Ovaling deformations of circular tunnels under ...

Tunnelling and Underground Space Technology Tunnelling and Underground Space Technology 20 (2005) 435–441

incorporating Trenchless Technology Research

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Ovaling deformations of circular tunnels under seismic loading, an update on seismic design and analysis of underground structures Youssef M.A. Hashash a

a,*

, Duhee Park b, John I.-Chiang Yao

c

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, MC-250, Urbana, IL 61801, United States b Department of Civil and Environmental Engineering, Hanyang University, Haengdang-Dong, Seoul, Korea c CH2M HIL, 13921 Park Center Road, Suite 600, Herndon, VA 201, United States Received 2 September 2004; received in revised form 28 January 2005; accepted 18 February 2005 Available online 21 April 2005

Abstract Two analytical solutions for estimating the ovaling deformation and forces in circular tunnels due to soil–structure interaction under seismic loading are widely used in engineering practice. This paper addresses an unresolved issue related to discrepancy between the two solutions. A comparison of the two solutions shows that the calculated forces and displacements are identical for the condition of full-slip between the tunnel lining and ground. However, the calculated lining thrusts differ by an order of magnitude when assuming no-slip between the tunnel lining and the ground. The analytical solutions are compared to numerical analyses of the no-slip condition using the finite element method to validate which of the two solutions provide the correct solution. Numerical analysis results agree with one of the analytical solution that provides a higher estimate of the thrust on the tunnel lining, thus highlighting the limitation of the other analytical solution. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Earthquake design; Seismic design; Tunnels; Underground structures

1. Introduction The component that has the most significant influence on the tunnel lining under seismic loading, except for the case of the tunnel being directly sheared by a fault, is the ovaling or racking deformations (Penzien, 2000). Studies suggest that, while ovaling may be caused by waves propagating horizontally or obliquely, vertically propagating shear waves are the predominant form of earthquake loading that causes these types of deformations (Wang, 1993). The ovaling deformation is commonly simulated as a two-dimensional, plane-strain

*

Corresponding author. Tel.: +1 217 333 6986; fax: +1 217 333 9464. E-mail addresses: [email protected] (Y.M.A. Hashash), dpark@ hanyang.ac.kr (D. Park), [email protected] (J.I.-C. Yao). 0886-7798/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2005.02.004

condition. Since the inertia effect can be relatively small, the ovaling deformation is further simplified as a quasistatic case and hence without the dynamic interaction. Wang (1993) and Penzien (2000) present closed form solutions to compute displacements and forces in the lining due to equivalent static ovaling deformations. The analytical solutions are frequently used in estimating the deformation and forces in tunnels. Hashash et al. (2001) identifies a significant discrepancy in the computed lining thrust between the Wang and Penzien solutions, as demonstrated in design example 3 of Hashash et al. (2001). This discrepancy has important implications as far as lining design and is of concern to many design engineers. For example, this discrepancy was an issue on two projects the first author was involved with including, (a) the Muni Metro Turnaround Project in downtown San Francisco, CA and (b) the Alameda

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Y.M.A. Hashash et al. / Tunnelling and Underground Space Technology 20 (2005) 435–441

Nomenclature shear strain maximum shear strain maximum free-field shear strain of soil or rock medium s simple shear stress of a soil element smax maximum shear stress D lateral deflection Ddfree-field free-field diametric deflection in non-perforated ground Ddlining lining diametric deflection Dd nlining lining diametric deflection under normal loading only ml PoissonÕs ratio of tunnel lining mm PoissonÕs ratio of soil or rock medium d diameter or equivalent diameter of tunnel lining h thickness of the soil deposit r radius of circular tunnel c cm cmax

Tubes Retrofit project connecting Oakland to Alameda island in the San Francisco Bay.

