ABSTRACT: Soft rock has the characteristic of time dependency, that is to say, creep and/or deterioration of strength. Ground-pressure toward tunnel lining due ...
Simulation of Tunnel Deformation by Considering Time Dependency of Rock Strength M. Nakagawa, M. Sato Geoscience Research Laboratory Co., Ltd., Osaka, Japan
Y. Jiang Nagasaki University, Nagasaki, Japan
Keywords: social overhead capital, maintenance, time dependency, strength deterioration
ABSTRACT: Soft rock has the characteristic of time dependency, that is to say, creep and/or deterioration of strength. Ground-pressure toward tunnel lining due to plasticity with deterioration of strength of the surrounding rock masses is expressed using a finite difference method, which is especially proposed by Cundall and is well expressed plastic flow and large-strain deformation. In this paper, the formulation proposed by Sato is adopted as the model of time-dependent deterioration of strength of the surrounding rocks and is applied for the expression of evolution of plastic zone. Furthermore, the difference of effects in the different time of reinforcement is investigated. This study is expected as one of the estimation methods of rational maintenance time.
1 Introduction The deformation and crack propagation can be seen in the lining concrete of road and/or rail way tunnels which are constructed in weathering and alteration rock zone as time passes in use of them. In the worst case, the accident which some lump of concrete fall down, has occurred in the recent years. In general, the main three causes (plastic ground pressure, loosening pressure, and unsymmetrical earth pressure) are listed as the factor. Especially, plastic ground pressure is defined as the pressure caused in the plastic zone which is formed in the ground around tunnel due to tunnel excavation. This plastic zone will spread around tunnel with time progression and as a result, loosen zone around tunnel will push toward the lining concrete. Thus, it is though that the deterioration of rock strength due to spread of plastic zone is time-dependent mechanical behaviour. It is important for tunnel design to estimate precisely the degree of damage, necessity of repair work, the essential maintenance and management, selection of rational reinforced method against damage and its effect. It is efficient that the mechanical investigation which mechanism of damage phenomenon is reflected appropriately is carried out in advance. For the purpose of this, it is necessary to develop the appropriate mechanical models for both rock material and lining concrete material to express deterioration of strength. Additionally it is necessary to apply appropriate numerical method that can be well-expressing the behavior in the mechanical model developed there. Based on the concept that the investigation based on time scale is important for the maintenance of Social Overhead Capital, in this paper, deformation simulation of a tunnel by considering time dependent deterioration of rock strength caused by plastic ground pressure is presented. First, the proposed mechanical model by considering time-dependent deterioration of rock strength and the
outline of the laboratory test to obtain the model parameters are described. Next, deformation simulations of tunnel are carried. The results about convergence, displacement of tunnel surface, evolution of both plastic zone and deterioration of rock strength around tunnel are obtained. Finally, the judgement of the rational maintenance time for tunnel lining is tried based on the results of the numerical analysis. It is expected that the approach presented in this paper could be applied for the judgement of the rational maintenance period and management method.
2 The model of time dependent deterioration of rock strength Time dependency of rock masses is in general treated as viscosity such as time delay of deformation macroscopically. In this case, rock masses are modelled as visco-elasticity in numerical analyses in general. However some interpretation that gradual progress of failure by stress corrosion and/or weathering is more dominant rather than visco-elastic behaviour which is finished initially for a short time is indicated as the mechanism of progress of gradual deterioration of rock strength from the results of tri-axial test and measurement. This is based on the idea that time delay of elasticity occurs under constant strength and large part of deformation observed in the measurements occurs due to gradual deterioration of strength. Large deformation of tunnel structures is caused from gradual deterioration of strength (i.e. stress erosion) of rock mass which is affected by underground water. Mechanical model of rock masses considering time-dependent deterioration of strength is shown by Sato (1983, 1984) based on above idea. This model is shown as below in Equation 1.
