research article reliability of foundation pile based on
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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 ยฉ 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
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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 ๐
๐and the estimated settlement of pile
head according to settlement formula of foundation pile is๐๐. The calculated settlement of pile head ๐
๐is 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 ๐
๐to the calculated one ๐
๐is used to definemodel
coefficient:
๐๐ =๐๐
๐๐
. (1)
Then, the uncertainty of calculationmodel of foundation pilecan be described through the mean value and the coefficientof variation of ๐
๐ .
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 ๐
๐ and coef-
ficient of variation were arrived: that is, ๐๐๐
= 1.25 andCOV๐๐
= 0.23.In addition, under the influence of site environment,
construction condition, measurement equipment, and testtechnology, the permissible limit of settlement ๐tol and themeasured settlement ๐
๐are of highly uncertain. Therefore,
the permissible limit of settlement ๐tol and the measured set-tlement ๐
๐are considered as random variables in reliability
analysis of settlement of foundation pile.Vertical settlement of pile head cannot exceeds the
permissible limit of settlement ๐tol under normal workingcondition. A function can be established according to the per-missible limit of settlement ๐tol and the measured settlement๐๐:
๐ (๐tol, ๐๐) = ๐tol โ ๐๐. (2)
A two-dimensional state space is established through formula(2), which takes the permissible limit of settlement ๐tol andthe measured settlement ๐
๐as the variables, wherein
(1) ๐(๐tol, ๐๐) = ๐tol โ ๐๐
= 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) ๐(๐tol, ๐๐) = ๐tol โ ๐๐> 0 indicates that foundation
pile is in a safe state and the representative domain isa security domain;
(3) ๐(๐tol, ๐๐) = ๐tol โ ๐๐ < 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 ๐tol and the measuredsettlement ๐
๐should 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
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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 ๐
๐of 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 ๐tol.
3. Calculation of Reliability Index
According to (2), if the measured settlement ๐๐is greater
than the permissible limit of settlement ๐tol, the foundationpile is invalid and cannot meet the design requirement offoundation pile engineering.
Let ๐๐and ๐ฝ 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
๐๐= Prob (๐tol < ๐
๐) = ฮฆ (โ๐ฝ) (3)
wherein ฮฆ(โ ) is the cumulative probability distribution func-tion of the standard normal distribution.
According to the existing research data, the permissiblelimit of settlement ๐tol, the measured settlement ๐
๐, and
model coefficient ๐๐ are assumed to obey the logarithmic nor-
mal distribution.Then the failure probability of a foundationpile can be written in
๐๐= Prob (๐tol < ๐
๐) = Prob(
๐tol๐๐
< 1)
= Prob [ln(๐tol๐๐
) < 0]
= Prob{ln[(๐tol๐๐
) ร (๐๐
๐๐
)] < 0}
= Prob{ln[(๐tol/๐๐)
๐๐
] < 0}
= Prob [ln (๐tol) โ ln (๐๐) โ ln (๐
๐ ) < 0]
= ฮฆ(โ๐๐๐
tolโ ๐๐๐
๐
โ ๐๐๐
๐
โ๐2
๐๐
tol+ ๐2๐๐
๐
+ ๐2๐๐
๐
)
(4)
wherein ๐๐๐
tol, ๐๐๐
๐
, and ๐๐๐
๐
are the mean value of ln(๐tol),ln(๐๐) and ln(๐
๐ ), respectively. While ๐
๐๐
tol, ๐๐๐
๐
, and ๐๐๐
๐
arestandard deviation of ln(๐tol), ln(๐๐), and ln(๐๐ ), respectively.
According to (3) and (4), the reliability index ๐ฝ offoundation pile based on settlement analysis is
๐ฝ =๐๐๐
tolโ ๐๐๐
๐
โ ๐๐๐
๐
โ๐2
๐๐
tol+ ๐2๐๐
๐
+ ๐2๐๐
๐
=
ln [๐๐tol/โ1 + COV2
๐tol] โ ln [๐
๐๐
/โ1 + COV2๐๐
] โ ln [๐๐๐
/โ1 + COV2๐๐
]
โln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
=
ln [(FS/๐๐๐
)โ(1 + COV2๐๐
) (1 + COV2๐๐
) / (1 + COV2๐tol)]
โln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
(5)
wherein ๐๐tol, ๐๐๐
, and ๐๐๐
are the mean value of variables ๐tol,๐๐, and ๐
๐ , respectively; COV
๐tol, COV
๐๐
, and COV๐๐
are thevariable coefficient of variables ๐tol, ๐๐, and ๐๐ , respectively;FS is safety coefficient defined by
FS =๐๐tol
๐๐๐
. (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),๐๐= ๐๐ ๐๐. (7)
Based on (7), we assume that ๐๐ and ๐๐are uncorrelated.The
statistical characteristic of the measured settlement ๐๐of pile
head can be estimated by
๐๐๐
= ๐๐๐
๐๐๐
,
COV๐๐
= โCOV2๐๐
+ COV2๐๐
.(8)
4. Parameter Sensitivity Analysis
According to (5), the settlement reliability of foundationpile is influenced by safety coefficient FS, the mean value๐๐๐
of calculation model coefficient, coefficient of variationCOV๐tol
of the permissible limit of settlement of founda-tion pile, coefficient of variation COV
๐๐
of the calculatedsettlement of foundation pile, and coefficient of variationCOV๐๐
of calculation model coefficient. Thus, it is difficult to
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4 Mathematical Problems in Engineering
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),
๐๐ฝ
๐COV๐tol
= โCOV๐tol
(1 + COV2๐tol)โln [(1 + COV2
๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
โ
COV๐tolln((FS/๐
๐๐
)โ(1 + COV2๐๐
) (1 + COV2๐๐
) / (1 + COV2๐tol))
(1 + COV2๐tol) (ln [(1 + COV2
๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)])3/2
.
