# Horizontal Bearing Capacity and Reliability of Piles in Coastal Soft Soil Considering the Time-Varying Characteristics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Time-Varying Model of Horizontal Loaded Pile in a Soft Soil Area

#### 2.1. Time-Varying Characteristics of the Soft Soil

_{0}) is the origin of cohesion, φ(t

_{0}) is the origin of internal friction angle, α

_{c}(t) is the attenuation factor of cohesion, and α

_{φ}(t) is the attenuation factor of the internal friction angle.

#### 2.2. The p-y Curve Model Considering Time-Varying Characteristics

_{p}p

_{u}(t). The relationship between p/p

_{u}(t) and the y/y

_{50}can be calculated as:

_{u}(t) is the ultimate soil resistance changing with time. The relationship can be shown as [27]:

_{u}= 1.212, b

_{u}= 0.6631/b. γ is the soil weight, b is the pile diameter, z is the distance from the calculation point to the pile top, and δ

_{1}is the passive earth pressure, which can be expressed as

_{p}, b

_{p}, and c

_{p}are undetermined parameters, which can be calculated as

_{p}is the lateral soil resistance softening ratio, which is related to the property of the soil and can be obtained by field test or laboratory test.

_{50}is the horizontal displacement when the soil reaction load reaches half of the ultimate resistance, which can be expressed as [27]

_{v}is the coefficient of compressibility of the soil.

#### 2.3. Time-Varying Characteristics of the Bending Stiffness of the Pile

_{p}I

_{P}is the bending stiffness of the corroded reinforced concrete pile. EI is the initial bending stiffness of a reinforced concrete pile. η is a reduction factor, that can be calculated as

_{i}is the cross-sectional area of the i-th tensile steel bar after corrosion, and η

_{i}is the stiffness reduction factor of the i-th steel bar caused by the corrosion. When the concrete cover of the pile body has no rust expansion crack, η

_{i}= 1. When the rust expansion crack appears, the expression of the η

_{i}can be calculated as

_{i}is the corrosion depth of the i-th steel bar. The corrosion depth (d) of steel under the action of chloride can be expressed as

_{corr}is the corrosion current density of a steel bar. Liu et al. [28,29] obtained the relationship between the chloride concentration on the surface of steel bar, the temperature, and concrete resistivity:

_{x}is the chloride concentration at position p from the concrete’s surface at time t, C

_{s}is the chloride concentration on the concrete’s surface, D is the effective diffusion coefficient of chloride in concrete, and erf(x) is the error function, which can be expressed as

## 3. Horizontal Bearing Capacity and Reliability Analysis of the Pile

#### 3.1. Analysis of Horizontal Bearing Capacity of the Pile

#### 3.2. Reliability Analysis

_{0}, b

_{i}, and c

_{i}are undetermined coefficients. When the random variable is n, the number of undetermined coefficients is 2n + 1.

_{i}is generally taken as the initial value point, and the experimental point (x

_{i}) is taken around the value of μX

_{i}. The relationship can be expressed as:

_{i}is the standard deviation. When f is greater than 0, the value is taken at about 2 [32].

_{i}) is obtained according to Equation (19), it is then iterated continuously to approach the real limit state surface (g(X) = 0).

_{Xi}is the sensitivity coefficient, and its expression can be written as

_{f}) and reliability index (β) can be expressed as:

- The initial iteration point is assumed, taking the mean (μX).
- The experimental point (x
_{i}) is selected around the mean point (μX). - The estimated values of the performance function are calculated at each expansion point (g
_{i}(i = 1,2…,2n + 1)). - The undetermined coefficients (a
_{0}, b_{i}, and c_{i}) are solved. - The reliability index (β) and the checkpoint (x*) are calculated.
- The estimated values of the performance function are calculated at the checking point (x*).
- The new (x) is obtained using interpolation.
- The steps (3) to (7) are repeated until |x − x*| < ε, and the values of ε are 1 × 10
^{−6}usually.

