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

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1 Mathematical Problem in Engineering Volume 2016, Article ID , 7 page Reearch Article Reliability of Foundation Pile Baed on Settlement and a Parameter Senitivity Analyi Shujun Zhang, 1,2 Luo Zhong, 1 and Zhijun Xu 3 1 Wuhan Univerity of Technology, School of Civil Engineering and Architecture, Wuhan , China 2 Nanyang Intitute of Technology, School of Civil Engineering, Nanyang , China 3 Henan Univerity of Technology, School of Civil Engineering and Architecture, Zhengzhou , China Correpondence hould be addreed to Shujun Zhang; -j-z668@163com Received 19 November 2015; Accepted 14 February 2016 Academic Editor: Egidiju R Vaidoga Copyright 2016 Shujun Zhang et al Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited Baed on the uncertainty analyi to calculation model of ettlement, the formula of reliability index of foundation pile i derived Baed on thi formula, the influence of coefficient of variation of the calculated ettlement at pile head, coefficient of variation of the permiible limit of the ettlement, coefficient of variation of the meaured ettlement, afety coefficient, and the mean value of calculation model coefficient on reliability i analyzed The reult indicate that (1) high reliability index can be obtained by increaing afety coefficient; (2) reliability index will be reduced with increaing of the mean value of calculation model coefficient and coefficient of variation of the permiible limit of ettlement; (3) reliability index will not alway monotonically increae or decreae with increaing of coefficient of variation of the calculated ettlement 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 hould be determined through the derived formula when value of other independent variable are determined 1 Introduction Analyi on reliability of foundation pile i one of the important reearch topic in the field of geotechnical engineering In order to make in-depth reearch to thi topic, expert and cholar have done a lot of work But mot of thee reearche areconductedbaedonbearingcapacityoffoundationpile [1, 2] Some theoretical method which are ued to analyze ettlement of ingle foundation pile and foundation pile group have emerged o far uch a load tranfer method, integral equation method, finite element method, approximate analyi method, and hybrid analyi approach [3] However, in thee reearche, parameter of oil property i baically conidered a a fixed value It i obviouly limited becaue the oil ma i variable in pace, there are many uncertaintie in ampletet,andthetetdataobtainedfromthetetingiteha very high variability Thu, it ha become a trend in analyi and reearch on ettlement of foundation pile to take into account ome variability parameter a variable and adopt uncertainty analyi method Bian [3], after analyzing the influence of patial variability of oil ma on ettlement of ingle foundation pile uing random finite element method, pointed out that ettlement of pile tip i influenced by uncertainty and randomne of parameter and then performed analyi to reliability of ettlement of foundation pile by taking thee uncertaintie into account Quek et al [4] performed analyi to reliability of ettlement of foundation pile uing deign drawing method baed on uncertain parameter, and they alo conidered relationhip between parameter a uncertaintie Zhang and Ng [5] invetigated probability ditribution of permiible limit of ettlement under erviceability limit tate of foundation pile uing a large quantity of data they collected, laying a certain foundation for deign of foundation pile under erviceability limit tate from deign baed on certainty to reliability deign Zhang and Phoon [6] took into account all the uncertain factor when they reearch reliability deign of foundation pile They conidered the meaured ettlement, etimated ettlement, and permiible limit of ettlement a random variable, thu giving the method for reliability

