A modified solution of radial subgrade modulus for a circular tunnel in elastic ground

A modified solution of radial subgrade modulus for a circular tunnel in elastic ground

Soils and Foundations 2014;54(2):225–232 The Japanese Geotechnical Society Soils and Foundations www.sciencedirect.com journal homepage: www.elsevier...

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Soils and Foundations 2014;54(2):225–232 The Japanese Geotechnical Society

Soils and Foundations www.sciencedirect.com journal homepage: www.elsevier.com/locate/sandf

A modified solution of radial subgrade modulus for a circular tunnel in elastic ground Dongming Zhanga, Hongwei Huanga,n, Kok Kwang Phoonb, Qunfang Huc a

Key Laboratory of Geotechnical and Underground Engineering of Minister of Education, Department of Geotechnical Engineering, Tongji University, Shanghai, China b Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, # 07-03, 1 Engineering Drive 2, Singapore 117576, Singapore c Shanghai Institute of Disaster Prevention and Relief, Shanghai, China Received 16 May 2013; received in revised form 19 August 2013; accepted 2 October 2013 Available online 13 April 2014

Abstract In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (kr). The widely used solution kr for a circular tunnel in elastic ground proposed by Wood (1975) was found to be applicable only when the tunnel radial deformation is oval-shaped. On the basis of the Wood's solution, this note presents a general solution for kr when the radial deformation of the tunnel is described by a Fourier series. This modified Wood's solution of kr using compatible stress functions is validated by a numerical example. The modified solution for the example shows good consistency with the original Wood's solution when the tunnel becomes an oval shape with deformations. The example indicates that the magnitude of kr is significantly affected by the distribution shape of the tunnel radial deformation. The value of kr is no longer a constant value around the tunnel when the tunnel deforms into a general shape described by a Fourier series. It is quite different from the value of kr for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for kr is illustrated by a design case using a bedded beam model. & 2014 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Tunnel lining; Winkler model; Radial subgrade modulus; Fourier series; Elastic analysis

1. Introduction

n

Corresponding author. Tel.: þ86 2165989273; fax: þ 86 2165985017. E-mail addresses: [email protected] (D. Zhang), [email protected] (H. Huang), [email protected] (K.K. Phoon), [email protected] (Q. Hu). Peer review under responsibility of The Japanese Geotechnical Society.

Recommended by guidelines for the structural design of tunnel linings, the bedded beam model based on Winkler springs is widely used by tunnel design engineers (RTRI, 1997; ITA, 2004; JSCE, 2007). In this model, however, it is not easy for engineers to determine the radial subgrade modulus (kr) appropriately (Duddeck and Erdmann, 1982; Mair, 2008; Gruebl, 2012). Table 1 shows values for kr recommended by Standards in China and Japan for shielddriven tunnels. Table 1indicates that even when the soil is of a specific type, engineers still have to select a value from

http://dx.doi.org/10.1016/j.sandf.2014.02.012 0038-0806 & 2014 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

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D. Zhang et al. / Soils and Foundations 54 (2014) 225–232

Nomenclature Am, Bm, am EC, v kr, krm m

Cm, Dm coefficients in stress function weight of the krm soil Young's modulus, Poisson's ratio radial subgrade modulus, the value for a general mode m shape mode of the tunnel radial deformation

the wide range of kr based on their own experience. Unfortunately, the structural behavior of the segmental lining and the joints have been found to be quite sensitive to the selected magnitude of kr (Lee et al., 2001). Hence, some analytical solutions of kr have been put forward as a rational method to determine the magnitude of this parameter (Arnau and Molins, 2011). To derive these analytical solutions for kr, the distribution shape of the radial deformation of the tunnel (hereafter referred to as the distribution shape) must be prescribed. Usually this distribution shape is assumed to be either circular (Sagaseta, 1987) or oval (Wood, 1975). However, a single distribution shape might not be sufficient to describe the actual behavior of the tunnel lining due to the complex soil-lining interactions. Hence, a solution of kr based on the single circular or oval shape would be ideal for tunnel designs. In this note, an analytical solution for kr for a more general distribution shape described by a Fourier series (Eq. (1)) is presented. ur ¼ ∑ um cos mθ

ur, uθ soil radial and tangential displacement um weight of a general mode m in Fourier series r, r0, r1 radial coordinate, tunnel radius, radius of the large circle in FEM mesh sr soil radial stress changes θ angle measured counterclockwise from the tunnel crown

