Mechanism for Buckling of Shield Tunnel Linings Under Hydrostatic Pressure

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MechanismforbucklingofshieldtunnelliningsunderhydrostaticpressureARTICLEinTUNNELLINGANDUNDERGROUNDSPACETECHNOLOGYJUNE2015ImpactFactor:1.59DOI:10.1016/j.tust.2015.04.012

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JianhongWangResearchanddevelopmentcenter,Nippoo12PUBLICATIONS15CITATIONS

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W.JZhangTianjinUniversity6PUBLICATIONS11CITATIONS

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XiangGuoTianjinUniversity28PUBLICATIONS167CITATIONS

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Availablefrom:W.JZhangRetrievedon:26June2015

li

. Tpan

100

55,

Received in revised form 30 March 2015Accepted 13 April 2015

The results show that radial joints have signicant impacts on the buckling behavior: the shield tunnel

compressive hoop force, which is required by the design of water-proof of joint. This eventually makes the exible joint be utilizedextensively and the circular shield tunnel can be designed moreeasily by the wide and thin segment (Kimura and Koizumi, 1999;Koizumi, 2000). Therefore, the structural stability of the tunnel lin-ings should be checked to avoid the buckling of circular tube underhydrostatic pressure. However, the current design specications of

apse in Februaryes of the concretemotivatedel linings

2001; Tamura and Hayashi, 2005; Wang et al.,Particularly, the recent shield tunnel collapse in Japan causeof ve lives, which is a very severe accident throughout thstruction history of the modern shield tunnel. Structural failureof segmental linings has been analyzed to be the most likely causeof undersea tunnel collapses (NBP, 2012). However, the failuremechanism of the tunnel linings has not been claried yet. Sincethe surrounding ground is dense and the external hydrostatic pres-sure is large, the buckling of segmental linings should be checkedas the necessary inuential factor in the collapse. Actually, the

Corresponding author at: School of Civil Engineering, Tianjin University, Tianjin300072, China.

Tunnelling and Underground Space Technology 49 (2015) 144155

Contents lists availab

ro

.econsidered as a principal design load, as it is almost equivalentto the acting load on the operating shield tunnel (Koyama, 2003;Mashino and Ishimura, 2003; Yahagi et al., 2005). Under highhydrostatic pressure, the tunnel linings are predominated by the

Kurashiki Undersea Tunnel (shield tunnel) coll2012 (NBP, 2012) are some typical failure exampllinings in recent years. These accidents haveresearchers to investigate the buckling of tunnhttp://dx.doi.org/10.1016/j.tust.2015.04.0120886-7798/Crown Copyright 2015 Published by Elsevier Ltd. All rights reserved.many(Croll,2014).d a losse con-In recent decades, the use of deep underground beneath seasand rivers has rapidly increased in order to meet the civic require-ments and improve the urban environment. Many new tunnel util-ities, including undersea and riverbed tunnels, have beenconstructed at progressively greater depths by shield tunnelingmethod, due to the congested uses of shallow ground and above-ground space in many large cities (Watanabe, 1990; JCRDB, 2006).

For such underwater tunnels, hydrostatic pressure should be

allowable stress method and ultimate state method. Equivalently,the structural stability of the whole tunnel linings is ignored.

Tunneling accidents usually have complicated causes and even-tually result in ground collapse, so that the structural problem ofthe tunnel linings is often underestimated and even ignored indesign practice and relevant research, especially for reinforcedconcrete linings. Heathrow Express Tunnel (NATM) collapse inOctober 1994 (HSE, 2000), Gerrards Cross Tunnel (Tesco tunnel,three pin arch) collapse in June 2005 (Wikipedia, 2005), andKeywords:Shield tunnelSegmental jointBucklingHydrostatic pressure

1. Introductionlinings with exible joints buckles in a single wave mode in the vicinity of K joint, while those with rigidjoints buckles in a multi-wave mode around the linings. Hydrostatic buckling strength is found toincrease with the exural rigidity of the radial joint and the thickness of segment increasing. This studyshows that ground support increases the buckling strength dramatically, while earth pressure reducesthe capacity to resist hydrostatic buckling. The tunnel linings during construction are found to be easierto buckle than that during operation. Meanwhile, the buckling of tunnel linings is studied by theoreticalanalysis of buried tube buckling.

Crown Copyright 2015 Published by Elsevier Ltd. All rights reserved.

the shield tunnel (JSCE, 2007) only check the material safety by theArticle history:Received 23 February 2014

In this paper, the effects of segmental joints, dimensions of segments, and ground conditions on bucklingof the shield tunnel linings under hydrostatic pressure are studied by analytical and numerical analysis.Mechanism for buckling of shield tunnelpressure

J.H. Wang a, W.J. Zhang b,c,, X. Guo d, A. Koizumi e, HaResearch & Development Center, Nippon Koei Co., Ltd., Ibaraki Prefecture 300-1259, Jab School of Civil Engineering, Tianjin University, Tianjin 300072, ChinacKey Laboratory of Transportation Tunnel Engineering, Ministry of Education, Sichuan 6d School of Mechanical Engineering, Tianjin University, Tianjin 300072, ChinaeDepartment of Civil and Environmental Engineering, Waseda University, Tokyo 169-85

a r t i c l e i n f o a b s t r a c t

Tunnelling and Underg

journal homepage: wwwnings under hydrostatic

anaka a

31, China

Japan

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und Space Technology

lsev ier .com/ locate / tust

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aus pu

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Air exhvacuum

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J.H. Wang et al. / Tunnelling and Undergexperimental study using the tunnel models with radial joints hasindicated that exible segment joints reduces the buckling load ofcylindrical shells signicantly (Wang and Koizumi, 2010).

