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A semi-analytic method for calculating non-probabilistic reliability index based on interval models
Applied Mathematical Modelling 31 (2007) 1362–1370 www.elsevier.com/locate/apm
A semi-analytic method for calculating non-probabilistic reliability index based on interval models
q
Jiang Tao, Chen Jian-Jun *, Xu Ya-Lan School of Mechanical and Electronic Engineering, Xidian University, P.O. Box 187, 710071 Xi’an, PR China Received 1 March 2005; accepted 20 February 2006 Available online 27 June 2006
Abstract This paper proves: (1) non-probabilistic reliability index of a structure exists merely at one of intersection points at which normalized failure surfaces of the structure intersects the straight lines passing not only through origin of an normalized infinite space but also through vertices of a symmetric convex polyhedron with its sym-center at the origin, and (2) the non-probabilistic reliability index equals to absolute value of the coordinate components of a particular intersection point. Based on a reduction of the feasible region, a semi-analytical method for calculating the reliability index is developed. The method proves to be simple and of practical significance, and has several advantages over the existing unconstrained multivariate nonlinear optimization approach. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Non-probabilistic reliability index; l1 norm; Convex polyhedron; Intersection point
1. Introduction It is beyond controversy that probability is a well established measure in safety and reliability engineering. However, a mass of experimental data are needed to characterize probability density functions for uncertainties existing in a mechanical model. In most situations, the necessary data are simply lacking, or only partial data are available. Studies [1,2] show that the usefulness of the results of probabilistic modeling may be questionable if the model is established on incomplete data. In the 1990s, Ben-Haim [3] and Elishakoff et al. [4] advocated that the bounded-convex set be a more reasonable measure to characterize parameter uncertainties in the case that probabilistic information needed is deficient. Because of a minor demand for probabilistic information, convex set serving as an uncertainty measurement has been widely used in engineering [5–8]. Based on the convex set theory, non-probabilistic reliability was firstly defined as the degree of uncertainty a structure can bear before its failure [9,10]. q *
The project is supported by the Natural Science Foundation of Shaanxi Province, China (No. D7010418). Corresponding author. E-mail address: [email protected] (C. Jian-Jun).
0307-904X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.02.013
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
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Successively, non-probabilistic reliability index was defined more perfectly as the minimum distance of a normalized failure surface from the origin of a normalized infinite space, and the distance is measured not in Euclidean norms (k Æ k2) but in l1 norms (k Æ k1) [6,11,12]. In this case, searching for the non-probabilistic reliability index in the normalized infinite space can be solved by posing it as an unconstrained multivariate nonlinear optimization problem [6,13]. Unfortunately, the algorithm adopted easily falls in a local optimal solution, or velocity of convergence is intolerably low. In this paper, the unconstrained multivariate nonlinear optimization problem is transformed into a semianalytical method, which is based on a mathematical proof in Section 3. The semi-analytical method derived by the mathematical proof proves to be simple and of practical significance, which is demonstrated by two examples. 2. Definition of non-probabilistic reliability index Suppose F(X) = 0 is failure surface of a structure, and X 2 Rn is an n-dimensional vector with bounded real components, namely X ¼ ðx1 ; x2 ; . . . ; xn Þ;
ð1Þ
in addition, xi 2 ½xli ; xui ;
i ¼ 1; 2; . . . ; n;
where xui and xli are respectively upper and lower bounds for the i-th interval variable, x1, x2, . . . , xn1 represent input variables effecting performances of the structure, such as Young’s modulus of elasticity, external load, etc. xn represents allowable responses of the structure, such as allowable stress, allowable bending moment, etc. Normalizing the n interval variables given in Eq. (1) by performing a operation xi ¼ xci þ xri di ;
i ¼ 1; 2; . . . ; n;
ð2Þ
(xci ¼ ðxli þ xui Þ=2 and xri ¼ ðxui xli Þ=2 are two characteristic quantities for xi), we can get a normalized vector represented by a notation d ¼ ðd1 ; d2 ; . . . ; dn Þ: Obviously, d is contained by a symmetric convex polyhedron Cnjdi j<1 ,fd : jdi j 6 1; i ¼ 1; 2; . . . ; ng with its symcenter at origin of a normalized space Cn ,fd : di 2 ð1; þ1Þ; i ¼ 1; 2; . . . ; ng. Substituting Eq. (2) into F(X) = 0 results a normalized failure surface of the structure defined in Cn . If we denote the normalized failure surface by GðdÞ ¼ Gðd1 ; d2 ; . . . ; dn Þ ¼ 0;
ð3Þ
the non-probabilistic reliability index of the structure can be defined as the minimum distance of the normalized failure surface from origin of Cn , and the distance is measured in l1 norms [6,11,12]. Therefore, the nonprobabilistic reliability index, we denote by a notation g, can be expressed as 8 > < Minimize g ¼ fmaxfjd1 j; jd2 j; . . . ; jdn jgg g ¼ minfkdk1 g Gðd1 ; d2 ; . . . ; dn Þ ¼ 0; ð4Þ , Subject to Subject to GðdÞ ¼ 0 > : di 2 ð1; 1Þ; i ¼ 1; 2; . . . ; n: Pm Pn If failure surface of a structure is a linear equation F ðXÞ ¼ i¼1 ai xi j¼mþ1 bj xj ¼ 0, it follows ! m n m n X X X X r r c c ai x i di bj x j dj þ ai x i bj xj ¼ 0: GðdÞ ¼ i¼1
j¼mþ1
i¼1
j¼mþ1