2. Problem statement Fig. 1 shows the problem geometry. A circular tunnel with radius r0 is embedded in a homogeneous isotropic infinite elastic ground. The tunnel is assumed to deform radially into the shape described by a Fourier series due to tunneling. With the prescribed radial displacement and the calculated stress change, the radial subgrade modulus (kr) can be obtained as follows: sr  kr ¼  ð2Þ ur r ¼ r 0 The assumptions made in this note are the same as those made by Wood (1975): (a) the plane strain condition is in a direction perpendicular to the cross section of the tunnel; (b) σr

σθ

ð1Þ

τrθ

m

Fourier series deformation shape ur=∑umcosmθ

r=r1

Besides the solution for an oval shape, Wood (1975) also presented a solution of kr for the distribution shape described by a general term of the above Fourier series. However, the Airy stress function used in the Wood (1975)'s solution of kr for the general term does not satisfy the strain compatibility equation identically. Wood (1975)'s solution is thus revised using compatible Airy stress functions to derive a complete solution of kr for a general distribution shape described in the form of a Fourier series. A numerical example presented by Bobet (2001) is adopted to validate the proposed analytical solution of kr. Finally, a design case is introduced to illustrate the applicability of the proposed solution of kr to the bedded beam model.

θ r=r0



ur

Undeformed tunnel shape

Homogeneous linear elastic soil: EC, v

Fig. 1. Geometry of the problem.

Table 1 Parameter kr recommended by standards in China and Japan. Type of soil

a

kr (MPa/m) (kr  2r0)b (MPa) a

Clayer or silty soils

Sandy soils

Very soft

Soft

Medium

Stiff

Very loose

Loose

Medium

Dense

3–15 0–4

15–30 4–15

30–150 15–46

4150 446

3–15 0–28

15–30

30–100

4 100 28–55

China Standard (Liu and Hou, 1997). Japan Standard (RTRI, 1997): radial subgrade modulus (kr)  tunnel diameter (2r0).

b

D. Zhang et al. / Soils and Foundations 54 (2014) 225–232

the interface between the tunnel lining and ground is assumed to have a small coefficient of friction, so that the shear stress at the interface can be neglected; and (c) the stress change and the displacement of the ground at infinity from the tunnel axis must be zero. 3. Wood (1975)'s solution On the basis of the work done by Morgan (1961), Wood (1975) refined a solution of kr for a distribution shape described by a general Fourier term (Eq. (3)), namely: ur ¼ um cos mθ;

mZ1

ð3Þ

To solve this two-dimensional problem, the Airy stress function method is adopted. The Airy stress function should satisfy both the differential equations of equilibrium and the compatibility equation. Thus, the solution of this two-dimensional problem is reduced to finding a compatible Airy stress function that only needs to satisfy the boundary conditions of the problem. The Airy stress function for Wood (1975)'s solution is ϕ ¼ ðcr  2 þ dÞ cos mθ

ð4Þ

The respective solutions for the radial stress change, radial displacement and tangential displacement, satisfying the equilibrium equations, the constitutive laws and the boundary conditions, are: sr ¼  cr  4 ðm2 þ 2  3m2 r 2 r 0 2 Þ cos mθ

