Tunnelling and Underground Space Technology 28 (2012) 124–134

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Tunnelling and Underground Space Technology

journal homepage: www.elsevier .com/ locate / tust

Tunnel structure analysis using the multi-scale modeling method

Cao Yuan a,⇑, Wang Puyong b, Jin Xianlong a, Wang Jianwei a, Yang Yanzhi a

a State Key Laboratory Mechanical System and Vibration, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai 200240, Chinab Shanghai Supercomputer Center, 585 Guo Shoujing RD, Shanghai 201203, China

a r t i c l e i n f o

Article history:Received 10 January 2011Received in revised form 15 September2011Accepted 6 October 2011Available online 8 November 2011

Keywords:Multi-scale modelingWater conveyance tunnelWater hammerStructure analysisFluid structure interaction

0886-7798/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.tust.2011.10.004

⇑ Corresponding author. Address: 225# Advanced MMechanical Engineering, Shanghai Jiao Tong UniverShanghai, China. Tel./fax: +86 21 34206099.

E-mail address: caoyuan@sjtu.edu.cn (Y. Cao).

a b s t r a c t

Structure analysis of the long tunnel is difficult due to the lack of available computing power. Water ham-mer simulation in the water conveyance tunnel is also complicated because of strong fluid structureinteractions (FSIs). In this paper, the multi-scale modeling method is used to simulate water hammerimpacts in the long tunnel. The method can not only yield water hammer simulations along the full tun-nel length, but also the detailed structural responses of the segment linings. In the proposed partitionedapproach, the structural field is solved with the finite-element program LS-DYNA. The fluid field is solvedwith the CFD software package FLUENT. The interaction between two physical fields is realized using ALEdescription. A practical case study is presented and the results are discussed in detail. The results provideus with a better understanding of water hammers and their effects on tunnel linings.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Buried tunnels have increased significantly with the increasingpopulation in metropolitan areas, and water conveyance tunnelsare becoming more important because of the development ofwater delivery projects. Such tunnels differ from other under-ground structures: the overburden caused by hydraulic transients,known as water hammer, can result in structural damage and con-tamination of potable water systems (Cesano et al., 2000; Ghidaouiet al., 2005). The usual procedure for estimating water hammerand the structural responses of the tunnel is to calculate the veloc-ity and pressure of the fluid using the classical Joukowski’s theory,and then apply them on a tunnel model to calculate the responses.Here, both the fluid and the tunnel model are oversimplified. Theliquid is typically represented in the structure model by its massonly, and the fluid structure interaction (FSI), which is hardly ig-nored in many cases (Wiggert and Tijsseling, 2001; Kochupillaiet al., 2005), is not taken into account.

Using numerical methods, many scholars studied tunnel stabil-ity and its responses due to different loads. Duhee Park et al. (2009)performed three-dimensional (3D) finite-element (FE) analysis tosimulate a 1000 m long tunnel response under spatially varyingground motion. Di Pilato et al. (2008) discussed the dynamicresponses of a submerged tunnel under seismic and hydrodynamic

ll rights reserved.

anufacturing BLD, School ofsity, 800# Dongchuan Road,

excitation. An example was analyzed regarding a design proposalfor the Messina Strait crossing, and the simulation results pointout some of the critical aspects of the tunnel behavior. Liu andWang (2010) applied 3D numerical method and sub-model tech-nology to simulate a bifurcated tunnel in detail. Using cutting edgetechnology, they intercepted the intersection from the originalmodel as a sub-model and refined its meshes. The initial boundaryconditions of the sub-model were simplified based on the finalresults of the original model.

In order to evaluate the mechanical characteristics of the tunnellining, Lee et al. (2001) presented an analytical solution for the pre-diction of internal forces and displacements of the segmental lin-ing. Aruga et al. (2007) gave a numerical method to calculate thedeformation and cracking of tunnel linings with reinforcing bars.

A review of the related technical literatures reveal that thenumerical methods often simplify either the detailed structuresof the tunnel, or the external loads and boundary conditions. Thesesimplifications are acceptable in most cases. But for structure anal-ysis in the water hammer process, these simplifications may leadto some inaccuracies. Such as it may underestimate the dynamicstress of the tunnel, and it cannot reflect the joint width variationbetween tunnel linings.

