Before the seismic analysis, initial conditions like the geostress field were obtained by using the dynamic relaxation. When the stress and deformation of the soil-tunnel system became stable, the equilibrium configurations were obtained and the initial calculations were considered to complete. Then, the seismic analysis was performed. The simulation results and the parallel performance are discussed in the following subsections. Due to space limitations, without specific explanation only the west line tunnel is discussed here, while the east line can be analyzed in the same way.
4.1. Results Comparison and Method Validation
A series of studies on the earthquake-induced dynamic responses of the horizontal-cylindrical liquid vessels have been carried out by Karamanos [
15,
16]. Numerical solutions about the sloshing frequencies, the impulsive masses and the convective masses of the horizontal-cylindrical liquid storage tanks with respect to different liquid depths under longitudinal and transverse seismic excitation are obtained. Besides, an “equivalent rectangular container” method is proposed, which means that the externally induced dynamic responses of a horizontal-cylindrical container with arbitrary liquid depth can be computed quite accurately by replacing the cylindrical container with an equivalent rectangular container which has the same free-surface dimensions and owns the same liquid volume [
15]. The effectiveness and correctness of this equivalence have been numerically [
38] and experimentally [
39] demonstrated. In this study this equivalence technique is adopted to demonstrate the practicability of the proposed ALE based FSI approach for the interaction between tunnel linings and inner water. Assuming that the inner diameter of the tunnel is
r and the free-surface height of the liquid is
h, based on the equivalence theorem, the height of the liquid (
heq) in the equivalent rectangular container can be calculated using Equation (18).
Figure 11 illustrates the layout of the equivalent relationship.
Figure 11.
Schematic layout of the equivalent relationship. r, h and heq mean the inner diameter of the tunnel, the free-surface height and the equivalent height, respectively.
Figure 11.
Schematic layout of the equivalent relationship. r, h and heq mean the inner diameter of the tunnel, the free-surface height and the equivalent height, respectively.
Once the cylindrical container is treated as its equivalent rectangular container, the equivalent masses can be calculated using the analytical solution proposed by Housner [
12]. For liquid storage containers with rectangular cross section, the hydrodynamic pressure (
) along the height direction can be expressed as Equation (19), where
can be expressed as Equation (20) and
means the vertical coordinate of the location where the pressure is calculated. The equivalent mass attached to each node can be obtained by using the steps as follows. Firstly, through the mapping of equivalent rectangular surfaces and the original cylinder surface, the transformation of the pressure on the equivalent rectangular face to that on the original cylindrical face is completed. Then, the total equivalent mass on an element can be calculated by dividing the product which can be obtained through multiplying the area of the element’s face bearing pressure by the pressure value, by the current acceleration
. At last, the obtained total equivalent mass is distributed to the four nodes owned by the element using Equation (21), where
is the area of the face with pressure owned by the element;
,
means the equivalent mass at each node. Once the procedure is programmed, the whole finite element model can be processed automatically.
where
Without loss of generality, the free-surface height of the liquid is assumed to be 1.5
r, where
r is the diameter of the tunnel lining which is 2.75 m for island and land part and 2.92 m for cross river part. The equivalent masses are calculated and applied by using the above-mentioned procedure.
Figure 12 illustrates the comparison of the lateral displacements under seismic excitation obtained by using the added mass method and the proposed ALE based fluid–structure interaction method at sections IS6 and RS6 (see
Figure 7). It is seen that the variation law of the displacement shows good agreement and the displacement values are also very close to each other. While, at some points the displacement values from these two methods cannot fit perfectly and discrepancy exists. This is because the proposed method considers the sloshing of the fluid which is neglected by the added mass method. Besides, during the derivation of the added mass method the structure is assumed to be the rigid body without flexibility [
40]; thus, it is considered that the results obtained from the proposed ALE based fluid–structure interaction method are closer to realistic conditions. Other sections show similar results. Thus, the correctness and reliability of the proposed ALE based fluid–structure interaction method for dealing with the coupling between the tunnel lining and inner water are verified.
Figure 12.
Contrast of the displacement responses using the proposed method and the added mass method at sections: (a) IS6; (b) RS6. IS6 and RS6 mean the chosen section at the island part and the cross-river part, respectively.
Figure 12.
Contrast of the displacement responses using the proposed method and the added mass method at sections: (a) IS6; (b) RS6. IS6 and RS6 mean the chosen section at the island part and the cross-river part, respectively.
