THE REASONABLE CROSS-SECTION SHAPE FOR THE TUNNEL … · The Hyperstatic Reaction Method has been given by Duddeck and Erdmann [8], Takano [4], Oreste [8]. Hyperstatic Reaction Method

http://www.iaeme.com/IJCIET/index.asp 871 [email protected]

International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 12, December 2018, pp. 871-880, Article ID: IJCIET_09_12_090

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=12

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication Scopus Indexed

THE REASONABLE CROSS-SECTION SHAPE

FOR THE TUNNEL FROM HANOI METRO

SYSTEM UNDER THE IMPACT OF THE

EARTHQUAKES

Gospodarikov Alexandr

Saint Petersburg Mining University, Saint Petersburg, Russia Federation

Thanh Nguyen Chi

Saint Petersburg Mining University, Saint Petersburg, Russia Federation

Hanoi University of Mining and Geology, Hanoi, Vietnam

ABSTRACT

The metro system has become an important part of the infrastructure of major cities

in the world. Hanoi is the capital of Vietnam, where has got very many people (there

are about 8 million people) and there are important works here. At present, the metro

system that has been designed and built in Hanoi. Hanoi is located near Song La - Dien

Bien fault and Red River fault with the strongest earthquake that has magnitude MW =

6.5 Richter. This paper presents the impact of the earthquake that has the strongest

magnitude that can affect to the tunnel of Hanoi metro system with the different cross-

sectional shapes of the tunnel (including circles, squares, and horseshoes). Based on

the results, the paper will comment and provide a choice of cross-sectional profiles

suitable for the tunnel from Hanoi metro system when the tunnel works under the impact

of earthquakes.

Key words: earthquake; horseshoe; circular; square; impact; tunnel.

Cite this Article: Gospodarikov Alexandr and Thanh Nguyen Chi, the Reasonable

Cross-Section shape for the Tunnel from Hanoi Metro System under the Impact of the

Earthquakes, International Journal of Civil Engineering and Technology, 9(12), 2018,

pp. 871–880

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=12

1. INTRODUCTION

At present, there are many types of tunnel cross sections used in design and construction. Each

of the cross-sectional shapes has different advantages and disadvantages and, when selected,

should be based on the advantages and disadvantages of these cross sections as well as on the

Gospodarikov Alexandr and Thanh Nguyen Chi


safety requirements of the tunnels. In this paper, the author mainly refers to the effect of

earthquakes on tunnel stability as the criterion for evaluating and selecting the right cross-

sectional shape for the tunnel. There are many methods of assessing the effects of earthquakes

on the tunnel including the methods of Wang (2003) [4], Penzien et al. (1998, 2000) [5, 9],

Oresten (2007) [5], Do NA (2014) [4, 5] with the circular tunnels, Wood (2004, 2005) [10, 11]

with the square tunnels. Based on the hydrogeological and geological conditions in the center

Hanoi, the paper used several methods to assess the impact of the strongest earthquake that that

can occur the area Hanoi and based on the results obtained select the appropriate cross-section

shape for the tunnel of Hanoi metro system in case of the tunnel lining is continuous.

2. CALCULATION METHODS

2.1. Method Wang [4, 5]

In 1993, Wang (1993) [4, 5] based on closed form solutions recommended by Peck et al,

proposed modified analytical equations in terms of axial force, bending moments and

displacements under external loading conditions. Wang [4, 5] given equations that be used to

calculation structural forces in the tunnel lining under seismic loading conditions.

In case full–slip at the soil–lining tunnel’s, bending moment M and forces normal T can be

written by Wang [4, 5]:

1 max

1cos 2(

6 (1 ) 4

ET K R

πγ θ

ν= ± +

+

(1)

2

1 max

1cos 2(

6 (1 ) 4

EM K R

πγ θ

ν= ± +

+

(2)

Where 1

12(1 )

2 5 6K

F

ν

ν

−=

+ − (3)

In case no-slip at the soil – the lining tunnel, only the normal forces T can be expressed by

Wang (1993) [4, 5]:

2 max

1cos 2(

2 (1 ) 4

ET K R

πγ θ

ν= ± +

+

(4)

Where

[ ]

[ ]

2

2

2

1(1 2 ) (1 2 ) (1 2 ) 2

215

(3 2 ) (1 2 ) 8 6 6 82

F C

K

F C C

ν ν ν

ν ν ν ν ν

− − − − − +

= +

− + − + − + + − (5)

( )21

(1 )(1 2 )

s

S

E RC

E t

ν

ν ν

−=

+ − (6)

2 3(1 )

6 (1 )

S

S S

E RF

E J

ν

ν

−=

+ (7)

2(1 )

EG

ν=

+ (8)

The Reasonable Cross-Section shape for the Tunnel from Hanoi Metro System under the Impact

of the Earthquakes


In 2013, Kouretzis el al. (2013) [5] proposed an equation of the maximum bending moment

in lining tunnel under no slip condition when effect dynamic loading, this to improve the

Wang’s method.