2 2 Mathematical Problem in Engineering deign under erviceability limit tate Bian et al [7] got formula of reliability of ingle pile under everal common deign method of foundation pile and analyzed the influence of deviation factor of thee everal foundation pile deign method on the bearing capacity reliability of foundation pile Bian et al [8] got formula for reliability of foundation pile under ultimate limit tate and erviceability limit tate and tudied the influence of randomne of allowable ettlement of pile head on reliability analyi reult under erviceability limit tate By combining failure probability of intact pile and pile which ha ediment at the bae; Li and Yan [9] obtained failure probability of ingle pile and got formula of deviation factor and coefficient of variation of foundation pile which ha ediment at the bae Itcanbeeenfromtheabovethatreearchonreliability of ettlement of foundation pile ha become a focu in the field of engineering However, the actual ettlement of foundation pile i mall and the difficulty for meaurement i a little high Meanwhile, the meaurement reult ha large deviation and dicretene All thee above mentioned would influence the reliability index and make the invetigation on the reliability index of foundation pile highly rich Although a lot of formula have been derived in the literature, and the effectivene of thee formula ha been proved through ome teting data, it i till neceary and beneficial to recognize enitive parameter in many affecting variable In thi paper, formula of reliability index of ettlement of foundation pile (ingle pile) i derived through the etablihment of limit tate equation baed on uncertainty analyi on ettlement model of foundation pile The influence of coefficient of variation of the calculated ettlement, coefficient of variation of the permiible limit of ettlement, and coefficient of variation of the meaured ettlement on reliability i tudied a well The reult of analyi can be ued a reference and guide for deign, contruction, and quality inpection of foundation pile engineering 2 Model Coefficient and Limit State Equation Load tranfer method i a commonly ued method to calculate ettlement of a foundation pile It divide the foundation pile into everal elatic unit It i aumed that each unit i connected with the oil ma (including pile tip) through nonlinear pring to imulate load tranfer mechanim between foundation pile and oil Nonlinearity of interaction between foundation pile and oil a well a tratification characteritic of oil can be correctly decribed through thi method, which i uitable for calculation of ettlement of ingle pile However, continuity of oil property i not fully conidered in thi method [10] A one of the difficult problem in foundation pile engineering, ettlement etimation of foundation pile i influenced by many uncertaintie, uch a phyical dimenion of pile body, pile formation technology, property of oil around the pile, and meaurement error [11] Thu, the calculation reult of ettlement of foundation pile i of high uncertainty Under vertical load, the meaured ettlement of pile head iaumedtobes m and the etimated ettlement of pile head according to ettlement formula of foundation pile i S p ThecalculatedettlementofpileheadS p i conidered a a random variable to reflect uncertainty of calculation model of ettlement In order to make quantitative evaluation to uncertainty of calculation model of ettlement, the ratio of the meaured ettlement S m to the calculated one S p i ued to define model coefficient: λ S m S p (1) Then, the uncertainty of calculation model of foundation pile can be decribed through the mean value and the coefficient of variation of λ Xu [12] collected tatic load tet data of 60 foundation pile with H-type teel frame and tudied it ettlement property by nonlinear load tranfer method After tatitical analyi, the mean value of calculation model λ and coefficient of variation were arrived: that i, μ λ 125 and COV λ 023 In addition, under the influence of ite environment, contruction condition, meaurement equipment, and tet technology, the permiible limit of ettlement S tol and the meaured ettlement S m are of highly uncertain Therefore, the permiible limit of ettlement S tol and the meaured ettlement S m are conidered a random variable in reliability analyi of ettlement of foundation pile Vertical ettlement of pile head cannot exceed the permiible limit of ettlement S tol under normal working condition A function can be etablihed according to the permiible limit of ettlement S tol and the meaured ettlement S m : g(s tol,s m )S tol S m (2) A two-dimenional tate pace i etablihed through formula (2), which take the