A modelling approach for joint rotations of segmental concrete tunnel linings

Tunnelling and Underground Space Technology 67 (2017) 61–67

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A modelling approach for joint rotations of segmental concrete tunnel linings Bo Tvede-Jensen, Morten Faurschou, Thomas Kasper

MARK

⁎

COWI A/S, Parallelvej 2, 2800 Kongens Lyngby, Denmark

A R T I C L E I N F O

A B S T R A C T

Keywords: Segmental lining TBM tunnel Joint rotations

This paper presents an approach to determine the nonlinear bending moment – rotation relations for longitudinal joints of segmental concrete tunnel linings with flat concrete contact areas based on the rules for confined concrete and partially loaded areas according to Eurocode 2. It is demonstrated that the resulting bending moment – rotation relations show better agreement with experimental data than other approaches from the literature. The proposed approach allows to establish the bending moment – rotation relations for both serviceability limit state (SLS) and ultimate limit state (ULS) and thereby a tunnel lining design consistent with Eurocode 2. The practical implications for the design concept of tunnel linings are discussed.

1. Introduction The effect of the joints in precast segmental concrete linings has been studied extensively in the literature. Muir Wood (1975) proposed a formula to account for the effect of the joints in uncoupled lining rings by an equivalent bending stiffness of a continuous ring. This formula is often used in engineering practice. Other correction factors to approximate jointed tunnel lining rings by continuous rings were proposed e.g. by Lee and Ge (2001). Blom (2002a, 2002b) and El Naggar and Hinchberger (2008) proposed analytical models for segmental tunnel linings with consideration of the joints. Explicit consideration of the joints by means of rotational springs in bedded beam models for tunnel linings has become wide-spread (e.g. Duddeck and Erdmann, 1982; ITA WG Research, 2000; Koyama, 2003; Grübl, 2006; Do et al., 2013). Three-dimensional shell-spring models with explicit consideration of the joints have been used e.g. by Klappers et al. (2006) and Arnau and Molins (2012). In beam and shell models, the joint behaviour is often modelled by nonlinear rotational springs based on the expressions by Leonhardt and Reimann (1965) and Janssen (1983) or Blom (2002a, 2002b). Several authors have investigated the effect of the joints by 2D and 3D continuum models of tunnel linings (e.g. Wittke, 2007; Chen and Mo, 2009; Arnau and Molins, 2011; El Naggar and Hinchberger, 2012). In this paper, the existing analytical expressions by Leonhardt and Reimann (1965) and Janssen (1983) and by Blom (2002a, 2002b) to describe the bending moment – rotation relation of the joints are briefly presented in Sections 2 and 3. A new analytical approach to describe the bending moment – rotation relation based on the rules for confined ⁎

concrete and partially loaded areas according to Eurocode 2 is presented in Section 4 and validated in Section 5 by comparison with experimental data and the aforementioned existing approaches. 2. Janssen's approach Based on the assumption of a joint deformation zone with a total length s equal to the joint height l (Fig. 1) such that (1)

Δs = εe1·l

and based on the assumption of linear elastic behaviour such that the stress at edge 1 is (2)

σe1 = Ec·εe1

Leonhardt and Reimann (1965) derived the following equation for the joint rotation φ as a function of the normal force N , the bending moment M , the Young's modulus of the concrete Ec and the joint height l

8N

φ=

M 2

9 (1−2 N ·l ) Ec·l

Eq. (3) applies to opened joints (joint opening lo > 0 and compressed joint height lc < l ), i.e. in case M > N ·l /6 . The authors demonstrated good agreement of this relation with various experimental data. Janssen (1983) completed the formulation

φ=

12M Ec·l 2

for

M⩽

Corresponding author. E-mail addresses: [email protected] (B. Tvede-Jensen), [email protected] (M. Faurschou), [email protected] (T. Kasper).

http://dx.doi.org/10.1016/j.tust.2017.04.019 Received 19 April 2016; Received in revised form 22 January 2017; Accepted 20 April 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.

(3)

N ·l 6

(4)

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B. Tvede-Jensen et al.

Sectional forces

N

Stresses

M

e1

Strains

e1

Centre line

e2

Stress trajectories

Δs

s=l

Joint deformation zone

Joint deformations

Fig. 2. Blom's approach.

lo

lc l t

⎛x ⎞ l x M = N −R1 ⎜ 1 + x2⎟−R2 2 ⎝3 ⎠ 2 2

Fig. 1. Janssen's approach.