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In this case, g of this structure equals to !, ! 8 m n m n > P P P < P ai xci bj xcj jai jxri jbj jxrj g¼ i¼1 j¼mþ1 i¼1 j¼mþ1 > : 0;
for
m P
ai xci þ
i¼1
n P
bj xcj > 0;
j¼mþ1
otherwise;
where g is obviously a linear combination of characteristic quantities for xi (i = 1, 2, . . . , n). In most cases, searching for g must be solved by posing it as an unconstrained multivariate nonlinear optimization problem indicated as Eq. (4). However, an optimization algorithm adopted can hardly find the ‘‘true’’ value of g due to the nonlinearity of G(d1,d2, . . . , dn) = 0, or velocity of convergence is intolerably slow owing to the infinite feasible region and the absence of constraints. Therefore, there is a need to find a reliable and efficient method for calculating g. Fortunately, a mathematical proof in Section 3 could serve as theoretical basis for the semi-analytic method developed in Section 4. 3. Mathematical proof In this section, the following two assertions will be proved: 1) g of an arbitrary structure merely exists at one of intersection points at which G(d1, d2, . . . , dn) = 0 intersect the straight lines passing not only through origin of Cn but also through vertices of Cnjdi j<1 . 2) g equals to the absolute value of components of the one intersection point whose components’ absolute value is the minimum one among that of all of the intersection points. Proof. As stated in the above, Cnjdi j<1 is a symmetric convex polyhedron with its symcenter at origin of Cn . Let us suppose that the vertices of Cnjdi j<1 and the origin of Cn be represented respectively by expressions P j ¼ fd : jdji j ¼ 1;
i ¼ 1; 2; . . . ; n;
j ¼ 1; 2; . . . ; 2n g
and
O ¼ fd : di ¼ 0;
i ¼ 1; 2; . . . ; ng;
n
where j and 2 are respectively sequence number and the number of the vertices. It is conceivable that there must exist 2n1 straight lines which pass not only through O but also through and Pj, and the straight lines can be written as d1 ¼ d2 ; . . . ; ¼ dn :
ð5Þ
Suppose d = (d1, d2, . . . , dn) be an arbitrary vector on the normalized failure surface, and then kdk1 and kdk2 satisfy an inequality as below [14] pffiffiffi kdk2 6 nkdk1 : ð6Þ Minimizing both sides of Eq. (6) results another inequality 1 min pffiffiffi kdk2 6 minfkdk1 g: n It is because d satisfies Eq. (3), the right side of Eq. (7) becomes the definition of g, that is, (sffiffiffiffiffiffiffiffiffiffiffiffi) n X 1 pffiffiffi min d2i 6 minfmaxfjd1 j; jd2 j; . . . ; jdn jgg ¼ g: n i¼1
ð7Þ
ð8Þ
Obviously, the equal-sign in Eq. (8) holds if and only if jd1 j ¼ jd2 j ¼ ¼ jdn j:
ð9Þ
It is conceivable that any point on the straight lines (Eq. (5)) possesses a characteristic which can be expressed by Eq. (9). Thus, we suppose there are p intersection points satisfy Eqs. (3) and (9), that is, there are in all p intersection points at which G(d1, d2, . . . , dn) = 0 intersect the straight lines (Eq. (5)), and those intersection points can be denoted by jde1 j ¼ jde2 j ¼ ¼ jden j ¼ we ;
e ¼ 1; 2; . . . ; p;
ð10Þ
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
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where e is an indicative of sequence number of the intersection points, and the value of p depends not only upon concrete expression of G(d1, d2, . . . , dn) = 0 but also upon dimensionality of d. Substitution of Eq. (10) into max{jd1j,jd2j, . . . , jdnj} gives maxfjde1 j; jde2 j; . . . ; jden jg ¼ we ;
e ¼ 1; 2; . . . ; p:
And then, substituting the equation above into Eq. (8), we get g ¼ minfw1 ; w2 ; . . . ; wp g:
ð11Þ
Obviously, Eq. (10) validates what the assertion (1) says, and the same conclusion as what is described in the assertion (2) can be drawn from Eq. (11). h 4. Procedure of the semi-analytic method Step 1. Normalize interval variables of the failure surface F(X) = 0, and we can get a normalized failure surface similar to Eq. (3). Step 2. Itemize 2n1 straight lines passing through both origin of Cn and vertices of Cnjdi j<1 in the following form d1 ¼ d2 ; . . . ; ¼ dn : Step 3. Substitute the straight lines obtained in Step 2 into the normalized failure surface gotten in step 1 sequentially, we can get 2n1 equations with one unknown. Step 4. If the equations are linear or quadratic, they can be solved manually. If degree of the equations is equal to or greater than 3, the equations can be solved by posing it as a single-variable optimization problem. This is the reason why we call our method ‘‘semi-analytic’’. After casting out all of complex roots of the equations, let us suppose the rest m real roots be denoted as fðdl1 ; dl2 ; . . . ; dln Þ : jdli j ¼ jdlj j ¼ wl ; i; j ¼ 1; 2; . . . ; ng;
l ¼ 1; 2; . . . ; m;
where l is the sequence number of real roots. Then, g of the structure is g ¼ minfw1 ; w2 ; . . . ; wm g: 5. Examples Example 1. Let (nx, ny) be an arbitrary point located the first quadrant of a normalized coordinate plan C2 ,fðdx ; dy Þ : dx ; dy 2 ð1; 1Þg shown in Fig. 1, and suppose the straight lines passing through (nx, ny) represent normalized failure surface of a structure, namely Gðdx ; dy Þ ¼ 0,dy ny kðdx nx Þ ¼ 0; where k 2 (1, +1) is slope of the normalized failure surfaces. Assume a set of straight lines passing through origin of C2 is dy = kdx, where k represents slope of those straight lines. n kn kðny knx Þ For k 5 k, dy = kdx intersects G(dx, dy) = 0 at ðdx ; dy Þ ¼ ykk x ; kk , and then
ny knx kðny knx Þ ¼ max ny knx ; kðny knx Þ : ; kðdx ; dy Þk1 ¼
kk kk kk kk
1
ðn kn Þ
kðny knx Þ
jkj 1 Because of ykk x ¼ jkkj jny knx j and kk
¼ jkkj jny knx j, the equation above is equivalent to 1 jkj jny knx j; jny knx j : kðdx ; dy Þk1 ¼ max ð12Þ jk kj jk kj
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δy
(ξ x ,ξ y )
k =1 η
k = −1
δx
η
O
C|δi |<1
k ≠1, k ≠ −1
Fig. 1. Linear normalized failure surface with two interval variables.