Here,
R=
dc = −λ R dt
( R ≤ 1.0)
dc = −λ dt
( R = 1. 0)
1
σ1 − σ 3 2c cos φ + sin φ (σ 1 + σ 3)
stress ratioσ/σt
The model has two strength parameters(cohesion c and internal friction angle φ). However the only deterioration of strength cohesion c is taken into consideration. Moreover, it is assumed that strength deteriorates with constant velocity after strength criteria is satisfied. Here, R indicates the degree of approach to Mohr-Coulomb yield criteria. λ prescribes rate of strength deterioration and is obtained from the experiment. This parameter is dependence of volume of void and/or cracks and degree of erosion in environment. The black dots in Figure 1 indicate the relation between stress ratio and elapsed time until failure conceptually. Here, the stress ratio indicates the ratio of constant loading stress( σ ) against unconfined compressive strength( σ t). λ is obtained by recurred several results of the test(black dots)
0.8 0.5 0.2 time(t)
Figure 1. Concept of stress ratio and elapsed time until failure.
with an exponential function. The rock material has characteristic that the rate of strength deterioration increases as stresses approach to strength criteria. Additionally, general tendency in the test that elapsed time until failure increases exponentially is shown as loading stress is smaller than the compressive strength in Figure 1.
3 Simulation of tunnel deformation Several simulations of tunnel deformation are shown here. The mechanism of time dependent deformation is caused by incremental plastic ground pressure due to deterioration of rock strength. Some considerations are also shown from the simulation results.
3.1 Simulation model Simulation of tunnel deformation is carried out by modelling a road tunnel which is constructed in soft rock mass. The tunnel has 100 m of overburden. Simulation model is shown in Figure 2. Stress on the tunnel surface release until 30% Shotcrete Steel arch support & Shotcrete
Invert
8.7m
Lining Stress release until 95% 95%
5.6m
Lining & Invert
Stress release until 100% 100%
Figure 2. Simulation model of a standard road tunnel and simulation stage. First, shotcrete and steel arch support are constructed after 30 % stress on the tunnel surface released without any supports so as to consider three dimensional effects around tunnel face. Next, lining and invert concrete are constructed after 95 % stress released for the same reason as the previous stage. Finally stress release is carried out until 100 %. Material properties of rock mass are shown in Table 1. Those are assumed as soft rock of D1 class of Japan Highway Public Corporation and modelled as elasto-plastic material based on Mohr-Coulomb yield criteria. It is assumed that deterioration of rock strength begins from the time of completion of tunnel construction and it is according to Sato’s model shown in Equation 1. The equation 1 is discretized by forward divided difference so as to express in real time as shown in Equation 2. Table 1. Material properties of soft rock mass. Density
Cohesion
Internal friction angle
υ
ρ (kg/m3)
c (MPa)
φ (deg)
0.35
2200.0
0.5
30.0
Modulus of elasticity
Poisson's rate
E (MPa) 100.0
2
c
t +1
= − λR∆t + c
dc = dt
t +1
c
−c ∆t
t
t
Table 2. Material properties of support and lining.
Tunnel suport
Shotcrete Steel arch support Lining & invert
Modulus of elasticity E （MPa） 4000.0
Poisson's rate υ
Second moment of inetia I 4 (m )
0.2
5
2.3×10 4 2.35×10
Cross-section area A 2 (m ) -4
63.53×10
4720.8×10
-8
0.2
Table 3. Analysis Cases. Condition Rock mass
Strength Contents deterioration
Lining Non Elasticity Strength deterioration
Reinforcement method
Elapsed years
Case No.
