On determining the undrained bearing capacity coefficients of variation for foundations embedded on spatially variable soil

Marcin Chwała

Open Access

On determining the undrained bearing capacity coefficients of variation for foundations embedded on spatially variable soil

Marcin Chwała

| Jun 30, 2020

Studia Geotechnica et Mechanica

Volume 42 (2020): Issue 2 (June 2020)

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Published Online: Jun 30, 2020

Page range: 125 - 136

Received: Dec 27, 2019

Accepted: Feb 19, 2020

DOI: https://doi.org/10.2478/sgem-2019-0037

Keywords
Spatial averaging, random fields, random bearing capacity, fluctuation scale, failure mechanism

© 2020 Marcin Chwała, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Failure geometry for a rough foundation base in the case of rectangular foundation. More details on failure geometry can be found in Chwała (2019a).

Value of bearing capacity during simulation within optimization procedure

Flow chart of the numerical algorithm. Detailed descriptions of the steps are in the text.

Comparison of bearing capacity variation coefficients obtained by the method proposed in this study (constant covariance matrix) with standard algorithm (individual covariance matrix, Chwała (2019a)). The relative differences between both approaches are below 5%, which is sufficient for the purpose of this study. A detailed description is in the text.

Mean value, standard deviation and variation coefficient of bearing capacity as a function of simulation number N. The square foundation is assumed with geometry parameters and fluctuation scales detailed in the figure.

Graph for reading Dp value for ratios θv ⁄ b ∈ [2.0,1.0] and θv ⁄ b∈ [0.25,0.125]. Note that the colour in the legend indicates θv ⁄ b and the line type (coloured on black in the legend) indicates θv ⁄ b.

Graph for reading Dp value for ratios θv ⁄ b ∈ [1.0,0.25]. Note that the colour in the legend indicates θv ⁄ b and the line type (coloured on black in the legend) indicates θv ⁄ b.

Graph for reading Dp value for an infinite horizontal fluctuation scale.

Illustration of example scenarios. Note that the colour in the legend indicates θv ⁄ b and the line type (coloured on black in the legend) indicates θv ⁄ b.

Exemplary usage of the graphs proposed in this study. Detailed descriptions are in the text.

No.	Scenario description					COV_p read from graphs [-]	COV_p determined by numerical analyses [-]	Difference [%]

	a [m]	b [m]	θ_v [m]	θ_h [m]	COV_{c_u} [-]
1	2.0	2.0	0.6	12.0	0.6	0.41	0.435	−6.1%
2	10.0	1.5	1.0	5.0	0.4	0.226	0.210	+7.1%
3	20.0	0.9	0.8	3.0	1.0	0.29	0.311	−7.2%
4	25.0	1.8	1.2	1.2	0.5	0.057	0.052	+8.7%
5	2.0	1.0	1.5	2.0	0.7	0.451	0.458	−1.5%
6	25.0	3.0	0.5	10.0	1.0	0.29	0.280	+3.4%
7	3.0	3.0	0.40	∞	0.24	0.13	0.130	0.0%
8	20.0	1.0	0.47	∞	0.51	0.40	0.403	−0.7%

Coefficients from Eq. (A.2)–Eq. (A.5) for rough and smooth foundation base. Note that the undrained shear strengths ci are defined individually for each dissipation region (for more details see Chwała, 2019a).

