A nonprobabilistic set model of structural reliability based on satisfaction degree of interval/Konstrukcinis netikimybinis aibes patikimumo modelis pagristas patikimu intervalo dydziu.

Huang, H.-Z. ; Wang, Z.L. ; Li, Y.F. 等

1. Introduction

Uncertainties in material properties, geometric dimensions, loads and other parameters are always unavoidable in engineering structural problems [1-4]. They have played a more and more important role in the structural reliability analysis. In order to obtain the objective of reliable design, the effects of the various uncertain parameters should be rationally considered and treated. The probabilistic models are widely used to describe the uncertainties and they have been proved very effective in the structural reliability problems [5-7]. However, it is difficult to estimate precise values of parameters to accurately define the probability distributions because of inaccurate and insufficient information. Once the assumption about the probability distributions is not satisfied, the structural reliability analysis seems doubtful and meaningless. Moreover, some researches [8-11] have also indicated that even small deviations from the real probability distributions may cause large errors in the reliability analysis.

The fuzzy set theory provides a useful complement of classic reliability theory, in which the probabilities of the system elements can be not certain. Cai [12] presented different forms of "fuzzy reliability theories". In some recent research, a general fuzzy multistate system model and corresponding reliability evaluation technique fuzzy universal generating function were proposed in [13] and [14], respectively, for dealing with the fuzziness of engineering systems. Similar with the probabilistic models, the membership functions of the uncertain parameters need to be established before carrying out structural reliability analysis with the fuzzy set theory. The nonprobabilistic reliability method and set model can be another direction for coping with the uncertain parameters. Although obtaining the precise probability distributions or membership functions of the uncertain parameters seems very difficult in many cases, the ranges or bounds of the uncertain parameters can be established relatively easily. Nowadays, there is not a precise method to find the precise intervals or the precise bounds of the uncertain parameters. However, one of the most feasible methods to find the approximate precise intervals or bounds of the uncertain parameters is "expert scoring method". For example, for a system uncertain variable x, several experts can given difference intervals or bounds for the variable. Sometimes, these intervals or bounds are not all the same, the method to handle these intervals or bounds are "average arithmetic". For example, there are n intervals scoring by n experts for an uncertainty variable x such as [x.sup.I.sub.1], [x.sup.I.sub.2], [x.sup.I.sub.3], ... [x.sup.I.sub.n]. The result interval of uncertainty variable x expressed as

[x.sup.I] = 1/n ([x.sup.I.sub.1], [x.sup.I.sub.2] + [x.sup.I.sub.3] + ... + [x.sup.I.sub.n].

Some researcher such as Ben-Haim [10, 11] proposed that it was more rational to describe the uncertain parameters with the set models instead of the probability models when the statistic information about the uncertain parameters is insufficient. Based on this idea, the concept of nonprobabilistic reliability based on the convex model theory was proposed clearly by Ben-Haim in 1994 [11]. In recent years, the nonprobabilistic reliability theory develops rapidly. Elishakoff [15] discussed the concept of nonprobabilistic reliability and pointed out that the reliability of structures should belong to an interval rather than a certain value. Through interval analysis [16], a nonprobabilistic model of structural reliability was proposed by Guo et al [17] which the reliability was measured as the minimum distance from the coordinate origin to the failure surface. Based on the interval interference model of stress and strength, Wang and Qiu [18] defined the nonprobabilistic reliability index as the ratio of the volume of safe region to the total volume of the region constructed by the basic interval variables. In addition, the nonprobabilistic approaches have already been effectively applied to many practical structure problems in presence of various uncertainties. For example, they were used in the analysis of shells with imperfections in [19, 20], stress concentration at a nearly circular hole with uncertain irregularities in [21] and sandwich plates subject to uncertain loads and initial deflections in [22].

In this paper a new nonprobabilistic set model for reliability assessment of structural system is proposed. Interval variables are used to represent the parameter uncertainty. The nonprobabilistic reliability of structure is defined as the satisfaction degree between the stress-interval and the strength-interval. The interval analysis based on the first-order Taylor series is used to calculate the corresponding reliability. The illustrative example is presented to demonstrate the technique.