after 10 years after 15 years
1 2 3 4 5
No repair Inner lining method
It is assumed that λ(input parameter of rate of deterioration strength) is 0.033(MPa/day) . Material properties of support and lining are decided referenced from the guide-line of Japan Highway Public Corporation as shown in Table 2. FLAC code developed by Cundall(1988) is applied in this simulation because the simulation of process of large deformation and collapse behaviour due to spread of plastic zone can be carried out stably. Time integration is carried out by using the equation 2 while time step ∆ t is adjusted so that equilibrium is kept to be satisfied during simulation. It is thought that the deformation of the surrounding rock masses due to strength deterioration can be treated as quasi-static phenomena and so the elasto-plastic numerical analysis with Mohr-Coulomb yield criteria based on time dependency is realized. Analysis cases are shown in Table 3. The necessity which deterioration of strength is introduced into the lining is investigated by comparing Case No.2 and Case No.3. Discrete crack model and Smeared crack model are proposed in the crack propagation of concrete lining. The former is that crack is treated microscopically and the latter is that crack is treated macroscopically. In this paper, strength deterioration of concrete lining is expressed by reducing the elastic modulus based on the theory of Smeared crack model. Based on above, the difference of increment of convergence displacement in time due to the difference of time when reinforcement is applied (10 years after and 15 years after construction). Convergence displacements are measured just after tunnel construction in this study.
3.2 Results of deformation analysis without reinforcement The progress of strength deterioration together with spread of plastic zone in the surrounding rock masses is shown in Figure 3. It is shown that the deterioration of strength progresses toward the direction of tunnel radius and plastic zone spreads over as the same way. Especially, the state which the deterioration degree of strength is higher at near tunnel surface is coincident with the phenomenon in enlarge construction of tunnel. The relations between convergence displacements and elapsed time for both without consideration of strength deterioration in lining (Case No.2) and with consideration of strength deterioration in lining (Case No.3) are shown in Figure 4. It can be seen that 25 cm of increment of convergence displacement is obtained after 20 years without any supports. On the other hand, any convergence displacement can not be seen in the case without strength reduction (Case No.2). The strength deterioration of linings is not considered in many analyses in practice. The results which can be seen little convergence displacement are obtained for this condition. Furthermore, about 12 cm of increment of convergence displacement is obtained in the case of strength deterioration in lining. Though some extent of restraint of convergence displacement can be seen, the increment of convergence displacement is obtained in progress of time. It can be understood that introduction of time-dependent deterioration of strength is necessary for not only surrounding rock but also tunnel lining since many cases of convergence are reported from many construction fields. Finally, change of increment of convergence displacement with time in Case No.1 can be seen in linear and this tendency is not the same as exponential change of convergence displacement in field measurements. It is thought that the rate of convergence displacement based on the equation 1 after yield is linear and plastic deformation is added in this process. It is necceary that the equation 1 has to be improved so as to realize more realistic modelling about strength deterioration of rock masses.
3.3 Results of deformation analysis with reinforcement The differences of convergence displacement due to different maintenance time are compared so as to clarify the effect of different maintenance time. The deformation analyses are carried out for both 10 years after and 15 years after construction based on the case No.3, which is modelled with strength deterioration for both rock mass and lining. Those results are shown in Figure 5. Inner 4 lining method as shown in Figure 6 (elastic modulus 3.1×10 MPa, thickness 15 cm) is adopted as reinforcement method since this method is adopted in general in Japan. This is modelled as elastic material and strength deterioration is not considered. From Figure 5, it can be seen that the convergence displacement is smaller in early maintenance, while the convergence displacement is larger in later maintenance. Though only Inner lining method is adopted in this paper, it is necessary to apply other maintenance methods. Moreover, the comparison of convergence displacement with different kind of reinforcement method is necessary by doing deformation analyses.