The purposes of the study are to: (1) evaluate water hammerwith consideration of the FSI and obtain the responses of the fulllength tunnel; and (2) analyze the mechanical characteristics ofthe segmental lining in water hammer. For the first purpose, a fullscale model, including the stratum, tunnel, and fluid, is necessary.For the second purpose, a segment lining model, including thedetailed structures, such as grooves, bolts, and so on, is required.

Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134 125

Therefore, there is a contradiction between the mesh dimensions,model scale, and affordable computational expense.

The development of the multi-scale modeling approach (Colellaet al., 2009, 2010; Nilakantan et al., 2010) may provide an alterna-tive method for overcoming such difficulty. This approach is basedon reducing the element number and complexity of the FE model.In this work, detailed lining models along the full-length tunnel arenot required, only several key sections of it, and other parts, whichare some distance away from these sections, are handled as ahomogeneous tube model with relative coarse mesh. This makesit possible to obtain responses along the tunnel, and, at the sametime, obtain the stress distribution and deformation of the seg-mental lining.

The paper is organized as follows: In Section 2 multi-scalemodeling method, computation workflow and ALE method areintroduced briefly. Section 3 gives practical water conveyancetunnel engineering and describes the numerical models in detail.Section 4 presents and discusses the simulation results andSection 5 gives some conclusions about this research.

2. Computation and modeling method

2.1. Multi-scale modeling

The tunnel length is usually several kilometers long, and may beeven longer than 10 km. Its layout is also complicated. In order tosimulate water hammer and understand the overall response ofthe tunnel, a full scale tunnel model is necessary. To reduce theFE model size, a homogeneous tube model is usually used as analternative (Wong and Larue, 1998; Bernaud et al., 2009). Thehomogeneous model is discretized with a relative coarse mesh,and detailed structures of the lining are ignored. A mesh sensitivitystudy should be conducted to determine a suitable mesh dimen-sion that balances computational time with accuracy of results.

The tunnel lining usually contains a number of detailed struc-tures, such as seal grooves, hand holes, iron castings, and bolts,among others. Lee’s work (2001) showed that these detailed struc-tures may affect the calculation results. Obviously, the mechanicalcharacteristics of tunnel linings cannot be very accurately repre-sented using a homogeneous model, and they need to be calculatedusing a segment lining model including these structures. This mod-el needs to be discretized with a relative fine mesh. With regard tothe 3D geometrical discretization of these detailed structures, thetypical size of the smaller elements should be of the same orderas the groove width. This inevitably leads to an oversized FE model,one that even a supercomputer would find difficult to solve.

When choosing multi-scale modeling, there is a tradeoffbetween accuracy and computational expense. In this work, themulti-scale modeling is defined as the model including multiplescale finite element formulations and mesh dimensions, and itcan also be referred to as the hybrid element analysis (HEA)(Nilakantan et al., 2010). One FE model of structures combinesthe homogeneous and segment lining models. The homogeneousand segment lining models are jointed through their interfaces.We refer to this FE model as a hybrid model. This approach candecreases the complexity and the element number of the finite ele-ment model.

The adoption of multi-scale models allows the achievement ofbalance between accuracy and fineness of modeling resolutionwith affordable computational expense. The multi-scale modelinguses different levels of details when describing different tunnel re-gions. The regions that are far from the concerned sections aremodeled using the homogeneous model. At the concerned sectionsof the tunnel, segment lining models are used, coupled on eitherside by the homogeneous model.

A tied interface is used between the two regions because it re-duces the need to match elements across the interface of jointedparts. The tied structure ensures that displacement and stress areconsecutive across the interface. Fig. 1 shows the tied interfacesused.

The interface of the homogeneous model is defined as a mastercontact surface, and one of the segment lining models is defined asa slave surface. During each computation step, the node force andmass on the slave surface are transformed to the node on the mas-ter surface, which is tied with the slave surface:

Df im ¼ /iðnc;gcÞfs; i ¼ 1;2;3;4 ð1Þ

Here, the subscript m and s represent the master and slave nodes,respectively; /i is the shape function of the node; ðnc; gcÞ is the con-tact point coordinates; f is the interface force.

