Multidimensional parallelepiped model a new type of non-probabilistic convex model for structural uncertainty analysis

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (015) Publshed onlne n Wley Onlne Lbrary (wleyonlnelbrary.com) Multdmensonal parallelepped model a new type of non-probablstc convex model for structural uncertanty analyss C. Jang*,, Q. F. Zhang, X. Han, J. Lu and D. A. Hu State Key Laboratory of Advanced Desgn and Manufacturng for Vehcle Body, College of Mechancal and Vehcle Engneerng, Hunan Unversty, Changsha Cty, Chna SUMMARY Non-probablstc convex models need to be provded only the changng boundary of parameters rather than ther exact probablty dstrbutons; thus, such models can be appled to uncertanty analyss of complex structures when expermental nformaton s lackng. The nterval and the ellpsodal models are the two most commonly used modelng methods n the feld of non-probablstc convex modelng. However, the former can only deal wth ndependent varables, whle the latter can only deal wth dependent varables. Ths paper presents a more general non-probablstc convex model, the multdmensonal parallelepped model. Ths model can nclude the ndependent and dependent uncertan varables n a unfed framework and can effectvely deal wth complex mult-source uncertanty problems n whch dependent varables and ndependent varables coexst. For any two parameters, the concepts of the correlaton angle and the correlaton coeffcent are defned. Through the margnal ntervals of all the parameters and also ther correlaton coeffcents, a multdmensonal parallelepped can easly be bult as the uncertanty doman for parameters. Through the ntroducton of affne coordnates, the parallelepped model n the orgnal parameter space s converted to an nterval model n the affne space, thus greatly facltatng subsequent structural uncertanty analyss. The parallelepped model s appled to structural uncertanty propagaton analyss, and the response nterval of the structure s obtaned n the case of uncertan ntal parameters. Fnally, the method descrbed n ths paper was appled to several numercal examples. Copyrght 015 John Wley & Sons, Ltd. Receved 5 May 01; Revsed 14 September 014; Accepted 19 December 014 KEY WORDS: multdmensonal parallelepped model; convex model; nterval analyss; ellpsodal model; non-probablstc uncertanty; parameter correlaton 1. INTRODUCTION Uncertanty wdely exsts n practcal engneerng problems, and t s commonly related to materal propertes, loads, boundary condtons, and other factors. The probablty model, n whch the dstrbutons of structural responses are obtaned based on statstcal technques [1 5], s most wdely used to quantfy uncertantes of these types. The probablty model has become the prncpal means for dealng wth uncertanty n engneerng and has been successfully appled to varous ndustral applcatons. In the processng of probablty models, a large number of samples are used to construct precse probablty dstrbutons for uncertan parameters; such samples, however, are not always avalable or are sometmes very costly to obtan for practcal reasons. Thus, n many cases, some assumptons regardng probablty dstrbutons have to be made when usng a probablty model. Nevertheless, research ndcates that even a small devaton of probablty dstrbutons from real values may result n an extremely large error n the uncertanty analyss [6]. In the early 1990s, Ben-Ham and Elshakoff [6 9] proposed a new knd of uncertanty analyss methodology based on a non-probablstc convex model. In ths method, t s assumed that the *Correspondence to: C. Jang, State Key Laboratory of Advanced Desgn and Manufacturng for Vehcle Body, College of Mechancal and Vehcle Engneerng, Hunan Unversty, Changsha Cty, Chna E-mal: jangc@hnu.edu.cn Copyrght 015 John Wley & Sons, Ltd.

2 C. JIANG ET AL. uncertanty of the parameters belongs to a convex set; thus, the uncertanty boundary can be obtaned based on a small number of samples nstead of an exact probablty dstrbuton. In addton, optmzaton methods are generally used n ths model to obtan the response nterval and thereby measure the degree of safety of structures. Because of ts weak dependence on sample number, the convex model approach s hghly sutable for uncertanty analyss of many complex engneerng problems. There are many excellent recent research fndngs n ths area. In [9], the probablty model and the convex model were compared. In other work, an uncertan trangle was adopted to descrbe the relatonshp of three uncertanty models - the probablty model, fuzzy sets, and the convex model [10]. A seres of numercal algorthms were developed to conduct uncertanty propagaton analyss for structural statc mechancs, egenvalues, and dynamcal problems [11 13]. A new nterval analyss technque was proposed to calculate the statc and dynamc responses of structures based on a frst-order Taylor nterval expanson [14]. An error estmaton method was proposed for nterval and subnterval analyss based on a second-order truncaton model [15]. A correlaton-analyzng technque based on a non-probablstc convex model was proposed as an effcent method for the constructon of multdmensonal ellpsodal convex models, and a covarance matrx was ntroduced to descrbe the correlaton among uncertan parameters [16]. The applcatons of convex model n engneerng mechancs also nclude non-lnear bucklng analyss of a column wth uncertan ntal mperfectons [17], stablty analyss of elastc bars on uncertan foundatons [18], boundary analyss of the structural responses of beams [19], uncertanty analyss n structural number determnaton n flexble pavement desgn [0], and so on. In recent years, the convex model has been appled to the relablty analyss of structures wth uncertanty, and some work n ths area has been publshed. By ntroducng the concept of the tradtonal frst-order relablty method nto problems wth convex models, a non-probablstc relablty ndex that represents a mnmal dstance n the standard convex space was defned [1, ]; an effcent soluton algorthm was further formulated for ths relablty ndex [3]. By placng the nonprobablstc relablty ndexes as constrants, several relablty-based optmzaton desgn methods were developed [4 6]. Based on the order relaton of ntervals, a nonlnear programmng method was proposed for desgn of structures wth nterval uncertanty [7, 8]. By ntegratng the probablty model and the convex model, the relablty analyss for a mxed uncertanty problem was also nvestgated [9 3]. Currently, two convex models, the nterval and the ellpsod, are prmarly used for analyss nvolvng non-probablstc convex models. In the nterval model, the fluctuatons of a sngle varable are descrbed through ts upper and lower boundares. The uncertanty doman s a multdmensonal box. For the ellpsod model, t s assumed that the parametrc uncertanty les wthn a multdmensonal ellpsod. The degree of uncertanty and the degree of correlaton of the varables are descrbed by the sze and shape of the ellpsod. In theory, the nterval model can deal only wth problems nvolvng ndependent varables, whle the ellpsod model can deal only wth correlated parametrc problems. However, many complex engneerng problems have mult-source uncertanty arsng from materal, load, geometrcal dmenson, and so on; modelng of these problems nvolves many uncertan parameters, some of whch are ndependent, whle others may have correlatons. Therefore, analyss based only on an nterval model or an ellpsodal model may result n the excessve expanson of the uncertanty doman, further resultng n a conservatve desgn. For ths reason, we need to develop a more general non-probablstc convex model that smultaneously takes nto account the ndependence and correlatons of the varables. Such a model would be expected to deal more effectvely wth the complex mult-source uncertanty problems, thus greatly expandng the scope of applcaton of the convex model method. Ths artcle proposes a novel non-probablstc convex model, the multdmensonal parallelepped model. A quanttatve descrpton of the degree of correlaton between any two varables was realzed by ntroducng a relevant angle. The proposed model can easly handle the uncertanty problem wth the coexstence of correlaton and ndependence and thus effectvely solve the complex ssues of mult-source uncertantes. The remander of ths artcle s organzed as follows: Secton ntroduces two types of conventonal convex models; Secton 3 proposes a new non-probablstc

