of California, 3 Mayapple Way, Irvine, CA 92612, USA
a r t i c l e
i n f o
Article history: Received 19 February 2008 Received in revised form 13 October 2008 Accepted 8 November 2008 Available online 25 December 2008 Keywords: Seismic analysis Soil-tunnel interaction Frictional contact Underground structures
a b s t r a c t The seismic analysis of underground structures requires a careful consideration of the important effect of shear strains in the soil due to vertically propagating horizontal shear waves. These strains result in ovaling deformations of circular tunnels or racking deformations of rectangular tunnels. Closed-form solutions as well as numerical analyses are used to characterize this soil-structure interaction problem. Many of these solutions assume full normal contact at the interface between the soil and tunnel lining. This work describes a numerical finite element study of soil-circular tunnel lining interaction with contact conditions that allow both limited slippage and separation to prevent development of potentially unrealistic normal tensile and tangential forces at the interface. The analyses highlight the significant limitations of widely used closed-form solutions in engineering practice. The finite element solutions demonstrate the need for realistic representation of the soil-tunnel interaction using numerical modeling approaches. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction While tunnels have generally performed better than above ground structures during earthquakes, damage to some of these important lifeline structures during previous earthquake events highlights the need to account for seismic loading in the design of underground structures. Hashash et al. (2001) provide a review of damage to underground structures and analytical approaches for seismic evaluations of these structures. In general, damage to tunnels is greatly reduced with increased overburden, and damage is greater in soils than in competent rock. Earthquake effects on underground structures can be grouped into ground failure and ground shaking. Ground failure constitutes conditions such as liquefaction, fault displacement, and slope stability. The collapse of the Daikai subway metro station in the 1995 Hyogo-ken Nambu earthquake in Kobe, Japan, was the first such significant collapse of an urban underground structure due to earthquake shaking, rather than ground instability. No seismic provisions were implemented in the 1962 design of this station as was pointed out by Toshio et al. (1998) and the complete collapse of this structure was due to loss of bearing capacity of the center columns. Giannakou et al. (2005) suggests that the failure of Bolu tunnel in Turkey, which was under construction during
* Corresponding author. Tel.: +1 217 333 6986. E-mail addresses: [email protected] (H. Sedarat), [email protected] (A. Kozak), [email protected] (Y.M.A. Hashash), [email protected] (A. Shamsabadi), [email protected] (A. Krimotat). 0886-7798/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2008.11.002
1999 Izmit, Turkey earthquake, was the result of the intensity of the seismic motions due to proximity of the seismic source to the tunnel. Three types of deformations characterize the response of underground structures to ground shaking: (1) axial compression and extension; (2) longitudinal bending; and (3) ovaling/racking. Ovaling/racking deformations have been the subject of many papers that describe closed-form solutions as well as numerical analysis approaches. For example, Kramer et al. (2007) describe detailed three-dimensional nonlinear ovaling analysis of a circular tunnel and Huo et al. (2006) developed closed-form solutions for rectangular linings. By far the best known closed-form solutions to compute liner forces subjected to seismic racking were developed for circular tunnels by Wang (1993) and Penzien and Wu (1998) and are widely used in engineering practice. Hashash et al. (2005) described the discrepancies between these two solutions and used numerical analyses under the same assumptions to better understand the differences between the two the solutions and their causes. The closed-form solutions are limited to the following assumptions: 1. In the direction normal to the lining, soil and lining are fully connected. 2. In the tangential direction, generally two cases are considered. These two cases are best known as: a. ‘‘full-slip” that assumes no tangential resistance transmitted from soil to the lining and b. ‘‘no-slip” that assumes full connection between soil and the lining.
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Fig. 1. Geometry and loading: (a) finite element model; (b) excavation load; (c) excavation force time function.
3. Tunnel is circular with uniform thickness and without any discontinuities. 4. Soil and lining are massless and linear elastic. 5. Plane strain conditions are postulated for soil and lining. 6. The effect of construction sequence is not considered. These assumptions are often not representative of actual conditions because 1. Soil and lining are not integrally connected in the direction normal to the lining. An exception would be when rock bolts or soil reinforcement elements are used. 2. ‘‘Full-slip” or ‘‘no-slip” conditions are idealizations of real contact of soil and lining. 3. The shape of the lining is rarely circular or rectangular with uniform thickness. 4. Soil around the lining may yield. 5. The construction sequence influences the state of stress at the lining interface with the ground. These deficiencies of the closed-form solutions are readily overcome through the use of numerical analysis techniques provided that proper characterizations of the physical system are implemented into the numerical models. Unfortunately, almost all the numerical analyses presented in known literature incorporate the same assumptions as the closed-form solutions, and therefore, have the same limited applicability. In engineering practice there is sometime heavy reliance on closed form solutions (Wang, 1993; Penzien and Wu, 1998) without full recognition of the limitations of these solutions. In this paper, a finite element based approach with rigorous characterization of the interface between soil and lining of circular tunnels through frictional contact is proposed. Although only circular tunnel results are reported in this paper, this does not result in any limitation of the proposed modeling approach and it is just a convenient way to compare with existing closed-form solutions. The behavior of the surrounding soil is assumed to be linear elastic, similar to the assumption in closed form solutions, in order to highlight the important role of modeling the soil-lining interface. In reality soil behavior is highly nonlinear and should be represented in a numerical analysis. The contact condition prevents development
of tensile normal stresses at the soil-tunnel interface. A Coulomb friction law is used to limit the tangential tractions. In this paper, the terms ‘‘normal stresses” and ‘‘tangential tractions” are used to describe normal and tangential components of the soil-tunnel interaction forces per unit area of the interface surface, respectively. The term ‘‘total tractions” is also used as a reference to vector summation of the normal and tangential components. The effect of frictional contact on the tunnel seismic response is examined for a range of soil and lining stiffness and friction coefficients. In all the cases, influence of gravity load and construction sequence on the state of stress of the tunnel is taken into account. The results of the analyses show that the more realistic modeling procedure using frictional contact to represent the interface between the tunnel liner and surrounding soil results in lining forces (due to seismically induced racking deformations) that are significantly different from those of the closed-form solutions. The study highlights the need for high fidelity tunnel and soil specific numerical modeling of seismic racking. It is not the intent of the study to develop alternative correlations. Analysis of tunnels anchored to the surrounding soil with means like rock bolts or soil reinforcement is outside the scope of this work. 2. Numerical modeling of tunnel ovaling with contact interface Fig. 1a shows the 2D finite element model used in this study and analyzed using the general purpose finite element program ADINA (2005). The soil continuum and the tunnel are represented with solid elements and beam elements of a unit width, respectively. The interface between soil and tunnel is modeled with frictional contact. The coefficient of friction is varied to examine its effect on the tunnel response. It is important to have realistic initial stress conditions prior to application of racking deformations due to seismic loading when the soil-tunnel interface is modeled with the frictional contact. The in situ state of stress is first established in the soil prior to tunnel construction. The overburden pressure is applied at the top of the mesh as shown in Fig. 1a. The soil initial stress is specified based on the assumed coefficient of lateral earth pressure K0. In order to account for the soil stress relaxation and arching during the tunnel excavation and construction, the widely used
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Table 1 Flexibility ratios*.