2. Interval variable and its operations

Before further discussion on the nonprobabilistic set model of structural reliability, a brief view of the definitions of the interval variable and its operations is provided. Assume that x denotes an uncertain parameter in the structural reliability problem, and it varies within a closed interval [x.sub.I] = [[x.bar], [bar.x]], then

x [member of] [x.sup.I] = [[x.bar], [bar.x]] (1)

is defined as an interval variable; [x.bar] and [bar.x] is the lower bound and upper bound of the interval [x.sub.I], respectively. Similar with the random variable, interval variable has its own center [x.sup.c] and radius [x.sup.r], which can be defined as follows

[x.sup.c] = x + [bar.x]/2 , [x.sup.r] = [bar.x] - x/2 (2)

According to Eq. (2), interval [x.sup.I] and interval variable x can be denoted in the following standardized form

[x.sup.I] = [x.sup.c] + [x.sup.r] [[DELTA].sup.I], x = [x.sup.c] + [x.sup.r][delta] (3)

where [[DELTA].sup.I] = [-1,1] is the standardized interval, [delta] [member of] [[DELTA].sup.I] is the standardized interval variable.

Let x [member of] [x.sup.I] = [[bar.x], [x.bar]] and y [member of] [y.sup.I] = [[bar.y], [y.bar]] be two interval variables, then the operations for [x.sup.I] + [y.sup.I], [x.sup.I] - [y.sup.I], [x.sup.I]/[y.sup.I] and [x.sup.I]/[y.sup.I] are obtained as [23]

[x.sup.I] + [y.sup.I] = [[bar.x], [x.bar]] + [[bar.y], [y.bar]] = [x.bar] + [y.bar], [bar.x] + [bar.y] (4)

[x.sup.I] - [y.sup.I] = [[bar.x], [x.bar]] - [[bar.y], [y.bar]] = [x.bar] + [y.bar], [bar.x] + [bar.y] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[x.sup.I] / [y.sup.I] = [[bar.x], [x.bar]] + [[bar.y], [y.bar]] = [[x.bar] x [bar.x]] x [1/[bar.y] 1/ [y.bar.] (7)

Supposed that I (R) denotes the sets of all closed real intervals. [x.sup.I.sub.i][member of] I(R), [x.sub.i] [member of] [x.sup.I.sub.i] (1,2,..., n) are arbitrary

interval variables which are independent with each other. The linear combination of these interval variables can be formed as follows

y = [n.summation over (i-1)] [a.sub.i][x.sub.i], 1 = 1,2,..., n (8)

where [a.sub.i] [member of] R are arbitrary real numbers. Because y is the linear combination of interval [x.sub.i], y is also an interval variable. If the center and radius of interval variables [x.sub.i] are denoted with [x.sup.c.sub.i] and [x.sup.r.sub.i], then the center and radius of interval variable y are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

3. Satisfaction degree of the relation [x.sup.I] [less than or equal to] [y.sup.I]

Different with the size relation of two real numbers, the size relation of two intervals is a kind of partial-order relation [24] which is usually denoted with the satisfaction degree of the two intervals. Here the concept of satisfaction degree of the relation [x.sup.I] [less than or equal to] < [y.sup.I] is actually a fuzzy set definition which represents the possibility that one interval is larger or smaller than the other. It is often used to compare intervals. Assumed that there are two intervals [x.sup.I] = [[x.bar], [bar.x]] and [y.sup.I] = [[y.bar], [bar.y]] , consider the related rectangle in the (x, y)--plane having the sides given by the two intervals. There are five case between [x.sup.I] [less than or equal to] [y.sup.I] which is expressed in Fig. 1. The area value of the set {(x, y): [x.bar] [less than or equal to] x [less than or equal to] [bar.x], [y.bar] [less than or equal to] y [less than or equal to] [bar.y]} can be computed as [[omega]([x.sup.I] x [omega]([y.sup.I]). The area value of shadow part can express as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where area ([x]) denotes the area value of shadow part.

