Research Article Reliability of Foundation Pile Based on ...

Research ArticleReliability of Foundation Pile Based on Settlement anda Parameter Sensitivity Analysis

Shujun Zhang,1,2 Luo Zhong,1 and Zhijun Xu3

1Wuhan University of Technology, School of Civil Engineering and Architecture, Wuhan 430070, China2Nanyang Institute of Technology, School of Civil Engineering, Nanyang 473004, China3Henan University of Technology, School of Civil Engineering and Architecture, Zhengzhou 450001, China

Correspondence should be addressed to Shujun Zhang; [email protected]

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.

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 ofthe permissible limit of the settlement, coefficient of variation of the measured settlement, safety coefficient, and the mean valueof calculation model coefficient on reliability is analyzed. The results indicate that (1) high reliability index can be obtained byincreasing safety coefficient; (2) reliability index will be reduced with increasing of the mean value of calculation model coefficientand coefficient of variation of the permissible limit of settlement; (3) reliability index will not always monotonically increase ordecrease with increasing of coefficient of variation of the calculated settlement and coefficient of variation of calculation modelcoefficient. To get a high reliability index, coefficient of variation of calculation model coefficient or value range of coefficient ofvariation of calculation model coefficient should be determined through the derived formula when values of other independentvariables are determined.

1. Introduction

Analysis on reliability of foundation pile is one of the impor-tant research topics in the field of geotechnical engineering.In order to make in-depth research to this topic, experts andscholars have done a lot of work. But most of these researchesare conducted based on bearing capacity of foundation pile[1, 2]. Some theoretical methods which are used to analyzesettlement of single foundation pile and foundation pilegroup have emerged so far such as load transfermethod, inte-gral equation method, finite element method, approximateanalysis method, and hybrid analysis approach [3]. However,in these researches, parameter of soil property is basicallyconsidered as a fixed value. It is obviously limited because thesoil mass is variable in space, there are many uncertainties insample test, and the test data obtained from the testing site hasvery high variability. Thus, it has become a trend in analysisand research on settlement of foundation pile to take intoaccount some variability parameters as variables and adoptuncertainty analysis method.

Bian [3], after analyzing the influence of spatial variabilityof soil mass on settlement of single foundation pile usingrandom finite element method, pointed out that settlementof pile tip is influenced by uncertainty and randomnessof parameter and then performed analysis to reliability ofsettlement of foundation pile by taking these uncertaintiesinto account. Quek et al. [4] performed analysis to reliabilityof settlement of foundation pile using design drawingmethodbased on uncertain parameters, and they also consideredrelationship between parameters as uncertainties. Zhang andNg [5] investigated probability distribution of permissiblelimit of settlement under serviceability limit state of founda-tion pile using a large quantity of data they collected, layinga certain foundation for design of foundation pile underserviceability limit state from design based on certainty toreliability design. Zhang and Phoon [6] took into account allthe uncertain factors when they research reliability design offoundation pile. They considered the measured settlement,estimated settlement, and permissible limit of settlementas random variables, thus giving the method for reliability

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 1659549, 7 pageshttp://dx.doi.org/10.1155/2016/1659549

2 Mathematical Problems in Engineering

design under serviceability limit state. Bian et al. [7] gotformula of reliability of single pile under several commondesignmethods of foundation pile and analyzed the influenceof deviation factor of these several foundation pile designmethods on the bearing capacity reliability of foundation pile.Bian et al. [8] got formula for reliability of foundation pileunder ultimate limit state and serviceability limit state andstudied the influence of randomness of allowable settlementof pile head on reliability analysis result under serviceabilitylimit state. By combining failure probability of intact pile andpile which has sediment at the base; Li and Yan [9] obtainedfailure probability of single pile and got formula of deviationfactor and coefficient of variation of foundation pile whichhas sediment at the base.

It can be seen from the above that research on reliabilityof settlement of foundation pile has become a focus inthe field of engineering. However, the actual settlement offoundation pile is small and the difficulty for measurementis a little high. Meanwhile, the measurement result has largedeviation and discreteness. All these above mentioned wouldinfluence the reliability index and make the investigation onthe reliability index of foundation pile highly rich. Althougha lot of formulas have been derived in the literature, and theeffectiveness of these formulas has been proved through sometesting data, it is still necessary and beneficial to recognizesensitive parameters inmany affecting variables. In this paper,formula of reliability index of settlement of foundation pile(single pile) is derived through the establishment of limit stateequation based on uncertainty analysis on settlement modelof foundation pile. The influence of coefficient of variationof the calculated settlement, coefficient of variation of thepermissible limit of settlement, and coefficient of variationof the measured settlement on reliability is studied as well.The result of analysis can be used as reference and guide fordesign, construction, and quality inspection of foundationpile engineering.