3. Finite element simulation with the closed-form solution assumptions
Lining thickness, t (m)
Soft soil, 0.1Es
Basic soil, Es
Stiff soil, 10Es
0.36 1.26
14.3 0.32
143 3.2
1430 32
*
The flexibility ratios are calculated according to Eq. (1) where R = 5m is the tunnel radius; Ec = 2.22E+04 MPa and mc = 0.3 are the modulus of elasticity and Poisson’s ratio of the lining, respectively; Es = 8.16E+02 MPa and ms = 0.3 are the modulus of elasticity and Poisson’s ratio of the soil, respectively.
b-method (Möller, 2006) is employed. The equivalent (excavation) load is applied at the tunnel perimeter (Fig. 1b) and reduced prior to lining installation as illustrated in Fig. 1c. The soil excavation and tunnel construction sequence are simulated in three steps. At time Step 1, the excavation load is in equilibrium with the initial in situ soil stresses and no displacements occur. From the first to the second time steps, soil stress relaxation is simulated by decreasing the excavation load by a factor b = 0.50. This b factor value is usually assumed and found to be conservative for the tunnel forces based on an extensive study reported by Möller (2006). The tunnel construction is simulated by activation of the lining members (beam finite elements) between the second and the third time steps. After time Step 3 the remaining excavation load is removed. Racking deformations due to seismic loading are then imposed as inverted triangular displacements along the lateral boundaries of the model and uniform lateral displacements along the top boundary (Fig. 1a). As the relative stiffness of soil and lining significantly affects the response of tunnel, the flexibility ratio F suggested by Wang (1993) is used to characterize the relative stiffness:
Es ð1 m2c ÞR3 F¼ 6Ec Ið1 þ ms Þ
ð1Þ
where Ec and mc are modulus of elasticity and Poisson’s ratio of the lining; Es and ms modulus of elasticity and Poisson’s ratio of the soil; R, radius of the tunnel; I, moment of inertia of the lining. The soil stiffness, the tunnel thickness, the friction coefficient as well as the coefficient of initial lateral earth pressure are varied in the parametric studies presented in the following sections. The soil and tunnel material properties are assumed to be linear elastic. Zero coefficient of friction represents a condition in which there is no tangential traction between the soil and lining and is analogous to the ‘‘full-slip” condition in closed form solutions. Nevertheless, as it will be shown later, due to the absence of the tensile normal contact stresses, the frictionless contact interface provides different tunnel response compared to that of the ‘‘full-slip” condition in closed form solutions. A coefficient of friction f = 1 corresponds to a friction angle of 45°.
A numerical simulation of the circular tunnel ovaling is performed with the closed-form solution assumptions (full normal connection and ‘‘no-slip” condition) of Wang (1993) and Penzien and Wu (1998) to establish a reference point for further studies. A 0.36 m thick tunnel in a soil with flexibility ratio of 143 and coefficient of lateral earth pressure equal to 1.0 (Table 1) is analyzed. After establishing initial stress conditions, racking deformations are applied corresponding to a soil shear strain equal to 0.5%. Table 2 lists the seismic increment of the lining thrust, shear force and bending moment due to ovaling deformations. The seismic increment is obtained by subtracting the forces computed at the end of tunnel construction from those at the end of racking analysis. Table 2 includes the corresponding values computed from the closed-form solutions of Wang (1993) and Penzien and Wu (1998). When using the same assumptions, the finite element analysis results in a lining thrust, moment and shear values which more closely match those of Wang’s closed-form solution (Table 2). This is in good agreement with the findings of Hashash et al. (2005). Although, Wang’s closed-form solution provides reasonable value of the thrust (discrepancy is just 2%), the bending moment value seems to be less accurate. The error (37%) is high because the bending moment is numerically small. The difference is about 100 kN-m/m compared to the thrust value of about 8000 kN/m. 4. Effect of frictional contact interface and limited circumferential slippage on the response of the tunnel The analysis in the previous section is repeated using a contact interface. The coefficient of friction at the contact surface is assumed to be 1.0. The lining sectional force seismic increments are compared in Table 2. While the shear and moment increments are similar to those from the ‘‘no-slip” analysis, the thrust increment is much smaller. The computed thrust increment is significantly different from both closed-form solutions. The diagram of the lining total thrust (due to gravity load and ovaling at soil shear strain equal to 0.5%) is quite different qualitatively and quantitatively from that for the ‘‘no-slip” analysis (Fig. 2). The contact interface results in compressive thrust of relatively small value (maximum compressive thrust = 3,745 kN/m in Fig. 2a), whereas the ‘‘no-slip” connection results in both tensile and compressive thrusts in the lining (maximum compressive thrust = 9,494 kN/m and maximum tensile thrust = 6,096 kN/m in Fig. 2b). The cause of this dissimilarity is the difference in the normal stresses and tangential tractions at the interface between soil and lining (Fig. 3). Since the finite element model with the contact condition prevents tensile (pulling) normal stresses and limits the
Table 2 Seismic increment of lining sectional forces at soil shear strain of 0.5%; F = 143, t = 0.36m, K0 = 1.0. Force/moment
Thrust increment, P (kN/m)
Finite element analysis
Closed-form solutions
Closed-form solution assumptions
Frictional contact, f = 1.0
Wang no-slip condition
Penzien & Wu ‘‘no-slip” condition
7795
1920
7988 (1.02)*
152 (0.02) 152 (1.21) 379 (1.36)
Shear force increment, V (kN/m)
126
132
Bending moment increment, M (kN-m/m)
278
303
*
380 (1.37)
(4.16)*
(1.25)
(0.08) (1.15) (1.25)
For the closed-form solutions, ratios to the finite element seismic increment forces and moments are given in parenthesis. The first number in the parenthesis corresponds to the numerical analysis with the closed-form solution assumptions whereas the second one corresponds to the finite element analysis with contact interface and friction coefficient equal to 1.0.
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Fig. 2. Lining total thrust at soil shear strain of 0.5%: (a) frictional contact (f = 1.0); (b) ‘‘no-slip” connection; displacement magnification factor = 20, F = 143, t = 0.36 m, K0 = 1.0.
Fig. 3. Soil-tunnel interaction tractions at soil shear strain of 0.5%: (a) contact interface (f = 1.0); (b) ‘‘no-slip” connection; displacement magnification factor = 20, F = 143, t = 0.36 m, K0 = 1.0.
Fig. 4. Soil maximum shear stress at soil shear strain of 0.5%: (a) contact interface (f = 1.0); (b) ‘‘no-slip” connection; F = 143, t = 0.36 m, K0 = 1.0.
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Fig. 5. Contact tractions, lining total thrust and bending moment at soil shear strain of 0.5% under different friction coefficients: (a) f = 0 and (b) f = 1.0; F = 143, t = 0.36 m, K0 = 1.0.
5. Effects of coefficient of friction, soil shear strain, and lining thickness on the tunnel response Distributions of contact total tractions, lining total thrust and bending moment are compared for two values of friction coefficient: 0 and 1.0 in Fig. 5a and b respectively. Other model parameters are F = 143, t = 0.36m, K0 = 1.0. At a soil shear strain of 0.5%, the frictionless contact (f = 0) analysis results in contact stresses that are normal to the tunnel (tangential component is zero) and relatively uniform: the deviation is between 28% and 14% from the mean value. Similarly the total thrust is relatively uniform along the tunnel circumference with a maximum deviation of ±6%. The high friction coefficient (f = 1.0) analysis results in large tangential traction with the maximum total traction 2.3 times larger than the maximum contact total traction for f = 0 analysis. Despite the fact that all normal contact components are compressive, this contact total traction distribution results in tensile and compressive thrust increments which lead to significantly non-uniform total thrust. The direction of the tangential tractions illustrates their essential contribution to the tunnel ovaling and lining thrust seismic increments. The tunnel ovaling is exclusively due to the normal contact stress distribution along the tunnel perimeter in the case of frictionless contact. Note that the friction coefficient increase does not significantly affect the lining bending moment whereby the maximum value decreases by 9%. The lining bending moment diagram is a function of the geometry or shape of the deformed tunnel which does not vary significantly due to the specific contact interface assumption. The nonlinear behavior of the thrust in the tunnel lining when interface is modeled with frictional contact can be further understood by varying the racking shear strains. In Fig. 6, the maximum
compressive thrust seismic increment versus soil shear strain is plotted for a set of friction coefficient values: 0, 0.5, 0.8, and 1.0. These correspond to analyses with F = 143, t = 0.36m, k0 = 1.0. At early stages of ovaling, the soil-tunnel interface is locked in the tangential direction which allows for a significant and rapid build up of thrust in the lining until the frictional limit resistance at the interface is reached. This transition occurs at a higher value of thrust for the higher value of the friction coefficient. Afterwards, some local slippage between the lining and the soil develops. The further increase in the thrust is due to the differential normal contact stress. At this stage, the rate of increase in the thrust is the same for all friction coefficients (Fig. 6). In the course of ovaling without friction (f = 0), thrust increases due to the normal contact pressure of the soil. In case of frictional contact (f > 0), the thrust increment is developed due to the normal and tangential traction at the soil-tunnel interface until friction limit resistance is reached and slippage occurs. From that moment on, further increase of the thrust is due to the same factor as in case of frictionless contact, namely, the normal contact pressure only. Fig. 7 compares the seismic thrust increment from the numerical analyses with the ‘‘full-slip” and ‘‘no-slip” closed-form solutions by
f = 1.0 2000
Thrust (kN/m)
tangential tractions, the total tractions are much lower (Fig. 3a). In the case of ‘‘no-slip” condition, total tractions at the soil-tunnel interface are much larger with both compressive and tensile (pushing and pulling) normal components (Fig. 3b). Consequently, the analysis with contact interface results in smaller regions in soil with high shear stresses compared with the ‘‘no-slip” condition (Fig. 4).
