Research Article Structural Reliability Assessment by Integrating Sensitivity Analysis and Support Vector Machine

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1 athematical Problems i Egieerig, Article ID 5869, 6 pages Research Article Structural Reliability Assessmet by Itegratig Sesitivity Aalysis ad Support Vector achie Shao-Fei Jiag, Da-Bao Fu, 2 ad Si-Yao Wu College of Civil Egieerig, Fuzhou Uiversity, Fuzhou 358, Chia 2 Fuzhou Plaig & Desig Research Istitute, Fuzhou 353, Chia Correspodece should be addressed to Shao-Fei Jiag; cejsf@63.com Received September 23; Accepted 23 December 23; Published 23 Jauary 24 Academic Editor: Orwa Jaber Housheya Copyright 24 Shao-Fei Jiag et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. To reduce the rutime ad esure eough computatio accuracy, this paper proposes a structural reliability assessmet method by the use of sesitivity aalysis (SA) ad support vector machie (SV). The sesitivity aalysis is firstly applied to assess the effect of radom variables o the values of performace fuctio, while the small-ifluece variables are rejected as iput vectors of SV. The, the traied SV is used to classify the iput vectors, which are produced by samplig the residual variables based o their distributios. Fially, the reliability assessmet is implemeted with the aid of reliability theory. A -bar plaar truss is used to validate the feasibility ad efficiecy of the proposed method, ad a performace compariso is made with other existig methods. The results show that the proposed method ca largely save the rutime with less reductio of the accuracy; furthermore, the accuracy usig the proposed method is the highest amog the methods employed.. Itroductio I recet years, a umber of structural reliability assessmet methods, icludig first-order reliability method (FOR) [], respose surface method (RS) [2], ad ote-carlo simulatio method (CS) [3], have bee developed ad applied to practical egieerig structures. Amog these methods, the FOR is usually used to directly estimate structural failure probability i the case of the explicit limit state fuctios. I cotrast, the RS ad CS are widely usedithecasethatthelimitstatefuctiosarecomplex ad implicit. The mai idea of RS is to trasfer the origial implicit limit state fuctio to a approximated explicit expressio, which will the be used for the assessmet of failure probability with the aid of FOR. However, i most cases, the hypothetical explicit expressios ca hardly be foud to represet precisely the origial oliear ad complexfuctios;thusrsusuallycausesauallowable error, eve a wrog assessmet result. CS ot oly is the mostprecisemethodforfailureprobabilityassessmet,but also solves theoretically all of reliability problems. However, CS is a time-cosumig process. It is suitable for solvig such problem whe structural failure probability is small, because a umber of samples are required for the purpose of obtaiig a reasoable result. To overcome the low-fidelity of RS ad low computig efficiecy of CS, several researchers have attempted to costructthelimitstatefuctiobasedotheitellectual techiques, such as artificial eural etworks [4, 5]adSV [6, 7]. Due to the strog small-samples learig ability ad geeralizatio capability of SV [8, 9], it has bee widely used for structural reliability aalysis i various fields. Hurtado ad Alvarez [] regarded reliability problems as model classificatio problems ad combied the SV ad FEA for the assessmet of structural failure probability. Hurtado [] used the statistical learig theory to prove the feasibility of SV i the applicatio of reliability problems. Ji et al. [2] combied RS with SV for structural reliability probability assessmet, ad the results showed that this method is more accurate ad efficiet i compariso with other covetioal methods. Guo ad Bai [3] itroducedtheleastsquares SV for regressio ito reliability aalysis to deal with huge computatioal cost ad huge space demads. The iput vectors of SV model are the variables ifluecig the structural reliability assessmet. For a largescale civil structure, its reliability is affected by a umber

2 2 athematical Problems i Egieerig of variables due to the complex service eviromet ad loadig situatios. If all of ifluecig variables are take ito cosideratio with o regard to their importace i the processofreliabilityassessmet,itwillicreasethesample size of iput variables, complicate the SV model, ad elarge the data storage memory demads while decreasig the classificatio accuracy (CA) of SV model. I fact, some iput variables have slight effect o the reliability assessmet results. Therefore, it is ecessary to elimiate the smallifluecig variables before assessig the structure reliability. Recetly, a series of SA techiques have bee developed ad studied for the purpose of quatifyig the importace of iput variables. These SA techiques are divided