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Efficient and conservative estimation reliability analysis of strip footing on spatially variable c - ϕ soil using random finite element limit analysis

Hubert Szabowicz

Hubert Szabowicz

Faculty of Civil Engineering, Wroclaw University of Science and TechnologyWroclaw, Poland
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Szabowicz, Hubert
, 
Marek Kawa

Marek Kawa

Faculty of Civil Engineering, Wroclaw University of Science and TechnologyWroclaw, Poland
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Kawa, Marek
 y 
Wojciech Puła

Wojciech Puła

Faculty of Civil Engineering, Wroclaw University of Science and TechnologyWroclaw, Poland
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Puła, Wojciech
  
27 feb 2025
Efficient and conservative estimation reliability analysis of strip footing on spatially variable c - ϕ soil using random finite element limit analysis's Cover Image
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Categoría del artículo: Original Study
Publicado en línea: 27 feb 2025
Páginas: 17 - 35
Recibido: 24 jul 2024
Aceptado: 12 dic 2024
DOI: https://doi.org/10.2478/sgem-2025-0002
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© 2025 Hubert Szabowicz et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1:

A graphical representation of the mesh adaptive procedure.
A graphical representation of the mesh adaptive procedure.

Figure 2:

Model geometry with boundary condition.
Model geometry with boundary condition.

Figure 3:

Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=0.0 m.
Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=0.0 m.

Figure 4:

Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=0.5 m.
Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=0.5 m.

Figure 5:

Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=1.0 m.
Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=1.0 m.

Figure 6:

Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=0.25 m.
Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=0.25 m.

Figure 7:

Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=0.5 m.
Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=0.5 m.

Figure 8:

Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=1.0 m.
Diagrams of mean value, standard deviation and coefficient of variation of bearing capacities of a shallow foundation as a function of the step of adaptive mesh refinement procedure for vertical scale of fluctuation θz=1.0 m.

Figure 9:

Diagrams of the relative difference in mean, standard deviation and coefficient of variation between the upper and lower bound estimations as a function of following mesh adaptation procedure steps for all considered foundation levels.
Diagrams of the relative difference in mean, standard deviation and coefficient of variation between the upper and lower bound estimations as a function of following mesh adaptation procedure steps for all considered foundation levels.

Figure 10:

Diagrams of mean value, standard deviation and coefficient of variation as a function of the horizontal scales of fluctuations θx for all considered foundation levels and vertical scales of fluctuations θz.
Diagrams of mean value, standard deviation and coefficient of variation as a function of the horizontal scales of fluctuations θx for all considered foundation levels and vertical scales of fluctuations θz.

Figure 11:

Diagrams of allowable loads and safety factors as a function of the horizontal scales of fluctuations θx with a constant vertical scale of fluctuation for all considered foundation levels.
Diagrams of allowable loads and safety factors as a function of the horizontal scales of fluctuations θx with a constant vertical scale of fluctuation for all considered foundation levels.

Allowable loads for β=3_8 obtained for lower LB, upper UB and mixed approach estimated distributions – D = 1_0 mD = 1_0 m_

Scales of fluctuations Design value Safety factor
Horizontal Vertical LB [kPa] UB [kPa] MIXED [kPa] LB [-] UB [-] MIXED [-]
θx =1.0 m θz =0.25 m 416.52 454.87 414.07 1.14 1.14 1.15
θz =0.5 m 405.84 432.65 403.93 1.20 1.19 1.20
θz =1.0 m 388.64 407.56 386.73 1.26 1.26 1.27
θx =2.0 m θz =0.25 m 405.40 441.97 402.59 1.18 1.17 1.19
θz =0.5 m 389.96 413.64 387.31 1.25 1.24 1.26
θz =1.0 m 366.12 381.61 363.75 1.35 1.34 1.36
θx =4.0 m θz =0.25 m 390.72 425.73 387.11 1.23 1.22 1.24
θz =0.5 m 370.82 393.48 368.37 1.32 1.31 1.33
θz =1.0 m 344.08 357.64 341.39 1.44 1.44 1.45
θx =8.0 m θz =0.25 m 379.37 413.29 375.19 1.27 1.26 1.28
θz =0.5 m 358.78 379.78 355.39 1.38 1.37 1.39
θz =1.0 m 323.79 336.16 320.73 1.55 1.54 1.56
θx =16.0 m θz =0.25 m 374.40 406.94 369.61 1.29 1.28 1.31
θz =0.5 m 347.94 367.66 344.07 1.43 1.42 1.45
θz =1.0 m 316.59 328.09 313.23 1.60 1.60 1.62