3.4 Total cost and incremental convergence displacement Based on the above results of deformation analysis, the time of reinforcement when the total cost 2 is the lowest is investigated. A unit cost of reinforcement of Inner lining method is 66000 yen/m . So the change of maintenance cost A (yen) per year is expressed in the next equation 3. In this 2 equation, B indicates the elapsed years and it is assumed that reinforcement area is 500 m . A = 972853 × B + 3814410
3
It is estimated that the cost is 1350 million yen in case of Inner lining method within 10 years, and 2330 million yen in case of Inner lining method within 15 years. In this reinforcement method, it is necessary to add 2945 million yen to estimate the total const as maintenance cost. This situation is shown in Figure 7. Based on this figure, the total cost and incremental convergence displacement
Strength c
Just after excavation
Strength c
After 10 years
Plastic zone
Strength c
After 15 years
Plastic zone
Strength c
After 20 years
Plastic zone
0.0 - 0.0 (MPa) 0.0 - 0.1 (MPa) 0.1 - 0.2 (MPa) 0.2 - 0.3 (MPa) 0.3 - 0.4 (MPa) 0.4 - 0.5 (MPa)
0.0 - 0.0 (MPa) 0.0 - 0.1 (MPa) 0.1 - 0.2 (MPa) 0.2 - 0.3 (MPa) 0.3 - 0.4 (MPa) 0.4 - 0.5 (MPa)
0.0 - 0.0 (MPa) 0.0 - 0.1 (MPa) 0.1 - 0.2 (MPa) 0.2 - 0.3 (MPa) 0.3 - 0.4 (MPa) 0.4 - 0.5 (MPa)
Plastic zone
0.0 - 0.0 (MPa) 0.0 - 0.1 (MPa) 0.1 - 0.2 (MPa) 0.2 - 0.3 (MPa) 0.3 - 0.4 (MPa) 0.4 - 0.5 (MPa)
Figure 3. Deterioration of rock strength and progressing plastic zone (without tunnel support).
Convergence displacement(cm)
60
no lining（Case No.１）
50
Elastic lining（Case No.２）
40
Yield deterioration lining （ ）
30 20 10 0 0
5
10 Elapsed years
15
20
Figure 4. Prediction of convergence displacement (according to the lining models).
Convergence displacement(cm)
25
20
15
10
No repair(Case No.３) after 10 years(Case No.4)
5
after 15 years(Case No.5) 0 0
5
10 Elapsed years
15
20
Figure 5. Prediction of convergence displacement (according to the reinforcement time).
Figure 6. Inner lining method.
Total repair cost（milion yen）
50 45
Repair(after 10 years)
40 Repair(after 15 years)
35 30 25 20 15 10 5 0 0
5
10 Elapsed years
15
20
Figure 7. Elapsed time vs. total maintenance cost.
Table 4. Total cost and increment of inner wall displacement after 20 years. Elapsed years
Total cost （milion yen）
Total convergence after construction (cm)
10
282
7.9
15
128
10.6
for 20 years utilization are shown in Table 4. From this table, it can be understood that the reinforcement after 10 years is more effective from the point of view of the restraint of convergence displacement. On the other hand, the total cost can be lower in the case of the reinforcement after 15 years. However incremental convergence displacement is 1.3 times than the latter. Social cost and reliability for the safety are involved in the actual maintenance cost other than initial cost and/or reinforcement cost. The total cost must be decided based on those matters.
4 Conclusions In this paper, the possibility to obtain any supportable information for judgment of maintenance is presented. More actual estimation system will be constructed by obtaining more actual measured data. Since the equation 3 is created from the limited data, accuracy of the equation will be improved by increasing data.
5 References Cundall, P.A. and Board M. 1988. A microcomputer program for modelling large-strain plasticity problems, Prepared for the 6th International Congress on Numerical Methods in Geomechanics, Innsbruck, Austria, 2101-2108. Japan Society of Civil Engineers. 2003. Tunnel Deformation Mechanism. Jiang, Y. 2003. Deformation Analysis and Database Development for Road Tunnel Maintenance, Reports of the Faculty of Engineering, Nagasaki University, 33-61, 115-122. Nakagawa, M. Jiang, Y. ,Esaki, T. 1997. Application of Large Strain Analysis for Estimation of Behaviour and Stability of Rock Mass. JSCE Journal. 575(Ⅲ-40): 93-104. Sato M., Takeda N., Kamemura K. 1983. Numerical analysis of rock behaviour considering time-dependent strength, Japanese Geotechnical Society, 817-820. Sato M., Kamemura K. 1984. A Study on Time Dependency of Rock Strength, Japanese Geotechnical Society, 783-784.