ito two classes: global SA methods ad local SA methods [4]. The local SA methods are usually based o differetial ad/or differece theory ad igore the probability distributios of variables, thus ot competet for aalyzig the radom variables i limit state fuctio. The global SA methods cosider ot oly the effects of the probability distributios of idividual iput variables o the output, but also the cotributio of the iteractio amog iput variables o the output. I the past decades, Sobol s SA method [5 2], as a ivaluable tool, has draw researchers attetio because it works well without simplifyig approximatios, eve for the fuctios with large umber of variables. I order to reduce the dimesio of iput samples, simplifyig the SV model i the case of esurig computatio accuracy, this paper presets a ovelty reliability aalysis method based o SA ad SV. The small-ifluece variables i the limit state fuctio are extracted i virtue of Sobol s SA method ad are rejected as iput vectors of SV model. The SV model is traied ad tested by samples of residual variables. The reliability assessmet is implemeted with the aidofreliabilitytheory.tovalidatetheapplicabilityad efficiecy of the proposed method, the reliability assessmet of a -bar plaar truss is employed. I additio, some comparisos are also carried out. 2. Structural Reliability Assessmet ethodologies 2.. Sobol s Global SA ethod. Sobol smethodisavariacebased global SA techique that has bee applied to assess the relative importace of iput variables o the output. It is able to decompose the variace of the output ito terms due to idividual iput variables ad terms due to the iteractios betwee iput variables. Cosider a square itegrable fuctio, f(x) = f(x,x 2,...,x ), as a fuctio of vector of iput variables x i,wherex Ω is the -dimesioal uit hypercube. If the iput variables are mutually idepedet the there exists a iterestig decompositio of f(x): f (x) =f + i= f i (x i )+ <i<j< +f,..., (x,...,x ), f ij (x i,x j )+ () where f isacostat.thetotalumberofsummadsi() is 2. If the followig coditio f i,...,i s dx k = (2) is imposed for k=i,...,i s,where i i s, the decompositio described i () isuique.oreover,all of summads are mutually orthogoal ad ca be obtaied with the aid of multiple itegral: f = Ω f (x) dx (3) f i (x i )= f + f ij (x i,x j )= f f i (x i ) f j (x j )+ f (x) dx i (4) f (x) dx (ij), (5) where x i is the set of iput variables except x i ad x (ij) is the setofiputvariablesexceptx i ad x j.thehigherdimesioal summads are similarly foud except for the last oe that is calculated usig (). Therefore, the partial variaces, D i,...,i s,represetigthe cotributio of each of summads to the total variace of output, ca be expressed as D i,...,i s = with the total variace equal to which ca also be expressed as f 2 i,...,i s dx i dx is (6) D= Ω f 2 (x) dx f 2 (7) D= D i,...,i s. (8) s= i < <i s The relative importace of iput variables is quatified by a set of idices, amely, first-order (S i ) ad total (TS i ) sesitivity idices. The former represets the cotributio of the idividual x i o the total variace without ay iteractios with other iput variables, while the latter refers to the cotributio of all iput variables. I additio, the s- order (s ) sesitivity idex, S is, represets the coupled cotributio of the iteractio amog s iput variables o the total variace. It is give by S is = D is D for is = i,...,i s. (9) I order to ivestigate the total sesitivity idex (TS i ) of iput variables x i,thetotalvariace,d, ca be divided ito two complemetary terms: D i ad D i,whered i deotes the variace due to all of iput variables except x i. Therefore, the

3 athematical Problems i Egieerig 3 total sesitivity idex (TS i ) of iput variable x i is expressed as TS i = D i D. () The variaces i (6) ca be calculated approximately by ote-carlo umerical itegratios, particularly whe the fuctio, f(x), is highly oliear ad/or implicit [6, 2]. The ote-carlo method approximatios for f, D, add i are give by D i = N f = N D = N N m= N m= N f(x () ( i)m,x() m= f(x m ) () f 2 (x m ) f 2 (2) im )f(x() ( i)m,x(2) im ) f 2, (3) where N is the sample size, x m is the samples i the - dimesioal uit hypercube, ad superscripts () ad(2) represet two differet samples, respectively. The x () im ad x(2) im deote that the iput variables x im usethesampledvalues i samples () ad(2), respectively. The x () ( i)m ad x(2) ( i)m represet cases whe all the iput variables except x im use the sampled values i samples ()ad(2), respectively. Usually, the iput variables whose total sesitivity idices are less tha.3 are cosidered to be slight of cotributio o the output of f(x) [22]. Therefore, it is reasoable to defie a threshold value of.5 to elimiate the small-ifluece iput variables SV. SV is a emergig machie learig techique that has bee successfully applied to patter classificatio ad regressio aalysis. It is based