Uncertain Structural Reliability Analysis Yi Miao School of Civil Engineering, Tongji University, Shanghai 200092, China 474989741@qq.com Abstract: The reliability of structure is already applied in some fields of structure design and evaluation, and the subjective uncertainty of structure is still not assessed mathematically and properly enough within current theories. This paper introduces the uncertainty theory to assess the subjective uncertainty of the structure, develops a specific equation to measure parallel system in structure, and gives two numerical examples including a space frame and a continuous beam. Keywords: uncertainty theory; uncertain measure; structural reliability 1 Introduction Reliability is usually used to analyze the structural model of uncertain factors influence on the results of the analysis. The idea was put forward very early that the design of the structure should make the summation of the initial construction costs and collapsed loss expectancy minimum, but there was no structural failure probability calculation method available at that time. In 1947, Freudenthal [1] laid the foundation of structural reliability. In 1969, a structural reliability index was defined by Cornell [2] as the ratio of the mean and standard deviation of structural function, with reliable indicator as a safety measure structure of uniform standard. He established the structure safety degree of second order moment model. From then on, the structural reliability theory stepped into practical stage. Hasofer and Lind [3] in 1974 put forward a new definition of structural reliability index. It was defined as the shortest distance from the origin to the limit state surface in a standard normal space, and the checking point is set to be the foot of the perpendicular from the origin to the curve. It solved the problem that different forms of equivalent function led to different reliable indicators. The consideration of probabilistic factors [4] mainly focuses on risk analysis of natural disasters and structural reliability analysis. In addition, there are many literatures studying the nonprobabilistic structural reliability. The requirements of data of nonprobabilistic model are relatively low. To deal with the lack of data to define probability model accurately, the nonprobabilistic reliability method is a better choice for reliability calculation [5 7]. Sometimes engineers or designers are just limited by subjective constraints, which means the available information is insufficient to determine the true state of things and the relation- Proceedings of the Twelfth International Conference on Information and Management Sciences, Kunming, China, August 3-9, 2013, pp. 230-234. ship of the data. Some researchers proposed to treat it as fuzzy variables [8]. But the fuzzy variables are not so fit for many nonprobabilistic cases. Till now, there has not been a proper method to estimate its effect. The engineers tend to adjust the data manually or to believe in experienced experts. Uncertainty theory which is based on normality, duality, subadditivity and product axioms was founded in 2007 by Liu [9] and refined in 2010 by Liu [10]. It is a branch of mathematics for modeling human uncertainty. This theory is specifically founded to deal with the subjective uncertainty. Uncertainty theory has become a powerful mathematical tool to deal with various problems under incomplete information, for instance uncertain control [11], uncertain differential equation [12], and uncertain programming [13], etc. As extensions, Liu [14] did research of uncertain reliability lifetime analysis of redundant system, and Zeng, Wen and Kang [15] proposed a metrics for products reliability. This paper is to study some structural reliability problems within the framework of uncertainty theory, and to discuss some probably applications of uncertain structural analysis. For this purpose, this paper is organized as follows. Section 2 recalls some basic concepts and properties about uncertainty theory. In Section 3, probably applying examples of structural analysis within uncertainty theory are discussed. At the end of this paper, a brief summary about this paper is given. 2 Uncertainty Theory In this section, some basic concepts about uncertainty theory such as uncertain measure, uncertain variable, uncertainty distribution, uncertain expected value and uncertain reliability are given. 2.1 Uncertain Measure Definition 2.1. (Liu [9]) Let Γ be a nonempty set. A collection L of subsets of Γ is a σ-algebra. Each element Λ in the σ-algebra L is called an event. If the function M on L which is subjected to (1) M{Γ = 1; (2) M{Λ + M{Λ c = 1 for each event Λ; (3) For every countable sequence of events {Λ i, we have { M Λ i M{Λ i ; then M is an uncertain measure, and (Γ, L, M) is an uncertainty space.
UNCERTAIN STRUCTURAL RELIABILITY ANALYSIS 231 Since we have the definitions of uncertain variable and uncertain measure, we must consider the product measure and uncertain arithmetic. In 2009, Liu [16] proposed the product axiom. Axiom 4. Let Γ k be nonempty sets on which M k are uncertain measures, k = 1, 2,..., n, respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L 1 L 2... L n satisfying { M Λ i = M{Λ i. k=1 In order to describe the uncertain phenomenon, Liu [9] gave the definition of uncertain variable. Definition 2.2. (Liu [9]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. ξ 1 (B) = {γ Γ ξ(γ) B (1) Definition 2.3. (Liu [16]) The uncertain variables ξ 1, ξ 2,..., ξ m are said to be independent if { m m M {ξ i B i = M{ξ i B i (2) t=1 for any Borel sets B 1, B 2,..., B m of real numbers. In order to characterize uncertain variables, in 2007, Liu [9] proposed the concept of uncertainty distribution. Then, in 2009, a sufficient and necessary condition for uncertainty distribution was proposed by Peng & Iwamura [17]. Definition 2.4. (Liu [9]) The uncertainty distribution Φ of an uncertain variable ξ is defined by for any real number x. Φ(x) = M{ξ x (3) Theorem 2.1. (Liu [10]) Let ξ 1, ξ 2,..., ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,..., Φ n, respectively. If the function f(x 1, x 2,..., x n ) is strictly increasing with respect to x 1, x 2,..., x m and strictly decreasing with x m+1, x m+2,..., x n then ξ = f(ξ 1,..., ξ m, ξ m+1,..., ξ n ) is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(ψ 1 1 (α),, Ψ 1 m (α), Ψ 1 m+1 (1 α),, Ψ 1 n (1 α)). Definition 2.5. (Liu [9]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E(ξ) = 0 M{ξ rdr 0 M{ξ rdr (4) provided that at least one of the two integrals is finite. Theorem 2.2. (Liu [9]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected values exists, then E(ξ) = 0 (1 Φ(x))dx 2.2 Uncertain Reliability Analysis 0 Φ(x)dx. (5) In 2010, Liu [18] proposed uncertain reliability analysis as a tool to deal with system reliability via uncertainty theory. Reliability index is defined as the uncertain measure that the system is working. Definition 2.6. (Liu [18]) Assume a system contains uncertain variables ξ 1, ξ 2,..., ξ n, and is working if and only if R(ξ 1, ξ 2,..., ξ n ) 0. Then the reliability index is Reliability = M{R(ξ 1, ξ 2,..., ξ n ) 0. Theorem 2.3. (Liu [18]) Assume ξ 1, ξ 2,..., ξ n are independent uncertain variables with uncertainty distributions Φ 1, Φ 2,..., Φ n, respectively, and R(x 1, x 2,..., x n ) is strictly increasing with respect to x 1, x 2,..., x m and strictly decreasing with x m+1, x m+2,..., x n. If some system is working if and only if R(ξ 1, ξ 2,..., ξ n ) 0, then the reliability index is where α is the root of Reliability = α R( 1 (1 α),, Φ 1 m (1 α), m+1 (α),, Φ 1 n (α)) = 0. (6) 2.3 Uncertain Structural Reliability Analysis The structural reliability index is defined as the uncertain measure that the resistance is larger than the load. According to the meaning of structural reliability index, it is determined by the resistance and the load. For each rod, if it fails, then we say the structure fails. Now, some theorems of basic structural reliability index are given below. Assume a structure contains uncertain variables ξ 1, ξ 2,..., ξ n, and is working if and only if R(ξ 1, ξ 2,..., ξ n ) 0 where R is the functional function of the structure, where ξ 1, ξ 2,..., ξ n are basic variables of the structure, which can be different load effects, the material parameter, the geometrical parameter, etc. Theorem 2.4. The structure is shown in Figure 1. The gravity of the object is an uncertain variable, and its distribution is Ψ. The resistances of each rods are β 1, β 2,..., β n, and the distributions of them are Φ 1, Φ 2,..., Φ n, respectively. The resistance of the structure is ν. Then the reliability index is α = α 1 α 2 α n, (7) where α 1, α 2,..., α n are the roots of the equations 1 (α 1) = Ψ 1 (1 α 1 ), 2 (α 2) = Ψ 1 (1 α 2 ),
