A response surface approach for structural reliability analysis using evidence theory

Advances in Engineering Software 69 (2014) 37–45

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A response surface approach for structural reliability analysis using evidence theory Z. Zhang, C. Jiang ⇑, X. Han, Dean Hu, S. Yu State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha City 410082, PR China

a r t i c l e

i n f o

Article history: Received 20 July 2013 Received in revised form 6 December 2013 Accepted 8 December 2013 Available online 21 January 2014 Keywords: Structural reliability Evidence theory Epistemic uncertainty Response surface Design of experiments Computational cost

a b s t r a c t Evidence theory employs a much more general and flexible framework to quantify the epistemic uncertainty, and thereby it is adopted to conduct reliability analysis for engineering structures recently. However, the large computational cost caused by its discrete property significantly influences the practicability of evidence theory. This paper proposes an efficient response surface (RS) method to evaluate the reliability for structures using evidence theory, and hence improves its applicability in engineering problems. A new design of experiments technique is developed, whose key issue is the search of the important control points. These points are the intersections of the limit-state surface and the uncertainty domain, thus they have a significant contribution to the accuracy of the subsequent established RS. Based on them, a high precise radial basis functions RS to the actual limit-state surface is established. With the RS, the reliability interval can be efficiently computed for the structure. Four numerical examples are investigated to demonstrate the effectiveness of the proposed method. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Uncertainties related to the material property, bounding condition, load, etc. widely exist in practical engineering problems. With intensive requirements of high product quality and reliability, understanding, identifying, controlling and managing various uncertainties have become imperative. Uncertainty refers to the difference between the present state of knowledge and the complete knowledge. Based on this view, uncertainty can be described as two distinct types – aleatory (random) and epistemic (subjective) uncertainty [1]. Aleatory uncertainty is irreducible and describes the inherent variability of a physical system, which can be modeled as random variables or processes using probability theory. Many probability-based reliability analysis techniques have been well established and successfully applied to varieties of industrial fields [2–5]. However, when data are scarce, the probability theory becomes not so useful because the key probability distributions cannot be obtained. In this case, the epistemic uncertainty will be involved. Epistemic uncertainty is defined as the lack of knowledge or information in some phases or activities of the modeling process. Therefore, it can be reduced with the collection of more information or an increase of knowledge. Some representative theories, including convex models [6–11], possibility theory [12–14], fuzzy sets [15] and evidence theory [16–21], can be used to deal with the epistemic uncertainty. ⇑ Corresponding author. Tel.: +86 731 88823325; fax: +86 731 88821748. E-mail address: [email protected] (C. Jiang). 0965-9978/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advengsoft.2013.12.005

Among the above theories for epistemic uncertainty, evidence theory employs a much more flexible framework with respect to the body of evidence and its measures [22]. Under some special situations, it can provide equivalent descriptions to the probability theory, convex models, possibility theory and fuzzy sets, respectively. Hence, in recent years evidence theory has been introduced to conduct reliability analysis and design for engineering structures and mechanical systems. Oberkampf and Helton [22] compared the similarities and differences between evidence theory and probability theory in reliability analysis through a simple algebraic function. Helton et al. [23] explored several approaches (probability model, evidence theory, possibility theory and interval analysis) in the representation of the uncertainty in model prediction and thereby gave a unified framework. Soundappan et al. [24] compared evidence theory with Bayesian theory in aspects of uncertainty modeling and decision making under epistemic uncertainty. Du [25] formulated a new reliability analysis model to handle the epistemic and aleatory mixed uncertainty. Tonon et al. [26] employed evidence theory to quantify the parameter uncertainty in rock engineering and whereby carried out a reliability-based design of tunnels. Through creating a multi-point approximation at a certain point on the limit-state surface, Bae et al. [27,28] proposed an efficient reliability analysis method for structures with epistemic uncertainty. Jiang et al. [29] proposed a structural reliability method using evidence theory by introducing a non-probabilistic reliability index approach. Agarwal et al. [30] proposed an evidence-theory-based multidisciplinary design optimization (EBDO) algorithm through a sequential approximate strategy. Alyanak