f = 0.8
f = 0.5 1000
f = 0.0 0
0
0.002
0.004
0.006
0.008
0.010
Soil Shear Strain Fig. 6. Seismic increment of lining thrust versus soil shear strain under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 143, t = 0.36 m, K0 = 1.0.
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Soil Shear Strain 0.005
0.010
15975
7988
Thrust (kN/m)
0.001
2000
1000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 7. Seismic increment of lining thrust at different soil shear strains: 0.1%, 0.5%, and 1.0%; F = 143, t = 0.36 m, K0 = 1.0. Fig. 9. Seismic increment of lining thrust at soil shear strain of 0.5% under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 3.23, t = 1.26 m, K0 = 1.0.
(F = 143) and 1.26m (F = 3.2), respectively (see Table 1). It is worth noting that a 1.26m lining is far too thick compared to the lining thickness commonly used in practice, however, it is used in this numerical study to account for wide variations in flexibility ratios. A value of K0 = 1.0, and friction coefficients 0, 0.5, 0.8, and 1.0 are used. In all cases, as the coefficient of friction increases, the thrust increment increases as well. However, the thicker lining develops much greater thrust seismic increment and is less sensitive to the soil-tunnel friction. 6. Parametric finite element analyses and closed-form solutions
Fig. 8. Seismic increment of lining thrust at soil shear strain of 0.5% under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 143, t = 0.36 m, K0 = 1.0.
Wang (1993) and Penzien and Wu (1998). Although, at a relatively small soil shear strain 0.1%, Wang’s ‘‘no-slip” closed-form solution provides a thrust increment value comparable with those from the numerical analyses, at larger shear strains it gives a much higher value (Fig. 7). The closed-form solution is linear, whereas the finite element model incorporates non-linearity of the soiltunnel interface. Therefore, neither of the two closed-form solutions provides estimates of the thrust that are comparable with those from the numerical analyses within a wide range of the soil shear strain values. Figs. 8 and 9 present computed thrust seismic increment distribution at a soil strain of 0.5% for the lining thickness of 0.36m
A series of numerical analyses are conducted as listed in Table 1 and Table 3 to further illustrate the influence of various factors on the computed response of the tunnel lining and the difference with the closed-form solutions. Tables 4 and 5 summarize the maximum and minimum seismic increments of the lining thrust, shear force, and bending moment obtained from numerical analyses and compared with those from the closed-form solutions at the soil shear strain equal to 0.5%. In addition, the lining diametric strain values are summarized in Table 6. The finite element thrust seismic increments are sensitive to various interface conditions and, in general, differ substantially from those of the closed-form solutions, whereas seismic increments of the shear and bending moment obtained from the numerical and closed-form solutions are within the same order of magnitude. As discussed earlier, the bending moment diagram corresponds to the deformed shape of the tunnel which does not vary significantly due to variation in contact condition. In some cases, the closed-form solutions (Penzien and Wu, 1998; Wang, 1993) correlate with the numerical results within tolerances acceptable for practical purposes, while in many other cases they do not. Figs. 10 and 11 present graphical comparisons of the maximum lining
Table 3 Analyses’ matrix. Case ID
Lining radius, R (m)
Lining thickness, t (m)
Flexibility ratio, F
Coefficient of friction, f
Coefficient of lateral earth pressure, K0
2B3 01 2B3 2B3 10 B3 01 B3 B3 10
5 5 5 5 5 5
0.36 0.36 0.36 1.26 1.26 1.26
14.3 143 1430 0.32 3.2 32
0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.8, 0.8, 0.8, 0.8, 0.8, 0.8,
1.0 1.0 1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0 1.0 1.0
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Table 4 Maximum/minimum seismic increment of lining sectional forces at soil shear strain of 0.5% under different flexibility ratios: 14.3, 143, and 1430; t = 0.36m. Flexibility ratio
Force/moment*
Closed-form solution Wang Full-slip
14.3
143
1430
Finite element solution with contact interface Penzien &Wu
No-slip
Full-slip
No-slip
K0
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
max P
69
931
69
136
min P
69
931
69
136
V
69
136
M
345
345
345
341
max P
76
7988
76
152
min P
76
7988
76
152
V
76
152
M
380
380
380
379
max P
77
36557
77
153
min P
77
36557
77
153
V
77
153
M
384
384
383
384
Coefficient of friction 0
0.5
0.8
1.0
91 113 72 80 130 122 299 301 217 231 53 58 132 130 332 334 460 464 310 311 132 132 344 346
681 843 628 813 125 120 276 263 811 1010 491 645 131 131 324 320 783 832 13 23 132 132 348 348
860 924 819 913 119 108 270 264 1240 1545 755 976 132 132 314 308 1024 1104 132 186 132 131 348 348
905 928 858 922 112 107 266 264 1550 1920 904 1160 132 132 309 303 1201 1307 218 280 132 131 347 348
*
The lining forces and bending moment are designated as follows: P is the thrust increment (kN/m), V is shear force increment (kN/m), M is bending moment increment (kNm/m). Negative thrust increment indicates tension.
Table 5 Maximum/minimum seismic increment of lining sectional forces at soil shear strain of 0.5% under different flexibility ratios: 0.32, 3.2, and 32; t = 1.26 m. Flexibility ratio
Force/moment*
Closed-form solution Wang Full-slip
0.32
32
No-slip
Full-slip
No-slip
K0
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
max P
571
1163
571
1034
min P
571
1163
571
1034
571
1034
V
3.2
Finite element solution with contact interface Penzien &Wu
M
2854
2854
2854
2585
max P
2271
9788
2271
4361
min P
2271
9788
2271
4361
V
2271
4361
M
11354
11354
11354
10903
max P
3234
66011
3234
6431
min P
3234
66011
3234
6431
V
3234
6431
M
16171
16171
16077
16171
Coefficient of friction 0
0.5
0.8
1.0
589 610 582 594 1161 1151 2852 2850 3416 3155 202 555 3934 3970 9388 9538 6593 6406 1420 1218 5863 5859 13,267 13,271
1033 1169 1150 1185 1110 1068 2694 2631 3713 3529 2277 2858 3926 3963 9206 9319 6834 6673 1748 1985 5950 5934 13,023 13,012
1160 1237 1187 1241 1077 1050 2632 2600 3958 3805 3437 4014 3905 3965 9120 9236 7066 6909 3415 3555 5916 5875 12,926 12,920
1219 1243 1225 1243 1060 1049 2611 2598 4103 3986 4039 4637 3907 3948 9086 9205 7218 7083 4200 4455 5840 5879 12,882 12,872
*
The lining forces and bending moment are designated as follows: P is the thrust increment (kN/m), V is shear force increment (kN/m), M is bending moment increment (kNm/m). Negative thrust increment indicates tension.