The following analysis will show: For k 2 (1, 1), which value does k equal to enables Eq. (12) to attain its minimum value ? Note that the minimum value of Eq. (12) is just the non-probabilistic reliability index g. For an intelligible argumentation, we divide the infinite open interval (1, 1) into four non-overlapping intervals (1, 1), [1, 0), [0, 1) and [1, 1). Upon omitting unimportant details, we are left with four equations equivalent to Eq. (12) as follows 1 1 1 1 For k 2 ð1; 1Þ; g ¼ min jny knx j; jny knx j; jny knx j; jny knx j ¼ jny knx j; 1 k k 1k 1k For k 2 ½1; 0Þ; g ¼ min jny knx j;
ð13Þ
1 1 1 1 jn knx j; jn knx j; jn knx j ¼ jn knx j; kþ1 y k y 1k y 1k y
ð14Þ
1 1 1 1 jn knx j; jny knx j; jn knx j ¼ jn knx j; ð15Þ For k 2 ½0; 1Þ; g ¼ min jny knx j; kþ1 y k 1k y kþ1 y 1 1 1 1 jny knx j; jny knx j; jny knx j ¼ jn knx j: For k 2 ½1; þ1Þ; g ¼ min jny knx j; kþ1 k k1 kþ1 y ð16Þ Eq. (13) and Eq. (14) indicate that g can be obtained if and only if k = 1. Eq. (15) and Eq. (16) indicate that g can be obtained if and only if k = 1. Moreover, g obtained which equals to absolute value of both lateral and vertical coordinates of the intersection point. Given (nx, ny) = (1.05, 4.5), Fig. 2 shows eight curves of Eq. (12) versus a continuous k, for k 2 (1, 1). An arrow in Fig. 2 indicates the direction in which k takes eight discrete values 40, 35, 30, 25, 20, 15, 10 and 5 in turn. Given (nx, ny) = (1.4, 7.6), Fig. 3 shows five curves of Eq. (12) versus a continuous k ,for k 2 [1, 0). An arrow in Fig. 3 indicates the direction in which k takes five discrete values 0.9, 0.7, 0.5, 0.3 and 0.1 in turn. Given (nx, ny) = (1.05, 4.5), Fig. 4 shows nine curves of Eq. (12) versus a continuous k, for k 2 [0, 1). An arrow in Fig. 4 indicates the direction in which k takes nine discrete values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 in turn. Given (nx, ny) = (3.7, 1.21), Fig. 5 shows seven curves of Eq. (12) versus a continuous k, for k 2 [1, +1). An arrow in Fig. 5 indicates the direction in which k takes seven discrete values 2, 3, 5, 7, 10, 20 and 50 in turn. Each curve in Figs. 2–5 illustrates that g can be obtained if and only if k = 1 or k = 1. If (nx, ny) is an arbitrary point in other quadrant of the normalized coordinate plan, we can draw the same conclusion as above that g can be obtained if and only if k = 1 or k = 1. Moreover, dy = dx and dy = dx are none other but the two straight lines that passing through both origin of C2 and vertices of a symmetric
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
2.4
1367
Eq(12)
2.2 2 1.8 1.6 1.4 1.2 1
k -3
-2
-1
0
2
1
Fig. 2. Eight curves of Eq. (12) versus a continuous k for k 2 (1, 1).
12
Eq (12)
11 10 9 8 7 6 5 4 -6
k -4
-2
0
2
4
6
Fig. 3. Five curves of Eq. (12) versus a continuous k for k 2 [1, 0).
5.5
Eq(12)
5 4.5 4 3.5 3 2.5 2 1.5 -3
k -2
-1
0
1
2
3
4
Fig. 4. Nine curves of Eq. (12) versus a continuous k for k 2 ([0, 1).
convex polyhedron C2jdi j<1 ,fd : jdx j 6 1; jdy j 6 1g. Here, origin of C2 is (dx = 0, dy = 0), and vertices of C2jdi j<1 are (dx = 1, dy = 1), (dx = 1, dy = 1), (dx = 1, dy = 1) and (dx = 1, dy = 1). Example 2. In this example, a comparison of the semi-analytic method with the unconstrained multivariate nonlinear optimization approach available in Reference [11] is made.
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Eq(12) 4.5
4
3.5
3
2.5
2
k -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Fig. 5. Seven curves of Eq. (12) versus a continuous k for k 2 [1,1).
b1
p2
p1 b2 Fig. 6. Socle beam.