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Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45

et al. [31] adopted a gradient projection technique to conduct a reliability-based design optimization (RBDO) for structures with epistemic uncertainty. Helton et al. [32] developed a samplingbased approach for sensitivity analysis of the uncertainty propagation problems using evidence theory. Mourelatos and Zhou [33] proposed a RBDO method based on evidence theory. Guo et al. [34] developed a RBDO method by combining evidence theory and interval analysis. Bai et al. [35] compared three metamodeling techniques for evidence-theory-based reliability analysis through six numerical examples. Despite the above achievements, presently evidence theory has been barely applied to conduct reliability analysis for complex engineering problems. One main reason is the high computational cost caused by the discontinuous nature of uncertainty quantification for the evidence variable [26]. Unlike the probability density function (PDF) in probability model, the uncertainty modeled by evidence theory is propagated through a discrete basic probability assignment (BPA), which cannot be expressed by any explicit function but generally described by a series of discontinuous subsets. This will in general lead to a combination explosion difficulty for a multidimensional problem. By using the response surface of the actual limit-state function, the high computational cost of evidence-theory-based reliability analysis can be significantly reduced. Some numerical methods [27,28] have been developed to reduce the computational cost by introducing the response surface technique, however, it seems not always an easy job to construct a sufficiently accurate response surface for a practical engineering problem using the existing methods. Therefore, to improve the applicability of evidence theory in practical applications, it seems necessary to develop some more robust and efficient reliability analysis methods. In this paper, a new response surface method is proposed to significantly improve the computational efficiency of evidencetheory-based reliability analysis, in which the analysis precision can be well guaranteed through a design of experiments technique. The remainder of this paper is organized as follows. The conventional reliability analysis using evidence theory is introduced in Section 2. An efficient algorithm is formulated to assess the reliability in Section 3. Four numerical examples are investigated in Section 4. Finally some conclusions are summarized in Section 5. 2. Conventional reliability analysis using evidence theory In this section, a simple problem is used to show the conventional reliability analysis using evidence theory, in which some fundamentals of evidence theory will also be introduced. Consider the following two-dimensional limit-state function:

gðXÞ ¼ g 0

ð1Þ

where X = (X1, X2) is the vector of two independent uncertain input parameters; g0 denotes an allowable value of the structural responses. For this problem, the safety region G is defined as:

G ¼ fg : gðXÞ P g 0 g

ð2Þ

In this paper, the uncertain parameters X will be described using evidence variables, and the reliability interval that X falls into the safety region G can be computed by two main steps. 2.1. Construction of joint BPA Evidence theory starts by defining a frame of discernment (FD) that is a set of mutually exclusive elementary subsets, which is similar to the sample space in probability theory. Here, the symbol X used to denote a parameter also represents its FD. All the possible subsets of X will form a power set X(X).

After defining the FD, a degree of belief is assigned to each subset based on the statistical data or the expert experience. It is called the basic probability assignment (BPA). The BPA is assigned through a mapping function m:X(X) ? [0, 1] which satisfies the following three axioms:

Axiom 1 : mðAÞ P 0 for any A 2 XðXÞ Axiom 2 : mðøÞ ¼ 0 X Axiom 3 : mðAÞ ¼ 1 A2XðXÞ

where m(A) characterizes the amount of ‘‘likelihood’’ that is assigned to the subset A. In this paper, we assume that the subsets A are all closed intervals instead of some other forms of sets. Each set A e X(X) satisfying m(A) > 0 is called focal element. Sometimes the information available for a parameter may come from multiple sources, for example several experts provide opinions for one event, then they should be aggregated by rules of combination [36]. Similar to joint probability in probability theory, the joint BPA is required in evidence theory when multiple uncertain variables are involved. Due to the independence among the parameters, the joint BPA mX can be obtained for a two-dimensional problem:

mX ðCÞ ¼



mX1 ðAÞ  mX 2 ðBÞ when C 2 A  B 0

otherwise

ð3Þ

where A e X(X1), B e X(X2), and C is the focal element of the Cartesian product A  B which can be defined as follows:

A  B ¼ fX ¼ ½X 1 ; X 2 ; X 1 2 A; X 2 2 Bg

ð4Þ

2.2. Computation of reliability interval Based on the joint BPA and the safety region, the reliability interval [Bel(G), Pl(G)] used to characterize the total degree of belief for the safety X e G can be calculated as below:

BelðGÞ ¼

X

mX ðCÞ

ð5Þ

C#G

PlðGÞ ¼

X

mX ðCÞ

ð6Þ

C\G – /

The belief measure Bel(G) and plausibility measure Pl(G) can be viewed as the lower and upper bounds of the probability measure, which bracket the true probabilistic reliability pr [33]:

BelðGÞ 6 pr 6 PlðGÞ

ð7Þ

In order to calculate the above two measures, whether C # G (the focal element C is entirely located inside the safety region G) or C \ G – £ (C is entirely or partially within the region G) should be determined [33]. Therefore, the extreme values of the limitstate function g over each focal element C should be computed:

½g min ; g max  ¼ ½min gðXÞ; max gðXÞ X2C

X2C

ð8Þ

To reduce the computational cost, the vertex method [37] can be used to calculate gmin and gmax approximately, in which only the vertex points of each focal element are checked. Through the above analysis it can be found that two main factors, namely the dimension of the problem and the number of the focal elements for each variable, determine the computational cost of the above reliability analysis. Suppose the dimension of the problem is n and the number of the focal elements for each variable is h, then hn focal elements in the joint FD will be involved. For each focal element 2n functional evaluations are required to calculate the extreme values of the limit-state function by using the vertex method, and thereby the total number of functional evaluations for the above reliability analysis will reach (2h)n.