8N

φ=

M

2

for

M>

9 (2 N ·l −1) Ec·l

N ·l 6

Finally, the maximum strain is determined as

εe1 = (5)

3. Blom's approach Blom (2002a, 2002b) proposed an approach based on the bi-linear stress-strain relation in order to consider the compressive strength of concrete fck . This approach represents a refinement compared to Janssen's approach as it considers a more advanced stress-strain relation (Fig. 2). The compressive strength fck and the ultimate strain εcu3 are basic material parameters as defined e.g. in Eurocode 2 (CEN, 2004). Using the same assumption of a joint deformation zone as in Janssen's approach, the rotation can be defined by

fck ·x1 2

and

R2 = N −R1

(10)

4. Proposed approach Compared to Blom's approach, the following further refinements are made in the proposed approach:

• The parabola-rectangle stress-strain relation is used to provide a better representation of the actual stress-strain relation of concrete. • An increase of the compressive strength and the strain limits is

(6)

considered according to the rules for partially loaded areas and confined concrete in Eurocode 2.

Then, the stress resultants R1 and R2 can be written as

R1 =

fck ·(x1 + x2 ) Ec·x1

The bending moment – rotation relation according to Blom's approach consists of three parts, the first part defined by Eq. (4), the second part defined by Eq. (5) until εe1 = fck / Ec , and the third part defined by Eq. (9) until εe1 according to Eq. (10) becomes εe1 = εcu3 . Blom's approach allows to determine both SLS bending moment – rotation relations based on fck as well as ULS bending moment – rotation relations based on fcd = fck / γc with the partial safety factor γc defined in Eurocode 2.

and derived corresponding expressions for the rotational stiffness. In the remainder of this paper, Eqs. (4) and (5) will be referred to as Janssen's approach.

f ·l f ·l ε ·l φ = c3 = ck ↔ x1 = ck x1 Ec·x1 Ec·φ

(9)

(7)

4.1. Eurocode basic equations

With R2 , the length x2 is obtained as

x2 =

R2 fck

As indicated in Fig. 1, the load transfer through a longitudinal joint represents the situation of a partially loaded area. For partially loaded areas, Section 6.7 Eq. (6.63) in Eurocode 2 allows to increase the compressive strength of concrete according to

(8)

Equilibrium of bending moments results in 62

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Ac1 ·f ⩽ 3.0·fck Ac0 ck

N

(11)

where fck ,p denotes the increased, characteristic compressive strength due to the partially loaded area, fck is the normal characteristic compressive strength, Ac0 is the loaded area and Ac1 is the maximum design distribution area with a similar shape to Ac0 . For a segment joint as shown in Fig. 1, this can be simplified to

t ·f ⩽ 3.0·fck lc ck

i

e1

(12)

σ2 ⩽ 0.05·fck

⎛ σ ⎞ fck ,c = fck ⎜1.125 + 2.5 2 ⎟ fck ⎠ ⎝

for

(13)

σ2 fck

(14)

l t

(15)

Fig. 4. Proposed approach.

σ2 =

(16)

where fck ,c and fck denote the confined and the unconfined characteristic compressive strength, respectively. The confinement stress is denoted by σ2 . The strains εc2,c and εc2 at which the maximum strength is reached and the ultimate strains εcu2,c and εcu2 are defined according to Fig. 3. fck , εc2 and εcu2 are basic material parameters defined for all concrete classes in Eurocode 2. The ULS design strength is defined as fcd ,c = fck ,c / γc , with the partial safety factor γc also defined in Eurocode 2.

σ2 =

for

εe2 < 0

N=

1

Unconfined

n

M=

2

3

(=

2

c2,c

cu2

fck ,c −fck 5

> 0.05·fck

(21)

(22)

(23)

εi < 0

for

for

0 ⩽ εi ⩽ εc2,c

εc2,c < εi ⩽ εcu2,c

(24) (25)

⎡ l⎤ ⎢⎣σi· ⎥⎦ n

(26)

⎡ l ⎛l l ⎞⎤ ⎢σi· ·⎜⎝ − ·(i−0.5) ⎟⎠ ⎥ ⎣ n 2 n ⎦

(27)

