Probabilistic Failure Estimation of an Oblique Loaded Footing Settlement on Cohesive Geomaterials with a Modified Cam Clay Material Yield Function
Abstract
:1. Introduction
2. Dynamic Soil–Pore–Fluid Interaction: The System of Equations and Its Numerical Solution
3. The Constitutive Material Yield Function
3.1. Plastic Yield Envelope and Bond Strength Envelope Mathematical Representation
4. Stochastic Processes, the Truncated Normal Distribution and Latin Hypercube Sampling
4.1. The Karhunen–Loeve Series and the Truncated Normal Variables
4.2. Latin Hypercube Sampling
- The interval [0,1] of the cumulative distribution function (CDF) is subdivided into equal subspaces
- A random number is chosen and through the inverse CDF a sample is acquired
- All the vectors are permuted in a random way and thus the vector realization X is composed
5. An Improvement of a Proposed Algorithm for the Determination of Failure Load in Ramp Dynamic Load Function
6. Numerical Tests on Stochastic Failure of Shallow Foundations with Random Linear and Non-Linear Material Properties
6.1. Description of the Problem
6.2. Presentation of the Results
6.2.1. Limit Failure Force and Displacement
6.2.2. Limit Stresses–Strains and Failure Spline
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Friction variables | |
Friction variable indicating the influence of possible vertical load in the lateral of | |
the foundation | |
Friction variable indicating the influence of the cohesion of the soil | |
Friction variable indicating the influence of the settlement dimensions | |
alongside with the total weight of the soil | |
Shape variables | |
Shape variable indicating the influence of possible vertical load in the lateral of | |
the foundation | |
Shape variable indicating the influence of the cohesion of the soil | |
Shape variable indicating the influence of the settlement dimensions | |
alongside with the total weight of the soil | |
Compressibility factor | |
c | Critical state line inclination |
k | Permeability in units |
Friction angle | |
Total mass matrix | |
Total damping matrix | |
Total stiffness matrix | |
Solid skeleton mass matrix | |
Density of the soil | |
Deformation matrix | |
Elasticity matrix | |
Solid skeleton damping matrix | |
Solid skeleton stiffness matrix | |
Unity matrix | |
Loading vector | |
Matrix of permeability in units | |
Shape functions for pore pressure | |
Shape functions for displacements | |
Saturation matrix | |
Coupling matrix | |
Permeability matrix | |
Equivalent forces due to external loading | |
Q | Variable for combining the influence of bulk moduli of fluid and solid skeleton in |
porous problems | |
Total stress tensor | |
Deviatoric component of the stress tensor | |
Hydrostatic component of the stress tensor | |
a | Halfsize of the bond strength envelope |
Deviatoric component of the stress point of the centre of the plastic yield envelope | |
Hydrostatic component of the stress point of the centre of the plastic yield envelope | |
Similarity factor between the plastic yield envelope and bond strength envelope | |
Generalized elliptic envelope | |
Plastic yield envelope (PYE) | |
F | Bond strength envelope (BSE) |
Specific volume of the soil | |
q | Von Mises stress |
e | Deviatoric strain measure |
Deviatoric component of the strain tensor | |
f | Random function |
Value of the random function at nodal points | |
Shape functions | |
Total number of shape functions | |
Truncated normal PDF | |
Standard normal PDF | |
Standard normal CDF | |
Standard deviation of the random variable before truncation | |