2. Analytical solutions of ovaling deformation of circular tunnel with soil–structure interaction The simplest form of estimating ovaling deformation is to assume the deformations in a circular tunnel to be identical to ‘‘free-field’’, thereby ignoring the tunnel– ground interaction. This assumption is appropriate when the ovaling stiffness of the lined tunnel is equal to that of the surrounding ground. The circular tunnel–ground shearing is then modeled as a continuous medium (referred to as non-perforated ground) without the presence of the tunnel (Fig. 1), in which the diametric strain for a circular section is calculated as:

t C El Em F Gm I K1 K2 Mmax R Rn Tmax Vmax

thickness of tunnel lining compressibility ratio of tunnel lining modulus of elasticity of tunnel lining modulus of elasticity of soil or rock medium flexibility ratio of tunnel lining shear modulus of soil or rock medium moment of inertia of the tunnel lining (per unit width) for circular lining full-slip lining response coefficient no-slip lining response coefficient maximum bending moment in tunnel crosssection due to shear waves lining-soil racking ratio lining-soil racking ratio under normal loading only maximum thrust in tunnel lining maximum shear force in tunnel cross-section due to shear waves

Dd free-field c ¼ max d 2 Eq: ð8Þ in Hashash et al. (2001).

ð1Þ

If the ovalling stiffness is very small compared to the surrounding ground, the tunnel distortion or diametric strain is calculated assuming an unlined tunnel (referred to as perforated ground): Dd free-field ¼ 2cmax ð1 mm Þ d Eq: ð9Þ in Hashash et al. (2001).

ð2Þ

This deformation is much greater in the case where the presence of the tunnel is included compared to the case where only the continuous ground deformation is assumed. In most cases the lining ground interaction has to be taken into account. As a first step, the relative stiffness of the tunnel to the ground is quantified by the compressibility and flexibility ratios (C and F), which are measures of the extensional and flexural stiffnesses (resistance to ovaling), respectively, of the medium relative to the lining (Hoeg, 1968; Peck et al., 1972): Em 1 m2l r C¼ El tð1 þ mm Þð1 2mm Þ Eq: ð19Þ in Hashash et al. (2001); ð3Þ Em 1 m2l r3 F ¼ 6El I ð1 þ mm Þ Eq: ð20Þ in Hashash et al. (2001);

Fig. 1. Free-field shear distortion of perforated (tunnel cavity is empty) and non-perforated ground (tunnel cavity is filled), circular shape (after Wang, 1993).

ð4Þ

where Em is the modulus of elasticity of the medium, I is the moment of inertia of the tunnel lining (per unit


width) for circular lining, r, t the radius and thickness of the tunnel lining respectively. In early studies of racking deformations, Peck et al. (1972), based on earlier work by Burns and Richard (1964) and Hoeg (1968), proposed closed-form solutions in terms of thrusts, bending moments, and displacements under external loading conditions. The response of a tunnel lining is expressed as functions of the compressibility and flexibility ratios of the structure, and the in-situ overburden pressure and at-rest coefficient of earth pressure of the soil. The solutions are developed for both full-slip and no-slip condition between the tunnel and the lining. Full-slip condition results in no tangential shear force. Wang (1993) reformulated the equations to adapt to seismic loadings caused by shear waves. The free-field shear stress replaces the in situ overburden pressure and the at-rest coefficient of earth pressure is assigned a value of (1) to simulate simple shear condition. The shear stress is further expressed as a function of shear strain. Assuming full-slip conditions, the diametric strain, the maximum thrust, and bending moment can be expressed as (Wang, 1993): Dd lining 2 ¼ K 1F Dd free-field 3 Eq: ð27Þ in Hashash et al. (2001); 1 Em T max ¼ K 1 rc 6 ð1 þ mm Þ max Eq: ð22Þ in Hashash et al. (2001); 1 Em M max ¼ K 1 r2 c 6 ð1 þ mm Þ max Eq: ð23Þ in Hashash et al. (2001);

Em rc 2ð1 þ mm Þ max

Eq: ð25Þ in Hashash et al. (2001);

ð9Þ

where ð6Þ

K2 ¼ 1 þ

F ½ð1 2mm Þ ð1 2mm ÞC 12 ð1 2mm Þ2 þ 2 F ½ð3 2mm Þ þ ð1 2mm ÞC þ C 52 8mm þ 6m2m þ 6 8mm

Eq: ð26Þ in Hashash et al. (2001). ð7Þ

where 12ð1 mm Þ 2F þ 5 6mm Eq: ð24Þ in Hashash et al. (2001).

