In this chapter, a new technique is proposed to find the probability density function (pdf) of a stress for a stochastic mechanical system. This technique is based on the combination of the Probabilistic Transformation Method (PTM) and the Finite Element Method (FEM) to obtain the pdf of the response. The PTM has the advantage of evaluating the probability density function pdf of a function with random variable, by multiplying the joint density of the arguments by the Jacobien of the opposite function. Thus, the “exact” pdf can be obtained by using the probabilistic transformation method (PTM) coupled with the deterministic finite elements method (FEM). In the method of the probabilistic transformation, the pdf of the response can be obtained analytically when the pdf of the input random variables is known. An industrial application on a plate perforated with random entries was analyzed followed by a validation of the technique using the simulation of Monte Carlo.
TopIntroduction
Fatigue crack growth is one of the most important factors in the design of the steel structures. Numerous experiment and researches have been performed for the prediction of fatigue crack growth (Kanninen et al. 1985). In the past, fatigue analysis was largely the domain of the development engineer, who used measurements taken from prototype components to predict the fatigue behavior. This gave rise to the traditional “Build it, Test It, Fix It” approach to fatigue design illustrated in Figure 1. This approach is known to be very costly as an iterative design cycle is centered on the construction of real prototype components. This inhibits the ability to develop new concepts and reduces confidence in the final product due to a low statistical sample of tests. It is also common to find early products released with ‘known’ defects or product release dates being delayed whilst durability issues were addressed. A more desirable approach is to conduct more testing based on computer simulations. Computational analysis can be performed relatively quickly and much earlier in the design cycle.
Figure 1. The build it, test it, fix it method of design
Confidence in the product is therefore improved because more usage scenarios can be simulated. It is not recommended, however, that these simulations completely replace prototype testing. It will always remain desirable to have prototype signoff tests to validate the analysis performed and improve our future modeling techniques. However, the number of prototype stages, and hence the total development time, can be reduced.
Stochastic Fatigue
The fatigue process of mechanical components under service loading is stochastic in nature. Life prediction and reliability evaluation is still a challenging problem despite extensive progress made in the past decades. A comprehensive review of early developments can be found in (Yao et al. 1986). Compared to fatigue under constant amplitude loading, the fatigue modeling under variable amplitude loading becomes more complex both from deterministic and probabilistic points of view. An accurate deterministic damage accumulation rule is required first, since the frequently used linear Palmgren-Miner’s rule may not be sufficient to describe the physics (Fatemi and Yang 1998). Second, an appropriate uncertainty modeling technique is required to include the stochasticity in both material properties and external loadings, which should accurately represent the randomness of the input variables and their covariance structures. In addition to the above difficulties, such a model should also be computationally and experimentally inexpensive. The last characteristic is the main reason for the popularity of simpler models despite their inadequacies.
Fatigue crack growth under random and variable amplitude loading has been analyzed by many different authors (Sunder and Prakash 1996, Schijve et al. 2004). Some of them pay special attention to the simulation of representative load histories of defined random loading processes (Van Dijk 1975, Chang 1981, and Schütz, 1989). Others analyze the effect of overloads and their distribution throughout the load history of the crack growth life (Wheatley et al. 1999, Lang and Marci 1999, Pommier and Freitas 2002).