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Physics Reports vol.50

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1. The fuzzy sets corresponding to z1 , z2 , . . , zd are generated by taking the z’s obtained by numerical analysis as midpoints of membership functions corresponding to a given structural damage. This strategy for selecting the midpoint ensures that the maximum degree of membership (μ = 1) for each fuzzy set occurs at the values of z since the Gaussian function is highest at the midpoint. The standard deviation of each set is initially fixed randomly within a prescribed range. 2. For each measurement delta corresponding to given fault, the degree of membership in the fuzzy set is calculated.

In those cases, a defuzzifier is used to calculate crisp values from fuzzy values. Thus, the defuzzifier converts the words back to numbers. The internal architecture of the fuzzy system thus operates with words, but the interface to the outside world can be through numbers because of the fuzzifier and the defuzzifier. We will now formally define the various fuzzy logic terms. Fuzzy Sets A fuzzy set F is defined on a universe of discourse U and is characterized by a degree of membership μ(x), which can take values between 0 and 1.

Aleatory or random uncertainty can be present because the material properties used in the deterministic mathematical model such as Young’s modulus are actually random variables. In other words, if we were to build ten beams and find the Young’s modulus for each beam, there would be some scatter in the data obtained. Besides random uncertainty, epistemic or model uncertainty may also be present. For instance, if Euler–Bernoulli beam theory is used to model a short thick beam, the results will not be accurate, as the effects of shear deformation and rotary inertia are neglected in the Euler– Bernoulli beam theory.