4.2. Dynamic Response Analysis
The seismic responses of a shield tunnel can be investigated in terms of three primary aspects, namely, extension/compression, longitudinal bending and ovaling [
9]. In this study, the extension/compression can be recognized through the axial forces and the bending behavior can be evaluated through the bending moments; the ovaling deformation can be evaluated by measuring the diameter variation of the tunnel. In total, 24 typical cross-sections including nine in the island part, 10 in the cross-river part and five in the land part (as shown in
Figure 7) are chosen as the concerned locations.
Figure 13 depicts the evolutions of dynamic internal forces and
Figure 14 illustrates the evolutions of dynamic internal moments. The tunnel suffers alternately positive and negative dynamic forces and moments. Time history of internal forces of other sections show comparable results. To get a total view of internal forces,
Figure 15 illustrates the variation of maximum axial forces along the tunnel and
Figure 16 depicts that of the maximum bending moments. The obtained results show that both the bending moments and the axial forces change along the tunnel. Different internal forces and moments have been observed among the island part, cross-river part and land part. The differences may be attributed to the non-uniform seismic input model, the winding and different spatial positions of the tunnel, the dissimilar distributions of soil strata and the diverse buried depths of the tunnel. The comparison illustrates the distributions of both the internal moments and the internal forces of the tunnel during and after the earthquake, which can be used to aid the design of connections of different rings, connections of different segments in a ring and water-proof measures.
Figure 13.
Dynamic integral axial forces at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part.
Figure 13.
Dynamic integral axial forces at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part.
Figure 14.
Dynamic integral transverse moments at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part.
Figure 14.
Dynamic integral transverse moments at sections RS5 and RS9. RS5 and RS9 mean the chosen sections at the cross-river part.
Figure 15.
Maximum integral axial forces along the tunnel.
Figure 15.
Maximum integral axial forces along the tunnel.
Figure 16.
Maximum integral transverse moments along the tunnel.
Figure 16.
Maximum integral transverse moments along the tunnel.
To evaluate the strength of the tunnel, the hoop stress is extracted and analyzed. As shown in
Figure 17, the local coordinate system is needed in each typical cross-section to obtain the hoop stress.
Figure 18 depicts the maximum compressive and tensile dynamic hoop stress distributions of cross-sections RS1 and RS5. It is shown that the distributions present similar patterns in different sections. The maximum stresses lie in the two sides of the ring, and the minimum stresses lie in the arc bottom and vault. Other sections show similar change laws.
Table 3 lists the maximum hoop stress for the chosen sections. The hoop stress also changes along the tunnel, and the larger stress appears at sections IS1, IS5, IS6 and IS9 in the island part, sections RS1 and RS10 in the cross-river part, and section LS1 in the land part. The maximum compressive stress and the maximum tensile stress are 8.36 MPa and 5.32 MPa, respectively. The larger stresses may be due to the ovaling deformation of the tunnel and the sharp stiffness changes between the work shafts and the tunnel rings. Since the maximum compressive and tensile hoop stresses are smaller than the corresponding concrete strength used, the tunnel meets the strength design requirements under the specific seismic excitation.
Figure 17.
Local coordinate system: (a) local cylindrical coordinate; (b) hoop and axial stress.
Figure 17.
Local coordinate system: (a) local cylindrical coordinate; (b) hoop and axial stress.
Figure 18.
Distributions of max dynamic hoop stresses around the tunnel lining during earthquake: (a) tensile; (b) compressive.
Figure 18.
Distributions of max dynamic hoop stresses around the tunnel lining during earthquake: (a) tensile; (b) compressive.
Table 3.
Maximum hoop stress of all control sections (MPa).
Table 3.
Maximum hoop stress of all control sections (MPa).