2

3 4 max(2 2 )2

K

RM K K τ= ± − −

(9)

max max max maxV Gτ ρ= ± (10)

Where max

τ -the maximum free field seismic shear stress, max

ρ -density of the surround

ground, max

G -the maximum ground shear modulus, max

V -the peak seismic velocity due to shear

wave propagation.

[ ] [ ]3

(1 2 )(1 C)F 0.5(1 2 )C 21

(3 2 ) (1 2 )C 0.5(5 6 ) (1 2 )C (6 8 )K

F

ν ν

ν ν ν ν ν

− − − − += +

− + − + − − + − (11)

[ ] [ ]

[ ] [ ]4

1 (1 2 )C F 0.5(1 2 )C 2

(3 2 ) (1 2 )C 0.5(5 6 ) (1 2 )C (6 8 )K

F

ν ν

ν ν ν ν ν

+ − − − −=

− + − + − − + − (12)

Where K1-full–slip lining response coefficient; K2-no–slip lining response coefficient; F -

flexibility ratio of tunnel lining respectively; Es-Young’s modulus of tunnel lining respectively;

C-compressibility ratio of tunnel lining respectively; νs- Poisson’s ratio of tunnel lining

respectively; R-tunnel radius respectively; Js-inertia moment of tunnel lining per unit length of

the tunnel (per unit width); t-thickness of tunnel lining respectively; ν-Poisson’s ratio of ground

mass surrounding tunnel; E- Young’s modulus of ground mass surrounding tunnel; G-shear

modulus of ground mass surrounding tunnel; I-moment of inertia of the lining tunnel; γmax-

maximum free–field shear strain; θ-angle measured counter-clockwise from spring line on the

right

2.2. Penzien’s Method [9]

Penzien et al. (1998, 2000) [4, 5, 9] developed similar analytical solutions for the thrust moment

and shear in tunnel lining. In Penzien’s method that has got two case of: the case of full-slip at

tunnel lining and soil layer surrounding and the case of no-slip condition. In case of full-slip

condition at the soil-lining, the normal forces T and bending moment M, that be defined by

equations:

3 2

12cos 2( )

(1 ) 4

n

s lining

S

E I dT

d

πθ

ν

∆= − +

− (13)

2 2

6cos 2( )

(1 ) 4

n

s lining

S

E I dM

d

πθ

ν

∆= − +

− (14)

3 2

24sin 2( )

(1 ) 4

n

s lining

S

E I dV

d

πθ

ν

∆= − +

− (15)

n n

lining free fieldd R d

−±∆ = ± ∆

(16)

4(1 )

1

n

nR

ν

α

−= ±

+ (17)



3 2

12 (5 6 )

(1 )

n s

S

E I

d G

να

ν

−=

− (18)

In the case of no-slip condition between the tunnel lining and soil surrounding, the

formulations are presented as:

max

2lining free field

dd R d R

γ−

±∆ = ± ∆ = (19)

3 2

24cos 2( )

(1 ) 4

s lining

S

E I dT

d

πθ

ν

∆= − +

− (20)

2 2

6cos 2( )

(1 ) 4

s lining

S

E I dM

d

πθ

ν

∆= − +

− (21)

4(1 )

1R

ν

α

−= ±

+ (22)

3 2

24 (3 4 )

(1 )

s

S

E I

d G

να

ν

−=

− (23)

2.3. HRM Method [1, 4, 7]

Figure. 1 Calculation scheme of tunnel lining with the HRM method. σv: vertical load in the model

tunnel – surrounding ground; σh: horizontal load in the model tunnel – surrounding ground; kn: normal

stiffness of the interaction springs; ks: tangential stiffness of the interaction springs; R: tunnel radius;