The satisfaction degree of the relation [x.sup.I] [less than or equal to] [y.sup.I] or reliability can be defined as

P([x.sup.I] [less than or equal to] [y.sup.I]) = area([x])/[omega]([x.sup.I]) X [omega]([y.sup.i]) (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where " P " means possibility, [omega] ([x.sup.I]) and [omega] ([y.sup.I]) denotes the width of interval [x.sup.I] and [y.sup.I] , respectively. That is [25]

[omega]([x.sup.I]) = [bar.x] - [x.bar], [omega]([y.sup.I]) = [bar.y] - [y.bar] (13)

It can be found according to Eq. (12) and Fig. 1 that P ([x.sup.I] [less than or equal to] [y.sup.I]) is equal to 1 for case 1 as interval [x.sup.I] is always smaller than interval [y.sup.I]. For case 5, P([x.sup.I] [less than or equal to] [y.sup.I]) is equal to 0 as interval [x.sup.I] is always larger than interval [y.sup.l] . For case 2 to 4, the value of P ([x.sup.I] [less than or equal to] [y.sup.I]) is between [0,1] as interval [x.sup.I] interferes with interval [y.sup.I] .

[FIGURE 1 OMITTED]

To sum up, the satisfaction degree of interval P ([x.sup.I] [less than or equal to] [y.sup.I]) has the following properties

(1) 0 [less than or equal to] P ([x.sup.I] [less than or equal to] [y.sup.I]) [less than or equal to] 1

(2) P([x.sup.I] [less than or equal to] [y.sup.I]) + P([x.sup.I] [greater than or equal to] [y.sup.I]) = 1

(3) if P ([x.sup.I] [less than or equal to] [y.sup.I] ) = P ([x.sup.I] [greater than or equal to] [y.sup.I]), then

P([x.sup.I] [less than or equal to] [y.sup.I]) = P([x.sup.I] [greater than or equal to] [y.sup.I]) = 0.5, and [x.sup.I] = [y.sup.I]

(4) if [x.sup.I] [less than or equal to] [y.sup.I], then P([x.sup.I] [less than or equal to] [y.sup.I]) = 1

(5) if [x.sup.I] [greater than or equal to] [y.sup.I], then P ([x.sup.I] [less than or equal to] [y.sup.I]) = 0.

4. Nonprobabilistic set model of structural reliability

As described in the introduction, structural reliability is subjected to many uncertain parameters. Therefore, the stress S and strength R of the structure can be denoted as the functions of these uncertain parameters

S = S ([X.sub.S]) = S ([x.sub.S1], [x.sub.S2],..., [x.sub.S1]) (14)

R = R ([X.sub.R]) = R ([x.sub.R1], [x.sub.R2],..., [x.sub.Rm]) (15)

where [X.sub.S] = {[x.sub.Si]}(i = 1,2, ..., l) is the parameter set impacting on the stress S , such as concentration forces, distribution forces, bending moments and so on. [X.sub.R] ={[x.sub.Ri]}(i = 1,2, ..., m) is the parameter set impacting on the strength R, such as material properties, geometric dimensions, surface cracks and so on. According to the basic idea of nonprobabilistic reliability presented by BenHaim, all the uncertain parameters are described with interval variables in this paper, which are

[x.sub.Si] [member of] [x.sup.L.sub.Si] = [[[x.bar.].sub.Si], [[bar.x].sub.Si]], (i =1,2,...,l) (16)

[x.sub.Ri] [member of] [x.sup.L.sub.Ri] = [[[x.bar.].sub.Ri], [[bar.x].sub.Ri]], (i =1,2,...,m) (17)

Based on Eqs. (2) and (3), the interval variables [x.sub.Si] and [x.sub.Ri] can be transformed into their standardized forms. That is

