Li et al 2021d - Revision history

Rimni at 10:45, 17 March 2022

2022-03-17T10:45:00Z

Rimni: /* 2.4 Solving procedures */

2022-03-17T10:43:00Z

‎2.4 Solving procedures

Rimni at 10:41, 17 March 2022

2022-03-17T10:41:07Z

Rimni at 10:40, 17 March 2022

2022-03-17T10:40:08Z

Rimni at 10:37, 17 March 2022

2022-03-17T10:37:38Z

Rimni at 10:35, 17 March 2022

2022-03-17T10:35:24Z

Rimni at 10:34, 17 March 2022

2022-03-17T10:34:02Z

Rimni: /* Author contributions */

2022-03-17T10:30:50Z

‎Author contributions

Rimni: /* References */

2022-03-17T10:28:04Z

‎References

Rimni at 09:40, 17 March 2022

2022-03-17T09:40:46Z

Show changes

@@ Line 41: / Line 41: @@
 It should be emphasized that all above-mentioned studies are limited to undisturbed soils. In fact, the existing field data and test data show that the disturbance effect on the soil around the pile is inevitable in the process of pile installation [15,16]. In general, in the process of pile installation, radial compression will occur to the soil around the pile, leading to the decrease of the radial permeability coefficient of the disturbed soil, which will lead to the decrease of the radial drainage performance of the soil. When the vertical drainage distance is long, the soil drainage path is mainly radial. In this case, the disturbance effect of soil should not be ignored.
-Similar to pile installation, the prefabricated vertical drains (i.e., PVDs) installation has been fully studied <sup>[17-20]</sup>. Compared with the smear effect by PVD installation [21-25], the disturbance effect induced by pile installation is much more significant due to its bigger radius. Thus, it is of significance to study the consolidation of soil around the pile considering disturbance effect. Additionally, the growth rate of pile bearing capacity could be overestimated by using the current consolidation model, which leads to some unknown risks (e.g., excessive settlement and insufficient bearing capacity) and reducing the accuracy of building’s reliability analysis [9,26,27]. Therefore, whether it is the study of consolidation theory of soil around pile or the subsequent analysis of pile bearing capacity, the study of consolidation theory considering disturbance effect is of great significance. Unfortunately, there is no an analytical solution to axisymmetric consolidation of soil considering the disturbance effect around a pipe pile.
+Similar to pile installation, the prefabricated vertical drains (i.e., PVDs) installation has been fully studied [17-20]. Compared with the smear effect by PVD installation [21-25], the disturbance effect induced by pile installation is much more significant due to its bigger radius. Thus, it is of significance to study the consolidation of soil around the pile considering disturbance effect. Additionally, the growth rate of pile bearing capacity could be overestimated by using the current consolidation model, which leads to some unknown risks (e.g., excessive settlement and insufficient bearing capacity) and reducing the accuracy of building’s reliability analysis [9,26,27]. Therefore, whether it is the study of consolidation theory of soil around pile or the subsequent analysis of pile bearing capacity, the study of consolidation theory considering disturbance effect is of great significance. Unfortunately, there is no an analytical solution to axisymmetric consolidation of soil considering the disturbance effect around a pipe pile.
 In order to better predict the dissipation process of excess pore pressure and the increase process of pile bearing capacity after piling, it is necessary to further optimize the existing consolidation theory. On the condition that the disturbance effect cannot be ignored, this paper presents an analytical solution to axisymmetric consolidation of soil considering the disturbance effect around a pipe pile on the basis of previous studies. Fourier transform and separation of variables are used to solve the problem, and the solution is calibrated by degradation analysis and FEM. Subsequently, the distribution of excess pore pressure along vertical and radial directions are analyzed by graphical interpretation. Moreover, the concept of the dimensionless average excess pore pressure at pile-soil interface is put forward and the influence of disturbance parameters on it is analyzed. Most importantly, the validity of the solution is demonstrated by comparing with test data.