EJ and EA: bending and normal stiffness of the tunnel lining [5, 8]

The Hyperstatic Reaction Method has been given by Duddeck and Erdmann [8], Takano

[4], Oreste [8]. Hyperstatic Reaction Method has content that is simulates the interaction

between the lining tunnel and environment material surrounding the tunnel lining by a number

of independent “Winkler” type springs [6]. In the Hyperstatic Reaction Method need to the

definition of the active loads that apply directly to the tunnel lining. In 2003, Mashimo and

Ishimura [5, 7] given different methods that be used to calculate for the active loads that apply

directly to the tunnel lining. When has got the effect of seismic loading, necessary to estimate

the active loads on the tunnel lining. Peinzen and Wu [5, 9] and Naggar et al. [2, 5] presented

R

EJ, EA


of the Earthquakes


equations, these can be used to the calculation of ovaling deformation of a circular tunnel

during a seismic due to the in-plane shear stresses caused by vertically propagating (notice that,

in case of the seismic load on the tunnel lining, all the external loads are rotated counter-

clockwise by 450 and the horizontal loads are in opposite directions). In this paper, the Matlab

program had used to the calculation for the internal forces on tunnel lining by Hyperstatic

Reaction Method.

2.4. Method Wood for the tunnel square

According Wang's method [5], Wood [5, 10, 11] had given the calculation method for the

rectangular tunnels. The content of Wood's method is defined by equations:

1

100

50 2

s s

s

s

HT T

G

ρ=

(24)

2

1

100

50 2

s s

s

s

Hu u

G

ρ=

(25)

1

s s

s

s

H

G

ρε ε=

(26)

si

ff

R∆

=∆

(27)

2

w

1

2 3 3241

2 2 (2 2 3 )

st

Hf

r jrP K

r j jr

∆= =

+ +−

+ + +

(28)

st

r

s

fF

f=

(29)

s

Hf

LG=

(30)

w6 (3 )

(2 )(2 )f

K j qM

H j j q q

∆ +=

− + +

(31)

w6 (3 )(2 )

1(2 )(2 )

r

K j q qM

H j j q q

∆ + += +

− + +

(32)

Where w

w

EIK

H= ; w

w

EIK

H= ; r

r

EIK

H= ;

w

rK

rK

= ; w1

r

Kq

r K= = ; E is Young’s modulus for

the structure material and Iw and Ir , If are the moments of inertia per unit length of the wall,

roof and floor respectively; Ts, us, εs , Gs are period, displacement strain and shear modulus of

soil layer; T1, u1 and ε1 are the reference soil layer values; sρ is soil density; G is shear modulus

in the soil at the level under consideration; fst is elastic flexibility of lining; R is the shear strain

deformation ratio; Fr is the flexibility ratio; Kw is wall stiffness; Kr is roof stiffness; Kf is floor

stiffness; q is ratio wall/roof stiffness; j is ratio floor/roof stiffness; r is ratio roof/wall stiffness;



Mr is the moment at the top corner of structure; Mf is the moment at the bottom corner of

structure.

2.5. 2D numerical method

There are three tunnel models produced by the 2D numerical method using Abaqus software,

including the tunnel model with the cross-section of tunnel is circular with radius R = 3.15 m,

the tunnel model cross-section is square with a size of 5.5mx5.5m and a tunnel model with the

cross-section is horseshoe-shaped with the radius of the dome R and the height of the column-

wall h, R=h=2.95 m. In this method, the case study is no-slip at the tunnel lining with soil and

regardless of the effect of gravity and groundwater conditions. These models are divided into

two zones. The stages of this method include:

- Phase 1. Setup the tunnel model, tunnel lining and soil layer surrounding with their

parameters;

- Phase 2. Setup boundary conditions, setting up the pressure on the tunnel lining and

model's boundary, assign to the peak ground acceleration of the earthquakes to the model with

layer soil by link conditions;

- Phase 3. Give the effects of earthquakes on the tunnel models and present the results

obtained of the models.

Note. The layered soil surrounding tunnel is elastic.

3. CALCULATION FOR THE TUNNEL LINING FROM HANOI

METRO SYSTEM

3.1. Characteristics of earthquakes, soil and tunnel in Hanoi metro system

The hydrogeological and geological conditions of Hanoi center these are as follows: the

groundwater level is three meters below the surface; the distance from the ground surface to

the bedrock is 48-50 m and there are usually six layers of soils in Hanoi center.