Fig. 6 shows a socle beam upon which are exerted two concentrated forces p1 2 [4.4, 5.6]KN and p2 2 [1.7, 2.3]KN. Distances of the two points of application from the clamped edge are also interval variables, namely b1 2 [1.8, 2.2]m and b2 2 [4.5, 5.5]m. If allowable ultimate bending moment, which we denote by a notation mcr, for this beam is an interval [32, 40]KNm, it is needed to work out g for this beam. Clearly, response bending moment for the beam, which we denote by mmax, equals to p1b1 + p2b2, and the beam comes to fail if jmmaxj P mcr. So that, the failure surface for the beam can be accordingly expressed by mcr p1 b1 p2 b2 ¼ 0: Normalizing each of interval variable in the equation above by substituting mcr ¼ 36 þ 4dm ; p1 ¼ 5 þ 0:6dp1 , p2 ¼ 2 þ 0:3dp2 , b1 ¼ 2 þ 0:2db1 and b2 ¼ 5 þ 0:5db2 into the equation in the above, we get the normalized failure surface of the beam written as 36 þ 4dm ð5 þ 0:6dp1 Þð2 þ 0:2db1 Þ ð2 þ 0:3dp2 Þð5 þ 0:5db2 Þ ¼ 0: Table 1 below lists 16 (251 = 16, and 5 is the dimensionality of normalized interval vector d ¼ ðdm ; dp1 ; db1 ; dp2 ; db2 Þ) straight lines that passing through both the origin (dm = 0, dp1 = 0 db1 = 0, dp2 = 0, db2 = 0) and vertices pj ¼ fðdm ; dp1 ; db1 ; dp2 ; db2 Þ : jdjm j ¼ jdjp1 j ¼ jdjb1 j ¼ jdjp2 j ¼ jdjb2 j ¼ 1; j ¼ 1; 2; . . . ; 32g: Substituting those straight lines into the normalized failure surface in turn yields 16 quadratic equations with one unknown. By using the formula giving roots of quadratic equation, we get 32 roots tabulated below the corresponding straight lines. Casting out the six complex roots, g for this beam can be obtained by 9 8 > = < 6:51; j 9:1026j; 1:74462075554244; j 33:9668j; 53:3333; j 10j; 2:3630; j 225:69j; > g ¼ min 64:8861; j 8:2195j; 2:5096; j 212:5j; 17:1575; j 3:4538j; 3:1125; j 19:0385j; > > ; : 220:91; 12:7651; 4:6422; 10; 5:9259; j 2:8497j; j 187:15j; 68:9292; 7:7374 g ¼ 1:74462075554244
dm ¼ db1 ¼ db1 ¼ dp2 ¼ db2 6.5100 9.1026 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 64.8861 8.2195 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 21.66+7.99i 21.66 7.99i dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 6.85 + 3.50i 6.85 3.50i
dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 1.74462075554244 33.9668 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 2.5096 212.50 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 220.91 220.91 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 10.0000 5.9259
dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 53.3333 10.0000 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 17.1575 3.4538 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 6.11 + 4.68i 6.11 4.68i dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 2.8497 187.15
dm ¼ dp1 2.3630 dm ¼ dp1 3.1125 dm ¼ dp1 12.7651 dm ¼ dp1 68.9292
¼ db1 ¼ dp2 ¼ db2 225.69 ¼ db1 ¼ dp2 ¼ db2 19.0385 ¼ db1 ¼ dp2 ¼ db2 4.6422 ¼ db1 ¼ dp2 ¼ db2 7.7374
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
Table 1 Straight lines and corresponding roots
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In Ref. [11], finding roots of unconstrained multivariate nonlinear optimization problem of the form Eq. (4) was carried out by using a modified Quasi–Newton algorithm written in Matlab, which was implemented on a PC with a Pentium(R) 2.40 GHz processor and a 256 Mbytes free RAM. When initial value of ðdm ; dp1 ; db1 ; dp2 ; db2 Þ was chosen as (5.0, 5.0, 5.0, 5.0, 5.0), and convergence criterion was chosen to be 0.01, the algorithm got g = 1.74 at the cost of 855 741 iterations and computational time 1847 s. When initial value was chosen as (20.0, 20.0, 20.0, 20.0, 20.0), iterations and computational time increased respectively to 98 547 921 times and 7194 s. However, g = 2.13, gotten in this case, is obviously a local optimal solution. This example illustrates that the algorithm converges very slowly once the initial value chosen is far from the true solution. However, it is not always easy to determine initial value that is close to the true solution, which leads that nonlinear multi-variable optimization technique is considerably expensive to employ. 6. Conclusions Based on a reduced feasible region, the existing unconstrained multivariate nonlinear optimization approach in solving non-probabilistic reliability index is transformed into a semi-analytical method. Objective functions of the latter are all equations in one unknown whose degree of complication is far below than that of the unconstrained multivariate nonlinear optimization approach. This is the reason why the semi-analytic method is more reliable and efficient than the existing approach. References [1] I. Ellishakoff, Essay on uncertainties in elastic and viscoelastic structures: from AMF reudenthal’s criticisms to modern convex modeling, Comput. Struct. 56 (6) (1995) 871–895. [2] R.G. Sexmith, Probability-based safety analysis-value and drawbacks, Struct. Saf. 21 (1999) 303–310. [3] Y. Ben-Haim, Convex models of uncertainty in radial pulse buckling of shells, J. Appl. Mech. 60 (3) (1993) 683. [4] I. Elishakoff, P. Elisseeff, et al., Non-probabilistic, convex-theoretic modeling of scatter in material properties, AIAA J. 32 (1994) 843–849. [5] R.L. Muhanna, R.L. Mullen, Uncertainty in mechanics problems-interval-based approach, J. Eng. Mech. 127 (2001) 557–566. [6] Denggang Wang, Jie Li, A reliable approach to compute the static response of uncertain structural system, Chinese J. Comput. Mech. 20 (6) (2003) 662–668. [7] Qiu Zhiping, Wang Xiaojun, Two non-probabilistic set-theoretical models for dynamic response and buckling failure measures of bars with unknown-but-bounded initial imperfections, Int. J. Solids Struct. 42 (3&4) (2005) 1039–1054. [8] Mc William, Stewart, Anti-optimization of uncertain structures using interval analysis, Comput. Struct. 79 (4) (2001) 421–430. [9] Y. Ben-Haim, A non-probabilistic concept of reliability, Struct. Saf. 14 (4) (1994) 227–245. [10] Y. Ben-Haim, A non-probabilistic measure of reliability of linear systems based on expansion of convex models, Struct. Saf. 17 (2) (1995) 91–109. [11] Shuxiang Guo, Zhenzhou Lu¨, A non-probabilistic model of structural reliability based on interval analysis, Chinese J. Comput. Mech. 18 (1) (2001) 56–60. [12] Shuxiang Guo, Zhenzhou Lu¨, Comparison between the non-probabilistic and probabilistic reliability methods for uncertain structure design, Chinese J. Comput. Mech. 20 (3) (2003) 12–18. [13] G. Alefeld, D. Claudio, The basic properties of interval arithmetic, its software realizations and some applications, Comput. Struct. 67 (1998) 3–8. [14] Tinsley John Oden, Applied Functional Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.