permiible limit of ettlement S tol and the meaured ettlement S m a the variable, wherein (1) g(s tol,s m ) S tol S m 0 mean that foundation pile reache the limit tate under the working vertical load, which i the critical tate It can be ued a the limit tate curve, dividing the entire tate pace into ecurity domain and failure domain; (2) g(s tol,s m )S tol S m >0indicate that foundation pile i in a afe tate and the repreentative domain i aecuritydomain; (3) g(s tol,s m )S tol S m <0mean that foundation pile i in a failure tate and the repreentative domain i a failure domain In order to effectively make reliability analyi of the ettlement with probability method, ditribution pattern of the permiible limit of ettlement S tol and the meaured ettlement S m hould be determined The cholar have made a lot of invetigation on thi topic Zhang and Xu [13] reearched the vertical ettlement of pile head of 149 foundation pile into which H-teel i inerted under tatic

3 Mathematical Problem in Engineering 3 load tet condition and analyzed relevant reult uing loadettlement model of foundation pile Zhang and Phoon [6] took logarithmic normal ditribution a the ditribution pattern of the meaured ettlement S m of pile head in their reearche Zhang and Ng [5] collected a huge ma of data about ettlement of foundation pile of bridge By uing tatitical analyi method, they found that a logarithmic normal ditribution can be ued for the permiible limit of ettlement S tol model coefficient λ areaumedtoobeythelogarithmicnormal ditribution Then the failure probability of a foundation pilecanbewrittenin p f Prob (S tol <S m )Prob ( S tol S m <1) Prob [ln ( S tol S m )<0] 3 Calculation of Reliability Index According to (2), if the meaured ettlement S m i greater than the permiible limit of ettlement S tol,thefoundation pile i invalid and cannot meet the deign requirement of foundation pile engineering Let p f and β repreent the failure probability of foundation pile due to ettlement and the correponding reliability index, repectively According to knowledge of mathematical tatitic and reliability theory [14, 15], relation between failure probability and reliability index read Prob {ln [( S tol S p ) ( S p S m < 0} Prob {ln [ (S tol/s p ) λ ]<0} Prob [ln (S tol ) ln (S p ) ln (λ )<0] Φ( μ S N μ tol S N p μ λ N ) σ 2 +σ 2 +σ 2 S N S N tol p λ N (4) p f Prob (S tol <S m )Φ( β) (3) wherein Φ( ) ithecumulativeprobabilityditributionfunction of the tandard normal ditribution According to the exiting reearch data, the permiible limit of ettlement S tol, the meaured ettlement S m,and are the mean value of ln(s tol ), ln(s p ) and ln(λ ),repectivelywhileσ S N, σ S tol p N,andσ λ N are wherein μ S N tol, μ S N p,andμ λ N tandard deviation of ln(s tol ),ln(s p ),andln(λ ),repectively According to (3) and (4), the reliability index β of foundation pile baed on ettlement analyi i β μ S N μ tol S N p μ λ N ln [μ Stol / 1+COV 2 S tol ] ln [μ Sp / 1+COV 2 S p ] ln [μ λ / 1+COV 2 λ ] σ 2 +σ 2 +σ 2 S N S N tol p λ N ln [(1 + COV 2 S tol ln [(FS/μ λ ) (1 + COV 2 S p )/(1+COV 2 S tol ln [(1 + COV 2 S tol (5) wherein μ Stol, μ Sp,andμ λ are the mean value of variable S tol, S p,andλ,repectively;cov Stol,,andCOV λ are the variable coefficient of variable S tol, S p,andλ,repectively; FS i afety coefficient defined by FS μ S tol μ (6) Sp In the invetigation of ettlement, Skempton and Mac- Donald [16] uggeted that the baic afety coefficient hould not be lower than 125 for differential ettlement and maximum ettlement and that the afety coefficient hould not be lower than 150 for building under torion Thu, reliability analyi of foundation pile baed on characteritic of ettlement hould ue higher afety coefficient According to (1), S m λ S p (7) Baedon(7),weaumethatλ and S p are uncorrelated The tatitical characteritic of the meaured ettlement S m of pile headcanbeetimatedby μ Sm μ λ μ Sp, COV Sm COV 2 S p + COV 2 λ 4 Parameter Senitivity Analyi According to (5), the ettlement reliability of foundation pile i influenced by afety coefficient FS, the mean value μ λ of calculation model coefficient, coefficient of variation COV Stol of the permiible limit of ettlement of foundation pile, coefficient of variation of the calculated ettlement of foundation pile, and coefficient of variation COV λ of calculation model coefficient Thu, it i difficult to (8)