The master node acceleration can be calculated according to thenode force and mass. The corresponding slave node accelerationcan be computed using interpolation:

ais ¼X4

j¼1

/jðnc;gcÞaji ð2Þ

where aj is the corresponding master node acceleration. The otherparameters are same with Eq. (1).

2.2. Simulation workflow

Tunnel structure analysis in water hammers is a complex prob-lem that requires a number of specialized tools. In this work, a par-titioned approach is selected in an attempt to solve all physicalfields involved by specialized, adapted, and tested single-field solv-ers. This fully partitioned approach splits the task into several sub-tasks, and allows the use of suitable software packages with theirown discretization for each subtask.

In the current approach, the initial geo stress field and struc-tural simulation are solved using LS-DYNA. The fluid part is solvedby the general purpose CFD code FLUENT, and the dynamic FSIanalysis is solved by coupling the ALE approach in LS-DYNA witha multi-scale modeling method. Fig. 2 provides a detailed view ofthe software environment and simulation workflow.

This approach includes the following steps: First, a full lengthmodel with large mesh dimensions is constructed. In this model,the tunnel is simplified as a homogeneous continuum tube. Thestrata are built according to the field survey. Then, initial computa-tion is carried out to determine the initial geo stress field and thetunnel form under the gravity. Afterwards, the steady flow field(i.e., normal operation condition) is calculated using CFD code. Thisflow field (pressure and velocity) is applied to the inner flow tocompute structural responses. The flow field is also used as the ini-tial condition of the water hammer simulation. From these calcula-tion results, several sections of the tunnel can be determined.

The segment lining model, with small mesh dimensions, is builtand used to replace the corresponding sections in the homoge-neous model. The new model combines two different mesh dimen-sion models, and is called a hybrid model. Finally, water hammersimulation is accomplished using FSI computations. The simulationresults can provide more accurate deformation and stress distribu-tion of the tunnel lining.

2.3. ALE kinematics

As validated by previous works, the FSI should be considered inthe water hammer simulation and structure analyses. There aremany successful applications of the Arbitrary Lagrangian Eulerian(ALE) method on FSI (Nomura, 1994; Bathe et al., 1999; Souliet al., 2000). Therefore, an attempt is made in this paper to studywater hammer with FSI using ALE method.

Fig. 1. Tied interface between two regions.

FE preprocessing homogeneous continuum

tunnel and soil models

Initial computation (LS-DYNA)initial geostress fieldtunnel form finding

CFD preprocessingflow field boundaries

CFD computation(FLUENT)steady flow field simulation

(pressure, velocity etc )

Inner water pressure and velocity

Normal operation condition simulation (LS-DYNA)

key positions determining

Finely meshed key position submodel and water model

FSI computation (LS-DYNA)water hammer simulation

Postprocessing

Water pressure and velocity as initial condition

Fig. 2. Software environment and simulation workflow.

126 Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134

The ALE method combines the advantages of the Lagrangianmethod and Eulerian method. In this section, we briefly outlinethe ALE approach used in the work. In water hammer analysis,the fluid model usually adopts the assumption of slight compress-ibility. This paper also adopts the same assumption. Thus, the massand momentum equations in ALE approach are:

@q@t

����vþ ci

@q@xiþ q

@v i

@xi¼ 0 ð3Þ

q@v i

@t

����vþ qcj

@v i

@xj¼ @rij

@xjþ qbi ð4Þ

Here, the v and X denote the ALE and Lagrangian coordinates; q isthe density, v is the fluid velocities, c is the ALE convective velocityand equals v � v̂ where v̂ is the velocity with respect to the ALEcoordinates; the subscript i represents the direction; bi representsthe body force.

The constitutive equation is introduced as:

rij ¼ �pdij þ sij ð5Þ

here deviatoric stress

sij ¼ kdijskk þ 2lsij;

where sij ¼ 12

@v i@xjþ @v j

@xi

� �.

The first term on the right-hand side of Eq. (5) is defined by theequation of state (EOS) in the simulation. For Newtonian fluid usedin the paper, the stress kdijskk due to the change of the volumenearly equals zero. Then the second term on the right hand sideof Eq. (5) only includes the shear stress. It describes the relation-ship between the shear stress and the shear strain rate. Eq. (5)can be rewritten as:

rij ¼ �pdij þ 2lsij ð6Þ

where l is kinetic viscosity and is a constant for Newtonian fluid.The constitutive equation of the fluid in the paper is composed

of the EOS and the material model. The EOS defines the volumetriccompression (or expansion) behavior of the fluid, and the material

(a) Tunnel and water (b) Tunnel and soil

Tunnel

Water

Fig. 3. Homogeneous model.

Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134 127

model defines the relationship between the shear stress and theshear strain rate.

3. Model description

The tunnel studied in this work is a 14 km long pressurizedwater conveyance tunnel. It conveys potable water for approxi-mately five million people. The average flow speed in the tunnelis about 1.8 m/s. The tunnel runs at a depth of 20–50 m from theground surface and across a river. The concrete segment lininghas a diameter of 5.5 m and a length of 1.5 m. This shield tunnelis composed of approximately 10,000 tunnel rings. Each ring con-sists of six segments, and these segments are jointed through bolts.The stagger-jointed assembling mode is adopted between rings. Inthe vertical direction, the tunnel line looks like a bow and the larg-est buried depth is located at the middle of the tunnel line.

The homogeneous tunnel model was discretized with hexahe-dron elements in the framework of the Lagrangian approach. In

(a) Segmenta

(b) Segmen

Fig. 4. Segment l

order to maintain good element aspect ratios, the annular lengthof the elements is 1/20 of the pipe circumference, the element axiallength is the width of the lining, and two elements are usedthrough the thickness. Good element aspect ratios are also impor-tant to maintain smooth contacting surfaces between tunnel andsoil. The tunnel is loaded and constrained by the strata. The equiv-alent tunnel model is shown in Fig. 3.

Fig. 4 displays the FE model of one segment lining. It was alsodiscretized with hexahedron elements. The total element numberof the lining model is about 24,000. This model includes a numberof detailed structures, such as seal grooves, hand holes, iron cast-ings, bolts, etc. The segment lining, whose constitutive relationshipis a linear elastic model, is simulated using an elastic materialmodel. The material properties are listed in Table 1.

The surrounding soil model was constructed according to thegeological exploration data, which included five layers of earthwith different thicknesses. To avoid calculation errors arising fromsize effects, the model should have enough range along the lateral

l lining

t view

ining model.

Table 1Parameters of the concrete model.

Density, q (kg/m3) Elastic modulus, EY (GPa) Poisson’s ratio, t

2500 32.5 0.2

Table 2Parameters of soil model.

Layers Density(kg/m3)

Poisson’sratio

Cohesionforce, c(kPa)

Internal frictionangle, / (Rad)

Elasticmodulus(MPa)

1 1897.96 0.26 9 0.471238894 182 1714.29 0.35 12 0.18325957 4.93 1836.73 0.32 18 0.314159263 8.64 1938.78 0.25 4 0.584685294 29.65 1959.18 0.24 2 0.610865233 40.4

128 Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134

direction. The calculation range of the geological model is120 m � 80 m (depth �width).

Although the boundary conditions of soil has little effect on thewater hammer simulation and the model boundaries are far awayfrom the tunnel, the boundaries widely accepted in earthquakesimulation were used. The bottom and side boundaries of the sur-rounding soil were modeled using viscous-elastic artificial bound-ary. It is implemented by distributing the parallel spring anddamper element in the normal and tangent direction on each finiteelement node of boundaries. The spring constant and the dampingcoefficient can be expressed as follows (Liu and Li, 2005):

Kb ¼ aGR

ð7Þ

Cb ¼ qc ð8Þ

where q is the mass density, G is the shear modulus, R is the dis-tance between the loading point and the boundary, c is the wavevelocity, parameter a is set according to the direction of the bound-ary, that is 1.33 for normal direction and 0.67 for tangent direction(Liu et al., 2006).

The normal interaction between tunnel and soil was simulatedby penalty-based surface–surface contact. The penalty method hasbeen applied in numerical simulations of soil–structure interaction(Ding et al., 2006).The penalty method introduces a coupling forcebetween the soil and the tunnel. The coupling force is proportionalto the depth of penetration and the contact rigidity. Fig. 3b showsthe tunnel and soil model.