Coeff.	Expression
m₁	c₁ cot β₂ + 2c₂₁(α₂ + β₂) + c₂ cot α₂
m₂	c₆ cot α₂ + 2c₂₄(α₂ + β₂) + c₅ cot β₂
m₃	c₈ cot α₂ + 2c₂₃(α₂ + β₂) + c₇ cot β₂
m₄	c₃ cot β₃ + 2c₂₂(α₃ + β₃) + c₄ cot α₃
m₅	c₁₀ cot α₃ + 2c₂₆(α₃ + β₃) + c₉ cot β₃
m₆	c₁₂ cot α₃ + 2c₂₅(α₃ + β₃) + c₁₁ cot β₃
m₇	c₁₆ cot α₁ + 2c₂₈(α₁ + β₁) + c₁₄ cot β₁
m₈	c₁₅ cot α₁ + 2c₂₇(α₁ + β₁) + c₁₃ cot β₁
m₉	c₂₀ cot α₄ + 2c₃₀(α₄ + β₄) + c₁₉ cot β₄
m₁₀	c₁₈ cot α₄ + 2c₂₉(α₄ + β₄) + c₁₇ cot β₄
n₁	$\sqrt{1 + \frac{b_{2}^{2}}{d_{1}^{2} {(sin β_{2})}^{2}}}$ \sqrt {1 + {{b_2^2 } \over {d_1^2 \left( {\sin \beta _2 } \right)^2 }}}
n₂	$\sqrt{1 + \frac{b_{2}^{2}}{d_{2}^{2} {(sin β_{2})}^{2}}}$ \sqrt {1 + {{b_2^2 } \over {d_2^2 \left( {\sin \beta _2 } \right)^2 }}}
n₃	$\sqrt{1 + \frac{b_{1}^{2}}{d_{1}^{2} {(sin β_{3})}^{2}}}$ \sqrt {1 + {{b_1^2 } \over {d_1^2 \left( {\sin \beta _3 } \right)^2 }}}
n₄	$\sqrt{1 + \frac{b_{1}^{2}}{d_{2}^{2} {(sin β_{3})}^{2}}}$ \sqrt {1 + {{b_1^2 } \over {d_2^2 \left( {\sin \beta _3 } \right)^2 }}}
n₅	$\sqrt{1 + \frac{d_{1}^{2}}{b_{1}^{2} {(sin β_{1})}^{2}}}$ \sqrt {1 + {{d_1^2 } \over {b_1^2 \left( {\sin \beta _1 } \right)^2 }}}
n₆	$\sqrt{1 + \frac{d_{1}^{2}}{b_{2}^{2} {(sin β_{1})}^{2}}}$ \sqrt {1 + {{d_1^2 } \over {b_2^2 \left( {\sin \beta _1 } \right)^2 }}}
n₇	$\sqrt{1 + \frac{d_{2}^{2}}{b_{1}^{2} {(sin β_{4})}^{2}}}$ \sqrt {1 + {{d_2^2 } \over {b_1^2 \left( {\sin \beta _4 } \right)^2 }}}
n₈	$\sqrt{1 + \frac{d_{2}^{2}}{b_{2}^{2} {(sin β_{4})}^{2}}}$ \sqrt {1 + {{d_2^2 } \over {b_2^2 \left( {\sin \beta _4 } \right)^2 }}}

Comparison of the results obtained in this study with those obtained by Simoes et al. (2014) by random finite limit analysis.

Scenario	Method	μ_{N_c} [-]	σ_{N_c} [-]	COV_{N_c} [-]
COV_{c_u}=0.5	RFLA (Simoes et al. 2014)	4.77	1.68	0.35
θ=8.0 m	This study	5.19	2.14	0.41
COV_{c_u}=1.0	RFLA (Simoes et al. 2014)	3.72	1.23	0.33
θ=2.0 m	This study	4.51	1.77	0.39

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On determining the undrained bearing capacity coefficients of variation for foundations embedded on spatially variable soil

Article Category: Research Article

Published Online: Jun 30, 2020

Page range: 125 - 136

Received: Dec 27, 2019

Accepted: Feb 19, 2020

DOI: https://doi.org/10.2478/sgem-2019-0037

Keywords
Spatial averaging, random fields, random bearing capacity, fluctuation scale, failure mechanism

© 2020 Marcin Chwała, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Exemplary usage of the graphs proposed in this study. Detailed descriptions are in the text.

Coefficients from Eq. (A.2)–Eq. (A.5) for rough and smooth foundation base. Note that the undrained shear strengths ci are defined individually for each dissipation region (for more details see Chwała, 2019a).

Comparison of the results obtained in this study with those obtained by Simoes et al. (2014) by random finite limit analysis.

On determining the undrained bearing capacity coefficients of variation for foundations embedded on spatially variable soil

Article Category: Research Article

Published Online: Jun 30, 2020

Page range: 125 - 136

Received: Dec 27, 2019

Accepted: Feb 19, 2020

DOI: https://doi.org/10.2478/sgem-2019-0037

KeywordsSpatial averaging, random fields, random bearing capacity, fluctuation scale, failure mechanism

© 2020 Marcin Chwała, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Exemplary usage of the graphs proposed in this study. Detailed descriptions are in the text.

Coefficients from Eq. (A.2)–Eq. (A.5) for rough and smooth foundation base. Note that the undrained shear strengths ci are defined individually for each dissipation region (for more details see Chwała, 2019a).

Comparison of the results obtained in this study with those obtained by Simoes et al. (2014) by random finite limit analysis.

Keywords
Spatial averaging, random fields, random bearing capacity, fluctuation scale, failure mechanism