ð5aÞ

ð1þ vÞcr  3  2 þ m2 þ vð4  m2 Þ 3EC  þ 9r 2 r 0 2 ðv 1Þm2 cos mθ

ð5bÞ

ð1þ vÞcr  3  2 m  16þ vð16  4m2 Þ 3E C m  þ 9r 2 r 0 2 ð2v 1Þm2 sin mθ

ð5cÞ

ur ¼

uθ ¼ 

By substituting Eqs. (5a) and (5b) into Eq. (2), the corresponding kr is kr ¼

3ðm2  1ÞE C ð1 þ vÞ½ð4m2  1Þ  ð4m2 þ 2Þvr 0

in Wood (1975)'s solution is not compatible for general values of m. This incompatibility of the Airy stress function adopted in the Muir Wood's solution is further demonstrated when an FEM analysis is used with the commercial finite element code ABAQUS™. This has been widely used for the numerical analysis for tunnels (Cui and Kimura, 2010). The finite element mesh, shown in Fig. 2, is a half ring cut from the infinite problem domain. The linear elastic soil is simulated by 3600 6-node quadratic triangle elements. The soil Young's modulus is assumed to be 20 MPa and different soil Poisson's ratios are studied. The inner arc is the tunnel boundary at r¼ r0 ¼ 3 m (BC-C) and the outer arc is the soil element at r¼ r1 ¼ 10 m (BC-A). The infinite problem domain away from the outer arc is simulated by 90 5-node infinite elements provided by the ABAQUS/Standard program. The length d in the infinite direction for each infinite element is set 2 times r1 (shown in Fig. 2). The boundary BC-B is an axis of symmetry. The displacement boundary conditions calculated by Eqs. (5b) and (5c) are applied on the tunnel boundary BC-C. With the calculated stress changes and the prescribed displacements, kr can be computed numerically from Eq. (2). Fig. 3 compares the numerical values of kr derived from the FEM with the analytical solutions (Eq. (6)) for different Poisson's ratios. The numerical values do not match the analytical solutions, except for the case where m ¼ 2. This inconsistency is likely to be caused by an incompatible Airy stress function, as noted previously. Hence, the Wood (1975)'s solution of kr is correct only when the tunnel deforms into an oval distribution shape. Fortunately, this oval shape might be a fairly satisfactory assumption in cases characterized by an absence of information about tunnel deformation. That explains why the solution proposed by Wood in the 1970s is still widely used today, especially in the preliminary design stage. However, the in-situ soil-lining interactions induced by tunneling can be highly complex due to various factors (Koyama, 2003; Sung et al., 2006; Shahin et al., 2011;

ð6Þ

From Eq. (6), Wood (1975) found that the parameter m has little effect on the magnitude of kr. As a result, he concluded that it is reasonable to assume kr is independent of the distribution shape and the error introduced by this assumption is acceptable when compared with the uncertainty associated with other soil parameters. However, by substituting the Airy stress function (Eq. (4)) into the strain compatibility equation,  2  2  ∂ 1∂ 1 ∂2 ∂ φ 1 ∂φ 1 ∂2 φ þ þ þ þ ∂r 2 r ∂r r 2 ∂θ2 ∂r 2 r ∂r r 2 ∂θ2 ¼ cr  6 cos mθð2  mÞð2 þ mÞð16 m2 þ 3m2 r 2 r 0 2 Þ ¼ 0 ð7Þ Eq. (7) cannot be identically satisfied except in special cases where m ¼ 2. In other words, the Airy stress function adopted

227

Fig. 2. Finite element mesh.

228

D. Zhang et al. / Soils and Foundations 54 (2014) 225–232

Huang et al., 2013). Even for the preliminary design in the greenfield, some of the analytical solutions (Bobet, 2001) have also indicated that there should be another shape mode, m, for the tunnel deformation. A single distribution shape (m ¼ 2) may well be insufficient to describe the actual behavior of the tunnel lining. Hence, a solution of kr for a distribution shape described by a general term of a Fourier series should be helpful in the design of the tunnel lining. In this case, a modified solution of kr found by adopting a compatible Airy stress function for the general Fourier term is presented and validated by the FEM results below.

4. Modified solution A compatible Airy stress function corresponding to Eq. (3) is chosen from the general stress function proposed by Timoshenko and Goodier (1970) as follows: ( φ¼

ðAm r  m þ Bm rm þ C m r 2  m þ Dm r 2 þ m Þ cos mθ;  1Þ ðA1 r  1 þ C 1 r ln r þ D1 r 3 Þ cos θ þ 2ðv 1  2v C 1 rθ sin θ;

m Z 2; m ¼ 1;

ð8Þ The general case for m Z 2 is first considered. Corresponding to the Airy stress function for m Z2 in Eq. (8), the modified solutions of the stress change and the displacement of the ground are   ð9aÞ sr ¼ ðm  1ÞCm r  m ðr 2 r 0 2  1Þm  2 cos mθ 6