Considering the great friction between the concrete and thesurrounding soil of the tunnel project, in addition to the normal

Initial computation Normal opera

Fig. 5. The load c

contact modeling, a classical Coulomb friction law is employed tomodel the friction contact behavior with an assumed friction coef-ficient l ¼ 0:5.

The fluid domain (water) was discretized with hexahedron ele-ments in the ALE description, shown in Fig. 3a. Although the lowMach number water flow generally could be regarded as an incom-pressible flow, the water compressibility is dominant in the waterhammer process. Then, the EOS which describes the relationshipbetween the deformation of fluid and the pressure was introducedhere. It relates the pressure with the volume variance rate at aphysical state. After comparing several simulation tests in a preli-minary phase, the Gruneisen equation was selected. It was used todetermine the pressure in a shock-compressed solid (Mitchell andNellis, 1981). By selecting appropriate parameters, it can also beused as the EOS of water. The equation can be written as:

p ¼q0C2l 1þ ð1� c0

2 Þl� a2 l

2� �

1� ðS1 � 1Þl� s2l2

lþ1� S3l3

ðlþ1Þ2þ ðc0 þ alÞE ð9Þ

where C is the speed of sound and is 1400 m/s in this research; S1,S2, and S3 are the coefficients of the slope of the us–up curve; c0 isthe Gruneisen gamma; a is the first-order volume correction toc0; l is the kinematic viscosity and equals 8.684E-4; and E is theinitial internal energy. The values S1, S2, S3, c0, and a are all inputconstant parameters, and their values are 1.192, �0.92, 0, 0.35,and 0 respectively (Cao and Jin, 2010).

The material model for soil includes the Drucker–Pragerelastic–plastic criterion, which may be expressed as:

F ¼ aJ1 þ ðJ02Þ

1=2 � k ¼ 0 ð10Þ

Here, J1 is the first invariant of stress tensor; J02 is the second invari-ant of stress deviator; and parameters a and k are calculated by:

a ¼ 2 sin /ffiffiffi3pð3� sin /Þ

; k ¼ 6c cos /ffiffiffi3pð3� sin /Þ

ð11Þ

where c is the cohesion force and / is the internal friction angle.They can be obtained from physical test results. All the parametersused in the material model are listed in Table 2.

As described in Section 2.2, the calculation process can bedivided to three steps: initial computation, normal operationsimulation, and FSI simulation (or water hammer simulation).The load combinations are different at each step, as Fig. 5 shows:(1) initial computation: Soil and structure weight + external waterpressure; (2) normal operation simulation: geo stress + soil andstructure weight + external water pressure + internal water pres-sure; and (3) FSI simulation: geo stress + soil and structureweight + external water pressure + internal water pressure + waterhammer pressure.

tion simulation FSI simulation

ombinations.

(a) Hoop stress (b) Deformation (100x magnification)

Fig. 6. Stress and deformation distribution.

Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134 129

The external water pressure refers to the ground water pres-sure. It was calculated according to the vertical distance betweenthe tunnel lining and water level. The internal water pressure isthe steady flow pressure in the tunnel and was calculated usingCFD code. The water hammer pressure was calculated using FSIcomputation. All the above pressure on each tunnel element wereconverted to nodal force and applied on each element node.

4. Results and discussion

4.1. Initial computation

In order to obtain the initial geostress field and the tunnel formunder gravity, initial computations were carried out. When the soiland tunnel deformation or the stress of the linings became stable,the computations were considered complete. The stress ratio wasselected based on the experience. It is about 0.4.

Fig. 6 shows the stress and deformation of one tunnel ringwhich locates at the middle of the tunnel. This ring is about 6,800 m from the tunnel entrance and the buried depth is about47 m. The stress and deformation contour is approximately sym-metric. The maximum and minimum stresses of the ring causedby external loads mainly lie at the arch bottom, vault, and twosides of the ring, and feature zonal distribution, as Fig. 6a shows.Furthermore, the maximal compressive stress appears on theexternal surface of the ring at the bottom and vault, while the max-imal tensile stress appears on the internal surface. The compressivestress is far larger than the tensile stress. Fig. 6b shows that themaximum and minimum deformations also distribute at the archbottom and vault, and present zonal distribution. Under theseloads, the tunnel ring becomes elliptical in shape, with its lateraldiameter larger than its vertical one.