Finally, using the assumption of a joint deformation zone according to Eq. (1), the joint rotation can be expressed as

(19)

φ = εe1−εe2

(28)

4.3. Determination of bending moment – rotation relation for a selected normal force By means of a systematic variation of εe1 and εe2 , strain distributions can be identified, for which the normal force N according to Eq. (26) corresponds to the selected normal force, for which the bending moment – rotation relation shall be determined. The data according to Eqs. (18)–(28) for each of these strain distributions can be saved and the bending moment – rotation relation can be plotted. A possible algorithm for this systematic variation is as follows. According to Eq. (12), the maximum possible value of fck ,c is 3fck and it follows from Eqs. (22) and (16) that the maximum possible values of σ2 and εcu2,c are

fcd,c

1 c2

for

⩽ 0.05·fck

(18)

fck,c

)

∑ i =1

Confined fck

∑ i =1

The confined compressive strength fck ,c is determined from Eq. (12). Based on Eqs. (13) and (14), the equivalent confinement stress σ2 can then be calculated as

= fck,c

5

The normal force N [kN/m] and bending moment M [kN m/m] in the joint are then obtained as

(20)

lo = l−lc

1

for

σi = fck ,c

(17)

εe2 ⩾ 0 for

2.5

fck ,c −fck

2⎞ ⎛ ⎛ ε ⎞ σi = fck ,c ⎜⎜1−⎜1− i ⎟ ⎟⎟ ⎝ ⎝ εc2,c ⎠ ⎠

Eq. (17) is based on εe1 and εe2 being positive for compression. In case of joint opening, εe2 is a negative fictive strain. The compressed joint height lc and the joint opening lo are

εe1 lc = ·l εe1−εe2

fck ,c −1.125·fck

n

εe2−εe1 ·(i−0.5) n

for

5

σi = 0

To determine the stress distribution and sectional forces in a joint based on a given strain distribution, the joint height l is discretised into n equal sections as shown in Fig. 4. Assuming a linear strain distribution characterised by the strains at the two edges of the joint εe1 and εe2 , the strain in the centre of each section i is calculated as

lc = l

fck ,c −fck

and the corresponding strains εc2,c and εcu2,c can be determined from Eqs. (15) and (16). The stress in the centre of each section i is calculated as

4.2. Application to segment joints

εi = εe1 +

lo

lc

σ2 > 0.05·fck

⎛ fck ,c ⎞2 ⎟ εc2,c = εc2 ⎜ ⎝ fck ⎠ εcu2,c = εcu2 + 0.2

Centre line

for

i=1

⎛ σ ⎞ fck ,c = fck ⎜1 + 5 2 ⎟ fck ⎠ ⎝

e2

i

For confined concrete, Section 3.1.9 in Eurocode 2 specifies the following compressive stress-strain relation with increased characteristic strength and strains

i

fck ,c =

M

i=n

fck ,p =

cu2,c

Fig. 3. Stress-strain relation for confined concrete according to Eurocode 2.

63

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B. Tvede-Jensen et al.

180

Table 1 Key parameters of the Elbe tunnel segment joint tests.

160 Value

Segment thickness t (mm) Joint height l (mm) Normal force N (kN/m) Compressive strength fc (MPa) Young's modulus Ec (MPa) εc2 (‰) εcu2 (‰) εcu3 (‰)

700 388 4000 38.1 37,000 2.0 3.5 3.5

ı (MPa)

Parameter

Proposed approach Janssen's approach Blom's approach

Point D

140 120 100

Point C

80 60

Point B

40

Point A

20 0

0

50

100

150 200 x (mm)

250

300

350

3 f 4 ck

(29)

Fig. 6. Stress distributions in the joint for point A to D in Fig. 5.

maxεcu2,c = εcu2 + 0.15

(30)

It is stated in Schreyer and Winselmann (1998), that the segments in the Elbe tunnel tests did not exhibit any damage even for a measured compressed joint height of only 18 mm. Assuming a rectangular stress distribution, this corresponds to a stress of 4 MN/0.018 m = 222 MPa. It can be seen in Fig. 6 that the proposed approach predicts a minimum compressed joint height of approximately 40 mm and a stress of 114 MPa at maximum bending moment. This means that the strength limitation of 3.0·fck in Eq. (11) according to Eurocode 2 appears to be conservative compared to this test.