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Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45

Therefore, the computational cost of conventional evidence-theory-based reliability analysis is very high.

  1; X  2; . . . ; Xi; . . . ; X n ; g X h i  j ¼ X L or X R ; j ¼ 1; 2; . . . ; n and j – i X i 2 X Li ; X Ri ; X j j

3. A response surface method As introduced above, the high computational cost significantly restricted the application of evidence theory in engineering problems. To solve this problem, the response surface (RS) technique can be adopted to approximate the black-box limit-state function. As indicated in [38–40], design of experiments (DOE) plays a significant role in improving the accuracy of the established RS. Therefore, in this paper, a new DOE technique is developed to construct a sufficiently accurate RS for evidence-theory-based reliability analysis. The evidence variables in practical engineering problems are generally continuous and the FD of each parameter can be expressed as an interval:

h i X i 2 X Ii ¼ X Li ; X Ri ;

corresponding lower or upper bounds. Thus the search of the control point actually will turn to a univariant root-finding problem:

i ¼ 1; 2; . . . ; n

ð9Þ

ð11Þ

where Xi denotes the only varying variable for the ith edge. This problem can be solved by many well established numerical methods. Here, the Newton’s method [41] is adopted due to its robust and fast convergence performance. At the first iteration, the center X Ci ¼ ðX Li þ X Ri Þ=2 of Xi is selected as the starting point. Assume that ðkÞ

ðkþ1Þ

X i has been obtained after the kth iteration, then X i dated [41]: ðkþ1Þ

Xi

ðkÞ

¼ Xi 

can be up-

   1; X  2 ; . . . ; X ðkÞ ; . . . ; X n g X i

ð12Þ

 1; X  2 ; . . . ; X ðkÞ ; . . . ; X nÞ g 0X i ðX i

1; X  2 ; . . . ; X ðkÞ ; . . . ; X  n Þ denotes the derivative of where g 0X i ðX i 1; X  2; . . . ; Xi; . . . ; X  n Þ at X ðkÞ , which can be calculated by the cengðX i

where I denotes the interval, L and R denote the lower and upper bound of interval, respectively. Thus, the uncertainty domain formed by all the FD intervals will become an n-dimensional polyhedron:

h i n   C X ¼ XX i 2 X Li ; X Ri ;

i ¼ 1; 2; . . . ; n

o

ð10Þ

Geometrically, the polyhedron is constituted by some edges, as shown in Fig. 1. For a practical engineering problem, the limit-state surface defined by g(X) = g0 generally intersects with the polyhedron and thereby it will also intersect with some edges of the polyhedron. In this paper, the intersection points are defined as control points. As shown in Fig. 1, these points are just located on the limitstate surface and they together confine the intersection part of the limit-state surface and the polyhedron. Generally, it is necessary to guarantee the RS precision on this intersection part since it determines the precision of the extreme analysis over the uncertainty domain. Therefore, the control points which together confine this part will have an important contribution to the accuracy of the subsequent established RS. Based on the above analysis, we propose an efficient reliability analysis method for evidence theory in the following text, which contains three main steps. Firstly, a new DOE technique is developed and then a radical basis functions (RBF) RS of the limit-state function is established. Finally, the reliability interval is efficiently obtained using the above RS.

tral difference method. The above iterations will be repeated until ðkþ1Þ

ðkÞ

ðkÞ

the convergence criteria kX i  X i k=kX i k 6 e1 and  1; X  2 ; . . . ; X ðkþ1Þ ; . . . ; X  n Þj 6 e2 are simultaneously satisfied, and a jgðX i corresponding control point can be obtained. Actually there are 2n(n  1) edges for an n-dimensional polyhedron. Thus, the above search process should be conducted for 2n(n  1) times. Assume that m control points XP are finally obtained, then the expansion sample points should be selected to guarantee the accuracy of the subsequently established RS. As shown in Fig. 2, the axial experimental design method [38,39] is adopted to obtain these points, where the central point of all control points, ð1Þ ð2Þ ðmÞ XC ¼ ðXP þ XP þ    þ XP Þ=m, and its surrounding axial points, XC ± hrX, are selected. Among them, rX = XI/2 denotes the interval radius of X and h is a sampling coefficient predetermined. 3.2. Construction of the response surface Compared with the polynomial functions, RBFs tend to achieve a better performance and the advantage becomes more obvious for high-order nonlinear problems. RBFs have been validated to be the best interpolation methods compared to others by using examples of different kinds of scattered data [42]. Therefore, in this paper, RBFs [43,44] are used to create a response surface (RS) g~ for the limit-state function:

g~ðXÞ ¼

ns X   wj U r j

ð13Þ

j¼1

3.1. Design of experiments (DOE) Obviously, on each edge of the uncertainty domain there is only one variable varying and the other variables are fixed at their