o the Vapik-Chervoekis dimesio of statistical learig theory ad the priciple of structural risk miimizatio; thus it has a better geeralizatio capability tha the covetioal classificatio methods. This is based o the priciple of empirical risk miimizatio. Suppose a set of traiig examples x ={x l l=,...,} are iput vectors i space x l R d with associated labels y l {,+} ( : label for class I, +: label for class II). Some kerel fuctios are used to map the iput samples to a high-dimesioal feature space so that the overlapped samples i the origial space become liearly separable i the feature space. Therefore, there exist hyperplaes separatig the positive examples o oe side ad the egative samples o the other side. The hyperplaes are give by y l (w T K(x, x l )+b) l=,...,, (4) where w ad b are the weight vector ad bias of hyperplae, respectively. Amog these separatig hyperplaes, the oe so-called optimal separatig hyperplae (OSH) separates all vectors without error ad the distace betwee the closest vectors to the hyperplae is maximal. The OSH is foud by miimizig w 2 uder costraits of (4). Therefore, the primal form of objective fuctio is mi w,b s.t. 2 wt w y l (w T K(x, x l )+b). (5) The Lagrage multipliers, α l, are employed to solve the above problem. Cosequetly, the optimal problem is rewritte as a dual form: max L (α) = s.t. α l, l= α l 2 w 2 = l l=α y 2 l y p K(x l, x p ) l=p= l= α l y l =. (6) I the case of liearly oseparable traiig data, by itroducig slack variables, ξ l,theobjectiveproblemisgive by mi w,b,ξ ξ l l= 2 wt w +C s.t. y l (w T K(x, x l )+b) ξ l, ξ l, (7) where C is the regularizig (margi) parameter that determies the trade-off betwee the maximizatio of the margi ad miimizatio of the classificatio error. Similarly, the correspodig dual problem is expressed as max L (α) = s.t. as l= α l C, α l 2 w 2 = l l=α y 2 l y p K(x l, x p ) l=p= l= α l y l =. (8) With the OSH foud, the decisio fuctio ca be writte where α l f (x) = sg ( l= y l α l K(x, x l)+b ), (9) ad b are the parameters of OSH, respectively ethodology for Structural Reliability Assessmet. The reliability assessmet based o SA ad SV ca be implemeted as follows. Step. Calculate the total sesitivity idices of each iput variable i the limit state fuctio, f(x), byote-carlo umerical itegratios ad elimiate those whose total sesitivity idices are less tha.5.

4 4 athematical Problems i Egieerig Variables 2 P Figure : -bar plaar truss. Table : Distributio types of radom variables. P 3 P 2 A i P P 2 P 3 (m 2 ) (N) (N) (N) ea value Coefficiet of variatio Distributio types Normal Normal Normal Normal Step 2. The samples used for the SA i Step are also selected as the trai samples for traiig SV model. It is oted that the colum of trai samples is the residual iput variable. The umber of trai samples is defied as N trai.thesetof failure ad ofailure samples are defied as class I ad class II, respectively. Use the trai samples ad associated classes to trai the SV model. Step 3. Produce the test samples of residual variables accordig to their distributios. The umber of test samples is N test (N test N trai ). The traied SV model is used to classify the test samples. Step 4. Cout the umber of samples located i class I. Cosequetly, the failure probability, P f,is P f = N f, (2) N test where N f is the umber of test samples located i class I. 3. Case Study: -Bar Plaar Truss 3.. Geeral Descriptio. A umerical -bar plaar truss has bee adopted to validate the proposed reliability assessmet method. The youg s modulus of each bar is E = 2. 8 KN/m 2. The sectioal area of each bar, A i (i=,...,), adloadsappliedtothetruss(asshowifigure ), P i (i =,...,3), were assumed to be radom variables. Their distributio types are listed i Table. The ultimate stregth of this material is assumed to be 48 Pa. Cosequetly, the limit state fuctio, f(x),cabe expressed as f (x) = 48 max {σ (A,...,A,P,...,P 3 ),..., σ (A,...,A,P,...,P 3 )}. (2) TS.5.5 A A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A P P 2 P 3 Variables Figure 2: Sesitivity idices of each variable SA of Iput Variables. The Sobol s method was employed to aalyze the cotributio of each variable o the output variace of limit state fuctio. Firstly, the Lati hypercube samplig techique [23] wasusedtoproducetwosetsof samples with size of 5 3. The, the values of f, D,ad D i arecalculatedbasedo() (3). Fially, the total sesitivity idices of each variable ca be obtaied by (). The SA resultsofiputvariablesareshowifigure 2. Itisfoud that the sesitivity idices of A 2, A 4, A 7, A 8, A 9,adA are less tha., which idicates that these variables have slight cotributio o limit state fuctio. Therefore, these variables are rejected as iput vectors of SV model Support Vector Classifier. It is oted that a total umber of 4 5 = 7 samples are calculated i the process of global SA i Sectio 3.2. These