Mathematical Problems in Engineering

Volume 2016, Article ID 1659549, 7 pages

http://dx.doi.org/10.1155/2016/1659549

## Reliability of Foundation Pile Based on Settlement and a Parameter Sensitivity Analysis

^{1}Wuhan University of Technology, School of Civil Engineering and Architecture, Wuhan 430070, China^{2}Nanyang Institute of Technology, School of Civil Engineering, Nanyang 473004, China^{3}Henan University of Technology, School of Civil Engineering and Architecture, Zhengzhou 450001, China

Received 19 November 2015; Accepted 14 February 2016

Academic Editor: Egidijus R. Vaidogas

Copyright © 2016 Shujun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on the uncertainty analysis to calculation model of settlement, the formula of reliability index of foundation pile is derived. Based on this formula, the influence of coefficient of variation of the calculated settlement at pile head, coefficient of variation of the permissible limit of the settlement, coefficient of variation of the measured settlement, safety coefficient, and the mean value of calculation model coefficient on reliability is analyzed. The results indicate that high reliability index can be obtained by increasing safety coefficient; reliability index will be reduced with increasing of the mean value of calculation model coefficient and coefficient of variation of the permissible limit of settlement; reliability index will not always monotonically increase or decrease with increasing of coefficient of variation of the calculated settlement and coefficient of variation of calculation model coefficient. To get a high reliability index, coefficient of variation of calculation model coefficient or value range of coefficient of variation of calculation model coefficient should be determined through the derived formula when values of other independent variables are determined.

#### 1. Introduction

Analysis on reliability of foundation pile is one of the important research topics in the field of geotechnical engineering. In order to make in-depth research to this topic, experts and scholars have done a lot of work. But most of these researches are conducted based on bearing capacity of foundation pile [1, 2]. Some theoretical methods which are used to analyze settlement of single foundation pile and foundation pile group have emerged so far such as load transfer method, integral equation method, finite element method, approximate analysis method, and hybrid analysis approach [3]. However, in these researches, parameter of soil property is basically considered as a fixed value. It is obviously limited because the soil mass is variable in space, there are many uncertainties in sample test, and the test data obtained from the testing site has very high variability. Thus, it has become a trend in analysis and research on settlement of foundation pile to take into account some variability parameters as variables and adopt uncertainty analysis method.

Bian [3], after analyzing the influence of spatial variability of soil mass on settlement of single foundation pile using random finite element method, pointed out that settlement of pile tip is influenced by uncertainty and randomness of parameter and then performed analysis to reliability of settlement of foundation pile by taking these uncertainties into account. Quek et al. [4] performed analysis to reliability of settlement of foundation pile using design drawing method based on uncertain parameters, and they also considered relationship between parameters as uncertainties. Zhang and Ng [5] investigated probability distribution of permissible limit of settlement under serviceability limit state of foundation pile using a large quantity of data they collected, laying a certain foundation for design of foundation pile under serviceability limit state from design based on certainty to reliability design. Zhang and Phoon [6] took into account all the uncertain factors when they research reliability design of foundation pile. They considered the measured settlement, estimated settlement, and permissible limit of settlement as random variables, thus giving the method for reliability design under serviceability limit state. Bian et al. [7] got formula of reliability of single pile under several common design methods of foundation pile and analyzed the influence of deviation factor of these several foundation pile design methods on the bearing capacity reliability of foundation pile. Bian et al. [8] got formula for reliability of foundation pile under ultimate limit state and serviceability limit state and studied the influence of randomness of allowable settlement of pile head on reliability analysis result under serviceability limit state. By combining failure probability of intact pile and pile which has sediment at the base; Li and Yan [9] obtained failure probability of single pile and got formula of deviation factor and coefficient of variation of foundation pile which has sediment at the base.

It can be seen from the above that research on reliability of settlement of foundation pile has become a focus in the field of engineering. However, the actual settlement of foundation pile is small and the difficulty for measurement is a little high. Meanwhile, the measurement result has large deviation and discreteness. All these above mentioned would influence the reliability index and make the investigation on the reliability index of foundation pile highly rich. Although a lot of formulas have been derived in the literature, and the effectiveness of these formulas has been proved through some testing data, it is still necessary and beneficial to recognize sensitive parameters in many affecting variables. In this paper, formula of reliability index of settlement of foundation pile (single pile) is derived through the establishment of limit state equation based on uncertainty analysis on settlement model of foundation pile. The influence of coefficient of variation of the calculated settlement, coefficient of variation of the permissible limit of settlement, and coefficient of variation of the measured settlement on reliability is studied as well. The result of analysis can be used as reference and guide for design, construction, and quality inspection of foundation pile engineering.

#### 2. Model Coefficient and Limit State Equation

Load transfer method is a commonly used method to calculate settlement of a foundation pile. It divides the foundation pile into several elastic units. It is assumed that each unit is connected with the soil mass (including pile tip) through nonlinear spring to simulate load transfer mechanism between foundation pile and soil. Nonlinearity of interaction between foundation pile and soil as well as stratification characteristic of soil can be correctly described through this method, which is suitable for calculation of settlement of single pile. However, continuity of soil property is not fully considered in this method [10].