Dimensionless k r k rr0 /E

5

v=0 (Wood(1975))

v=0 (FEM)

v=0.35 (Wood(1975))

v=0.35(FEM)

v=0.49 (Wood(1975))

v=0.49(FEM)

4

uθ ¼

0

3

4 5 Mode parameter m

6

7

Fig. 3. Validation of Wood (1975)'s solution of kr. 7 6 5

v=0 (Modified)

v=0 (FEM)

v=0.35 (Modified)

v=0.35 (FEM)

v=0.49 (Modified)

v=0.49 (FEM)

4 3 2 1 0

2

3

4 5 Mode parameter m

6

Fig. 4. Validation of modified solution of kr.

7

  ð1þ vÞC m  mðm  1Þr 20  ðm þ 1Þ½m  4ð1 vÞr 2 1 þ m ðm þ 1ÞEC r  sin mθ ð9cÞ

Eq. (10) is also validated using an FEM analysis. The finite element mesh and the soil properties are the same as those used in the FEM analysis for the validation of the Wood (1975)'s solution in Fig. 2. Following the previous procedure, the displacements calculated by Eqs. (9b) and (9c) are first applied on the tunnel boundary BC-C. Then the stress changes are calculated and the numerical results of kr can be obtained from Eq. (2). Fig. 4 shows the results of kr both from the numerical analysis and from Eq. (10). The numerical results are clearly in good agreement with the analytical solutions from Eq. (10), indicating that the modified solution of kr for the distribution shape described by a general term of Fourier series is correct. In Fig. 4, both the analytical and numerical results show a significant effect of the mode parameter m on the magnitude of kr. Corresponding to the Airy stress function for the case where m ¼ 1 in Eq. (8), solutions of the stress change and the displacement of the ground are  2  r 0 ð1 2vÞþ r 2 ð3  2vÞ C 1 cos θ ð11aÞ sr ¼ r 3 ð2v  1Þ

uθ ¼ 2

ð9bÞ

By substituting Eqs. (9a) and (9b) into Eq. (2), the kr is given by sr  ðm2  1ÞEC kr ¼  ¼ ð10Þ ur r ¼ r0 ð1 þ vÞ½2mð1  vÞþ 1  2vr 0

ur ¼

2

  ð1þ vÞC m  mðm  1Þr 20  ðm þ 1Þ½m þ 2ð1  2vÞr 2 1 þ m ðm þ 1ÞE C r  cos mθ

3

1

Dimensionless k r k rr0 /E

ur ¼

 C 1 ð1 þ vÞ  2  2 r 0 r ð1  2vÞþ ln rð8v  6Þ cos θ 2ð1 2vÞE C

ð11bÞ

 C 1 ð1 þ vÞ  2  2 r 0 r ð1  2vÞþ 2½ ln rð3 4vÞþ 1 sin θ 2ð1 2vÞE ð11cÞ

It is interesting to note that the solutions of soil displacement (Eqs. (11b) and (11c)) have a term expressed by the logarithm of the radial coordinate. In this case, the amount of displacement increases as the distance from the tunnel axis increases, which is in conflict with the assumption (c). Mathematically, this is a consequence of the unbalanced boundary condition when the tunnel deforms into the mode shape when m ¼ 1, but it has no physical meaning in reality. To avoid this confliction, Bobet (2001) introduced a characteristic dimension to make the displacement of the ground arbitrarily equal to zero for rZ characteristic dimension. The adoption of the characteristic dimension is reasonable because in actual cases there is always a stiff layer below the tunnel where displacement should be zero. The effect of the depth of the stiff layer on the soil stress is not significant as long as the depth of the layer is