Fig. 7. Flow field com

4.2. Normal operation simulation

The flow in the tunnel is a gravity flow. The computation modelof the flow field is shown in Fig. 7. The hydraulic pressure at theentrance is about 0.13 MPa (Pressure P1) and the pressure at theexit is about 0.108 MPa (Pressure P2). The viscosity of water is8.7 e–4 Pa s. The Reynolds number is more than 9e6. The flow isturbulent flow. In this work, the inner water pressure was calcu-lated as the hydrostatic pressure. The friction between the flowand the tunnel wall was ignored. The flow field computation wasaccomplished by commercial CFD codes. After the computation,the pressure and velocity were imposed on the fluid finite elementnodes in the homogeneous model as boundary conditions. Basedon the results of initial computations, the authors then carriedout normal operation simulations.

We analyzed and compared results from more than 30 crosssections. Fig. 8 shows hoop stress variations along the tunnel line.The results indicate that the geological conditions change along thetunnel, and the ground stress significantly affects the hoop stress ofthe tunnel rings. The compressive stress is still much larger thanthe tensile stress.

Fig. 8 also compares the hoop stresses of normal operation andinitial conditions. The inner water pressure varies along the tunneland changes the tunnel stress. Because the inner water pressurecounteracts partial external loads, the compressive stress de-creases but the tensile stress increases. Compared to externalloads, the internal pressure is not significantly large.

After careful comparison, eight cross sections were selectedaccording to their stress values and geological conditions. Theyare all numbered and shown in Fig. 8. Segment lining models wereused on these sections to obtain more precise and detailed resultsduring FSI simulation.

putation model.

(a) Tensile stress

(b) Compressive stress

Fig. 8. Hoop stress variations along the tunnel.

(a) Eight rings of segment lining (b) Cutaway view of the hybrid model

Fig. 9. Segment lining model and hybrid model.

130 Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134

4.3. FSI simulation

4.3.1. Hybrid modelAccording to the St. Venant’s theory, accurate results may be

obtained if the intersection interfaces are located in regions far

from the concerned sections. Therefore, each cross section iscomposed of eight rings of the segment lining, as Fig. 9a shows.In all, there are 64 rings of segment linings in the hybrid model(8 rings � 8 cross sections). The cut edge separated from the origi-nal coarse model is the segment lining model’s boundary, as Fig. 9b

Fig. 11. Water pressure curve at the entrance of the tunnel.

Fig. 12. Relationship curve between water pressure and valve closing time.

Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134 131

shows. In this work, the results of the two middle rings are used forstructure analysis.

4.3.2. Water hammer simulationWater hammer was produced by a rapid valve closure at exit.

The valve closing times were 1, 30, 60, 120, and 240 s. The otherboundary conditions are same with the normal operation.

The initial conditions were divided to fluid part and structurepart (soil and tunnel). The initial conditions of fluid part refer tothe initial water pressure and flow velocity. The steady flow fieldbefore water hammer was calculated using FLUENT, as shown inFig. 2. Both the pressure and flow velocity of each fluid elementwas taken out from CFD computation results and applied on thecorrespondence fluid element or the correspondence node in thehybrid model. The initial conditions of structure parts came fromthe simulation results of normal operation.

Fig. 10 shows water pressure curves at three different positionswhen the closing time is 1 s. It could be seen that the maximumpressures occurred at different time along the tunnel. The maxi-mum at the exit was earlier than it at the entrance. Obviously,the water hammer impact traveled in the form of wave fromdownstream to upstream. This speed can be calculated accordingto the water hammer period, and it is about 800–900 m/s. Thisspeed is less than that obtained from Korteweg formula (about1100 m/s), Eq. (12). This may be due to multiple factors attributedto the boundary constraint conditions and FSI.

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKf

q

1þ Kf DeE

vuut ð12Þ

Here, a is the wave speed, Kf is the bulk modulus of water, q iswater density, D is the diameter of the tunnel, E is the elastic mod-ulus of the tunnel lining, and e is the thickness of the lining.

