A formula of evaluating structural safety based on fuzzy set theory

Vietnam Journal of Mechanics, VAST, Vol. 33, No. 3 (2011), pp. 162 – 169 A FORMULA OF EVALUATING STRUCTURAL SAFETY BASED ON FUZZY SET THEORY Le Xuan Huynh, Le Cong Duy National University of Civil Engineering, Vietnam Abstract. This article presents an approach to assess safety levels of structures. A new formula for determining the fuzzy reliability of structures is proposed for the case where the set of loading effect and set of structural durability are general fuzzy sets. Illustration example concerning the bending strength evaluation of a simple-beam structure, is pre- sented with the choice of triangular fuzzy sets for loading effect and structural durability. Key words: Fuzzy logic and application, fuzzy reliability. 1. INTRODUCTION Data and models encountered in natural sciences and engineering are more or less characterized by uncertainty. The uncertainty models can be investigated by the fuzzy theory. Fuzzy theory has been known since 1965. Professor Lofti A. Zadeh had the first article presenting the fuzzy sets and fuzzy logic. It is firstly applied in electronic engineer- ing, next in fields of computer sciences and control techniques. And since 1970, fuzzy set theory has been applied and developed to the fields of civil engineering and computational mechanics. In order to assess the safety of construction structures based on fuzzy models, now, there are two trends for approaching. The first trend, based on fuzzy probability theory is to establish methods to define the fuzzy reliability of structures [1]. Using the second trend, researchers propose different formulas to calculate the fuzzy reliability based on fuzzy set theory and random interference model [2-7]. This article presents a new for- mula to define the fuzzy reliability following the second trend. Numerical examples are compared with some formulas for estimating the safety of structures. In [2], we present a formula to evaluate safety levels and/or failure levels of struc- tures, in cases where loading effect and structural durability are two triangular fuzzy sets. Based on the mathematics of fuzzy logic, the failure ratio (FR) of structures is defined by the formulation: FR = (ωR + ωS)/(ΩS+ ΩR), and the safety ratio (SR) of structures can be inferred: SR = 1 - FR, where ΩR, ΩS is in turn full area of R˜, S˜; and (ωR + ωS) is the area of intersection part of R˜ and S˜. A problem of the formula in [2] is finding the (ωR + ωS), however in some cases it is very difficult. N.D. Xan [4] uses fuzzy-random interferential model. Set of response of structure S˜ is described as a triangular fuzzy set with membership function µ(x), and ability of structure R is described as a random variable with standard distributed density function f(x). Fuzzy A formula of evaluating structural safety based on fuzzy set theory 163 incredibility ∼ Pf is defined by formula: ∼ Pf = ∫ µ(x).f(x)dx, and fuzzy reliability ∼ PS can be inferred: ∼ Ps = 1− ∼ Pf . Changing characteristic of S and R, Kwan Ling Lai [5] uses random-fuzzy interferen- tial model. Set of response of structure S is described as a random variable with standard distributed density function f(x), and Ability of structure R is described as a triangular Fuzzy set R˜ with membership function µ(x). Fuzzy incredibility ∼ Pf is defined by formula- tion: ∼ Pf = ∫ f(x).