Control Sections | IS1 | IS2 | IS3 | IS4 | IS5 | IS6 | IS7 | IS8 | IS9 | RS1 | RS2 | RS3 |
---|
Maximum compressive stress | 4.00 | 1.99 | 1.89 | 1.83 | 3.91 | 5.69 | 2.07 | 1.88 | 3.80 | 4.76 | 4.20 | 2.95 |
Maximum tensile stress | 3.14 | 2.38 | 2.24 | 1.85 | 3.52 | 3.57 | 1.99 | 1.87 | 1.61 | 5.32 | 3.95 | 3.15 |
Control sections | RS4 | RS5 | RS6 | RS7 | RS8 | RS9 | RS10 | LS1 | LS2 | LS3 | LS4 | LS5 |
Maximum compressive stress | 4.05 | 4.55 | 4.58 | 4.65 | 5.39 | 5.27 | 8.36 | 6.17 | 5.52 | 4.09 | 3.82 | 1.71 |
Maximum tensile stress | 3.29 | 3.76 | 3.71 | 3.07 | 2.04 | 2.92 | 5.20 | 4.22 | 3.47 | 1.77 | 3.09 | 2.11 |
The ovaling describes the cross-sectional distortion of the tunnel lining and can be characterized by the change of the diameter. Thus, the ovaling here is defined by both the horizontal and the vertical diameter variations. Assuming that
represents the horizontal and vertical variation rate of the tunnel lining, it can be expressed as
, where
and
are the diameter of the tunnel before and after the deformation, respectively.
Figure 19 represents the diameter variation of the tunnel lining at section RS6. Unlike the uniform excitation, pure shear deformation is not observed. This may be due to the non-uniform seismic input and the sophisticated SSI.
Figure 20 shows the maximum variation of diameter for all cross-sections along the tunnel. The cross sections IS5, RS6 and LS5 bear larger deformations, which is partially due to the deeper buried depth. It is known that the ovaling deformation will cause additional dynamic stress concentrations in the tunnel linings and may cause damage to the tunnel during earthquakes. However, for this tunnel, from the aforementioned discussion, it can be stated that the hoop stress is below the strength limitation. Furthermore, from the discussion in this section it can be noted that the maximum ovaling deformation is less than 0.3%, which also satisfies the design requirements [
41].
Figure 19.
History over time of diameter change rate at section RS6. RS6 depicts the chosen control sections in the cross-river part.
Figure 19.
History over time of diameter change rate at section RS6. RS6 depicts the chosen control sections in the cross-river part.
Figure 20.
Normalized maximum values of ovaling deformation at different cross sections (horizontal (+), vertical (−)).
Figure 20.
Normalized maximum values of ovaling deformation at different cross sections (horizontal (+), vertical (−)).
4.3. Domain Decomposition Comparison and Parallel Performance Evaluation
All simulations were carried out using the proposed MRCB (modified Recursive Coordinate Bisection) based parallel computing method, which was embedded in the LS-DYNA optimized to run on the Dawning 5000 A at the Shanghai Supercomputer Center. The evaluation of the developed parallel method applied in the seismic analysis of the specific water conveyance tunnel will be presented in this section.
Figure 21.
Description of domain decomposition topology for 16 subdomains using: (a) recursive coordinate bisection (RCB) method; (b) modified recursive coordinate bisection (MRCB) method.
Figure 21.
Description of domain decomposition topology for 16 subdomains using: (a) recursive coordinate bisection (RCB) method; (b) modified recursive coordinate bisection (MRCB) method.
To evaluate the proposed MRCB domain decomposition method (DDM), calculations were carried out with 1, 2, 4, 8, 16 and 32 cores using RCB and MRCB, separately. The topologies of domain decomposition with 16 subdomains for the dynamic model using these two DDMs are shown in
Figure 21. From this figure, one can easily find that the topologies from these two DDMs are different, more specifically, by using RCB the model is split in both
x and
y direction along which the model is longest, while by using MRCB the model is divided only in the,
x direction along which the coupling (contact and FSI) load is distributed. Thus the MRCB may lead to a more balanced distribution of coupling loads from a qualitative perspective.
For further quantitative analyses of the differences between these two domain decomposition methods,
Table 4 lists the load details (number of nodes participating in calculation) of each subdomain using these two methods, respectively. As shown in this table, for each subdomain of both these two DDMs, the total number of nodes (N
tol) is almost equal. However, the subdomains obtained from RCB have a very unbalanced distribution of nodes involved in contact (N
con) and FSI (N
fsi), in contrast with those obtained from MRCB. As is known to all, the calculation overheads of nodes participating in contact or FSI far outweigh those of common nodes. Therefore, for the traditional RCB method, the loads of each partition cannot achieve balance, and cores processing partitions with heavier loads will take extra time when cores processing partitions with less loads are idle and waiting. As a result, the expensive computing resources are wasted and the total calculation time increases unnecessarily. On the contrary, the nodes in contact and the FSI of the partitions from the MRCB method are well distributed, and the number of these nodes in every subdomain is approximately equal. Therefore, the total loads in every subdomain are balanced. Besides, the running time of cores processing each subdomain is almost the same, so the computing resources are used fully and reasonably, and the total calculation time decreases greatly.