A semi-analytic method for calculating non-probabilistic reliability index based on interval models
q
Jiang Tao, Chen Jian-Jun *, Xu Ya-Lan School of Mechanical and Electronic Engineering, Xidian University, P.O. Box 187, 710071 Xi’an, PR China Received 1 March 2005; accepted 20 February 2006 Available online 27 June 2006
Abstract This paper proves: (1) non-probabilistic reliability index of a structure exists merely at one of intersection points at which normalized failure surfaces of the structure intersects the straight lines passing not only through origin of an normalized infinite space but also through vertices of a symmetric convex polyhedron with its sym-center at the origin, and (2) the non-probabilistic reliability index equals to absolute value of the coordinate components of a particular intersection point. Based on a reduction of the feasible region, a semi-analytical method for calculating the reliability index is developed. The method proves to be simple and of practical significance, and has several advantages over the existing unconstrained multivariate nonlinear optimization approach. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Non-probabilistic reliability index; l1 norm; Convex polyhedron; Intersection point
1. Introduction It is beyond controversy that probability is a well established measure in safety and reliability engineering. However, a mass of experimental data are needed to characterize probability density functions for uncertainties existing in a mechanical model. In most situations, the necessary data are simply lacking, or only partial data are available. Studies [1,2] show that the usefulness of the results of probabilistic modeling may be questionable if the model is established on incomplete data. In the 1990s, Ben-Haim [3] and Elishakoff et al. [4] advocated that the bounded-convex set be a more reasonable measure to characterize parameter uncertainties in the case that probabilistic information needed is deficient. Because of a minor demand for probabilistic information, convex set serving as an uncertainty measurement has been widely used in engineering [5–8]. Based on the convex set theory, non-probabilistic reliability was firstly defined as the degree of uncertainty a structure can bear before its failure [9,10]. q *
The project is supported by the Natural Science Foundation of Shaanxi Province, China (No. D7010418). Corresponding author. E-mail address: [email protected] (C. Jian-Jun).
0307-904X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.02.013
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
1363
Successively, non-probabilistic reliability index was defined more perfectly as the minimum distance of a normalized failure surface from the origin of a normalized infinite space, and the distance is measured not in Euclidean norms (k Æ k2) but in l1 norms (k Æ k1) [6,11,12]. In this case, searching for the non-probabilistic reliability index in the normalized infinite space can be solved by posing it as an unconstrained multivariate nonlinear optimization problem [6,13]. Unfortunately, the algorithm adopted easily falls in a local optimal solution, or velocity of convergence is intolerably low. In this paper, the unconstrained multivariate nonlinear optimization problem is transformed into a semianalytical method, which is based on a mathematical proof in Section 3. The semi-analytical method derived by the mathematical proof proves to be simple and of practical significance, which is demonstrated by two examples. 2. Definition of non-probabilistic reliability index Suppose F(X) = 0 is failure surface of a structure, and X 2 Rn is an n-dimensional vector with bounded real components, namely X ¼ ðx1 ; x2 ; . . . ; xn Þ;
ð1Þ
in addition, xi 2 ½xli ; xui ;
i ¼ 1; 2; . . . ; n;
where xui and xli are respectively upper and lower bounds for the i-th interval variable, x1, x2, . . . , xn1 represent input variables effecting performances of the structure, such as Young’s modulus of elasticity, external load, etc. xn represents allowable responses of the structure, such as allowable stress, allowable bending moment, etc. Normalizing the n interval variables given in Eq. (1) by performing a operation xi ¼ xci þ xri di ;
i ¼ 1; 2; . . . ; n;
ð2Þ
(xci ¼ ðxli þ xui Þ=2 and xri ¼ ðxui xli Þ=2 are two characteristic quantities for xi), we can get a normalized vector represented by a notation d ¼ ðd1 ; d2 ; . . . ; dn Þ: Obviously, d is contained by a symmetric convex polyhedron Cnjdi j<1 ,fd : jdi j 6 1; i ¼ 1; 2; . . . ; ng with its symcenter at origin of a normalized space Cn ,fd : di 2 ð1; þ1Þ; i ¼ 1; 2; . . . ; ng. Substituting Eq. (2) into F(X) = 0 results a normalized failure surface of the structure defined in Cn . If we denote the normalized failure surface by GðdÞ ¼ Gðd1 ; d2 ; . . . ; dn Þ ¼ 0;
ð3Þ
the non-probabilistic reliability index of the structure can be defined as the minimum distance of the normalized failure surface from origin of Cn , and the distance is measured in l1 norms [6,11,12]. Therefore, the nonprobabilistic reliability index, we denote by a notation g, can be expressed as 8 > < Minimize g ¼ fmaxfjd1 j; jd2 j; . . . ; jdn jgg g ¼ minfkdk1 g Gðd1 ; d2 ; . . . ; dn Þ ¼ 0; ð4Þ , Subject to Subject to GðdÞ ¼ 0 > : di 2 ð1; 1Þ; i ¼ 1; 2; . . . ; n: Pm Pn If failure surface of a structure is a linear equation F ðXÞ ¼ i¼1 ai xi j¼mþ1 bj xj ¼ 0, it follows ! m n m n X X X X r r c c ai x i di bj x j dj þ ai x i bj xj ¼ 0: GðdÞ ¼ i¼1
j¼mþ1
i¼1
j¼mþ1
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In this case, g of this structure equals to !, ! 8 m n m n > P P P < P ai xci bj xcj jai jxri jbj jxrj g¼ i¼1 j¼mþ1 i¼1 j¼mþ1 > : 0;
