On the Loads for Strength Design of Cutterhead of Full Face Rock Tunnel Boring Machine
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Abstract
Cutterhead loads are the key mechanical parameters for the strength design of the full face hard rock tunnel boring machine (TBM). Due to the brittle rock-breaking mechanism, the excavation loads acting on cutters fluctuate strongly and show some randomness. The conventional method that using combinations of some special static loads to perform the strength design of TBM cutterhead may lead to strength failure during working practice. In this paper, a three-dimensional finite element model for coupled Cutterhead–Rock is developed to determine the cutterhead loads. Then the distribution characteristics and the influence factors of cutterhead loads are analyzed based on the numerical results. It is found that, as time changes, the normal and tangential forces acting on cutters and the total torque acting on the cutterhead approximately distribute log normally, while the total thrusts acting on the cutterhead approximately show a normal distribution. Furthermore, the statistical average values of cutterhead loads are proportional to the uniaxial compressive strength (UCS) of cutting rocks. The values also change with the penetration and the diameter of cutterhead following a power function. Based on these findings, we propose a three-parameter model for the mean of cutterhead loads and a method of generating the random cutter forces. Then the strength properties of a typical cutterhead are analyzed in detail using loads generated by the new method. The optimized cutterhead has been successfully applied in engineering. The method in this paper may provide a useful reference for the strength design of TBM cutterhead.
Keywords
TBM cutterhead Strength design Numerical simulation Three-parameter model Random cutter forces1 Introduction
TBM is widely used in the construction of various types of hard rock tunnels. The working performance of cutterhead, which is the key tunneling component of TBM, directly affects the efficiency and safety of construction [1, 2]. Due to the complex geology, such as crumbly strata, super hard rock and jointed rock, cutterhead bears large thrust, strong torque and impact cutter loads. Additionally the brittle rock-breaking mechanism and the inappropriate operating parameters may also result in severe vibration of cutterheads of TBM, which would lead to serious strength failure. Therefore, analyzing the strength property with a certain static load, which is commonly used in the current TBM design, may be unable to fulfill the strength requirement of cutterhead accurately in working condition. Thus, how to impose those loads is essential for the strength design of TBM cutterhead.
However, restricted by the closed and complex underground environment, it is difficult to observe the cutterhead loads directly in tunneling process. Thus, in the early years, researches on cutterhead loads are mainly performed experimentally and theoretically focusing on the cutting forces of the single cutter. Roxborough et al. [3] analyzed cutting forces on a wedge-shaped cutter taking into account the contact area and the UCS of rock. Rostami [4] put forward Colorado School of Mines (CSM) force prediction model based on linear cutting machine (LCM). It’s one of the most commonly used formulas in estimating cutter wear and optimizing cutter layout [5, 6, 7]. Using the broken theory of interaction of compression and shearing, Xia et al. [8], established a three-axis force rotary cutting mechanical model of disc cutter. Based on dense core theory, Huo et al. [9] proposed a multi-stage rock fragmentation load prediction model. In addition, with a full-scale rotary cutting machine (RCM), Geng et al. [10] investigated the cutting forces of TBM gage cutters and found that the cutting forces of the gage cutter were lower than those of the normal cutter. Upon this finding, a design scheme of multi-stage cutterhead was proposed [11]. Recently, with the increasing computational capability and the development of sophisticated material models, numerical methods are widely used to better analyze cutting forces. Xia et al. [12], established a finite element model for the rock fragmentation of the center cutter, and studied the formation mechanism and change law of the side force on the center cutter. Tan et al. [13], simulated the construction process of concrete rolling by a disc cutter and pointed out that the three-axis force on the cutter obviously presented the characteristic of a step change. Furthermore, the rock-breaking process by multi cutters was simulated, and the effect of cutter spacing on the cutting forces were discussed [14, 15].
Taking advantage of the cutting forces on the single cutter, some researchers calculated the total loads acting on the cutterhead. Based on the CSM model mentioned above, Liu et al. [6], calculated the cutterhead loads by linear superposition of cutter forces. Using the same method, Zhou et al. [16], proposed a new model to predict the total thrust acting on TBM. However, due to the neglecting of the great differences of cutting forces on different cutters, these models could only provide a relatively rough estimation of the average loads. In addition to these simplified calculation model, the available engineering data were employed for studying the cutterhead loads. Farmer et al. [17], analyzed the variation of cutterhead loads with the rock strength, Nelson et al. [18] analyzed a large number of TBM project data and found that the advancing distance per revolution had a significant effect on thrust. Upon the statistical results of 262 TBMs manufactured after 1985, Ates et al. [19] concluded that cutterhead torque was highly correlated with the TBM diameter, and also related with rock strength and geological discontinuities. Analysis of these studies indicates that cutterhead loads are very complicated mechanical quantities, they depend largely on the geological conditions, operating parameters and structural features.