The characteristics of the tunnel of the Hanoi metro system in Hanoi center [3, 5]: The

tunnels were a depth of between 15 and 20m below the ground surface. The tunnel lining is

made reinforced concrete with characteristics are: El = 35500 MPa; Poisson’s ratio νl = 0.15;

Lining thickness tl = 0.35 m.

The strongest earthquakes that can occur in the Hanoi center that have characteristics [4,

5]: highest magnitude MW=6.5 Richter; distance from the epicenter of the earthquake to Hanoi

is 20 to 50 km and peak ground acceleration amax=0.2g.

Table 1 The properties of the soil in Hanoi Centre [3, 4, 5]

Number

of soil

layers

Elastic

module, E,

MPa

Poisson’s

ratio, µ Thickness of

layer (d), m

Measured SPT

blow count, N

Density of the

soil, ρ, g/cm3

1 9.25 0.41 4.6 2 1.75

2 7.68 0.38 1.1 1 1.76

3 15.3 0.35 11.8 3 1.81

4 35.02 0.33 12.5 7 1.78

5 53.9 0.32 11.0 10 1.83

6 65 0.3 7.0 12 1.86


of the Earthquakes


Figure 2 Data of El Centro earthquake [5]

3.2. Results and Discussions

In case of the circular tunnel [4, 5, 6]:

Figure 3 Result calculation of model tunnel’s by 2D numerical method [4, 6 ]

Figure 4 State stress on the tunnel lining in 2D numerical method [4, 6]

Figure 5 Strain of the tunnel lining in 2D numerical method [4, 6]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000

Acc

eler

atio

n,

g

Time, s



Figure 6 Bending Moment M and Normal force T in the lining tunnel by HRM [4, 6]

Table 2 Summary of analysis results

The internal forces in lining tunnel HRM method 2D numerical method

M (kN.m/m) 106.64 -

T (kN/m) 190.34 -

σ (MPa) 5.771 4.213

Figure 7 State stress on the square tunnel lining and the horseshoe tunnel lining by 2D numerical

method

Figure 8 State stress on the square tunnel lining and the horseshoe tunnel lining by 2D numerical

method

-150

-100

-50

0

50

100

150

0 60 120 180 240 300 360

Ben

din

g M

om

ent

M (

kN

m/m

)

θ (degree)

0

20

40

60

80

100

120

140

160

180

200

0 60 120 180 240 300 360

No

rmal

fo

rce

T (

kN

/m)

θ (degree)


of the Earthquakes


Figure 9 Strain of the square tunnel lining and the horseshoe tunnel lining by 2D numerical method

Table 3 Analysis results of calculation 2D numerical method for the different tunnel models

Tunnel cross-sectional shape

Stress on the

tunnel lining

(MPa)

Difference with the

reference (%)

The circular tunnel (reference case) 5.721 -

The square tunnel 6.912 20.818

The horseshoe tunnel 15.23 171.281

Based to the results in Table 2 and Figures 3, 4, 5, 6. In case of the circular tunnel, under

the impact of the strongest earthquake that can occur in the Hanoi center by use to 2D numerical

method with Abaqus program then a difference between the maximum stress value on the

tunnel lining with the maximum stress value in the tunnel lining of the Hyperstatic Reaction

Method is 5.799%, this difference is not large and can conclude about the accuracy of the

Hyperstatic Reaction Method when this method was used for the circular tunnel. With the

results obtained in Figures 7, 8, 9 and in Table 3, it is possible to give the following conclusions:

The stress values appearing in the tunnel lining under the influence of the strongest earthquakes

that can affect the Hanoi center in the case of the tunnels have different cross-sectional shape

with very large differences. By selecting the circular tunnel that is the reference case, the

difference in stress on the tunnel lining when the tunnel cross-section is square with the

standard case is 20.818% The difference in stress on the tunnel lining when the tunnel has

horseshoe cross-section with the reference case is 171.281%.