µ(x)dx, and Fuzzy reliability ∼ PS can be inferred: ∼ Ps = 1− ∼ Pf . Formulas in [4] and [5] are the approximate formulas, one member in the integral is a fuzzy set, other member is a random variable. Functions µ(x) and f(x) are not of the same measurement, so these formulas [4], [5] give approach results. In the follow example, these formulas will not be used. N.V. Pho [6] uses interferential model which is similar to random model. Set of response of structure S˜ and set of ability of structure R˜ are described as fuzzy sets with membership functions are triangular models. The formula consider the difference set M˜ = R˜− S˜ with membership function µ(m), then the author convert area of graphs set of µ(m) into new membership, area of which is equal to unit. The fuzzy incredibility ∼ Pf is calculated by the left-part area of the vertical axis of graphs of new membership function, and fuzzy reliability ∼ PS is the right-part area of the vertical axis of graphs of new membership function this mean ∼ Ps = 1− ∼ Pf . Weimin Dong et all [7] directly uses set S˜ and set R˜ with corresponding triangu- lar membership functions µ(s) and µ(r) . Fuzzy failure possibility (FP) is calculated as formulation: FP = h/2, and fuzzy safety possibility (SP) is calculated as: SP =1 - h/2, where h is the ordinate of intersectional point between two curves µ(s) and µ(r). Formula in [7] shows the way of calculating approximately, it only considers the height h of the intersectional part area but hasn’t calculated its width base c. Based on fuzzy interferential model, in this article, authors propose a new formula- tion of safety assessment for structures, named: "Formula of area ratio". 2. FORMULATION OF AREA RATIO The formulation for calculating fuzzy reliability of structures is established based on the idea of fuzzy interferential model, comparing the set of loading effect S˜i with the set of structural durability R˜i. Consider set S˜i and set R˜i as fuzzy sets, in the real numbers field, with corresponding membership function µS˜i(x) and µR˜i(x) which in the general forms (Fig. 1). For safety evaluate, comparing the set S˜i with the set R˜i. We consider the difference set M˜i = R˜i− S˜i. By the fuzzy interval analysis algorithm or α - level optimization of the extension principle [1], we define the membership functions µM˜i(x) of fuzzy set M˜i, can be as follow ( Fig. 2). In the Fig. 2a, we see that the membership function µM˜i(x) is fully on the left of the vertical axis, this mean set of loading effect S˜i > set of structural durability R˜i, member 164 Le Xuan Huynh, Le Cong Duy Fig. 1. Membership functions of set Si a) and of set Ri b) Fig. 2. Cases of the set M˜i of structure is entirely failure or we say the failure ratio (FR) of the structural member is 100%. On the contrary, in the Fig. 2b we see the membership function µM˜i(x) is fully on the right of the vertical axis, this mean set of loading effect S˜i < set of structural durability R˜i, member of structure is entirely safety or we say the safety ratio (SR) of the structural member is 100% Generally, in the Fig. 2c, we see that the membership function µM˜i(x) of fuzzy set M˜i has a part on the left and another part on the right of the vertical axis. This mean state of the structural member S˜i is not entirely safety, or we say it has a failure part, corresponding to the left-area from the vertical axis, and the safety part correspond to the right-area from the vertical axis. So the reliability (P is) of the i-th structural member can be defined by a formulation proposed as follows: P is = ω1 ΩM = b∫ 0 µM˜i(x)dx b∫ a µM˜i(x)dx (1) and the incredibility (P if ) of structural member : P if = ω2 ΩM = 0∫ a µM˜i(x)dx b∫ a µM˜i(x)dx = 1− P is (2) where ω1 is the right-area, ω2 is the left-area from the vertical axis, and ΩM = (ω1 + ω2) is the full area of graph µM˜i(x). A formula of evaluating structural safety based on fuzzy set theory 165 We see that P is + P i f = 1. After determining the reliabilityP is of i th member of structural system, we can define the reliability of structural system by electric net schema, or follow reliability interval: n∏ i=1 P is ≤ Ps ≤ min(P 1 s , P 2 s , ..., P n s ) = P i s min . 