## 4. Case Study

^{−12}m

^{2}/s, and the surface temperature of rebar is 296 K. It is assumed that cohesion (c), internal friction angle (φ), concrete cover (B), and chloride concentration (C

_{s}) obey the normal distribution, the mean values are 20 kPa, 5°, 65 mm, and 8 kg/m

^{3}, respectively, and the coefficient of variation is 0.1.

#### 4.1. Analysis of Time-Varying Characteristics of the Horizontal Bearing of the Pile

^{3}and 8 kg/m

^{3}, respectively [1,33]. The relationship between the bending stiffness reduction factor (η) and time under different chloride concentrations is shown in Figure 6. The relationship between the horizontal displacement of the pile and the maximum bending moment changing with service time is shown in Figure 7.

^{3}[30], the initial corrosion time of the pile is the 34th year and 22nd year with the conditions of chloride concentration of 4 kg/m

^{3}and 8 kg/m

^{3}, respectively.

^{3}as an example, the initial horizontal displacement of 4.3 mm increases to 5.4 mm, 6.1 mm, 6.8 mm, and 7.8 mm, at the service times of 25 years, 50 years, 75 years, and 100 years, respectively. The displacement increases by 81.4% in the 100th year. Furthermore, the greater the chloride concentration on the pile surface, the greater the horizontal displacement. When the service time is 50 years, and the chlorine ion concentration of the pile surface is 0 kg/m

^{3}, 4 kg/m

^{3}, and 8 kg/m

^{3}, the corresponding horizontal displacement is 5.5 mm, 5.8 mm, and 6.0 mm, respectively.

^{3}. Since the chloride concentration on the surface of the steel bar in the pile is less than the critical value, the bending stiffness of the pile shows unchanging characteristics. The maximum bending moment and the horizontal displacement are equal to that of the chloride concentration of 0 kg/m

^{3}.

^{3}, the maximum bending moment of the pile increases with the increase in service time. If the maximum bending moment at the initial state is 855 kN·m, at the service time of 25 years, 50 years, 75 years, and 100 years, that changes to 894 kN·m, 899 kN·m, 06 kN·m, and 913 kN·m, respectively. The maximum bending moment increases by only 6.7% after 100 years of service.

#### 4.2. Time-Varying Reliability Analysis of a Pile under Horizontal Displacement Failure Mode

#### 4.2.1. Impact of Time-Varying Characteristics on Safety Margin

^{4}groups of random variables obeying the normal distribution were extracted and substituted into the limit state function expression. The frequency distribution curve for horizontal displacement failure mode is shown in Figure 8.