@@ Line 382: / Line 382: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>A_{2mn}=\displaystyle\frac{a_1\left(\sin{\alpha }_nH-\sin{\alpha }_nh_0\right)}{{\alpha }_n{}^2}\displaystyle\frac{\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}{\delta }_1+{\delta }_2}{\displaystyle\frac{R_2^2\left({\beta }_{2mn}r_{d}\right)}{R_1^2\left({\beta }_{1mn}r_{d}\right)}{\delta }_3+{\delta }_4}</math>
+| <math>A_{2mn}=\displaystyle\frac{a_1\left(\sin{\alpha }_nH-\sin{\alpha }_nh_0\right)}{{\alpha }_n{}^2}\displaystyle\frac{\displaystyle\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}{\delta }_1+{\delta }_2}{\displaystyle\frac{R_2^2\left({\beta }_{2mn}r_{d}\right)}{R_1^2\left({\beta }_{1mn}r_{d}\right)}{\delta }_3+{\delta }_4}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (22)

@@ Line 382: / Line 382: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>A_{2mn}=\frac{a_1\left(\sin{\alpha }_nH-\sin{\alpha }_nh_0\right)}{{\alpha }_n{}^2}\frac{\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}{\delta }_1+{\delta }_2}{\frac{R_2^2\left({\beta }_{2mn}r_{d}\right)}{R_1^2\left({\beta }_{1mn}r_{d}\right)}{\delta }_3+{\delta }_4}</math>
+| <math>A_{2mn}=\displaystyle\frac{a_1\left(\sin{\alpha }_nH-\sin{\alpha }_nh_0\right)}{{\alpha }_n{}^2}\displaystyle\frac{\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}{\delta }_1+{\delta }_2}{\displaystyle\frac{R_2^2\left({\beta }_{2mn}r_{d}\right)}{R_1^2\left({\beta }_{1mn}r_{d}\right)}{\delta }_3+{\delta }_4}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (22)

@@ Line 308: / Line 308: @@
 | <span id='ZEqnNum741519'></span>  <math>\left\{ \begin{array}{c}
 C_{d}{\beta }_{1mn}^2=C_{h}{\beta }_{2mn}^2\\
-\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}=\frac{k_{h}{\beta }_{2mn}P_2\left({\beta }_{2mn}r_{d}\right)}{k_{d}{\beta }_{1mn}P_1\left({\beta }_{1mn}r_{d}\right)}
+\displaystyle\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}=\displaystyle\frac{k_{h}{\beta }_{2mn}P_2\left({\beta }_{2mn}r_{d}\right)}{k_{d}{\beta }_{1mn}P_1\left({\beta }_{1mn}r_{d}\right)}
 \end{array}\right.</math>
 |}
@@ Line 324: / Line 324: @@
 |-
 | <span id='ZEqnNum775505'></span>  <math>\begin{array}{l}
-{\overline{u}}_1=\displaystyle\sum_{m=1}^{\infty }A_{2mn}\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}R_1\left({\beta }_{1mn}r\right){e}^{-\left(C_{d}{\beta }_{1mn}^2+C_{v}{\alpha }_n^2\right)t}\\
+{\overline{u}}_1=\displaystyle\sum_{m=1}^{\infty }A_{2mn}\displaystyle\frac{R_2\left({\beta }_{2mn}r_{d}\right)}{R_1\left({\beta }_{1mn}r_{d}\right)}R_1\left({\beta }_{1mn}r\right){e}^{-\left(C_{d}{\beta }_{1mn}^2+C_{v}{\alpha }_n^2\right)t}\\
 {\overline{u}}_2=\displaystyle\sum_{m=1}^{\infty }A_{2mn}R_2\left({\beta }_{2mn}r\right){e}^{-\left(C_{h}{\beta }_{2mn}^2+C_{v}{\alpha }_n^2\right)t}
 \end{array}</math>