Comparing to the cutter forces which have been well studied using experimental, numerical and theoretical methods, the cutterhead loads are relatively less-investigated, with only a few qualitative results presented. However, the cutterhead loads are more important to TBM design. They are more than the summary of the cutter forces. Currently, there is still no quantitative model for cutterhead loads of TBM that comprehensively reflects the effects of geological conditions, equipment structures, and operational status on the loads. In addition, the distribution characteristics of the cutterhead loads are not clear yet. To obtain the cutterhead loads for effective strength design, in this paper, a three-dimensional finite element model for coupled Cutterhead–Rock is established firstly. Then the distribution characteristics and the influencing factors of the cutterhead loads are investigated based on the simulation results. And a three-parameter model for the mean of cutterhead loads and the method of generating the random cutter forces are proposed. Finally, a new method is developed to analyze the strength for TBM cutterhead.
2 Simulation of Cutterhead Loads in TBM Tunneling
2.1 Material Model of Rock
Generally speaking, cutterhead loads depend on the counterforce of rock in tunneling process. So, in order to obtain the cutterhead loads, the tunneling process should be simulated firstly. To describe the mechanical behavior of rock in the tunneling process, the rock material model needs contain three parts, namely, a constitutive equation describing the stress-strain relationship, a damage law describing the degradation of rock stiffness, and separation criteria describing the separation of rock fragments.
Fracture occurs when all of the nodes at any element meet the separation criteria. Once fracture occurs, the fractured elements are deleted from the mesh.
2.2 Cutterhead–Rock Interaction
During the tunneling process, cutters roll through the excavation face. Rock-breaking is induced by the interaction between cutterhead and rock. As the detailed modeling process of Cutterhead–Rock interaction has been elaborated in our previous article [24], only the key steps is to be introduced in brief description as follows.
The contact pair algorithm was employed to define the Cutterhead–Rock interaction. It enforces contact constraints using a penalty contact method, which searches for slave node penetrations in the current configuration, including node-into-face, node-into-analytical rigid surface, and edge-into-edge penetrations. The surface of each cutter and the node-based surface of the rock were set as an independent non-smooth contact pair. The former was defined as the master surface due to its high stiffness and the latter was defined as the slave surface. When the master surface moves across the deformable slave surface, the tangential behavior is described by the Coulomb friction model.
Due to the relatively high stiffness, the cutterhead was modeled as a rigid body with a reference point. Then, the motion of cutterhead can be controlled by applying advancing speed and rotary speed on the reference point. Additionally, simulating the cutterhead using the rigid body feature makes it possible to calculate the thrust and torque of cutterhead using the resulting reaction force and moment acting on the reference point [25].
3 Characteristics of Cutter Loads
Material parameters of rock
Parameter | Value |
---|---|
Density (kg/mm^{3}) | 2.5×10^{−6} |
Young’s modulus (MPa) | 22500 |
Poisson’s ratio | 0.3 |
Friction angle (°) | 46.5 |
Dilation angle (°) | 46.5 |
UCS (MPa) | 112 |
3.1 Random Cutter Forces
Firstly, based on the numerical model introduced in Section 2, the forces on each cutter are obtained from the simulation of the tunneling process. Then, the type and parameters of the distribution of cutter forces are analyzed by statistical method.
Additionally, performing Kolmogorov–Smirnov test on the forces of both center cutters and gage cutters, results show that forces on all cutters approximately obey the lognormal distribution. Meanwhile, the mean and standard deviation of each cutter’s forces were obtained. Based on the distribution form, the mean and the standard deviation, the random forces on each cutter can be generated by using the Latin Hypercube Sampling method.
3.2 Characteristics of Cutterhead Loads
3.3 A Three-Parameter Model for the Mean of Cutterhead Loads
In the stage of initial design, the cutterhead loads should be estimated according to the geological conditions and construction parameters. And upon the estimated loads, the main structural parameters of the cutterhead could be determined. To estimate cutterhead loads, it is necessary to investigate the effects of different parameters on cutterhead loads.
This model provides the quantitative relationship between the mean of cutterhead loads and the core parameters of geology, operation and structure. Moreover, it is a reasonable non-dimensional model from the perspective of dimensional analysis.