4 4 Mathematical Problem in Engineering determine ettlement reliability of foundation pile However, the influence of each parameter on reliability index can be recognized through parameter enitivity analyi, and then high reliability domain and low reliability domain can be obtained, which can be ued a reference in engineering 41 Influence of Coefficient of Variation of the Permiible Limit of Settlement According to (5), COV Stol COV Stol (1 + COV 2 S tol ) ln [(1 + COV 2 S tol COV Stol ln ((FS/μ λ ) (1 + COV 2 S p )/(1+COV 2 S tol )) (1 + COV 2 S tol )(ln [(1 + COV 2 S tol ) 3/2 (9) According to the cope of independent variable and formula (5), / COV Stol i contantly maller than 0 Thu, the reliability index will alway decreae with the increae of the coefficient of variation of the permiible limit of ettlement of foundation pile Figure 1, baed on formula (5), how the influence of the permiible limit of ettlement of foundation pile on reliability index in the circumtance of a et of pecial parameter, wherein FS 30, μ λ 125,andCOV λ 023 Itcanbeeenthat,withtheincreaeofthecoefficient of variation of the permiible limit of ettlement, reliability index gradually decreae, which i conitent with the law given by (9) In the particular condition given in Figure 1, a greater reliability index can be obtained when coefficient of variation COV tol of the permiible limit of ettlement i between 00 and 04, and coefficient of variation of the meaured ettlement i near zero In thi domain, however, reliability index will be ignificantly changed when COV tol change within Influence of Coefficient of Variation of Calculation Model Coefficient In a imilar way, from (5) COV λ COV λ (1 + COV 2 λ ) ln [(1 + COV 2 S tol COV λ ln ((FS/μ λ ) (1 + COV 2 S p )/(1+COV 2 S tol )) (1 + COV 2 S p )(ln [(1 + COV 2 S tol ) 3/2 (10) It can be een from formula (10) that the ign of / COV λ i related to the value of the independent variable: that i, the reliability index can increae and decreae with the increae of the coefficient of variation of calculation model coefficient The reliability index reache it maximum value when the value of COV λ meet 2 β/ 2 COV λ 0Inthereal engineering,(10)canbeuedaaguidanceforobtaininghigh reliability index For example, the ign of / COV λ can be determined uing formula (10) when the afety coefficient FS, the mean value μ λ of calculation model coefficient and coefficient of variation COV tol of the permiible limit of ettlement are known Subequently, thi reult can be ued to determine the range of independent variable (coefficient of variation COV λ of calculation model coefficient and coefficient of variation of the meaured ettlement) which can make the reliability reache the deired level According to (5), the influence of coefficient of variation of calculation model coefficient on reliability index under thi et of pecial circumtance can be obtained taking COV λ a 00, 02, 04, 06, 08, and 10 in equence, FS 30, μ λ 125, and COV tol 0583, and taking coefficient of variation of the meaured ettlement a independent variable The calculation reult i a hown in Figure 2 It can be een from Figure 2 that reliability index doe not continuouly decreae with increaing of coefficient of variation of calculation model coefficient Reliability index increae with increaing of coefficient of variation of calculation model coefficient when coefficient of variation of the meaured ettlement i greater than 09 Thi i conitent with the law given by (10) At the ame time, when COV λ change within [0, 1], reliabilityindexionlyinthevariation

5 Mathematical Problem in Engineering 5 region [106, 132] Thi indicate that the change of coefficient of variation of calculation model coefficient doe not ignificantly influence reliability index under the particular circumtance of thi et of parameter repreentative 43 Influence of Coefficient of Variation of the Meaured Settlement From (5), (1 + COV 2 S p ) ln [(1 + COV 2 S tol ln ((FS/μ λ ) (1 + COV 2 S p )/(1+COV 2 S tol )) (1 + COV 2 S p )(ln [(1 + COV 2 S tol ) 3/2 (11) Itcanbeeenfrom(11)thattheignof/ i related to the value of the independent variable: that i, reliability index can increae and decreae with the increae of coefficient of variation of the meaured ettlement Thi reult i imilar to the value of / COV λ In real engineering, the ign of / can be determined uing (11) when the afety coefficient FS, the mean value μ λ of calculation model coefficient, and coefficient of variation COV tol of the permiiblelimitofettlementareknownthireultcan alo be ued to determine the range of independent variable (coefficient of variation COV λ of calculation model coefficient and coefficient of variation of the meaured ettlement) which can make the reliability reache the deired level It can alo be een from the particular cae hown in Figure 1 and 2 that coefficient of variation of the meaured ettlement ha influence on the reliability index and that the reliability index can increae and decreae with theincreaeof, which i conitent with the law given by (11) 44 Influence of Safety Coefficient According to (5), FS 1 FS ln [(1 + COV 2 S tol (12) Since the independent variable in formula (12) are greater than 0, we can eaily know that / FS >0,whichmean reliability index will alway increae with the increae of the afety coefficient Figure 3, baed on (5), how the influence of the afety coefficient on the reliability index in the circumtance of a et of pecial parameter: that i, COV λ 023, μ λ 125,and COV tol 0583 It can be een that the law given by Figure 3 i conitent with that by (12) and that the reliability index alway increae with afety coefficient 45 Influence of the Mean Value of Calculation Model Coefficient According to (5), μ λ 1 μ λ ln [(1 + COV 2 S tol (13) In a imilarly way, independent variable in formula (13) are greater than 0, and thu / μ λ <0,whichmeanthatthe increae of the mean value of calculation model coefficient reduce the reliability index According to (5), the influence of the mean value of calculation model coefficient on reliability index under thi particular condition can be obtained taking COV λ 023, COV tol 0583, FS 30, and 05,ahownin Figure4Itcanbeeenthatreliabilityindexcontinuouly reduce with the increae of the mean value of calculation model coefficient, which i conitent with the concluion obtained through (13) 5 Analyi of Dicuion Reult It can be een from (9) and Figure 1 that the lower the coefficient of variation of the permiible limit of ettlement i, the greater the reliability index i, and ideal reliability index can be obtained when coefficient of variation of the permiible limit of ettlement take value from 0 to 02 It indicate that it i eaier to obtain high reliability index when dicretene of the permiible limit of ettlement i mall In real engineering, it i very important engineering value to try to obtain accurate permiible limit of ettlement The reult hown in (12) and Figure 3 indicate that reliabilityindexcanbeimprovedwhenafetycoefficienti increaed, which mean that it ha a clear poitive ignificance to improve reliability index of engineering uing greater afety coefficient in engineering practice Formula (13) and Figure 4 how that high reliability index canbeobtainedwithalowmeanvalueofcalculationmodel coefficient Thi indicate that it i eaier to obtain high reliability index when the ratio of the meaured ettlement

6 6 Mathematical Problem in Engineering β 3 2 β COV tol 00 COV tol 02 COV tol COV tol 06 COV tol 08 COV tol 10 Figure 1: Relation between reliability index β and coefficient of variable COV tol FS 30 FS 35 FS 40 FS 45 FS 50 Figure 3: Relation between reliability index β and coefficient of variable β β COV λ 00 COVλ 02 COVλ 04 COV λ 06 COVλ 08 COVλ 10 Figure 4: Relation between reliability index β and the mean value of calculation model coefficient μ λ Figure 2: Relation between reliability index β and coefficient of variable COV λ value to the calculated ettlement value i mall, which obviouly agree with the logic and the practical engineering It mean that we hould try our bet to obtain detailed ettlement obervation data to reduce the ratio of the meaured ettlement value to the calculated ettlement value and finally improve the reliability index Formula (10) and (11) indicate that partial derivative of the reliability index to the coefficient of variation of the calculated ettlement and the coefficient of variation of calculation modelcoefficientcanbegreatthan0orlethan0,which mean that reliability index will not alway monotonically increae or decreae with increaing of coefficient of variation of the calculated ettlement and coefficient of variation of calculation model coefficient Intead, it increae ometime and then decreae in the other cae Thi can be een from the particular circumtance hown in Figure 1 and 2 Therefore, to get a high reliability index, coefficient of variation of calculation model coefficient or value range of coefficient of variation of calculation model coefficient hould be determined through the derived formula when value of other independent variable are determined