#### 4.2.2. Influence of Parameter Variability on the Reliability of the Pile

#### 4.2.3. Influence of Parameter Distribution Mode on the Reliability of the Pile

## 5. Conclusions

- (1)
- The time-varying characteristics of soft soil and bending stiffness have a significant influence on the horizontal bearing capacity of the pile. In the early stages, it is mainly affected by the creep of soft soil, and in the later stages, the attenuation of the bending stiffness of the pile becomes the main influencing factor. With the increase in service time, the change value of the maximum bending moment of the pile decreases, the maximum horizontal displacement increases nonlinearly, the concentrated area of safety margin of horizontal displacement decreases gradually, and the reliability index of the horizontal displacement failure mode decreases gradually.
- (2)
- The larger the coefficient of variation of random variables, the lower the reliability index under horizontal displacement failure mode. Among the random variables, the internal friction angle has the greatest influence on the reliability index. When the coefficient of variation of the internal friction angle is greater than 0.6, the reliability index gradually tends to be stable. When the service time only reaches the 25th year, the influence of the parameters of steel corrosion damage has a slight effect on the reliability of the pile.
- (3)
- The reliability index is impacted significantly by the distribution types of random variables under the failure mode of horizontal displacement. When the soil parameters obey the extreme value type I distribution, the corresponding reliability index is greater than that of the log-normal distribution and the normal distribution. If the target reliability index is used to predict the working life of the pile, the result obtained from the extreme value type I distribution is greater than that of the log-normal distribution and the normal distribution.
- (4)
- Finally, the reliability calculation model for the laterally loaded pile is established. However, the above research results are based on my hypothesis and can provide some reference for similar construction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Shao, W.; Shi, D.; Jiang, J.; Chen, Y. Time-dependent lateral bearing behavior of corrosion-damaged RC pipe piles in marine environments. Constr. Build. Mater.
**2017**, 157, 676–684. [Google Scholar] [CrossRef] - Zhuang, N.; Zhou, Y.; Ma, Y.; Liao, Y.; Chen, D. Corrosion activity on CFRP-strengthened RC piles of high-pile wharf in a simulated marine environment. Adv. Mater. Sci. Eng.
**2017**, 2017, 7185452. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.Z.; Ge, H.B.; Wu, L.J. Time-variant reliability analysis of bank slope of open type wharf on piles considering the effect of soft soil creep on strength. Mar. Sci.
**2018**, 42, 88–97. [Google Scholar] - Matlock, H. Correlations for design of laterally loaded piles in soft clay. In Proceedings of the 11th Offshore Technology Conference, Houston, TX, USA, 21 April 1970; pp. 577–594. [Google Scholar]
- Sullivan, W.R.; Reese, L.C.; Fenske, C.W. Unified method for analysis of laterally loaded piles in clay. In Numerical Methods in Offshore Piling; Thomas Telford Publishing: London, UK, 1980; pp. 135–146. [Google Scholar]
- Fu, D.; Zhang, Y.; Aamodt, K.K.; Yan, Y. A multi-spring model for monopile analysis in soft clays. Mar. Struct.
**2020**, 72, 102768. [Google Scholar] [CrossRef] - Yang, K.; Liang, R. Numerical solution for laterally loaded piles in a two-layer soil profile. J. Geotech. Geoenviron.
**2006**, 132, 1436–1443. [Google Scholar] [CrossRef] - Conte, E.; Troncone, A.; Vena, M. Nonlinear three-dimensional analysis of reinforced concrete piles subjected to horizontal loading. Comput. Geotech.
**2013**, 49, 123–133. [Google Scholar] [CrossRef] - He, Z.J.; Lei, H.C.; Xia, Z.Q.; Zhao, L.H. Analysis of settlement and internal force displacement of single pile in multilayer soft soil foundation. Rock Soil Mech.
**2020**, 41, 655–666. [Google Scholar] - Basack, S.; Karami, M.; Karakouzian, M. Pile-soil interaction under cyclic lateral load in loose sand: Experimental and numerical evaluations. Soil Dyn. Earthq. Eng.
**2022**, 162, 107439. [Google Scholar] [CrossRef] - Cheng, X.; El Naggar, M.H.; Lu, D.; Wang, P.; Tu, W. A cyclic p-y elastoplastic model applied to lateral loaded pile in soft clays. Can. Geotech. J.
**2022**. [Google Scholar] [CrossRef] - Basack, S.; Nimbalkar, S. Measured and predicted response of pile groups in soft clay subjected to cyclic lateral loading. Int. J. Geomech.
**2018**, 18, 04018073. [Google Scholar] [CrossRef] - Muszyński, Z.; Rybak, J. Horizontal displacement control in course of lateral loading of a pile in a slope. IOP Conf. Ser. Mater. Sci. Eng.
**2017**, 245, 032002. [Google Scholar] [CrossRef] - Chen, C.F.; Qin, H.J. Stability analysis of slope considering the time and depth effect of strength parameters. J. Huanan Univ.