Fig. 2. Sign convention for force components in circular lining (after Penzien, 2000).

T max ¼ K 2 smax r ¼ K 2 ð5Þ

437

K1 ¼

ð8Þ

The sign convention for the above force components in circular lining is shown in Fig. 2. According to various studies, slip at the interface is only possible for tunnels in soft soils or cases of severe seismic loading intensity. For most tunnels, the interface condition is between full-slip and no-slip, so both cases should be investigated for critical lining forces and deformations. However, full-slip assumptions under simple shear may cause significant underestimation of the maximum thrust, so it has been recommended that the no-slip assumption of complete soil continuity be made in assessing the lining thrust response (Schwartz and Einstein, 1980):

ð10Þ

Note that no solution is developed for calculating diametric strain and maximum moment under no-slip condition. It is recommended that the solutions for full-slip condition be used for no-slip condition. The more conservative estimates of the full-slip condition is considered to offset the potential underestimation due to pseudo-static representation of the dynamic problem (Wang, 1993). Penzien and Wu (1998) and Penzien (2000) developed similar analytical solutions for thrust, shear, and moment in the tunnel lining due to racking deformations. Assuming full slip condition, solutions for thrust, moment, and shear in circular tunnel linings caused by soil–structure interaction during a seismic event are expressed as (Penzien, 2000): d Dd nlining ¼ Rn Dd free-field ¼ Rn cmax 2 modified after Eq: ð29Þ in Hashash et al. (2001); ð11Þ

438


T max ¼

12El IDd nlining 3

d ð1

m2l Þ

¼

6El IRn cmax d 2 ð1 m2l Þ

a¼

modified after Eq: ð30Þ in Hashash et al. (2001); ð12Þ M max ¼

6El IDd nlining 2

d ð1

m2l Þ

¼

24El IDd nlining 3

d ð1

m2l Þ

¼

12El IRn cmax d 2 ð1 m2l Þ

modified after Eq: ð32Þ in Hashash et al. (2001). ð14Þ The lining-soil racking ratio under normal loading only is defined as: Rn ¼

4ð1 mm Þ ðan þ 1Þ

Eq: ð33Þ in Hashash et al. (2001);

ð15Þ

where an ¼

12El Ið5 6mm Þ d 3 Gm ð1 m2l Þ

Eq: ð34Þ in Hashash et al. (2001).

ð16Þ

In the case of no slip condition, the formulations are presented as: d Dd lining ¼ RDd free-field ¼ R cmax 2 modified after Eq: ð35Þ in Hashash et al. (2001); ð17Þ T max ¼

24El IDd lining 12El IRcmax ¼ 2 d 3 ð1 m2l Þ d ð1 m2l Þ

modified after Eq: ð36Þ in Hashash et al. (2001); ð18Þ M max

6El IDd lining 3El IRcmax ¼ 2 ¼ 2 dð1 m2l Þ d ð1 ml Þ

modified after Eq: ð37Þ in Hashash et al. (2001); ð19Þ V max ¼

24El IDd lining 12El IRcmax ¼ 2 3 2 d ð1 ml Þ d ð1 m2l Þ

modified after Eq: ð38Þ in Hashash et al. (2001);

ð22Þ

The two analytical solutions are compared by calculating the forces and displacements for three typical circular tunnels with concrete lining. The dimension and material properties of the lined tunnels are given in Tables 1 and 2. The soil properties selected for Case 1 of Table 1 are identical to design example 3 presented in Hashash et al. (2001). Three combinations of YoungÕs modulus and PoissonÕs ratio are used. Maximum applied shear strain is 0.00252 as calculated in design example 3 of Hashash et al. (2001). The calculated forces (maximum thrust and moment) and displacement (racking ratio), for both full-slip and no-slip conditions, are listed in Table 3. The comparisons show that the calculated forces and displacements are identical for the full-slip assumption for all three cases. However, PenzienÕs solutions result in much lower estimates of maximum thrusts compared to WangÕs solutions assuming no-slip condition. PenzienÕs solutions result in approximately doubling the values of thrust for no-slip compared to full-slip condition. WangÕs solution provides a much larger value for thrust. For Case 1, thrust for no-slip is almost 16 times higher than for full-slip. PenzienÕs maximum moment and racking ratios are almost identical for full-slip and no-slip conditions. This contradicts WangÕs conclusion that full-slip condition results in more conservative estimates.