The aim of this paper is to study the stability of deep shieldtunnel linings under hydrostatic pressure. The buckling of thesegmental linings will be analytically and numerically investi-gated to clarify its structural failure mechanism. The followingfactors will be identied: (a) the rotational rigidity of radialjoints; (b) the number and orientation of radial joints; (c) thethickness and width of segment; and (d) the overburden depthand ground support. In addition, the examination of stress willbe performed to check the material safety by allowable stressmethod, and the buckling of tunnel linings will also be examinedby the analytical solutions of Winkler model and the elasticcontinuum model.

Buckled

(a) Free cylinder

(single w

Rigid joint (multi-wave

Fig. 1. Experimental prole and buckling of

(a) Load condition 1 and 2 (LC1 and LC2, in construction)

Groundwater pressure

grouting materibefore complehardening

Void

Shield machine Groundwater a

Loosing ground

Load condition 1 Load condition 2

Fig. 2. Load conditions considered in buckling aimen

Sand

ted by the mp

oint

d Space Technology 49 (2015) 144155 1452. Overview of cylinder buckling theories

2.1. Relevant buckling theories

Buckling of a free cylindrical shell under external pressure wasstudied by Levy (1884) and Timoshenko and Gere (1961). Bucklingof a tube encased shell in rigid cavity was investigated by Cheney(1971), Amstutz (1970), Jacobsen (1974), Glock (1977), El-Sawyand Moore (1998), and El-Sawy (2001). The former was usuallyclassied into the multi-wave buckling theory, while the latter intothe single-wave buckling theory. The free buckling strength of abeam ring can be estimated by the following equation:

Prcr 3EIBR3

1

mode

(b) Buried cylinder

ave)

)

cylindrical shells with one exible joint.

(b) Load condition 3 (LC3 in operation)

al te

nd earth pressures

Segmental linings

Stable ground

grouting material before complete hardening

nalysis during construction and operation.

where R, E, I, and B (commonly B = unit length) are mean radius,Youngs modulus, second moment of inertia, and width, respec-tively. Considering the effect of Poisson ratio, the buckling formulafor an innite long pipe can be obtained as follows:

Pcr E41 v2tR

32

(a) load condition 1 (LC1) (b) load condition

Pw

pe1+g

g

External pressure (Pw)

Self-weight (g)

Earth

Fig. 3. Load

146 J.H. Wang et al. / Tunnelling and Undergrounwhere t is thickness and v is Poisson ratio.Glock (1977) presented a simple solution to evaluate the buck-

ling strength of a pipe encased in rigid cavity, i.e.,

PGcr E

1 v2t2R

2:23

The works of introducing imperfection were done by Boot(1998), El-Sawy and Moore (1997), and El-Sawy (2013).

However, for a circular tube embedded in soil medium, the the-ories of buried structure buckling should be applied. Stevens(1952), Flgge (1962), Luscher (1966) and Chelapati and Allgood(1972) presented a multi-wave buckling solution, using the

KBHwB

AA

H0

Sea

Shield Tunnel (RC segment)

K

B

A

B

ADo=4950mm

Sandy s=18 kN/m3

v=0.495 E=50 MN/m2

Fig. 4. Prole of tunnel and ground conditions.Winkler spring model for ground to resist both inward and out-ward deformation; the critical hoop force (Ncr) resulting in theinstability can be formulated by

Ncr 2 EIKBR 1=2

4

where K is modulus of subgrade reaction.The single wave buckling of a ring in Winkler medium was

studied by Falter (1980) and Gumbel (1983); and the correspond-ing solution was expressed by Eq. (5), in which only the effects ofground resistance to outward deformation is considered,

Ncr EIKBR 1=2

5

On the other hand, Cheney (1976) and Moore and Booker(1985) proposed an elastic continuum theory for multi-wave buck-ling, using the continuum model to simulate the surroundingground. When the interface between structure and ground issmooth, the critical hoop force can be formulated by

Ncr 1:2 EIB 1=3 Es

1 m2s

2=36

where Es and vs are Youngs modulus and Poisson ratio of theground, respectively. Both inward and outward deformation areassumed to be resisted by ground. However, the linear bucklingsolution using the elastic continuum model is applicable only when

2 (LC2) (c) load condition 3 (LC3)

qe1

qe2Pw

pe1+g

g

qe1

qe2 Ground support (K)

pressure

pe1 pe1

modes.

d Space Technology 49 (2015) 144155hoop force is generated by ground rather than by hydrostatic pres-sure (Moore, 1989). In addition, all theories of buried buckling canonly be applied to the thin-walled tube.

The relation between modulus of subgrade reaction K and elas-tic modulus of ground Es can be expressed by an empirical equa-tion (Wood, 1975; Kimura and Koizumi, 1999)

K 3Es1 v5 6vR 7

As for the studies on buckling of tunnel linings, Croll (2001) pro-posed several models on buckling failure and indicted that bothlinings pull-off and bending failure resulted in the degenerationof tunnel rigidity. Tamura and Hayashi (2005) identied the buck-ling failure phenomenon by replacing the tunnel with some thin-walled aluminum pipes and also the ground with some aluminumrods. Finally, they proposed a two-dimensional numerical solutionby a frame model of elastic beams and rotational springs. Blom(2002) studied the local instability of segmental linings anddemonstrated the snap-through problem when linings was sub-jected to the uniform tangential compressive load and ovalisationload. Meanwhile, the analytical solution using Eulers theory was

load condition 1 (LC1) is the loading state of a tunnel linings just

Table 1Material properties of RC segment.