**2009**, 36, 1–6. [Google Scholar] - Zhu, X.; Zi, G.; Cao, Z.; Cheng, X. Combined effect of carbonation and chloride ingress in concrete. Constr. Build. Mater.
**2016**, 110, 369–380. [Google Scholar] [CrossRef] - Florea, M.V.A.; Brouwers, H.J.H. Modelling of chloride binding related to hydration products in slag-blended cements. Constr. Build. Mater.
**2014**, 64, 421–430. [Google Scholar] [CrossRef] [Green Version] - Loser, R.; Lothenbach, B.; Leemann, A.; Tuchschmid, M. Chloride resistance of concrete and its binding capacity—Comparison between experimental results and thermodynamic modeling. Cem. Concr. Compos.
**2010**, 32, 34–42. [Google Scholar] [CrossRef] - Feng, D.C.; Xie, S.C.; Li, Y.; Jin, L. Time-dependent reliability-based redundancy assessment of deteriorated RC structures against progressive collapse considering corrosion effect. Struct. Saf.
**2021**, 89, 102061. [Google Scholar] [CrossRef] - Kozubal, J.; Steshenko, D.; Wyjadlowski, M. Lateral loaded pile durability in case of chloride corrosion. Int. Multidiscip. Sci. GeoConf. SGEM
**2016**, 3, 155–162. [Google Scholar] - Kozubal, J.; Wyjadłowski, M.; Steshenko, D. Probabilistic analysis of a concrete column in an aggressive soil environment. PLoS ONE
**2019**, 14, e0212902. [Google Scholar] [CrossRef] [Green Version] - Dang, H.V.; Trestian, R.; Bui-Tien, T.; Nguyen, H.X. Probabilistic method for time-varying reliability analysis of structure via variational Bayesian neural network. In Structures; Elsevier: Amsterdam, The Netherlands, 2021; Volume 34, pp. 3703–3715. [Google Scholar]
- Xu, Q.; Shi, D.; Shao, W. Service life prediction of RC square piles based on time-varying probability analysis. Constr. Build. Mater.
**2019**, 227, 116824. [Google Scholar] [CrossRef] - Wang, J.; Ji, H.G.; Wang, J.J.; Zhang, Z.J. Residual Life Predication of Reinforced Concrete Elements Based on Time-Varying Reliability. In Advanced Materials Research; Trans Tech Publications Ltd.: Stafa-Zurich, Switzerland, 2011; Volume 163, pp. 3258–3262. [Google Scholar]
- Wu, Y.; Miao, F.; Li, L.; Xie, Y.; Chang, B. Time-varying reliability analysis of Huangtupo Riverside No. 2 Landslide in the Three Gorges Reservoir based on water-soil coupling. Eng. Geol.
**2017**, 226, 267–276. [Google Scholar] [CrossRef] - Deepthi, D.; Sivakumar, B.G.L.; Lekshmi, S. Time-Dependent Reliability Analysis of Pavement Structures under Fatigue Loading. In Geotechnical Safety and Risk V; IOS Press: Amsterdam, The Netherlands, 2015; pp. 358–363. [Google Scholar]
- Liu, T.H.; Lin, T. Soft Rock Engineering Design Theory and Construction Practice; China Construction Industry Press: Beijing, China, 2001; pp. 135–141. [Google Scholar]
- Zhang, S.Y. Study on P-Y Curves of Laterally Loaded Pile under Static Loading. Master’s Thesis, Hehai University, Nanjing, China, 2001. [Google Scholar]
- Liu, Y.; Weyers, R.W. Modeling the Dynamic Corrosion Process in Chloride Contaminated Concrete Structures. Cem. Concr. Res.
**1998**, 28, 365–379. [Google Scholar] [CrossRef] - Liu, Y. Modeling the Time-To-Corrosion Cracking of the Cover Concrete in Chloride Contaminated Reinforced Concrete Structures. Ph.D. Thesis, Virginia Polytechnic Insitute and State University, Blacksburg, VA, USA, 1996. [Google Scholar]
- Ma, Y.L.; Zhang, A.L. Durablity life prediction of concrete structure based on the regulated reliability index under chloride environment. China Civ. Eng. J.
**2006**, 39, 36–41. [Google Scholar] - Yin, P.B.; Nie, D.L.; Yang, Z.H.; He, W.; Jia, W.W. The p-y curve and computation method of the horizontal bearing capacity of piles in sloping ground. Chin. J. Rock Mech. Eng.
**2018**, 37, 996–1003. [Google Scholar] - Zhang, M. Analysis of Structural Reliability; Science Press: Beijing, China, 2009; pp. 116–118. [Google Scholar]
- Song, H.W.; Lee, C.H.; Ann, K.Y. Factors influencing chloride transport in concrete structures exposed to marine environments. Cement Concrete Comp.
**2008**, 30, 113–121. [Google Scholar] [CrossRef] - Vu, K.A.; Stewart, M.G. Predicting the likelihood and extent of reinforced concrete corrosion-induced cracking. J. Struct. Eng.
**2005**, 131, 1681–1689. [Google Scholar] [CrossRef] - Dianty, M.A.; Yahaya, A.S.; Ahmad, F. Probability distribution of engineering properties of soil at telecommunication sites in Indonesia. Int. J. Sci. Res. Knowl.
**2014**, 2, 143–150. [Google Scholar] - Langejan, A. Some aspects of the safety factor in soil mechanics, considered as a problem of probability. In Proceedings of the Sixth International Conference on Soil Mechanics and Foundation Engineering, Montreal, QC, Canada, 8–15 September 1965; pp. 500–502. [Google Scholar]
- Yan, C.F.; Liu, D.Y.; Zhang, J.H.; Zhu, K.S. The susceptibility analysis of reliability for the probability distribution types of parameters in strength criterion. Chin. J. Rock Mech. Eng.
**1999**, 18, 36–39. [Google Scholar]

