Three Dimensional Numerical Simulation of Immersed Tunnel

Seismic Response Based on Elastic-plastic FEM1

Jin Xianlonga, Guo Yizhib and Ding Junhongc

High Performance Computing Center, Shanghai Jiao Tong University, Shanghai, China 200030, ajxlong@sjtu.edu.cn,

byizhiguo@sjtu.edu.cn,

cJunhongDing@hotmail.com

Keywords: elastic-plastic; numerical simulation; immersed tunnel; seismic response;

Abstract. Using the high performance computer SGI Onyx3800, this paper introduced finite element

method in simulating the seismic response of this immersed tunnel in Shanghai. First, consisted of the

immersed tunnel, soil and river, a large three dimension solid model was constructed by CAD

software according to actual size and geologic circumstance. Second, the finite element model was

finished in the CAE software, and the number of nodes and element in final finite element model

exceeded 1.2 million and 1 million respectively. Considering the complicity and the diversity of

different parts in the model, several material models with distinctive constitution relationships were

brought in the research. Contact between two parts also was considered in the research. Such contact

regions exist between soil and tunnels segment, GINA rubber gasket and tunnels segment, and

between vibration isolation rubber cushions and shear key. All these contact style were treated as

surface-to-surface style, and penalty function method was used to solve such problems. The ultimate

calculation adopted explicit dynamic algorithm and parallel domain decomposition method. The

result reveals the weak joints in this immersed tunnel under Tangshan seismic waves. It could also

provide data and references for the aseismatic design of immersed tunnel and its flexible joints.

Introduction

Ever-increasing populations of Shanghai, density of transportation and need for storage capacity have

led, inevitably, to an increased use of underground facilities. The second biggest immersed tunnel in

the world has been opened to traffic in Shanghai. Worldwide experience indicates that underground

facilities are economically effective in the framework of national and municipal economics [1].

However, because the East China including Shanghai is adjacent to the Circum- Pacific region and the

local sedimentary soil affects greatly on the seismic response of soil layers, it is necessary to study the

earthquake-resistant ability of the immersed tunnel under the Huangpu River.

Two methods are often used to analyze the dynamic response of the immersed tunnel. One method

is multi-mass-spring system method, which divides the soil into several strips and each strip serves as

a mass-spring system. The tunnel is treated as beams connected by linear springs in longitudinal and

lateral directions. Many immersed tunnel in Japan and the Pearl River tunnel in Guangzhou of China

were analyzed all using this method [2,3]. Another method is the finite element method. Limited to

the need of large memory capacity and considerable computing time, this method has not been widely

used until recently. Based on high performance computer, this paper introduced finite element method

in simulating the seismic response of this immersed tunnel in Shanghai.

Finite Element Model

Consisted of the immersed tunnel, soil and river, a large three dimension solid model is constructed

by CAD software according to actual size and geologic circumstance. To decrease the effect of

boundary conditions farthest, the whole size of this soil is determined to be 1670m×1110m×280m.

1 This project is supported by National Natural Science Foundation of China (No. 62073048).

Key Engineering Materials Vols. 274-276 (2004) pp 661-666Online available since 2004/Oct/15 at www.scientific.net© (2004) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/KEM.274-276.661

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Then, the finite element model is finished in the CAE software. For the immersed tunnel segment,

GINA rubber and vibration isolation rubber, the 8-nodes solid element is used, while 4-nodes solid

element is adopted for the soil (Fig.1, Fig.2). At last, the number of nodes and element in whole finite

element model exceeds 1.2 million and 1 million respectively. Due to the fine finite element grid, the

results precision is guaranteed.

Fig.1 Finite element model of the soil Fig.2 Finite element model of immersed tunnel

In Fig. 2, Ei expressed tunnel segment i, and Ji expressed flexible joint i. When establishing the

finite element model, contact and material nonlinearity are considered in this paper.

Contact. Contact is dynamic ineraction between different parts. Such dynamic interaction exists

between the soil and immersed tunnel, GINA rubber gasket and tunnel segment, and between

vibration isolation rubber cushions and shear key (Fig.3). From a finite element point of view, the

contact entities, i.e. nodes and elements in contact, must be accurately detected. LS-DYNA has a very

efficient contact search algorithm. For simplicity, the bodies in contact are decomposed in two parts:

the slave and the master, and it is assumed that slaves entities must not go through master surfaces, i.e.

slaves nodes must not penetrate master segments, where a segment is a shell surface or a side-surface

of a volume element. The slave body usually corresponds to the finest mesh body or the softest

material body. Here, the tunnel segment is slave body, and the soil is master body. All these contact

style were treated as automic surface-to-surface style. In order to compute the contact forces, i.e. the

forces resulting from impenetrability assumption, a mathematical method is used: the penalty

function method. It consists in introducing fictitious springs between contact entities. The main point

of this method is the choice of the spring stiffness, which drives the strength of the contact

phenomenon. LS-DYNA uses special formulas to establish contact stiffness that is a function of the

geometric and material features of contact entities.

Fig.3 Contact entities in immersed tunnel joints Fig.4 Description of yield surface

Material Nonlinearity. Considering the complicity and the diversity of different parts in the

model, several material models with distinctive constitution relationships are brought in the research.