Concrete

Youngs modulus Poisson ratio Compressive strength Allowable stressEc (N/mm2) m rc (N/mm2) rca (N/mm2) Es (N/mm2) m fy (N/mm2) rsa (N/mm2)

3.3 104 0.17 42.0 16

Table 2Earth pressures.

Items Ground 1(basic)

Ground2

Ground3

Overburden thickness H0 (m) 10 5 15Vertical earth pressure pe1 (kN/

m2)80.00 40.00 120.00

Lateral earth pressure qe1 (kN/m2) 40.00 20.00 60.00Lateral earth pressure qe2 (kN/m2) 60.00 40.00 80.00Sea depth Hw (m) Hw0 = 18 m (basic), variableSelf-weight g (kN/m2) 3.12 (t = 120 mm), 4.16 (t = 160 mm),

5.20 (t = 200 mm)

J.H. Wang et al. / Tunnelling and Undergrounrecommended. Although the buckling failure was discussed in theabove investigations, the effect of the radial joints has not beenaddressed.

Wang and Koizumi (2010) conducted experiments of a thin-walled cylindrical shell with longitudinal joints to investigate thebuckling of shield tunnel linings under hydrostatic pressure. Itwas found that the free cylindrical shell with exible longitudinaljoints buckles in a single-wave mode, while that with rigid jointsbuckles in a multi-wave mode, as shown in Fig. 1(a). It was foundthat the buckling strength decreases as the exural rigidity of jointdecreases. Furthermore, they veried that the buried cylindricalshell with exible joints can buckle under external air pressure,in a similar deformed mode with the free cylindrical shell, asshown in Fig. 1(b); but its buckling strength is increased by about26% than the free one. The experiments have claried that theradial joint and ground support have important inuence on the

Notice: soilwater separated method in load calculation, and the coefcient oflateral earth pressure k = 0.5.buckling strength of shield tunnel linings, although the specimensused the thin-walled cylindrical shells (R/t > 150) with one radialjoint, rather than a segmental linings (R/t < 20) with multi-joints.Besides, the inuence of imperfection of inward deformation at

However, each ring can be simply considered as an independent

No joint (Eq.(2)) J1(n=1) J2(n=2)

J3(n=3) J4(n=4) J5(n=5)

No.1

=13.8

No.1

No.2=41.5

No.1

No.2

No.3 No.4

=152.3

No.1

No.2

No.3

=124.6

No.1

No.2

No.3 No.4

No.5

=69.2

Fig. 5. Number and location of joints.structure in buckling analysis, because the hydrostatic pressuremainly causes a circumferential compression. Therefore, the buck-ling investigation only needs to be focused on a ring of the tunnellinings. Three-dimensional joint-shell model is employed, consid-leaving the shield tail. The buckling strength under LC1 can bestudied by Eq. (2). The load condition 2 (LC2) is the loading stateof the grouting material before complete hardening; and the loadcondition 3 (LC3) is the loading state of the support and loadsapplied on the tunnel linings during operation. The correspondingloading models can be dened based on the current design spec-ications (JSCE, 2007), as shown in Fig. 3. Model 1 in Fig. 3 will beused to simulate LC1; model 2 will be used to simulate LC2 byconsidering the earth pressure, because the earth pressure actson the segmental linings through the soft grouting material;and model 3 is used to simulate LC3 by adding the ground sup-port beside the LC2, considering the completion of ground settle-ment. For LC3, the buckling investigation should consider theinstability of the tunnel linings under earth pressure and hydro-static pressure.

The conventional load conditions and the corresponding loadmodels are dened, considering the complexity and uncertaintyof the real load conditions involving the ground and construction.Moreover, this study mainly focuses on the hydrostatic buckling,although the effect of the ground support and the earth pressureon the buckling strength are investigated.

2.3. Numerical analysis model of segmental linings

Shield tunnel linings are assembled by the segments withstraight or staggered joints. The staggered conguration canenhance the integrity of the tunnel linings, and the effect is consid-ered by a factor to reduce the moment of the radial joints andincrease the moment of the adjacent segments in structural design.joint have been identied by numerical analysis, and the goodagreement with experiments has been obtained (Wang andKoizumi, 2010).

2.2. Buckling analysis considering load conditions

Shield tunnel linings in the life cycle will undergo many load-ing states during construction, while three typical load conditionsshould be selected when examining the stability, as shown inFig. 2: load condition 1 and 2 corresponding to construction per-iod, and load condition 3 corresponding to operation period. The

2.1 105 0.17 345 200Reinforcement (SD345)

Youngs modulus Poisson ratio Yield strength Allowable stress

d Space Technology 49 (2015) 144155 147ering the locations of radial joints of bolts and treating the seg-ments as the cylindrical shell.

3. Numerical analysis

A two-dimensional beam-spring model is used to check thesafety of the tunnel linings based on material strength. Extensivestudies are performed to investigate the effects of joints, segmentdimensions and ground conditions on the buckling strength of tun-nel linings, based on three-dimensional joint-shell model.

ive tor

y

ounTable 3Parameters and cases in numerical analysis.