Allowable loads for β=3_8 obtained for lower LB, upper UB and mixed approach estimated distributions – D=0_0 m_

Scales of fluctuations Design value Safety Factor
Horizontal Vertical LB [kPa] UB [kPa] MIXED [kPa] LB [-] UB [-] MIXED [-]
θx =1.0 m θz =0.25 m 196.99 206.12 191.09 1.28 1.30 1.32
θz =0.5 m 181.53 186.61 176.61 1.40 1.42 1.44
θz =1.0 m 168.33 171.25 165.75 1.53 1.54 1.56
θx =2.0 m θz =0.25 m 185.03 192.94 178.46 1.37 1.39 1.42
θz =0.5 m 167.21 171.17 161.95 1.53 1.55 1.58
θz =1.0 m 150.77 153.06 148.29 1.72 1.73 1.75
θx =4.0 m θz =0.25 m 176.14 183.70 169.00 1.45 1.47 1.51
θz =0.5 m 156.28 160.01 151.00 1.66 1.68 1.71
θz =1.0 m 140.51 142.40 137.87 1.88 1.89 1.91
θx =8.0m θz =0.25 m 171.18 179.21 163.95 1.51 1.54 1.58
θz =0.5 m 153.78 157.81 148.49 1.71 1.74 1.78
θz =1.0 m 132.59 134.16 129.61 2.03 2.05 2.08
θx =16.0 m θz =0.25 m 170.89 178.54 162.62 1.52 1.55 1.60
θz =0.5 m 150.67 155.24 145.06 1.77 1.79 1.84
θz =1.0 m 133.86 135.50 130.80 2.05 2.06 2.10

Allowable loads for β=3_8 obtained for lower LB, upper UB and mixed approach estimated distributions – D = 0_5 mD = 0_5 m_

Scales of fluctuations Design value Safety factor
Horizontal Vertical LB [kPa] UB [kPa] MIXED [kPa] LB [-] UB [-] MIXED [-]
θx =1.0 m θz =0.25 m 318.66 341.82 315.19 1.17 1.17 1.19
θz =0.5 m 304.96 319.53 301.46 1.25 1.25 1.26
θz =1.0 m 289.25 299.34 285.95 1.33 1.33 1.34
θx =2.0 m θz =0.25 m 307.40 328.85 303.02 1.22 1.22 1.24
θz =0.5 m 289.56 302.10 285.45 1.32 1.32 1.34
θz =1.0 m 267.29 275.22 263.59 1.44 1.44 1.46
θx =4.0 m θz =0.25 m 294.26 314.12 288.81 1.28 1.28 1.31
θz =0.5 m 273.07 284.18 268.20 1.41 1.41 1.43
θz =1.0 m 247.97 254.68 244.02 1.56 1.57 1.59
θx =8.0 m θz =0.25 m 285.32 304.16 279.22 1.33 1.33 1.36
θz =0.5 m 263.68 274.16 258.67 1.47 1.48 1.50
θz =1.0 m 231.18 237.32 227.17 1.70 1.70 1.73
θx =16.0 m θz =0.25 m 281.51 299.53 275.09 1.35 1.36 1.38
θz =0.5 m 256.46 266.60 251.48 1.52 1.53 1.55
θz =1.0 m 228.21 234.00 224.17 1.74 1.75 1.77

The mean values, standard deviations and coefficients of variation for the lower and upper bound of bearing capacity estimated based on results from the last (fourth) step of the adaptive mesh procedure – D=0_5m_