thrust seismic increments for the tunnel lining thicknesses of 0.36 m and 1.26 m, respectively, at a soil strain of 0.5% and K0 = 1.0. Except for the case of F = 1430, the lining thrust seismic
increment increases with the flexibility ratio for a constant lining stiffness. The thrust seismic increment increases as the thickness of the lining increases for a constant soil stiffness. This can be
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H. Sedarat et al. / Tunnelling and Underground Space Technology 24 (2009) 482–490 Table 6 Seismic increment of diagonal strain of the lining at soil shear strain of 0.5%. Flexibility ratio
Closed-form solution
Finite element solution with frictional interface
Wang
Penzien & Wu
Full-slip (%)
No-slip (%)
Full-slip (%)
No-slip (%)
K0
Coefficient of friction 0 (%)
0.5 (%)
0.8 (%)
1.0 (%)
14.3
0.63
0.63
0.63
0.62
143
0.69
0.69
0.69
0.69
1430
0.70
0.70
0.70
0.70
0.32
0.12
0.12
0.12
0.11
3.2
0.47
0.47
0.47
0.45
32
0.67
0.67
0.67
0.66
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
0.59 0.58 0.64 0.66 0.64 0.64 0.13 0.13 0.56 0.54 0.63 0.63
0.55 0.53 0.63 0.63 0.64 0.64 0.12 0.12 0.54 0.52 0.63 0.63
0.53 0.52 0.63 0.63 0.64 0.64 0.12 0.12 0.53 0.51 0.63 0.63
0.53 0.52 0.63 0.62 0.64 0.64 0.12 0.12 0.52 0.50 0.63 0.63
Flexibility Ratio 36557 Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
1430
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
143 7988
Thrust (kN/m)
14.3
2000
1000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 10. Seismic increment of lining thrust at soil shear strain of 0.5% under different flexibility ratios: 14.3, 143, and 1430; t = 0.36 m, K0 = 1.0.
Flexibility Ratio 3.2
32 66011
9788
Thrust (kN/m)
0.32
6000
The effect of coefficient of lateral earth pressure appears to be significant only for the 0.36 m thick lining and flexibility ratio equal to 143. The soil diametric strain values obtained from the finite element analyses are somewhat below those of the closed-form solutions except for the case of the stiff lining (t = 1.26 m, F = 3.2). The lining thrust seismic increments of the ‘‘full-slip” closedform solutions are comparable to those of the finite element model with the soil-tunnel contact interface only for the 1.26 m thick lining and flexibility ratios equal to 0.32 and 3.2 (Fig. 11). This is also true when Wang’s ‘‘no-slip” closed-form solution is used for the soft soil (F = 14.3 in Fig. 10 and F = 0.32 in Fig. 11). On the other hand, the lining thrust seismic increments of Penzien and Wu’s ‘‘no-slip” closed-form solution seem closer to the numerical analysis values for the thick lining (Fig. 11). For other cases there is no clear correlation between the closed-form solutions and finite element analyses with the soil-tunnel contact interface. The comparisons suggest that the thrust seismic increment in the lining is dependent on several factors including relative soil and lining stiffness and the soil-tunnel interface conditions. The available closed-form solutions are ill-equipped to represent these conditions in a general way and may overestimate or underestimate the lining forces. Therefore, the closedform solutions should be used with caution even as first order estimates. Although the characteristics of the soil-tunnel interface significantly affect the level of the thrust in the tunnel lining, the tunnel lining diagonal strain is mostly controlled by the flexibility ratio (Table 6).
4000
7. Summary and conclusions 2000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 11. Seismic increment of lining thrust at soil shear strain of 0.5% under different flexibility ratios: 0.32, 3.2, and 32; t = 1.26 m, K0 = 1.0.
clearly observed when comparing thrust values obtained from cases with flexibility ratios 14.3, 143, and 1430 in Fig. 10 with those for flexibility ratios 0.32, 3.2, and 32 in Fig. 11, respectively.
This paper highlighted the importance of a more realistic representation of the soil-tunnel interface when performing seismic racking analysis of underground structures. The contact interface incorporated into a finite element model prevents development of unrealistic normal tensile stresses and controls build up of the tangential tractions between the soil and tunnel lining. As a result, the lining thrust increments due to the tunnel ovaling are significantly different from those of widely used closed-form solutions. Bending moments are less sensitive to the specific assumption regarding the soil-tunneling interface. The frictional contact is a nonlinear phenomenon and cannot be handled by linear closedform solutions. It is shown that even with a very high value of friction coefficient (f = 1.0), the contact interface limits the tangential tractions and provides much lower thrust levels than ‘‘no-slip” closed-form solutions. In some of the cases presented in this paper,
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the closed-form solutions correlate with the numerical results within tolerances acceptable for practical purposes, while in many other cases they do not. While it is not the intention of this study to find out when the closed-form solutions match with numerical analyses, it is quite reasonable that a finite element technique with proper definition of the soil-tunnel interface be always used to simulate seismic response of the tunnels. Although, only circular tunnels of a constant thickness in a uniform soil have been analyzed in this study, many tunnels have a horse-shoe shape with variable thickness in a multilayered soil profile. All these complicating factors are beyond the limiting assumptions of the available closed-form solutions but might be easily addressed with a numerical approach. During a seismic event, the soil and tunnel might undergo both shear and normal strains. Since the lining thrust is sensitive to both normal stresses and tangential tractions at the soil-tunnel lining interface and since the tangential tractions depend on the value of normal contact stresses (Coulomb friction law), the thrust due to seismic racking deformations could be underestimated. For a comprehensive estimation of the seismic response of tunnels, both shear and normal (vertical and horizontal) motions might have to be applied to the soil and tunnel in the course of a time history analysis. The nonlinear behavior of soil is another important aspect in soil-tunnel lining interaction that can be accounted for in finite element simulation of a seismic event and will be considered in future studies.
References ADINA, 2005. ADINA User Interface. Command Reference Manual. Volume 1: ADINA Solids & Structures Model Definition. Report ARD 05–2. October 2005. ADINA System version 8.3, ADINA R&D Inc. Giannakou, A., Nomikos, P., Anastasopoulos, I., Sofianos, A., Gazetas, G., YioutaMitra, 2005. Sesimic behaviour of tunnels in soft soil: parametric numerical study and investigation on the causes of failure of the Bolu Tunnel (Duzce, Turkey, 1999). In: Yücel Erdem & Tülin Solak (Eds.), Underground Space Use: Analysis of the Past and Lessons for the Future, Taylor & Francis Group, London, ISBN: 04 1537 452 9. Hashash, Y.M.A., Hook, J.J., Schmidt, B., Yao, J.I.-C., 2001. Seismic design and analysis of underground structure. Tunneling and Underground Space Technology 16, 247–293. Hashash, Y., Park D., M.A., Yao, J.I.C., 2005. Ovaling Deformations of Circular Tunnels Under Seismic Loading, an Update on Seismic Design and Analysis of Underground Structures. Elsevier, Tunneling and underground space Technology. Huo, H., Bobet, A., Fernandez, G., Ramirez, J., 2006. Analytical solution for deep rectangular structures subjected to far-field shear stresses. Tunnelling and Underground Space Technology 21 (6), 613–625. Kramer, G.J.E., Sedarat, H., Kozak, A., 2007. Seismic response of pre-cast tunnel linings. In: Proc. 2007 Rapid Excavation and Tunneling Conference, Toronto, ON, 2007. Möller, S., 2006. Tunnel induced settlements and structural forces in linings. Ph.D. Thesis,Universität Stuttgart, Germany. Penzien, J., Wu, C.L., 1998. Stresses in linings of bored tunnels. International Journal of Earthquake Engineering Structural Dynamics 27, 283–300. Toshio, Umehara, Hiroomi, Iiada, Susumu, Nakamura, Youichi, Yamahara, 1998. Restoration of the collapsed Subway Station due to Hyogoken-Nanbu Earthquake, January 17, 1995. In: Arsenio Negro, Jr. & Argimiro Alvarez Ferreira (Eds.), Tunnels and Metropolises, c 1998, Balkema, Rotherdam, ISBN: 90 5410 936X. Wang, J.-N., 1993. Seismic Design of Tunnels: A State-of-the-Art Approach, Monograph, monograph 7, Parsons. Brinckerhoff Quade and Douglas Inc., New York.
