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Investigation of the soil deformation around laterally loaded deep foundations with large diameters

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Abstract

In projects such as highway bridges and offshore wind farms, understanding the kinematics of soil layers and their displacement is important for accurately predicting the behavior of caissons and monopiles under lateral loads. To better predict the distribution of soil deformation around the foundations, this paper presents an extensive usage of the energy-based variational method in laterally loaded deep foundations (caissons or monopiles) with very large diameters. By adding an assumption of soil deformation in multilayered soils, the displacement distribution around the foundations can be better described. The responses of deep foundations were obtained by minimizing the potential energy and virtual work of the foundation-soil system. After that, static lateral loading tests on two caissons with the same embedment depth of 36 m and the same diameter of 6 m were performed, and the displacement field in soils and the deflection profiles of the caissons were measured. Results show that the lateral displacement and the decay distribution fields calculated from the theory in this paper agree with the measured data from the static loading tests and the corresponding finite differential method (FDM) results. At last, a series of case studies performed by MATLAB and FDM analyses were also conducted to study the influence of uz on the response of foundations, as well as the influence of foundation parameters and soil modulus on the value of ϕz.

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Data availability

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The study presented herein is supported by the National Natural Science Foundation of China (NO.52201324, 52078128), and the Natural Science Foundation of the Jiangsu Higher Education Institution of China (22KJB560015). The authors are grateful for their support.

Author information

Authors and Affiliations

  1. School of Civil Engineering and Architecture, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Xiaojuan Li & Mingxing Zhu

  2. Key Laboratory for RC and PRC Structure of Education Ministry, School of Civil Engineering, Southeast University, Nanjing, 210096, Jiangsu, China

    Guoliang Dai & Wenbo Zhu

  3. School of Architecture and Civil Engineering, Jiangsu Open University, Nanjing, 210036, China

    Fan Zhang

Authors
  1. Xiaojuan Li
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  2. Guoliang Dai
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  3. Mingxing Zhu
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  4. Wenbo Zhu
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  5. Fan Zhang
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Contributions

XL: Conceptualization, Methodology, Software, Results analysis, Writing original draft. GD: Software, Validation, Funding acquisition, Project administration. MZ: Conceptualization, Writing review and editing, Resources, Formal analysis. WZ: Software, Results analysis, Making scientific figures. FZ: Investigation, Making scientific figures, Validation.

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Correspondence to Xiaojuan Li.

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Supplementary file1 (DOCX 59 KB)

Appendix

Appendix

The matrixes of Kϕr, Kϕθ, Kϕz, Am, Bm, Cm, P, E, Q, G, M and N are given below:

$$\begin{aligned} {\mathbf{K}}_{{\phi _{{\mathbf{r}}} }} & = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ 0 \hfill & . \hfill & . \hfill & 1 \hfill & 0 \hfill & . \hfill \\ 0 \hfill & . \hfill & . \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{1} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{2} }}{{rp}}} \right)^{2} } \right)} \hfill & {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{1} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{2} }}{{rp}}} \right)^{2} } \right)} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{1} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{2} }}{{rp}}} \right)^{2} } \right)} \hfill \\ {} \hfill & . \hfill & . \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]_{{4n \times 4n}} \\ \end{aligned}$$
(22)
$${\mathbf{A}}_{{\mathbf{m}}} = \left[ {\begin{array}{*{20}c} 1 \\ 1 \\ . \\ 1 \\ 0 \\ 0 \\ . \\ . \\ . \\ . \\ 0 \\ \end{array} } \right]_{1 \times 4n}$$
(23)
$${\mathbf{B}}_{{\text{m}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & . & . & . & . & . & . & . & . & 0 \\ 0 & 0 & 0 & . & . & . & . & . & . & . & 0 \\ . & . & . & . & . & . & . & . & . & . & . \\ 0 & . & . & 0 & 0 & . & . & . & . & . & . \\ 0 & . & . & { - \frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{1} }}{r}} \right)^{2} } & {\frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & { - \frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{1} }}{r}} \right)^{2} } & {\frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & { - \frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{1} }}{r}} \right)^{2} } & {\frac{{\gamma_{3}^{2} }}{r}\frac{1}{2\Delta r}} \\ {} & {} & {} & {} & {} & {} & . & . & 0 & 0 & 0 \\ \end{array} } \right]_{4n \times 4n}$$
(24)
$${\mathbf{C}}_{{\text{m}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & . & . & . & . & . & . & . & . & 0 \\ 0 & 0 & 0 & . & . & . & . & . & . & . & 0 \\ . & . & . & . & . & . & . & . & . & . & . \\ 0 & . & . & 0 & 0 & . & . & . & . & . & . \\ 0 & . & . & {\frac{{\gamma_{0}^{2} }}{2\Delta r}} & 0 & { - \frac{{\gamma_{0}^{2} }}{2\Delta r}} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {\frac{{\gamma_{0}^{2} }}{2\Delta r}} & 0 & { - \frac{{\gamma_{0}^{2} }}{2\Delta r}} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {\frac{{\gamma_{0}^{2} }}{2\Delta r}} & 0 & { - \frac{{\gamma_{0}^{2} }}{2\Delta r}} \\ {} & {} & {} & {} & {} & {} & . & . & 0 & 0 & 0 \\ \end{array} } \right]_{4n \times 4n}$$
(25)
$$\begin{aligned} {\mathbf{K}}_{{\phi _{\theta } }} & = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ 0 \hfill & . \hfill & . \hfill & 1 \hfill & 0 \hfill & . \hfill \\ 0 \hfill & . \hfill & . \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{4} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{5} }}{{r_{p} }}} \right)^{2} } \right)} \hfill & {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{4} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{5} }}{{r_{p} }}} \right)^{2} } \right)} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & . \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} . \hfill & . \hfill & . \hfill & . \hfill & 0 \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{2}{{\Delta r^{2} }} + \left( {\frac{{\gamma _{4} }}{r}} \right)^{2} + \left( {\frac{{\gamma _{5} }}{{r_{p} }}} \right)^{2} } \right)} \hfill & {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill \\ . \hfill & . \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right] \\ \end{aligned}$$
(26)
$${\mathbf{P}} = \left[ {\begin{array}{*{20}c} 1 \\ 1 \\ . \\ 1 \\ 0 \\ 0 \\ . \\ . \\ . \\ . \\ 0 \\ \end{array} } \right]_{1 \times 4n}$$
(27)
$${\mathbf{E}} = \left[ {\begin{array}{*{20}c} 0 & 0 & . & . & . & . & . & . & . & . & 0 \\ 0 & 0 & 0 & . & . & . & . & . & . & . & 0 \\ . & . & . & . & . & . & . & . & . & . & . \\ 0 & . & . & 0 & 0 & . & . & . & . & . & . \\ 0 & . & . & {\frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{4} }}{r}} \right)^{2} } & { - \frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {\frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{4} }}{r}} \right)^{2} } & { - \frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {\frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} & { - \left( {\frac{{\gamma_{4} }}{r}} \right)^{2} } & { - \frac{{\gamma_{6}^{2} }}{r}\frac{1}{2\Delta r}} \\ {} & {} & {} & {} & {} & {} & . & . & 0 & 0 & 0 \\ \end{array} } \right]_{4n \times 4n}$$
(28)
$${\mathbf{Q}} = \left[ {\begin{array}{*{20}c} 0 & 0 & . & . & . & . & . & . & . & . & 0 \\ 0 & 0 & 0 & . & . & . & . & . & . & . & 0 \\ . & . & . & . & . & . & . & . & . & . & . \\ 0 & . & . & 0 & 0 & . & . & . & . & . & . \\ 0 & . & . & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 \\ {} & {} & {} & {} & {} & {} & . & . & 0 & 0 & 0 \\ \end{array} } \right]_{4n \times 4n}$$
(29)
$$\begin{aligned} {\mathbf{K}}_{{\phi _{z} }} & = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & {} \hfill & . \hfill \\ {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{1}{{\Delta r^{2} }} + \gamma _{8}^{2} + \frac{1}{{r^{2} }}} \right)} \hfill & {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & 0 \hfill & . \hfill \\ 0 \hfill & {} \hfill & 0 \hfill & { - \frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{\Delta r}}} \hfill & {\frac{1}{{\Delta r^{2} }}} \hfill \\ 0 \hfill & . \hfill & . \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{\Delta r}} - (\gamma _{8} ^{2} + \frac{1}{{r^{2} }})} \hfill & { - \frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{\Delta r}}} \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & 0 \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill & . \hfill \\ 0 \hfill & . \hfill & . \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right. \\ & \quad \left. {\begin{array}{*{20}l} . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill \\ {\frac{1}{{\Delta r^{2} }}} \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill \\ . \hfill & . \hfill & . \hfill & . \hfill \\ {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{1}{{\Delta r^{2} }} + \gamma _{8} ^{2} + \frac{1}{{r^{2} }}} \right)} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & 0 \hfill \\ 0 \hfill & {\frac{1}{{\Delta r^{2} }} + \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill & { - \left( {\frac{1}{{\Delta r^{2} }} + \gamma _{8} ^{2} + \frac{1}{{r^{2} }}} \right)} \hfill & {\frac{1}{{\Delta r^{2} }} - \frac{1}{r}\frac{1}{{2\Delta r}}} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]_{{4n \times 4n}} \\ \end{aligned}$$
(30)
$${\mathbf{G}} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ . \\ 0 \\ 0 \\ 0 \\ . \\ . \\ . \\ . \\ 0 \\ \end{array} } \right]_{1 \times 4n}$$
(31)
$${\mathbf{M}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & . & . & . & . & . & . \\ { - \frac{{\gamma_{7}^{2} }}{2\Delta r}} & {\frac{{\gamma_{7}^{2} }}{r}} & {\frac{{\gamma_{7}^{2} }}{2\Delta r}} & 0 & . & . & . & . & . \\ 0 & { - \frac{{\gamma_{7}^{2} }}{2\Delta r}} & {\frac{{\gamma_{7}^{2} }}{r}} & {\frac{{\gamma_{7}^{2} }}{2\Delta r}} & 0 & . & . & . & . \\ 0 & 0 & 0 & { - \gamma_{7}^{2} \frac{1}{\Delta r} + \frac{1}{r}\gamma_{7}^{2} } & {\gamma_{7}^{2} \frac{1}{\Delta r}} & . & . & . & . \\ {} & . & . & . & {} & . & . & . & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & {} & { - \frac{{\gamma_{7}^{2} }}{2\Delta r}} & {\frac{{\gamma_{7}^{2} }}{r}} & {\frac{{\gamma_{7}^{2} }}{2\Delta r}} & 0 \\ . & . & . & . & . & {} & { - \frac{{\gamma_{7}^{2} }}{2\Delta r}} & {\frac{{\gamma_{7}^{2} }}{r}} & {\frac{{\gamma_{7}^{2} }}{2\Delta r}} \\ 0 & . & . & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]_{4n \times 4n}$$
(32)
$${\mathbf{N}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & . & . & . & . & . & . \\ 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & 0 & . & . & . & . & . \\ 0 & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & 0 & . & . & . & . \\ 0 & 0 & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & . & . & . & . \\ {} & . & . & . & {} & . & . & . & . \\ . & . & . & . & . & . & . & . & . \\ . & . & . & . & {} & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 & 0 \\ . & . & . & . & . & {} & 0 & { - \frac{{\gamma_{7}^{2} }}{r}} & 0 \\ 0 & . & . & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]_{4n \times 4n}$$
(33)