Table 1 Soil properties used in the analysis Soil parameter

Value

Case 1

YoungÕs modulus (Em) PoissonÕs ratio (m) YoungÕs modulus (Em) PoissonÕs ratio (m) YoungÕs modulus (Em) PoissonÕs ratio (m)

Case 2 Case 3

312,000 kN/m2 0.3 312,000 kN/m2 0.49 185,400 kN/m2 0.49

Table 2 Tunnel lining properties used in the analysis

ð20Þ

Lining parameter

Value

ð21Þ

YoungÕs modulus (El) Area (per unit width) Moment of inertia (I) Lining thickness (t) Weight PoissonÕs ratio (m)

24,800,000 kN/m2 0.3 m2/m 0.00225 m4/m 0.3 m 0 0.2

where 4ð1 mm Þ ða þ 1Þ Eq: ð39Þ in Hashash et al. (2001);

Eq: ð40Þ in Hashash et al. (2001).

2.1. Comparison of closed form solutions including soil structure interaction

3El IRn cmax dð1 m2l Þ

modified after Eq: ð31Þ in Hashash et al. (2001); ð13Þ V max ¼

24El Ið3 4mm Þ d 3 Gm ð1 m2l Þ

R¼


439

Table 3 Calculated forces and stress using the analytical solutions Wang

Penzien

% Difference

Full slip

No slip

Full slip

No slip

Full slip

No slip

Case 1 Racking ratio Maximum thrust (Tmax) Maximum moment (Mmax) Maximum shear (Vmax)

2.58 62.94 188.81 –

2.58 1045.38 188.81 –

2.58 62.94 188.81 125.87

2.55 124.64 186.95 124.64

0 0 0

1.2 738.7 1.0

Case 2 Racking ratio Tmax Mmax Vmax

1.92 46.83 140.48 –

1.92 813.59 140.48 –

1.92 46.83 140.48 46.83

1.92 93.60 140.40 93.60

0 0 0

0.0 769.2 0.8


1.84 44.99 134.97 –

1.84 507.21 134.97

1.84 44.99 134.97 44.99

1.84 89.90 134.85 89.90

0 0 0 0

0.0 464.2 41.5

Note that the solutions of Case 1 are different from the results calculated in Hashash et al. (2001). Minor errors in the previous calculations (maximum applied strain should have been 0.00252 instead of 0.0021 and moment of inertia of the lining should be 0.00225 m4/m instead of 0.0023 m4/m) are corrected in this paper.

3. Numerical analysis to evaluate analytical solutions A series of numerical analyses are performed using the finite element code PLAXIS (PLAXIS-B.V., 2002) to evaluate the analytical solutions for ovaling deformations of circular tunnels. The numerical analysis uses first principle for the solution of a boundary va-

lue problem and is expected to give the correct solution. The analyses use assumptions identical to those of the analytical solution; (a) plane-strain condition, (b) ground and lining are linear elastic and mass-less materials. Shear loading is applied at the upper ends of the boundaries to simulate pure shear condition. In PLAXIS, only no-slip condition between the tunnel lining and ground is simulated. The numerical analysis solution is first verified by analyzing non-perforated and perforated ground. The computed ovaling deformations are nearly identical to those obtained from Eqs. (1) and (2). The results of the numerical analyses of tunnel– ground interaction for Cases 1–3 are presented and compared to analytical solutions in Table 4 and

Table 4 Comparison of analytical solution with numerical solution Numerical

Wang

No-slip

No-slip

Numerical vs. Wang (%)

Penzien

Numerical vs. Penzien (%)


2.18 1050 158.87 105.98

2.58 1045.38 188.81 –

15.5 0.4 15.9

2.55 124.64 186.95 124.64

14.5 742.4 15.0 15.0


1.86 820.86 138.89 95.28

1.92 813.59 141.57 –

3.1 0.9 1.9

1.92 93.60 140.40 93.60

3.1 777.0 1.1 1.8


1.82 511.28 133.43 90.38

1.84 507.21 134.97

1.1 0.8 1.1

1.84 89.90 134.85 89.90

1.1 468.7 1.1 0.5

No-slip

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Y.M.A. Hashash et al. / Tunnelling and Underground Space Technology 20 (2005) 435–441 1400