maxσ2 =

For M = 0 , the uniform strain εM=0 in the joint can be determined using Eqs. (12), (15) and (24). Starting from a strain distribution εe1 = εe2 = εM =0 , εe1 is varied in a loop from εM=0 to maxεcu2,c in a chosen number of steps. For each value of εe1, εe2 is varied to match N according to Eq. (26) with the selected normal force. For determination of the ULS bending moment – rotation relation, fck ,c in Eqs. (24) and (25) should be replaced by fcd ,c = fck ,c / γc . 5. Verification and discussion of the proposed approach

5.2. Dutch segment joint tests

5.1. Elbe tunnel tests

Hordijk and Gijsbers (1996) carried out segment joint rotation tests for different normal force levels. The key parameters are shown in Table 2. For the simulation of these tests, measured values of strength and Young's modulus are available and have been used. Tests were carried out with and without bolts in the joints. The authors concluded that the influence of the bolts on the bending moment – rotation relation was insignificant (Fig. 7). In the following, the experimental data of the tests without bolts are compared with the proposed approach (Fig. 8), with Janssen's approach (Fig. 9) and with Blom's approach (Fig. 10). For each normal force, two tests were carried out with opposite bending direction. An exception is the test with 4400 kN/m normal force, for which only one test was made. Hordijk and Gijsbers observed that the initial rotational stiffness was lower than expected for low normal force levels and increased with increasing normal force. This could be explained by the fact that the surfaces of the joints were not in full contact for low normal forces. This observation is reflected in Figs. 8–10. The initial stiffness from all three modelling approaches is very similar and agrees well with the observed initial stiffness for higher normal force levels. It can further be observed in Figs. 8–10, that the proposed approach provides the best agreement with the experimental data. Similar to the observation made for the Elbe tunnel tests, Blom's approach provides less good agreement compared to both the proposed approach and Janssen's approach due to the fact that it does not consider the increased compressive strength

Schreyer and Winselmann (1998) presented the results of a segment joint rotation test for the 4th Elbe tunnel tube. The key parameters are shown in Table 1. For a B45 concrete used in the test, the German code DIN1045 specifies a Young's modulus of 37 GPa. The 20 cm cube strength of a B45 has been converted into an equivalent cylinder strength of 45 MPa/1.18 = 38.1 MPa. For the Elbe tunnel test, both the proposed approach and Janssen's approach provide very good agreement with the experimental data (Fig. 5). The stress distributions in the joint for point A to D on the bending moment – rotation curve in Fig. 5 according to the different approaches are shown in Fig. 6. According to the proposed approach, the joint starts to open at a bending moment of 251 kNm/m (point A) and has opened to half the joint height at a bending moment of 512 kNm/m (point B). Points C and D are defined by a joint rotation of 10 and 35‰, respectively. Although Blom's approach represents a refinement compared to Janssen's approach, it shows less good agreement with the experimental data due to the fact that it does not consider the increased compressive strength (the maximum moment is too low) and increased strain limits (the maximum joint rotation is too low) for partially loaded areas and confined concrete. It can be observed in Fig. 6 that there is good agreement between the different approaches up to point B (joint half open) and that the difference becomes larger for larger joint rotations (point C and D). Bending moment M (kNm/m)

800

Point C: ij = 10 ‰

700 600

Point B: M = 512 kNm/m

500

Table 2 Key parameters of the Dutch segment joint tests.

Point D: ij = 35 ‰

400 300

Point A: M = 251 kNm/m

200 100 0

0

5

10

Experimental data Proposed approach Janssen's approach Blom's approach

15 20 25 Joint rotation ij (‰)

30

35

Fig. 5. Bending moment – rotation relation: Comparison of Elbe tunnel experimental data with the proposed approach, Janssen's approach and Blom's approach.

64

Parameter

Value

Segment thickness t (mm) Joint height l (mm) Normal force N (kN/m) Compressive strength fc (MPa) Young's modulus Ec (MPa) εc2 (‰) εcu2 (‰) εcu3 (‰)

350 158 200–4400 57 32,000 2.2 3.0 3.0

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300

250

Without bolts N = 4400 kN/m With bolts, positive bending With bolts, negative bending

200

N = 3000 kN/m

Bending moment M (kNm/m)

Bending moment M (kNm/m)

300

N = 2300 kN/m 150

N = 1600 kN/m

100

N = 900 kN/m

50 0

+

N = 200 kN/m

+ 0

1

2

3 4 5 6 7 Joint rotation ij (‰)

8

9

10

Experimental data Blom's approach

250

N = 3000 kN/m

200 150

N = 2300 kN/m N = 1600 kN/m

100

N = 900 kN/m 50 N = 200 kN/m 0

0

1

2

Fig. 7. Summary plot of all experimental data, with and without bolts in the joints.

