Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D
Abstract
:1. Introduction
2. Statement of the Problem and Modelling Technique
3. Comparison for Model Validation
4. Results and Discussion
5. Design Equations
6. Conclusions
- The stability number, N, increases with an increase in all of the investigated parameters of B/L, H/B, m, and re. The increases can be either in a linear or nonlinear relationship. The linear relationship was obtained for all investigated cases except for cases with smaller values of m, where a nonlinear relationship exists between N and B/L.
- The failure patterns of unsupported rectangular excavation in anisotropic and heterogeneous clays are either in a toe-failure mode (for small values of H/B, i.e., H/B = 0.5, 1) or a face-failure mode (for large values of H/B > 1) due to the effect of the strength gradient ratio m. For large values of m > 4, the failure modes are predominately the face-failure mode. The variation in the anisotropic ratio, re, does not seem to affect the failure size of the unsupported rectangular excavation problem.
- A new equation for predicting the stability number, N, of the unsupported rectangular excavation in anisotropic and heterogeneous clays is proposed. With the coefficient of determination (R2) being 99.99%, the proposed equation is highly accurate and useful for practical uses.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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m | re = 1 | re = 0.9 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
H/B | B/L | H/B | B/L | |||||||||
1 | 2/3 | 1/2 | 1/4 | 1/8 | 1 | 2/3 | 1/2 | 1/4 | 1/8 | |||
0 | 0.5 | 4.559 | 4.372 | 4.234 | 3.959 | 3.860 | 0.5 | 4.299 | 4.128 | 4.027 | 3.764 | 3.460 |
1 | 5.291 | 4.953 | 4.677 | 4.153 | 3.955 | 1 | 4.958 | 4.661 | 4.420 | 3.931 | 3.759 | |
2 | 6.420 | 5.969 | 5.553 | 4.637 | 4.125 | 2 | 5.968 | 5.560 | 5.201 | 4.384 | 3.917 | |
3 | 7.170 | 6.707 | 6.279 | 5.135 | 4.362 | 3 | 6.657 | 6.243 | 5.849 | 4.833 | 4.137 | |
4 | 7.770 | 7.280 | 6.862 | 5.606 | 4.632 | 4 | 7.216 | 6.774 | 6.388 | 5.262 | 4.378 | |
4 | 0.5 | 9.259 | 9.008 | 8.758 | 8.362 | 8.255 | 0.5 | 8.749 | 8.495 | 8.288 | 7.870 | 7.898 |
1 | 15.758 | 14.648 | 13.891 | 12.832 | 12.437 | 1 | 14.778 | 13.806 | 13.189 | 12.151 | 11.775 | |
2 | 30.562 | 28.105 | 26.095 | 22.663 | 21.168 | 2 | 28.283 | 26.251 | 24.639 | 21.455 | 19.996 | |
3 | 45.966 | 42.212 | 39.161 | 33.462 | 30.330 | 3 | 42.365 | 39.288 | 36.776 | 31.691 | 28.730 | |
4 | 61.106 | 56.160 | 52.214 | 44.704 | 39.942 | 4 | 56.528 | 52.402 | 48.986 | 42.238 | 37.914 | |
12 | 0.5 | 18.213 | 17.662 | 17.175 | 16.735 | 16.561 | 0.5 | 17.156 | 16.730 | 16.444 | 15.785 | 15.516 |
1 | 34.964 | 32.841 | 31.468 | 29.458 | 28.713 | 1 | 32.745 | 31.043 | 29.716 | 27.916 | 27.309 | |