From figures, it is also noted that the maximum pressure and itsduration gradually decreased and shortened from downstream toupstream along the tunnel. The pressure peak at the exit of thetunnel was about 5 MPa and could sustained dozens of seconds.The peak value at the entrance was about 3.5 MPa, and thendropped almost instantaneously. Other simulation results showthat the duration of the peak value also prolonged with the valveclosing time. So, although water hammer is probably highly tran-sient, in terms of the tunnel length and valve closing time, it isquite reasonable to approximate the peak flow period as steadyin the tunnel.

In general, the valve closing time will not be less than dozens ofseconds. Therefore, the analysis of other closing times is moremeaningful. Fig. 11 shows the water pressure time history at theentrance of the tunnel. The peak pressure occurs at different clos-ing times. The maximum pressure is inversely related with theclosing time. Fig. 12 shows the relationship curve, and it is not alinear one. When the closing time increases from 30 to 60 s, themaximum pressure decreases more than 60% (from 0.9 to

(a) Tunnel exit (b) Middl

Fig. 10. Water hamme

0.35 MPa). When the closing time is 240 s, the peak pressure isnearly equal that of the operational pressure. This means that ifthe closing time exceeds 240 s, the water hammer impact vanishes(Fig. 12).

4.3.3. Structure analysesIn order to evaluate the multi-scale model and analyze struc-

tural responses in water hammer, we carried out FSI computationsusing the hybrid and homogeneous models, respectively. The valveclosing time was 60 s. As described in the previous section, thepeak pressure can be maintained for some time; therefore, thestructural responses presented and discussed are treated as a stea-dy process.

Taking cross section #2 as an example, Fig. 13 compares thehoop stresses of the hybrid and homogeneous models. Except forthe stress values, the stress distribution of the homogeneous modelis similar to the results of normal operation. The stress distribution

e of tunnel (c) Tunnel entrance

r pressure curves.

Fig. 13. Stress distributions of the hybrid model and an equivalent model in water hammer.

Fig. 14. Comparison of tensile stress increments between the hybrid model and an equivalent model.

Fig. 15. Segment joint number diagrams.

132 Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134

of the segment lining model does not present zonal distributionbecause of its stagger-jointed assembly. Maximum stress appearson positions opposite the segment joint of an adjacent ring. Inaddition, the stress values of the hybrid model are larger thanthose of the homogeneous one. This is because the homogeneousmodel is a simplified homogeneous tube and some small structuresare ignored in it, such as grooves. As well, the small mesh dimen-

sions of the segment lining model may contribute to the increase instress values. The assembly discrepancy of the segment liningmodel also may affect stress. Discrepancies may lead to stress con-centrations on some detailed structures, such as the sharp cornerof the seal groove.

The increase in tensile stress and joint width increment areused to study tunnel stability in water hammer. Here, the stress

Table 3Joint width increments at cross section #2.

No. Increment (mm) No. Increment (mm)

D11

0.04 D21

0.04

D12

0.62 D22

0.03

D13

0.03 D23

0.57

D14

0.05 D24

0.14

D15

0 D25

0

D16

0 D26

0.03

Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134 133

increase (SIn, increase percentage) is defined by the followingequation:

SIn ¼ ðrw � rnÞ=rn � 100% ð13Þ

where rw and rn are the maximal tensile stresses in water ham-mer and in the normal operation process, respectively.

Fig. 14 compares the tensile stress increases of the two modelsat eight cross sections. The variations of these increases along thetunnel are similar. This proves that, except for the difference in val-ues, the results of the hybrid model are consistent with those of thehomogeneous model. The plot also indicates that stress increasesof the hybrid model are less than those of the homogeneous one.This may be due to their higher tensile stress values in normaloperation process.

As discussed earlier, the water hammer pressure at the exit ismaximal along the tunnel. Therefore, tunnel rings at cross section#8 withstand the greatest impact and inner pressure. As a result,SIn at #8 is the largest. Among the other sections, the stress in-creases at cross section #4 and #5 are the largest. The reason istheir lower stress values in normal operation process. Both thestress values at #7 and #8 are the largest in all cross sections. Gen-erally, these values in water hammer are no more than 5 MPa andacceptable for tunnel structure. In addition, there are no remark-able changes in stress distributions of the tunnel rings duringwater hammer.The joint width increment is also studied in this pa-per. The increment is calculated and indexed by Di

j ¼ Dw � Dn.Here, Dw and Dn are the joint width increments in water hammerand normal operation process, respectively. The superscript i de-notes the ring number (i = 1, 2), and the subscript j representsthe joint number in the ring (j = 1–6). There are six joints in eachtunnel ring. To facilitate analysis, the joints between segmentsare sequentially numbered and shown in Fig. 15.