N = 4400 kN/m

350

N = 3000 kN/m

200

Bending moment M (kNm/m)

Bending moment M (kNm/m)

Experimental data Proposed approach

N = 2300 kN/m

150

N = 1600 kN/m 100 N = 900 kN/m 50 0

N = 200 kN/m 0

1

2

3 4 5 6 7 Joint rotation ij (‰)

8

9

10

9

10

250 200 150 100 50 0

2

4

6 8 10 12 14 16 18 20 Joint rotation ij (‰)

Fig. 11. Extrapolation of the Dutch segment joint experimental data.

300

Bending moment M (kNm/m)

8

Experimental data Trendlines

300

0

Fig. 8. Comparison between the Dutch segment joint experimental data and the proposed approach.

Experimental data Janssen's approach

250 200

N = 4400 kN/m

In Janssen's approach, linear elastic material behaviour is assumed. Thus, there is no limit for the magnitude of the compressive stresses in the joint. The maximum bending moment according to Janssen's approach therefore always corresponds to a very (infinitely) narrow stress triangle at the joint edge with a maximum stress such that the integral of the triangle corresponds to the normal force. Consequently, the maximum bending moment according to Janssen is simply M = N ·l /2 , i.e. for the Dutch segment joint M = N ·0.079 m (Fig. 12). In Blom's approach and in the proposed approach, the compressive stresses in the joint are limited to the compressive strength. Therefore, at the maximum bending moment, the necessary compressed joint height lc to transfer the normal force increases with increasing normal force (Fig. 13). As a consequence, the lever arm of the normal force decreases and the maximum bending moment decreases after having reached a peak (Fig. 12). Compared to Blom's approach, the proposed approach considers an increased compressive strength due to partially loaded area (Fig. 13). As a consequence, the peak value of the maximum bending moment is higher in the proposed approach and is reached at a higher normal force (Fig. 12). After having reached the peak value, the maximum bending moment decreases more rapidly in the proposed approach compared to Blom's approach. This can be explained by the fact that the transfer of higher normal forces requires a larger compressed joint height lc , which according to Eq. (12) leads to a decrease of the compressive strength in the proposed approach (Fig. 13). In contrast to that, the compressive strength in Blom's approach is constant. Finally, when the normal force fully utilises the

N = 3000 kN/m N = 2300 kN/m

150

N = 1600 kN/m

100

N = 900 kN/m 50 N = 200 kN/m 0

3 4 5 6 7 Joint rotation ij (‰)

Fig. 10. Comparison between the Dutch segment joint experimental data and Blom's approach.

300 250

N = 4400 kN/m

0

1

2

3 4 5 6 7 Joint rotation ij (‰)

8

9

10

Fig. 9. Comparison between the Dutch segment joint experimental data and Janssen's approach.

(the maximum moment is too low) and increased strain limits (the maximum joint rotation is too low) for partially loaded areas and confined concrete. The test specimens exhibited no visible cracks or only hairline cracks within the tested range of rotations. If the normal force is varied, the maximum possible bending moments for the Dutch segment joint are shown in Fig. 12. The experimental maximum bending moments in Fig. 12 are based on extrapolations of the experimental data as shown in Fig. 11. 65

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1000

700

Bending moment M (kNm/m)

800

max M (kNm/m)

200

Experimental data Proposed approach Janssen's approach Blom's approach

900

600 500 400 300 200 100 0

0

2500

Fig. 12. Maximum bending moment as a function of normal force: Comparison between Dutch segment joint experimental data, the proposed approach, Janssen's approach and Blom's approach.