3 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS convex model; Secton 4 gves an affne coordnate transformaton for the suggested convex model; Secton 5 apples the model to uncertanty propagaton analyss of structures; Secton 6 conducts the numercal analyss cases; and Secton 7 states the conclusons.. CONVENTIONAL NON-PROBABILISTIC CONVEX MODELS In theory, the buldng of convex models s not unque as long as the uncertanty doman belongs to a convex set. Because of the advantages of smplcty of expresson and ease of use, the nterval model and ellpsodal model have become the two most commonly used types of models n ths feld [33, 34]. In nterval model, the varaton range of an uncertan parameter s expressed through an nterval. Only the lower and upper bounds of the range are requred to be specfed, and there s no need to know the exact probablty dstrbuton of the parameter. Assume a bounded closed nterval I.R/. The upper and lower bounds of an uncertan varable X R are X U and X L, respectvely. Then, the nterval model can be expressed as or X I.R/ D¹XjX L 6 X 6 X U º (1) X X L ;X U () In the nterval model, the mdpont of the nterval s X C D X L C X ı U, and the nterval radus s X W D X U X Lı.Forn uncertan varables, the correspondng nterval model s h X X L ; X U ; X X L ;X R ; D 1; ; :::; n : (3) The uncertanty doman usng an nterval model s a multdmensonal box. As shown n Fgure 1, the uncertan doman of the two-dmensonal problem s a rectangle wth the margnal ntervals X1 I and X I of the two parameters. A margnal nterval represents the change range for a sngle varable. In an ellpsodal model, the uncertanty doman of the n-dmensonal varables X belongs to an ellpsod X X C T M X X C 6 1 (4) n whch 3 g 11 g 1 ::: g 1n g 1 g ::: g n M D : : : : 5 g n1 g n ::: g nn (5) Fgure 1. Interval model.

4 C. JIANG ET AL. Fgure. Ellpsodal model. M s a symmetrc postve defnte matrx and s called the characterstc matrx of the ellpsodal model. It determnes the sze and orentaton of the ellpsod. For the two-dmensonal problem shown n Fgure, the uncertanty doman s an ellpse, and X1 I and X I represent the margnal ntervals of the two parameters. The ellpsodal model could deal wth the correlaton of the varables. For a dscusson on how to defne ts correlaton and precsely construct a multdmensonal ellpsodal model, one can refer to our recent work [16]. Although the nterval model and the ellpsodal model can both effectvely solve many uncertanty problems, they possess some nadequaces. In theory, the nterval model can deal wth the problem wth only ndependent varables, whle the ellpsodal model can process problems wth only dependent varables. However, many practcal engneerng problems have mult-source uncertantes related to materal, load, geometrcal dmenson, and other factors and thus have many uncertan parameters. Some parameters are ndependent, whle others mght be dependent to each other. If ether the nterval model or the ellpsodal model s used separately to conduct the analyss, t may result n the excessve expanson of the uncertanty doman, leadng to a conservatve desgn. For ths purpose, we need to develop a more general non-probablstc convex model that could take nto account both the ndependence and correlaton of the varables. Usng ths model, the mportant mult-source uncertanty problems wll be handled more easly, thus effectvely expandng the scope of applcaton of the convex model method. 3. NON-PROBABILISTIC CONVEX MODEL BASED ON A MULTIDIMENSIONAL PARALLELEPIPED In ths secton, a more general convex model, namely multdmensonal parallelepped model that could contan the ndependent and dependent uncertan varables n a unfed framework, wll be presented. As long as nformaton on the varables margnal ntervals and the correlaton between any two varables s gven, the uncertanty doman can then be constructed. In geometry, the uncertan doman that s bult n our model s a multdmensonal parallelepped, whch s stll a convex set. For better understandng, examples of the constructon of the gven convex model n two dmensons, three dmensons, and n dmensons are gven Two-dmensonal problem As shown n Fgure 3, for a two-dmensonal problem, our multdmensonal parallelepped model wll degenerate nto a parallelogram. In ths model, a quadrlateral sde s set to be parallel to the abscssa axs. In theory, the parallelogram s requred to be obtaned from expermental samples of the parameters. X1 I andx I are the respectve ranges of the two varables, namely ther margnal ntervals, and X1 W and X W are the nterval rad of the uncertan varables X 1 and X, respectvely. We can fnd that under the fxed X1 W and X W, the angle 1 actually could depct the correcton