[x.sub.Si] = [x.sup.x.sub.Si] + [[[x.sup.r.sub.Si], [delta], (i =1,2,...,l) (18)

[x.sub.Ri] = [x.sup.x.sub.Ri] + [[[x.sup.r.sub.Ri], [delta], (i =1,2,...,m) (19)

where [x.sup.x.sub.Si] and [x.sup.x.sub.Si] are the center and radius of the interval variables [x.sup.I.sub.Si]; [x.sup.c.sub.Ri] and [x.sup.r.sub.Ri] are the center and radius of the interval variables [x.sub.Ri]; [[delta].sub.i] [[DELTA].sup.I] = [-1,1] are the standardized interval variables.

Because the stress S and strength R are functions of these interval variables respectively, they will vary within some closed intervals [S.sup.I] and [R.sup.I] . In order to obtain the upper bounds and the lower bounds of the intervals [S.sup.I] and [R.sup.I] , Eqs. (14) and (15) can be respectively expanded at the center [x.sup.c.sub.Si] and [x.sup.c.sub.Ri] of the uncertain interval variables [x.sub.Si] and [x.sub.Ri] by using the first-order Taylor series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [partial derivative]S/[partial derivative][x.sub.Sl], (i = 1,2,...,l) is the first-order partial derivative of the stress S at the center [x.sup.x.sub.Si]; [partial derivative]R/[partial derivative][x.sub.Rm], (i = 1,2,...,m) is the first-order partial derivative of the strength R at the center [x.sup.c.sub.Ri]. Substituting Eqs. (18) and (19) into Eqs. (20) and (21) respectively, Eqs. (20) and Eq. (21) can be rewritten as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

According to Eqs. (8), (9) and (22), the center [S.sup.c] and radius [S.sup.r] of the interval [S.sup.I] can be determined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Therefore, stress-interval [S.sup.I] the of structure is

[S.sup.I] [approximately equal to] [[S.sup.c] - [S.sup.r], [S.sup.c] + [S.sup.r]] (25)

According to Eqs. (8), (9) and (23), the center [R.sup.c] and radius [R.sup.r] of the interval [R.sup.I] can be determined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Therefore, strength-interval [R.sup.I] of the structure is

[R.sup.I] [approximately equal to] [[R.sup.c] - [R.sup.r], [R.sup.c] + [R.sup.r]] (27)

According to the stress-strength interference model, the reliability criterion of structure design is that the stress of the structure is less than or equal to the strength of the structure. Therefore, based on the principle of satisfaction degree of interval, a nonprobabilistic reliability of the structure can be defined as the satisfaction degree between the stress-interval [S.sup.I] and the strength-interval [R.sup.I] . For the definition of the satisfaction degree of the relation [x.sup.I] [less than or equal to] [y.sup.I] in Eq. (11), there are also five cases between [S.sup.I] [less than or equal to] [R.sup.I] as same as the [x.sup.I] [less than or equal to] [y.sup.I] which shown in Fig. 2. The satisfaction degree of the relation [S.sup.I] [less than or equal to] [R.sup.I] or reliability becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

By the definition of the satisfaction degree of the relation [S.sup.I] < [R.sup.I], the value of P ([x]) varies from 0 to 1. When P ([x]) is equal to 1, it means that the stress-interval [S.sup.I] is absolutely smaller than the strength-interval [R.sup.I] and the structure is in the state of safety which is denoted by case 1 in Fig. 2. When P ([??]) is equal to 0, it means that the stress-interval [S.sup.I] is absolutely larger than the strength-interval [R.sup.I] and the structure is in the state of failure which is denoted by case 3 in Fig. 2. When P ([??]) is equal to some value between 0 and 1, it means that the stress-interval S' interfered with the strength-interval [R.sup.I] and the structure may be safety or may be failure.