Furthermore, data collected from one TBM tunneling project (the Water Supply Project in the middle of Jilin Province, China) was used to verify the effectiveness of the presented model. The engineering scenario is the construction of stake numbers from 21050 m to 21400 m. The major relevant geological strata identified in this area include sandstone, tuff, and granite, whose UCS ranges from 55 MPa to 130 MPa. The diameter of the cutterhead used in this project is 8 m. In addition, the operating parameters used in this analysis were automatically recorded by the machine during tunneling, the penetration range is 6.2 mm to 16 mm.
As for the thrust, the estimated value is a little smaller than the in-situ data. The in-situ thrust refers to the equipment’s total thrust, which includes the rock-breaking force acting on the cutterhead, the friction force of the shield, and the traction force of the subsequent equipment. While the estimated thrust only refers to the cutterhead’s rock-breaking force, which is the major component of the total thrust. Therefore, the estimated thrust is smaller than the in-situ data. According to our calculation, the estimated cutterhead thrust accounts for 74.3% of the in-situ total thrust, which is consistent with engineering practice. As for the torque, the estimated value shows a good agreement with the measured data, and the mean of the relative error is 17.10%.
4 Case Applications
Based on the results of load research, the strength properties of a cutterhead used in the Water Supply Project in the middle of Jilin Province, was analyzed comprehensively. Firstly, in the stage of initial design, the static extreme loads evaluated by Eq. (8) were used for analyzing the structural strength preliminarily. Then, in the stage of detailed design, the failure probability of the cutterhead’s strength was calculated under the random cutter forces, and the key factors of strength failure were analyzed. Finally, the dynamic loads obtained from tunneling simulation were employed to check the strength properties of the cutterhead with dynamical effects.
Obviously, under the static extreme loads, the high stress appears in the stiffened panels of the cutterhead. The maximum equivalent stress is about 81 MPa, which is lower than the yield stress of Q345. It indicates that the design of the main structural parameters of the cutterhead meet the basic strength requirements.
Based on the analysis of the load characteristics, the rock-breaking forces on the cutters change in randomness, and approximately obey the lognormal distribution. Thus, in the stage of detailed design, it’s necessary to calculate the failure probability of the cutterhead’s strength under the random cutter forces. Meanwhile, the key factors of strength failure should be analyzed.
Distribution parameters of cutter forces
Installed radius (m) | F_{R} (kN) | F_{N} (kN) | Correlation coefficient | Installed radius (m) | F_{R} (kN) | F_{N} (kN) | Correlation coefficient | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Standard deviation | Mean | Standard deviation | Mean | Standard deviation | Mean | Standard deviation | ||||
0.06 | 3.83 | 3.68 | 59.29 | 41.46 | 0.72 | 2.35 | 16.46 | 12.06 | 224.04 | 185.60 | 0.88 |
0.15 | 6.75 | 4.23 | 79.75 | 51.40 | 0.85 | 2.42 | 15.64 | 13.69 | 218.16 | 150.95 | 0.73 |
0.24 | 9.13 | 6.57 | 118.51 | 81.67 | 0.93 | 2.50 | 15.19 | 12.25 | 211.94 | 160.71 | 0.84 |
0.33 | 9.79 | 8.06 | 133.57 | 84.12 | 0.94 | 2.57 | 16.14 | 15.67 | 225.13 | 198.63 | 0.79 |
0.42 | 6.12 | 5.84 | 93.97 | 64.38 | 0.90 | 2.65 | 17.20 | 13.82 | 239.94 | 158.01 | 0.80 |
0.51 | 9.32 | 8.42 | 133.21 | 99.06 | 0.93 | 2.72 | 16.63 | 14.97 | 226.37 | 164.36 | 0.83 |
0.60 | 8.66 | 7.14 | 127.76 | 99.80 | 0.93 | 2.80 | 15.32 | 13.99 | 213.76 | 150.75 | 0.82 |
0.69 | 7.00 | 5.04 | 108.18 | 74.40 | 0.91 | 2.87 | 15.72 | 15.72 | 213.75 | 162.11 | 0.88 |
0.77 | 7.28 | 6.28 | 110.17 | 86.57 | 0.92 | 2.95 | 16.13 | 13.73 | 225.06 | 152.62 | 0.89 |
0.85 | 9.99 | 5.53 | 142.58 | 105.52 | 0.94 | 3.02 | 16.58 | 11.48 | 231.35 | 136.40 | 0.87 |