Parameters andcases

Tunnel lining

Outerdiameter

Thickness ofsegment

Width ofsegment

Joint effectrigidity fac

D0 (m) t (m) B (m) g

Basic 4.95 0.16 1.2 01Extensive 0.12, 0.20 0.9, 1.5 01

RyRz

Rxz

148 J.H. Wang et al. / Tunnelling and Undergr3.1. Tunnel condition and study cases

A 4.95 m outer diameter (O.D.) underwater shield tunnelassembled by staggered conguration is selected, as shown inFig. 4. The shield tunnel ring is composed of ve reinforced con-crete at segments with two A-segments (aA = 83.1), two B-seg-ments (aB = 83.1) and a K-segment (ak = 27.7). The basic groundconditions are overburden H0 = 10 m, sea water level Hw = 18 m,density cs = 18 kN/m3, Youngs modulus Es = 50 MN/m2, andPoisson ratio v = 0.495. The coefcient of lateral earth pressurek = 0.5 and modulus of subgrade reaction K = 20 MN/m3 are taken,based on in-situ ground condition and shield tunnel design speci-cation for the dense sand (JSCE, 2007). The dimensions of the seg-ment are t = 160 mm in thickness and B = 1200 mm in width.Segmental joints are assumed to have a variable exural rigidity.The material properties of concrete and reinforcement are shownin Table 1.

B

A

A

B

K

Segment/radialjoint

Segm

Shell elemen

Ground

(a) FE model of a ring lining (c) Jo

Shell element

x Segm

Shell (b)

Fig. 6. FE model and join

Table 4Rotational rigidity of segment joint kh (kN m/rad).

Segment Rotational rigidity of segment joint kh (kN m/rad)

t (mm) B (m) g = 1.0E6 g = 0.05 g = 0.1120 0.9 8.67E+01 4.00E+02 5.20E+02

1.2 1.16E+02 5.33E+02 6.94E+021.5 1.44E+02 6.66E+02 8.67E+02

160 0.9 2.06E+02 9.48E+02 1.23E+031.2 2.74E+02 1.26E+03 1.64E+031.5 3.43E+02 1.58E+03 2.06E+03

200 0.9 4.01E+02 2.47E+03 3.21E+031.2 5.35E+02 2.47E+03 3.21E+031.5 6.69E+02 3.09E+03 4.01E+03Ground

exural Lateral earth pressurefactor

Modulus of subgradereaction

Overburdendepth

k K (MN/m3) H (m)

0.5 20 10.00.5 0200 5.0, 15.0

Nodal ties (x,y,z,Rx,Ry)& Rotation spring (Rz) Nodal ties (x,y,z,Rx,Ry)

d Space Technology 49 (2015) 144155The above three load conditions are used to study the buck-ling behavior of the tunnel linings. The earth pressures shownin Table 2 are used to check the stability of the tunnel liningsunder variable hydrostatic pressure and calculated based onthe assumption of soilwater separated. Extensive studies areperformed to investigate the effects of joints, segment dimen-sions and ground conditions on the buckling strength of tunnellinings. Four joint patterns arranged by adding each joint inthe clockwise direction are used to clarify the interaction ofjoints, as shown in Fig. 5. The effect of segment dimension isinvestigated in cases of three widths (B = 900, 1200 and1500 mm) and three thickness (t = 120, 160 and 200 mm).Additionally, the inuences of earth pressure are studied bycomparing three overburden thickness of H0 = 5, 10 and 15 m,and that of ground supports using the modulus of subgrade reac-tion K = 0200 MN/m3. The dimensions of tunnel linings, groundparameters are summarized in Table 3.

ent A Segment B

tGround reaction spring

int modeling details (LC3)

Nodal ties (x,y,z,Rx,Ry)& Rotation spring (Rz)

ent A Segment B

elementJoint modeling details (LC1and LC2)

t modeling details.

g = 0.2 g = 0.5 g = 0.7 g = 0.9 g = 1.0

7.44E+02 1.73E+03 3.36E+03 1.14E+04 1.00E+089.92E+02 2.30E+03 4.48E+03 1.52E+04 1.00E+081.24E+03 2.88E+03 5.60E+03 1.89E+04 1.00E+08



tio

rounBending moment

(a) Internal forces calculated using equa

Compressive stress

Maximum 19.53Hoop force 822.4

Location 0Minimum -15.17

Location875.2289

(kN)( )

(kNm)Hoop force

(kN)( )

(kNm)

J.H. Wang et al. / Tunnelling and Underg3.2. Numerical model

Numerical model of the segments uses thick-shell elements,which are four-node elements with global displacements and rota-tions of degrees of freedom. All meshes are ensured to be suf-ciently small and have a good aspect ratio of about 1 in order toachieve better convergence. Generally, segment joints are modeledby springs to describe the relation between force and displace-ment, with each node six degrees of freedom. Considering thatbuckling is mainly determined by exural rigidity, the segmentjoints are modeled by rotational springs and nodal ties. The niteelement analysis and details of the joint modeling are illustratedin Fig. 6. The rigidity kh of the rotational spring is estimated by(Wang and Koizumi, 2010)

kh 0:30loggEIR

8

so that joints with an effective exural rigidity factor (non-dimen-sional parameter g) within the range 01 can be investigated. Thecalculated results of rotational rigidity are shown in Table 4.