3. EXAMPLE OF APPLICATION Consider a simple example in order to test and to illustrate the proposed formula. In this example, the membership functions of S and R are triangular types, the most commonly used in engineering practice. A reinforced-concrete beam is shown in Fig. 3, where As =3φ18 = 7, 63cm2. Loads are triangular-fuzzy numbers ( Fig. 4). The problem is to evaluate strength safety level of the beam. q A B AsC h = 4 0 c m b=25cm l=4m 2m 2m Fig. 3. Reinforced-concrete beam; loads, cross section Fig. 4. Fuzzy loads membership functions 3.1. Determination of membership functions of fuzzy moment at C Based on structural methods, we apply the principle of load-contribution to define value of fuzzy moment at C as in Fig. 5 : M˜C = l 4 P˜ + l2 8 q˜ (3) 166 Le Xuan Huynh, Le Cong Duy Fig. 5. Membership function of fuzzy moment at C Application of fuzzy interval analysis to determine the membership functions of fuzzy moment at C. 3.2. Construction of membership functions of fuzzy moment of designing section Variable matrix and moment matrix of section. Designing section parameters: b = 25 cm, h = 40 cm, and area of steel rods As = 3φ18 = 7.63 cm2, using method of fuzzy linear regression to define membership function of moment of designing section [M˜ ]. Using method of fuzzy linear regression, we need to construct a variable matrix by changing values of random parameters with possible amplitude is ±5% of mean value. Changing values of random parameters, we have a matrix as follows (see Table 1): Fuzzy linear regression function. Equation of fuzzy linear regression function [9] is expressed: y = [M˜ ] = γ1x1 + γ2x2 + γ3x3 + γ4x4 + γ5x5 + γ6x6 (4) y = [M˜ ] = γ1.b+ γ2.h+ γ3.AS + γ4.b 2 + γ5.h 2 + γ6.A 2 S (5) in which γj(aj, cj) is a fuzzy component, we replace values of fuzzy component into equation (11) to have value of y = [M˜ ], show the membership function of [M˜ ]. Based on Fuzzy Lest-Squares Linear Regression [9] to define the value of aj: a = [xT .x]−1[xT .y] (6) where x is a variable matrix (nxk) = (27x6) and y is a moment matrix (nx1) = (27x1). Matrix [c] is defined from restrain condition: total of errors of cj is minimized. So, we have a linear-optimal problem: Z = n∑ i=1 cT |xi| T → min (7) subjected to:   aTxTi − (1−H)c T |xi| T ≤ yi aTxTi + (1−H)c T |xi| T ≥ yi cj ≥ 0, j = 1, k (8) A formula of evaluating structural safety based on fuzzy set theory 167 Table 1. Values of random parameters Variable matrix Moment matrix (y) b (cm) h (cm) As (cm)2 b2 (cm)2 h2 (cm)2 A2S (cm) 4 M (kG.m) 23.75 38 7.2485 564.0625 1444 52.54075 5415.252 23.75 38 7.63 564.0625 1444 58.2169 5630.546 23.75 38 8.0115 564.0625 1444 64.18413 5838.867 23.75 40 7.2485 564.0625 1600 52.54075 5799.475 23.75 40 7.63 564.0625 1600 58.2169 6034.592 23.75 40 8.0115 564.0625 1600 64.18413 6262.697 23.75 42 7.2485 564.0625 1764 52.54075 6184.383 23.75 42 7.63 564.0625 1764 58.2169 6439.397 23.75 42 8.0115 564.0625 1764 64.18413 6687.364 25 38 7.2485 625 1444 52.54075 5478.174 25 38 7.63 625 1444 58.2169 5700.265 25 38 8.0115 625 1444 64.18413 5915.733 25 40 7.2485 625 1600 52.54075 5862.757 25 40 7.63 625 1600 58.2169 6104.71 25 40 8.0115 625 1600 64.18413 6340.003 25 42 7.2485 625 1764 52.54075 6247.99 25 42 7.63 625 1764 58.2169 6509.877 25 42 8.0115 625 1764 64.18413 6765.067 26.25 38 7.2485 689.0625 1444 52.54075 5535.104 26.25 38 7.63 689.0625 1444 58.2169 5763.345 26.25 38 8.0115 689.0625 1444 64.18413 5985.279 26.25 40 7.2485 689.0625 1600 52.54075 5920.012 26.25 40 7.63 689.0625 1600 58.2169 6168.151 26.25 40 8.0115 689.0625 1600 64.18413 6409.946 26.25 42 7.2485 689.0625 1764 52.54075 6305.54 26.25 42 7.63 689.0625 1764 58.2169 6573.643 26.25 42 8.0115 689.0625 1764 64.18413 6835.37 Where c = [ c1 c2 cj ... ck] T ] , yi is a moment vector (row i) of matrix y, xi is a variable matrix (row i) of matrix x, and H is fuzzy threshold which has value ∈ [0, 1]. We assume H = 0.5 and change (7), (8) into: Z = 6∑ j=1 cj ( 27∑ i=1 xij ) → min (9) subjected to:   −cT |xi| T ≤ (yi − a TxTi )/0, 5 = bi −cT |xi| T ≥ (aTxTi − yi)/0, 5 = −bi cj ≥ 0, j = 1, 6 (10) Using Matlab 7.0.4 to solve problem (9), (10): Zmin = 2, 457. Membership function of fuzzy moment of designing section. 