Reliability Prediction of Tunnel Roof with a Nonlinear Failure Criterion
Abstract
:1. Introduction
2. Methodology
2.1. Reliability Analysis Methods
2.1.1. FORM
2.1.2. RSM
- (a)
- Sampling points are chosen around the mean value . Usually, mean value points with are selected to evaluate the performance function , in which is the sampling range factor.
- (b)
- Altogether coefficients of Equation (5) can be obtained by solving the set of linear equations. Thus, a tentative response surface is generated.
- (c)
- Calculating the reliability index and corresponding design points by FORM and Equation (1). In this computation, is subject to the constraint that .
- (d)
- Repeating steps (a)–(c) until or converges. Besides the first trial, new sampling points may be selected around the tentative design points concerning the interpolation method.
2.1.3. MCS
2.2. Kinematic Analysis of Tunnels Roofs with Hoek-Brown Criterion
2.3. Performance Functions of Roof Collapse
3. Results and Discussion
3.1. Reliability Analysis of Rectangular Tunnels
3.1.1. Reliability Index and Failure Probability
3.1.2. Sensitivity Analysis
3.1.3. Influence of Coefficient of Variation
3.2. Reliability Analysis of Circular Tunnels
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Nomenclature
coefficients in response surface method | |
material constant | |
coefficients in response surface method | |
material constant | |
covariance matrix | |
energy dissipation density of the internal forces | |
sampling range factor. | |
failure region | |
collapsing curve | |
first derivative of | |
original probability density function ordinate at | |
original non-normal CDF evaluated at | |
performance function | |
performance function of a rectangular tunnel | |
performance function of a circular tunnel | |
weight of failure block | |
height of the collapsing block | |
half width of the collapsing block | |
unit vector | |
number of samples | |
total energy dissipation at the impending collapse | |
work rate done by weight | |
work rate of the pore pressure | |
work rate of supporting force | |
failure probability | |
support pressure | |
pore water coefficient |
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Random Variable | Mean Value | Distribution Type | |
---|---|---|---|
Case 1 | 100 | normal | |
B | 0.7 | normal | |
0.2 | normal | ||
normal | |||
Case 2 | 100 | lognormal | |
B | 0.7 | lognormal | |
0.2 | lognormal | ||
lognormal |
Case 1 | Case 2 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Design Point | Reliability Index | Design Point | Reliability Index | |||||||||
FORM | RSM | MCS | FORM | RSM | MCS | |||||||
60 | 104.255 | 0.686 | 0.199 | 0.442 | 0.442 | 0.433 | 103.150 | 0.679 | 0.197 | 0.420 | 0.420 | 0.433 |
70 | 109.965 | 0.664 | 0.198 | 1.122 | 1.122 | 1.107 | 110.640 | 0.659 | 0.196 | 1.116 | 1.116 | 1.130 |
80 | 113.933 | 0.648 | 0.198 | 1.691 | 1.691 | 1.673 | 117.598 | 0.642 | 0.195 | 1.720 | 1.720 | 1.738 |
90 | 116.649 | 0.635 | 0.197 | 2.171 | 2.171 | 2.154 | 124.120 | 0.628 | 0.195 | 2.253 | 2.253 | 2.269 |
100 | 118.462 | 0.627 | 0.197 | 2.579 | 2.579 | 2.566 | 130.280 | 0.616 | 0.194 | 2.731 | 2.731 | 2.770 |
110 | 119.618 | 0.621 | 0.197 | 2.929 | 2.929 | 2.919 | 136.130 | 0.605 | 0.194 | 3.164 | 3.164 | 3.160 |
120 | 120.296 | 0.618 | 0.197 | 3.230 | 3.230 | 3.226 | 141.713 | 0.596 | 0.193 | 3.559 | 3.559 | 3.633 |
130 | 120.628 | 0.616 | 0.196 | 3.491 | 3.491 | 3.486 | 147.061 | 0.588 | 0.193 | 3.923 | 3.923 | 3.911 |
140 | 120.710 | 0.616 | 0.196 | 3.719 | 3.719 | 3.707 | 152.203 | 0.580 | 0.192 | 4.260 | 4.260 | 4.244 |
60 | 0.641 | −0.308 | −0.050 |
70 | 0.592 | −0.304 | −0.048 |
80 | 0.549 | −0.296 | −0.046 |
90 | 0.511 | −0.284 | −0.043 |
100 | 0.477 | −0.271 | −0.041 |
110 | 0.447 | −0.257 | −0.038 |
120 | 0.419 | −0.243 | −0.036 |
Mean value | 0.520 | −0.280 | −0.043 |
Random Variables | Normal/Lognormal Distribution | |
---|---|---|
Mean Value | Coefficient of Variation | |
(MPa) | 0.1 | 0.15 |
(MPa) | 10 | 0.15 |
0.5 | 0.15 | |
0.7 | 0.15 | |
(kN/m3) | 25 | 0.15 |
0.2 | 0.15 | |
0.15 |
Support Pressure (kPa) | Sample Size | COV of Failure Probability | |
---|---|---|---|
Normal Distribution | Lognormal Distribution | ||
45 | 0.0098 | 0.0097 | |
55 | 0.0087 | 0.0090 | |
65 | 0.0113 | 0.0122 | |
75 | 0.0117 | 0.0140 | |
85 | 0.0214 | 0.0340 |
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Yang, X.; Long, J. Reliability Prediction of Tunnel Roof with a Nonlinear Failure Criterion. Mathematics 2023, 11, 937. https://doi.org/10.3390/math11040937
Yang X, Long J. Reliability Prediction of Tunnel Roof with a Nonlinear Failure Criterion. Mathematics. 2023; 11(4):937. https://doi.org/10.3390/math11040937
Chicago/Turabian StyleYang, Xin, and Jiangping Long. 2023. "Reliability Prediction of Tunnel Roof with a Nonlinear Failure Criterion" Mathematics 11, no. 4: 937. https://doi.org/10.3390/math11040937
APA StyleYang, X., & Long, J. (2023). Reliability Prediction of Tunnel Roof with a Nonlinear Failure Criterion. Mathematics, 11(4), 937. https://doi.org/10.3390/math11040937