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Technical Note
Contact interface in seismic analysis of circular tunnels Hassan Sedarat a, Alexander Kozak a, Youssef M.A. Hashash b,*, Anoosh Shamsabadi c, Alex Krimotat a a b c
SC Solutions, 1261 Oakmead Parkway, Sunnyvale, CA 94085, USA University of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana, IL 61801, USA Caltrans, Department of Transportation, State of California, 3 Mayapple Way, Irvine, CA 92612, USA
a r t i c l e
i n f o
Article history: Received 19 February 2008 Received in revised form 13 October 2008 Accepted 8 November 2008 Available online 25 December 2008 Keywords: Seismic analysis Soil-tunnel interaction Frictional contact Underground structures
a b s t r a c t The seismic analysis of underground structures requires a careful consideration of the important effect of shear strains in the soil due to vertically propagating horizontal shear waves. These strains result in ovaling deformations of circular tunnels or racking deformations of rectangular tunnels. Closed-form solutions as well as numerical analyses are used to characterize this soil-structure interaction problem. Many of these solutions assume full normal contact at the interface between the soil and tunnel lining. This work describes a numerical finite element study of soil-circular tunnel lining interaction with contact conditions that allow both limited slippage and separation to prevent development of potentially unrealistic normal tensile and tangential forces at the interface. The analyses highlight the significant limitations of widely used closed-form solutions in engineering practice. The finite element solutions demonstrate the need for realistic representation of the soil-tunnel interaction using numerical modeling approaches. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction While tunnels have generally performed better than above ground structures during earthquakes, damage to some of these important lifeline structures during previous earthquake events highlights the need to account for seismic loading in the design of underground structures. Hashash et al. (2001) provide a review of damage to underground structures and analytical approaches for seismic evaluations of these structures. In general, damage to tunnels is greatly reduced with increased overburden, and damage is greater in soils than in competent rock. Earthquake effects on underground structures can be grouped into ground failure and ground shaking. Ground failure constitutes conditions such as liquefaction, fault displacement, and slope stability. The collapse of the Daikai subway metro station in the 1995 Hyogo-ken Nambu earthquake in Kobe, Japan, was the first such significant collapse of an urban underground structure due to earthquake shaking, rather than ground instability. No seismic provisions were implemented in the 1962 design of this station as was pointed out by Toshio et al. (1998) and the complete collapse of this structure was due to loss of bearing capacity of the center columns. Giannakou et al. (2005) suggests that the failure of Bolu tunnel in Turkey, which was under construction during
* Corresponding author. Tel.: +1 217 333 6986. E-mail addresses: [email protected] (H. Sedarat), [email protected] (A. Kozak), [email protected] (Y.M.A. Hashash), [email protected] (A. Shamsabadi), [email protected] (A. Krimotat). 0886-7798/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2008.11.002
1999 Izmit, Turkey earthquake, was the result of the intensity of the seismic motions due to proximity of the seismic source to the tunnel. Three types of deformations characterize the response of underground structures to ground shaking: (1) axial compression and extension; (2) longitudinal bending; and (3) ovaling/racking. Ovaling/racking deformations have been the subject of many papers that describe closed-form solutions as well as numerical analysis approaches. For example, Kramer et al. (2007) describe detailed three-dimensional nonlinear ovaling analysis of a circular tunnel and Huo et al. (2006) developed closed-form solutions for rectangular linings. By far the best known closed-form solutions to compute liner forces subjected to seismic racking were developed for circular tunnels by Wang (1993) and Penzien and Wu (1998) and are widely used in engineering practice. Hashash et al. (2005) described the discrepancies between these two solutions and used numerical analyses under the same assumptions to better understand the differences between the two the solutions and their causes. The closed-form solutions are limited to the following assumptions: 1. In the direction normal to the lining, soil and lining are fully connected. 2. In the tangential direction, generally two cases are considered. These two cases are best known as: a. ‘‘full-slip” that assumes no tangential resistance transmitted from soil to the lining and b. ‘‘no-slip” that assumes full connection between soil and the lining.
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483
Fig. 1. Geometry and loading: (a) finite element model; (b) excavation load; (c) excavation force time function.
3. Tunnel is circular with uniform thickness and without any discontinuities. 4. Soil and lining are massless and linear elastic. 5. Plane strain conditions are postulated for soil and lining. 6. The effect of construction sequence is not considered. These assumptions are often not representative of actual conditions because 1. Soil and lining are not integrally connected in the direction normal to the lining. An exception would be when rock bolts or soil reinforcement elements are used. 2. ‘‘Full-slip” or ‘‘no-slip” conditions are idealizations of real contact of soil and lining. 3. The shape of the lining is rarely circular or rectangular with uniform thickness. 4. Soil around the lining may yield. 5. The construction sequence influences the state of stress at the lining interface with the ground. These deficiencies of the closed-form solutions are readily overcome through the use of numerical analysis techniques provided that proper characterizations of the physical system are implemented into the numerical models. Unfortunately, almost all the numerical analyses presented in known literature incorporate the same assumptions as the closed-form solutions, and therefore, have the same limited applicability. In engineering practice there is sometime heavy reliance on closed form solutions (Wang, 1993; Penzien and Wu, 1998) without full recognition of the limitations of these solutions. In this paper, a finite element based approach with rigorous characterization of the interface between soil and lining of circular tunnels through frictional contact is proposed. Although only circular tunnel results are reported in this paper, this does not result in any limitation of the proposed modeling approach and it is just a convenient way to compare with existing closed-form solutions. The behavior of the surrounding soil is assumed to be linear elastic, similar to the assumption in closed form solutions, in order to highlight the important role of modeling the soil-lining interface. In reality soil behavior is highly nonlinear and should be represented in a numerical analysis. The contact condition prevents development
of tensile normal stresses at the soil-tunnel interface. A Coulomb friction law is used to limit the tangential tractions. In this paper, the terms ‘‘normal stresses” and ‘‘tangential tractions” are used to describe normal and tangential components of the soil-tunnel interaction forces per unit area of the interface surface, respectively. The term ‘‘total tractions” is also used as a reference to vector summation of the normal and tangential components. The effect of frictional contact on the tunnel seismic response is examined for a range of soil and lining stiffness and friction coefficients. In all the cases, influence of gravity load and construction sequence on the state of stress of the tunnel is taken into account. The results of the analyses show that the more realistic modeling procedure using frictional contact to represent the interface between the tunnel liner and surrounding soil results in lining forces (due to seismically induced racking deformations) that are significantly different from those of the closed-form solutions. The study highlights the need for high fidelity tunnel and soil specific numerical modeling of seismic racking. It is not the intent of the study to develop alternative correlations. Analysis of tunnels anchored to the surrounding soil with means like rock bolts or soil reinforcement is outside the scope of this work. 2. Numerical modeling of tunnel ovaling with contact interface Fig. 1a shows the 2D finite element model used in this study and analyzed using the general purpose finite element program ADINA (2005). The soil continuum and the tunnel are represented with solid elements and beam elements of a unit width, respectively. The interface between soil and tunnel is modeled with frictional contact. The coefficient of friction is varied to examine its effect on the tunnel response. It is important to have realistic initial stress conditions prior to application of racking deformations due to seismic loading when the soil-tunnel interface is modeled with the frictional contact. The in situ state of stress is first established in the soil prior to tunnel construction. The overburden pressure is applied at the top of the mesh as shown in Fig. 1a. The soil initial stress is specified based on the assumed coefficient of lateral earth pressure K0. In order to account for the soil stress relaxation and arching during the tunnel excavation and construction, the widely used
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Table 1 Flexibility ratios*.