2. Model Coefficient and Limit State Equation

Load transfer method is a commonly used method tocalculate settlement of a foundation pile. It divides thefoundation pile into several elastic units. It is assumedthat each unit is connected with the soil mass (includingpile tip) through nonlinear spring to simulate load transfermechanism between foundation pile and soil. Nonlinearityof interaction between foundation pile and soil as well asstratification characteristic of soil can be correctly describedthrough this method, which is suitable for calculation ofsettlement of single pile. However, continuity of soil propertyis not fully considered in this method [10].

As one of the difficult problems in foundation pileengineering, settlement estimation of foundation pile is influ-enced by many uncertainties, such as physical dimension ofpile body, pile formation technology, property of soil aroundthe pile, and measurement error [11]. Thus, the calculationresult of settlement of foundation pile is of high uncertainty.Under vertical load, the measured settlement of pile headis assumed to be 𝑆

𝑚and the estimated settlement of pile

head according to settlement formula of foundation pile is𝑆𝑝. The calculated settlement of pile head 𝑆

𝑝is considered as

a random variable to reflect uncertainty of calculation modelof settlement.

In order to make quantitative evaluation to uncertaintyof calculation model of settlement, the ratio of the measuredsettlement 𝑆

𝑚to the calculated one 𝑆

𝑝is used to definemodel

coefficient:

𝜆𝑠=𝑆𝑚

𝑆𝑝

. (1)

Then, the uncertainty of calculationmodel of foundation pilecan be described through the mean value and the coefficientof variation of 𝜆

𝑠.

Xu [12] collected static load test data of 60 foundationpiles with H-type steel frame and studied its settlementproperty by nonlinear load transfer method. After statisticalanalysis, the mean value of calculation model 𝜆

𝑠and coef-

ficient of variation were arrived: that is, 𝜇𝜆𝑠

= 1.25 andCOV𝜆𝑠

= 0.23.In addition, under the influence of site environment,

construction condition, measurement equipment, and testtechnology, the permissible limit of settlement 𝑆tol and themeasured settlement 𝑆

𝑚are of highly uncertain. Therefore,

the permissible limit of settlement 𝑆tol and the measured set-tlement 𝑆

𝑚are considered as random variables in reliability

analysis of settlement of foundation pile.Vertical settlement of pile head cannot exceeds the

permissible limit of settlement 𝑆tol under normal workingcondition. A function can be established according to the per-missible limit of settlement 𝑆tol and the measured settlement𝑆𝑚:

𝑔 (𝑆tol, 𝑆𝑚) = 𝑆tol − 𝑆𝑚. (2)

A two-dimensional state space is established through formula(2), which takes the permissible limit of settlement 𝑆tol andthe measured settlement 𝑆

𝑚as the variables, wherein

(1) 𝑔(𝑆tol, 𝑆𝑚) = 𝑆tol − 𝑆𝑚

= 0 means that foundationpile reaches the limit state under the working verticalload, which is the critical state. It can be used as thelimit state curve, dividing the entire state space intosecurity domain and failure domain;

(2) 𝑔(𝑆tol, 𝑆𝑚) = 𝑆tol − 𝑆𝑚> 0 indicates that foundation

pile is in a safe state and the representative domain isa security domain;

(3) 𝑔(𝑆tol, 𝑆𝑚) = 𝑆tol − 𝑆𝑚 < 0means that foundation pileis in a failure state and the representative domain is afailure domain.

In order to effectively make reliability analysis of thesettlement with probability method, distribution pattern ofthe permissible limit of settlement 𝑆tol and the measuredsettlement 𝑆

𝑚should 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 149foundation piles into which H-steel is inserted under static

Mathematical Problems in Engineering 3

load test condition and analyzed relevant results using load-settlement model of foundation pile. Zhang and Phoon [6]took logarithmic normal distribution as the distributionpattern of the measured settlement 𝑆

𝑚of pile head in their

researches. Zhang and Ng [5] collected a huge mass ofdata about settlement of foundation pile of bridge. By usingstatistical analysis method, they found that a logarithmicnormal distribution can be used for the permissible limit ofsettlement 𝑆tol.

