UBC Theses and Dissertations
On the probabilistic design of critical engineering components. Agrawal, Avinash Chandra
The present study investigates the reliability approach to the design of a critical mechanical component and applies this approach to several design problems. For the design of a critical component the combination of maximum loads and minimum material strength is selected for the design. Under the probabilistic approach, maximum load and minimum material strength values are considered as random variables having Extreme Value density functions of Type I (maximum) and Type III (minimum), respectively. With this combination of probability density functions for the material strength and the load, a closed form solution does not seem to be feasible for either the probability density function of the safety factor or for the probability of failure of the design. Consequently, numerical evaluations are made for the probability density function of the safety factor Ʋ the probability of failure Pf and the mean value of the safety factor, ῡ , for a set of parameter values of the density functions for the maximum load and the minimum strength. The effect of changing these parameter values on the probability of failure is studied. An important feature of the design of a critical component from the reliability approach, in general, is that the reliability statement implies a specific "mission" time of operation for the component. This is due to the dependence of the value of certain parameters of E.V. models on the length of time over which the extreme value measurements are taken. Three design models are considered under the reliability approach for a given load and material strength, and reliability specification. The parameters defining the extreme value density functions of load and material strength are assumed to be given. In model 1, the problem of designing a single critical mechanical component subjected to purely axial loads is considered for a given single material. The failure criterion in such a case is assumed to be separation and the output of the design process is a cross section area A of the component with a specified reliability over a corresponding period of operating time. This cross section area is considered as a statistical constant in order to avoid additional mathematical complexity. In model 2, the design problem of the first model is extended by considering more than one material available for the design. The design problem thus considered is one of selecting one among various alternative materials on the basis of some design criterion such as minimum weight, etc. The method consists of calculating the design cross section area for each material available and then calculating the value of the design criterion for each design. The material which optimizes the value of the design criterion becomes the choice for that design. It is observed that for a given load distribution and various available materials, the design cross section area is a function of the ratio of the mean strength to its standard deviation and not a function of the mean value of strength alone. It is, therefore, considered logical to take this ratio of the mean strength to the standard deviation as the measure of the quality of the material and express the cost of material (dollars per lb.) as a function of this ratio. This is in contrast with the conventional design approach where the cost of material is considered as a function of a single value of strength of material only. Model 3 considers the design problem of making a choice from among several materials on the basis of the economic criterion of minimum cost. The cost of material is considered as a function of the ratio of mean strength to standard deviation, as mentioned earlier. In the absence of data required to assess this functional relationship, a linear relationship is assumed. Another cost factor, the cost of safety factor values, is introduced. This cost is a measure of the margin of safety provided in the design for each component. As the safety factor is a random variable in the probabilistic approach, tools such as statistical decision theory and utility theory are used to obtain the cost of design for each material. These cost values are then weighted with respect to the probability density function of the safety factor. The expected overall cost of the design is then evaluated for each material and the given load distribution, such that the desired reliability value is attained. The material corresponding to the minimum value of the expected overall cost is selected as the optimal choice of the designer.
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