A, B, | Normalized coordinates of the subspace of the truncated PDF |
limits and x respectively | |
Karhunen–Loeve random field | |
Number of subintervals in the Latin hypercube Sampling | |
Mean value of the random field | |
Random vector created by the Latin hypercube sampling | |
, , | Total number of eigenvalues and eigenfunctions |
respectively | |
b | Correlation length |
Covariance function | |
Load factor causing failure of the body at exactly the time which ends | |
the rampload function | |
T | Time which the rampload function ends |
Symmetrization factor of the stochastic process | |
Trial load factor of step n causing failure | |
Time of failure at the generalized load factor | |
Maximum trial load factor which causes safety | |
Initial trial load factor causing failure | |
Initial trial load factor which causes safety | |
Equivalent forces of the shallow foundation | |
Obliquity angle | |
Dimensions of the total finite element mesh | |
Geostatic stresses in vertical direction and directions x and y respectively | |
Inclination of isotropic compression line for the respective normally | |
consolidated clay | |
Initial halfisize of the ellipse | |
Residual halfisize of the ellipse | |
OCR | Overconsolidation ratio |
G | Shear modulus |
Bulk modulus | |
Specific weight | |
Initial specific volume of the soil | |
Displacement vector in direction x | |
Compressibilty factor at depth = 0 | |
Compressibilty factor at maximum depth | |
Ratio of the compressibility factors measured at depth = 0 and at | |
maximum depth | |
Mean value of ratio R | |
Compressibilty factor at maximum depth when the ratio R has its | |
mean value | |
Linear distribution over depth for the compressibility factor | |
Constant distribution over depth for the compressibility factor | |
Mean value of | |
Random variable case for the critical state line inclination | |
Deterministic case for the critical state line inclination | |
Mean value of the friction angle | |
Standard deviation of the friction angle | |
Mean value of the permeability | |
Coefficient of variation of the friction angle | |
Mean value of the compressibility factor in the random field representation | |
Mean value of the critical state line inclination in the random | |
field representation | |
Mean value of the permeability in the random field representation | |
Standard deviation of the compressibility factor in the random field | |
representation | |
Standard deviation of the critical state line inclination in the random field | |
representation | |
Standard deviation of the permeability in the random field representation | |
Mean values of the results | |
Coefficient of variation of the results | |
M | Maximum values of the results |
Minimum values of the results | |
N | Total settlement force |
Horizontal displacement at failure | |
Vertical displacement at failure | |
Volumetric stress at failure | |
Von Mises stress at failure | |
Volumetric strain at failure | |
Deviatoric strain at failure | |
Percentage plastic volumetric strains at failure | |
Percentage plastic deviatoric strains at failure |
Appendix A
N | ||||||||||
- | - | - | - | - | - | - | - | |||
668.94 | 667.19 | 645.75 | 643.56 | 809.81 | 805.00 | 759.06 | 755.13 | |||
0.02 | 0.04 | 0.01 | 0.04 | 0.03 | 0.06 | 0.01 | 0.06 | |||
M | 700.00 | 721.00 | 658.00 | 700.00 | 861.00 | 882.00 | 770.00 | 833.00 | ||
651.00 | 609.00 | 637.00 | 588.00 | 777.00 | 714.00 | 756.00 | 665.00 | |||
1.08 | 1.18 | 1.03 | 1.19 | 1.11 | 1.24 | 1.02 | 1.25 | |||
N | ||||||||||
- | - | - | - | - | - | - | - | |||
1120.90 | 1121.30 | 1030.30 | 1029.90 | 1330.40 | 1331.80 | 1225.40 | 1226.80 | |||
