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Sensitivity and Reliability Analysis of a Self Anchored Suspension Bridge Jianhui Li1 Aiqun Li2 and Maria Q Feng F ASCE3 Abstract This paper presents a sensitivity and reliability analysis of a self anchored suspension bridge by applying a new hybrid method proposed by the authors based on integration of the Latin hypercube sampling technique LHS artifi cial neural network ANN fi rst order reliability method FORM Pearson s linear correlation coeffi cient PLCC and Monte Carlo simulation with important sampling technique MCS IS The framework consists of three stages of analysis 1 selection of training validation and test datasets for establishing an ANN model by the LHS technique 2 formulation of a performance function from the well trained ANN model and 3 sensitivity analysis using PLCC identifi cation of the most probabilistic failure point based on FORM and estimation of the failure probability using the MCS IS technique Upondemonstrationofitseffi ciencythroughanalysisofa12 storyframestructure themethodisappliedtosensitivityandreliability analysis of the Jiangxinzhou Bridge a fi ve span self anchored suspension bridge in which both structural parameters and external loads are considered as random variables The analysis identifi ed a number of structural parameters as well as external loads that have a signifi cant infl uenceonstructuralserviceabilityandsafety DOI 10 1061 ASCE BE 1943 5592 0000424 2013AmericanSocietyofCivilEngineers CEDatabasesubjectheadings Sensitivityanalysis Structuralreliability Sampling Neuralnetworks MonteCarlomethod Simulation Suspension bridge Author keywords Sensitivity analysis Reliability analysis Latin hypercube sampling Artifi cial neural networks First order reliability method Monte Carlo simulation Important sampling Implicit performance function Self anchored suspension bridge Introduction The self anchored suspension bridge has recently gained popularity in bridge engineering in China because of 1 the appealing aes thetics 2 the relatively small size of the substructure and 3 the development of construction and analysis methods The structural analysis of self anchored suspension bridges has been the subject ofconsiderableresearch Lietal 2009 butthesestudieswerebased on the assumption of complete determinacy of structural parame ters referred to as deterministic analysis In practice however uncertainties in loading and structural parameters are unavoidable which are often associated with geometric properties material mechanical properties and load magnitude and distribution De terministic analysis cannot provide complete information regarding structural behavior of self anchored suspension bridges There fore structural analysis and reliability assessment considering the uncertainties need to be performed from a stochastic and probabi listic viewpoint Thus far compared with static and dynamic re sponse analyses very few studies have appeared in the literature regarding the sensitivity and reliability analysis of self anchored suspension bridge structures Because serviceability is an important consideration in thedesign ofself anchored suspension bridges the reliability of such bridges under serviceability limit conditions should be properly evaluated Sensitivityandreliabilityanalysisofacomplexstructurerequires methods that are not only accurate but also computationally effi cient For a self anchored suspension bridge the structural response is computed by a numerical procedure such as fi nite element FE analysis Its performance function is not available as an explicit closed form function of the input variables A considerable amount of research has been devoted to handle implicit performance func tions including the numerical simulation method the response surface approach and other analytical methods incorporated in the formulation of the FE method and the boundary element method The Monte Carlo simulation MCS is the simplest numerical simulation technique providing an important general solution pro cedure that is limited only by the affordable sample size However this approach has not received an overwhelming acceptance be cause of the excessive computational effort that is required Al thoughseveralsamplingtechniques Olssonetal 2003 Ditlevesen et al 1990 Ayyub and Chia 1992 also referred to as variance reduction techniques can reduce the computational time to a certain extent the computational cost is still unendurable when applying these methods to a self anchored suspension bridge with many variables and considerably small failure probabilities The response surface method RSM is another widely used method in structural reliability analysis Two such response surface functions have been developed in the literature including the polynomial function and the approximate function constructed based on an ar tifi cialneuralnetwork ANN model Elhewyetal 2006 compared theaccuracyandeffi ciencyofpolynomial basedRSMandthatofthe ANN basedRSM ItisconcludedthattheANN basedRSMismore accurate and effi cient than the polynomial based RSM However it is generally believed that the ANN based model would require much more number of experiments than the polynomial based 1Lecturer College of Civil Engineering Nanjing Forestry Univ Nanjing 210037 P R China E mail subway96 2Professor College of Civil Engineering Southeast Univ Nanjing 210096 P R China E mail aiqunli 3Professor Dept of Civil Engineering and Engineering Mechanics Columbia Univ New York NY 10027 corresponding author E mail mqf2101 columbia edu Note ThismanuscriptwassubmittedonApril8 2011 approved on July 5 2012 published online on July 21 2012 Discussion period open until January 1 2014 separate discussions must be submitted for individ ualpapers ThispaperispartoftheJournalofBridgeEngineering Vol 18 No 8 August1 2013 ASCE ISSN 1084 0702 2013 8 703 711 25 00 JOURNAL OF BRIDGE ENGINEERING ASCE AUGUST 2013 