2 | 69.931 | 65.641 | 62.571 | 57.045 | 53.981 | 2 | 65.470 | 61.913 | 59.186 | 53.975 | 51.219 | |
3 | 104.996 | 98.597 | 93.842 | 85.662 | 80.372 | 3 | 98.151 | 92.897 | 88.782 | 81.116 | 76.146 | |
4 | 140.088 | 131.336 | 125.132 | 114.290 | 107.172 | 4 | 130.918 | 124.024 | 118.326 | 108.284 | 101.830 | |
25 | 0.5 | 32.254 | 31.504 | 30.613 | 30.072 | 29.207 | 0.5 | 30.536 | 29.713 | 29.120 | 28.351 | 28.198 |
1 | 64.231 | 61.253 | 59.346 | 56.412 | 55.183 | 1 | 60.475 | 57.966 | 56.147 | 53.315 | 51.845 | |
2 | 128.460 | 122.685 | 118.597 | 111.345 | 107.223 | 2 | 120.981 | 115.931 | 112.269 | 104.690 | 101.445 | |
3 | 192.627 | 183.920 | 178.077 | 167.298 | 160.784 | 3 | 181.659 | 174.002 | 168.429 | 158.561 | 152.492 | |
4 | 256.720 | 245.260 | 237.102 | 223.242 | 214.300 | 4 | 241.932 | 231.924 | 224.580 | 211.538 | 203.318 | |
100 | 0.5 | 111.392 | 109.320 | 107.744 | 106.143 | 105.742 | 0.5 | 105.461 | 103.419 | 101.948 | 100.684 | 100.869 |
1 | 223.302 | 218.275 | 215.000 | 209.382 | 206.320 | 1 | 211.114 | 206.404 | 203.672 | 198.231 | 196.114 | |
2 | 446.620 | 436.122 | 429.972 | 415.484 | 411.352 | 2 | 423.437 | 413.457 | 407.257 | 396.090 | 389.648 | |
3 | 671.124 | 654.846 | 645.300 | 627.836 | 617.555 | 3 | 632.489 | 620.067 | 611.184 | 594.366 | 585.180 | |
4 | 894.322 | 873.216 | 859.562 | 837.478 | 823.428 | 4 | 845.674 | 827.874 | 813.880 | 793.196 | 780.282 |
m | re = 0.8 | re = 0.7 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
H/B | B/L | H/B | B/L | |||||||||
1 | 2/3 | 1/2 | 1/4 | 1/8 | 1 | 2/3 | 1/2 | 1/4 | 1/8 | |||
0 | 0.5 | 3.998 | 3.880 | 3.807 | 3.531 | 3.255 | 0.5 | 3.696 | 3.571 | 3.503 | 3.273 | 3.043 |
1 | 4.594 | 4.327 | 4.129 | 3.697 | 3.525 | 1 | 4.206 | 3.983 | 3.807 | 3.415 | 3.257 | |
2 | 5.502 | 5.126 | 4.825 | 4.103 | 3.668 | 2 | 5.006 | 4.694 | 4.418 | 3.790 | 3.398 | |
3 | 6.120 | 5.741 | 5.400 | 4.518 | 3.885 | 3 | 5.576 | 5.232 | 4.935 | 4.155 | 3.590 | |
4 | 6.608 | 7.654 | 5.868 | 4.896 | 4.100 | 4 | 6.010 | 5.658 | 5.350 | 4.494 | 3.788 | |
4 | 0.5 | 8.192 | 8.020 | 7.753 | 7.399 | 7.252 | 0.5 | 7.574 | 7.352 | 7.212 | 6.878 | 6.741 |
1 | 13.733 | 12.929 | 12.311 | 11.406 | 11.087 | 1 | 12.572 | 11.952 | 11.463 | 10.533 | 10.203 | |
2 | 26.032 | 24.326 | 22.942 | 20.133 | 18.797 | 2 | 23.642 | 22.283 | 21.126 | 18.623 | 17.413 | |
3 | 38.835 | 36.234 | 34.224 | 29.712 | 26.985 | 3 | 35.222 | 33.110 | 31.422 | 27.500 | 25.028 | |
4 | 51.836 | 48.330 | 45.558 | 39.574 | 35.560 | 4 | 46.896 | 44.168 | 41.810 | 36.482 | 32.908 | |
12 | 0.5 | 16.080 | 15.921 | 15.174 | 14.900 | 14.827 | 0.5 | 14.856 | 14.759 | 14.108 | 13.673 | 13.539 |
1 | 30.498 | 29.064 | 27.867 | 26.213 | 25.641 | 1 | 28.015 | 26.842 | 25.829 | 24.231 | 23.713 | |