0.04 | 0.10 | 0.01 | 0.09 | 0.05 | 0.10 | 0.02 | 0.09 | |||
M | 1239.00 | 1358.00 | 1057.00 | 1218.00 | 1491.00 | 1624.00 | 1274.00 | 1456.00 | ||
1043.00 | 931.00 | 1008.00 | 861.00 | 1232.00 | 1106.00 | 1190.00 | 1022.00 | |||
1.19 | 1.46 | 1.05 | 1.41 | 1.21 | 1.47 | 1.07 | 1.42 | |||
- | - | - | - | - | - | - | - | |||
0.0018 | 0.0019 | 0.0029 | 0.0029 | 0.0051 | 0.0051 | 0.0094 | 0.0094 | |||
0.21 | 0.25 | 0.04 | 0.11 | 0.25 | 0.26 | 0.03 | 0.04 | |||
M | 0.0026 | 0.0028 | 0.0031 | 0.0035 | 0.0075 | 0.0076 | 0.0101 | 0.0102 | ||
0.0010 | 0.0010 | 0.0027 | 0.0024 | 0.0024 | 0.0024 | 0.0087 | 0.0088 | |||
2.67 | 2.73 | 1.16 | 1.44 | 3.16 | 3.18 | 1.15 | 1.16 | |||
- | - | - | - | - | - | - | - | |||
0.0121 | 0.0122 | 0.0207 | 0.0207 | 0.0186 | 0.0187 | 0.0307 | 0.0307 | |||
0.23 | 0.25 | 0.04 | 0.07 | 0.22 | 0.25 | 0.04 | 0.08 | |||
M | 0.0173 | 0.0182 | 0.0222 | 0.0233 | 0.0263 | 0.0278 | 0.0332 | 0.0352 | ||
0.0060 | 0.0062 | 0.0188 | 0.0185 | 0.0095 | 0.0098 | 0.0278 | 0.0271 | |||
2.87 | 2.95 | 1.18 | 1.26 | 2.76 | 2.84 | 1.19 | 1.30 | |||
- | - | - | - | - | - | - | - | |||
0.0257 | 0.0257 | 0.0409 | 0.0408 | 0.0432 | 0.0431 | 0.0655 | 0.0652 | |||
0.23 | 0.25 | 0.06 | 0.09 | 0.23 | 0.25 | 0.07 | 0.10 | |||
M | 0.0370 | 0.0382 | 0.0454 | 0.0475 | 0.0617 | 0.0645 | 0.0745 | 0.0773 | ||
0.0127 | 0.0130 | 0.0358 | 0.0362 | 0.0216 | 0.0220 | 0.0555 | 0.0563 | |||
2.91 | 2.94 | 1.27 | 1.31 | 2.86 | 2.93 | 1.34 | 1.37 | |||
- | - | - | - | - | - | - | - | |||
0.0419 | 0.0421 | 0.0621 | 0.0621 | 0.0311 | 0.0312 | 0.0460 | 0.0461 | |||
0.22 | 0.25 | 0.07 | 0.11 | 0.22 | 0.25 | 0.07 | 0.11 | |||
M | 0.0586 | 0.0632 | 0.0701 | 0.0749 | 0.0435 | 0.0465 | 0.0520 | 0.0554 | ||
0.0216 | 0.0224 | 0.0525 | 0.0533 | 0.0161 | 0.0166 | 0.0390 | 0.0399 | |||
2.71 | 2.82 | 1.33 | 1.41 | 2.71 | 2.80 | 1.33 | 1.39 |
N | , | , | ||||||||
- | - | - | - | - | - | - | - | |||
693.47 | 691.40 | 674.89 | 672.91 | 694.26 | 692.14 | 675.75 | 673.60 | |||
0.02 | 0.05 | 0.01 | 0.05 | 0.02 | 0.04 | 0.01 | 0.05 | |||
M | 724.10 | 751.64 | 683.38 | 733.44 | 722.97 | 751.66 | 688.53 | 733.60 | ||
674.89 | 629.52 | 668.17 | 607.46 | 675.09 | 636.80 | 665.44 | 617.92 | |||
1.07 | 1.19 | 1.02 | 1.21 | 1.07 | 1.18 | 1.03 | 1.19 | |||
N | , | , | ||||||||
- | - | - | - | - | - | - | - | |||
798.42 | 794.09 | 746.87 | 743.03 | 798.99 | 794.52 | 747.81 | 744.16 | |||
0.03 | 0.05 | 0.02 | 0.06 | 0.03 | 0.05 | 0.02 | 0.06 | |||
M | 850.39 | 868.48 | 751.43 | 816.38 | 850.59 | 868.20 | 760.42 | 816.63 | ||
769.53 | 710.21 | 702.08 | 659.51 | 773.25 | 714.13 | 699.54 | 669.53 | |||
1.11 | 1.22 | 1.07 | 1.24 | 1.10 | 1.22 | 1.09 | 1.22 | |||
N | , | , | ||||||||
- | - | - | - | - | - | - | - | |||
1040.50 | 1035.90 | 989.56 | 988.71 | 1041.30 | 1031.20 | 989.05 | 988.57 | |||
0.03 | 0.07 | 0.01 | 0.08 | 0.03 | 0.07 | 0.01 | 0.09 | |||
M | 1079.10 | 1163.90 | 1005.90 | 1175.60 | 1099.20 | 1163.90 | 1007.10 | 1175.60 | ||
937.26 | 896.72 | 964.87 | 832.49 | 932.40 | 894.52 | 963.24 | 830.55 | |||
1.15 | 1.30 | 1.04 | 1.41 | 1.18 | 1.30 | 1.05 | 1.42 | |||
N | , | , | ||||||||
- | - | - | - | - | - | - | - | |||
1137.00 | 1158.50 | 1158.50 | 1152.90 | 1140.30 | 1143.60 | 1156.60 | 1150.90 | |||
0.06 | 0.10 | 0.02 | 0.08 | 0.06 | 0.09 | 0.02 | 0.08 | |||
M | 1262.50 | 1435.60 | 1204.30 | 1308.60 | 1265.40 | 1434.10 | 1205.60 | 1302.20 | ||
1039.70 | 1037.50 | 1120.40 | 974.57 | 1016.70 | 1001.00 | 1115.90 | 973.22 | |||
1.21 | 1.38 | 1.07 | 1.34 | 1.24 | 1.43 | 1.08 | 1.34 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0018 | 0.0018 | 0.0028 | 0.0028 | 0.0018 | 0.0018 | 0.0028 | 0.0028 | |||