5 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Fgure 3. Two-dmensonal parallelogram model: (a) Postve correlaton and (b) negatve correlaton. Fgure 4. Three-dmensonal parallelepped model. of the two varables, and hence, t s defned as correlaton angle n ths paper. 1 s a value n the followng 1 arctan X W X W 1 ; arctan X W X W 1 When 1 < 90 ı, X 1 and X are postvely correlated, as shown n Fgure 3(a). When 1 D 90 ı, X 1 and X are ndependent of each other, and the parallelogram model degenerates to the nterval model. When 1 >90 ı, the varables X 1 and X are n negatve correlaton, as shown n Fgure 3(b). 3.. Multdmensonal problem For a three-dmensonal problem, the uncertanty doman s a parallelepped, as shown n Fgure 4. Smlarly, one surface of the parallelepped s set to be parallel to the X-Y coordnate plane. The margnal ntervalsx1 I ;XI,andX 3 I represent the uncertanty range for each varable, whle the correlaton angles 1, 13,and 3 descrbe the dependence of any two varables. For example, when only X 1 and X have correlaton among the three varables, the uncertanty doman wll be a parallelepped, as shown n Fgure 5. For a general n-dmensonal problem, the uncertanty doman wll be a multdmensonal parallelepped. X I ; D 1; ; :::; n represent the margnal ntervals of the varables. The degree of correlaton of any two varables X and X j can be depcted by ther correlaton angle j. In addton, for convenence of descrpton, we defne the correlaton coeffcent j between the two varables as follows j D (6) X W j X W tan j (7)

6 C. JIANG ET AL. Fgure 5. Parallelepped model wth correlaton only between X 1 and X. Fgure 6. Parallelepped models under some specal correlaton cases: (a) j D 1, (b) j D 0, and(c) j D 1. n whch j arctan X W j X W ; arctan X W j X W and the range of the correlaton coeffcent j s Œ 1; 1. As shown n Fgure 6, when j D arctan X j W, the correlaton coeffcent X W j D 1, and X and X j are totally lnearly postvely correlated. When j D 90 ı, the correlaton coeffcent j D 0, and the two parameters are ndependent. When j D arctan X j W, the correlaton X W coeffcent j D 1, and the two parameters are totally lnearly negatvely correlated. From the aforementoned analyss, t can be seen that n the stuaton n whch the correlaton coeffcents of all the varables are 0, the multdmensonal parallelepped model degenerates nto the tradtonal nterval model; that s, the nterval model s actually a specal case of the parallelepped model Constructon of the uncertanty doman From the aforementoned dscusson, t can be seen that n our model, as long as the margnal ntervals of all the varables and the correlaton angles or correlaton coeffcents between any two

7 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS varables are known, a multdmensonal parallelepped that represents the uncertanty doman of the parameters can be bult. In practcal engneerng problems, the changng ranges of the varables, that s, ther margnal ntervals, are often not hard to obtan based on the experence or exstng knowledge of the problems concerned. For example, we know that gven a partcular machnng accuracy, the dmenson of a product wll le wthn the nterval of a fxed tolerance. In addton, n many cases, nformaton on the correlaton between varables can also be obtaned based on our experence. For example, n a gven structure, we know that the parameters of materals, loads, and geometrc dmensons arse from dfferent uncertanty sources and are usually not correlated wth one another. Stochastc process analyss of an ocean wave s heght and wnd speed usually shows strong correlaton between these two parameters and thus should be gven a relatvely large correlaton coeffcent. However, f we lack experence wth a partcular problem, the buldng of the uncertanty doman wll be based entrely on the samples avalable. In the followng, we provde an entrely sample-based method to create a multdmensonal parallelepped convex model for a general uncertanty problem. Assume a structure wth n-dmensonal uncertan varables X, D 1; ; :::; n, andtherearem test samples X.r/ ;rd1; ; :::; m. Then, the process of buldng the convex model s as follows: Step 1: Take any two uncertan varables X and X j, wth j. Step : Extract the values of X and X j from the sample X.r/ to obtan a two-dmensonal sample set X.r/ ;X.r/ j ;rd1; ; :::; m. Step 3: In the X -X j two-dmensonal varable space, establsh a parallelogram envelopng all the samples X.r/ ;X.r/ j ;rd1; ; :::; m, thus obtanng the margnal ntervals X I and Xj I of the varables, the correlaton angle j, and correlaton coeffcent j. Step 4: Repeat the aforementoned steps for any two uncertan varables and obtan the margnal ntervals and correlaton coeffcents of all the uncertan varables. Step 5: Based on the obtaned margnal ntervals and correlaton coeffcents of all the varables, buld the multdmensonal parallelepped convex model. The bult multdmensonal parallelepped model should satsfy: (1) ts changng range along each axs s equal to the correspondng margnal nterval and () the angle of each two edges of the parallelepped model s equal to correspondng correlaton angle. For a multdmensonal problem, partcularly n cases of hgh-dmensonal parameters, t s relatvely dffcult to buld a parallelepped model drectly through the samples. However, n the aforementoned method, ths dffculty can be effectvely elmnated through decomposng the complex n-dmensonal problem nto n.nc1/ smple two-dmensonal problems. Just usng the margnal ntervals and correlaton angles that can be easly obtaned through the samples, a multdmensonal uncertanty doman can be bult, whch geometrcally s a parallelepped. In addton, t should be noted that n many practcal cases, the margnal ntervals of some parameters can be drectly known based on the engneers experence or the exstng knowledge of the problem concerned. Thus, only ther correlaton angles need to be obtaned usng the samples by means of the aforementoned procedure. 4. AFFINE COORDINATE TRANSFORMATION FOR THE CONVEX MODEL In the aforementoned analyss, we constructed a multdmensonal parallelepped model for the descrpton of the uncertanty of parameters. However, the doman formed by a multdmensonal parallelepped can hardly be expressed mathematcally by explct functons such as the boundares of the nterval model and the quadratc functon of the ellpsodal model. Ths results n dffculty n performng subsequent structural uncertanty analyses, such as uncertanty propagaton analyss and relablty analyss. In ths secton, we ntroduce the use of affne coordnates [35, 36], whch provdes an effectve soluton to the problem. In the affne coordnates of our formulaton, the angle between the axes wll no longer be constant (equal to 90 ı ); rather, t wll be equal to the correlaton angle between the correspondng varables. Through affne transformaton, the parallelepped model