For LC1 and LC2, four nodes in the middle of the model alongthe longitudinal and circumferential directions are xed.

0 500 1000

-50

0

50

100

150

200

250

100m50m28m

Basic ground condition

Tensile stress

Steel bar

Allowable compressive streng

Allowable co

Hoo

p st

ress

(N/m

m2 )

Hydrostatic pressure at tu

(b) Maximum and minimu

Fig. 7. Stress examining results in the basic designHoop force

n method for basic ground conditions Maximum 882.5 (kN)Location 180 ( )

d Space Technology 49 (2015) 144155 149Considering the Winkler method has been widely used in currenttunnel design practice and the target load is the dominant ground-water, the Winkler method is applied in LC3, by modeling theground support as a number of independent springs to resist theradial deformation. In this model, the rigidity of spring is takenas zero for the inward deformation and as K for the outwarddeformation.

3.3. Buckling analysis approach

The analysis is performed by a nite element analysis softwarepackage of Marc (MSC, 2005). Analysis can be done by an elasticbuckling analysis as an eigenvalue problem or a nonlinear bucklinganalysis by performing eigenvalue analysis in each increment as anonlinear problem. For both geometrical (large deformation) andmaterial (elasto-plastic material) nonlinear problems, the criticalload should be estimated by an auto-incremental analysis basedon arc-length method (Riks, 1979). In this study, although the tun-nel linings can be easily considered elastic, the auto-incrementalanalysis is applied to consider the effects of joints, earth pressureand nonlinear ground support. Criselds arc-length method(Criseld, 1981, 1983) is employed and described by a governing

1500 2000 2500

ca=16 N/mm2

Hw+H0(m)200m150m

Concrete

th of steel bar

mpressive strength of concrete

nnel crown Pw ( kN/m2)

sa=200 N/mm2

m stress in concrete lining

condition under variable hydrostatic pressure.

oun0

500

1000

1500

2000

2500

3000

3500

Pcr=2295 kN/m2(Eq.(2)) =1.0

=0,0.05, 0.1,0.2

=0.5=0.7

LC1 LC2 LC3

Buc

klin

g st

reng

th P

cr (k

Pa)

Rotational rigidity of joint K (kNm/rad)

=0.9

(a) Relation of buckling critical pressure androtation rigidities of the joints

0 10000 20000 30000 40000 50000 60000

150 J.H. Wang et al. / Tunnelling and Undergrequation, Eq. (9), and increment control equation, Eq. (10). The for-mer governs the forcedisplacement relation, while the latterimplicitly denes the load increment size.

Kfdg kffg fOg 9

fDdgTfDdg Dk2 Dr2 10where k is the incremental load factor, K the tangent-stiffnessmatrix, f the load vector, d the displacement vector, and r the arc-length control parameter. After written into the load incrementmatrix form by adding step n, Eq. (10) becomes:

KnDO dn ffgfdngT kn

" #Ddn

Dkn

( ) P

n

Q n

( )11

where

fdng Xn1i1

fDdig kn Xn1i1

fDkig

fPng kO kn

ffg KDO dnfDO dng

grade reaction are discussed.

the external pressure.

Fig. 8 indicates that tunnel linings most easily buckle under LC2.

0.0 0.2 0.4 0.6 0.8 1.0 1.20

500

1000

1500

2000

2500

3000

3500

Pcr=2295 kN/m2(Eq.(2))

Buc

klin

g st

reng

th P

cr (k

Pa)

Effective flexural rigidity factor of joint

LC1 LC2 LC3

(b) Relation of buckling critical pressure and

flexural rigidity factors of the joints

Fig. 8. Buckling strength versus rigidity of joint for ground conditions(B = 1200 mm, t = 160 mm, n = 5, K = 20 MN/m3).By comparing LC2 and LC1, it can be found that the rigidity of jointsignicantly affects the buckling strength and the earth pressuredecreases the buckling strength of tunnel linings. This can beexplained by the initial imperfections such ovalisation, inducedby earth pressure. On the other hand, LC3 corresponds to the high-est buckling strength, although earth pressure is taken intoaccount. In addition, the variation of exural rigidity of the jointhas a slight effect on the buckling strength when the effective ex-ural rigidity factor g is larger than 0.2. Accordingly, the surround-4.2. Effect of load conditions

The effects of load conditions on buckling of the shield tunnellinings are veried, by comparing the results of three load condi-tions. The relation between the buckling strength and rotationalrigidity kh and effective exural rigidity factor g of the joint shownin Fig. 8. The buckling mode congurations are plotted in Fig. 9,with the effective exural rigidity factor g of 0, 0.7, and 1.0, corre-sponding to the hinge joint, the general bolt joint, and non-joint.4.1. Check of safety of tunnel linings

The internal forces of tunnel linings under design loads areexamined using the allowable stress method. Here, the stresschanges with the variable hydrostatic pressure. The internal forcesare calculated by the equations in the structural design guideline(JSCE, 2007).

For the basic design condition, the bending moment and thehoop force are shown in Fig. 7(a). The relations between stressand hydrostatic pressure are plotted in Fig. 7(b), in which the stressexamination for design condition is also presented.