662 Advances in Engineering Plasticity and Its Applications

The Drucker-Prager material model is used for the soil (Fig.4), and its yield criteria expression is:

( ) 021'

21 =−+= kJJF α . (1)

where, 1J is the first invariant of stress tensor, and '

2J is the second invariant of stress deviator. The

expressions for α and k are:

)sin3(3

sin2

φ

φα

±= . (2)

)sin3(3

cos6

φ

φ

±=

ck . (3)

The flexible joints comprised Mooney-Rivlin rubber model for GINA rubber gasket and vibration

isolation rubber cushions [4]. Its strain energy density function is defined as:

( ) ( )33 201110 −+−= ICICW . (4)

where, C10 and C01 are material constants, and can be gained from experiments. I1 and I2 are invariants

of right Cauchy-Green Tensor C.

Elastic-Plastic model for the concrete tunnel, the horizontal and vertical shear key, and the spring

model for pulling rope. The main parameters for those material models such as Mooney-Rivlin rubber

model were directly derived from experiment results. The following tables list all material parameters

for immersed tunnel.

Material type Mass density (g/cm3 ) Poisson’s ratio Elastic modulus (GPa)

Concrete tunnels segment 2.50 0.167 31.5

GINA rubber gasket 1.14 0.499 --------

vibration isolation rubber 1.14 0.499 --------

Steel shear key 7.85 0.26 201

Tab.1 The material parameters of the concrete, GINA rubber gasket and vibration isolation rubber

Layer number Shear modulus

(MPa)

Poisson’s

modulus

Mass density

(g/cm3)

Cohension

(c/KPa)

Angle of internal

friction

1 10.00 0.30 1.70 22.3 22.02

2 14.35 0.40 1.92 17 21

3 14.35 0.35 1.86 4 25.4

4 30.28 0.45 1.74 13 14.2

5 61.45 0.35 1.86 4 20.7

6 38.59 0.45 1.72 10.5 7.1

7 58.74 0.40 1.77 11.0 8.8

8 77.82 0.35 1.85 4 23.3

9 80.06 0.47 1.99 27 14.1

10 83.13 0.47 2.00 25 15.5

11 115.00 0.36 1.92 8.5 21.4

12 92.75 0.47 1.79 14 14

Tab.2 The material parameters of the soil (partially)

Algorithms

Introducing three-dimensional and nonlinear modeling method, the number of nodes and elements in

whole finite element model exceeds 1.2 million and 1 million respectively. Thus, the immersed tunnel

seismic response analysis can be considered as the transient response computing of a grand scale and

nonlinear system. However, there are some difficulties: it needs great capacity data exchange and

storage for the transient response computing of a grand scale and nonlinear system. Consequently the

serial algorithm and software used in earthquake safety evaluation cannot complete this task. In

Key Engineering Materials Vols. 274-276 663

addition, contact detection and computation require rather complex procedures and make up for a lot

of CPU time.

To solve the difficulties above, explicit algorithm, parallel domain decomposition method and

high performance computer are used in this analysis.

Explicit Algorithm. Under seismic loads, the dynamic equation of three-dimensional finite

element system can be defined as:

)()()()( tPtKxtxCtxM =++ &&& . (5)

where, xxx ,, &&& is nodes acceleration, velocity and displacement vector respectively. ),( txMM = is

mass matrix, ),( txKK = is stiffness matrix, and ),,( txxCC &= is damping matrix. P is the exterior

loads of the whole system.

The central difference method are adopted to solve Eq.5:

2)()()()( 21212121 nnnnn txtttxtx &&&&−+−+ ∆+∆+= . (6)

)()()( 21211 +++ ∆+= nnnn txttxtx & . (7)

in Eq.6 and Eq.7, )( 21+ntx& is nodes velocity at 21+nt , and )( 1+ntx is nodes coordinate at 1+nt .

Time Step Control: during the solution we loop through the elements to update the stresses and the

right hand side force vector. We also determine a new time step size by taking the minimum value

over all elements.

{ }N

nttttt ∆⋅⋅⋅∆∆∆⋅=∆ + ,,,,min 321

1 α . (8)

where N is the number of elements. For stability reasons the scale factor α is typically set to a value

of 0.9 (default) or some smaller value.

When establishing the finite element model, solid elements are used in this paper. A critical time

step size, et∆ , is computed from:

( )( )( )2122CPPLt ee ++=∆ . (9)

where P is a function of the bulk viscosity coefficients, Le is a characteristic length.

Hourglass Control: explicit dynamic algorithm usually adopts one-point integration. The biggest

disadvantage to one-point integration is the need to control the zero energy modes which arise, called

hourglassing modes. Undesirable hourglass modes tend to have periods that are typically much

shorter than the periods of the structural response, and they are often observed to be oscillatory.

In order to control the purely numerical deformations, hourglass resisting forces are added for

cases when they are excited. Then, in the mechanical energy balance, it appears an hourglass energy

that is linked to the hourglass resisting forces against formation of hourglass modes.