200 Wang

Wang (Analytical solutions)

Case 1

Penzien 1000 Case 2 800 600

Case 3

M 200 0

Case 2

Case 3 0

200

Case 1

Penzien

150

Case 3

Case 2

100

max

1:1 Line

400

T

max

(Analytical solutions)

1200

1:1 Line 50

Case 1

400 600 800 1000 T (Numerical solution)

0

1200 1400

0

50 M

max

max

100 150 (Numerical solution)

200

Racking ratio (Analytical solutions)

3 Wang 2.5

Case 1

Penzien

2 Case 3

Case 2

1.5 1:1 Line

1 0.5 0

0

0.5 1 1.5 2 2.5 Racking ratio (Numerical solution)

3

Fig. 3. Comparison of calculated forces and racking ratios of numerical solution (x-axis) and analytical solutions (y-axis).

Fig. 3. The calculated maximum moments and racking ratios agree well with both analytical solutions of Case 2 and 3, but differ by approximately 15% for Case 1. The maximum axial thrusts from numerical analyses result in almost perfect match with WangÕs solutions, whereby the differences are within 1% for all three cases. However, the difference between the numerical and PenzienÕs solutions are significant. The difference is higher than 700% for Cases 1 and 2, in which PenzienÕs solutions highly underestimate the thrust for all three cases. The comparisons clearly demonstrate that the WangÕs solution provides a realistic estimate of the thrust in the tunnel linings for the no-slip condition. It is recommended that the PenziensÕs solution not be used for no-slip condition.

4. Summary and conclusions Two available analytical solutions to compute induced forces and deformations due to ovaling deformation of a circular tunnel are presented. The solutions provide identical results for the condition of full-slip between the tunnel lining and the ground but differ in values of the calculated thrust for the

condition of no-slip. This discrepancy is a source of confusion in the design of circular tunnel lining. Two-dimensional finite element analyses are performed to validate which of the two analytical solutions provide the correct solution. Comparison with numerical analysis demonstrates that one of the solutions significantly underestimates the thrust in the tunnel lining for the condition of no-slip and should not be used for that condition.

References Burns, J.Q., Richard, R.M., 1964. Attenuation of stressses for buried cylinders. In: Proceedings of the Symposium on Soil–Structure Interaction. University of Arizona, Tempe, AZ. Hashash, Y.M.A., Hook, J.J., Schmidt, B., Yao, J.I.-C., 2001. Seismic design and analysis of underground structure. Tunn. Undergr. Sp. Technol. 16, 247–293. Hoeg, K., 1968. Stresses against underground structural cylinders. J. Soil Mech. Found. Div. 94 (SM4). Peck, R.B., Hendron, A.J., Mohraz, B., 1972. State of the art in soft ground tunneling. In: The Proceedings of the Rapid Excavation and Tunneling Conference. American Institute of Mining, Metallurgical, and Petroleum Engineers, New York, NY, pp. 259–286. Penzien, J., 2000. Seismically induced racking of tunnel linings. Int. J. Earthquake Eng. Struct. Dynamics 29, 683–691.

Y.M.A. Hashash et al. / Tunnelling and Underground Space Technology 20 (2005) 435–441 Penzien, J., Wu, C.L., 1998. Stresses in linings of bored tunnels. Int. J. Earthquake Eng. Struct. Dynamics 27, 283–300. PLAXIS-B.V., 2002. PLAXIS: Finite element Package for Analysis of Geotechnical Structures, Delft, Netherland. Schwartz, C.W., Einstein, H.H., 1980. Improved Design of Tunnel Supports, vol. 1. Simplified Analysis for Ground–

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structure Interaction in Tunneling. UMTA-MA-06-010080-4, Urban Mass Transit Transportation Administration, MA. Wang, J.N., 1993. Seismic Design of Tunnels: A State-of-the-art Approach. Parsons Brinckerhoff Quade & Douglas, Inc., New York, NY, Monograph 7.