5. Illustrative example

Gears are widely used in many practical engineering systems. The gear transmission system plays an important role in modern industry. However, in the process of gear meshing, contact stress will be produced which causes pitting. Systems including gears meshing shocks with the increase of the pitting, which will lead to the decrease of the transmission efficiency and accuracy. Therefore, contact fatigue analysis is necessary and important for increasing the reliability of gear transmission. In this section, the nonprobabilistic reliability of the contact fatigue of a pair of spur gear meshing of a reducer is calculated. Main parameters of the gear pairs used in the example are described as: modulus m = 4mm; tooth number of two gear are [z.sub.1] = 14, [z.sub.2] = 47; torques are [T.sub.1] = 353 Nm, [T.sub.2] = 1180 Nm; rotation speed are [n.sub.1] = 76.5 r/min, [n.sub.2 ]= 22.8 r/min; pitch diameters are [d.sub.1] = 56.57 mm, [d.sub.2] = 189.89 mm respectively; width of the tooth b = 46 mm; material of the pinion: 20MnTiB, HRC = 56~62; material of the gear: 40Cr, HRC = 50~56; life of the reducer: 1000 h.

[FIGURE 2 OMITTED]

According to reference [26], the calculated contact stress [[sigma].sub.H] is denoted by the formula

[[sigma].sub.H] = [Z.sub.E] [absolute value of [F.sub.t][K.sub.O][K.sub.V][K.sub.S] [K.sub.H]/[Z.sub.R]/[bd.sub.1] [Z.sub.I] (29)

where [Z.sub.E] is an elastic coefficient; [F.sub.t] is the transmitted tangential load; [K.sub.o] is the overload factor; [K.sub.V] is the dynamic factor; [K.sub.S] is the size factor; [K.sub.H] is the load-distribution factor; b is the width of the tooth; [d.sub.1] is the pitch diameter of the pinion; [Z.sub.R] is the surface condition factor; [Z.sub.I] is the geometry factor.

According to the nonprobabilistic reliability model presented in this paper, all the parameters in Eq. (29) are described with interval variables.

By means of Eq. (23), the center and radius of the calculated contact stress [[sigma].sub.H] are

[[sigma].sup.c.sub.H] = 1350.04 MPa, [[sigma].sup.r.sub.H] = 118.05 MPa (30)

According to reference [26], the contact fatigue strength [[sigma].sub.HS] is denoted by the formula

[[sigma].sub.HS] = [[sigma].sub.HP][Z.sub.N][Z.sub.W]/[S.sub.H][Y.sub.[theta]] (31)

where [[sigma].sub.HP] is the surface fatigue strength; [S.sub.H] is the AGMA factor of safety; [Z.sub.N] is the stress cycle life factor; [Z.sub.W] is the hardness ratio factor; [Y.sub.[theta]] is the temperature factor. Similarly, all the parameters in Eq. (31) are described with interval variables.

The center and radius of interval variables in Eqs. (29) and (31) are expressed in Table [27].

By means of Eq. (26), the center and radius of the contact fatigue strength [[sigma].sub.HS] are

[[sigma].sup.c.sub.HS] = 1661.33 MPa, [[sigma].sup.r.sub.HS] = 207.67 MPa (32)

Thus, from the relation [[sigma].sup.I.sub.H] [less than or equal to] [[sigma].sup.I.sub.HS] shown in Fig. 3, the satisfaction degree of the relation [[sigma].sup.I.sub.H [less than or equal to] [[sigma].sup.I.sub.HS] or the reliability of the contact fatigue is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[FIGURE 3 OMITTED]

From Eq. (33), the satisfaction degree of the relation [[sigma].sup.I.sub.H] [less than or equal to] [[sigma].sup.I.sub.HS] or the reliability is very close to 1. It indicates that the gear transmission of the reducer is very reliable. If all the parameters in the example are of uniform distribution, for example, [[sigma].sub.HP] follows the uniform distribution [1330.3, 1660.1], from the Monte Carlo simulation, the reliability R [approximately equal to] 1, Obviously, the nonprobabilistic reliability is a little smaller than the probabilistic reliability and it means that if the calculated result by nonprobabilistic approach is thought to be reliable, the calculated result by probabilistic approach is absolutely reliable. From the result there is a conclusion that the method proposed in the paper is not as same as the probabilistic reliability method which assumes that all the variables are of uniform distribution. The nonprobabilistic method is more conservative than probabilistic method because there is no human assumption for system parameters distribution.