3. Calculation of Reliability Index

According to (2), if the measured settlement 𝑆𝑚is greater

than the permissible limit of settlement 𝑆tol, the foundationpile is invalid and cannot meet the design requirement offoundation pile engineering.

Let 𝑝𝑓and 𝛽 represent the failure probability of founda-

tion pile due to settlement and the corresponding reliabilityindex, respectively. According to knowledge of mathematicalstatistics and reliability theory [14, 15], relation betweenfailure probability and reliability index reads

𝑝𝑓= Prob (𝑆tol < 𝑆

𝑚) = Φ (−𝛽) (3)

wherein Φ(⋅) is the cumulative probability distribution func-tion of the standard normal distribution.

According to the existing research data, the permissiblelimit of settlement 𝑆tol, the measured settlement 𝑆

𝑚, and

model coefficient 𝜆𝑠are assumed to obey the logarithmic nor-

mal distribution.Then the failure probability of a foundationpile can be written in

𝑝𝑓= Prob (𝑆tol < 𝑆

𝑚) = Prob(

𝑆tol𝑆𝑚

< 1)

= Prob [ln(𝑆tol𝑆𝑚

) < 0]

= Prob{ln[(𝑆tol𝑆𝑝

) × (𝑆𝑝

𝑆𝑚

)] < 0}

= Prob{ln[(𝑆tol/𝑆𝑝)

𝜆𝑠

] < 0}

= Prob [ln (𝑆tol) − ln (𝑆𝑝) − ln (𝜆

𝑠) < 0]

= Φ(−𝜇𝑆𝑁

tol− 𝜇𝑆𝑁

𝑝

− 𝜇𝜆𝑁

𝑠

√𝜎2

𝑆𝑁

tol+ 𝜎2𝑆𝑁

𝑝

+ 𝜎2𝜆𝑁

𝑠

)

(4)

wherein 𝜇𝑆𝑁

tol, 𝜇𝑆𝑁

𝑝

, and 𝜇𝜆𝑁

𝑠

are the mean value of ln(𝑆tol),ln(𝑆𝑝) and ln(𝜆

𝑠), respectively. While 𝜎

𝑆𝑁

tol, 𝜎𝑆𝑁

𝑝

, and 𝜎𝜆𝑁

𝑠

arestandard deviation of ln(𝑆tol), ln(𝑆𝑝), and ln(𝜆𝑠), respectively.

According to (3) and (4), the reliability index 𝛽 offoundation pile based on settlement analysis is

𝛽 =𝜇𝑆𝑁

tol− 𝜇𝑆𝑁

𝑝

− 𝜇𝜆𝑁

𝑠

√𝜎2

𝑆𝑁

tol+ 𝜎2𝑆𝑁

𝑝

+ 𝜎2𝜆𝑁

𝑠

=

ln [𝜇𝑆tol/√1 + COV2

𝑆tol] − ln [𝜇

𝑆𝑝

/√1 + COV2𝑆𝑝

] − ln [𝜇𝜆𝑠

/√1 + COV2𝜆𝑠

]

√ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)]

=

ln [(FS/𝜇𝜆𝑠

)√(1 + COV2𝑆𝑝

) (1 + COV2𝜆𝑠

) / (1 + COV2𝑆tol)]


𝑆𝑝

) (1 + COV2𝜆𝑠

)]

(5)

wherein 𝜇𝑆tol, 𝜇𝑆𝑝

, and 𝜇𝜆𝑠

are the mean value of variables 𝑆tol,𝑆𝑝, and 𝜆

𝑠, respectively; COV

𝑆tol, COV

𝑆𝑝

, and COV𝜆𝑠

are thevariable coefficient of variables 𝑆tol, 𝑆𝑝, and 𝜆𝑠, respectively;FS is safety coefficient defined by

FS =𝜇𝑆tol

𝜇𝑆𝑝

. (6)

In the investigation of settlement, Skempton and Mac-Donald [16] suggested that the basic safety coefficient shouldnot be lower than 1.25 for differential settlement and maxi-mum settlement and that the safety coefficient should not belower than 1.50 for buildings under torsion. Thus, reliabilityanalysis of foundation pile based on characteristic of settle-ment should use higher safety coefficient.