@@ Line 142: / Line 142: @@
 ===2.4 Solving procedures===
-According to the boundary condition of Eq.<span id='cite-ZEqnNum798795'></span>[[#ZEqnNum798795|(5)]], the finite Fourier sine transform is performed on the governing equation (i.e., Eq.<span id='cite-ZEqnNum571777'></span>[[#ZEqnNum571777|(1)]]) and its remaining solving conditions (i.e., Eq.<span id='cite-ZEqnNum823648'></span>[[#ZEqnNum823648|(2)]]-<span id='cite-ZEqnNum201641'></span>[[#ZEqnNum201641|(4)]]), which yields
+According to the boundary condition of Eq.<span id='cite-ZEqnNum798795'></span>[[#ZEqnNum798795|(5)]], the finite Fourier sine transform is performed on the governing equation (i.e., Eq.<span id='cite-ZEqnNum571777'></span>[[#ZEqnNum571777|(1)]]) and its remaining solving conditions (i.e., Eqs.<span id='cite-ZEqnNum823648'></span>[[#ZEqnNum823648|(2)]]-<span id='cite-ZEqnNum201641'></span>[[#ZEqnNum201641|(4)]]), which yields
 {| class="formulaSCP" style="width: 100%; text-align: center;"

← Older revision		Revision as of 10:35, 17 March 2022
Line 701:		Line 701:


−	In [[#img-4\|Figure 4]], the influence of permeability coefficient ratio <math>k_{d}/k_~~\mbox~~{h}</math> on the excess pore pressure under deep and shallow foundations, where <math>k_d/k_{h}</math> reflects the disturbance degree of pile installation. When <math>k_{d}/k_h=1</math>, this solution can degenerate into the solution of homogeneous soil around a pile [12]. Whether it's shallow foundation (i.e., case of <math>H=5</math>m) or deep foundation (i.e., case of <math>H = 20</math>m), the dissipation rate of the excess pore pressure gradually decreases with decreases in <math>k_{d}/k_{h}</math>. That indicates that <math>u_{ave}\left(t\right)</math> could be overestimated at the anytime without considering the disturbance effect. The dissipation rate of the average excess pore pressure at pile-soil interface decreases obviously with the increase of <math>k_{d}/k_{h}</math>, which indicated that soil disturbance has a significant hindering effect on <math>u_{ave}\left(t\right)</math>. By comparing Case of shallow foundation and Case of deep foundation, it is clearly known that the disturbance effect has a greater influence on deep foundation than on shallow foundation. Moreover, the difference of drainage performance of deep and shallow foundations is gradually obvious with the increase of <math>k_{d}/k_{h}</math>.	+	In [[#img-4\|Figure 4]], the influence of permeability coefficient ratio <math>k_{d}/k_{h}</math> on the excess pore pressure under deep and shallow foundations, where <math>k_d/k_{h}</math> reflects the disturbance degree of pile installation. When <math>k_{d}/k_h=1</math>, this solution can degenerate into the solution of homogeneous soil around a pile [12]. Whether it's shallow foundation (i.e., case of <math>H=5</math>m) or deep foundation (i.e., case of <math>H = 20</math>m), the dissipation rate of the excess pore pressure gradually decreases with decreases in <math>k_{d}/k_{h}</math>. That indicates that <math>u_{ave}\left(t\right)</math> could be overestimated at the anytime without considering the disturbance effect. The dissipation rate of the average excess pore pressure at pile-soil interface decreases obviously with the increase of <math>k_{d}/k_{h}</math>, which indicated that soil disturbance has a significant hindering effect on <math>u_{ave}\left(t\right)</math>. By comparing Case of shallow foundation and Case of deep foundation, it is clearly known that the disturbance effect has a greater influence on deep foundation than on shallow foundation. Moreover, the difference of drainage performance of deep and shallow foundations is gradually obvious with the increase of <math>k_{d}/k_{h}</math>.

