<p>This paper aims to conduct a reliability analysis methodology for the serviceability limit state based on the consolidation settlement of a shallow footing resting on soft clayey soil. In this study, uncertainty quantification is adopted to find the appropriate characteristic distribution for the consolidation parameters, such as the initial void ratio, compression index, and saturated density. The three parameters are mutually dependent; hence, a best fit copula model is adopted to predict the dependence structure combining all these parameters. Monte Carlo simulation based probabilistic analysis is performed based on a joint probability distribution model constructed from the optimal copula function and characteristic distribution of the consolidation parameters. The realistic application of this methodology is presented to obtain the settlement under a shallow footing. Consequently, the probability of exceedance of a settlement (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>) corresponding to the permissible limit as prescribed by the Indian Standard code is obtained. A comparative study between the reliability analysis considering and not considering the dependence structure within the simulated data is presented. It is concluded that the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> considering the dependence structure exceeds the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> without considering the dependence structure. A discussion on the impact of dependence structure on varying loading intensity to find <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> is elucidated in turn. It is observed that upto a loading intensity of 60 kPa, the effect of dependence structure does not have any considerable effect on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>. Beyond the loading intensity of 60 kPa, the copula approach gives a higher <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> value and beyond 100 kPa, the increase in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40999_2025_1075_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> is very minimal.</p>

Reliability Analysis for Consolidation Settlement of a Shallow Footing Based on Copula Model

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发表日期：

2025年2月15日

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Samayika Senapati,
Ashim Kanti Dey,
Subhrajit Dutta

摘要

This paper aims to conduct a reliability analysis methodology for the serviceability limit state based on the consolidation settlement of a shallow footing resting on soft clayey soil. In this study, uncertainty quantification is adopted to find the appropriate characteristic distribution for the consolidation parameters, such as the initial void ratio, compression index, and saturated density. The three parameters are mutually dependent; hence, a best fit copula model is adopted to predict the dependence structure combining all these parameters. Monte Carlo simulation based probabilistic analysis is performed based on a joint probability distribution model constructed from the optimal copula function and characteristic distribution of the consolidation parameters. The realistic application of this methodology is presented to obtain the settlement under a shallow footing. Consequently, the probability of exceedance of a settlement ( $P_{f}$ $P_{f}$ ) corresponding to the permissible limit as prescribed by the Indian Standard code is obtained. A comparative study between the reliability analysis considering and not considering the dependence structure within the simulated data is presented. It is concluded that the $P_{f}$ $P_{f}$ considering the dependence structure exceeds the $P_{f}$ $P_{f}$ without considering the dependence structure. A discussion on the impact of dependence structure on varying loading intensity to find $P_{f}$ $P_{f}$ is elucidated in turn. It is observed that upto a loading intensity of 60 kPa, the effect of dependence structure does not have any considerable effect on $P_{f}$ $P_{f}$ . Beyond the loading intensity of 60 kPa, the copula approach gives a higher $P_{f}$ $P_{f}$ value and beyond 100 kPa, the increase in $P_{f}$ $P_{f}$ is very minimal.