8 C. JIANG ET AL. n the orgnal parameter space can be converted to the nterval model n the affne space, thus greatly facltatng subsequent structural uncertanty analyss and calculaton Two-dmensonal problem As shown n Fgure 7, the margnal ntervals x 1 x1 I D x1 L;xU 1 and x x I D x L;xU and the correlaton± angle 1 of the parallelogram model are known. A new Cartesan coordnate system O 0 I e 1 ; e s defned by makng ts orgn and the center of the parallelogram concde. And the affne coordnate system ¹O 00 I e 1 ; e º s establshed accordng to the parallelogram model. As shown n Fgure 8, the center of the parallelogram s taken as the orgn; axs e 1 s parallel to the bottom edge of the parallelogram, that s, n the same drecton as axs e 1 n the Cartesan coordnate system, whle axs e s parallel to another sde of the parallelogram. Obvously, through the affne coordnates ¹O 00 I e 1 ; e º, the correlaton between two varables can be transformed nto the ncluded angle between the two axes n the affne coordnates. Wth respect to the Cartesan coordnate system ¹O 0 I e 1 ; e º and the affne coordnate system ¹O 00 I e 1 ; e º, accordng to the prncples of the affne coordnate system [35], t can be assumed that e 1 D a 11 e 1 C a 1e e D a 1e 1 C a (8) e where a 11 ;a 1 ;a 1 and a are weght coeffcents. From the condton that the parallelogram bottom edge s parallel to the axs e 1, we can conclude that a 1 D 0. Consderng the condton that e 1?e, Equaton (8) can be rewrtten n matrx form as e 1 e D e 1 e A (9) Fgure 7. The mdpont shft n the coordnate system. Fgure 8. The establshment of an affne coordnate system.

9 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Fgure 9. Rad of the margnal ntervals n dfferent coordnate systems: (a) In orgnal coordnates and (b) n affne coordnates. In whch A s the coordnate transformaton matrx A T 1 0 D cos 1 sn 1 : (10) Combnng the coordnate translaton, we obtan the mappng relaton between the ntal parametrc space and the fnal affne space T 00 T x1 x D A x 1 x00 C x1 RCxL 1 x R CxL T (11) As shown n Fgure 9, after transformaton of the affne coordnate system, the rad of the margnal ntervals n the orgnal Cartesan coordnate system and the ones n the affne coordnates wll change and have the followng relaton x W 00 1 x W 00 T D A 1 x1 W x W T (1) n whch x1 W and xw are the margnal ntervals of the uncertan parameters n the orgnal space, and x W 00 1 and x W 00 are the ones n the affne space. 4.. Multdmensonal problem In n-dmensonal parallelepped model, as for the two-dmensonal case, the correlaton angle between varables can be transformed nto the angle between the correspondng axes n affne coordnates. In a smlar way as the two-dmensonal case, we can obtan the weght coeffcents of the coordnate transformaton, whch are the elements of matrx A T 8 0.j > / ˆ< a j D s ˆ: jp 1 cos k a m a jm md1 1 ajj.j < / jp 1 ld1 a l.j D / For smplcty of expresson, the aforementoned equaton uses k to represent angle j,nwhch.n j /.j 1/ the subscrpt k D C. j/. Combnng the coordnate translaton, the mappng relaton of the ntal parameter space and the fnal affne space for an n-dmensonal problem can be expressed as T 00 T T x1 x ::: x n D A x 1 x00 ::: x00 n C x1 RCxL 1 x RCxL ::: xr n CxL n (14) (13)