	<div id='img-4'></div>		<div id='img-4'></div>

@@ Line 407: / Line 407: @@
 |-
 | style="text-align: center;" | <math>\begin{array}{ccc}
-m_\mbox{v}={10}^{-2}\mbox{ }\mbox{m}^2/\mbox{kN} & {\gamma }_\mbox{w}=10\mbox{kN}/\mbox{m}^3 & k_\mbox{v}=2\times {10}^{-9}\mbox{m}/\mbox{s}\\
+m_{v}={10}^{-2}{m}^2/{kN} & {\gamma }_{w}=10{kN}/{m}^3 & k_{v}=2\times {10}^{-9}{m}/{s}\\
-r_0=0.25\mbox{ m} & r_\mbox{p}=1.2\times r_\mbox{d} & r_\mbox{e}=20\times r_\mbox{0}\\
+r_0=0.25{ m} & r_{p}=1.2\times r_{d} & r_{e}=20\times r_{0}\\
-H=20\mbox{ m} & h_0=0.5\mbox{ m} & a_1=5\mbox{kPa}/\mbox{m}
+H=20{ m} & h_0=0.5{ m} & a_1=5{kPa}/{m}
 \end{array}</math>
 |}

@@ Line 760: / Line 760: @@
 Conceptualization: Ping Li; writing—original draft preparation: Ping Li and Zhijian Chen; writing—review and editing: Yi Ding.
 <!--===Informed Consent Statement===
@@ Line 773: / Line 772: @@
 The authors declared that they have no conflicts of interest to this work.<br/>
 <br/>-->
 ==References==
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">