for
m P
ai xci þ
i¼1
n P
bj xcj > 0;
j¼mþ1
otherwise;
where g is obviously a linear combination of characteristic quantities for xi (i = 1, 2, . . . , n). In most cases, searching for g must be solved by posing it as an unconstrained multivariate nonlinear optimization problem indicated as Eq. (4). However, an optimization algorithm adopted can hardly find the ‘‘true’’ value of g due to the nonlinearity of G(d1,d2, . . . , dn) = 0, or velocity of convergence is intolerably slow owing to the infinite feasible region and the absence of constraints. Therefore, there is a need to find a reliable and efficient method for calculating g. Fortunately, a mathematical proof in Section 3 could serve as theoretical basis for the semi-analytic method developed in Section 4. 3. Mathematical proof In this section, the following two assertions will be proved: 1) g of an arbitrary structure merely exists at one of intersection points at which G(d1, d2, . . . , dn) = 0 intersect the straight lines passing not only through origin of Cn but also through vertices of Cnjdi j<1 . 2) g equals to the absolute value of components of the one intersection point whose components’ absolute value is the minimum one among that of all of the intersection points. Proof. As stated in the above, Cnjdi j<1 is a symmetric convex polyhedron with its symcenter at origin of Cn . Let us suppose that the vertices of Cnjdi j<1 and the origin of Cn be represented respectively by expressions P j ¼ fd : jdji j ¼ 1;
i ¼ 1; 2; . . . ; n;
j ¼ 1; 2; . . . ; 2n g
and
O ¼ fd : di ¼ 0;
i ¼ 1; 2; . . . ; ng;
n
where j and 2 are respectively sequence number and the number of the vertices. It is conceivable that there must exist 2n1 straight lines which pass not only through O but also through and Pj, and the straight lines can be written as d1 ¼ d2 ; . . . ; ¼ dn :
ð5Þ
Suppose d = (d1, d2, . . . , dn) be an arbitrary vector on the normalized failure surface, and then kdk1 and kdk2 satisfy an inequality as below [14] pffiffiffi kdk2 6 nkdk1 : ð6Þ Minimizing both sides of Eq. (6) results another inequality 1 min pffiffiffi kdk2 6 minfkdk1 g: n It is because d satisfies Eq. (3), the right side of Eq. (7) becomes the definition of g, that is, (sffiffiffiffiffiffiffiffiffiffiffiffi) n X 1 pffiffiffi min d2i 6 minfmaxfjd1 j; jd2 j; . . . ; jdn jgg ¼ g: n i¼1
ð7Þ
ð8Þ
Obviously, the equal-sign in Eq. (8) holds if and only if jd1 j ¼ jd2 j ¼ ¼ jdn j:
ð9Þ
It is conceivable that any point on the straight lines (Eq. (5)) possesses a characteristic which can be expressed by Eq. (9). Thus, we suppose there are p intersection points satisfy Eqs. (3) and (9), that is, there are in all p intersection points at which G(d1, d2, . . . , dn) = 0 intersect the straight lines (Eq. (5)), and those intersection points can be denoted by jde1 j ¼ jde2 j ¼ ¼ jden j ¼ we ;
e ¼ 1; 2; . . . ; p;
ð10Þ
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where e is an indicative of sequence number of the intersection points, and the value of p depends not only upon concrete expression of G(d1, d2, . . . , dn) = 0 but also upon dimensionality of d. Substitution of Eq. (10) into max{jd1j,jd2j, . . . , jdnj} gives maxfjde1 j; jde2 j; . . . ; jden jg ¼ we ;
e ¼ 1; 2; . . . ; p:
And then, substituting the equation above into Eq. (8), we get g ¼ minfw1 ; w2 ; . . . ; wp g:
ð11Þ
Obviously, Eq. (10) validates what the assertion (1) says, and the same conclusion as what is described in the assertion (2) can be drawn from Eq. (11). h 4. Procedure of the semi-analytic method Step 1. Normalize interval variables of the failure surface F(X) = 0, and we can get a normalized failure surface similar to Eq. (3). Step 2. Itemize 2n1 straight lines passing through both origin of Cn and vertices of Cnjdi j<1 in the following form d1 ¼ d2 ; . . . ; ¼ dn : Step 3. Substitute the straight lines obtained in Step 2 into the normalized failure surface gotten in step 1 sequentially, we can get 2n1 equations with one unknown. Step 4. If the equations are linear or quadratic, they can be solved manually. If degree of the equations is equal to or greater than 3, the equations can be solved by posing it as a single-variable optimization problem. This is the reason why we call our method ‘‘semi-analytic’’. After casting out all of complex roots of the equations, let us suppose the rest m real roots be denoted as fðdl1 ; dl2 ; . . . ; dln Þ : jdli j ¼ jdlj j ¼ wl ; i; j ¼ 1; 2; . . . ; ng;
l ¼ 1; 2; . . . ; m;
where l is the sequence number of real roots. Then, g of the structure is g ¼ minfw1 ; w2 ; . . . ; wm g: 5. Examples Example 1. Let (nx, ny) be an arbitrary point located the first quadrant of a normalized coordinate plan C2 ,fðdx ; dy Þ : dx ; dy 2 ð1; 1Þg shown in Fig. 1, and suppose the straight lines passing through (nx, ny) represent normalized failure surface of a structure, namely Gðdx ; dy Þ ¼ 0,dy ny kðdx nx Þ ¼ 0; where k 2 (1, +1) is slope of the normalized failure surfaces. Assume a set of straight lines passing through origin of C2 is dy = kdx, where k represents slope of those straight lines. n kn kðny knx Þ For k 5 k, dy = kdx intersects G(dx, dy) = 0 at ðdx ; dy Þ ¼ ykk x ; kk , and then
ny knx kðny knx Þ ¼ max ny knx ; kðny knx Þ : ; kðdx ; dy Þk1 ¼
kk kk kk kk
1
ðn kn Þ
kðny knx Þ
jkj 1 Because of ykk x ¼ jkkj jny knx j and kk
¼ jkkj jny knx j, the equation above is equivalent to 1 jkj jny knx j; jny knx j : kðdx ; dy Þk1 ¼ max ð12Þ jk kj jk kj
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δy
(ξ x ,ξ y )
k =1 η
k = −1
δx
η
O
C|δi |<1
k ≠1, k ≠ −1
Fig. 1. Linear normalized failure surface with two interval variables.