7 Mathematical Problem in Engineering 7 6 Concluion In thi paper, baed on analyi to ettlement of foundation pile,limittateequationareetablihedformulaforreliability index of foundation pile ettlement i derived, and a parameter enitivity analyi i conducted The following concluion can be arrived: (1) With the increae of the coefficient of variation of the permiible limit of ettlement, the reliability index gradually reduce (2) Gradual increae of the reliability index can be obtained by the increae of the afety coefficient (3) The reliability index alway decreae with the increae of the mean value of calculation model coefficient (4) Reliability index will not alway monotonically increae or decreae with the increae of the coefficient of variation of the calculated ettlement and the coefficient of variation of calculation model coefficient Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper Acknowledgment Thank are due to the National Natural Science Foundation of China ( ), High-Level Talent Fund Program of Henan Univerity of Technology (2013BS010), and Henan Baic and Cutting-Edge Technology Reearch Plan Program ( ) for upporting the preent work [8] X-Y Bian, J-J Zheng, and Z-J Xu, Reliability analyi of erviceability limit tate of foundation pile conidering uncertaintie of parameter and model, RockandSoilMechanic,vol 35,no11,pp ,2014(Chinee) [9] D-Q Li and L-L Yan, Reliability evaluation of bored pile conidering poibility of toe debri preence, Rock and Soil Mechanic,vol29,no1,pp ,2008 [10] M-H Zhao, D Zou, and X-J Zou, Settlement calculation of pile group by load tranfer method, Engineering Mechanic, vol 23, no 7, pp , 2006 [11] M-H Zhao, D Zou, and X-J Zou, Settlement calculation of pile foundation with elevated cap by load tranfer method, Chinee Rock Mechanic and Engineering,vol24,no 13, pp , 2005 [12] Y Xu, Calibration of Settlement Analyi Model for Single Pile and Pile Group, Hong Kong Univerity of Science and Technology, Hong Kong, 2006 [13] L Zhang and Y Xu, Settlement of building foundation baed on field pile load tet, in Proceeding of the 16th International Conference on Soil Mechanic and Geotechnical Engineering (ICSMGE 05), pp , Oaka, Japan, September 2005 [14] A H-S Ang and W H Tang, Probability Concept in Engineering: Emphai on Application to Civil and Environmental Engineering, John Wiley & Son, New York, NY, USA, 2nd edition, 2007 [15]GBBaecherandJTChritian,Reliability and Statitic in Geotechnical Engineering, JohnWiley&Son,Chicheter,UK, 2003 [16] A W Skempton and D H MacDonald, The allowable ettlement of building, Proceeding of the Intitution of Civil Engineer,vol3,no5,pp ,1956 Reference [1] E Momeni, R Nazir, D Jahed Armaghani, and H Maizir, Prediction of pile bearing capacity uing a hybrid genetic algorithm-baed ANN, Meaurement,vol57,pp ,2014 [2] F Milad, T Kamal, H Nader, and O E Erman, New method for predicting the ultimate bearing capacity of driven pile by uing Flap number, KSCE Civil Engineering, vol19, no 3, pp , 2015 [3] XYBian,Reliability Analyi and Deign of Pile Foundation for Serviceability Limit State, Huazhong Univerity of Science and Technology, Wuhan, China, 2014 (Chinee) [4] S T Quek, Y K Chow, and K K Phoon, Further contribution to reliability-baed pile ettlement analyi, Geotechnical Engineering, vol 118, no 5, pp , 1992 [5] LMZhangandAMYNg, Probabiliticlimitingtolerable diplacement for erviceability limit tate deign of foundation, Geotechnique,vol55,no2,pp ,2005 [6] L Zhang and K Phoon, Serviceability conideration in reliability-baed foundation deign, Foundation Analyi and Deign: Innovative Method,GSP153,pp ,2006 [7] X-Y Bian, J-J Zheng, and Z-J Xu, Reliability analyi of bearing capacity of foundation pile conidering pile foundation deign approach, Chinee Geotechnical Engineering, vol35,no2,pp ,2013(Chinee)

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