D. Zhang et al. / Soils and Foundations 54 (2014) 225–232

approximately 5 times the tunnel radius (Bobet, 2001). In addition, the arching effect of the soils for a deep tunnel also makes the ground behavior insignificant above the tunnel for a distance of characteristic dimension (Chevalier and Otani, 2011). Based on Bobet's suggestion, the characteristic dimension in this note is considered to be 10 times the tunnel radius. Eqs. (11a) and (11b) is modified as follows (r r characteristic dimension): sr ¼

ur ¼

uθ ¼

  C  ð1  2vÞr 20 r  3  ð10r 0 Þ  3 2v  1  þ ð3  2vÞ½r  1  ð10r 0 Þ  1  cos θ

ð12aÞ

  Cð1 þ vÞ  ð1  2vÞr 20 r  2  ð10r 0 Þ  2 2ð1  2vÞE  þ ð8v  6Þln ½rð10rÞ  1  cos θ

ð12bÞ

  Cð1 þ vÞ  ð1  2vÞr 20 r  2  ð10r 0 Þ  2 2ð1  2vÞE  þ 2ð3  4vÞln ½rð10r 0 Þ  1  cos θ

ð12cÞ

By substituting Eqs. (12a) and (12b) into Eq. (2), the solution of kr for m ¼ 1 then becomes kr ¼

9ð411 422vÞE C 5ð1 þ vÞ½99 þ 600 ln 10 ð198 þ 800 ln 10Þvr 0

5. Numerical example The following distribution shape presented by Bobet (2001) is adopted to validate the analytical solution presented in Eq. 16: ur0 ¼  u0  u1 cos θ  u2 cos 2θ þ u3 cos 3θ

u0 u0

u3

r0 u2

u3 u2

Underformed shape

m=2

Bobet (2001)'s shape

u3

u3 m=3

Fig. 5. The Bobet (2001)'s distribution shape.

ð15Þ

By substituting Eqs. (1) and (15) into Eq. (2), it follows that the solution of kr for any arbitrary distribution shape described by a Fourier series is 1

∑ krm um cos mθ 1 sr  kr ¼  ¼ ∑ am krm ¼ m ¼10 ur r ¼ r 0 m¼0 ∑ um cos mθ

m=1

u2

ð16Þ

2.0 Modified solution Wood(1975)'s solution FEM (Modified solution) FEM (Wood (1975)'s solution)

1.5 Dimensionless k r k rr0/Ec

m¼0

u1

m=0

sr ¼ kr0 u0 þ k r1 u1 cos θ þ ⋯þ krm um cos mθ þ ⋯ 1

u1

u0

Bobet (2001)'s general shape u0:u1 :u2 :u3 =190:77:83:1

ð13Þ

So far, the distribution shape of tunnel radial deformation has only been described by a single term of the Fourier series. However, because of the complexity of real tunnel deformations, the Fourier series (Eq. (1)) is preferred to describe the distribution shape. By invoking superposition, the soil stress change corresponding to Eq. (1) is

ð17Þ

A specific Bobet's shape is chosen and plotted in Fig. 5(a). Fig. 5(b)–(e) are the shapes corresponding to a single mode parameter m (m ¼ 0, 1, 2 and 3) in Bobet's shape. The Young's modulus of the soil is assumed to be 20 MPa and its Poisson's ratio is assumed to be 0.3. The tunnel radius is 3.0 m. For comparison purposes, both Wood (1975)'s solution (Eq. (6)) and the modified solutions developed in the present study (Eqs. (10) and (13)) are used to formulate the krm in Eq. (16). The analytical solutions of kr for the Bobet's shape are obtained by using Eq. (16) and plotted against the angle θ in Fig. 6. The validations of these analytical solutions are carried out by using a similar approach adopted for the validation of Eqs. (6) and (10). The displacements for each

The case in which the mode parameter m is equal to zero, which represents a circular distribution shape, also needs to be considered to cover all the terms in a Fourier series. The solution of kr for this circular shape has already been solved (Sagaseta, 1987) and is given by sr  EC ¼ ð14Þ kr ¼  ur r ¼ r0 ð1þ vÞr 0

¼ ∑ krm um cos mθ

229

max krr0/E (Single m=3)

Wood (1975)'s Solution

1.0 Modified Solution 0.5

min krr0/E (Single m=1)

m¼0

where krm is the kr for the distribution shape described by a general Fourier term (Eqs. (10), (13) and (14)), and am is the weight of krm, expressed by (um/ur)cos mθ.