ı (MPa)

140 120

N = 7500 kN/m

N = 13404 kN/m

80 60 40 0

N = 4400 kN/m 0

20

40

60 80 x (mm)

100

120

120 100 80

SLS lower N=2000 kN/m SLS/ULS upper N=2000 kN/m ULS lower N=2000 kN/m ULS lower N=2700 kN/m ULS upper N=2700 kN/m

60 40 20 0

2

4 6 Joint rotation ij (‰)

8

10

used in the Dutch tests for a normal force of 2000 kN/m with estimated lower and upper characteristic values of the compressive strength of the fck ,0,05 = fcm −8 MPa = 49 MPa fck ,0,95 = fcm + concrete and 8 MPa = 65 MPa . For ULS verifications, ULS bending moment – rotation relations should be used. ULS bending moment – rotation relations based on the lower design concrete strength provide the largest joint rotations. In case of coupled rings, they also provide the largest coupling forces and peak moments in the segments. With the proposed approach, it can be verified that the joints are able to transfer the ULS hoop forces considering a realistic bending moment. It should be noted that the exceedance of the normal force capacity of the joints represents a true ULS failure. Furthermore, it can be verified that the joints do not exhibit unacceptable rotations, leading to unacceptable tunnel lining deformations, which in the (theoretical) extreme case may also represent a true ULS failure e.g. due to snapthrough problems of the lining. ULS calculations form the basis for splitting verifications at the joints. It should generally be noted that adequate reinforcement at the joints needs to be provided for the rules of partially loaded areas and confined concrete to be valid. For the ULS segment bending verification of linings with uncoupled rings, ULS bending moment – rotation relations based on the upper concrete strength may be used, as this results in the largest bending moments in the segments. As an example, Fig. 14 also presents the ULS bending moment – rotation relations for the segments used in the Dutch tests. The lower and upper ULS relations are based on fcd,0,05 and fck,0,95, respectively. In ULS, it is distinguished in Fig. 14 between N as a favourable effect (N = 1.0·2000 = 2000 kN/m) and N as an unfavourable effect (N = 1.35·2000 = 2700 kN/m). The designer needs to consider that for the different ULS verifications, bending moment – rotation relations either based on N as a favourable effect or based on N as an unfavourable effect may be governing.

100

20

140

Fig. 14. SLS and ULS bending moment – rotation relations for the Dutch segment joint.

Proposed approach Blom's approach

N = 11000 kN/m

160

160

0

5000 7500 10000 12500 15000 Normal force N (kN/m)

180

180

140

Fig. 13. Stress distributions in the Dutch segment joint at maximum bending moment for different normal forces according to the proposed approach and Blom's approach.

strength over the whole joint height, no bending moment can be transferred. The maximum normal force that can be transferred across the segment joint according to Blom's approach is 57 MPa·158 mm = 9006 kN/m (Fig. 12). According to the proposed approach, it is 57 MPa· 350 mm/158 mm ·158 mm = 13404 kN/m (Figs. 12 and 13). 6. Implications for tunnel lining design The proposed approach provides the most realistic representation of the stress-strain behaviour of the concrete at the joints. In Section 5, it could be shown that it provides the best agreement with experimental data both in terms of the bending moment – rotation relation and in terms of the maximum bending moment as a function of normal force. It allows a detailed study of the behaviour of the tunnel ring considering the actual stress conditions in the joints and therefore provides the basis for an engineering assessment of SLS and ULS joint design in compliance with Eurocode 2. The proposed approach may be used to verify the joints of segmental linings in accordance with Eurocode 2 as follows: For SLS verifications, SLS bending moment – rotation relations should be used. For deformation and water tightness verifications of the tunnel lining, it is conservative to base the bending moment – rotation relations on the lower characteristic value of the concrete strength. For SLS bending moments and crack widths in the segments of linings with uncoupled rings, the upper characteristic value of the concrete strength would be conservative. For SLS bending moments and crack widths in the segments of linings with coupled rings, the lower characteristic value of the concrete strength may be conservative. This is motivated by the fact that peak moments in coupled rings increase with decreasing flexural stiffness of the rings (Grübl, 2006). As an example, Fig. 14 presents the SLS bending moment – rotation relations for the segments

7. Conclusions The proposed modelling approach accounts for the nonlinear material behaviour of concrete according to Eurocode 2 based on simple mathematical expressions, which can easily be incorporated into the lining design. It provides a rational approach to consider and verify the joints in tunnel linings in compliance with Eurocode 2, both in SLS and ULS. It allows detailed studies of the stresses in the joints based on simple beam models or shell models with nonlinear rotational springs representing the joints. Adequate reinforcement at the joints needs to be provided for the rules of partially loaded areas and confined concrete to be valid. The reinforcement needs to provide sufficient robustness to avoid local failure mechanisms for the stress conditions in the vicinity 66