2 | 60.747 | 57.799 | 55.401 | 50.671 | 47.946 | 2 | 55.688 | 53.260 | 51.218 | 46.939 | 44.337 | |
3 | 91.143 | 86.838 | 83.121 | 76.007 | 71.582 | 3 | 83.531 | 79.896 | 76.892 | 70.436 | 66.102 | |
4 | 121.484 | 115.712 | 110.738 | 101.452 | 95.462 | 4 | 111.290 | 106.608 | 102.410 | 93.984 | 88.160 | |
25 | 0.5 | 28.631 | 27.955 | 27.377 | 26.578 | 25.636 | 0.5 | 26.442 | 25.683 | 25.257 | 24.657 | 24.445 |
1 | 56.445 | 54.317 | 52.686 | 50.016 | 49.108 | 1 | 51.882 | 50.107 | 48.741 | 46.161 | 45.508 | |
2 | 112.831 | 108.652 | 105.268 | 99.169 | 95.067 | 2 | 103.890 | 100.505 | 97.455 | 91.487 | 87.891 | |
3 | 169.331 | 162.911 | 157.857 | 148.704 | 142.529 | 3 | 155.756 | 150.548 | 146.213 | 137.873 | 132.387 | |
4 | 225.288 | 231.924 | 210.434 | 198.498 | 190.700 | 4 | 207.658 | 200.882 | 194.796 | 183.884 | 176.652 | |
100 | 0.5 | 98.804 | 97.060 | 95.754 | 94.274 | 92.989 | 0.5 | 91.379 | 89.479 | 88.734 | 87.369 | 82.820 |
1 | 197.710 | 193.955 | 190.910 | 183.274 | 183.608 | 1 | 182.879 | 179.463 | 176.887 | 171.749 | 169.835 | |
2 | 395.618 | 387.909 | 382.134 | 372.104 | 364.822 | 2 | 365.556 | 359.051 | 353.148 | 340.475 | 337.916 | |
3 | 594.897 | 581.319 | 573.506 | 558.114 | 548.117 | 3 | 549.108 | 538.401 | 530.817 | 517.062 | 502.274 | |
4 | 794.012 | 775.738 | 763.674 | 744.168 | 731.800 | 4 | 733.122 | 719.498 | 706.896 | 689.530 | 677.690 |
m | re = 0.6 | re = 0.5 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
H/B | B/L | H/B | B/L | |||||||||
1 | 2/3 | 1/2 | 1/4 | 1/8 | 1 | 2/3 | 1/2 | 1/4 | 1/8 | |||
0 | 0.5 | 3.317 | 3.256 | 3.181 | 2.955 | 2.771 | 0.5 | 2.944 | 2.882 | 2.820 | 2.628 | 2.319 |
1 | 3.774 | 3.586 | 3.450 | 3.117 | 2.969 | 1 | 3.314 | 3.155 | 3.029 | 2.758 | 2.631 | |
2 | 4.479 | 4.207 | 3.977 | 3.444 | 3.097 | 2 | 3.901 | 3.674 | 3.495 | 3.050 | 2.743 | |
3 | 4.979 | 4.679 | 4.421 | 3.753 | 3.254 | 3 | 4.328 | 4.085 | 3.857 | 3.309 | 2.898 | |
4 | 5.366 | 5.048 | 4.784 | 4.054 | 3.440 | 4 | 4.666 | 4.390 | 4.188 | 3.564 | 3.052 | |
4 | 0.5 | 6.823 | 6.675 | 6.550 | 6.294 | 6.027 | 0.5 | 6.038 | 5.960 | 5.753 | 5.559 | 5.491 |
1 | 11.358 | 10.816 | 10.409 | 9.624 | 9.369 | 1 | 9.967 | 9.521 | 9.226 | 8.561 | 8.277 | |
2 | 21.138 | 19.958 | 19.048 | 16.942 | 15.852 | 2 | 18.563 | 17.571 | 16.751 | 15.037 | 14.072 | |
3 | 31.406 | 29.540 | 28.229 | 24.951 | 22.737 | 3 | 27.335 | 25.931 | 24.728 | 22.184 | 20.210 | |
4 | 41.728 | 39.450 | 37.670 | 33.366 | 29.942 | 4 | 36.506 | 34.454 | 32.832 | 29.342 | 26.568 | |
12 | 0.5 | 13.475 | 13.208 | 12.811 | 12.563 | 11.985 | 0.5 | 11.915 | 11.702 | 11.428 | 11.133 | 10.896 |
1 | 25.262 | 24.302 | 23.432 | 22.134 | 21.623 | 1 | 22.330 | 21.468 | 20.794 | 19.667 | 19.228 | |