According to (1),𝑆𝑚= 𝜆𝑠𝑆𝑝. (7)

Based on (7), we assume that 𝜆𝑠and 𝑆𝑝are uncorrelated.The

statistical characteristic of the measured settlement 𝑆𝑚of pile

head can be estimated by

𝜇𝑆𝑚

= 𝜇𝜆𝑠

𝜇𝑆𝑝

,

COV𝑆𝑚

= √COV2𝑆𝑝

+ COV2𝜆𝑠

.(8)

4. Parameter Sensitivity Analysis

According to (5), the settlement reliability of foundationpile is influenced by safety coefficient FS, the mean value𝜇𝜆𝑠

of calculation model coefficient, coefficient of variationCOV𝑆tol

of the permissible limit of settlement of founda-tion pile, coefficient of variation COV

𝑆𝑝

of the calculatedsettlement of foundation pile, and coefficient of variationCOV𝜆𝑠

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 berecognized through parameter sensitivity analysis, and thenhigh reliability domain and low reliability domain can beobtained, which can be used as reference in engineering.

4.1. Influence of Coefficient of Variation of the Permissible Limitof Settlement. According to (5),

𝜕𝛽

𝜕COV𝑆tol

= −COV𝑆tol

(1 + COV2𝑆tol)√ln [(1 + COV2

𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)]

−

COV𝑆tolln((FS/𝜇

𝜆𝑠

)√(1 + COV2𝑆𝑝

) (1 + COV2𝜆𝑠

) / (1 + COV2𝑆tol))

(1 + COV2𝑆tol) (ln [(1 + COV2

𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)])3/2

.

(9)

According to the scope of independent variable and formula(5), 𝜕𝛽/𝜕COV

𝑆tolis constantly smaller than 0. Thus, the

reliability index will always decrease with the increase of thecoefficient of variation of the permissible limit of settlementof foundation pile.

Figure 1, based on formula (5), shows the influence ofthe permissible limit of settlement of foundation pile onreliability index in the circumstance of a set of specialparameters, wherein FS = 3.0, 𝜇

𝜆𝑠

= 1.25, and COV𝜆𝑠

= 0.23.It can be seen that, with the increase of the coefficient

of variation of the permissible limit of settlement, reliability

index gradually decreases, which is consistent with the lawgiven by (9). In the particular condition given in Figure 1,a greater reliability index can be obtained when coefficientof variation COVtol of the permissible limit of settlement isbetween 0.0 and 0.4, and coefficient of variationCOV

𝑆𝑝

of themeasured settlement is near zero. In this domain, however,reliability index will be significantly changed when COVtolchanges within 0.0–0.4.

4.2. Influence of Coefficient of Variation of Calculation ModelCoefficient. In a similar way, from (5)

𝜕𝛽

𝜕COV𝜆𝑠

=COV𝜆𝑠

(1 + COV2𝜆𝑠

)√ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)]

−

COV𝜆𝑠

ln((FS/𝜇𝜆𝑠

)√(1 + COV2𝑆𝑝

) (1 + COV2𝜆𝑠

) / (1 + COV2𝑆tol))

(1 + COV2𝑆𝑝

) (ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)])3/2

.

(10)

It can be seen from formula (10) that the sign of 𝜕𝛽/𝜕COV𝜆𝑠

is related to the value of the independent variables: thatis, the reliability index can increase and decrease with theincrease of the coefficient of variation of calculation modelcoefficient. The reliability index reaches its maximum valuewhen the value of COV

𝜆𝑠

meets 𝜕2𝛽/𝜕2COV𝜆𝑠

= 0. In the realengineering, (10) can be used as a guidance for obtaining highreliability index. For example, the sign of 𝜕𝛽/𝜕COV

𝜆𝑠

canbe determined using formula (10) when the safety coefficientFS, the mean value 𝜇

𝜆𝑠

of calculation model coefficient andcoefficient of variation COVtol of the permissible limit ofsettlement are known. Subsequently, this result can be usedto determine the range of independent variable (coefficientof variation COV

𝜆𝑠

of calculation model coefficient andcoefficient of variation COV

𝑆𝑝

of the measured settlement)which can make the reliability reaches the desired level.

