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

Hubert Szabowicz; Marek Kawa; Wojciech Puła

doi:10.2478/sgem-2025-0002

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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

Studia Geotechnica et Mechanica

Volume 47 (2025): Issue 1 (March 2025)

By: Hubert Szabowicz, Marek Kawa and Wojciech Puła

Open Access

|Feb 2025

Figures & Tables

A graphical representation of the mesh adaptive procedure.

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=0.5 m.

Upper (red lines) and lower (blue lines) bound of bearing capacities of a shallow foundation D=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 *θ_z*=0.25 m.

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 mean value, standard deviation and coefficient of variation as a function of the horizontal scales of fluctuations *θ_x* for all considered foundation levels and vertical scales of fluctuations *θ_z*.

Table 1:

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
Scales of fluctuations		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
θx =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0 m	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
θx =16.0 m	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

Table 2:

Scales of fluctuations		Lower-bound estimation			Upper-bound estimation
Scales of fluctuations		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
θx =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0 m	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
θx =16.0 m	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

Table 3:

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
Scales of fluctuations		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
θx =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0 m	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
θx =16.0 m	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

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

Table 4:

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 [-]
θ_x =1.0 m	θz =0.25 m	196.99	206.12	191.09	1.28	1.30	1.32
θ_x =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0m	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
θx =16.0 m	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

Table 5:

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
θx =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0 m	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
θx =16.0 m	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

Table 6:

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
θx =1.0 m	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
θx =2.0 m	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
θx =4.0 m	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
θx =8.0 m	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
θx =16.0 m	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

References

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DOI: https://doi.org/10.2478/sgem-2025-0002 | Journal eISSN: 2083-831X (formerly 0137-124X) | Journal ISSN: 0137-6365

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Language: English

Page range: 17 - 35

Submitted on: Jul 24, 2024

Accepted on: Dec 12, 2024

Published on: Feb 27, 2025

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Keywords:

Strip Fundation,

Random field,

Random Finite Element Limit Analysis,

Related subjects:

Physics,

Mechanics and fluid dynamics

© 2025 Hubert Szabowicz, Marek Kawa, Wojciech Puła, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 47 (2025): Issue 1 (March 2025)

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

Figures & Tables

Figure 1:

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Figure 5:

Figure 6:

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Figure 9:

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Table 1:

Table 2:

Table 3:

Figure 11:

Table 4:

Table 5:

Table 6:

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