Table 3 gives joint width increments at cross section #2. It canbe seen that most joint widths increase in water hammer, andthese joints are on both sides of the tunnel lining. The incrementsof the joints on the vault or at the bottom of the arch are zero. Thejoints of other cross sections also show similar results.

Fig. 16. Maximal joint width incre

The maximal joint width increments are different at differentcross sections. Fig. 16 compares the values of eight cross sections.The joint width increments at cross section #6 and #7 are the larg-est. The reason is that the water hammer impact is the greatest atdownstream. The results indicate that the damage caused by waterhammer may occur in the downstream. Due to this reason, the tun-nel rings at cross section #8 were fastened using some methods indesign phase. As a result, its width increment is small. The jointwidth increments at cross section #4 and #5 are the smallestamong all cross sections. These two cross sections both are in themiddle of the tunnel and the buried depth is about 45–50 m. Thetunnel rings are under the large pressure of soil and ground water.By comparing Figs. 13 and 15, it is interesting to note that wherethe joint width increment is small, there is a large tunnel stress in-crease. It means that the tunnel ring partially releases stressesthorough joint opening.

5. Conclusions

Water hammers are dangerous for long water conveyance tun-nels. They may result in tunnel damage and contamination of po-table water systems. Numerical simulations are an effectivemethod of evaluating the tunnel stability in water hammers. How-ever, due to limitations in computing power, it is difficult to solvethe numerical model of a long tunnel with much detailedstructures.

In this study, a multi-scale modeling method is presented.Based on this method, simulation of the long tunnel can beachieved, and calculation of the deformation and stress of tunnellinings can be performed. Moreover, a partitioned approach usedin the paper allows the use of the suitable software packages withtheir own discretization for each subtask. The ALE formulation isemployed to describe the FSI between the inner water and the tun-nel. Conclusions are as follows:

(1) Under the soil weight and ground water pressure, the tunnelring becomes elliptical in shape, with its lateral diameter lar-ger than its vertical one. The compressive stress of tunnel ringis much larger than its tensile stress. Compared to externalloads, the pressure of tunnel flow is not significantly large.

(2) The simulation results of homogeneous and hybrid modelsare basically similar. However the stress on tunnel rings isnot exactly consistent. The results of the homogeneous modelshow that the maximum and minimum stresses present azonal distribution, and mainly distribute at the arch bottom,vault, and two sides of the ring. In the results of the segmentlining model, stress distribution does not present the samedistribution because of its stagger-jointed assembly.

ments at eight cross sections.

134 Y. Cao et al. / Tunnelling and Underground Space Technology 28 (2012) 124–134

Maximum tensile stress usually appears at the positionsopposite the segment joint of the adjacent ring. Moreover,the latter stress value is higher than the former’s. Becausethe homogeneous model is a simplified model and the seg-ment lining model is closer to reality, authors think the latterresults are more reasonable.

(3) The water hammer impact caused by valve closure travelsfrom downstream to upstream. The wave speed of waterhammer is lower than that obtained from Korteweg formula.This may be due to the boundary constraint conditions andFSI between water and the tunnel. The closing time has aremarkable effect on the pressure peak and its duration. Itshows that the water hammer impact is inversely relatedwith the closing time. When the closing time is long enough,the water hammer impact may vanish completely.

Both the pressure peak and its duration gradually decrease fromdownstream to upstream. As a result, if there is damage, it is morelikely to occur in the downstream. The main risk caused by waterhammer is the width increments of joints between segments in-stead of the segment failure. The ground water may flow in andcontaminate the portable water. Therefore some special methodsshould be considered to reduce the risk. Such as, lengthen the valveclosing time caused by accidents and apply more large pre-tighten-ing force on the bolt connecting segments.

Acknowledgements

This work is supported by National Science Found of China (Nos.11072150 and 61073088) and the Fund of State Key LaboratoryMechanical System and Vibration (Shanghai Jiao Tong University,No. MSV-MS201107).

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