Keywords

26.1 Introduction

In the structural reliability analysis, a great deal of uncertain factors need to be dealt with reasonably. Research [1, 2] shows that the uncertainty of the variables can be divided into two forms to describe. One is the probabilistic form, which uses the probability distribution function or membership functions to describe the uncertainty. The other is the non-probabilistic form, which uses convex set or interval to describe the uncertainty. Commonly, it is recognized that the probabilistic form is more accurate and convenient in describing the variables with abundant experimental data, and the non-probabilistic form is more accurate and objective in describing the variables with a few of experimental data [3]. Corresponding to these two conditions, two reliability analysis methods, namely the probabilistic reliability method and the non-probabilistic reliability method, were presented by the researchers. The probabilistic reliability method can get more accurate results when the experimental data of variables are sufficient, while the non-probabilistic method can get more reliable results when the experimental data of variables are in lack [4, 5]. However, in an engineering condition, some variables may have sufficient data and the other may have a few of data. It is impossible to apply only one method, probabilistic method or non-probabilistic method, to solve this problem. Therefore, the research of the hybrid probabilistic and non-probabilistic model is very important.

The paper [6] established the hybrid probabilistic and non-probabilistic model of structural reliability analysis, in which the reliability is calculated by establishing two-level limit state equations. However, the model cannot solve complicated nonlinear limit state problem, and neglect the influence of the fuzzy variables, which may also exist in some engineering. Aiming to these problems, the original hybrid probabilistic and non-probabilistic model was improved and the hybrid probabilistic, fuzzy, and non-probabilistic model was presented in this paper.

26.2 The Improved Probabilistic and Non-Probabilistic Reliability Model

26.2.1 The Strategy

According to the definition of interval model, interval variable can be assigned any value within the interval bound. When the interval variable is assigned one fixed value within the defined bound, the hybrid probabilistic and non-probabilistic model degenerates into the probabilistic model. Supposing that we assign each value in the bound to the interval variable and calculate the corresponding random reliability, respectively, the bound of the reliability can be easily confirmed. To accomplish the object, the optimization method and enumeration method can be applied to solve the problem. However, these methods are computationally expensive. When the number of variables is large and the interval variables take the majority, or the problem analysis, such as finite element analysis, is complicated, the calculation cost will become higher.

As we know, in the reliability analysis, the random reliability method is more mature and the treatment of random variables is easier than that of interval variables. Therefore, if the interval variables can be replaced with the random variables, the calculation cost will be reduced and the calculation process will become easy. Then, the key to improve computational efficiency is determining one method to transform interval variables into random variables. Here, interval variables are transformed into random variables based on the maximum entropy principle [7].

26.2.2 The Maximum Entropy Principle

According to the information theory, the entropy is used to measure the uncertainty of the phenomenon. When the interval bound of the variable is known, the uniform distribution has the maximum entropy [8]. That is to say the uniform distribution represents that the probability distribution within the interval bound is completely unknown and the degree of uncertainty is the largest at the moment.

In the engineering practice, quite a number of uncertain parameters are geometric parameters and material parameters. To improve the reliability, the designers and producers will try their best to make the geometric dimension and material property as accurate as possible. Based on the fact, the paper [9] points out it is credible that the distribution of uncertainty under this condition is between the normal distribution and uniform distribution. According to the maximum entropy principle, it is recognized conservatively that these parameters, whose tolerance can be controlled through the technology or human factors, obey the uniform distribution in the defined intervals.

26.2.3 The Procedure

The procedure of the improved probabilistic and non-probabilistic reliability model is as follows:

  1. Step 1.

    Divide the variables into interval variables and random variables. And transform those interval variables, whose tolerance range can be controlled in the engineering project, into random variables under uniform distribution in the defined interval.

  2. Step 2.

    Apply the optimization method [10] or enumeration method to calculate the reliability bound. Here only introduce the enumeration method as follows. Firstly, divide each interval variable into n subintervals and get n + 1 endpoint values. Secondly, combine the endpoint values of these interval variables and calculate the corresponding reliability of each combination with the random reliability method. At last, choose the maximum and minimum of the results as the upper and lower limits of the reliability.