0.20 | 0.24 | 0.05 | 0.12 | 0.20 | 0.24 | 0.05 | 0.12 | |||
M | 0.0024 | 0.0026 | 0.0030 | 0.0034 | 0.0024 | 0.0026 | 0.0030 | 0.0034 | ||
0.0010 | 0.0010 | 0.0026 | 0.0022 | 0.0010 | 0.0010 | 0.0026 | 0.0021 | |||
2.54 | 2.67 | 1.18 | 1.53 | 2.49 | 2.61 | 1.16 | 1.59 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0054 | 0.0054 | 0.0100 | 0.0100 | 0.0054 | 0.0054 | 0.0101 | 0.0100 | |||
0.27 | 0.27 | 0.04 | 0.05 | 0.27 | 0.27 | 0.05 | 0.05 | |||
M | 0.0081 | 0.0082 | 0.0106 | 0.0108 | 0.0083 | 0.0084 | 0.0109 | 0.0111 | ||
0.0024 | 0.0025 | 0.0088 | 0.0089 | 0.0024 | 0.0024 | 0.0088 | 0.0089 | |||
3.30 | 3.34 | 1.20 | 1.21 | 3.41 | 3.45 | 1.24 | 1.25 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0123 | 0.0123 | 0.0214 | 0.0213 | 0.0123 | 0.0123 | 0.0214 | 0.0213 | |||
0.26 | 0.28 | 0.04 | 0.07 | 0.26 | 0.28 | 0.05 | 0.07 | |||
M | 0.0180 | 0.0189 | 0.0228 | 0.0241 | 0.0182 | 0.0191 | 0.0230 | 0.0242 | ||
0.0051 | 0.0051 | 0.0189 | 0.0191 | 0.0051 | 0.0051 | 0.0188 | 0.0190 | |||
3.56 | 3.70 | 1.21 | 1.26 | 3.58 | 3.75 | 1.22 | 1.27 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0175 | 0.0177 | 0.0317 | 0.0316 | 0.0184 | 0.0186 | 0.0317 | 0.0315 | |||
0.27 | 0.27 | 0.04 | 0.08 | 0.23 | 0.23 | 0.05 | 0.07 | |||
M | 0.0267 | 0.0271 | 0.0341 | 0.0361 | 0.0263 | 0.0269 | 0.0342 | 0.0358 | ||
0.0089 | 0.0090 | 0.0286 | 0.0280 | 0.0112 | 0.0113 | 0.0286 | 0.0282 | |||
3.02 | 2.99 | 1.19 | 1.29 | 2.35 | 2.38 | 1.20 | 1.27 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0276 | 0.0276 | 0.0439 | 0.0438 | 0.0277 | 0.0276 | 0.0439 | 0.0438 | |||
0.23 | 0.25 | 0.07 | 0.10 | 0.23 | 0.25 | 0.07 | 0.10 | |||
M | 0.0398 | 0.0416 | 0.0496 | 0.0520 | 0.0400 | 0.0418 | 0.0499 | 0.0523 | ||
0.0136 | 0.0139 | 0.0382 | 0.0386 | 0.0136 | 0.0139 | 0.0380 | 0.0388 | |||
2.93 | 3.00 | 1.30 | 1.35 | 2.94 | 3.02 | 1.31 | 1.35 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0457 | 0.0456 | 0.0687 | 0.0685 | 0.0457 | 0.0456 | 0.0688 | 0.0686 | |||
0.23 | 0.25 | 0.09 | 0.11 | 0.23 | 0.25 | 0.09 | 0.11 | |||
M | 0.0655 | 0.0683 | 0.0799 | 0.0835 | 0.0656 | 0.0684 | 0.0803 | 0.0839 | ||
0.0228 | 0.0232 | 0.0546 | 0.0558 | 0.0228 | 0.0232 | 0.0544 | 0.0556 | |||
2.87 | 2.94 | 1.46 | 1.50 | 2.88 | 2.95 | 1.48 | 1.51 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0439 | 0.0439 | 0.0657 | 0.0657 | 0.0439 | 0.0438 | 0.0657 | 0.0657 | |||
0.25 | 0.28 | 0.08 | 0.12 | 0.25 | 0.28 | 0.08 | 0.12 | |||
M | 0.0628 | 0.0675 | 0.0754 | 0.0807 | 0.0626 | 0.0672 | 0.0752 | 0.0806 | ||
0.0186 | 0.0189 | 0.0547 | 0.0558 | 0.0186 | 0.0187 | 0.0544 | 0.0555 | |||
3.37 | 3.57 | 1.38 | 1.45 | 3.36 | 3.60 | 1.38 | 1.45 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
0.0305 | 0.0309 | 0.0492 | 0.0490 | 0.0321 | 0.0323 | 0.0491 | 0.0489 | |||
0.26 | 0.26 | 0.07 | 0.11 | 0.22 | 0.23 | 0.07 | 0.10 | |||
M | 0.0459 | 0.0468 | 0.0564 | 0.0601 | 0.0448 | 0.0461 | 0.0564 | 0.0589 | ||
0.0155 | 0.0159 | 0.0420 | 0.0422 | 0.0194 | 0.0196 | 0.0420 | 0.0423 | |||
2.95 | 2.95 | 1.34 | 1.42 | 2.31 | 2.35 | 1.34 | 1.39 |
N | ||||||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
652.06 | 642.80 | 658.60 | 720.63 | 711.95 | 736.82 | |||
0.07 | 0.05 | 0.03 | 0.06 | 0.06 | 0.03 | |||
M | 714.77 | 700.04 | 699.92 | 795.46 | 788.87 | 764.27 | ||
566.54 | 569.01 | 610.76 | 617.93 | 625.67 | 683.02 | |||
1.26 | 1.23 | 1.15 | 1.29 | 1.26 | 1.12 | |||
N | ||||||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
926.81 | 905.85 | 943.85 | 1069.60 | 1044.70 | 1086.30 | |||
0.11 | 0.08 | 0.06 | 0.12 | 0.09 | 0.06 | |||