10 C. JIANG ET AL. Accordngly, the relatonshp between the rad of the parameter ntervals before and after the affne transformaton can be represented as follows x W 00 1 x W 00 ::: x W 00 T n D A 1 x1 W x W ::: xn W T (15) 5. APPLICATION TO STRUCTURAL UNCERTAINTY PROPAGATION ANALYSIS In structural analyss, uncertantes n parameters such as materal propertes, loads, boundares, and other factors wll lead to uncertantes n structural responses such as dsplacement, stress, and stran. Ths represents an uncertanty propagaton problem. Through the analyss of uncertanty propagaton, t s possble to determne the quanttatve nfluence of the degree of ntal uncertanty on the structural response. Thus, relablty desgn can be conducted. In the convex model method, the parameter uncertanty belongs to a bounded set. Therefore, the structural response usually belongs to an nterval. In the convex model, performng an accurate calculaton of ths response nterval s the man task of uncertanty propagaton analyss. Assume that the uncertan parameters for a structure are X ; D 1; ; :::; n. The structural response functon s then g.x/ ; X X I ; (16) In Equaton (16), s the uncertanty doman, and n ths paper, t comprses a multdmensonal parallelepped. The sze and shape of the uncertanty doman are decded by the margnal ntervals X I and the correlaton angles of the parameters. Frst, through the affne transformaton Equaton (14), the parallelepped convex model s transformed to the affne coordnates. Accordngly, the response functon G n the affne space can be generated g.x/ D g T X 00 D G X 00 ; X (17) where T s the transformaton functon determned by Equaton (14). 00 represents the uncertanty doman n the affne space. As shown n Fgure 9, n the affne space, the uncertanty doman becomes an nterval model n whch the center pont s the orgn of the affne coordnate system. The parameters nterval rad are X W 00 ; D 1; ; :::; n. Therefore, the exstng structural uncertanty propagaton methods such as [11 15, 37], whch are based on the conventonal nterval analyss, can be drectly used. Consderng that the level of uncertanty of parameters s generally small, n order to decrease the amount of calculaton, especally for some complex engneerng problems, the response functon can be expanded to the frst-order Taylor seres G.X 00 / D G.X C 00 / C C 00 / 00 X 00 X C 00 n whch the deployment pont X C 00 s the affne coordnate followng equaton 00 The value of X 00 X C C 00 / 00 nx j C j can be explctly obtaned from Equaton (14). belongs to an nterval below h X 00 X C 00 X W 00 ;X W 00 (18) can be calculated by the ; (19) ;D 1; ; :::; n (0)

11 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Thus, applyng the nterval extenson [37] to Equaton (18), an explct expresson can be obtaned for the response nterval; the lower and upper bounds wll be 8 ˆ< G L.X / D G.X C 00 P / n 00 / ˇX W D1 ˆ: G R.X / D G.X C 00 P / C n 00 / (1) ˇ ˇX W 00 The rad of the ntervals n the affne space are X W 00 ;D 1; ; :::; n, whch can be obtaned from Equaton (15) NUMERICAL EXAMPLES AND DISCUSSIONS In ths secton, three numercal examples are analyzed. In the frst problem, subgrade settlement predcton problem was ntroduced to descrbe the process of modelng of multdmensonal parallelepped model. In the second problem of the explct functon, a comparson between the multdmensonal parallelepped model proposed n ths paper, the conventonal nterval model, and the ellpsodal model was conducted. In the thrd and fourth examples, parallelepped models were used n the uncertanty modelng and propagaton analyss of statcs and dynamcs problems, respectvely. Addtonally, the thrd example orgnates from a real engneerng problem of the roll steerng characterstcs The model constructon of engneerng problems Analyss for a subgrade settlement problem. Subgrade settlement s an mportant ndcator of road safety and a frequent cause of the road traffc accdents. Presently, some methods have been developed to mprove the accuracy and relablty of the subgrade settlement predcton. However, most of them are based on the sngle-pont montorng mode and thus could only study the local deformaton, whle correlaton of the settlements at dfferent ponts could not be obtaned. Subgrade settlement s a complex systematc process, n whch the deformaton of a montorng pont wll be effected by other montorng ponts on the same subgrade cross-secton. Thus, to mprove the predcton accuracy, t s often necessary to use the mult-pont montorng mode and then nvestgate the correlaton of the deformaton amounts from dfferent montorng ponts. Based on the reference [38], a practcal two-way four-lane hghway s nvestgated here. As shown n Fgure 10, three montorng ponts A, B,andC were arranged n the left half secton of the road, and three dsplacement sensors were nstalled to test ther responses. Ten groups of measured data for the montorng ponts were gven n Table I [38], n whch a measurng perod of 15 days was adopted. In ths problem, we expect to analyze the correlatons between the settlements of the three montorng ponts and furthermore gve a quanttatve descrpton for ther whole uncertanty doman. There are only 10 groups of data; thus, the tradtonal probablty approach seems not Fgure 10. Arrangement of the montorng ponts (cm) [38].