Table 4.
Comparison of domain decomposition results obtained by different approaches.
Table 4.
Comparison of domain decomposition results obtained by different approaches.
Subdomain | RCB Method | MRCB Method |
---|
Ntol | Ncon | Nfsi | Ntol | Ncon | Nfsi |
---|
SD1 | 120,880 | 8828 | 5515 | 120,037 | 11,252 | 5790 |
SD2 | 122,157 | 11,548 | 6188 | 121,002 | 11,067 | 5736 |
SD3 | 117,797 | 17,873 | 10,480 | 119,524 | 11,133 | 5736 |
SD4 | 116,319 | 0 | 0 | 119,577 | 11,074 | 5736 |
SD5 | 121,399 | 14,339 | 6200 | 119,399 | 11,159 | 5736 |
SD6 | 120,709 | 11,708 | 7334 | 119,378 | 10,871 | 5736 |
SD7 | 118,680 | 14,120 | 6580 | 119,355 | 11,062 | 5735 |
SD8 | 122,386 | 11,596 | 7206 | 119,342 | 11,059 | 5736 |
SD9 | 121,623 | 12,545 | 6240 | 119,217 | 11,059 | 5736 |
SD10 | 118,561 | 7497 | 4197 | 119,106 | 11,060 | 5735 |
SD11 | 120,636 | 19,267 | 9257 | 118,645 | 11,058 | 5736 |
SD12 | 116,330 | 0 | 0 | 119,413 | 11,059 | 5734 |
SD13 | 118,108 | 11,736 | 5740 | 119,657 | 11,058 | 5736 |
SD14 | 121,415 | 12,996 | 5514 | 120,054 | 11,063 | 5736 |
SD15 | 120,924 | 16,128 | 8800 | 121,230 | 11,071 | 5739 |
SD16 | 115,600 | 6719 | 2578 | 118,588 | 10,795 | 5736 |
Table 5 shows the time consumption and parallel performance of different approaches with different numbers of cores to perform a 20 s real-time seismic analysis, where
,
and
represent total computation time, speedup and parallel efficiency, respectively. As shown in the table, MRCB presents a more balanced load distribution and a higher speed up and parallel efficiency compared to those of RCB. Therefore, the total computation time decreases.
Table 5.
Scalability performance of different methods.
Table 5.
Scalability performance of different methods.
Number of Cores | RCB Method | MRCB Method |
---|
Ti (s) | Su | Ef (%) | Ti (s) | Su | Ef (%) |
---|
1 | 3,299,460 | 1 | 100 | 3,299,460 | 1 | 100 |
2 | 1,729,320 | 1.91 | 95.4 | 1,693,620 | 1.95 | 97.4 |
4 | 900,240 | 3.67 | 91.6 | 882,720 | 3.74 | 93.4 |
8 | 523,920 | 6.3 | 78.7 | 505,500 | 6.53 | 81.6 |
16 | 322,056 | 10.24 | 64 | 282,352 | 11.69 | 73 |
32 | 182,210 | 18.11 | 56.6 | 159,923 | 20.63 | 64.5 |
The comparison of speedup for the parallel dynamic analysis is also shown in
Figure 22. The increment of speedup for both RCB and MRCB is not linearly proportional to the increment of cores involved in computation, which means that the parallel efficiency drops when the number of processor cores grow. Compared with RCB, the speedup of MRCB is larger and closer to the theoretical one and this superiority increases as the number of cores grows.
Figure 22.
Speedup of different parallel schemes.
Figure 22.
Speedup of different parallel schemes.
4.4. Parametric Study
Different from common tunnels, water conveyance tunnels require special consideration. The interaction between inner water and the tunnel lining must be examined [
23,
42], as it will affect the dynamic responses of the tunnel during earthquakes. In this study, the tunnel with full water, corresponding to normal operation conditions, is considered. However, there are also other conditions like the modification condition with an empty tunnel and the water hammer condition. With the intention of comparing and evaluating the effects of the inner water on seismic responses of the tunnel, the double-line tunnel with no water inside under the same traveling seismic excitation is also analyzed using the proposed parallel algorithm. The parameters are the same as those illustrated in
Section 3 except for the removed inner water.