M | 1104.00 | 1031.60 | 1056.80 | 1287.50 | 1213.10 | 1231.30 | ||
727.98 | 745.38 | 823.83 | 840.41 | 855.21 | 945.59 | |||
1.52 | 1.38 | 1.28 | 1.53 | 1.42 | 1.30 | |||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
0.0031 | 0.0030 | 0.0031 | 0.0121 | 0.0114 | 0.0118 | |||
0.15 | 0.12 | 0.05 | 0.10 | 0.08 | 0.06 | |||
M | 0.0040 | 0.0038 | 0.0034 | 0.0139 | 0.0127 | 0.0135 | ||
0.0022 | 0.0023 | 0.0027 | 0.0093 | 0.0098 | 0.0104 | |||
1.79 | 1.62 | 1.24 | 1.50 | 1.29 | 1.29 | |||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
0.0254 | 0.0240 | 0.0250 | 0.0374 | 0.0355 | 0.0370 | |||
0.11 | 0.08 | 0.05 | 0.12 | 0.10 | 0.05 | |||
M | 0.0301 | 0.0278 | 0.0283 | 0.0447 | 0.0423 | 0.0417 | ||
0.0191 | 0.0203 | 0.0228 | 0.0273 | 0.0299 | 0.0346 | |||
1.58 | 1.37 | 1.24 | 1.64 | 1.42 | 1.20 | |||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
0.0551 | 0.0528 | 0.0549 | 0.0889 | 0.0861 | 0.0898 | |||
0.10 | 0.07 | 0.06 | 0.08 | 0.07 | 0.06 | |||
M | 0.0647 | 0.0595 | 0.0609 | 0.0987 | 0.0955 | 0.1001 | ||
0.0426 | 0.0457 | 0.0484 | 0.0750 | 0.0730 | 0.0751 | |||
1.52 | 1.30 | 1.26 | 1.32 | 1.31 | 1.33 | |||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||
0.0845 | 0.0810 | 0.0850 | 0.0634 | 0.0611 | 0.0639 | |||
0.11 | 0.09 | 0.07 | 0.11 | 0.08 | 0.07 | |||
M | 0.1000 | 0.0931 | 0.0946 | 0.0756 | 0.0704 | 0.0712 | ||
0.0670 | 0.0680 | 0.0720 | 0.0493 | 0.0516 | 0.0533 | |||
1.49 | 1.37 | 1.31 | 1.53 | 1.36 | 1.34 |
- | - | - | - | - | - | - | - | |||
59.81 | 59.66 | 59.71 | 59.55 | 62.71 | 62.59 | 62.80 | 62.68 | |||
1.14 × 10−3 | 0.03 | 6.08 × 10−4 | 0.03 | 8.38 × 10−4 | 0.02 | 1.10 × 10−3 | 0.02 | |||
59.21 | 58.36 | 59.41 | 57.96 | 62.43 | 61.51 | 62.30 | 61.23 | |||
- | - | - | - | - | - | - | - | |||
727.14 | 727.03 | 628.98 | 628.49 | 831.76 | 831.56 | 712.12 | 712.67 | |||
0.05 | 0.11 | 0.01 | 0.10 | 0.06 | 0.11 | 0.01 | 0.09 | |||
663.98 | 595.68 | 623.40 | 516.65 | 758.50 | 686.22 | 706.17 | 585.51 | |||
- | - | - | - | - | - | - | - | |||
102.67 | 102.51 | 99.19 | 98.97 | 86.15 | 86.04 | 82.39 | 82.27 | |||
0.01 | 0.01 | 2.17 × 10−3 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | |||
101.49 | 100.96 | 98.88 | 96.99 | 85.32 | 85.38 | 81.32 | 81.22 | |||
- | - | - | - | - | - | - | - | |||
1054.80 | 1054.90 | 947.37 | 947.81 | 1104.70 | 1104.30 | 991.54 | 991.90 | |||
0.03 | 0.11 | 2.41 × 10−3 | 0.11 | 0.03 | 0.11 | 2.30 × 10−3 | 0.11 | |||
994.62 | 839.67 | 943.22 | 754.34 | 1042.10 | 888.32 | 987.30 | 796.09 | |||
- | - | - | - | - | - | - | - | |||
7.68 | 7.68 | 14.42 | 14.36 | 8.76 | 8.74 | 16.60 | 16.52 | |||
0.22 | 0.25 | 0.01 | 0.07 | 0.22 | 0.24 | 0.01 | 0.06 | |||
- | - | - | - | - | - | - | - | |||
5.50 | 5.51 | 11.20 | 11.19 | 5.90 | 5.91 | 11.93 | 11.92 | |||
0.24 | 0.26 | 4.02 × 10−3 | 0.05 | 0.24 | 0.26 | 4.26 × 10−3 | 0.04 | |||
- | - | - | - | - | - | - | - | |||
8.86 | 8.86 | 16.90 | 16.85 | 8.38 | 8.36 | 15.92 | 15.84 | |||
0.23 | 0.25 | 0.01 | 0.06 | 0.23 | 0.25 | 0.01 | 0.05 | |||
- | - | - | - | - | - | - | - | |||
10.34 | 10.36 | 19.45 | 19.42 | 10.29 | 10.32 | 19.21 | 19.21 | |||
0.22 | 0.24 | 0.01 | 0.05 | 0.22 | 0.24 | 0.01 | 0.05 | |||
- | - | - | - | - | - | - | - | |||
0.32 | 0.32 | 0.25 | 0.25 | 0.28 | 0.28 | 0.21 | 0.20 | |||
0.06 | 0.10 | 0.02 | 0.07 | 0.08 | 0.13 | 0.02 | 0.12 | |||
- | - | - | - | - | - | - | - | |||
0.12 | 0.12 | 0.08 | 0.08 | 0.10 | 0.10 | 0.07 | 0.07 | |||
0.12 | 0.15 | 0.01 | 0.08 | 0.11 | 0.17 | 0.01 | 0.12 | |||
- | - | - | - | - | - | - | - | |||
0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.02 | |||
0.06 | 0.25 | 0.03 | 0.25 | 0.08 | 0.30 | 0.02 | 0.31 | |||