6. Conclusions

1. For the structural reliability analysis, the stress and strength are the function of several interval variables. The approximations S [approximately equal to] S([x.sup.c.sub.S1], [x.sup.c.sub.S2], ..., [x.sup.c.sub.Sl]) + [l.summation over (i=1)] [partial derivative]S/[partial derivative][x.sub.Si] [x.sup.r.sub.Si][delta] and R [approximately equal to] R([x.sup.c.sub.R1], [x.sup.c.sub.R2], ..., [x.sup.c.sub.Rm]) + [m.summation over (i=1)] [partial derivative]R/[partial derivative][x.sub.Ri] [x.sup.r.sub.Ri][delta] for the stress and strength are implemented with the first order Taylor series to guarantee the computational efficiency and accuracy of the reliability analysis.

2. Comparison of results between the proposed nonprobabilistic method and the probabilistic method has shown that the reliability by using the proposed nonprobabilistic method (R = 0.9989) is a little smaller than using the probabilistic method (R [approximately equal to] 1). Hence it is reliable with the proposed nonprobabilistic method.

Acknowledgements

This research was partially supported by the National Natural Science Foundation of China under the contract number 50775026, and the Specialized Research Fund for the Doctoral Program of Higher Education of China under the contract number 20090185110019.

Received September 13, 2010

Accepted January 17, 2011

References

[1.] Elishakoff, I. 1995. Essay on uncertainties in elastic and viscoelastic structures: from A M Freudenthal's criticisms to modern convex modeling, Computers & Structures, vol.56, No.6: 871-895.

[2.] Wang, Z.; Huang, H.Z.; Du, X. 2010. Optimal design accounting for reliability, maintenance, and warranty, Journal of Mechanical Design, Transactions of the ASME, vol.132, No.1: 011007-1-011007-8.

[3.] Povilionis, A.; Bargelis, A. 2010. Structural optimization in product design process, Mechanika 1(81): 66-70.

[4.] Medekshas, H. 2008. Effect of elevated temperature and welding on low cycle fatigue strength of titanium alloys, Mechanika 2(70): 5-10.

[5.] Frudenthal, A.M. 1947. Safety of structures, Transactions ASCE, vol.125: 112-117.

[6.] Chirstensen, P.T.; Baker, M.J. 1982. Structural Reliability Theory and Its Application. Berlin: Springer-Verlag.

[7.] Franqopol, D.M. 2002. Progress in probabilistic mechanics and structural reliability, Computers & Structures, vol.80, No.12: 1025-1026.

[8.] Ben-Haim, Y. 1993. Convex models of uncertainty in radial pulse buckling of shells, Journal of Applied Mechanics, vol.60, No.3: 683-688.

[9.] Ben-Haim, Y. 1995. A non-probabilistic measure of reliability of linear systems based on expansion of convex models, Structural Safety, vol.17, No.2: 91-109.

[10.] Ben-Haim, Y. 1996. Robust Reliability in the Mechanical Sciences. Berlin: Springer-Verlag.

[11.] Ben-Haim, Y. 1994. A non-probabilistic concept of reliability, Structural Safety, vol.14, No.4: 227-245.

[12.] Cai, K.Y. 1996. Introduction to fuzzy Reliability. Kluwer Academic Publishers.

[13.] Liu, Y.; Huang, H.Z. 2010. Reliability assessment for fuzzy multi-state systems, International Journal of Systems Science, vol.41, No.4: 365-379.

[14.] Ding, Y.; Lisnianski, A. 2008. Fuzzy universal generating functions for multi-state system reliability assessment, Fuzzy Sets & Systems, vol.159, No.3: 307-324.

[15.] Elishakoff, I. 1995. Discussion on: A non-probabilistic concept of reliability, Structural Safety, vol.17, No.3: 195-199.