X2 X

X(1) p

R 2

Control points

X

Limit-state surface

Edge

Uncertainty domain

X 1L

X

R 1

XC XC h

Control X(4) P points X

X(3) P

X 2L

o

o

(1) P

Expansion sample points

X(2) P

(2) p

Limit-state surface X

X 2R

Focal element

X 2L

Edge

X2

Edge

X 1L

Edge X 1R

X1

X1

Fig. 1. Control points in evidence-theory-based reliability analysis.

Fig. 2. The expansion sample points allocated based on the control points.

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Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45

where ns denotes the number of samples; wj, j = 1, 2, . . ., ns denote the coefficients of the linear combinations, respectively. U(rj) is the radial basis function with respect to the Euclidean distance between the estimation point X and the sample point Xj, where rj = ||X  Xj||. Here, the Gaussian radial basis function is adopted [43]:



UðrÞ ¼ exp r2 =c2



ð14Þ

where c is a positive constant.

Table 1 BPA structures with 4 subintervals per parameter (the mathematical problem). X1

X2

X3

Interval

BPA (%)

Interval

BPA (%)

Interval

BPA (%)

[2.0, [2.5, [3.0, [3.5,

10.0 40.0 40.0 10.0

[2.0, [2.5, [3.0, [3.5,

10.0 40.0 40.0 10.0

[2.0, [2.5, [3.0, [3.5,

10.0 40.0 40.0 10.0

2.5] 3.0] 3.5] 4.0]

2.5] 3.0] 3.5] 4.0]

2.5] 3.0] 3.5] 4.0]

3.3. Computation of Bel and Pl By using the RS created above, the reliability interval [Bel, Pl] in Eqs. (5) and (6) can be efficiently calculated. Instead of computing the extreme values of the actual limit-state function g(X) over each focal element C, Eq. (8) can be accomplished only based on g~ðXÞ:

 ½g min ; g max  ¼ ming~ðXÞ; maxg~ðXÞ x2C

x2C

ð15Þ

Since the above extreme analysis is based on the explicit func~ðXÞ rather than some time-consuming simulation models, the tion g belief and plausibility measures can then be quickly obtained and hence a high efficiency can be ensured for the evidence-theorybased reliability analysis. Our method can be summarized as follows: Step 1: Collect all uncertain variables X = (X1, X2, . . ., Xn). Step 2: Conduct the design of experiments. Firstly, search the control points XP for the single-variable function given in Eq. (11) by using the Newton’s method in Eq. (12). Secondly, add the expansion sample points based on XP. Step 3: Construct a RBF response surface g~ðXÞ by Eq. (13) using the above samples. Step 4: Conduct extreme analysis over each focal element based on g~ðXÞ and whereby efficiently obtain the belief and plausibility measures of the structural reliability. From the above procedure it can be found that the computational cost of our method is mainly determined by the search of control points. For each search process, it is essentially a simple problem of finding root for a single-variable function, which needs only several iterations to converge. Thus, the computational cost caused by this process is minor compared to the conventional method. Also, this computational efficiency will become higher when the number of focal elements for each parameter increases, since the proposed method is not affected by the number of focal elements while the conventional method is significantly affected by this factor.

Table 2 BPA structures with 8 subintervals per parameter (the mathematical problem). X1

X2

X3

Interval

BPA (%)

Interval

BPA (%)

Interval

BPA (%)

[2.0, 2.25] [2.25, 2.5] [2.5, 2.75] [2.75, 3.0] [3.0, 3.25] [3.25, 3.5] [3.5, 3.75] [3.75, 4.0]

4 6 15 25 25 15 6 4

[2.0, 2.25] [2.25, 2.5] [2.5, 2.75] [2.75, 3.0] [3.0, 3.25] [3.25, 3.5] [3.5, 3.75] [3.75, 4.0]

4 6 15 25 25 15 6 4

[2.0, 2.25] [2.25, 2.5] [2.5, 2.75] [2.75, 3.0] [3.0, 3.25] [3.25, 3.5] [3.5, 3.75] [3.75, 4.0]

