Weibull distributions

This continuous random variable distribution is named after W. Weibull, a Swedish mechanical engineering professor, who developed it in the early 1950s [12]. The distribution can be used to represent many different physical phenomena, and its probability density fimction is defined by 

By substituting Equation (2.47), into Equation (2.22), we get the following cumulative distribution function 

It is to be noted that exponential and Rayleigh distributions are the special cases of this distribution for = 1 and b = 2, respectively. 

Using Equation (2.47) and Equation (2.25), we obtain the following equations for the distribution mean value  

The normal distribution is a widely used continuous random variable distribution, and sometimes it is called the Gaussian distribution after Carl Friedrich Gauss (1777-1855), a German mathematician. The probability density function of the distribution is expressed by 

By inserhng Equation (2.31) into Equation (2.23), we obtain the following cumulahve distribution function  

Finally, it is to be noted that at s = 1, a Weibull distribution becomes an exponential distribution. 

This continuous random variable probability distribution was developed in the early 1950s by Walliodi Weibull, a Swedish professor in mechanical engineering [15]. The probability density function for the distribution is defined by 

By inserting Equation 2.45 into Equation 2.19, we obtain the following equation for the cumulative distribution function  

In 1951, Weibull [5] proposed the use of a mathematical function that, by changing the value of its three parameters, a, and x can cover many shapes and can approximate to a normal distribution under certain conditions. The Weibull distribution is defined by  

When j6 = 1, the function reduces to an exponential form and when p 3.5 (with a= 1 and x = 0) approximately a normal distribution. It is not necessary to calculate these 

When making carbon fiber, the product can deviate due to  

Variability between tows produced at the same time, e.g. strength. 

Variability with time, e.g. humidity, ambient temperature, night shift. 

In all previous subsections the problem of detennining the parameters characteristic of a population of data or of a sample of data has been addressed with the use of a bell shaped distribution or normal distribution. Not always experimental data are effectively distributed in a symmetrical fashion. This, for instance, is the case of fatigue life at stress amplitudes close to fatigue Umit. A more general distribution function was introduced in 1939 by a Swedish engineer and researcher Weibull [7]. His probability density function in its simplest form (two parameters function) is defined as 

From the probability density function it is derived the cumulative probability function P x), that is the probability that an element of the population assumes a value less or equal to a given x. 

In a bi-logarithmic scale Eq. 4.38 is represented by a line whose slope is m and the known term is mln(Xo). In practice, the steps to determine the Weibull parameters m and Xg are the following first, list the experimental results x,- (i = 1, 2, 3. n) in ascending order second, assign a rank to each results. 

The rank is indicating the failure probability in percent associated to each result, i.e., the frequency. Since the entire population is never tested, we can only estimate the rank of the sample population. In this respect, several options are available. One of the most common estimate is the median rank, i.e., that rank that is as likely to err on the high side, with respect to the median value, as on the low side. The following equation gives a very good approximation of the estimated median ranks for any sample size n 

A valid alternative to the median rank is the mean rank, which is given by the equation 

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Weibull analysis Weibull distribution Weibull function Weibullmodulus... 

D. I. Gibbons and L. C. Vance, M Simulation Study of Estimators for the Parameters and Percentiles in the Two-Parameter Weibull Distribution, General Motors Research Publication No. GMR-3041, General Motors, Detroit, Mich., 1979. 

Fig. 18.3. (a) The Weibull distribution function, (b) When the modulus, m, changes, the survival probability changes os shown. 

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

The above process above could also be performed for the 3-parameter Weibull distribution to compare the correlation coefficients and determine the better fitting distributional model. Computer-based techniques have been devised as part of the approach to support businesses attempting to determine the characterizing distributions... 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

As ean be seen from the above equation, for brittle materials like glass and eeramies, we ean seale the strength for a proposed design from a test speeimen analysis. In a more useful form for the 2-parameter Weibull distribution, the probability of failure is a funetion of the applied stress, L. 

Gn L) is often difficult to determine for a given load distribution, but when is large, an approximation is given by the Maximum Extreme Value Type I distribution of the maximum extremes with a scale parameter, 0, and location parameter, v. When the initial loading stress distribution,/(L), is modelled by a Normal, Lognormal, 2-par-ameter Weibull or 3-parameter Weibull distribution, the extremal model parameters can be determined by the equations in Table 4.11. These equations include terms for the number of load applications, n. The extremal model for the loading stress can then be used in the SSI analysis to determine the reliability. 

Therefore, the loading stress CDF ean be represented by a 3-parameter Weibull distribution ... 

The analysis of the frequeney data is shown in Table 4.12. Note the use of the Median Rank equation, eommonly used for both Weibull distributions. Linear reetifieation equations provided in Appendix X for the 2-parameter Weibull model are used to... 

Table 4.12 Analysis of load frequency data and plotting positions for the 2-parameter Weibull distribution...

We ean use a Monte Carlo simulation of the random variables in equation 4.83 to determine the likely mean and standard deviation of the loading stress, assuming that this will be a Normal distribution too. Exeept for the load, F, whieh is modelled by a 2-parameter Weibull distribution, the remaining variables are eharaeterized by the Normal distribution. The 3-parameter Weibull distribution ean be used to model... 

In the problem here, the loading stress is a Normal distribution and the strength is a 3-parameter Weibull distribution. Beeause the Normal distribution s CDF is not in elosed form, the 3-parameter Weibull distribution ean be used as an approximating distribution when [3 = 3.44. The parameters for the 3-parameter Weibull distribution. 

The final reliability formulation for the interference of two 3-parameter Weibull distributions subjected to multiple load applications is given in equation 4.84 ... 

Note the parameters for the 3-parameter Weibull distribution, xo and 6, ean be estimated given the mean, /i, and standard deviation, cr, for a Normal distribution (assuming [3 = 3.44) by ... 

Many distributions can be represented in closed form except for the Normal and Lognormal types. The CDF for these distributions can only be determined numerically. For example, the 3-parameter Weibull distribution s CDF is in closed form, where ... 

Therefore, for the 3-parameter Weibull distribution, the inverse CDF with respect to u is ... 

Suppose again that both the stress and strength distributions of interest are of the Normal type, where the loading stress is given as L A (350,40) MPa and the strength distribution is S A (500, 50) MPa. The Normal distribution eannot be used with the integral transform method, but ean be approximated by the 3-parameter Weibull distribution where the CDF is in elosed form. It was determined above that the loading stress parameters for the 3-parameter Weibull distribution were ... 

Similarly, the strength parameters for the Weibull distribution ean be approximated by the same method to ... 

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