- | - | - | - | - | - | - | - | |||
0.17 | 0.18 | 0.13 | 0.13 | 0.19 | 0.19 | 0.14 | 0.14 | |||
0.11 | 0.11 | 0.01 | 0.01 | 0.11 | 0.11 | 0.01 | 0.02 | |||
X | Y | Z | Probability | X | Y | Z | Probability | |||
- | 3.21 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 3.21 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 3.21 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 3.21 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
X | Y | Z | Probability | X | Y | Z | Probability | |||
- | 1.79 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 1.79 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 1.79 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 | ||
- | 1.79 | 2.21 | 3.79 | 100.00 | 1.79 | 2.21 | 3.79 | 100.00 |
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
58.77 | 58.61 | 58.53 | 58.37 | 58.76 | 58.60 | 58.51 | 58.35 | |||
1.88× 10−3 | 0.03 | 9.13× 10−4 | 0.03 | 2.21× 10−3 | 0.03 | 1.88× 10−3 | 0.03 | |||
58.55 | 57.80 | 58.29 | 57.68 | 58.35 | 57.95 | 57.98 | 57.55 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
61.97 | 61.85 | 62.46 | 62.35 | 61.96 | 61.85 | 62.45 | 62.35 | |||
1.2× 10−3 | 0.02 | 0.03 | 0.04 | 1.38× 10−3 | 0.02 | 0.03 | 0.04 | |||
61.55 | 60.80 | 59.33 | 58.85 | 61.46 | 60.03 | 60.15 | 59.96 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
651.89 | 647.68 | 553.58 | 551.63 | 645.94 | 640.38 | 553.13 | 551.47 | |||
0.06 | 0.08 | 0.29 | 0.31 | 0.08 | 0.10 | 0.29 | 0.31 | |||
506.76 | 524.91 | 7.05 | 7.12 | 487.21 | 452.06 | 7.06 | 7.12 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
585.67 | 631.72 | 660.34 | 654.98 | 554.99 | 598.62 | 658.64 | 652.31 | |||
0.13 | 0.13 | 0.01 | 0.08 | 0.30 | 0.31 | 0.01 | 0.08 | |||
432.41 | 474.54 | 650.04 | 547.02 | 3.80 | 0.19 | 649.08 | 544.82 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
104.93 | 104.73 | 102.52 | 102.28 | 105.04 | 104.83 | 102.64 | 102.39 | |||
4.43× 10−3 | 0.01 | 3.17× 10−3 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | |||
104.24 | 102.94 | 101.86 | 99.49 | 103.74 | 103.25 | 101.04 | 100.26 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
85.96 | 85.86 | 83.44 | 83.34 | 86.03 | 85.93 | 83.52 | 83.41 | |||
1.71× 10−3 | 2.30× 10−3 | 0.02 | 0.02 | 0.01 | 4.78× 10−3 | 0.02 | 0.02 | |||
85.74 | 85.43 | 82.12 | 81.71 | 85.33 | 85.35 | 81.32 | 81.38 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
1023.70 | 1019.40 | 880.24 | 877.83 | 1019.70 | 1015.50 | 880.04 | 877.72 | |||
0.02 | 0.09 | 0.23 | 0.26 | 0.03 | 0.09 | 0.23 | 0.26 | |||
980.69 | 825.87 | 83.43 | 83.99 | 935.75 | 825.01 | 82.94 | 83.48 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
1001.00 | 1032.30 | 972.18 | 970.95 | 958.26 | 977.07 | 971.70 | 969.90 | |||
0.04 | 0.13 | 3.28× 10−3 | 0.10 | 0.20 | 0.24 | 3.26× 10−3 | 0.10 | |||
943.32 | 862.25 | 965.64 | 780.05 | 233.89 | 235.83 | 965.46 | 779.38 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
7.45 | 7.45 | 13.92 | 13.87 | 7.45 | 7.45 | 13.91 | 13.87 | |||
0.22 | 0.25 | 3.87× 10−3 | 0.07 | 0.22 | 0.25 | 3.90× 10−3 | 0.07 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
8.53 | 8.52 | 15.49 | 15.40 | 8.53 | 8.52 | 15.49 | 15.40 | |||
0.22 | 0.24 | 0.19 | 0.20 | 0.22 | 0.24 | 0.19 | 0.20 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
5.32 | 5.31 | 10.02 | 9.99 | 5.29 | 5.32 | 10.02 | 9.99 | |||
0.27 | 0.28 | 0.36 | 0.37 | 0.27 | 0.28 | 0.36 | 0.37 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
5.11 | 5.35 | 11.53 | 11.51 | 4.68 | 4.88 | 11.53 | 11.49 | |||
0.35 | 0.31 | 3.19× 10−3 | 0.04 | 0.62 | 0.60 | 3.48× 10−3 | 0.04 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