232 YI MIAO respectively.. n (α n ) = Ψ 1 (1 α n ), (8) Figure 2: Parallel System Figure 1: Series System Proof. The resistance of the structure β is β 1 β 2 β n, and the load of each rod is ν. So the functional function of this structure can be expressed as R(β 1, β 2,..., β n, ν) = β 1 β 2 β n ν. If and only if R 0, it works. Then the reliability index α is the root of 1 (1 α) Φ 1 2 Let α i be the roots of (1 α) Φ 1 n (1 α) = Ψ 1 (α). i (1 α i ) = Ψ 1 (α i ), for i = 1, 2,..., n, respectively. Then the reliability index of the structure must be the reliability index of one of the rods. it means that there exists i, 1 i n, subject to α = α i, and we have i (1 α) = Ψ 1 (α). According to the property of the distribution functions, and Ψ 1 are all increasing functions. The reliability index α is the minimum of the reliability index of each rod, that is α = α 1 α 2 α n. Apparently, it is not enough to do analysis just using the series theorem above. We have to meet another type of structure, the parallel structure. The parallel structure is different from other parallel system. It may be applied more in analysis of existed structure and other fields. Theorem 2.5. The structure is shown in Figure 2. And the whole rods can work under plastic stage. The gravity of the object is an uncertain variable ν, and its distribution is Ψ. The resistances of each rods are β 1, β 2,..., β n, and the distributions of them are Ψ 1, Ψ 2,..., Ψ n, respectively. The resistance of the structure is β. The reliability index α of the system is the root of the equation i (α) Ψ 1 (1 α) = 0. (9) Proof. The material of the rods being under plastic stage means the strain and the stress are no longer linear. It leads to the stress distribution of each rod being not available by stress analysis, neither is the reliability of each rod. While one rod has been reaching the limit load, the strain of the rod will keep increase without stress increasing. Hence the limit state of such structure means all the rods reach the limit state together. Since the resistances of rods are β 1, β 2,..., β n, the total resistance force of the system is β 1 + β 2 + + β n. It can be inferred that the functional function R(β 1, β 2,..., β n, ν) of this system is R = β i ν < 0. The system works if and only if R 0. According to (6), the reliability index α of the system can be expressed as the root of the equation 3 Numerical Examples i (α) Ψ 1 (1 α) = 0. The structure design is based on the limit state of the structure. The so-called structure limit state is defined as: if the entire structure or part of the structure is over a particular state, the structure does not meet the requirement of the design rules of a particular function, then this particular state is called the limit state [1]. In structural design, consideration should be given to all the corresponding limit state, to ensure that the structure has enough safety, durability and suitability. For a certain structural system, certain structural mechanics tool is enough to analyze the stress state of the structure. But in fact, even in the case of certain structure style, the stress of the structure or the resistance is not as imaged to be a certain amount. Uncertainty needs to be taken into account and evaluated. Example 3.1. The structure is shown in the Figure 3. All the joints are articulated. The side length of the square grid is 5m and height is 2.5m. The stiffness of each rod EA = 10 5 kn. The external force of the system is uncertain force ν just vertical downward, and its distribution is Φ. The resistances of rods are β 1, β 2,..., β 9, and the distributions of them are Ψ 1, Ψ 2,..., Ψ 9, respectively. The resistance of the structure is β. In order to discuss conveniently, the distributions Ψ i are
UNCERTAIN STRUCTURAL RELIABILITY ANALYSIS 233 Then according to the calculate rules of linear distribution, α 1 = α 2 = α 3 = α 4 b 1 0.354a 0 = 0 (b 1 a 1 ) + 0.354(b 0 a 0 ) 1 = 0.900, α 5 = α 6 = α 7 = α 8 b 6 0.433a 0 = 0 (b 6 a 6 ) + 0.433(b 0 a 0 ) 1 = 0.833, Figure 3: A Typical Element in Space Structure α 9 = 0 = 0.875. b 9 a 0 (b 9 a 9 ) + (b 0 a 0 ) 1 assumed to be linear uncertainty distributions L(a i, b i ), i = 1, 2,..., 9; and Φ is a linear uncertainty distribution L(a 0, b 0 ). Table 1: Data of Example 3.1 i a i b i 0 2 7 1,2,3,4 2 5 5,6,7,8 2 6 9 6 9 Such style of structure is widely used in grid structure and reticulated shell structure as a single element. Set up internal force of the lever i suffered for T i. It is inferred within structural mechanics that T 1 = T 2 = T 3 = T 4 = 0.3537ν, T 5 = T 6 = T 7 = T 8 = 0.433ν, T 9 = ν. It can be expressed as T i = t i ν, i = 1, 2,..., 9. The failure mode of each rod is β i T i 0, and the reliability of each rod is the root of the equation Ψ 1 i (1 α i ) = t i (α i ). According to Theorem 2.4, we have α = α 1 α 2 α 9. So, the reliability index of the structure in Figure 3.1 is α = α 1 α 2 α 9 = 0.833. Example 3.2. The structure is shown in Figure 4. Joints 1,3,5 are articulated, and Joint 7 is fixed. The length of each rod is showed in the figure and l = 2m. The external force q is uncertain force ν just vertical downward, and its distribution is Φ. The bending moments of joints are M 1, M 2,..., M 7. Since joint 1 is free to rotate, M 1 = 0. The other distributions of the limit resistances are Ψ 2, Ψ 3,..., Ψ 7, respectively. In