According to (5), the influence of coefficient of variationof calculationmodel coefficient on reliability index under thisset of special circumstance can be obtained taking COV

𝜆𝑠

as0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 in sequence, FS = 3.0, 𝜇

𝜆𝑠

= 1.25,and COVtol = 0.583, and taking coefficient of variation of themeasured settlement as independent variable.The calculationresult is as shown in Figure 2.

It can be seen from Figure 2 that reliability index doesnot continuously decrease with increasing of coefficient ofvariation of calculation model coefficient. Reliability indexincreases with increasing of coefficient of variation of calcula-tionmodel coefficient when coefficient of variation COV

𝑆𝑝

ofthe measured settlement is greater than 0.9.This is consistentwith the law given by (10). At the same time, when COV

𝜆𝑠

changes within [0, 1], reliability index is only in the variation


region [1.06, 1.32]. This indicates that the change of coeffi-cient of variation of calculation model coefficient does notsignificantly influence reliability index under the particularcircumstance of this set of parameter representative.

4.3. Influence of Coefficient of Variation of the MeasuredSettlement. From (5),

𝜕𝛽

𝜕COV𝑆𝑝

=COV𝑆𝑝

(1 + COV2𝑆𝑝

)√ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)]

−

COV𝑆𝑝

ln((FS/𝜇𝜆𝑠

)√(1 + COV2𝑆𝑝

) (1 + COV2𝜆𝑠

) / (1 + COV2𝑆tol))

(1 + COV2𝑆𝑝

) (ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)])3/2

.

(11)

It can be seen from (11) that the sign of 𝜕𝛽/𝜕COV𝑆𝑝

isrelated to the value of the independent variables: that is,reliability index can increase and decreasewith the increase ofcoefficient of variation of themeasured settlement.This resultis similar to the value of 𝜕𝛽/𝜕COV

𝜆𝑠

. In real engineering,the sign of 𝜕𝛽/𝜕COV

𝑆𝑝

can be determined using (11) whenthe safety coefficient FS, the mean value 𝜇

𝜆𝑠

of calculationmodel coefficient, and coefficient of variation COVtol of thepermissible limit of settlement are known. This result canalso be used to determine the range of independent variable(coefficient of variation COV

𝜆𝑠

of calculation model coef-ficient and coefficient of variation COV

𝑆𝑝

of the measuredsettlement) which canmake the reliability reaches the desiredlevel.

It can also be seen from the particular case shown inFigures 1 and 2 that coefficient of variation COV

𝑆𝑝

of themeasured settlement has influence on the reliability indexand that the reliability index can increase and decrease withthe increase of COV

𝑆𝑝

, which is consistent with the law givenby (11).

4.4. Influence of Safety Coefficient. According to (5),

𝜕𝛽

𝜕FS

=1

FS√ln [(1 + COV2𝑆tol) (1 + COV2

𝑆𝑝

) (1 + COV2𝜆𝑠

)]

.(12)

Since the independent variables in formula (12) are greaterthan 0, we can easily know that 𝜕𝛽/𝜕FS > 0, which meansreliability index will always increase with the increase of thesafety coefficient.

Figure 3, based on (5), shows the influence of the safetycoefficient on the reliability index in the circumstance of a setof special parameter: that is, COV

𝜆𝑠

= 0.23, 𝜇𝜆𝑠

= 1.25, andCOVtol = 0.583. It can be seen that the law given by Figure 3is consistent with that by (12) and that the reliability indexalways increases with safety coefficient.

4.5. Influence of the Mean Value of Calculation Model Coeffi-cient. According to (5),

𝜕𝛽

𝜕𝜇𝜆𝑠

= −1

𝜇𝜆𝑠


𝑆𝑝

) (1 + COV2𝜆𝑠

)]

.(13)

In a similarly way, independent variables in formula (13) aregreater than 0, and thus 𝜕𝛽/𝜕𝜇

𝜆𝑠

< 0, which means that theincrease of the mean value of calculation model coefficientreduces the reliability index.

According to (5), the influence of the mean value ofcalculation model coefficient on reliability index under thisparticular condition can be obtained taking COV

𝜆𝑠

= 0.23,COVtol = 0.583, FS = 3.0, and COV

𝑆𝑝

= 0.5, as shown inFigure 4. It can be seen that reliability index continuouslyreduces with the increase of the mean value of calculationmodel coefficient, which is consistent with the conclusionobtained through (13).