12 C. JIANG ET AL. Table I. Measured data of the montorng ponts [38]. Settlement(mm) No. Days Pont A Pont B Pont C Table II. Results from the two-dmensonal analyss n A-B parameter space. Montorng ponts Lower bound (mm) Upper bound (mm) Correlaton angle( ı ) A B Table III. Results from the two-dmensonal analyss n A-C parameter space. Montorng ponts Lower bound (mm) Upper bound (mm) Correlaton angle( ı ) A C Table IV. Results from the two-dmensonal analyss n B-C parameter space. Montorng ponts Lower bound (mm) Upper bound (mm) Correlaton angle( ı ) B C applcable here. The proposed parallelepped convex model s then used to deal wth these expermental data. We frst created three parallelograms fttng the samples n the three two-dmensonal parameter spaces, and the correspondng analyss results were gven n Tables II IV. Syntheszng these results, the margnal ntervals and correlaton angles of the dsplacements at the three montorng ponts can be obtaned, as A Œ11:35 mm; 30:78 mm, B Œ8:69 mm;4:75 mm, C Œ1:03 mm;7:16 mm, AB D 4:39 ı, BC D 41:51 ı,and AC D 45:53 ı, respectvely. Here, we also use A, B, and C to represent the dsplacements of the three montorng ponts. The three parallelograms fttng the measured data n the two-dmensonal spaces are llustrated n Fgure 11(a) (c). It can be observed that all the parallelograms seem thn and long, whch means that for these problems, the settlements of these three montorng ponts are sgnfcantly correlated wth each other. Furthermore, the correlatons of the three settlements seem at a smlar level. Based on all the margnal ntervals and correlaton angles, the whole uncertanty doman of the subgrade settlements at the three montorng ponts can be created as shown n Fgure 11(d), whch geometrcally s a parallelepped. It can be found that the parallelepped has a good fttng to the 10 samples. Also, through the parallelepped, the correlatons between each par of settlements can be well descrbed. Based on ths uncertanty doman, some mportant structural uncertanty analyss such as uncertanty propagaton and relablty desgn can then be carred out for the subgrade settlement problem.

13 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS B / mm (a) A / mm C / mm (c) C / mm (b) B / mm A / mm 6 4 C / mm (d) B / mm A / mm 10 Fgure 11. The multdmensonal parallelepped model for the subgrade settlement predcton problem Analyss for a ple load test problem. In ple foundaton desgn, research studes have ndcated that model uncertanty s an mportant factor. We consder a ple load test problem, n whch the ple load test reports were collected from varous plng companes n South Afrca [39]. In order to evaluate the measured or nterpreted ultmate capactes of all the test ples n the database, a combnaton of Chn s extrapolaton procedure and Davsson s falure crteron was employed.

14 C. JIANG ET AL. And a and b are the hyperbolc curve-fttng parameters for the load-settlement curve, and they are not ndependent. A dataset for D-C (drven ples n cohesve sols) wth 57 data (two among the orgnal 59 samples are abnormal ponts, and hence, they were removed), and a dataset for B-NC (bored ples n noncohesve sols) wth 31 data (two among the orgnal 33 samples are abnormal ponts, and hence, they were removed) as shown n Table V and VI, are used for llustraton [39]. In ths problem, we expect to analyze the correlaton between the hyperbolc curve-fttng parameters and furthermore gve a quanttatve descrpton for ther whole uncertanty doman. The proposed parallelepped convex model s then used to deal wth these expermental data, and we can create parallelogram (for a two-dmensonal problem) fttng the samples n the two-dmensonal parameter spaces. For D-C, the margnal ntervals and correlaton angle of the varables a and b can be obtaned: a Œ 0:8; 7:78, b Œ0:61; 0:95, ab D 176:6 ı. And for B-NC, the margnal ntervals and correlaton angle of the varables a and b can be obtaned: a Œ 1:44; 11:58, b Œ0:44; 0:99, ab D 176:4 ı. The correspondng analyss results are gven n Table VII and VIII, and the uncertanty domans are llustrated n Fgure 1(a) and (b) (a parallelogram doman, respectvely). When adoptng the nterval model, the correlaton of the varables s not taken nto account. The uncertanty domans are constructed wth the margnal ntervals a Œ0:57; 6:93, b Œ0:61; 0:95 for D-C and a Œ0:06; 10:93, b Œ0:44; 0:99 for B-NC, respectvely, whch are also llustrated n Fgure 1(a) and (b) (a rectangular doman, respectvely). Fgure 1 shows that the uncertanty domans from nterval model are much larger than those from parallelepped model. Wth the much more compact uncertanty doman of the parameters obtaned accordng to the parallelepped model, the subsequent uncertanty analyss for the ple load test problem thus wll be more precse and useful. Table V. Hyperbolc curve-fttng parameters a and b for ples (D-C) [37]. No. a b No. a b

15 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Table VI. Hyperbolc curve-fttng parameters a and b for ples (B-NC) [37]. No. a b No. a b Table VII. The margnal ntervals and correlaton angle of a and b for ples (D-C). Parameters Lower bound Upper bound Correlaton angle( ı ) a b Table VIII. The margnal ntervals and correlaton angle of a and b for ples (B-NC). Parameters Lower bound Upper bound Correlaton angle( ı ) a b a (a) D-C b b a (b) B-NC Fgure 1. The multdmensonal parallelepped model and nterval model for the Hyperbolc curve-fttng parameters: (a) D-C and (b) B-NC. 6.. An analytcal functon problem Consder the followng response functon g.x/ D X 1 X 3X 1 C X X 3 C 4X C 5X 3 X 1 X 3 C 10 ()

16 C. JIANG ET AL. Table IX. The frst sample case data. No. x 1 x x 3 No. x 1 x x In ths functon, the margnal ntervals of the three uncertan parameters are known as X1 I D Œ 5; 5 ; X I D Œ 4; 4 and X 3 I D Œ 8; 8. It s supposed there are 54 samples of the uncertan parameters, as shown n Table IX. In the followng, uncertanty modelng and propagaton analyss wll be conducted usng the nterval, ellpsod, and parallelepped models, and ther results wll be compared Interval model. When adoptng the nterval model, the correlaton of the varables s not taken nto account. The uncertanty doman s a cubod composed of the margnal ntervals X1 I ;XI, and X3 I. In uncertanty propagaton analyss, the response nterval can be approxmately obtaned drectly n the orgnal parameter space based on a frst-order Taylor expanson 8 ˆ< g L.X/ D g.x C P / n / ˇˇˇX D1 ˆ: g R.X/ D g.x C P / C n (3) / ˇˇˇX 6... Ellpsodal model. In ths model, all samples are frst ftted usng the mnmum volume method [40]. The ellpsodal model s then obtaned as follows X 1 3 T 0 1 C 0:0049 0:0540 0:0419 0:0008 X 0:0540 A 6 4 0:0419 0:0550 0: X 1 1 C 0:0049 X 0:0540 A 6 1 (4) X 3 0:0076 0:0008 0:0010 0:0075 X 3 0:0076 Thus, the changng bounds of the response functon over the ellpsod uncertanty doman can be calculated n the orgnal parameter space. It can be obtaned by the followng equaton [37] D1