The following analysis will show: For k 2 (1, 1), which value does k equal to enables Eq. (12) to attain its minimum value ? Note that the minimum value of Eq. (12) is just the non-probabilistic reliability index g. For an intelligible argumentation, we divide the infinite open interval (1, 1) into four non-overlapping intervals (1, 1), [1, 0), [0, 1) and [1, 1). Upon omitting unimportant details, we are left with four equations equivalent to Eq. (12) as follows 1 1 1 1 For k 2 ð1; 1Þ; g ¼ min jny knx j; jny knx j; jny knx j; jny knx j ¼ jny knx j; 1 k k 1k 1k For k 2 ½1; 0Þ; g ¼ min jny knx j;
ð13Þ
1 1 1 1 jn knx j; jn knx j; jn knx j ¼ jn knx j; kþ1 y k y 1k y 1k y
ð14Þ
1 1 1 1 jn knx j; jny knx j; jn knx j ¼ jn knx j; ð15Þ For k 2 ½0; 1Þ; g ¼ min jny knx j; kþ1 y k 1k y kþ1 y 1 1 1 1 jny knx j; jny knx j; jny knx j ¼ jn knx j: For k 2 ½1; þ1Þ; g ¼ min jny knx j; kþ1 k k1 kþ1 y ð16Þ Eq. (13) and Eq. (14) indicate that g can be obtained if and only if k = 1. Eq. (15) and Eq. (16) indicate that g can be obtained if and only if k = 1. Moreover, g obtained which equals to absolute value of both lateral and vertical coordinates of the intersection point. Given (nx, ny) = (1.05, 4.5), Fig. 2 shows eight curves of Eq. (12) versus a continuous k, for k 2 (1, 1). An arrow in Fig. 2 indicates the direction in which k takes eight discrete values 40, 35, 30, 25, 20, 15, 10 and 5 in turn. Given (nx, ny) = (1.4, 7.6), Fig. 3 shows five curves of Eq. (12) versus a continuous k ,for k 2 [1, 0). An arrow in Fig. 3 indicates the direction in which k takes five discrete values 0.9, 0.7, 0.5, 0.3 and 0.1 in turn. Given (nx, ny) = (1.05, 4.5), Fig. 4 shows nine curves of Eq. (12) versus a continuous k, for k 2 [0, 1). An arrow in Fig. 4 indicates the direction in which k takes nine discrete values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 in turn. Given (nx, ny) = (3.7, 1.21), Fig. 5 shows seven curves of Eq. (12) versus a continuous k, for k 2 [1, +1). An arrow in Fig. 5 indicates the direction in which k takes seven discrete values 2, 3, 5, 7, 10, 20 and 50 in turn. Each curve in Figs. 2–5 illustrates that g can be obtained if and only if k = 1 or k = 1. If (nx, ny) is an arbitrary point in other quadrant of the normalized coordinate plan, we can draw the same conclusion as above that g can be obtained if and only if k = 1 or k = 1. Moreover, dy = dx and dy = dx are none other but the two straight lines that passing through both origin of C2 and vertices of a symmetric
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
2.4
1367
Eq(12)
2.2 2 1.8 1.6 1.4 1.2 1
k -3
-2
-1
0
2
1
Fig. 2. Eight curves of Eq. (12) versus a continuous k for k 2 (1, 1).
12
Eq (12)
11 10 9 8 7 6 5 4 -6
k -4
-2
0
2
4
6
Fig. 3. Five curves of Eq. (12) versus a continuous k for k 2 [1, 0).
5.5
Eq(12)
5 4.5 4 3.5 3 2.5 2 1.5 -3
k -2
-1
0
1
2
3
4
Fig. 4. Nine curves of Eq. (12) versus a continuous k for k 2 ([0, 1).
convex polyhedron C2jdi j<1 ,fd : jdx j 6 1; jdy j 6 1g. Here, origin of C2 is (dx = 0, dy = 0), and vertices of C2jdi j<1 are (dx = 1, dy = 1), (dx = 1, dy = 1), (dx = 1, dy = 1) and (dx = 1, dy = 1). Example 2. In this example, a comparison of the semi-analytic method with the unconstrained multivariate nonlinear optimization approach available in Reference [11] is made.
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Eq(12) 4.5
4
3.5
3
2.5
2
k -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Fig. 5. Seven curves of Eq. (12) versus a continuous k for k 2 [1,1).
b1
p2
p1 b2 Fig. 6. Socle beam.