It is noted that the calculation precision of the enumeration method will be improved with the increment of the value of n, but the calculation burden will be also increased with it. Therefore, both the calculation precision and the calculation cost should be given consideration in determining the value of \( n \). If the number of interval variables is \( m \), there will be \( (n + 1)^{m} \) combinations of the endpoint values and \( (n + 1)^{m} \) reliability values correspondingly.

26.3 The Hybrid Probabilistic, Fuzzy, and Non-Probabilistic Model

In the above section, the hybrid probabilistic and non-probabilistic model was discussed. However, this model has two limitations. (1) In addition to random variables and interval variables, there are still fuzzy variables existing in some engineering practice. So it is necessary to consider the influence of the fuzzy variables in the hybrid reliability model. (2) The handling method to distinguish failure and safety based on the limit state function is rough, because the transformation from the safety to the failure is usually a gradual process and there is a fuzzy domain in the transition process [11]. In this section, such a new reliability model was established that is suitable for the situation in which random variables, fuzzy variables, and interval variables coexist.

26.3.1 Fuzziness and the Handling Method of Fuzzy Variables

Because the concept does not have the clear extension, fuzziness is such a kind of uncertainty that can not give the clear definition and evaluation standard to certain things. The most common method to study fuzzy reliability theory is to describe the fuzzy random phenomena with the fuzzy random variables and transform the fuzzy reliability problem into the conventional random reliability problem.

At present, there are usually two methods to transform fuzzy variables into random variables [12]. (1) \( \lambda \)-cut set method. Its computation is complicated and there is not a fixed standard to choose the probability distribution in the cut sets, which will influence the failure probability greatly. (2) The transformation of fuzzy variables to equivalent random variables based on entropy equality [13]. Its computational cost is small, and the influence brought by the probability distribution form is avoided. Its detailed content is as follows:

The entropy of variables is a measure of the uncertainty. As two different kinds of uncertainty, randomness and fuzziness can be measured with the entropy, respectively. The measure of the randomness is called probabilistic entropy, and the measure of the fuzziness is called fuzzy entropy.

According to the definition of probabilistic entropy given by Shannon, the probabilistic entropy is expressed as:

$$ H_{x} = - \int\limits_{R} {f(x)} \ln f(x)\text{d}x $$
(26.1)

where \( f(x) \) is the probability density function of the random variable \( x \).

According to the definition of the non-probabilistic entropy given by DeLuca and Termini, the fuzzy entropy is expressed as:

$$ G_{x}^{{\prime }} = - \int\limits_{R} {f^{{\prime }} (x)} \ln f^{{\prime }} (x)\text{d}x $$
(26.2)

and

$$ f^{{\prime }} (x) = \frac{\mu (x)}{{\int_{R} {\mu (x){\text{d}}x} }} $$
(26.3)

where \( \mu (x) \) is the membership function of the fuzzy variable.

From the basic concept of entropy, it is known that the fuzzy uncertainty can be transformed into random uncertainties, and the transformation must be on the basis of the invariance of the measures, namely that the premise of the transformation is entropy equality. Therefore, according to the formulas (26.1) and (26.2), the transformation equation from the fuzzy entropy to the equivalent random entropy is established as:

$$ H_{x} = G{}_{x}^{{\prime }} .$$
(26.4)

When

$$ f(x) = f^{{\prime }} (x) = \frac{\mu (x)}{{\int_{R} {\mu (x){\text{d}}x} }} $$
(26.5)
$$ H_{x} = G{}_{x}^{\prime }. $$
(26.6)

Therefore, \( f(x) \) is called the probability density function of the equivalent random variable, which is the transformation of the fuzzy variable.