5. Analysis of Discussion Result

It can be seen from (9) and Figure 1 that the lower thecoefficient of variation of the permissible limit of settlementis, the greater the reliability index is, and ideal reliabilityindex can be obtained when coefficient of variation of thepermissible limit of settlement takes value from 0 to 0.2. Itindicates that it is easier to obtain high reliability index whendiscreteness of the permissible limit of settlement is small. Inreal engineering, it is very important engineering value to tryto obtain accurate permissible limit of settlement.

The result shown in (12) and Figure 3 indicates thatreliability index can be improved when safety coefficient isincreased, whichmeans that it has a clear positive significanceto improve reliability index of engineering using greatersafety coefficient in engineering practice.

Formula (13) and Figure 4 show that high reliability indexcan be obtained with a low mean value of calculation modelcoefficient. This indicates that it is easier to obtain highreliability index when the ratio of the measured settlement


𝛽

0

1

2

3

4

5

0.2 0.4 0.6 0.8 1.00.0

S𝑝COV

tol = 0.0COV

tol = 1.0COVtol = 0.8COVtol = 0.6COV

tol = 0.4COVtol = 0.2COV

Figure 1: Relation between reliability index 𝛽 and coefficient ofvariable COVtol.

0.2 0.4 0.6 0.8 1.00.01.0

1.1

𝛽 1.2

1.3

1.4

S𝑝COV

= 0.0COV𝜆𝑠= 0.2COV𝜆𝑠= 0.4COV𝜆𝑠

= 0.6COV𝜆𝑠= 0.8COV𝜆𝑠= 1.0COV𝜆𝑠

Figure 2: Relation between reliability index 𝛽 and coefficient ofvariable COV

𝜆𝑠

.

value to the calculated settlement value is small, whichobviously agrees with the logic and the practical engineering.It means that we should try our best to obtain detailed set-tlement observation data to reduce the ratio of the measuredsettlement value to the calculated settlement value and finallyimprove the reliability index.

Formulas (10) and (11) indicate that partial derivative ofthe reliability index to the coefficient of variation of the calcu-lated settlement and the coefficient of variation of calculationmodel coefficient can be great than 0 or less than 0, which

𝛽

FS = 3.0 FS = 3.5FS = 4.0 FS = 4.5FS = 5.0

0.1 0.3 0.5 0.7 0.9 1.1−0.1COVS𝑝

0.0

0.5

1.0

1.5

2.0

2.5

Figure 3: Relation between reliability index 𝛽 and coefficient ofvariable COV

𝑆𝑝

.

0.5 1.0 1.5 2.00.0𝜇𝜆𝑠

0

1

2

𝛽

4

5

3

Figure 4: Relation between reliability index 𝛽 and the mean valueof calculation model coefficient.

means that reliability index will not always monotonicallyincrease or decrease with increasing of coefficient of variationof the calculated settlement and coefficient of variation ofcalculation model coefficient. Instead, it increases sometimesand then decreases in the other cases. This can be seenfrom the particular circumstance shown in Figures 1 and2. Therefore, to get a high reliability index, coefficient ofvariation of calculation model coefficient or value range ofcoefficient of variation of calculationmodel coefficient shouldbe determined through the derived formula when values ofother independent variables are determined.


6. Conclusion

In this paper, based on analysis to settlement of foundationpile, limit state equations are established. Formula for reli-ability index of foundation pile settlement is derived, anda parameter sensitivity analysis is conducted. The followingconclusion can be arrived:

(1) With the increase of the coefficient of variation of thepermissible limit of settlement, the reliability indexgradually reduces.

(2) Gradual increase of the reliability index can beobtained by the increase of the safety coefficient.

(3) The reliability index always decreases with theincrease of the mean value of calculation modelcoefficient.

(4) Reliability index will not always monotonically in-crease or decrease with the increase of the coefficientof variation of the calculated settlement and the coef-ficient of variation of calculation model coefficient.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Thanks are due to the National Natural Science Foundationof China (51178165), High-Level Talent Fund Program ofHenan University of Technology (2013BS010), and HenanBasic and Cutting-Edge Technology Research Plan Program(132300410024) for supporting the present work.

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