4 6 15 25 25 15 6 4

and seven expansion sample points are allocated for the case a = 2. They are subsequently used to construct a RBF response surface. In this paper, both the proposed method and the conventional method given in Section 2 are used. The conventional method is directly based on the actual limit-state function, and its results are used as reference ones to test the accuracy of the present method. By synthesizing the results under different a, the complementary cumulative belief function (CCBF) and complementary cumulative plausibility function (CCPF) are obtained as shown in Fig. 4. It can be found that the CCBF and CCPF results are all staircase curves which are resulted from the discrete property of BPA in evidence theory. Besides, when comparing the results under 4-subinterval and 8-subinterval cases, it can be found that with the increase of the subintervals for each parameter, the gap between the belief measure and the plausibility measure will become narrower. In other words, the increasing information will lead to a lower level of epistemic uncertainty. Most importantly, the results of the proposed method are very close to those of the reference ones and in most cases they are even exactly the same, which indicates a fine accuracy of the proposed method. For instance, in the 8-subinterval case, the belief measure results of the present method are exactly the same as the reference ones, and the largest deviation of the plausibility measure is only 2.26% which occurs at a = 2. Edge of uncertainty domain

4. Numerical examples and discussion 4.5

4.1. A mathematical problem

Control points Expansion sample points

4

The following limit-state function is considered: 3.5

X3

ðX 1 þ X 2 þ X 3  6Þ2 gðX 1 ; X 2 ; X 3 Þ ¼ a  þ ðX 1  4Þ2 3 þ ðX 2  4Þ2 þ ðX 3  4Þ

ð16Þ

The FDs of X1, X2 and X3 are all [2, 4]. Two different cases that the BPA structure contains 4 and 8 subintervals are investigated, as shown in Tables 1 and 2. Firstly, we make the allowable value a vary and a series of different limit-state functions are generated. For each limit-state function, the Newton’s method is employed to search the control point on each edge, according to which the expansion sample points are also deployed. As shown in Fig. 3, three control points

3 2.5 Actual limit-state surface

2 1.5 1.5

1.5 2.5

2

3 2

2.5

3

X2

3.5 3.5

4

4.5

4.5

4

X1

Fig. 3. Samples obtained from the DOE method when a = 2 (the mathematical problem).

41

Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

Bel,Pl

Bel, Pl

0.6 0.5

0.5

0.4

0.4

0.3

0.3 0.2

0.2 Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0.1 0

-6

-4

-2

0

2

4

6

8

10

Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0.1 0 12

-6

-4

-2

0

2

4

6

α

α

(a) 4 subintervals

(b) 8 subintervals

8

10

12

Fig. 4. Reliability analysis results (the mathematical problem).

On the other hand, the computational cost for the conventional method and our method is compared in Table 3. In this paper, the functional evaluation number is used to reflect the computational cost of the algorithm, but not the computational time. For the 4subinterval case, the conventional method requires 512 functional evaluations, which are 5–8 times of those needed by our method. For the 8-subinterval case, the functional evaluations of the proposed method remain unchanged, while that of the conventional method rises to 4096, which is 40–70 times of the former. Thus, through the comparison of the functional evaluations, the proposed method can be found to be very efficient, especially when the evidence variables have more focal elements.

z F2

y

F1 θ2

θ1

d

P T

x

t

L2

L1 Fig. 5. A cantilever tube [45].

4.2. A cantilever tube Consider the cantilever tube [45] as shown in Fig. 5, which is subjected to external forces F1, F2, P and torsion T. The limit-state function is defined as the difference between the yield strength Sy and the maximum stress on the top surface of the tube at the origin rmax:

g ¼ Sy  rmax

ð17Þ

Table 3 Comparison of functional evaluation number (computational cost) between the proposed method and conventional method (the mathematical problem).

a

6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12

The conventional method

The proposed method

4-Subinterval case

8-Subinterval case

4 and 8-Subinterval cases

512 512 512 512 512 512 512 512 512 512 512 512 512 512 512 512 512 512 512

4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096

61 67 67 68 68 68 68 80 84 88 90 96 92 92 95 98 100 107 109

Table 4 BPA structures with 4 subintervals per parameter (the cantilever tube). t (mm)

h1, h2

d (mm)

Interval

BPA (%)

Interval

BPA (%)

Interval

BPA (%)

[4.0, [4.5, [5.0, [5.5,

10.0 40.0 40.0 10.0

[2p/9, 9p/36] [9p/36, 10p/36] [10p/36, 11p/36] [11p/36, p/3]

10.0 40.0 40.0 10.0

[40, [41, [42, [43,

10.0 40.0 40.0 10.0

4.5] 5.0] 5.5] 6.0]

41] 42] 43] 44]

Table 5 BPA structures with 8 subintervals per parameter (the cantilever tube). t (mm)

h1, h2

d (mm)

Interval

BPA (%)

Interval

BPA (%)

Interval

BPA (%)

[4.0, 4.25] [4.25, 4.5] [4.5, 4.75] [4.75, 5.0] [5.0, 5.25] [5.25, 5.5] [5.5, 5.75] [5.75, 6.0]