26.3.2 The Procedure

It is assumed that uncertain factors influencing the reliability are random variable \( x \) with the probability density function \( f_{x} (x) \), fuzzy variable \( y \) with the membership function \( \mu (y) \), and interval variable \( z \) with the upper limit \( z^{u} \) and the lower limit \( z^{l} \). Also, assume that the variables are independent, the limit state function is \( M = g(x,y,z) \), and the fuzziness of the failure state is expressed by the membership \( \mu_{{\tilde{\Omega }_{f} }} (M) \) of the state variable \( M \) attributed to fuzzy failure domain \( \tilde{\Omega }_{f} \). Then, the reliability can be calculated as follows:

  1. Step 1.

    Transform fuzzy variables into equivalent random variables and derive the probability density functions of the equivalent random variables as follows:

$$ f_{y} (y) = \frac{\mu (y)}{{\int_{R} {\mu (y){\text{d}}y} }}. $$
(26.7)
  1. Step 2.

    Regard the interval variables as certainty and calculate the probability density function \( f(M) \)of the state variable \( M \) according to \( f_{x} (x) \) and \( f_{y} (y) \).

  2. Step 3.

    According to the calculation steps of the improved probabilistic and non-probabilistic model, consider the uncertainty of interval variables and calculate the failure probability based on the formula (26.8). Through comparing different values of the failure probability, which are, respectively, corresponding to different valuations of the interval variable, the bound of the failure probability can be derived.

$$ P_{f} = \int\limits_{ - \infty }^{ + \infty } {f(M)\mu_{{\tilde{\Omega }_{f} }} (M)dM} $$
(26.8)

However, when the forms of \( \mu_{{\tilde{\Omega }_{f} }} (M) \) and \( f(M) \)are complicated, it is difficult and even impossible to apply the analytical method to calculate the failure probability. The numerical simulation method, such as Monte Carlo method, will be a better choice.

In the two-state model, the Monte Carlo method compares the state variable with 0 and records one realization of the limit state function when the state variable is less than 0. However, in the fuzzy-state model, it is not suitable to compare the state variable with 0 and the Monte Carlo method needs some revisions. In order to facilitate the presentation, assume that the membership of the state variable attributed to fuzzy failure domain is minor type trapezium distribution, as shown in Fig. 26.1, and its specific expression is:

$$ \mu_{{\tilde{\Omega }_{f} }} (M) = \left\{ {\begin{array}{*{20}l} \text{1} & {M < a_{1} } \\ {\frac{{a_{2} - M}}{{a_{2} - a_{1} }}} & {a_{1} \le M \le a_{2} } \\ \text{0} & {M > a_{2} } \\ \end{array} } \right. $$
(26.9)
Fig. 26.1
figure 1

Minor type trapezium distribution

Full size image

The Monte Carlo method is revised as follows:

  1. Step 1.

    Like the traditional Monte Carlo method, produce the corresponding pseudo-random numbers according to the probability distribution function of variables.

  2. Step 2.

    Substitute these pseudo-random numbers into the state variable and derive its value \( M_{i} \). Then use the variable \( m \) to record the times of the realization of the limit state function and give it an initial value of 0. If \( M_{i} < a_{1} \), add 1 to \( m \). If \( M_{i} > a_{2} \), remain the value of \( m \). If \( a_{1} \le M_{i} \le a_{2} \), add \( \frac{{a_{2} - M_{i} }}{{a_{2} - a_{1} }} \) to \( m \).

  3. Step 3.

    Repeat steps 1 and 2 until completing the predetermined number of cycles K and finally derive the failure probability \( P_{f} = m/K \).

26.4 Conclusion

The hybrid probabilistic and non-probabilistic reliability model was improved to solve the reliability problem with any complicated limit state function. And based on the improved model, the hybrid probabilistic, fuzzy, and non-probabilistic model was established, which not only considers the effect of the fuzzy variables but also the effect of the fuzzy failure state. Therefore, the model can effectively solve the reliability analysis problem that random variables, fuzzy variables, and interval variables coexist. However, usually it is difficult to apply analytical method to calculate the integral function of the reliability. To solve the problem, Monte Carlo method was modified and applied, which can reduce the analytical complexity greatly.