Figure 23 shows the comparison of the largest and smallest first order principal stresses of section RS5 during the earthquake. It is shown that the tunnel with inner water suffers higher stresses than those of the empty tunnel, which means that the existence of inner water may make the tunnel more vulnerable during earthquakes. These results may be attributed to the added mass, the sloshing of the inner water and the FSI between inner water and the tunnel. Other sections show similar trends. Therefore, the simulation of the water conveyance tunnel with filled water included by this paper’s study may give more reasonable guidelines for the tunnel design.
Figure 23.
Effects of the inner water on dynamic responses evaluated by principal stress: (a) the largest one; (b) the smallest one.
Figure 23.
Effects of the inner water on dynamic responses evaluated by principal stress: (a) the largest one; (b) the smallest one.
The deformation joints embedded in the tunnel can prevent the tunnel from being damaged by non-uniform subsidence and accidental external forces. They can also alleviate the stress concentrations caused by sharp changes of structure and stiffness when the tunnel is subjected to earthquakes. For this specific tunnel, three deformation joints are embedded to connect the tunnel lining ring and the work shaft. The layout of deformation joints is shown in
Figure 8d. The effects of the deformation joints on the dynamic responses of the tunnel under seismic excitation are studied here using the established three-dimensional full scale model.
Figure 24 illustrates the modulus of the maximum dislocation displacement of each deformation joint, namely,
(see
Figure 8d). For brevity, in this Figures S1–S5 represent the water sluice work shaft, middle work shaft, Changxing island work shaft, Pudong work shaft and Pudong #5 conduit work shaft, respectively; LE and RI mean the left side and the right side of each work shaft along the tunnel from S1 to S5, respectively; a, b and c denote the first, second and third deformation joint in the direction away from each work shaft, respectively (see
Figure 8d). For instance, S1RIa means the first deformation joint at the right side of the water sluice work shaft. As shown in
Figure 24, the maximum dislocation displacements of deformation joints situated in the same location at both two sides of a work shaft are close to each other; and for deformation joints at one side of a work shaft, the smaller the distance between the deformation joint and the work shaft is, the bigger the dislocation displacement becomes. Besides, the maximum dislocation displacement occurs at S3RIa and the maximum value is about 3 mm.
Figure 24.
Modulus of maximum dislocation displacement of each deformation joint.
Figure 24.
Modulus of maximum dislocation displacement of each deformation joint.
To further evaluate the function of the deformation joints, cases with and without deformation joints are considered.
Table 6 shows the largest compressive stresses and the largest tensile stresses of the lining at two sides of each work shaft with and without deformation joints during earthquakes. It can be seen that with deformation joints the tensile and compressive stresses decrease and the maximum decrement appears at S3RI (right of the Changxing island work shaft) with a value of about 16.7%. By comparing with
Figure 24, it is found that this is the exact location of the maximum dislocation displacement. Thus, it can be perceived that the dislocation of the deformation joints absorbs energy and causes the corresponding stresses to decrease. Therefore, the existence of deformation joints makes the shear deformation of tunnel linings occur at two sides of the deformation joints, and thus partially releases the stress of the tunnel linings near work shafts.
Table 6.
Effect of the deformation joints on maximum stresses of the tunnel.
Table 6.
Effect of the deformation joints on maximum stresses of the tunnel.
Location | Largest Tensile Stress | Largest Compressive Stress |
---|
Without | With | Change (%) | Without | With | Change (%) |
---|
S1RI | 1.698 | 1.548 | −8.83 | 1.716 | 1.610 | −6.18 |
S2LE | 1.208 | 1.125 | −6.87 | 1.222 | 1.157 | −5.32 |
S2RI | 1.266 | 1.176 | −7.11 | 1.526 | 1.429 | −6.36 |
S3LE | 1.894 | 1.718 | −9.29 | 2.861 | 2.538 | −11.29 |
S3RI | 2.634 | 2.194 | −16.7 | 2.793 | 2.349 | −15.9 |
S4LE | 3.027 | 2.661 | −12.09 | 2.841 | 2.383 | −16.12 |
S4RI | 2.548 | 2.278 | −10.6 | 2.838 | 2.568 | −9.51 |
S5LE | 2.589 | 2.348 | −9.31 | 2.730 | 2.487 | −8.9 |