Scales of fluctuations Lower-bound estimation Upper-bound estimation
Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%) Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%)
θx =1.0 m θz =0.25 m 374.15 15.73 4.20 401.09 16.79 4.19
θz =0.5 m 379.86 21.81 5.74 398.32 22.95 5.76
θz =1.0 m 383.55 28.25 7.36 397.40 29.39 7.39
θx =2.0 m θz =0.25 m 375.96 19.79 5.26 402.26 21.20 5.27
θz =0.5 m 381.39 27.42 7.19 398.58 28.83 7.23
θz =1.0 m 384.83 36.53 9.49 397.09 37.92 9.55
θx =4.0 m θz =0.25 m 376.99 24.40 6.47 403.00 26.23 6.51
θz =0.5 m 384.16 34.17 8.90 400.92 35.96 8.97
θz =1.0 m 387.88 45.13 11.64 399.40 46.74 11.70
θx =8.0 m θz =0.25 m 380.13 28.46 7.49 405.97 30.59 7.53
θz =0.5 m 388.19 39.09 10.07 404.65 41.01 10.14
θz =1.0 m 393.19 54.23 13.79 404.47 56.00 13.85
θx =16.0 m θz =0.25 m 380.82 30.02 7.88 406.25 32.29 7.95
θz =0.5 m 390.91 42.87 10.97 407.12 44.85 11.02
θz =1.0 m 397.74 57.36 14.42 408.76 59.19 14.48

The mean values, standard deviations and coefficients of variation for the lower and upper bound of bearing capacity estimated based on results from the last (fourth) step of the adaptive mesh procedure – D=0_0 m_

Scales of fluctuations Lower-bound estimation Upper-bound estimation
Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%) Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%)
θx =1.0 m θz =0.25m 252.11 16.25 6.44 267.60 18.24 6.82
θz =0.5 m 254.02 22.25 8.76 264.53 24.05 9.09
θz =1.0 m 258.28 28.77 11.14 264.19 29.80 11.28
θx =2.0 m θz =0.25 m 253.51 20.82 8.21 268.67 23.19 8.63
θz =0.5 m 255.50 28.19 11.03 265.44 30.29 11.41
θz =1.0 m 259.00 36.39 14.05 264.32 37.49 14.18
θx =4.0 m θz =0.25 m 255.01 24.58 9.64 270.63 27.30 10.09
θz =0.5 m 258.76 33.90 13.10 268.74 36.19 13.47
θz =1.0 m 263.80 43.09 16.34 269.03 44.39 16.50
θx =8.0 m θz =0.25 m 258.78 27.83 10.75 275.21 30.71 11.16
θz =0.5 m 263.71 36.93 14.01 274.16 39.31 14.34
θz =1.0 m 269.69 49.62 18.40 275.12 51.20 18.61
θx =16.0 m θz =0.25 m 260.23 28.48 10.94 277.44 31.81 11.46
θz =0.5 m 254.02 39.49 14.81 278.15 42.09 15.13
θz =1.0 m 274.18 50.94 18.58 279.80 52.57 18.79

The mean values, standard deviations and coefficients of variation for the lower and upper bound of bearing capacity are estimated based on results from the last (fourth) step of the adaptive mesh procedure – D=1_0 m_

Scales of fluctuations Lower-bound estimation Upper-bound estimation
Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%) Mean value (kPa) Standard deviation (kPa) Coefficient of variation (%)
θx =1.0 m θz =0.25 m 476.12 16.68 3.50 517.23 17.42 3.37
θz =0.5 m 485.51 22.77 4.69 514.61 23.37 4.54
θz =1.0 m 491.12 30.03 6.11 512.43 30.66 5.98
θx =2.0 m θz =0.25 m 478.43 20.74 4.34 518.28 21.61 4.17
θz =0.5 m 487.44 28.43 5.83 514.32 29.29 5.69
θz =1.0 m 493.39 38.40 7.78 511.90 39.23 7.66
θx =4.0 m θz =0.25 m 478.85 25.47 5.32 518.15 26.62 5.14
θz =0.5 m 490.31 35.75 7.29 516.22 36.59 7.09
θz =1.0 m 496.35 47.37 9.54 513.48 48.38 9.42
θx =8.0 m θz =0.25 m 481.79 30.08 6.24 520.81 31.47 6.04
θz =0.5 m 494.37 41.32 8.36 519.78 42.54 8.18
θz =1.0 m 501.01 56.89 11.35 517.61 58.12 11.23
θx =16.0 m θz =0.25 m 482.99 32.12 6.65 521.34 33.73 6.47
θz =0.5 m 497.53 46.35 9.32 522.34 47.79 9.15
θz =1.0 m 507.52 62.26 12.27 523.68 63.66 12.16
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