0.0 0.0

22.5

45.0

67.5 90.0 112.5 The angle θ

135.0

157.5

Fig. 6. kr for Bobet (2001)'s distribution shape.

180.0

230

D. Zhang et al. / Soils and Foundations 54 (2014) 225–232

single term in Eq. (16) are superposed and applied to the tunnel boundary. The numerical values of kr are calculated and plotted in Fig. 6. In Fig. 6, the modified solutions are consistent with the corresponding numerical results. Wood (1975)'s solutions do not agree with the corresponding numerical results. It is also interesting to find that the kr at the tunnel springline (θ ¼ 90º) calculated by Wood (1975)'s solution are exactly the same as those calculated by modified solutions. The reason is that cos θ and cos 3θ vanish when θ is equal to 90º. The deformation shape (Eq. (17)) is made up of two terms (i.e. u0 and u2 cos 2θ) at tunnel springline. As mentioned in relation to Fig. 3, Wood, (1975)'s solution is correct for the case where m ¼ 2. Hence, the same value of kr is calculated by these two solutions only at the tunnel springline (i.e. θ ¼ 90º) in Fig. 6. However, the modified solution presented in this note should be used to determine the kr for an arbitrary distribution shape described by a Fourier series. The results from Fig. 6 also show that kr for the distribution shape described by a Fourier series will not be constant around the tunnel, which is different from the kr for the shape described by a single Fourier term (see the dash lines in Fig. 6). The variation of kr can be attributed to the fact that the weight (am) varies with the angle θ around the tunnel according to the definition of am in Eq. (16). 6. Application to a design case It is well recognized from the numerical example that the kr can be significantly affected by the mode parameter m and the corresponding weight um in the Fourier series. However, the values for the parameters m and um are usually unknown at the design stage. In that case, the values for m and um can be determined by an iterative process of applying the bedded beam model (BBM) repeatedly. Following the traditional calculation procedure of the BBM, the lining displacements are first calculated and then are best fitted by a trial Fourier series. Then from Eq. (16), this best fitted Fourier series can be used to determine the kr for next step of calculation. The only difference between calculation steps is the kr used in the BBM. The iterative process can be stopped until the error between two successive steps is below the tolerance level. Note that the soil loads acting on the lining should remain unchanged during the iterative process, which is also the basic assumption when using the BBM. A simple design case has been chosen to demonstrate this iterative process. In this application case, the tunnel lining has a density of 2.5 kg/m3, an outer radius r0 of 6 m and a thickness of 0.50 m. The cover depth from the ground surface to the tunnel crown is 20 m. The water table is at the ground surface. The saturated soil gravity is 18 kN/m3. The soil has a Young's modulus of 40 MPa and a Poisson's ratio of 0.33. The lateral earth pressure coefficient is 0.7. The corresponding bedded beam model for this case is illustrated in Fig. 7. The loads acting on the lining, e.g., vertical total pressure (p1, p2), lateral total pressure (q1, q2) and dead load (pg), are determined by following the ITA guideline (ITA, 2004). The kr around the tunnel at the initial step is set to zero. The trial Fourier series is

p1 q1

q1 pg r0 Tunnel Lining

q2

q2

Spring

p2 Fig. 7. The bedded beam model (BBM) for the application case.