order to discuss conveniently, the distributions Ψ i (i = 2, 3,..., 7) are assumed to be linear uncertainty distributions L(a i, b i ), i = 2,..., 7; and Φ is linear uncertainty distribution L(a 0, b 0 ). Table 2: Data of Example 3.2 i a i b i 0 0 1 2,3 0.5 1 4,5 0 1 6,7 1 2 Based on the structural mechanics, it is inferred that the continuous beam under loads of same direction can only be damaged independently in each span, instead of being damaged of compositely. Hence this continuous beam has only 3 different limit states, just in each span. This example shows how the combination of series and parallel system works. In each span, the limit state works as a parallel system, and as a series system as a whole. In the first span, according to the virtual work principle,we have ql = M 3 0.5l + 2M 2 0.5l. Then R 1 = 2M 3 + 4M 2 ql 2 0. (10) In a similar way, in the second span and the third span, R 2 = 4M 3 + 8M 4 + 2M 5 ql 2 0, (11)
234 YI MIAO Figure 4: A Continuous Beam R 3 = 8 9 M 5 + 16 9 M 6 + 8 9 M 7 ql 2 0. (12) Then according to Theorems 2.4 and 2.5, the reliability of the span is α = α 1 α 2 α 3 where α 1, α 2, α 3 are the roots of the equation 4Ψ 1 2 (1 α 1) + 2Ψ 1 3 (1 α 1) (α 1 ) = 0, (13) 4Ψ 1 3 (1 α 2)+8Ψ 1 4 (1 α 2)+2Ψ 1 5 (1 α 2) (α 2 ) = 0, (14) 8 9 Ψ 1 5 (1 α 3)+ 16 9 Ψ 1 6 (1 α 3)+ 8 9 Ψ 1 7 (1 α 3) (α 3 ) = 0 (15) respectively. According to the operation law of uncertain variables, we have α 1 = α 2 = 6b 2 4a 0 6(b 2 a 2 ) + 4(b 0 a 0 ) = 0.8571, 4b 3 + 8b 4 + 4b 5 4a 0 4(b 3 a 3 ) + 8(b 4 a 4 ) + 4(b 5 a 5 ) + 4(b 0 a 0 ) = 0.8889. Similarly, α 3 = 0.8235. So, the reliability of the structure in Figure 4 is α = α 1 α 2 α 3 = 0.8235. Then it is easily concluded that the third span is the most dangerous span. 4 Conclusion The reliability of the structure could be used to do structural design based on limit state, do structural optimization and analyze the life-time of the structure. Since there was a lot of subjective uncertainties in engineering practice, the uncertainty theory would be of great practical significance in structural analysis. In this paper, the resistance and the load of a structure were treated as uncertain variables and the reliability index was treated as the uncertain measure of the event that the residence is larger than the load. Two typical structure styles were discussed and two basic theorems were concluded and attempted. Acknowledgments The work was supported by the National Natural Science Foundation of China Grant No.61273004 and No.91224008. References [1] Freudenthal AM. The Safety of the Structures. Transactions of the American Society of Civil Engineers, 1947, 112:125-159. [2] Cornell CA. A Probability-based Structural Code. Journal of American Concrete Institute, 1969, 66:975-985. [3] Hasofer MA & Lind NC. Exact and Invariant Second-moment Format. Journal of Engineering Mechanics Division, 1974, 100(1):111-121. [4] Alfredo HA & Wilson H. Probability Concepts in Engineering Planning and Design. New York: Wiley, 1975. [5] Ben-Haim Y. A Non-Probabilistic Concept of Reliability. Structural Safety, 1994, 14(4): 227-245. [6] Ben-Haim Y. A Non-Probabilistic Measure of Reliability of Linear Systems Based on Expansion of Convex Models. Structural Safety, 1995, 17(2):91-109. [7] Ben-Haim Y. Robust Reliability in the Mechanical Sciences. Berlin: Springer-Verlag, 1996. [8] Marano GC & Quaranta G. A New Possibilistic Realiability Index Definition. Acta Mechanica, 2010, 210:291-303. [9] Liu B. Uncertainty Theory, 2nd edition. Berlin: Springer-Verlag, 2007. [10] Liu B. Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Berlin:Springer-Verlag, 2010. [11] Zhu Y, Uncertain Optimal Control with Aplication to a Portfolio Selection Model, Cybernetics and Systems, 2010, 41:535-547. [12] Chen X & Liu B. Existence and Uniqueness Theorem for Uncertain Differential Equations. Fuzzy Optimization and Decision Making, 2010, 9:69-81. [13] Liu B. Theory and Practice of Uncertain Programming (2nd ed.). Berlin: Springer, 2009. [14] Liu W, Reliability Analysis of Redundant System with Uncertain Lifetimes, Information: An International Interdisciplinary Journal, 2013, 16(2):881-888. [15] Zeng, Wen, and Kang, Belief Reliability: A New Metrics for Products Reliability, Fuzzy Optimization and Decision Making, 2013, 12(1):15-27. [16] Liu B. Some Research Problems in Uncertainty Theory. Journal of Uncertain Systems, 2009, 3(1):3-10. [17] Peng and Iwamura, A Sufficient and Necessary Condition of Uncertainty Distribution, Journal of Interdisciplinary Mathematics, 2010, 13(3):277-285. [18] Liu B. Uncertain Risk Analysis and Uncertain Reliability Analysis. Journal of Uncertain Systems, 2010, 4(3):163-170.