4. CONCLUSION

In this paper, tunnel models with different cross-sectional shapes have been constructed

together with the properties of tunnel lining and surrounding environments. Using the 2D

numerical method with Abaqus software and the Hyperstatic Reaction Method, the paper

presents the internal forces that appear on the lining of tunnel models when these tunnels work

under the influence of the strongest magnitude earthquake that can affect the Hanoi center. This

paper studied and compared the internal forces results obtained on the tunnel lining in each

case and given the following conclusions:

1. In all case studies: the circular tunnel, the square tunnel, and the horseshoes tunnel,

tunnels from the metro system of Hanoi operate stably under the influence of the strongest

magnitude earthquake that can affect the Hanoi-the maximum value stress occurring on the

tunnel lining under the impact of the strongest earthquake that can affect Hanoi (maximumσ =

15.281 MPa) is less than the limited stress value of material tunnel’s ( limitσ = 22 Mpa);

2. In three cases of the cross-sectional shape of the tunnel that has been studied, including

circles, squares and horseshoes, under the effect of the same earthquake, the maximum value



stress appeared on the circular tunnel lining is the smallest value stress appeared on the tunnel

lining in three cases ( maximumσ = 5.72 MPa), the stress for the case of a tunnel whose cross-

section is square has a value of maximumσ = 6.912 MPa, and the maximum stress occurs on the

horse-shaped tunnel lining that is the greatest value stress in three cases of maximumσ = 15.281

MPa. From this result, it is possible to select the reasonable shape of the tunnel from the Hanoi

metro system when the tunnel has been affected by the earthquake that is the circular tunnel.

ACKNOWLEDGEMENTS

This work was supported by the Saint Petersburg Mining University. We thank two anonymous

reviewers for their comments that were very valuable for revising the manuscript.

REFERENCES

[1] Do, N. A., Dias, D., Oreste, P. P., & Djeran-Maigre, I (2014a),“2D tunnel numerical

investigation - the influence of the simplified excavation method on tunnel behavior”.

Geotechnical and Geological Engineering, 32(1), pp. 43-58.

[2] Do, N. A., Dias, D., Oreste, P. P., & Djeran-Maigre, I (2014d), “The behaviour of the

segmental tunnel lining studied by the Hyperstatic Reaction Method”, European Journal

of Environmental and Civil Engineering, 18(4), pp. 489-510.

[3] Gospodarikov Alexandr, Thanh Nguyen Chi (2017), “Liquefaction possibility of soil layers

during earthquake in Hanoi”, International Journal of GEOMATE, Vol. 13, Issue 39, pp.

148-155.

[4] Gospodarikov Alexandr, Thanh Nguyen Chi (2018a), “The impact of earthquakes of tunnel

linings: A case study from the Hanoi metro system”, International Journal of GEOMATE, Vol.

14, Issue 41, 2018, pp. 151-158.

[5] Gospodarikov Alexandr, Thanh Nguyen Chi (2018b), “Behaviour of segmental tunnel

linings under the impact of earthquakes: A case study from the tunnel of Hanoi metro

system”, International Journal of GEOMATE, Jan., Vol.15, Issue 48, 2018, pp. 91-98.

[6] Gospodarikov Alexandr, Thanh Nguyen Chi (2018c), “Different behaviour of circular and

rectangular tunnels under the impact of earthquakes: a case study from the tunnel of Hanoi

metro system”, International Journal of GEOMATE, Jan., Vol.15, Issue 51, 2018, pp. 217-

224.

[7] Möller, S (2006), “Tunnel induced settlements and structural forces in linings”. Ph.D.

dissertation, Stuttgart University.

[8] Naggar, H. E., & Hinchberger, S. D (2008), “An analytical solution for jointed tunnel

linings in elastic soil or rock”. Canadian Geotechnical Journal, 45, pp. 1572-1593.

[9] Oreste, P. P (2007), “A numerical approach to the hyperstatic reaction method for the

dimenshioning of tunnel supports”, Tunnelling and Underground space technology, 22, pp.

185-205.

[10] Penzien J., Wu, C (1998), “Stresses in linings of bored tunnels”, Journal of Earthquake

Eng. Structural Dynamics 27, pp. 283–300.

[11] Wood, J.H., (2004), “Earthquake design procedures for rectangular underground

structures”. Project Report to Earthquake Commission, EQC Project No 01/470, Rev B July

2004.№ 01/470. New Zealand.

[12] Wood, J.H (2005), “Earthquake design of rectangular underground structures”. NZSEE

Conference, New Zealand, pp. 39-47.

Documents

THE REASONABLE CROSS-SECTION SHAPE FOR THE TUNNEL … · The Hyperstatic Reaction Method has been given by Duddeck and Erdmann [8], Takano [4], Oreste [8]. Hyperstatic Reaction Method