703 J Bridge Eng 2013 18 703 711 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 14 14 Copyright ASCE For personal use only all rights reserved model to build an effi cient RSM model There is room for further improvementineffi ciency whichisanimportantissuewhenapplying the ANN based RSM method to a real complex bridge structure In addition the results of all these ANN based methods are highly dependentonthenumberandqualityofinputdatasetsforestablishing a well trained ANN model Generally speaking to achieve a valid ANN model thedataselected forestablishing itmustbe representative of the overall design space The simplest and widely used procedure to generate input datasets is random sampling which has been widely studied and discussed Hurtado and Barbat 1998 A signifi cant dis advantage of the random sampling method is in its lack of guarantee that the selected dataset covers the entire design space uniformly especially when the number of variables is large and the number of input dataset is relatively small Therefore in some cases the ANN model cannot be well trained by these selected input datasets gen erated by the random method To overcome this shortcoming a new hybridmethodisdevelopedinthisstudyandappliedtothesensitivity and reliability analysis of a self anchored suspension bridge Theaccuracyandcomputationaleffi ciencyofthemethodarefi rst demonstrated through analysis of a 12 story frame structure Then the method is applied to sensitivity and reliability analysis of the Jiangxinzhou Bridge a self anchored suspension bridge in which both structural parameters and external loads are considered as ran domvariables Themostinfl uentialrandomvariablesonthereliability of the self anchored suspension bridge are identifi ed through a sen sitivity analysis and the bridge reliability under a normal service ability state is assessed Proposed Method The proposed method aims to improving the computational effi ciency of the structural reliability analysis without sacrifi cing accu racy through integration of the Latin hypercube sampling technique LHS ANN Pearson s linear correlation coeffi cient PLCC fi rst order reliability method FORM and Monte Carlo simulation with important sampling technique MCS IS techniques Latin Hypercube Sampling Technique The LHS technique was fi rst proposed for computer experiments which is designed to accurately recreate the input distribution through sampling in fewer iterations in comparison with simple random sampling McKay et al 1979 LHS proceeds as follows 1 The probability distribution of an input factor is split into N nonoverlapping intervals of equal marginal probability 1 N 2 Inthefi rstiteration oneoftheseintervalsisselectedrandomly 3 Withinthisinterval oneobservationischosenrandomlyandis now the fi rst sample of this specifi c input factor 4 In the second and the rest of the N iterations one interval that hasnotbeenchosenisrandomlyselected andtheprocedureis repeated from Step 3 to obtain the N samples Because there areN samplesgenerated eachoftheN intervalswillhavebeen sampled from only once and 5 Thewholeprocedureisrepeatedforeachofthek inputfactors In addition a method for generating the Latin hypercube and random samples from correlated variables has been developed by Iman and Conover 1982 Artifi cial Neural Network A multilayer feed forward ANN structure trained by back propagation is adopted in this study The processing units in these ANNsarearrangedinlayers i e aninputlayer anoutputlayer and a number of hidden layers as shown in Fig 1 Suppose an ANN model with one hidden layer where the input layer contains I nodes the hidden layer contains J nodes and the output layer contains K nodes Then the output ykcan be expressed as yk f2 bok P J j 1 bjk f1 aoj P I i 1 aij xi 1 where f1and f25 transfer functions or activation functions xi5 input aijand bjk i51 2 I j51 2 J k51 2 K 5 weights and aojand bok5 deviations In this study a three layer ANN model was built with a sigmoid transfer function at the hidden layer and a linear transfer function at output layer Hornik et al 1989 showed that the multilayer feed forward networks with as few as a single hidden layer and an ap propriately smooth hidden layer activation function are capable of arbitrarily accurate approximation to an arbitrary function and its derivatives Extensive studies have been made on ANN issues such as the number of hidden neurons the training algorithm and avoid anceoflocalminima Hiroseetal 1991 HurtadoandAlvarez2001 Apotentialproblemwhentrainingaback propagationneuralnetwork is that the network can overfi t on the training set but not generalize well to new data outside the training set This can be prevented by usinganearlystoppingtechnique DemuthandBeale2000 inwhich the available data are divided into three subsets the training set validation set andtest set Inthetraditional randommethod allinput datasetsarerandomlyselectedbyarandomgeneratoranddividedinto the three subsets A signifi cant disadvantage is the lack of guarantee that the selected dataset covers the entire design space uniformly especially when the number of variables is large and the number of input dataset is relatively small Therefore in some cases the ANN model cannot be well trained by these selected input datasets gen erated by the random method In the proposed method three datasets are created by the LHS technique dependently any of which is guaranteedtoberelativelyuniformlydistributedovereachdimension thus overcoming the aforementioned disadvantage Pearson s Linear Correlation Coeffi cient The Pearson product moment correlation coeffi cient is a dimen sionless index which is invariant to linear transformations of either variable Pearson s correlation coeffi cient Sr sensitivity factor can be calculated from Sr P n i Xi2Xaver Yi2Yaver ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi P n i Xi2Xaver 2 s ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi P n i Yi2Yaver 2 s 2 where Xiand Yi5 input and output of variable i respectively and Xaverand Yaver5 average values of Xiand Yi respectively Fig 1 General structure of an ANN 704 JOURNAL OF BRIDGE ENGINEERING ASCE AUGUST 2013 J Bridge Eng 2013 18 703 711 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 14 14 Copyright ASCE For personal use only all rights reserved The value of the Srranges from 21 to 1 The negative values stand for opposite trends betweeninput variables andoutput results The higher the correlation the more signifi cant the input parameter is in determining the output response To visualize the relative sensitivity of the random variables a new sensitive factor Svis defi ned as Sv S2 r 100 P n i 1 Sv 100 3 First Order Reliability Method The FORM is formulated as an optimization problem with the equality constrains in U space which is defi ned as the performance function minimizekUk S tG U 0 4 where the optimum point on the failure surface is called the most probable point MPP u G U 50 and the fi rst order safety re liability index bs5jju G U 50jj EitheranMPPsearchalgorithmthatisspecifi callydevelopedfor the fi rst order reliability analysis or general optimization algorithms canbeusedtosolveEq 4 TheHasoferLindandRackwitzFiessler HLRF method Rackwitz and Flessler 1978 is used in this paper because of its simplicity and effi ciency Monte Carlo Simulation with Important Sampling Monte Carlo simulation can be stated as follows in structural re liability analysis problems Pf G x 0 fx x dx 1 N P N j 1 I xj 5 where fx x 5 joint probability of failure for all random variables and I xj 5 indicator for successful and unsuccessful simulations Importance sampling IS is generally recognized as the most effi cient reduction technique It had been proven by Papadrakakis et al 1996 that the use of IS can further improve the performance for the prediction of the probability of failure The key idea of this technique is to obtain a nonnegative sampling density located in the neighborhood of the most probable failure point The normal dis tribution function based on MPP is used as the IS function in this study Using the IS technique Eq 5 can be expressed as Pf 1 N P N j 1 I xj f x x j gx xj 6 where gx x 5 IS function Procedure of the Proposed Integrated Method The proposed method represents an integration of the techniques presented previously that involves the following procedure illus trated in Fig 2 1 Applying the LHS technique to select suitable training val idation and test samples from random variables as input datasets 2 Performing structural analysis to obtain deterministic struc tural responses as target datasets 3 Determining the architecture of the ANN model and relative algorithms 4 Training the ANN model with the input and target datasets 5 Extracting the explicit formula of the approximate perfor mance function based on Eq 1 from the well trained ANN model and 6 Applying PLCC to analyze parameter sensitivity based on the extracted formula FORM to obtain the most probabilistic failure point and MCS IS to predict the structural failure probability To apply the proposed method to structural sensitivity and re liabilityanalysis aMATLAB basedprogramisdevelopedbasedon the fl owchart The program is structured in a modular form so that it can be easily applied to different types ofstructures andcommercial software thus facilitating practical applications Application to a Frame Structure A 12 story frame structure as shown in Fig 3 is selected as an example to demonstrate the effi ciency and accuracy of the proposed method The frame consists of 84 members with 104 degrees of freedom The basic random variables as described in Table 1 in clude the column and beam cross sectional areas A1 A2 A3 A4 and A5and the wind load P All the variables are assumed to be Fig 2 Flowchart of the proposed structural reliability analysis procedure JOURNAL OF BRIDGE ENGINEERING ASCE AUGUST 2013 705 J Bridge Eng 2013 18 703 711 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 14 14 Copyright ASCE For personal use only all rights reserved uncorrelated The Young s modulus of all the members is assumed to be deterministic and is equal to 2 03107kN m2 The moments of inertia of the beam and the column correlate with the cross sectional areas as follows Ii giA2 i i 1 2 5 7 where Ii5 moments of inertia and gi 5 coeffi cients whose values are listed in Table 1 Jian 2006 Theperformancefunctionforthestructuralsafetymaybedefi ned as g X 0 0962uA A1 A2 A3 A4 A5 P 8 whereuA5 horizontal displacement at node A as a function of basic random variables In this equation g X 0 indicates failure TheresponsevariableuAisdependentontherandomvariablesand the deterministic structural parameters which cannot be expressed as aclosed formfunction Instead itisevaluatedusingFEanalysis The explicit performance function of this structure obtained by the pro posed method is d g x 20 070K1 0 151K2 0 020K320 046 9 in which K1 1 1 e25 596A122 738A220 791A3214 950A425 639A5 5 168 10 25P 7 192 K2 1 1 e20 875A120 276A220 223A320 946A420 281A5 2 330 10 25P20 182 K3 1 1 e0 381A122 009A221 054A329 213A425 175A526 599 25P 5 415 To evaluate the contribution of the input parameters to response variable uA a sensitivity analysis is carried out by the developed program Histograms of the sensitivity factors are shown in Fig 4 It can be seen that the load P followed by A4and A1 is the most signifi cant infl uence factor with respect to response variable uA In comparison parameters A2 A3 and A5have minimal contribution to the response Therefore only parameters P A1 and A4are con sideredasvariableswhiletheothersaredeterministicinthefollowing analysis The explicit performance function of this situation is d g x 20 021K120 045K2 0 026 10 in which K1 1 1 e1 523A1 3 094A429 417 10 25P 0 308 K2 1 1 e3 252A1 7 411A425 433 10 25P20 108 Performance of the method is compared between the conven tional