Design Using Weibull Distribution

Weibull distribution can also be used to find the failure probability of a component, apart from its use to determine the failure probability against applied stress. This is done by integrating the Weibull distribution for the material with the stress distribution for the component. Finite element analysis is useful to do this. Comparison is made between the material strength data in the form of a Weibull probability curve and a finite element in the component. This is repeated for all the elements in the component. Thereafter, the probabilities of all the elements are added. This sum gives the probability of failure of the component. 

Probabilistic design allows designing closer to the properties of the material. That means a material can be used even at a high localized stress. In this approach, peak stresses and stress distributions cannot be adequately defined. Also, the true strength-flaw size distribution cannot be defined adequately. This is because local heat transfer conditions, the effects of geometry, precise loads, load application angles, and so on, are not accurately defined for the calculation of mechanical stresses. 

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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. 

The Normal (or Gaussian) and Weibull distribution functions are most commonly used in engineering design. The normal distribution is usually expressed in one of the following two functional forms ... 

Weibull Distribution. This empirical statistical distribution function describes the scatter in strength values of brittle materials. It is used to assign mechanical properties to brittle materials in probabilistic terms, and to define design requirements in terms of strength and reliability. If p (aj is the probability that the material will fail at a measured critical stress then In In [1 - p (Oc)] ... 

Gumbel-distribution (Type I), Frechet-distribution (Type II), and Weibull-distribution (Type III). However, in these studies, a method for making practical use of the extreme environmental conditions estimated for a structure design was not considered. 

The methods for estimating Weibull parameters are also used in the estimation of statistically-based design values, namely A-basis and B-basis material properties (or simply A-basis and B-basis values), which are the 95% confidence lower bounds on the first and tenth percentiles of a Weibull distribution, respectively. They are of great interest to the engineer in the design of structural and mechanical components however, recent research on their estimation has remained limited [1, 4, 5, 19, 20]. 

The Weibull distribution was first formulated in detail by Walloddi Weibull in 1951, and thus it bears his name. It more accurately describes the distribution of life data, such as fatigue endurance, compared to other statistical distributions, such as the normal distribution which fits better for hardness and tensile strength. Weibull analysis is particularly effective in life prediction. It can provide reasonably accurate failure analyses and failure predictions with few data points, and therefore facilitates cost-effective and efficient component testing. Weibull analysis is widely used in many machine design... 

The use of Weibull plots for design purposes has to be handled with extreme care. As with all extrapolations, a small uncertainty in the slope can result in large uncertainties in the survival probabilities, and hence to increase the confidence level, the data sample has to be sufficiently large N > 100). Furthermore, in the Weibull model, it is implicitly assumed that the material is homogeneous, with a single flaw population that does not change with time. It further assumes that only one failure mechanism is operative and that the defects are randomly distributed and are small relative to the specimen or component size. Needless to say, whenever any of these assumptions is invalid, Eq. (11.23) has to be modified. For instance, bimodal distributions that lead to strong deviations from a linear Weibull plot are not uncommon. 

The probabilistic approach considers the flaw distribution and stress distribution in the material. This approach will be useful when high stresses and their complex distributions are present. In these situations, both empirical and deterministic designs have limitations. The flaw distribution can be characterized by the Weibull approach [1]. This approach is based on the weakest link theory. This theory says that a given volume of a ceramic under a uniform stress will fail at the most severe flaw. Thus, the probability of failure F is given by the following equation ... 

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