8.97 | 8.97 | 17.20 | 17.15 | 8.98 | 8.97 | 17.21 | 17.16 | |||
0.23 | 0.25 | 0.01 | 0.06 | 0.23 | 0.25 | 0.01 | 0.06 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
8.38 | 8.37 | 15.30 | 15.22 | 8.39 | 8.37 | 15.32 | 15.24 | |||
0.23 | 0.25 | 0.19 | 0.20 | 0.23 | 0.25 | 0.19 | 0.20 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
10.30 | 10.30 | 18.95 | 18.91 | 10.33 | 10.29 | 18.96 | 18.92 | |||
0.25 | 0.27 | 0.21 | 0.21 | 0.24 | 0.27 | 0.21 | 0.21 | |||
, | , | |||||||||
- | - | - | - | - | - | - | - | |||
9.65 | 9.61 | 19.78 | 19.67 | 9.73 | 9.72 | 19.76 | 0.02 | |||
0.24 | 0.25 | 0.01 | 0.04 | 0.23 | 0.23 | 0.01 | 0.04 | |||
X | Y | Z | Probability | |||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
X | Y | Z | Probability | |||||||
-- | 3.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 3.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 3.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 3.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 3.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
X | Y | Z | Probability | |||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 93.75 | ||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
X | Y | Z | Probability | |||||||
-- | 1.79 | 2.21 | 3.79 | 81.25 | ||||||
1.79 | 1.79 | 3.79 | 12.50 | |||||||
1.79 | 3.21 | 3.79 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 93.75 | ||||||
1.79 | 3.21 | 3.79 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 87.50 | ||||||
1.79 | 3.21 | 3.79 | 6.25 | |||||||
0.21 | 0.21 | 0.21 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 81.25 | ||||||
1.79 | 2.79 | 3.79 | 12.50 | |||||||
0.21 | 0.21 | 0.21 | 6.25 | |||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
-- | 1.79 | 2.21 | 3.79 | 100.00 |
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
57.92 | 57.68 | 58.29 | 60.38 | 59.67 | 61.37 | |||||
0.04 | 0.03 | 0.01 | 0.11 | 0.10 | 0.05 | |||||
56.95 | 56.85 | 57.67 | 48.89 | 49.76 | 57.69 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
512.39 | 497.14 | 573.23 | 621.49 | 607.80 | 634.77 | |||||
0.32 | 0.31 | 0.04 | 0.11 | 0.08 | 0.04 | |||||
6.84 | 6.92 | 509.42 | 493.61 | 519.47 | 566.88 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
103.22 | 104.88 | 103.65 | 101.89 | 103.44 | 92.96 | |||||
0.03 | 0.03 | 0.02 | 0.29 | 0.28 | 0.22 | |||||
98.04 | 96.97 | 99.64 | 80.98 | 78.60 | 81.08 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
842.46 | 839.72 | 919.65 | 946.08 | 942.54 | 957.25 | |||||
0.27 | 0.26 | 0.04 | 0.14 | 0.10 | 0.04 | |||||
84.38 | 84.52 | 821.08 | 710.68 | 753.96 | 858.87 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
14.55 | 13.17 | 14.14 | 14.04 | 12.58 | 15.37 | |||||
0.21 | 0.24 | 0.08 | 0.28 | 0.30 | 0.17 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
9.97 | 8.74 | 11.06 | 12.09 | 10.79 | 11.65 | |||||
0.57 | 0.51 | 0.09 | 0.22 | 0.29 | 0.09 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
18.25 | 16.63 | 17.67 | 15.88 | 14.41 | 16.25 | |||||
0.20 | 0.23 | 0.07 | 0.27 | 0.28 | 0.16 | |||||
b = 2 m | b = 4 m | b = 8 m | b = 2 m | b = 4 m | b = 8 m | |||||
19.99 | 17.83 | 20.30 | 20.77 | 18.95 | 20.10 | |||||
0.24 | 0.27 | 0.07 | 0.20 | 0.24 | 0.07 | |||||
X | Y | Z | Probability | |||||||
b = 2 m | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
b = 4 m | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
b = 8 m | 3.21 | 2.21 | 3.79 | 100.00 | ||||||
X | Y | Z | Probability | |||||||
b = 2 m | 3.79 | 2.21 | 3.79 | 43.75 | ||||||
4.79 | 2.21 | 0.21 | 18.75 | |||||||