3. Finite element simulation with the closed-form solution assumptions
Lining thickness, t (m)
Soft soil, 0.1Es
Basic soil, Es
Stiff soil, 10Es
0.36 1.26
14.3 0.32
143 3.2
1430 32
*
The flexibility ratios are calculated according to Eq. (1) where R = 5m is the tunnel radius; Ec = 2.22E+04 MPa and mc = 0.3 are the modulus of elasticity and Poisson’s ratio of the lining, respectively; Es = 8.16E+02 MPa and ms = 0.3 are the modulus of elasticity and Poisson’s ratio of the soil, respectively.
b-method (Möller, 2006) is employed. The equivalent (excavation) load is applied at the tunnel perimeter (Fig. 1b) and reduced prior to lining installation as illustrated in Fig. 1c. The soil excavation and tunnel construction sequence are simulated in three steps. At time Step 1, the excavation load is in equilibrium with the initial in situ soil stresses and no displacements occur. From the first to the second time steps, soil stress relaxation is simulated by decreasing the excavation load by a factor b = 0.50. This b factor value is usually assumed and found to be conservative for the tunnel forces based on an extensive study reported by Möller (2006). The tunnel construction is simulated by activation of the lining members (beam finite elements) between the second and the third time steps. After time Step 3 the remaining excavation load is removed. Racking deformations due to seismic loading are then imposed as inverted triangular displacements along the lateral boundaries of the model and uniform lateral displacements along the top boundary (Fig. 1a). As the relative stiffness of soil and lining significantly affects the response of tunnel, the flexibility ratio F suggested by Wang (1993) is used to characterize the relative stiffness:
Es ð1 m2c ÞR3 F¼ 6Ec Ið1 þ ms Þ
ð1Þ
where Ec and mc are modulus of elasticity and Poisson’s ratio of the lining; Es and ms modulus of elasticity and Poisson’s ratio of the soil; R, radius of the tunnel; I, moment of inertia of the lining. The soil stiffness, the tunnel thickness, the friction coefficient as well as the coefficient of initial lateral earth pressure are varied in the parametric studies presented in the following sections. The soil and tunnel material properties are assumed to be linear elastic. Zero coefficient of friction represents a condition in which there is no tangential traction between the soil and lining and is analogous to the ‘‘full-slip” condition in closed form solutions. Nevertheless, as it will be shown later, due to the absence of the tensile normal contact stresses, the frictionless contact interface provides different tunnel response compared to that of the ‘‘full-slip” condition in closed form solutions. A coefficient of friction f = 1 corresponds to a friction angle of 45°.
A numerical simulation of the circular tunnel ovaling is performed with the closed-form solution assumptions (full normal connection and ‘‘no-slip” condition) of Wang (1993) and Penzien and Wu (1998) to establish a reference point for further studies. A 0.36 m thick tunnel in a soil with flexibility ratio of 143 and coefficient of lateral earth pressure equal to 1.0 (Table 1) is analyzed. After establishing initial stress conditions, racking deformations are applied corresponding to a soil shear strain equal to 0.5%. Table 2 lists the seismic increment of the lining thrust, shear force and bending moment due to ovaling deformations. The seismic increment is obtained by subtracting the forces computed at the end of tunnel construction from those at the end of racking analysis. Table 2 includes the corresponding values computed from the closed-form solutions of Wang (1993) and Penzien and Wu (1998). When using the same assumptions, the finite element analysis results in a lining thrust, moment and shear values which more closely match those of Wang’s closed-form solution (Table 2). This is in good agreement with the findings of Hashash et al. (2005). Although, Wang’s closed-form solution provides reasonable value of the thrust (discrepancy is just 2%), the bending moment value seems to be less accurate. The error (37%) is high because the bending moment is numerically small. The difference is about 100 kN-m/m compared to the thrust value of about 8000 kN/m. 4. Effect of frictional contact interface and limited circumferential slippage on the response of the tunnel The analysis in the previous section is repeated using a contact interface. The coefficient of friction at the contact surface is assumed to be 1.0. The lining sectional force seismic increments are compared in Table 2. While the shear and moment increments are similar to those from the ‘‘no-slip” analysis, the thrust increment is much smaller. The computed thrust increment is significantly different from both closed-form solutions. The diagram of the lining total thrust (due to gravity load and ovaling at soil shear strain equal to 0.5%) is quite different qualitatively and quantitatively from that for the ‘‘no-slip” analysis (Fig. 2). The contact interface results in compressive thrust of relatively small value (maximum compressive thrust = 3,745 kN/m in Fig. 2a), whereas the ‘‘no-slip” connection results in both tensile and compressive thrusts in the lining (maximum compressive thrust = 9,494 kN/m and maximum tensile thrust = 6,096 kN/m in Fig. 2b). The cause of this dissimilarity is the difference in the normal stresses and tangential tractions at the interface between soil and lining (Fig. 3). Since the finite element model with the contact condition prevents tensile (pulling) normal stresses and limits the
Table 2 Seismic increment of lining sectional forces at soil shear strain of 0.5%; F = 143, t = 0.36m, K0 = 1.0. Force/moment
Thrust increment, P (kN/m)
Finite element analysis
Closed-form solutions
Closed-form solution assumptions
Frictional contact, f = 1.0
Wang no-slip condition
Penzien & Wu ‘‘no-slip” condition
7795
1920
7988 (1.02)*
152 (0.02) 152 (1.21) 379 (1.36)
Shear force increment, V (kN/m)
126
132
Bending moment increment, M (kN-m/m)
278
303
*
380 (1.37)
(4.16)*
(1.25)
(0.08) (1.15) (1.25)
For the closed-form solutions, ratios to the finite element seismic increment forces and moments are given in parenthesis. The first number in the parenthesis corresponds to the numerical analysis with the closed-form solution assumptions whereas the second one corresponds to the finite element analysis with contact interface and friction coefficient equal to 1.0.
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Fig. 2. Lining total thrust at soil shear strain of 0.5%: (a) frictional contact (f = 1.0); (b) ‘‘no-slip” connection; displacement magnification factor = 20, F = 143, t = 0.36 m, K0 = 1.0.
Fig. 3. Soil-tunnel interaction tractions at soil shear strain of 0.5%: (a) contact interface (f = 1.0); (b) ‘‘no-slip” connection; displacement magnification factor = 20, F = 143, t = 0.36 m, K0 = 1.0.
Fig. 4. Soil maximum shear stress at soil shear strain of 0.5%: (a) contact interface (f = 1.0); (b) ‘‘no-slip” connection; F = 143, t = 0.36 m, K0 = 1.0.
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Fig. 5. Contact tractions, lining total thrust and bending moment at soil shear strain of 0.5% under different friction coefficients: (a) f = 0 and (b) f = 1.0; F = 143, t = 0.36 m, K0 = 1.0.