**Figure 3.**Diagram of a cross-section of reinforced concrete pile and reinforcement corrosion mechanism.

**Figure 7.**Stress and deformation diagrams for the pile under different chloride concentrations. (

**a**) Relationship curve between the maximum horizontal displacement of the pile and service time. (

**b**) Relationship curve between the maximum bending moment of the pile shaft and service time.

**Figure 8.**The influence of time-varying characteristics on the safety margin frequency distribution of horizontal displacement. (

**a**) PDF curve. (

**b**) CDF curve.

**Figure 9.**Influence of the variation coefficient of a parameter on the reliability index. (

**a**) Relationship curve between cohesion and time-varying reliability. (

**b**) Relationship curve between internal friction angle and time-varying reliability. (

**c**) Relationship curve between the concrete cover and time-varying reliability. (

**d**) Relationship curve between chloride concentration and time-varying reliability.

Layer | Depth/m | Cohesive/kPa | Internal Friction Angle/° | Soil Weight/kN/m^{3} | Coefficient of Compressibility/MPa |
---|---|---|---|---|---|

Silty soft soil | 0–14.5 | 20 | 5.0 | 8.9 | 0.51 |

Plastic muddy loam mixed with silt | 14.5–21.8 | 34 | 4.0 | 7.1 | 0.86 |

Medium sand mixed with clay | 21.8–26.9 | 51 | 3.5 | 9.0 | 0.10 |

Containing clay gravel | >26.9 | \ | 40 | 18.0 | 0.18 |

Service Time t/a | Quadratic Response Surface Method | Monte Carlo Method | Error 1/% |
---|---|---|---|

0 | 9.60 | 9.12 | 5.26 |

25 | 8.41 | 8.11 | 3.70 |

50 | 7.60 | 7.91 | −3.92 |

75 | 6.23 | 5.91 | 5.41 |

100 | 4.04 | 4.32 | −6.48 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yin, P.; Wang, K.; Chen, L.; Zhang, Y.; Yang, K.; Wang, J.
Horizontal Bearing Capacity and Reliability of Piles in Coastal Soft Soil Considering the Time-Varying Characteristics. *J. Mar. Sci. Eng.* **2023**, *11*, 247.
https://doi.org/10.3390/jmse11020247

**AMA Style**

Yin P, Wang K, Chen L, Zhang Y, Yang K, Wang J.
Horizontal Bearing Capacity and Reliability of Piles in Coastal Soft Soil Considering the Time-Varying Characteristics. *Journal of Marine Science and Engineering*. 2023; 11(2):247.
https://doi.org/10.3390/jmse11020247

**Chicago/Turabian Style**

Yin, Pingbao, Kang Wang, Lu Chen, Yongjie Zhang, Kaibo Yang, and Jie Wang.
2023. "Horizontal Bearing Capacity and Reliability of Piles in Coastal Soft Soil Considering the Time-Varying Characteristics" *Journal of Marine Science and Engineering* 11, no. 2: 247.
https://doi.org/10.3390/jmse11020247