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Duddeck, H., Erdmann, J., 1982. Structural design models for tunnels. In: Jones, M.J. (Ed.), Tunnelling'82. Institution of Mining and Metallurgy, London, pp. 83–92. El Naggar, H., Hinchberger, S.D., 2008. An analytical solution for jointed tunnel linings in elastic soil or rock. Can. Geotech. J. 45, 1572–1593. El Naggar, H., Hinchberger, S.D., 2012. Approximate evaluation of stresses in degraded tunnel linings. Soil Dyn. Earthq. Eng. 43, 45–57. Grübl, F., 2006. Modern design aspects of segmental lining. International Seminar on Tunnels and Underground Works LNEC, Lisbon 29–30 of June 2006. Hordijk, D.A., Gijsbers, F.B.J., 1996. Laboratoriumproeven Tunnelsegmenten. Report Number 96-CON-R0708/03. TNO-Bouw, Delft. ITA WG Research, 2000. Guidelines for the design of shield tunnel lining. Tunn. Undergr. Space Technol. 15 (3), 303–331. Janssen, P., 1983. Tragverhalten von Tunnelausbauten mit Gelenktübbings. PhD thesis Technische Universität Braunschweig. Klappers, C., Grübl, F., Ostermeier, B., 2006. Structural analyses of segmental lining – coupled beam and spring analyses versus 3D-FEM calculations with shell elements. Tunn. Undergr. Space Technol. 21 (3–4) 6 pages. Koyama, Y., 2003. Present status and technology of shield tunneling method in Japan. Tunn. Undergr. Space Technol. 18, 145–159. Lee, K.M., Ge, X.W., 2001. The equivalence of a jointed shield-driven tunnel lining to a continuous ring structure. Can. Geotech. J. 38, 461–483. Leonhardt, F., Reimann, H., 1965. Betongelenke. Versuchbericht, Vorschläge zur Bemessung und konstruktiven Ausbildung. Deutscher Ausschuss für Stahlbeton DAfStb, Heft 175. Muir Wood, A.M., 1975. The circular tunnel in elastic ground. Géotechnique 25 (1), 115–127. Schreyer, J., Winselmann, D., 1998. Eignungsprüfungen für die Tübbingauskleidung der 4. Röhre des Elbtunnels – Suitability tests for the segmental lining for the 4th Elbe tunnel tube, Hamburg. Tunnel 98 (2), 30–37. Wittke, W., 2007. Stability analysis and design for mechanized tunnelling. Geotechnical engineering in research and practice, WBI Print 6, < www.wbionline.de > .

of the joints. The modelling approach for the joints also influences the segment bending verification. The presented approach is relevant mainly for the design of tunnel linings for the permanent situation. The design approach is valid independently of the soil-structure interaction modelling approach applied in the analysis. It should be noted that temporary situations during transport and handling, lining installation and TBM advance as well as geometrical constraints with respect to gasket arrangement often affect or even govern the required lining thickness. References Arnau, O., Molins, C., 2011. Experimental and analytical study of the structural response of segmental tunnel linings based on an in situ loading test. Part 2: Numerical simulation. Tunn. Undergr. Space Technol. 26, 778–788. Arnau, O., Molins, C., 2012. Three dimensional structural response of segmental tunnel linings. Eng. Struct. 44, 210–221. Blom, C.B.M., 2002a. Design philosophy of concrete linings for tunnels in soft soils. PhD thesis Technische Universiteit Delft. Blom, C.B.M., 2002b. Background document “Lining behaviour – analytical solutions of coupled segmented rings in soil”. Technische Universiteit Delft. Chen, J.S., Mo, H.H., 2009. Numerical study on crack problems in segments of shield tunnel using finite element method. Tunn. Undergr. Space Technol. 24, 91–102. CEN European Committee for Standardization, 2004. Eurocode 2: Design of concrete structures – Part 1–1: General rules and rules for buildings (EN1992-1-1). Do, N.A., Dias, D., Oreste, P., Djeran-Maigre, I., 2013. 2D numerical investigation of segmental tunnel lining behaviour. Tunn. Undergr. Space Technol. 37, 115–127.

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