5. Effects of coefficient of friction, soil shear strain, and lining thickness on the tunnel response Distributions of contact total tractions, lining total thrust and bending moment are compared for two values of friction coefficient: 0 and 1.0 in Fig. 5a and b respectively. Other model parameters are F = 143, t = 0.36m, K0 = 1.0. At a soil shear strain of 0.5%, the frictionless contact (f = 0) analysis results in contact stresses that are normal to the tunnel (tangential component is zero) and relatively uniform: the deviation is between 28% and 14% from the mean value. Similarly the total thrust is relatively uniform along the tunnel circumference with a maximum deviation of ±6%. The high friction coefficient (f = 1.0) analysis results in large tangential traction with the maximum total traction 2.3 times larger than the maximum contact total traction for f = 0 analysis. Despite the fact that all normal contact components are compressive, this contact total traction distribution results in tensile and compressive thrust increments which lead to significantly non-uniform total thrust. The direction of the tangential tractions illustrates their essential contribution to the tunnel ovaling and lining thrust seismic increments. The tunnel ovaling is exclusively due to the normal contact stress distribution along the tunnel perimeter in the case of frictionless contact. Note that the friction coefficient increase does not significantly affect the lining bending moment whereby the maximum value decreases by 9%. The lining bending moment diagram is a function of the geometry or shape of the deformed tunnel which does not vary significantly due to the specific contact interface assumption. The nonlinear behavior of the thrust in the tunnel lining when interface is modeled with frictional contact can be further understood by varying the racking shear strains. In Fig. 6, the maximum
compressive thrust seismic increment versus soil shear strain is plotted for a set of friction coefficient values: 0, 0.5, 0.8, and 1.0. These correspond to analyses with F = 143, t = 0.36m, k0 = 1.0. At early stages of ovaling, the soil-tunnel interface is locked in the tangential direction which allows for a significant and rapid build up of thrust in the lining until the frictional limit resistance at the interface is reached. This transition occurs at a higher value of thrust for the higher value of the friction coefficient. Afterwards, some local slippage between the lining and the soil develops. The further increase in the thrust is due to the differential normal contact stress. At this stage, the rate of increase in the thrust is the same for all friction coefficients (Fig. 6). In the course of ovaling without friction (f = 0), thrust increases due to the normal contact pressure of the soil. In case of frictional contact (f > 0), the thrust increment is developed due to the normal and tangential traction at the soil-tunnel interface until friction limit resistance is reached and slippage occurs. From that moment on, further increase of the thrust is due to the same factor as in case of frictionless contact, namely, the normal contact pressure only. Fig. 7 compares the seismic thrust increment from the numerical analyses with the ‘‘full-slip” and ‘‘no-slip” closed-form solutions by
f = 1.0 2000
Thrust (kN/m)
tangential tractions, the total tractions are much lower (Fig. 3a). In the case of ‘‘no-slip” condition, total tractions at the soil-tunnel interface are much larger with both compressive and tensile (pushing and pulling) normal components (Fig. 3b). Consequently, the analysis with contact interface results in smaller regions in soil with high shear stresses compared with the ‘‘no-slip” condition (Fig. 4).
f = 0.8
f = 0.5 1000
f = 0.0 0
0
0.002
0.004
0.006
0.008
0.010
Soil Shear Strain Fig. 6. Seismic increment of lining thrust versus soil shear strain under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 143, t = 0.36 m, K0 = 1.0.
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Soil Shear Strain 0.005
0.010
15975
7988
Thrust (kN/m)
0.001
2000
1000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 7. Seismic increment of lining thrust at different soil shear strains: 0.1%, 0.5%, and 1.0%; F = 143, t = 0.36 m, K0 = 1.0. Fig. 9. Seismic increment of lining thrust at soil shear strain of 0.5% under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 3.23, t = 1.26 m, K0 = 1.0.
(F = 143) and 1.26m (F = 3.2), respectively (see Table 1). It is worth noting that a 1.26m lining is far too thick compared to the lining thickness commonly used in practice, however, it is used in this numerical study to account for wide variations in flexibility ratios. A value of K0 = 1.0, and friction coefficients 0, 0.5, 0.8, and 1.0 are used. In all cases, as the coefficient of friction increases, the thrust increment increases as well. However, the thicker lining develops much greater thrust seismic increment and is less sensitive to the soil-tunnel friction. 6. Parametric finite element analyses and closed-form solutions
Fig. 8. Seismic increment of lining thrust at soil shear strain of 0.5% under different friction coefficients: 0, 0.5, 0.8, and 1.0; F = 143, t = 0.36 m, K0 = 1.0.
Wang (1993) and Penzien and Wu (1998). Although, at a relatively small soil shear strain 0.1%, Wang’s ‘‘no-slip” closed-form solution provides a thrust increment value comparable with those from the numerical analyses, at larger shear strains it gives a much higher value (Fig. 7). The closed-form solution is linear, whereas the finite element model incorporates non-linearity of the soiltunnel interface. Therefore, neither of the two closed-form solutions provides estimates of the thrust that are comparable with those from the numerical analyses within a wide range of the soil shear strain values. Figs. 8 and 9 present computed thrust seismic increment distribution at a soil strain of 0.5% for the lining thickness of 0.36m
A series of numerical analyses are conducted as listed in Table 1 and Table 3 to further illustrate the influence of various factors on the computed response of the tunnel lining and the difference with the closed-form solutions. Tables 4 and 5 summarize the maximum and minimum seismic increments of the lining thrust, shear force, and bending moment obtained from numerical analyses and compared with those from the closed-form solutions at the soil shear strain equal to 0.5%. In addition, the lining diametric strain values are summarized in Table 6. The finite element thrust seismic increments are sensitive to various interface conditions and, in general, differ substantially from those of the closed-form solutions, whereas seismic increments of the shear and bending moment obtained from the numerical and closed-form solutions are within the same order of magnitude. As discussed earlier, the bending moment diagram corresponds to the deformed shape of the tunnel which does not vary significantly due to variation in contact condition. In some cases, the closed-form solutions (Penzien and Wu, 1998; Wang, 1993) correlate with the numerical results within tolerances acceptable for practical purposes, while in many other cases they do not. Figs. 10 and 11 present graphical comparisons of the maximum lining
Table 3 Analyses’ matrix. Case ID
Lining radius, R (m)
Lining thickness, t (m)
Flexibility ratio, F
Coefficient of friction, f
Coefficient of lateral earth pressure, K0
2B3 01 2B3 2B3 10 B3 01 B3 B3 10
5 5 5 5 5 5
0.36 0.36 0.36 1.26 1.26 1.26
14.3 143 1430 0.32 3.2 32
0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.8, 0.8, 0.8, 0.8, 0.8, 0.8,
1.0 1.0 1.0 1.0 1.0 1.0
1.0 1.0 1.0 1.0 1.0 1.0
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Table 4 Maximum/minimum seismic increment of lining sectional forces at soil shear strain of 0.5% under different flexibility ratios: 14.3, 143, and 1430; t = 0.36m. Flexibility ratio
Force/moment*
Closed-form solution Wang Full-slip
14.3
143
1430
Finite element solution with contact interface Penzien &Wu
No-slip
Full-slip
No-slip
K0
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
max P
69
931
69
136
min P
69
931
69
136
V
69
136
M
345
345
345
341
max P
76
7988
76
152
min P
76
7988
76
152
V
76
152
M
380
380
380
379
max P
77
36557
77
153
min P
77
36557
77
153
V
77
153
M
384
384
383
384
Coefficient of friction 0
0.5
0.8
1.0
91 113 72 80 130 122 299 301 217 231 53 58 132 130 332 334 460 464 310 311 132 132 344 346
681 843 628 813 125 120 276 263 811 1010 491 645 131 131 324 320 783 832 13 23 132 132 348 348
860 924 819 913 119 108 270 264 1240 1545 755 976 132 132 314 308 1024 1104 132 186 132 131 348 348
905 928 858 922 112 107 266 264 1550 1920 904 1160 132 132 309 303 1201 1307 218 280 132 131 347 348
*
The lining forces and bending moment are designated as follows: P is the thrust increment (kN/m), V is shear force increment (kN/m), M is bending moment increment (kNm/m). Negative thrust increment indicates tension.