As one of the difficult problems in foundation pile engineering, settlement estimation of foundation pile is influenced by many uncertainties, such as physical dimension of pile body, pile formation technology, property of soil around the pile, and measurement error [11]. Thus, the calculation result of settlement of foundation pile is of high uncertainty. Under vertical load, the measured settlement of pile head is assumed to be and the estimated settlement of pile head according to settlement formula of foundation pile is . The calculated settlement of pile head is considered as a random variable to reflect uncertainty of calculation model of settlement.

In order to make quantitative evaluation to uncertainty of calculation model of settlement, the ratio of the measured settlement to the calculated one is used to define model coefficient:Then, the uncertainty of calculation model of foundation pile can be described through the mean value and the coefficient of variation of .

Xu [12] collected static load test data of 60 foundation piles with H-type steel frame and studied its settlement property by nonlinear load transfer method. After statistical analysis, the mean value of calculation model and coefficient of variation were arrived: that is, and .

In addition, under the influence of site environment, construction condition, measurement equipment, and test technology, the permissible limit of settlement and the measured settlement are of highly uncertain. Therefore, the permissible limit of settlement and the measured settlement are considered as random variables in reliability analysis of settlement of foundation pile.

Vertical settlement of pile head cannot exceeds the permissible limit of settlement under normal working condition. A function can be established according to the permissible limit of settlement and the measured settlement :A two-dimensional state space is established through formula (2), which takes the permissible limit of settlement and the measured settlement as the variables, wherein(1) means that foundation pile reaches the limit state under the working vertical load, which is the critical state. It can be used as the limit state curve, dividing the entire state space into security domain and failure domain;(2) indicates that foundation pile is in a safe state and the representative domain is a security domain;(3) means that foundation pile is in a failure state and the representative domain is a failure domain.

In order to effectively make reliability analysis of the settlement with probability method, distribution pattern of the permissible limit of settlement and the measured settlement should be determined. The scholars have made a lot of investigations on this topic. Zhang and Xu [13] researched the vertical settlement of pile head of 149 foundation piles into which H-steel is inserted under static load test condition and analyzed relevant results using load-settlement model of foundation pile. Zhang and Phoon [6] took logarithmic normal distribution as the distribution pattern of the measured settlement of pile head in their researches. Zhang and Ng [5] collected a huge mass of data about settlement of foundation pile of bridge. By using statistical analysis method, they found that a logarithmic normal distribution can be used for the permissible limit of settlement .

#### 3. Calculation of Reliability Index

According to (2), if the measured settlement is greater than the permissible limit of settlement , the foundation pile is invalid and cannot meet the design requirement of foundation pile engineering.

Let and represent the failure probability of foundation pile due to settlement and the corresponding reliability index, respectively. According to knowledge of mathematical statistics and reliability theory [14, 15], relation between failure probability and reliability index readswherein is the cumulative probability distribution function of the standard normal distribution.

According to the existing research data, the permissible limit of settlement , the measured settlement , and model coefficient are assumed to obey the logarithmic normal distribution. Then the failure probability of a foundation pile can be written inwherein , , and are the mean value of , and , respectively. While , , and are standard deviation of , , and , respectively.

According to (3) and (4), the reliability index of foundation pile based on settlement analysis iswherein , , and are the mean value of variables , , and , respectively; , , and are the variable coefficient of variables , , and , respectively; is safety coefficient defined by

In the investigation of settlement, Skempton and MacDonald [16] suggested that the basic safety coefficient should not be lower than 1.25 for differential settlement and maximum settlement and that the safety coefficient should not be lower than 1.50 for buildings under torsion. Thus, reliability analysis of foundation pile based on characteristic of settlement should use higher safety coefficient.

According to (1),Based on (7), we assume that and are uncorrelated. The statistical characteristic of the measured settlement of pile head can be estimated by

#### 4. Parameter Sensitivity Analysis

According to (5), the settlement reliability of foundation pile is influenced by safety coefficient FS, the mean value of calculation model coefficient, coefficient of variation of the permissible limit of settlement of foundation pile, coefficient of variation of the calculated settlement of foundation pile, and coefficient of variation of calculation model coefficient. Thus, it is difficult to determine settlement reliability of foundation pile. However, the influence of each parameter on reliability index can be recognized through parameter sensitivity analysis, and then high reliability domain and low reliability domain can be obtained, which can be used as reference in engineering.

##### 4.1. Influence of Coefficient of Variation of the Permissible Limit of Settlement

According to (5),According to the scope of independent variable and formula (5), is constantly smaller than 0. Thus, the reliability index will always decrease with the increase of the coefficient of variation of the permissible limit of settlement of foundation pile.

Figure 1, based on formula (5), shows the influence of the permissible limit of settlement of foundation pile on reliability index in the circumstance of a set of special parameters, wherein FS = 3.0, = 1.25, and = 0.23.