0.93 | 4.82 | 3.27 | 68.33 | 35.50 | 0.84 | 3.10 | 15.52 | 11.34 | 216.49 | 134.13 | 0.89 |
1.01 | 8.02 | 4.90 | 111.03 | 87.50 | 0.88 | 3.17 | 16.70 | 12.08 | 233.02 | 93.43 | 0.78 |
1.09 | 8.54 | 5.04 | 120.50 | 75.77 | 0.85 | 3.25 | 16.38 | 13.44 | 228.54 | 101.30 | 0.53 |
1.17 | 9.58 | 5.45 | 142.95 | 90.64 | 0.91 | 3.32 | 15.82 | 9.64 | 220.66 | 96.38 | 0.60 |
1.25 | 8.06 | 4.79 | 118.86 | 77.32 | 0.88 | 3.40 | 14.51 | 6.65 | 202.35 | 66.29 | 0.76 |
1.33 | 8.20 | 6.51 | 120.25 | 98.35 | 0.93 | 3.47 | 16.09 | 13.07 | 231.47 | 159.16 | 0.91 |
1.41 | 12.18 | 7.88 | 167.75 | 121.60 | 0.91 | 3.53 | 14.04 | 3.70 | 199.78 | 50.09 | 0.81 |
1.49 | 9.00 | 6.18 | 129.32 | 97.59 | 0.89 | 3.59 | 13.55 | 4.38 | 177.51 | 53.85 | 0.68 |
1.57 | 11.84 | 7.64 | 167.60 | 121.59 | 0.89 | 3.65 | 12.71 | 3.56 | 169.37 | 57.27 | 0.91 |
1.65 | 9.28 | 6.99 | 134.84 | 110.49 | 0.87 | 3.71 | 12.64 | 6.64 | 150.59 | 71.38 | 0.92 |
1.73 | 11.56 | 8.30 | 165.24 | 129.01 | 0.93 | 3.77 | 12.25 | 7.03 | 155.68 | 57.63 | 0.84 |
1.81 | 12.97 | 8.89 | 188.38 | 133.50 | 0.89 | 3.82 | 10.60 | 5.94 | 126.32 | 65.55 | 0.85 |
1.89 | 13.55 | 10.31 | 192.50 | 154.94 | 0.89 | 3.86 | 9.54 | 5.09 | 115.85 | 104.10 | 0.93 |
1.97 | 12.22 | 8.96 | 173.71 | 135.71 | 0.87 | 3.89 | 8.17 | 7.62 | 97.29 | 82.03 | 0.91 |
2.05 | 12.41 | 10.37 | 173.08 | 132.13 | 0.94 | 3.92 | 8.56 | 8.00 | 101.98 | 84.78 | 0.92 |
2.12 | 13.49 | 10.16 | 188.15 | 144.91 | 0.89 | 3.95 | 9.03 | 5.22 | 107.55 | 75.49 | 0.92 |
2.20 | 16.16 | 12.71 | 233.83 | 181.26 | 0.95 | 3.96 | 8.73 | 8.68 | 104.01 | 77.60 | 0.93 |
2.27 | 15.66 | 11.03 | 218.51 | 155.96 | 0.92 | 3.97 | 7.97 | 4.31 | 94.94 | 72.29 | 0.75 |
5 Conclusions
To determine the cutterhead loads in TBM tunneling, a three-dimensional finite element model for Cutterhead–Rock interaction was developed. Based on the simulation results, the distribution characteristics of cutterhead loads and cutter forces were analyzed. Moreover, the effects of different parameters on cutterhead loads were discussed.
Results reveal that the cutter forces present the characteristics of a step change during the rock breaking process, and the responses of the forces show some randomness. The normal and tangential forces on cutters approximately have a lognormal distribution. And the tangential force is highly correlated with the normal force.
The cutterhead loads fluctuate strongly during the excavation, the thrusts are approximately distributed normally, while the torques approximately show the lognormal distribution. In addition, the cutterhead loads are proportional to the UCS of rock, they also change with the penetration and the diameter of cutterhead following a power function. Based on these results, a dimensionless three-parameter model for the mean of cutterhead loads was proposed and its effectiveness is verified by comparing the estimated loads and the in-situ data.
A systematic method of strength analysis of the cutterhead was then developed based on the simulated loads. In the stage of initial design, the static extreme loads evaluated by Eq. (8) could be used to analyze the structural strength preliminarily. In the stage of detailed design, the random cutter forces generated by Monte Carlo method should be imposed to calculate the failure probability of the cutterhead’s strength and to find the key factors of strength failure. Finally, the dynamic loads simulated under typical geology and operating parameters should be employed to check the strength properties with dynamical effects of the cutterhead.
Notes
Authors’ Contributions
ZC conceived and designed the research methods and processes; MH was in charge of numerical simulation and data analysis; CQ assisted with writing the manuscript. All authors read and approved the final manuscript.
Funding
Supported by National Basic Research Program of China (973 Program, Grant No. 2013CB035042) and the National Natural Science Foundation of China (Grant No. 11672202).
Competing Interests
The authors declare that they have no competing interests.
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