Table 5 Maximum/minimum seismic increment of lining sectional forces at soil shear strain of 0.5% under different flexibility ratios: 0.32, 3.2, and 32; t = 1.26 m. Flexibility ratio
Force/moment*
Closed-form solution Wang Full-slip
0.32
32
No-slip
Full-slip
No-slip
K0
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
max P
571
1163
571
1034
min P
571
1163
571
1034
571
1034
V
3.2
Finite element solution with contact interface Penzien &Wu
M
2854
2854
2854
2585
max P
2271
9788
2271
4361
min P
2271
9788
2271
4361
V
2271
4361
M
11354
11354
11354
10903
max P
3234
66011
3234
6431
min P
3234
66011
3234
6431
V
3234
6431
M
16171
16171
16077
16171
Coefficient of friction 0
0.5
0.8
1.0
589 610 582 594 1161 1151 2852 2850 3416 3155 202 555 3934 3970 9388 9538 6593 6406 1420 1218 5863 5859 13,267 13,271
1033 1169 1150 1185 1110 1068 2694 2631 3713 3529 2277 2858 3926 3963 9206 9319 6834 6673 1748 1985 5950 5934 13,023 13,012
1160 1237 1187 1241 1077 1050 2632 2600 3958 3805 3437 4014 3905 3965 9120 9236 7066 6909 3415 3555 5916 5875 12,926 12,920
1219 1243 1225 1243 1060 1049 2611 2598 4103 3986 4039 4637 3907 3948 9086 9205 7218 7083 4200 4455 5840 5879 12,882 12,872
*
The lining forces and bending moment are designated as follows: P is the thrust increment (kN/m), V is shear force increment (kN/m), M is bending moment increment (kNm/m). Negative thrust increment indicates tension.
thrust seismic increments for the tunnel lining thicknesses of 0.36 m and 1.26 m, respectively, at a soil strain of 0.5% and K0 = 1.0. Except for the case of F = 1430, the lining thrust seismic
increment increases with the flexibility ratio for a constant lining stiffness. The thrust seismic increment increases as the thickness of the lining increases for a constant soil stiffness. This can be
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H. Sedarat et al. / Tunnelling and Underground Space Technology 24 (2009) 482–490 Table 6 Seismic increment of diagonal strain of the lining at soil shear strain of 0.5%. Flexibility ratio
Closed-form solution
Finite element solution with frictional interface
Wang
Penzien & Wu
Full-slip (%)
No-slip (%)
Full-slip (%)
No-slip (%)
K0
Coefficient of friction 0 (%)
0.5 (%)
0.8 (%)
1.0 (%)
14.3
0.63
0.63
0.63
0.62
143
0.69
0.69
0.69
0.69
1430
0.70
0.70
0.70
0.70
0.32
0.12
0.12
0.12
0.11
3.2
0.47
0.47
0.47
0.45
32
0.67
0.67
0.67
0.66
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
0.59 0.58 0.64 0.66 0.64 0.64 0.13 0.13 0.56 0.54 0.63 0.63
0.55 0.53 0.63 0.63 0.64 0.64 0.12 0.12 0.54 0.52 0.63 0.63
0.53 0.52 0.63 0.63 0.64 0.64 0.12 0.12 0.53 0.51 0.63 0.63
0.53 0.52 0.63 0.62 0.64 0.64 0.12 0.12 0.52 0.50 0.63 0.63
Flexibility Ratio 36557 Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
1430
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
143 7988
Thrust (kN/m)
14.3
2000
1000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 10. Seismic increment of lining thrust at soil shear strain of 0.5% under different flexibility ratios: 14.3, 143, and 1430; t = 0.36 m, K0 = 1.0.
Flexibility Ratio 3.2
32 66011
9788
Thrust (kN/m)
0.32
6000
The effect of coefficient of lateral earth pressure appears to be significant only for the 0.36 m thick lining and flexibility ratio equal to 143. The soil diametric strain values obtained from the finite element analyses are somewhat below those of the closed-form solutions except for the case of the stiff lining (t = 1.26 m, F = 3.2). The lining thrust seismic increments of the ‘‘full-slip” closedform solutions are comparable to those of the finite element model with the soil-tunnel contact interface only for the 1.26 m thick lining and flexibility ratios equal to 0.32 and 3.2 (Fig. 11). This is also true when Wang’s ‘‘no-slip” closed-form solution is used for the soft soil (F = 14.3 in Fig. 10 and F = 0.32 in Fig. 11). On the other hand, the lining thrust seismic increments of Penzien and Wu’s ‘‘no-slip” closed-form solution seem closer to the numerical analysis values for the thick lining (Fig. 11). For other cases there is no clear correlation between the closed-form solutions and finite element analyses with the soil-tunnel contact interface. The comparisons suggest that the thrust seismic increment in the lining is dependent on several factors including relative soil and lining stiffness and the soil-tunnel interface conditions. The available closed-form solutions are ill-equipped to represent these conditions in a general way and may overestimate or underestimate the lining forces. Therefore, the closedform solutions should be used with caution even as first order estimates. Although the characteristics of the soil-tunnel interface significantly affect the level of the thrust in the tunnel lining, the tunnel lining diagonal strain is mostly controlled by the flexibility ratio (Table 6).
4000
7. Summary and conclusions 2000
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
Wang Full-Slip Penzien Full-Slip Contact - f=0.0 Contact - f=0.5 Contact - f=0.8 Contact - f=1.0 Wang No-Slip Penzien No-Slip
0
Fig. 11. Seismic increment of lining thrust at soil shear strain of 0.5% under different flexibility ratios: 0.32, 3.2, and 32; t = 1.26 m, K0 = 1.0.
clearly observed when comparing thrust values obtained from cases with flexibility ratios 14.3, 143, and 1430 in Fig. 10 with those for flexibility ratios 0.32, 3.2, and 32 in Fig. 11, respectively.
This paper highlighted the importance of a more realistic representation of the soil-tunnel interface when performing seismic racking analysis of underground structures. The contact interface incorporated into a finite element model prevents development of unrealistic normal tensile stresses and controls build up of the tangential tractions between the soil and tunnel lining. As a result, the lining thrust increments due to the tunnel ovaling are significantly different from those of widely used closed-form solutions. Bending moments are less sensitive to the specific assumption regarding the soil-tunneling interface. The frictional contact is a nonlinear phenomenon and cannot be handled by linear closedform solutions. It is shown that even with a very high value of friction coefficient (f = 1.0), the contact interface limits the tangential tractions and provides much lower thrust levels than ‘‘no-slip” closed-form solutions. In some of the cases presented in this paper,
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the closed-form solutions correlate with the numerical results within tolerances acceptable for practical purposes, while in many other cases they do not. While it is not the intention of this study to find out when the closed-form solutions match with numerical analyses, it is quite reasonable that a finite element technique with proper definition of the soil-tunnel interface be always used to simulate seismic response of the tunnels. Although, only circular tunnels of a constant thickness in a uniform soil have been analyzed in this study, many tunnels have a horse-shoe shape with variable thickness in a multilayered soil profile. All these complicating factors are beyond the limiting assumptions of the available closed-form solutions but might be easily addressed with a numerical approach. During a seismic event, the soil and tunnel might undergo both shear and normal strains. Since the lining thrust is sensitive to both normal stresses and tangential tractions at the soil-tunnel lining interface and since the tangential tractions depend on the value of normal contact stresses (Coulomb friction law), the thrust due to seismic racking deformations could be underestimated. For a comprehensive estimation of the seismic response of tunnels, both shear and normal (vertical and horizontal) motions might have to be applied to the soil and tunnel in the course of a time history analysis. The nonlinear behavior of soil is another important aspect in soil-tunnel lining interaction that can be accounted for in finite element simulation of a seismic event and will be considered in future studies.
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