168 Le Xuan Huynh, Le Cong Duy Table 2. Values of regression coefficients γj 1 2 3 4 5 6 aj -0.1232 0.0179 0.0453 0.0035 0.0023 0.038 cj 0.0027 0 0 0 0 0.0004 From values of γj(aj, cj) and mean value of b, h, AS, b 2, h2, A2S , we have the value of fuzzy moment [M˜ ]: R˜ = [M˜ ] = (−0.1232; 0.0027).b+ (0.0179; 0).h+ (0.0453; 0).AS + (0.0035; 0).b 2+ +(0.0023; 0).h2+ (0.038; 0.0004).A2S = (6.061381; 0.090787). where a = 6.061381 is the central value of fuzzy moment [M˜ ], c = 0.090787 is the amplitude of fuzzy moment [M˜ ]. Triangular-fuzzy number of moment [M˜ ] is show in Fig. 6. X(T.m) Fig. 6. Membership function of fuzzy moment of designing section Based on values of a and c we have the membership function of fuzzy moment [M˜ ]: µ (x) ([M˜ ]) =   1− |x− 6.061381| 0.090787 when 5.970594≤ x ≤ 6.152168 0 when x ≤ 5.970594 and x ≥ 6.152168 (11) 3.3. Reliability assessment of beam at section C After determining two membership functions of two fuzzy sets M˜c and [M˜ ], we evaluate the reliability of beam by new formula (Formula of area ratio), and compare with formulas in [2], [6], [7]. The Table 3 shows the results of the reliability of structure by different formulas. Table 3. Results of formulas New Formula Formula [2] Formula [6] Formula [7] PS Pf PS Pf PS Pf PS Pf 0,999621 0,000379 0,999621 0,000379 0,999611 0,000389 0,982513 0,017487 A formula of evaluating structural safety based on fuzzy set theory 169 3.4. Comparison The numerical resuls calculating by new formula and by the formula in [2] are the same. The difference between new formula and the formula[6] is very small. In this example, the result calculated by [7] is less than 17% compared with the result obtained by the new formula. 4. CONCLUSIONS Results obtained by the new formula and the formula [2] are the same. However, the new formula is applied easily in cases we can’t define the intersectional part area of two sets S˜ and R˜, while the formula in [2] is not applicable. It can be seen the "Formula of area ratio" is more general for evaluating safety of structures. Because this formula is established on the base of analysing not only the height ordinate h but also calculates to the width base of intersectional part area, while the formula [7] only considers the height h. That is explained why the result obtained by the formula [7] less 17%. The new formula uses fuzzy difference set M˜ = R˜− S˜ its height is equal to unit, it mean normal fuzzy set, and so reflects exactly the essence of fuzzy numbers, while in the formula [6] height h is not equal to unit. So the formula proposed is believed for assessment of safety of structures. REFERENCES [1] Bernd Moller, Wolfgang Graf, Michael Beer, Safety Assessment of Structure in View of Fuzzy Randomness, Institute of Structural Analysis, Dresden University of Technology, Dresden Ger- many, (2003). [2] Le Xuan Huynh, Fuzzy Set Theory application for Assessment of Safety of Structures, Pro- ceedings of the Seventh National Congress on Mechanics, (2006) (in Vietnamese). [3] Le Xuan Huynh, Application ability of Fuzzy Sets Theory for Quality Assessment of Con- struction Projects, Journal of Science Technology, National University of Civil Engineering Vietnam, 1 (2007) (in Vietnamese). 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