Fig. 7 indicates that stresses in both concrete and reinforcementsteel bars are far smaller than the allowable strength. Furthermore,the entire tunnel linings in compression can be observed under thedesign condition, as well as the compressive stress increasing withthe hydrostatic pressure. The design results verify that the liningsis safe in material strength, and the hoop compressive force is morepredominant than the bending moment. Therefore, consideringthat the buckling is mainly induced by compressive hoop force,the structural stability of tunnel linings should be checked underQ n 12

r2 fdngTfdng kn2h i

where {DO} and kO are the n 1 step-displacement vector and loadfactor, and {P(n)} and Q(n) are residual errors. However, when buck-ling occurs with the sudden development of nonlinearity, it shouldbe ensured that the arc-length remains sufciently small prior tothe occurrence of buckling. The buckling strength is estimatedwhen the tangent rigidity approaches zero.

4. Results and discussions of numerical analysis

Numerical results are presented to investigate the safety of tun-nel linings based on material strength, and the buckling of tunnellinings under three load conditions, where the maximum hydro-static pressure acting on the crown of tunnel is used as the criticalpressure of the bucking strength. The effects of joints, as well as thesegment dimension, on buckling are studied. Meanwhile, groundconditions including the overburden depth and modulus of sub-

d Space Technology 49 (2015) 144155ing ground support dramatically enhances the linings stabilitybecause the tunnel linings supported by hard ground (K = 20 MN/m3) becomes a statically-indeterminate structure.

= 0

roun = 0 LC

1

J.H. Wang et al. / Tunnelling and UndergOn comparing Fig. 8(a) and (b), the relation between bucklingstrength and rotational rigidity kh is more complex than that ofthe effective exural rigidity factor g. The buckling strength dropsdramatically when kh varies from 3.59 104 to 1.06 104 kN m/rad, corresponding to g = 0.9 and 0.7, respectively. It can be foundthat the use of an effective exural rigidity factor is a valid meansto simply and clearly describe the relation between the bucklingstrength and the exural rigidity of joints. Therefore, the abovenumerical analysis is reasonable since the numerical results underLC1 agree well with the calculated results of Eq. (2), which wasderived for cylindrical shells without joints (g = 1) under the exter-nal pressure.

From Fig. 9, buckling deformation presents a good explanationfor the buckling strength changing with the exural rigidity ofthe joints and load conditions. For the buckling of tunnel liningswith hinge joints, the buckling mode is quite different from thegeneral ovalisation (two-wave) mode of cylindrical shell. The buck-led deformation mainly occurs close to joints of K-segment, while asingle-wave buckling mode occurs under all load conditions.Actually, this phenomenon (shown in Fig. 1) has been veried inexperimental studies (Wang and Koizumi, 2010; Wang et al.,

LC2

LC3

Fig. 9. Buckling deformation (t = 160.7 Non joint ( = 1.0)

d Space Technology 49 (2015) 144155 1512014). However, as the joint becomes more rigid, the bucklingmode changes from a single-wave mode to two-wave mode underLC1 and LC2, but the buckling of linings under LC3 shows a high-order buckling mode having four waves. This can explain whythe critical load of the linings under LC3 is always larger than thatunder LC1 and LC2.

The above results show that shield tunnel linings is most likelyto buckle during construction when the hydrostatic/grouting pres-sure and earth pressure (LC2) act on the tunnel linings.

4.3. Effect of joints on buckling

The interactional effect of joints on buckling should be claried,since the shield tunnel linings is assembled by segments and joints.The relations between buckling strength and joint number areplotted in Fig. 10 in the conditions of different effective exuralrigidity.

From Fig. 10, it is shown that the buckling strength of tunnellinings with rigid joints (g = 1) has no variation, while the bucklingstrength of other linings declines as the number of joints increasesunder both LC1 and LC2. The decreasing of buckling strength due to

mm, B = 1.2 m, K = 20 MN/m3).

is smaller due to the second joint, while the reduction of the buck-ling strength becomes larger with the joint number increasing. Thereason lies in the buckling mechanism that tunnel linings buckleslocally near the K-segment for hinge joints, but buckles in a two-wave mode for moderately-rigid joint. Generally, the effect of thejoints under LC1 and LC2 have the same tendency. Therefore, con-sidering the shield tunnel linings are assembled by segments andjoints, the analysis of buckling strength must take the interactioneffects of the number, the location, and exural rigidity of jointsinto consideration.

4.4. Effect of segment dimension

In elastic buckling of a ring without joint, the dimension andYoungs modulus can determine the buckling strength. As forshield tunnel linings, the effects of thickness and width of segmenton the buckling strength should be investigated. The bucklingstrengths of designed linings under LC1 and LC2 are plotted inFigs. 11 and 12, respectively, considering variation of exural rigid-ity of joint. Furthermore, to clarify the relation between bucklingstrength and thickness described in Eq. (2), the buckling strengths

ound Space Technology 49 (2015) 1441550

500

1000

1500

2000

2500

3000

1111

463.2 463.2362.8

121.2

2287

=0 =0.1 =0.2

Pcr=2295 kN/m2(Eq.(2))

=1.0=0.5

=0.7

Buc

klin

g st

reng

th P

cr (k

Pa)

Number of joints (n)

=0.9 Eq.(2)

(a) LC1

1 2 3 4 5

3000 =0 =0.1 =0.2=1.0

=0.5=0.7 =0.9 Eq.(2)