in the form of Eq. (17), for which the mode weights (u0, u1, u2 and u3) need to be determined. Fig. 8 shows the iterative process of determining the mode weights um and the kr in the form of the results against the angle θ around the tunnel perimeter. In total, ten iterative steps are conducted for this case. For each step, the radial displacements ur and the corresponding kr are plotted in the left and right panels of Fig. 8, respectively. From the left panel of Fig. 8, it is evident that the trial Fourier series (Eq. (17)) describes the “true” distribution shape calculated by BBM with remarkable accuracy. The coefficient of determination (R2) for the trial Fourier series is 1.00 for each step. The ur and kr obtained at step 10 are regarded as the object values for this design case. From a visual inspection, it is found in Fig. 8 that the difference between the results of step 2 and the object results (step 10) are not significant. It implies that error can be accepted even at step 2 in engineering practice. From the results of ur shown in step 10, the ellipse mode shape (m ¼ 2) is the dominant mode with 68% in the Fourier series. However, it is also obvious that this single shape (m ¼ 2) does not describe the true distribution shape accurately. It is possible that the error may be in the range of 50% when compared with the true ur at the crown and invert, as shown in Fig. 8. Correspondingly, the kr at step 10 shown in Fig. 8 is quite different from the kr for the single mode shape (m ¼ 2). It is also interesting to find that singular points for kr were observed near the location where ur is close to zero. In that case, calculated negative kr around a singular point should be set to zero due to the nonnegativity for kr. The effect of the subgrade reaction at these points on the results is negligible due to the small magnitude of ur. The convergence of the error between two successive steps is analyzed for these ten steps. The relative error for ur and kr at the tunnel crown are both plotted against the step number N in Fig. 9. Numerically, it is evident that convergence occurs rapidly. In this case, the error of the results at the 3rd step is smaller than 10% and at the 6th step it is smaller than 0.1%. This proves that the iteration method is very efficient to determine the values for mode parameters um and the kr.

D. Zhang et al. / Soils and Foundations 54 (2014) 225–232 2.00

100

BBM Fourier (R2=1.00 Object (Step 10)

ur (mm)

50 25

kr

The angle θ

0 0.0

Object (Step 10)

1.50

krr0/Ec

75

1.00

22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0

-25

0.50

-50 -75

u0:u1:u2:u3=2 : -4 : 84 : -10

-100

The angle θ

0.00 0.0

30

45.0

67.5

90.0 112.5 135.0 157.5 180.0

kr Object (Step 10)

The angle θ 0 22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0

krr0/Ec

1.50

10

0.0

22.5

2.00

BBM Fourier Object (Step 10)

20

ur (mm)

231

1.00

-10 0.50 -20

u0:u1:u2:u3=5 : -9 : 66 : -20 -30

The angle θ

0.00 0.0

30

45.0

67.5

90.0 112.5 135.0 157.5 180.0

2.00 BBM Object (Step 10) m=2

Object (Step 10) 1.50

10

The angle θ

0 0.0

krr0/Ec

20

ur (mm)

22.5

1.00

22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0

m=2

-10 0.50 -20

u0:u1:u2:u3=5 : -9 : 68 : -18

-30

The angle θ

0.00 0.0

22.5

45.0

67.5

90.0 112.5 135.0 157.5 180.0

Fig. 8. The ur and kr during the iterative process for the application case.

1.0E+01

Ur_Error

1.0E+00

kr_Error

Error

1.0E-01 1.0E-02 1.0E-03 1.0E-04

Errorn=(Rn-Rn-1)/Rn

1.0E-05 1.0E-06 0

1

2

3

4

5

6

7

8

9

10

Iterative Step N Fig. 9. The convergence of the relative error for ur and kr.

7. Conclusion To determine the analytical solution of kr, it is necessary to prescribe the distribution shape of tunnel radial deformation. In this note, a general distribution shape, described by a Fourier series, is considered for the solution of kr. The Wood (1975)'s solution of kr is found to be applicable in cases where the deformation shape is oval only (one special Fourier term). A modified solution using a compatible Airy stress function is

presented and validated using a numerical example. The example demonstrates that apart from parameters of the elastic properties (Ec, ν) and tunnel radius (r0), kr can also be significantly affected by the mode parameter m, the weight um of the corresponding mode m in Fourier series and the angle θ measured from the tunnel crown. The value of kr is no longer constant around the tunnel when the distribution shape is described by a Fourier series. This is different from the value of kr for a distribution shape described by a single Fourier term, which is independent of θ. Finally, a simple design case was presented using the bedded beam model to show the applicability of the proposed solution of kr. It should be noted that this modified solution, like other elastic solutions, is restricted to certain geotechnical situations where the soil deformation is relatively small. Fortunately, the lining deformation for tunnels driven by modern shield techniques tends to be quite small. This allows for the application of elastic solutions of kr over a wider range of ground conditions. Acknowledgments This study was substantially supported by the National Basic Research Program of China (2011CB013800), Natural Science Foundation Committee Program (51278381) and Shanghai

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