samples are also regarded as traiig data of SV but the small-ifluece variables were rejected as the iput vector of SV model. Therefore, the size of trai samples is 7 7. AsmetioediSectio 2.3, classiad classiiareitroducedtodescribethefailureadofailure sample set. The SV model is traied by use of the trai samples ad correspodig sample labels. Gaussia radial basisfuctioisascertaiedaskerelfuctioofthesv model. The value of pealty term, C, ad kerel parameter, σ, are determied as 2 6 ad.2. A total umber of 2 test samples are costructed ad iput ito the traied SV model. The classificatio results of test samples are show i Table 2.ItisshowiTable 2 that a total umber of = 46 test samples are classified ito class I. Cosequetly, the correspodig structural failure probability is 7.3% Comparisos ad Discussios Computatio Precise. It is observed i Table 2 that the total CA of test samples is 96.34%, showig a excellet classificatio capability. However, it is also oted that the CA of

5 athematical Problems i Egieerig 5 Table 2: Classificatio results. Classificatio umber for differet classes ethod Samples Class Number of samples CA (%) Total CA (%) I II I 6 Trai SA ad SV II I 396 Test II I 6 Trai SV II I 396 Test II Table 3: Failure probabilities evaluated by differet methods. ethod CS RS SV SA ad SV Failure probability (%) Error (%) Amout of FEA class I is oly 72.3%. The mai reasos of this pheomeo ca be summed as two aspects: (a) the OSH i SV model is the approximate limit state fuctio, ad the samples early the limit state fuctio maybe misclassified. However, it is see that the umber of misclassificatio samples i class I is 389, almost equal to the umber of misclassificatio samples (399). It idicates that misclassificatio samples have a slight effect o the assessmet results; (b) the sample umber of class I is far less tha that of class II. Whe the same umber of misclassificatio samples shows up i classes I ad II, the CA of class I drops more rapidly tha that of class II. I order to validate the applicability, other three structure reliability assessmet methods (i.e., RS, CS, ad SV) areemployedtoevaluatethestructuralfailureprobability. The failure probabilities evaluated by these four methods are listed i Table 3. Usually, the results from CS are regarded as the exact solutio. It is foud that the result evaluated by the proposed method is closest to that by CS, idicatig that it is the most precise method amog these methods employed except CS. The classificatio result of SV model is also listed i Table 2. It is obvious that i total the CA of test samples is 95.28%, slightly less tha the proposed method (96.6%). The mai reaso is that the small-ifluece iput variables affect the shape of OSH, thus causig more distortio of OSH tha that of the proposed method Computatio Efficiecy. The amout of FEA required by each method is also listed i Table 3. It is foud that the least amout of FEA required is RS, which oly eeds 27 times. However, its accuracy is the worst. The amout of FEA required by the proposed method ad SV model are farlessthathecs,whiletherelativeerrorsare.72% ad 3.%, respectively. However, the size of test samples is 2 3, two times that of the proposed method (2 7). This idicates that the proposed method ca largely reduce the rutime i case of esurig computatio precise i compariso with the SV model. It ca be predicted that the proposed method ca largely reduce the data storage memory requiremets for the reliability assessmet of a more complex structures. 4. Coclusive Remarks I this study, a ovelty reliability assessmet method based o SA ad SV has bee developed ad successfully applied forreliabilityassessmetofa-barplaartruss.theresults show that the proposed method ot oly reduces data storage memory requiremets with eough computatio accuracy, but also has a better assessmet capability i compariso with other methods. The proposed assessmet method itegratig both SA ad SV is proved to be a successful example. However, it should be oted as well that our success i the proposed method was oly achieved through umerical simulatios, ad more field tests should be doe to testify its feasibility ad efficiecy i practice. Coflict of Iterests We declare that we do ot have ay commercial or associative iterest that represets a coflict of iterest i coectio with the work submitted. Ackowledgmets The work is supported by the Natioal Natural Sciece Foudatio of Chia (os ad ), the Ph.D. Programs Foudatio of iistry of Educatio (o ), ad the Natioal 2th Five-Year Research Program of Chia (o. 22BAJ4B5), Chia. Refereces [] S. ahadeva, Probability, Reliability, ad Statistical ethods i Egieerig Desig,Wiley,2. [2] C. G. Bucher ad U. Bourgud, A fast ad efficiet respose surface approach for structural reliability problems, Structural Safety,vol.7,o.,pp.57 66,99.

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