152 J.H. Wang et al. / Tunnelling and Undergrthe increasing of joint number vary with the exural rigidity of thejoint. The more exible the joint is, the smaller buckling strengthis. In addition, it is found that the joints near the crown, spring-lineand invert of the tunnel dramatically reduce the buckling strength,compared with exible joints, and the buckling strength decreaseswith the joint number nonlinearly. As shown in Fig. 10(a), thebuckling strength of tunnel linings with hinge joints under LC1decreases from 2287 kPa to 1111 kPa and 463.2 kPa when thenumber of joints of K-segment increases from zero to one andtwo, from 463.2 kPa to 362.8 kPa when the number of joints ofK-segment increases from three to four and the fourth joint islocated in the invert, and from 362.8 kPa to 121.2 kPa when thenumber of joints of K-segment increases from four to ve andthe fth joint is located in the spring-line. Similarly, for the tunnellinings with hinge joints under LC2, Fig. 10(b) shows that the buck-ling strength decreases from 1265 kPa to 629.6 kPa and 287.2 kPawhen the number of joints of K-segment increases from zero toone and two, from 274.7 kPa to 114.1 kPa when the number ofjoints of K-segment increases from three to four, and from114.1 kPa to 0 when the number of joints of K-segment increasesfrom four to ve. However, if the joint is moderately rigid, thebuckling strength decreases with the increase of the joint numberrather than the joint location. For instance, in the case of a jointwith exural rigidity g = 0.7, the reduction of the buckling strength

of tunnel linings with thickness t1 = 120 and t2 = 200 mm are calcu-lated by equations Pcr1 = Pcr2(t1/t2)3 and Pcr3 = Pcr2(t3/t2)3 and plot-ted in Fig. 11. Here, Pcr2 is the buckling strength for t2 = 160 mm.

1 2 3 4 50

500

1000

1500

2000

2500

629.6

287.2 274.7114.1

0

1265

Pcr=2295 kN/m2(Eq.(2))

Buc

klin

g st

reng

th P

cr (k

Pa)

Number of joints (n)

(b) LC2

Fig. 10. Buckling strength versus effective exural rigidity factor with the numberof variations of joint.

1000

2000

3000

4000

5000

Pcr=4483 kN/m2(Eq.(2))

Pcr=968 kN/m2(Eq.(2))

Pcr=2295 kN/m2(Eq.(2))

LC1_t120 LC1_t160 LC1_t200 Pcr1_Cal. Pcr2_Base Pcr2_Cal.

Buc

klin

g st

reng

th P

cr (k

Pa)0

Effective flexural rigidity factor of joint(a) LC1

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.20

600

1200

1800

2400

3000


LC2_t120 LC2_t160 LC2_t200 Pcr1_Cal. Pcr2_Base Pcr3_Cal.

Buc

klin

g st

reng

th P

cr (k

Pa)

(b) LC2 Fig. 11. Buckling strength versus effective exural rigidity factor with the varia-tions of the segment thickness (B = 1200 mm, H0 = 10 m).

the buckling resistance and that the buckling strength increasesas the thickness of segment increases under LC1 and LC2. In addi-

0.0 0.2 0.4 0.6 0.8 1.0 1.20

500

1000

1500

2000

2500Load condition 2


LC1_B900 LC1_B1200 LC1_B1500 LC2_B900 LC2_B1200 LC2_B1500

Buc

klin

g st

reng

th P

cr (k

Pa)

Load condition 1

0.1 1 10 100 10000

500

1000

1500

2000

2500

3000

3500

4000

4500 =0 =0.2=1.0=0.5

=0.7

Buc

klin

g st

reng

th P

cr (k

Pa)

Modulus of subgrade reaction K (MN/m3)

=0.9

J.H. Wang et al. / Tunnelling and Underground Space Technology 49 (2015) 144155 153tion, the calculated buckling strengths of tunnel linings with thick-ness t1 = 120 mm and t3 = 200 mm agree well with the numericalresults under LC1, while they are different from the numericalresults under LC2. From Fig. 11(b), it is shown that the numericalresults become smaller for thin segment and larger for thick seg-ment, comparing with the calculated results under LC2. Thisimplies that the imperfection induced by the earth pressureincreases as the thickness of segment decreases, leading todecreasing of buckling strength. However, it can be found inFig. 12 that the width of the segment has only a slight effect onthe buckling strength of tunnel linings under LC1 and LC2.

Generally, the relation between buckling strength and thicknesscan be described by Eq. (2), as well as the width of the segment, ifthe tunnel linings is mainly subjected to the hydrostatic pressure.

4.5. Effect of ground conditionFig. 11 shows that the segment thickness substantially affects

Fig. 12. Buckling strength versus effective exural rigidity factor with the varia-tions of the segment width (H0 = 10 m, t = 160 mm).The effect of the ground condition on the buckling strength isinvestigated. Figs. 13 and 14 show the buckling loads change with

0.0 0.2 0.4 0.6 0.8 1.0 1.20

300

600

900

1200

1500

1800


H0=5 H0=10 H0=15

Buc

klin

g st

reng

th P

cr (k

Pa)

Load condition 2

Fig. 13. Buckling strength versus effective exural rigidity factor with the varia-tions of the overburden H0 (t = 160 mm, B = 1200 mm, LC2).overburden depth under LC2 and the modulus of subgrade reactionunder LC3, respectively.