Parallel Domain Decomposition Method. In this analysis, the parallel algorithm uses the domain

decomposition technique; for example, the mesh of a model is partitioned into subdomains and, in

most cases, each domain is assigned to one processor. Processors exchange boundary data by using

explicit communication tools, specifically, MPI (message passing interface).

Recursive Coordinate Bisection (RCB) Method: RCB is attractive as a dynamic load-balancing

algorithm because it implicitly produces incremental partitions. In RCB, the computational domain is

first divided into two regions by a cutting plane orthogonal to one of the coordinate axes so that half

the workload is in each of the sub-regions. The splitting direction is determined by computing in

which coordinate direction the set of objects is most elongated, based upon the geometric locations of

the objects. The sub-regions are then further divided by recursive application of the same splitting

algorithm until the number of sub-regions equals the number of processors. Although this algorithm

was first devised to cut into a number of sets which is a power of two, the set sizes in a particular cut

needn't be equal. By adjusting the partition sizes appropriately, any number of equally-sized sets can

be created. If the parallel machine has processors with different speeds, sets with nonuniform sizes

can also be easily generated.

664 Advances in Engineering Plasticity and Its Applications

Fig.5 Three-dimensional model partitioning description of the immersed tunnel

Because too much contact exits in 8 flexible joints in the immersed tunnel, every sub-region

includes a flexible joint in order to achieve load balance in this partition (Fig.5).

Results

A high performance computer SGI Onyx3800 (Tab.3) completed the final calculation.

Number of CPUs Clock frequency (MHz) Memory (GB) Hard disk (TB) Perk performance (GFLOPS)

64 500 32 4.3 64

Tab.3 The technical parameters of SGI Onyx3800

Fig.6 Tangshan seismic acceleration Fig.7 The maximal relative displacement of all joints

Tangshan seismic acceleration recorded in Beijing Restaurant in 1976 (Fig.6) is used in this

analysis, and the sampling interval is 0.01 second. The maximal acceleration is 65.9 cm/s2. According

to the field safety requirement of the immersed tunnel in Shanghai, the amplitude modulation is

applied to Tangshan seismic acceleration and its maximal acceleration changes to be 100 cm/s2.

Because the vibration for underground structures is affected greatly by the excited direction of seismic

wave, three cases are considered in results analysis: seismic wave excites in X-axis direction, Y-axis

direction and Z-axis direction (in Fig.1).

The results reveal the weak joints in this immersed tunnel form Fig.7, which are J2 and J3. The

reason is that the second tunnel segment goes beyond the riverbed and it has less covering soil. Thus,

the second tunnel segment has less restriction under seismic acceleration, and the maximal relative

displacement comes into being at J2 and J3.

Key Engineering Materials Vols. 274-276 665

(a) Interaction between tunnels and the soil (b) Displacement between two tunnels segments

Fig.8 The strain nephogram of immersed tunnel under Tangshan seismic wave

Fig.8 is about the displacement nephogram of this immersed tunnel when Tangshan seismic wave

excites in X-axis direction.

Conclusion

It is insured that the harmonious units are used in this analysis when defining material parameters,

because wrong units would affect the response and contact stiffness of the materials. The material

data in finite element model should be precise, and the reason is that the precision of most nonlinear

dynamic problems depends on the quality of material data. The initial contact between two contact

entities is not permitted, and there is no overlap in contact areas. The repetitive contact should not be

defined between two same parts.

According to the results, the deformation style and the deformation zones show the earthquake-

resistant ability of this immersed tunnel and the joints that need more attention in the future. It could

also provide data and references for the aseismatic design of immersed tunnel.

References

[1] F. Kirzhner and G. Rosenhouse: Numerical Analysis of Tunnel Dynamic Response to Earth

Motions. Tunnelling and Underground Space Technology, Vol. 15, No. 3 (2000), p.249

[2] Kiyomiya O.: Earthquake-resistant Design Features of Immersed Tunnels in Japan. Tunnelling

and Underground Space Technology, Vol. 20, No. 4 (1995), p.463

[3] Yan Songhong, Gao Bo and Pan Changshi: Dynamic Property Analysis on Joint for Submerged

Tunnel Under Earthquake. Chinese Journal of Rock Mechanics and Engineering, Vol. 22, No. 2

(2003), p.286

[4] Wei Yintao, Yang Tingqing and Du Xingwen: On the Large Deformation on Rubber-like

Materials: Constitutive Laws and Finite Element Method. ACTA MECHANICA SOLIDA SINICA,

Vol. 20, No. 4 (1999), p.281

666 Advances in Engineering Plasticity and Its Applications

Advances in Engineering Plasticity and Its Applications 10.4028/www.scientific.net/KEM.274-276 Three Dimensional Numerical Simulation of Immersed Tunnel Seismic Response Based on Elastic-

Plastic FEM 10.4028/www.scientific.net/KEM.274-276.661

DOI References

[2] Kiyomiya O.: Earthquake-resistant Design Features of Immersed Tunnels in Japan. Tunnelling nd

Underground Space Technology, Vol. 20, No. 4 (1995), p.463

doi:10.1016/0886-7798(95)00033-U