17 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS 8 q ˆ< g L.X/ D g.x C / rg T M -1 rg q ˆ: g R.X/ D g.x C / C rg T M -1 rg (5) where M s the characterstc matrx of the aforementoned ellpsod. rg C 1 T s the gradent vector of the response functon wth respect to the uncertan C Multdmensonal parallelepped model. Usng the method descrbed n Secton 3.3, all sample ponts can be ftted. In ths way, the parallelepped model for the uncertanty doman, n whch the correlaton angles of the parameters are 1 D 60 ı, 13 D 90 ı,and 3 D 90 ı, s obtaned. Ths ndcates that only X 1 and X have correlaton; thus, ths problem has both dependent parameters and ndependent parameters. The coordnate transformaton matrx A s 1 cos 60 ı cos 90 ı A D 4 0 sn 60 ı cos 90ı cos 60 ı cos 90 ı sn 60 ı D p (6) From Equaton (18), the mappng relaton between the orgnal parameter space and the affne space s X X X 00 X C 1 X 00 1 X A B D X 00 C 6 p A D B 0 X 00 p C B A 3 X 00 C A : (7) X 3 X X 00 3 X 00 3 Thus, the expresson of the response functon n the affne space can be explctly obtaned for ths problem as p p G.X 00 / D 3X C X 00 1 X X1 X C 4 X 00 C p 3 3 p! X 00 3 C 1 X 00 X C5X3 C10 (8) The margnal ntervals of the parameters n the affne space are X C p 4 ;5 4 p ;X 00 p 8 ; p ;X 00 3 Œ 8; 8 : (9) 3 Accordng to Equaton (1), the uncertanty nterval of the response functon can eventually be solved Comparson of the results. In ths secton, the geometrc shape and sze of the uncertanty doman constructed from three convex models wll be compared. For smplcty, the uncertanty doman s projected onto the two-dmensonal parameter spaces, as shown n Fgure 13. It can be seen from the fgure that, n the X 1 X parameter space, the ellpsodal and parallelepped models both reflect the correlaton of the parameters. However, n the spaces of X 1 X 3 and X X 3,the ellpsodal model was not able to deal wth the problem of ndependent parameters, thus enlargng the uncertanty doman to a great extent, whereas the parallelepped model was able to envelope all the samples well. In the spaces of X 1 X 3 and X X 3, due to the fact that the parameters do not exhbt correlaton wth each other, the nterval and parallelepped models have the same uncertanty domans. In the X 1 X space, because the nterval model cannot handle the correlaton, t sgnfcantly expands the uncertanty doman, whle the parallelepped model fts all the samples very well. Fgure 14 shows three-dmensonal graphcs depctng the uncertanty domans from the three models. It can be seen that the uncertanty doman obtaned from the parallelepped model s sgnfcantly smaller than that obtaned by ether of the other two methods.

18 C. JIANG ET AL. Fgure 13. Projectons of the uncertanty domans n two-dmensonal parameter spaces: (a) X 1 X plane, (b) X 1 X 3 plane, and (c) X X 3 plane. Fgure 14. Three-dmensonal uncertanty domans from three convex models.

19 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Through the foregong analyss, we found that, for problems wth ether correlaton or ndependence, the nterval and ellpsodal model are both lkely to construct a too-large uncertanty doman, resultng n an ultra-conservatve desgn. The parallelepped model, however, can handle both dependent varables and ndependent varables and thus can construct a more compact uncertanty doman, resultng n more reasonable uncertanty analyss results. The uncertanty propagaton analyss results based on the aforementoned convex models are shown n Table X. The results show that, among these three methods, the response nterval obtaned by the parallelepped model s the narrowest and s contaned n the results of both the nterval model and the ellpsodal model. Ths occurs because the parallelepped model constructs a more compact doman of uncertanty, so that the calculaton results are not as conservatve as those produced by the nterval and the ellpsodal models. In the aforementoned example, the margnal ntervals of the parameters do not change. We also consdered three other knds of sample dstrbutons, as shown n Table XI XIII; the three convex models were used separately for analyzng these samples. When uncertanty modelng s conducted usng the multdmensonal parallelepped model, 13 and 3 are both 90 ı n all three sample cases, whle the correlaton angle 1 takes on values 5:5 ı, 75 ı,and90 ı, respectvely. The correspondng correlaton coeffcents 1 are 0.614, 0.14, and 0.000, respectvely. Fgure 15 shows the geometry of the uncertanty doman under dfferent correlaton angles. It can be seen that dfferent correlaton angles 1 drectly result n uncertanty domans of dfferent shapes and szes. Wth a smaller 1, the parallelepped s more flat, and the volume s also smaller. As 1 gradually approaches 90 ı,the shape of the uncertanty doman gradually becomes a cube, and ts volume gradually ncreases. The results of uncertanty propagaton analyses for the aforementoned cases are shown n Table XIV and Fgure 16. From the calculaton results of the ellpsodal and parallelepped models, t can be Table X. Uncertanty propagaton analyss results for the frst sample case. Model Response Interval model Ellpsodal model Parallelepped model Response nterval [ 61.00, 81.00] [ 51.5, 7.06] [ 47.14, 67.14] Radus of nterval Table XI. The second sample case data. No. x 1 x x 3 No. x 1 x x