2 | 50.092 | 48.210 | 46.566 | 42.644 | 40.585 | 2 | 44.187 | 42.254 | 41.138 | 37.943 | 35.950 | |
3 | 75.329 | 72.156 | 69.764 | 64.122 | 60.272 | 3 | 66.371 | 63.263 | 61.695 | 56.939 | 53.634 | |
4 | 100.316 | 96.282 | 92.980 | 85.524 | 80.114 | 4 | 88.410 | 84.236 | 82.204 | 75.902 | 71.372 | |
25 | 0.5 | 24.129 | 23.467 | 23.007 | 22.585 | 22.364 | 0.5 | 21.192 | 20.744 | 20.493 | 19.967 | 19.757 |
1 | 47.000 | 45.646 | 44.432 | 42.214 | 41.395 | 1 | 41.437 | 40.252 | 39.182 | 37.430 | 36.636 | |
2 | 93.925 | 91.067 | 88.624 | 83.521 | 80.038 | 2 | 82.907 | 80.487 | 78.697 | 74.134 | 70.683 | |
3 | 141.065 | 136.667 | 132.840 | 125.496 | 120.269 | 3 | 124.328 | 120.735 | 118.287 | 111.372 | 107.109 | |
4 | 188.000 | 182.288 | 177.256 | 167.254 | 160.868 | 4 | 165.788 | 160.782 | 157.210 | 148.552 | 141.700 | |
100 | 0.5 | 83.089 | 81.711 | 79.615 | 79.360 | 75.411 | 0.5 | 73.433 | 72.459 | 71.454 | 70.765 | 69.833 |
1 | 165.317 | 163.306 | 160.071 | 156.043 | 154.564 | 1 | 147.068 | 144.681 | 142.492 | 139.251 | 137.323 | |
2 | 332.289 | 326.784 | 322.154 | 313.455 | 305.386 | 2 | 294.210 | 289.922 | 285.932 | 278.151 | 272.439 | |
3 | 500.166 | 489.830 | 483.281 | 470.489 | 462.878 | 3 | 440.799 | 433.592 | 428.594 | 416.861 | 408.983 | |
4 | 664.460 | 653.052 | 644.744 | 627.270 | 617.136 | 4 | 587.558 | 579.212 | 571.564 | 556.762 | 545.928 |
Constant Coefficients | re | |||||
---|---|---|---|---|---|---|
0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |
a1 | 3.16697 | 3.3617 | 3.73044 | 4.1027 | 3.81764 | 4.28435 |
a2 | 1.35068 | 1.52276 | 1.66728 | 1.80244 | 1.92334 | 2.02793 |
a3 | −0.69739 | −0.55146 | −0.64589 | −1.47779 | −0.09287 | −0.55003 |
a4 | 0.46763 | 0.6014 | 0.72694 | 0.99956 | 1.01361 | 1.18849 |
a5 | 1.96752 | 2.24958 | 2.55588 | 3.08414 | 3.30362 | 3.39274 |
a6 | 0.92331 | 0.99065 | 1.14742 | 1.57205 | 1.5951 | 1.54619 |
a7 | −0.07001 | −0.07579 | −0.081989 | −0.12147 | −0.13622 | −0.13032 |
a8 | −0.05667 | −0.05068 | −0.06182 | −0.10530 | −0.10229 | −0.09884 |
R2 | 99.99% | 99.99% | 99.99% | 99.99% | 99.99% | 99.99% |
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Lai, V.Q.; Shiau, J.; Keawsawasvong, S.; Seehavong, S.; Cabangon, L.T. Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D. Buildings 2022, 12, 1425. https://doi.org/10.3390/buildings12091425
Lai VQ, Shiau J, Keawsawasvong S, Seehavong S, Cabangon LT. Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D. Buildings. 2022; 12(9):1425. https://doi.org/10.3390/buildings12091425
Chicago/Turabian StyleLai, Van Qui, Jim Shiau, Suraparb Keawsawasvong, Sorawit Seehavong, and Lowell Tan Cabangon. 2022. "Undrained Stability of Unsupported Rectangular Excavations: Anisotropy and Non-Homogeneity in 3D" Buildings 12, no. 9: 1425. https://doi.org/10.3390/buildings12091425