Fig. 6 shows a socle beam upon which are exerted two concentrated forces p1 2 [4.4, 5.6]KN and p2 2 [1.7, 2.3]KN. Distances of the two points of application from the clamped edge are also interval variables, namely b1 2 [1.8, 2.2]m and b2 2 [4.5, 5.5]m. If allowable ultimate bending moment, which we denote by a notation mcr, for this beam is an interval [32, 40]KNm, it is needed to work out g for this beam. Clearly, response bending moment for the beam, which we denote by mmax, equals to p1b1 + p2b2, and the beam comes to fail if jmmaxj P mcr. So that, the failure surface for the beam can be accordingly expressed by mcr p1 b1 p2 b2 ¼ 0: Normalizing each of interval variable in the equation above by substituting mcr ¼ 36 þ 4dm ; p1 ¼ 5 þ 0:6dp1 , p2 ¼ 2 þ 0:3dp2 , b1 ¼ 2 þ 0:2db1 and b2 ¼ 5 þ 0:5db2 into the equation in the above, we get the normalized failure surface of the beam written as 36 þ 4dm ð5 þ 0:6dp1 Þð2 þ 0:2db1 Þ ð2 þ 0:3dp2 Þð5 þ 0:5db2 Þ ¼ 0: Table 1 below lists 16 (251 = 16, and 5 is the dimensionality of normalized interval vector d ¼ ðdm ; dp1 ; db1 ; dp2 ; db2 Þ) straight lines that passing through both the origin (dm = 0, dp1 = 0 db1 = 0, dp2 = 0, db2 = 0) and vertices pj ¼ fðdm ; dp1 ; db1 ; dp2 ; db2 Þ : jdjm j ¼ jdjp1 j ¼ jdjb1 j ¼ jdjp2 j ¼ jdjb2 j ¼ 1; j ¼ 1; 2; . . . ; 32g: Substituting those straight lines into the normalized failure surface in turn yields 16 quadratic equations with one unknown. By using the formula giving roots of quadratic equation, we get 32 roots tabulated below the corresponding straight lines. Casting out the six complex roots, g for this beam can be obtained by 9 8 > = < 6:51; j 9:1026j; 1:74462075554244; j 33:9668j; 53:3333; j 10j; 2:3630; j 225:69j; > g ¼ min 64:8861; j 8:2195j; 2:5096; j 212:5j; 17:1575; j 3:4538j; 3:1125; j 19:0385j; > > ; : 220:91; 12:7651; 4:6422; 10; 5:9259; j 2:8497j; j 187:15j; 68:9292; 7:7374 g ¼ 1:74462075554244
dm ¼ db1 ¼ db1 ¼ dp2 ¼ db2 6.5100 9.1026 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 64.8861 8.2195 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 21.66+7.99i 21.66 7.99i dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 6.85 + 3.50i 6.85 3.50i
dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 1.74462075554244 33.9668 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 2.5096 212.50 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 220.91 220.91 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 10.0000 5.9259
dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 53.3333 10.0000 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 17.1575 3.4538 dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 6.11 + 4.68i 6.11 4.68i dm ¼ dp1 ¼ db1 ¼ dp2 ¼ db2 2.8497 187.15
dm ¼ dp1 2.3630 dm ¼ dp1 3.1125 dm ¼ dp1 12.7651 dm ¼ dp1 68.9292
¼ db1 ¼ dp2 ¼ db2 225.69 ¼ db1 ¼ dp2 ¼ db2 19.0385 ¼ db1 ¼ dp2 ¼ db2 4.6422 ¼ db1 ¼ dp2 ¼ db2 7.7374
J. Tao et al. / Applied Mathematical Modelling 31 (2007) 1362–1370
Table 1 Straight lines and corresponding roots
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In Ref. [11], finding roots of unconstrained multivariate nonlinear optimization problem of the form Eq. (4) was carried out by using a modified Quasi–Newton algorithm written in Matlab, which was implemented on a PC with a Pentium(R) 2.40 GHz processor and a 256 Mbytes free RAM. When initial value of ðdm ; dp1 ; db1 ; dp2 ; db2 Þ was chosen as (5.0, 5.0, 5.0, 5.0, 5.0), and convergence criterion was chosen to be 0.01, the algorithm got g = 1.74 at the cost of 855 741 iterations and computational time 1847 s. When initial value was chosen as (20.0, 20.0, 20.0, 20.0, 20.0), iterations and computational time increased respectively to 98 547 921 times and 7194 s. However, g = 2.13, gotten in this case, is obviously a local optimal solution. This example illustrates that the algorithm converges very slowly once the initial value chosen is far from the true solution. However, it is not always easy to determine initial value that is close to the true solution, which leads that nonlinear multi-variable optimization technique is considerably expensive to employ. 6. Conclusions Based on a reduced feasible region, the existing unconstrained multivariate nonlinear optimization approach in solving non-probabilistic reliability index is transformed into a semi-analytical method. Objective functions of the latter are all equations in one unknown whose degree of complication is far below than that of the unconstrained multivariate nonlinear optimization approach. This is the reason why the semi-analytic method is more reliable and efficient than the existing approach. References [1] I. Ellishakoff, Essay on uncertainties in elastic and viscoelastic structures: from AMF reudenthal’s criticisms to modern convex modeling, Comput. Struct. 56 (6) (1995) 871–895. [2] R.G. Sexmith, Probability-based safety analysis-value and drawbacks, Struct. Saf. 21 (1999) 303–310. [3] Y. Ben-Haim, Convex models of uncertainty in radial pulse buckling of shells, J. Appl. Mech. 60 (3) (1993) 683. [4] I. Elishakoff, P. Elisseeff, et al., Non-probabilistic, convex-theoretic modeling of scatter in material properties, AIAA J. 32 (1994) 843–849. [5] R.L. Muhanna, R.L. Mullen, Uncertainty in mechanics problems-interval-based approach, J. Eng. Mech. 127 (2001) 557–566. [6] Denggang Wang, Jie Li, A reliable approach to compute the static response of uncertain structural system, Chinese J. Comput. Mech. 20 (6) (2003) 662–668. [7] Qiu Zhiping, Wang Xiaojun, Two non-probabilistic set-theoretical models for dynamic response and buckling failure measures of bars with unknown-but-bounded initial imperfections, Int. J. Solids Struct. 42 (3&4) (2005) 1039–1054. [8] Mc William, Stewart, Anti-optimization of uncertain structures using interval analysis, Comput. Struct. 79 (4) (2001) 421–430. [9] Y. Ben-Haim, A non-probabilistic concept of reliability, Struct. Saf. 14 (4) (1994) 227–245. [10] Y. Ben-Haim, A non-probabilistic measure of reliability of linear systems based on expansion of convex models, Struct. Saf. 17 (2) (1995) 91–109. [11] Shuxiang Guo, Zhenzhou Lu¨, A non-probabilistic model of structural reliability based on interval analysis, Chinese J. Comput. Mech. 18 (1) (2001) 56–60. [12] Shuxiang Guo, Zhenzhou Lu¨, Comparison between the non-probabilistic and probabilistic reliability methods for uncertain structure design, Chinese J. Comput. Mech. 20 (3) (2003) 12–18. [13] G. Alefeld, D. Claudio, The basic properties of interval arithmetic, its software realizations and some applications, Comput. Struct. 67 (1998) 3–8. [14] Tinsley John Oden, Applied Functional Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.