where \(\gamma_{0} = \sqrt {\frac{{\int\limits_{0}^{\infty } {(\lambda_{s} - G_{s} )w\frac{dw}{{dz}}dz} }}{{\int\limits_{0}^{\infty } {(\lambda_{s} + 2G_{s} )wwdz} }}},\) \(\gamma_{1} = \frac{{\lambda_{s} + 3G_{s} }}{{\lambda_{s} + 2G_{s} }},\) \(\gamma_{2} = r_{p} \sqrt {\frac{{\int\limits_{0}^{\infty } {G_{s} \frac{dw}{{dz}}\frac{dw}{{dz}}dz} }}{{\int\limits_{0}^{\infty } {(\lambda_{s} + 2G_{s} )wwdz} }}},\)\(\gamma_{3} = \frac{{G_{s} + \lambda_{s} }}{{\lambda_{s} + 2G_{s} }},\) \(\gamma_{4} = \frac{{\lambda_{s} + 3G_{s} }}{{G_{s} }},\) \(\gamma_{5} = rp\sqrt {\frac{{\int\limits_{0}^{\infty } {G_{s} \frac{dw}{{dz}}\frac{dw}{{dz}}dz} }}{{\int\limits_{0}^{\infty } {G_{s} wwdz} }}},\) \(\gamma_{6} = \frac{{G_{s} + \lambda_{s} }}{{G_{s} }},\) \(\gamma_{7} = \sqrt {\frac{{\int\limits_{0}^{\infty } {(\lambda_{s} - G_{s} )w\frac{dw}{{dz}}dz} }}{{\int\limits_{0}^{\infty } {G_{s} wwdz} }}},\) \(\gamma_{8} = \sqrt {\frac{{\int\limits_{0}^{\infty } {(\lambda_{s} + 2G_{s} )\frac{dw}{{dz}}\frac{dw}{{dz}}dz} }}{{\int\limits_{0}^{\infty } {G_{s} wwdz} }}}.\)

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Li, X., Dai, G., Zhu, M. et al. Investigation of the soil deformation around laterally loaded deep foundations with large diameters. Acta Geotech. 19, 2293–2314 (2024). https://doi.org/10.1007/s11440-023-02012-1

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