20 C. JIANG ET AL. Table XII. The thrd sample case data. No. x 1 x x 3 No. x 1 x x Table XIII. The fourth sample case data. No. x 1 x x 3 No. x 1 x x

21 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Fgure 15. Uncertanty domans under dfferent correlatons of the parameters: (a) 1 D 40 ı,(b) 1 D 5:5 ı,(c) 1 D 67:5 ı,(d) 1 D 75 ı,(e) 1 D 80 ı, and (f) 1 D 90 ı. Table XIV. Uncertanty propagaton analyss results for dfferent sample cases. Case Model Response nterval of case Response nterval of case 3 Response nterval of case 4 Interval model [ 61.00,81.00] [ 61.00,81.00] [ 61.00,81.00] Ellpsodal model [ 50.85,70.07] [ 63.7,83.30] [ 69.55,90.47] Parallelepped [ 4.58,6.58] [ 54.57,74.57] [ 61.00,81.00] Model 1 D 5:5 ı 1 D 75 ı 1 D 90 ı seen that the correlaton of parameters X 1 and X has a sgnfcant effect on the varaton of response functon n ths example. Wth the correlaton decreasng (.e., 1 becomng larger), the response ntervals calculated by both the ellpsodal and parallelepped model show a progressvely larger trend. However, because the nterval model cannot descrbe the correlaton of parameters, only a constant analyss results can be generated n all three cases. In the fourth case, all the parameters are not correlated, and the parallelepped model results are consstent wth the nterval model results. However, because of the nablty of the ellpsodal model to deal wth ndependent varables, the model gves a wder range of the response. In addton, t should be noted that the parallelepped model gves the narrowest range of the response functon n all cases, and ts result are ncluded by the nterval model and ellpsodal model results. Ths further shows that the uncertanty doman of the parallelepped model s more compact than the ones of the other two models and that t generally could provde more reasonable and effectve uncertanty analyss results.

22 C. JIANG ET AL Fgure 16. The bounds of the response functon from three convex models wth dfferent parametrc correlatons. Fgure 17. A 10-bar alumnum truss A 10-bar truss structure A well-known 10-bar alumnum truss structure [41], shown n Fgure 17, s used as an example. The length of the vertcal and horzontal bars s L D 360 n. The cross-sectonal area of the bar s A D 10 n. The elastc Modulus of the materal s E D 10 4 ks. A vertcal load F 1 s appled on node 4. A vertcal force F and a horzontal force F 3 are appled on node. Loads F 1, F,andF 3 are all uncertan parameters. A multdmensonal parallelepped model s used to buld the uncertanty doman. Suppose the margnal ntervals are known to be F1 I D Œ90; 110 kps, F I D Œ90; 110 kps, and F3 I D Œ190; 10 kps, respectvely. All the three correlaton angles of the parameters have the same value. In ths example, the response nterval of the vertcal dsplacement ı of node wll be analyzed. Standard structural mechancs analyss gves [41], ı D " 6X D1 N 0 N A C p X10 D7 N 0N # L A E n whch N ;D 1; ; :::; 10 are the axal forces n the bar, as follows N 1 D F p p p N 8;N D N 10;N 3 D F 1 F C F 3 N 8 (30)

23 A NON-PROBABILISTIC CONVEX MODEL FOR STRUCTURAL UNCERTAINTY ANALYSIS Table XV. Response nterval of the vertcal dsplacement ı for cases wth dfferent parametrc correlatons. Correlaton angle Response nterval 50 ı 55 ı 60 ı 65 ı 70 ı 75 ı 80 ı 85 ı 90 ı Lower bound (n) Upper bound (n) Radus of nterval (n)

24 C. JIANG ET AL. N 4 D F C F 3 p p p p N 10;N 5 D F N 8 N 10;N 6 D N 10 N 7 D p.f 1 C F / C N 8 ;N 8 D a b 1 a 1 b a 11 a a 1 a 1 N 9 D p F C N 10 ;N 10 D a 11b a 1 b 1 a 11 a a 1 a 1 a 11 D 1 C 1 C 1 C p C p! A 1 A 3 A 5 A 7 a D A 8 1 A C 1 C 1 A 4 A5 C 1 p C p A6 A 9 A 10 L E ;a 1 D a 1 D! L E L A 5 E (31) b 1 D F A 1 F 1 C F F 3 A 3 F A 5 b D p.f3 F / A 4 p! F 4F A 5 A 7 p.f1 C F / A 7 L E ;! p L E N 0 s the value of N when F 1 D F 3 D 0 and F D 1. Durng the structural uncertanty propagaton analyss, a set of dfferent parametrc correlatons are taken nto account. The computatonal results by usng the present parallelepped model are gven n Table XV and Fgure 18. The results show that for the 10 bar truss, the correlaton between the three load parameters also has a sgnfcant nfluence on the vertcal dsplacement ı of node. Under dfferent correlaton angles, the response nterval of ı shows a relatvely large fluctuaton. In addton, when the correlaton angle ncreases, the response radus of ı frst exhbts a downtrend and then a rsng trend. In all nne stuatons, ı has a mnmum response radus equal to ı W D n when D 60 ı Fgure 18. The relatonshp between the response nterval of vertcal dsplacement ı and the parametrc correlaton angle.