[16.] Moore, R.E. 1979. Methods and Applications of Interval Analysis. London: Prentice-Hall, Inc.

[17.] Guo, S.X. Lv, Z.Z.; Feng, Y.S. 2001. A nonprobabilistic model of structural reliability based on interval analysis, Chinese Journal of Computational Mechanics, vol.18, No.1: 56-60 (in Chinese).

[18.] Wang, X.J.; Qiu Z.P.; Wu, Z. 2007. Non-probabilistic set-based model for structural reliability, Chinese Journal of Mechanics, vol.39, No.5: 641-646 (in Chinese).

[19.] Ben-Haim, Y. 1993. Convex models of uncertainty in radial pulse buckling of shells, Journal of Applied Mechanics, ASME, vol.60: 683-688.

[20.] Elishakoff, I.; Ben-Haim,Y. 1990. Dynamics of a thin cylindrical shell under impact with limited deterministic information on its initial imperfections, Structural Safety, vol.8, No.1: 103-112.

[21.] Givoli, D.; Elishakoff, I. 1992. Stress concentration at a nearly circular hole with uncertain irregularities, Journal of Applied Mechanics, ASME, vol.59: 65-71.

[22.] Adali, S.; Richter A.; Verijenko, V.E. 1995. Nonprobabilistic modeling and design of sandwich plates subject to uncertain loads and initial deflections, Int. J. Engng Sci., vol.33, No.6: 855-866.

[23.] Ferson, S.; Kreinovich, V.; Hajagoset, J. 2007. Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty. Sandia National Laboratories, Report SAND2007-0939.

[24.] Liu, X.W.; Da, Q.L. 1999. A satisfactory solution for interval number linear programming, Chinese Journal of Systems Engineering, vol.14, No.2: 123-128 (in Chinese).

[25.] Jiang, C.; Han, X.; Liu, G.R. 2007. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval, Computer Methods in Applied Mechanics and Engineering, vol.169, No.49: 4791-4800.

[26.] Shigley, E.J.; Mischke, C.R. 2001. Mechanical Engineering Design, 6th Edition. New York: McGraw-Hill.

[27.] Standards of the American Gear Manufacturers Association. 1990. Arlington, VA: AGMA.

H.-Z. Huang, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R.China, E-mail: hzhuang@uestc.edu.cn

Z. L. Wang, University of Electronic Science and Technology ofChina, Chengdu, Sichuan, 611731, P.R.China, E-mail: wzhonglai@uestc.edu.cn

Y. F. Li, University of Electronic Science and Technology ofChina, Chengdu, Sichuan, 611731, P.R.China, E-mail: lyfkjxy@163.com

B. Huang, University of Electronic Science and Technology ofChina, Chengdu, Sichuan, 611731, P.R.China, E-mail: bohuang@uestc.edu.cn

N. C. Xiao, University of Electronic Science and Technology ofChina, Chengdu, Sichuan, 611731, P.R.China, E-mail: ncxiao@uestc.edu.cn

L. P. He, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R.China, E-mail: hlping0621@163.com

Table
Center and Radius of uncertain parameters

Uncertain parameters                        Center     Radius

[Z.sub.E] ([square root of MPa])            189.8       17.1
[F.sub.t] (N)                               12480       1248
[K.sub.O]                                     1         0.01
[K.sub.V]                                    1.04       0.04
[K.sub.S]                                    1.00       0.01
[K.sub.H]                                   1.496       0.40
b (mm)                                        46        0.01
[d.sub.1] (mm)                              56.57       0.01
[Z.sub.R]                                    1.02       0.02
[Z.sub.I]                                    1.07       0.01
[[sigma].sub.HP] ([square root of MPa])     1495.2     164.9
[S.sub.H]                                    1.35       0.03
[Z.sub.N]                                    1.5        0.04
[Z.sub.W]                                    1.00       0.02
[Y.sub.[theta]]                              1.00       0.01