4 6 15 25 25 15 6 4

[2p/9, 17p/72] [17p/72, 18p/72] [18p/72, 19p/72] [19p/72, 20p/72] [20p/72, 21p/72] [21p/72, 22p/72] [22p/72, 23p/72] [23p/72, p/3]

4 6 15 25 25 15 6 4

[40, 40.5] [40.5, 41] [41, 41.5] [41.5, 42] [42, 42.5] [42.5, 43] [43, 43.5] [43.5, 44]

4 6 15 25 25 15 6 4

where

rmax ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2x þ 3s2zx

ð18Þ

The normal stress rx is calculated by:

rx ¼

P þ F 1 sinðh1 Þ þ F 2 sinðh2 Þ Mc þ A I

ð19Þ

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Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45 1

1

0.9

0.9

0.8

0.7

0.7

0.6

0.6

Bel,Pl

Bel,Pl

0.8

0.5

0.5

0.4

0.4

0.3

0.3 0.2

0.2 Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0.1 0 77.5

90

100

110

120

130

Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0.1 0 77.5

137.5

90

100

110

120

Sy

Sy

(a) 4 subintervals

(b) 8 subintervals

130

137.5

Fig. 6. Reliability analysis results (the cantilever tube).

Table 6 Comparison of functional evaluation number (computational cost) between the proposed method and conventional method (the cantilever tube). Sy

77.5 80 82.5 85 87.5 90 92.5 95 97.5 100 102.5 105 107.5 110 112.5 115 117.5 120 122.5 125 127.5 130 132.5 135 137.5

The conventional method

The proposed method

4-Subinterval case

8-Subinterval case

4 and 8-Subinterval cases

4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096 4096

65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536 65536

154 153 149 159 159 152 160 166 175 175 185 186 189 184 194 199 209 203 208 215 231 235 234 237 242

Table 7 BPA structures with 4 subintervals per parameter (the 10bar aluminum truss). A1, A2, A3, A4, A5, A6 Interval

BPA (%)

[2.0, 6.0] [6.0, 10.0] [10.0, 14.0] [14.0, 18.0]

10.0 40.0 40.0 10.0

Table 8 BPA structures with 8 subintervals per parameter (the 10bar aluminum truss). A1, A2, A3, A4, A5, A6 Interval

BPA (%)

[2.0, 4.0] [4.0, 6.0] [6.0, 8.0] [8.0, 10.0] [10.0, 12.0] [12.0, 14.0] [14.0, 16.0] [16.0, 18.0]

4 6 15 25 25 15 6 4

where the first term is the normal stress due to the axial forces, and the second term is the normal stress due to the bending moment M, which is given by:

M ¼ F 1 L1 cos h1 þ F 2 L2 cos h2

ð20Þ

and

A¼

ph 4

ph

I¼

64

c¼

2

2

d  ðd  2tÞ 4

d  ðd  2tÞ

i

4

i

d 2

ð21Þ ð22Þ

ð23Þ

The torsional stress szx at the same point is calculated by: Fig. 7. A 10-bar aluminum truss [46].

szx ¼

Td 4I

ð24Þ

43

Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45

1

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Bel,Pl

Bel,Pl

0.9

1 Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0.55

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Bel results of our method Pl results of our method Bel results of conventional method Pl results of conventional method

0 0.55

0.7

0.8

0.9

1

1.1

1.2

dy

dy

(a) 4 subintervals

(b) 8 subintervals

1.3

1.4

1.5

Fig. 8. Reliability analysis results (the 10-bar aluminum truss).

Table 9 Comparison of functional evaluation number (computational cost) between the proposed method and conventional method (the 10-bar aluminum truss). dy

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

The conventional method

The proposed method

4-Subinterval case

8-Subinterval case

4 and 8-Subinterval cases

262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144 262144

16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216 16777216

1290 1307 1281 1244 1234 1185 1245 1265 1243 1253 1293 1270 1330 1355 1295 1383 1449 1471 1494 1499

cases that the BPA structure contains 4 and 8 subintervals are investigated and similar BPA assignments are given in Tables 4 and 5. Through computing the reliability under different yield strengths Sy, the CCBF, CCPF for both methods can then be obtained as shown in Fig. 6. It can be found that under both cases the staircase curves of CCBF and CCPF from the present method agree with those of the reference ones very well, which once again implies a good accuracy of the present method. Table 6 gives a comparison of the computational cost, which also shows that the proposed method has a high efficiency. The functional evaluation number of the conventional method is much larger than that of the proposed one. For the 4-subinterval case, when Sy = 90 the functional evaluation number for the conventional method is 4096, which, however, is only 152 for our method. For the same Sy at the 8subinterval case, this number sharply increases to 65536 for the conventional method, while it is still only 152 for the proposed method. 4.3. A 10-bar truss Consider the well-known 10-bar aluminum truss [46] as shown in Fig. 7. The truss is subjected to two vertical forces F1 = 100 kip and F2 = 120 kip, and a horizontal force F3 = 400 kip. Its length L of the vertical and horizontal bars is 360 in and the modulus of elasticity E is 15,000 ksi. The cross-sectional area of member j = 1, 2, . . ., 10 is denoted as Aj, among them, A7, A8, A9, A10 are 10 in2. The limit-state function is defined as the difference between the allowable displacement dy and the vertical displacement d2 of joint 2:

g ¼ dy  d2 ðA1 ; A2 ; A3 ; A4 ; A5 ; A6 Þ

ð25Þ

where d2 is computed through:

d2 ¼

6 X Ni N0 i

i¼1

Ai

! 10 pffiffiffiX Ni N 0i L þ 2 E Ai i¼7

ð26Þ

And the axial forces Nj, j = 1, 2, . . ., 10 can be easily obtained from the equilibrium and compatibility equations: Fig. 9. A five-degrees-of-freedom vehicle single track model.

In this problem, t, h1, h2 and d are treated  as evidence  variables and their FDs are [4.0 mm, 6.0 mm], 29p ; p3 , 29p ; p3 , [40 mm, 44 mm], respectively. Just like the first example, two different

pffiffiffi pffiffiffi 8 N1 ¼ F 2  2=2N8 ; N2 ¼  2=2N10 > > p ffiffiffi pffiffiffi > > > þ F 3  2=2N8 ; N4 ¼ F 2 þ F 3  2=2N10 > < N3 ¼ F 1  2F pffiffiffi2 pffiffiffi pffiffiffi N5 ¼ F 2  2=2N8  2=2N10 ; N6 ¼  2=2N10 > pffiffiffi > > > N7 ¼ 2ðF 1 þ F 2 Þ þ N8 ; N8 ¼ ða22 b1  a12 b2 Þ=ða11 a22  a12 a21 Þ > > pffiffiffi : N9 ¼ 2F 2 þ N10 ; N10 ¼ ða11 b2  a21 b1 Þ=ða11 a22  a12 a21 Þ ð27Þ

44

Z. Zhang et al. / Advances in Engineering Software 69 (2014) 37–45

where pffiffiffi pffiffiffi 8 a11 ¼ ð1=A1 þ 1=A3 þ 1=A5 þ 2 2=A7 þ 2 2=A8 ÞL=2E > > > > > > < a12 ¼ a21 ¼ L=2A5 E pffiffiffi pffiffiffi a22 ¼ ð1=A2 þ 1=A4 þ 1=A6 þ 2 2=A9 þ 2 2=A10 ÞL=2E > pffiffiffi > > b ¼ ðF =A  ðF þ 2F  F Þ=A  F =A  2pffiffiffi > 2ðF 1 þ F 2 Þ=A7 Þ 2L=2E 1 2 1 1 2 3 3 2 5 > > pffiffiffi pffiffiffi : b2 ¼ ð 2ðF 3  F 2 Þ=A4  2F 2 =A5  4F 2 =A9 ÞL=2E

ð28Þ N 0i

is obtained by assuming F1 = F3 = 0 and F2 = 1 in Eq. (27). In this problem, 6 parameters, namely the cross-sectional areas Aj, j = 1, 2, . . ., 6, are treated as evidence variables, and they possess the same FD of [2.0 in2,18.0 in2]. Two different cases that the BPA structure contains 4 and 8 subintervals are investigated and the detailed BPA assignments are given in Tables 7 and 8. The CCBF, CCPF for the present method and the conventional method under different dy are obtained as shown in Fig. 8. Similar phenomenon can be found that the results of the present method correspond to those of the reference ones very well, which means that the present method has a good precision. Additionally, Table 9 gives a comparison of the computational cost between these two methods, from which we can see that under both the 4-subinterval and 8-subinterval cases the computational cost of the present method is much smaller than that of the conventional method. Thus, the present method has a much higher efficiency. 4.4. Application to a five-degrees-of-freedom vehicle single track model As an important evaluation for the vehicle performance, ride comfort has attracted more and more attention due to its importance to improve the comfort of the drivers and also the grade of a vehicle. Here, a five-degrees-of-freedom vehicle single track model as shown in Fig. 9 is investigated. Ride comfort can be assessed by the weighted vibration level which is converted from the acceleration experienced by the drivers [47]. In this application we will carry out a reliability analysis for the weighted vibration level under a given road surface roughness. Besides, as indicated in [48], the suspension stiffness and damp as well as the vehicle speed have significant impact on the ride comfort. Therefore, the equal stiffness k3 of the front and rear suspensions, the equal damp c3 of the front and rear suspensions as well as the vehicle speed ua are treated as evidence variables in this problem. Their FDs are [1400 Ns/m, 1600 Ns/m], [16,000 N/m, 18,000 N/m] and [10 m/s, 30 m/s] respectively. The limit state function can be defined as the allowable weighted vibration level L0aw and its real value Law:

g ¼ L0aw  Law ðk3 ; c3 ; ua Þ

ð29Þ

This model includes five degrees of freedom (DOF), namely two vertical DOFs for the front wheel and the rear wheel, one vertical DOF and one pitching DOF for the vehicle body, and one vertical DOF for the driver. Two cases of L0aw are considered and the analysis results are given in Table 10. When L0aw ¼ 110 dB, a reliability interval [Bel, Pl] = [0.1, 0.5] is obtained, which indicates that the vehicle’s weighted vibration level during driving cannot satisfy the given allowable value, thus the drivers will feel very uncomfortable. The functional evaluation number in this case is only 166,

Table 10 Reliability analysis results (the five-degrees-of-freedom vehicle single track model). Allowable weighted vibration level (L0max ) (dB)

Bel

PL

Functional evaluation number

110 116

0.1 0.95

0.5 1

166 175

which is acceptable for practical application. When we use a looser design requirement L0aw ¼ 116 dB, the reliability interval [Bel, Pl] = [0.95, 1.0] is obtained. It means that the given weighted vibration level can be satisfied in most cases regardless of the variation of the above three uncertain parameters. The evaluation number in this case is only 175. 5. Conclusions In this paper, a response surface method is developed for the evidence-theory-based reliability analysis to resolve its low efficiency problem and hence expands its engineering application range. In the proposed method, a new DOE technique is developed, which includes the search of the important control points and the deployment of expansion sample points. Based on them, a high precise RBF RS is established. Thus, the present method has a fine accuracy. Also, the obtained RBF RS is used to calculate the reliability interval in evidence theory, which guarantees the high computational efficiency of the proposed method. In the numerical examples, different cases of the allowable values are investigated and the results are compared with those obtained from the conventional reliability analysis. The results show that the proposed method has a fine efficiency and also good accuracy. However, for some complex problems with a large dimension of uncertainty variables, the computational cost still seems to be a challenge to our approach. To further improve the efficiency for this class of problems will be our future work. Acknowledgements This work is supported by the National Outstanding Youth Science Foundation of China (51222502), the National Science Foundation of China (11172096), the Fok Ying-Tong Education Foundation, China (131005) and the program for Century Excellent Talents in University (NCET-11-0124). References [1] Hoffman FO, Hammonds JS. Propagation of uncertainty in risk assessment: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 1994;14(5):707–12. [2] Hasofer AM, Lind NC. Exact and invariant second-moment code format. ASME J Eng Mech Div 1974;100:111–21. [3] Rackwitz R, Fiessler B. Structural reliability under combined random load sequences. Comput Struct 1978;9(5):489–94. [4] Hohenbichler M, Rackwitz R. Non-normal dependent vectors in structural safety. ASME J Eng Mech Div 1981;107(6):1227–38. [5] Breitung KW. Asymptotic approximation for multinormal integrals. ASCE J Eng Mech 1984;110(3):357–66. [6] Ben-Haim Y, Elishakoff I. Convex models of uncertainties in applied mechanics. Amsterdam: Elsevier Science Publisher; 1990. [7] Elishakoff I, Elisseeff P, Glegg S. Non-probabilistic convex-theoretic modeling of scatter in material properties. AIAA J 1994;32:843–9. [8] Qiu ZP, Ma LH, Wang XJ. Ellipsoidal-bound convex model for the non-linear buckling of a column with uncertain initial imperfection. Int J Nonlin Mech 2006;41(8):919–25. [9] Jiang C, Han X, Lu GY, Liu J. Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Method Appl Mech 2011;200(33–36):2528–46. [10] Luo YJ, Kang Z, Li A. Structural reliability assessment based on probability and convex set mixed model. Comput Struct 2009;87(21–22):1408–15. [11] Gao W, Song CM, Tin-Loi F. Probabilistic interval analysis for structures with uncertainty. Struct Saf 2010;32(3):191–9. [12] Zadeh L. Fuzzy set as a basis for a theory of possibility. Fuzzy Set Syst 1978;1:3–28. [13] Klir GJ, Wierman MJ. Uncertainty-based information-elements of generalized information theory. Heidelberg: Physica-Verlag; 1999. [14] Klir GJ. Generalized information theory: aims, results, and open problems. Reliab Eng Syst Saf 2004;85(1–3):21–38. [15] Zadeh L. Fuzzy sets. Inform Control 1965;8:338–53. [16] Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc Ser B Stat Methodol 1977;9(1):1–38. [17] Shafer G. A mathematical theory of evidence. NJ: Princeton; 1976.

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