(9)
According to the scope of independent variable and formula(5), ๐๐ฝ/๐COV
๐tolis 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, ๐
๐๐
= 1.25, and COV๐๐
= 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
๐๐
of themeasured settlement is near zero. In this domain, however,reliability index will be significantly changed when COVtolchanges within 0.0โ0.4.
4.2. Influence of Coefficient of Variation of Calculation ModelCoefficient. In a similar way, from (5)
๐๐ฝ
๐COV๐๐
=COV๐๐
(1 + COV2๐๐
)โln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
โ
COV๐๐
ln((FS/๐๐๐
)โ(1 + COV2๐๐
) (1 + COV2๐๐
) / (1 + COV2๐tol))
(1 + COV2๐๐
) (ln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)])3/2
.
(10)
It can be seen from formula (10) that the sign of ๐๐ฝ/๐COV๐๐
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
๐๐
meets ๐2๐ฝ/๐2COV๐๐
= 0. In the realengineering, (10) can be used as a guidance for obtaining highreliability index. For example, the sign of ๐๐ฝ/๐COV
๐๐
canbe determined using formula (10) when the safety coefficientFS, the mean value ๐
๐๐
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
๐๐
of calculation model coefficient andcoefficient of variation COV
๐๐
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
๐๐
as0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 in sequence, FS = 3.0, ๐
๐๐
= 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
๐๐
ofthe measured settlement is greater than 0.9.This is consistentwith the law given by (10). At the same time, when COV
๐๐
changes within [0, 1], reliability index is only in the variation
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Mathematical Problems in Engineering 5
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),
๐๐ฝ
๐COV๐๐
=COV๐๐
(1 + COV2๐๐
)โln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
โ
COV๐๐
ln((FS/๐๐๐
)โ(1 + COV2๐๐
) (1 + COV2๐๐
) / (1 + COV2๐tol))
(1 + COV2๐๐
) (ln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)])3/2
.
(11)
It can be seen from (11) that the sign of ๐๐ฝ/๐COV๐๐
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 ๐๐ฝ/๐COV
๐๐
. In real engineering,the sign of ๐๐ฝ/๐COV
๐๐
can be determined using (11) whenthe safety coefficient FS, the mean value ๐
๐๐
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
๐๐
of calculation model coef-ficient and coefficient of variation COV
๐๐
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
๐๐
of themeasured settlement has influence on the reliability indexand that the reliability index can increase and decrease withthe increase of COV
๐๐
, which is consistent with the law givenby (11).
4.4. Influence of Safety Coefficient. According to (5),
๐๐ฝ
๐FS
=1
FSโln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
.(12)
Since the independent variables in formula (12) are greaterthan 0, we can easily know that ๐๐ฝ/๐FS > 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
๐๐
= 0.23, ๐๐๐
= 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),
๐๐ฝ
๐๐๐๐
= โ1
๐๐๐
โln [(1 + COV2๐tol) (1 + COV2
๐๐
) (1 + COV2๐๐
)]
.(13)
In a similarly way, independent variables in formula (13) aregreater than 0, and thus ๐๐ฝ/๐๐
๐๐
< 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
๐๐
= 0.23,COVtol = 0.583, FS = 3.0, and COV
๐๐
= 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
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6 Mathematical Problems in Engineering
๐ฝ
0
1
2
3
4
5
0.2 0.4 0.6 0.8 1.00.0
S๐COV
tol = 0.0COV
tol = 1.0COVtol = 0.8COVtol = 0.6COV
tol = 0.4COVtol = 0.2COV
Figure 1: Relation between reliability index ๐ฝ and coefficient ofvariable COVtol.
0.2 0.4 0.6 0.8 1.00.01.0
1.1
๐ฝ 1.2
1.3
1.4
S๐COV
= 0.0COV๐๐ = 0.2COV๐๐ = 0.4COV๐๐
= 0.6COV๐๐ = 0.8COV๐๐ = 1.0COV๐๐
Figure 2: Relation between reliability index ๐ฝ and coefficient ofvariable COV
๐๐
.
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
๐ฝ
FS = 3.0 FS = 3.5FS = 4.0 FS = 4.5FS = 5.0
0.1 0.3 0.5 0.7 0.9 1.1โ0.1COVS๐
0.0
0.5
1.0
1.5
2.0
2.5
Figure 3: Relation between reliability index ๐ฝ and coefficient ofvariable COV
๐๐
.
0.5 1.0 1.5 2.00.0๐๐๐
0
1
2
๐ฝ
4
5
3
Figure 4: Relation between reliability index ๐ฝ 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.
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Mathematical Problems in Engineering 7
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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