@@ Line 794: / Line 794: @@
 [9] Li L., Li J.P.,  Sun D.A.,  Zhang L.X. Time-dependent bearing capacity of a jacked pile: An analytical approach based on effective stress method. Ocean Engineering, 143:177-185, 2017.
-[10]  Randolph M.F.,  Wroth C.P. An analytical solution for the consolidation around a driven pile. International Journal for Numerical Analytical Methods in Geomechanics, 3(3):217-229, 1979.
+[10] Randolph M.F.,  Wroth C.P. An analytical solution for the consolidation around a driven pile. International Journal for Numerical Analytical Methods in Geomechanics, 3(3):217-229, 1979.
 [11] Guo W.D. Visco-elastic consolidation subsequent to pile installation. Computers and Geotechnics, 26(2):113-144, 2000.
@@ Line 800: / Line 800: @@
 [12]  Gao Z.K.,  Shi J.Y. Consolidation solution of soil around single-pile after pile sinking. Rock and Soil Mechanics, 29(4):979-982, 2008.
-[13]  Wang S.H.,  Ni P.P., Chen Z.,  Mei G.X. Consolidation solution of soil around a permeable pipe pile. Marine Georesources Geotechnology,   20(9):1097-1105, 2020.
+[13]  Wang S.H.,  Ni P.P., Chen Z.,  Mei G.X. Consolidation solution of soil around a permeable pipe pile. Marine Georesources Geotechnology,  20(9):1097-1105, 2020.
 [14] Chen Z., Xiao T.,  Feng J.X., Ni P.P,  Chen D.Q.,  Mei G.X., Chen Y.F. Mathematical characterization of pile-soil interface boundary for consolidation analysis of soil around permeable pipe pile. Canadian Geotechnical Journal,  58(9):1277-1288, 2020.
@@ Line 816: / Line 816: @@
 [20] Nazir R., Moayedi H., Subramaniam P., Ghareh S. Ground improvement using SPVD and RPE. Arabian Journal of Geosciences, 10(23):1-21, 2017.
-[21] RD Holtz, M B Jamiolkowski, R Lancellotta, R Pedroni. Prefabricated vertical drains: design and performance; 1991.
+[21]  Holtz R.D.,  Jamiolkowski M.B., Lancellotta R., Pedroni R. Prefabricated vertical drains: design and performance. Construction Industry Research & Information Assoc., Boston, Mass.: Butterworth, 1991.
-[22] J Wang, Y L Yang, H T Fu, Y Q Cai, X Q Hu, X M Lou, Y W Jin. Improving consolidation of dredged slurry by vacuum preloading using prefabricated vertical drains (PVDs) with varying filter pore sizes. Canadian Geotechnical Journal. 2020;57(2):294-303.
+[22] Wang J.,  Yang Y.L.,  Fu H.T.,  Cai Y.Q.,  Hu X.Q.,  Lou X.M.,  Jin Y.W. Improving consolidation of dredged slurry by vacuum preloading using prefabricated vertical drains (PVDs) with varying filter pore sizes. Canadian Geotechnical Journal, 57(2):294-303, 2020.
-[23] Y B Deng, G B Liu, B Indraratna, C Rujikiatkamjorn, K H Xie. Model test and theoretical analysis for soft soil foundations improved by prefabricated vertical drains. International Journal of Geomechanics. 2017;17(1):04016045.
+[23]  Deng Y.B.,  Liu G.B., Indraratna B., Rujikiatkamjorn C.,  Xie K.H. Model test and theoretical analysis for soft soil foundations improved by prefabricated vertical drains. International Journal of Geomechanics, 17(1):04016045, 2017.
-[24] Z Chen, P P Ni, G X Mei, Y F Chen. Semi-analytical solution for consolidation of ground with partially penetrating PVDs under the free-strain condition. Journal of Engineering Mechanics. 2020;147(2):04020148.
+[24] Chen Z.,  Ni P.P., Mei G.X.,  Chen Y.F. Semi-analytical solution for consolidation of ground with partially penetrating PVDs under the free-strain condition. Journal of Engineering Mechanics, 147(2):04020148, 2020.
-[25] Y Tian, W B Wu, G S Jiang, M H El Naggar, G X Mei, P P Ni. Analytical solutions for vacuum preloading consolidation with prefabricated vertical drain based on elliptical cylinder model. Computers Geotechnics. 2019;116:103202.
+[25] Tian Y.,  Wu W.B.,  Jiang G.S.,  El Naggar M.H., Mei G.X., Ni P.P. Analytical solutions for vacuum preloading consolidation with prefabricated vertical drain based on elliptical cylinder model. Computers Geotechnics, 116:103202, 2019.
-[26] L O Soderberg. Consolidation theory applied to foundation pile time effects. Géotechnique. 1962;12(3):217-225.
+[26]  Soderberg L.O. Consolidation theory applied to foundation pile time effects. Géotechnique, 12(3):217-225, 1962.
-[27] Z L Luo, F H Dong. Statistical investigation of bearing capacity of pile foundation based on Bayesian reliability theory. Advances in Civil Engineering. 2019;2019.
+[27]  Luo Z.L.,  Dong F.H. Statistical investigation of bearing capacity of pile foundation based on Bayesian reliability theory. Advances in Civil Engineering, 2019:9858617, 2019.
-[28] B M Lehane, D R Gill. Displacement fields induced by penetrometer installation in an artificial soil. International Journal of Physical Modelling in Geotechnics. 2004;4(1):25-36.
+[28]  Lehane B.M.,  Gill D.R. Displacement fields induced by penetrometer installation in an artificial soil. International Journal of Physical Modelling in Geotechnics, 4(1):25-36, 2004.
-[29] H de Chaunac, A Holeyman. Numerical analysis of the set-up around the shaft of a closed-ended pile driven in clay. Géotechnique. 2018;68(4):332-344.
+[29] de Chaunac H., Holeyman A. Numerical analysis of the set-up around the shaft of a closed-ended pile driven in clay. Géotechnique, 68(4):332-344, 2018.
-[30] Q M Chen, M N Haque, M Abu-Farsakh, B A Fernandez. Field investigation of pile setup in mixed soil. Geotechnical Testing Journal. 2014;37(2):268-281.
+[30]  Chen Q.M.,  Haque M,N., Abu-Farsakh M.,  Fernandez B.A. Field investigation of pile setup in mixed soil. Geotechnical Testing Journal, 37(2):268-281, 2014.
-[31] P P Ni, S Mangalathu, G X Mei, Y L Zhao. Laboratory investigation of pore pressure dissipation in clay around permeable piles. Canadian Geotechnical Journal. 2018;55(9):1257-1267.
+[31] Ni P.P., Mangalathu S., Mei G.X., Zhao Y.L. Laboratory investigation of pore pressure dissipation in clay around permeable piles. Canadian Geotechnical Journal, 55(9):1257-1267, 2018.