4.79 | 0.21 | 0.21 | 12.50 | |||||||
3.21 | 1.79 | 3.79 | 12.50 | |||||||
3.21 | 2.21 | 3.79 | 12.50 | |||||||
b = 4 m | 3.79 | 2.21 | 3.79 | 56.25 | ||||||
4.79 | 0.21 | 0.21 | 18.75 | |||||||
3.21 | 1.79 | 3.79 | 12.50 | |||||||
3.21 | 2.21 | 3.79 | 12.50 | |||||||
b = 8 m | 3.79 | 2.21 | 3.79 | 75 | ||||||
3.21 | 2.21 | 3.79 | 12.5 | |||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
4.79 | 2.21 | 0.21 | 6.25 | |||||||
X | Y | Z | Probability | |||||||
b = 2 m | 1.79 | 2.21 | 3.79 | 87.50 | ||||||
1.79 | 1.79 | 3.79 | 6.25 | |||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
b = 4 m | 1.79 | 2.21 | 3.79 | 68.75 | ||||||
1.79 | 1.79 | 3.79 | 25.00 | |||||||
4.79 | 0.21 | 0.21 | 6.25 | |||||||
b = 8 m | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
X | Y | Z | Probability | |||||||
b = 2 m | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
b = 4 m | 1.79 | 2.21 | 3.79 | 100.00 | ||||||
b = 8 m | 1.79 | 2.21 | 3.79 | 100.00 |
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Bisection Algorithm | Bisection Algorithm | Bisection Algorithm | Proposed Algorithm | Absolute Percentage Difference | |
---|---|---|---|---|---|
Initial value of failure stress (kPa) | 5000 | 5000 | 5000 | 5000 | |
Initial value of safety stress (kPa) | 1000 | 2000 | 3000 | - | |
Convergence tolerance | 0.01 | 0.01 | 0.01 | 0.01 | |
Number of trials for convergence | 6 | 5 | 5 | 3 | |
Displacement of failure at convergence (m) | 0.03054 | 0.03072 | 0.03054 | 0.03072 | 0.58 |
Load of failure at convergence (kPa) | 4484.38 | 4478.52 | 4484.38 | 4477.50 | 0.15 |
Computational time (mins) | 750 | 833 | 663 | 302 | 54.45 |
Kpa | Kpa | OCR | |||||
---|---|---|---|---|---|---|---|
10 | 1600 | 400 | 4 | 1.627 | 0.75 | 0.05 | 20 |
c | Abbreviation | |
---|---|---|
Constant | Deterministic | -- |
Constant | Random | -- |
Linear | Deterministic | -- |
Linear | Random | -- |
c | k | Abbreviation | |
---|---|---|---|
Constant | Deterministic | Deterministic | --- |
Constant | Random | Deterministic | --- |
Linear | Deterministic | Deterministic | --- |
Linear | Random | Deterministic | --- |
Constant | Deterministic | Random | --- |
Constant | Random | Random | --- |
Linear | Deterministic | Random | --- |
Linear | Random | Random | --- |
Random Field, b = 2 | Random Field, b = 2 | Random Field, b = 2 | --- |
Random Field, b = 4 | Random Field, b = 4 | Random Field, b = 4 | --- |
Random Field, b = 8 | Random Field, b = 8 | Random Field, b = 8 | --- |
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Savvides, A.-A.; Papadrakakis, M. Probabilistic Failure Estimation of an Oblique Loaded Footing Settlement on Cohesive Geomaterials with a Modified Cam Clay Material Yield Function. Geotechnics 2021, 1, 347-384. https://doi.org/10.3390/geotechnics1020017
Savvides A-A, Papadrakakis M. Probabilistic Failure Estimation of an Oblique Loaded Footing Settlement on Cohesive Geomaterials with a Modified Cam Clay Material Yield Function. Geotechnics. 2021; 1(2):347-384. https://doi.org/10.3390/geotechnics1020017
Chicago/Turabian StyleSavvides, Ambrosios-Antonios, and Manolis Papadrakakis. 2021. "Probabilistic Failure Estimation of an Oblique Loaded Footing Settlement on Cohesive Geomaterials with a Modified Cam Clay Material Yield Function" Geotechnics 1, no. 2: 347-384. https://doi.org/10.3390/geotechnics1020017
APA StyleSavvides, A. -A., & Papadrakakis, M. (2021). Probabilistic Failure Estimation of an Oblique Loaded Footing Settlement on Cohesive Geomaterials with a Modified Cam Clay Material Yield Function. Geotechnics, 1(2), 347-384. https://doi.org/10.3390/geotechnics1020017