Fig. 13 shows that the earth pressure induced by overburdencan reduce the buckling strength of tunnel linings. This can beexplained by the fact that imperfections in the tunnel liningsbecome numerous as the earth pressure increases. On the otherhand, from Fig. 14 it can be found that the buckling strengthincreases as the modulus of subgrade reaction becomes larger.The effect of ground support on buckling strength changes withthe joint rigidity and modulus of subgrade reaction when theground reaction K is less than 50 MN/m3. If the ground is moder-ately rigid with modulus of subgrade reaction above 50 MN/m3,the exural rigidity of joint has a slight effect on the bucklingstrength.

5. Theoretical verication of buckling of tunnel linings

The buckling of tunnel linings under design condition is inves-tigated by the buried tube buckling theories. Buckling strength iscalculated by Eqs. (4)(6), corresponding to the linear multi-wave

Fig. 14. Buckling strength versus modulus of subgrade reaction with the variationsof g (t = 160 mm, B = 1200 mm, H0 = 10 m, LC3).and single-wave theories based on the Winkler model, and themulti-wave theory based on the elastic continuum model, respec-tively. The elastic modulus E is evaluated from the assumed

0.1 1 10 100 1000102

103

104

105

106 Max. hoop force N (Hw=18 m) Max. hoop force N (Hw=190 m) Eq.(4)_Winkler_M.W Eq.(5)_Winkle_S.W. Eq.(6)_Continuum_M.W.

Crit

ical

hoo

p fo

rce

Ncr

(kN

)

Modulus of subgrade reaction K (MN/m3)

Fig. 15. Critical hoop force versus modulus of subgrade reaction based on theburied tube buckling theories (t = 160 mm, B = 1200 mm, H0 = 10 m, Hw = 18 m).

Levy, M., 1884, Mememoire sur un nouveau cas integrable du problem de

ounsubgrade reaction factor by Eq. (7). Meanwhile, the maximumdesign hoop forces in various modulus of subgrade reaction are cal-culated by the equation in design specication of tunnel linings.The calculated results are presented in Fig.15.

In Fig. 15, it can be found that the buckling under effectiveground stress does not occur, except that the modulus of subgradereaction is smaller than 0.1 MN/m3, which means the ground hasYoungs modulus E less than 250 KN/m2. Such ground is quite softso that it is not proper to construct a tunnel. Besides, the criticalhoop force in elastic continuum theory is larger than that calcu-lated by the single-wave Winkler model, while it is smaller thanthat of multi-waveWinkler model for soft ground and it is inversedfor hard ground. The comparison results between the above theo-ries are consistent with the analytical results in Gumbel (1983) andMoore (1989). Moreover, the buckling will not occur for thegroundwater level Hw = 190 m. From the above discussion, it canbe concluded that the operating tunnel linings will not buckleunder earth pressure and hydrostatic pressure, even though themaximum hoop forces are over-evaluated by considering allweight of overburden.

6. Conclusions and recommendations

Buckling of tunnel linings under external pressure has beeninvestigated by analytical and numerical analysis. The bucklingbehavior under different load conditions has been claried, as wellas the effects of joint, dimensions of segment, and the ground con-dition. The following conclusions can be drawn based on the aboveanalysis:

(1) Buckling of tunnel linings can be affected by the thickness ofthe segment, the joint and ground condition. Buckling occursin the vicinity of the K-segment joint under low hydrostaticpressure in a single-wave mode, when the tunnel linings isassembled by thin segments and exible joints. The bucklingfailure will occur in the tunnel linings during construction,particularly for the segmental rings just leaving the shieldtail.

(2) The buckling of tunnel linings occurs more easily as the seg-mental joints becomes moderately exible. Segment widthhas a slight effect while the thickness can dramaticallyreduce the buckling resistance. The ground condition sub-stantially affects the buckling strength of tunnel linings:the softer ground is, the larger earth pressure acting on thetunnel linings, the tunnel linings buckle easily.

Conclusively, radial joint as the structural imperfection of circu-lar tube, greatly affects the buckling behavior; the effect of groundcondition can be considered due to the ovalisation deectioninduced by earth pressure and subgrade reaction.

As recommendations, the following issues should be paid moreattention to during the design and construction of a shield tunnel:

(1) For the design of a shield tunnel subjected to high hydro-static pressure, the buckling failure should be checked ratherthan only the material failure, especially for the case of theloading state of the grouting material before complete hard-ening during construction stage.

(2) For the analysis of buckling, numerical solution should takethe effects of joints, segment dimensions, and ground condi-tions into account.

(3) The use of exible segment joint should be avoided when atunnel is subjected to high hydrostatic pressure, and the

154 J.H. Wang et al. / Tunnelling and Undergrapplication of circular shape retainer is recommended, aswell as the rapidly-hardening grouting material during tun-nel construction.However, since the buckling of shield tunnels is an importantbut rare structural problem, external experimental studies shouldbe conducted, especially when considering that the buckling oftunnel linings always results in a disaster during construction ofshield tunnel. Validation by experiments should provide moreinsight in this phenomenon of tunnel engineering. The publicationof this paper will be helpful in stimulating research interest on thebuckling of this concrete linings of shield tunnels.

Acknowledgments

Dr. J.H. Wang acknowledges the support from Deep TunnelTechnical Research Committee (Japan). Dr. W.J. Zhang acknowl-edges the supports from National Natural Science Foundation ofChina (Grant no. 51378342) and Ph.D. Program Foundation of theMinistry of Education of China (Grant no. 20120032120050). Dr.X. Guo acknowledges the support from National Natural ScienceFoundation of China (Grant nos. 11102128 and 11372214).

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