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Introduction to Contact Mechanics Part 5

Tham khảo tài liệu 'introduction to contact mechanics part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.2 Basic Statistics 63 x F x f u 4.2c u -w where u in Eq. 4.2c is a dummy variable and takes on all values of x for which u x. The cumulative distribution function F x always increases with increasing values of x. Now if the random variable X is a continuous variable then the probability that X takes on a particular value x is zero. However the probability that X lies between two different values of x say a and b is by definition given by b p a x b J f x dx 4.2d a w where f x 0 and f x 1. -w Note that it is the area under the curve of f x that gives the probability as shown in a in Fig. 4.2.3. For the continuous case the value of f x at any point is not a probability. Rather f x is called the probability density function. A cumulative distribution F x for the continuous case gives the probability that X takes on some value x and can be found from x P X x F x f u J f u du -w -w 4.2e where u is a dummy variable which takes on all values between minus infinity and x. The value of F x approaches 1 with increasing x as shown in Fig. 4.2.3 b . Equations 4.2d and 4.2e satisfy the basic rules of probability. Fig. 4.2.3 a Probability density function and b cumulative probability distribution function for a continuous random variable X. 64 Statistics of Brittle Fracture 4.3 Weibull Statistics 4.3.1 Strength and failure probability Consider a chain that consists of n links carrying a load W as shown in Fig. 4.3.1. Because of the load a stress ơa is induced in each link of the chain. Let the tensile strength of each link be represented by a continuous random variable S. The value of S may in principle take on all values from -TO to to but in the present work we may assume that links only fail in tension and hence S 0 or more realistically S ơu where ơu 0 and is a lower limiting value of tensile strength. All links are said to have a tensile strength equal to or greater than ơu. For distributions involving continuous random variables as in the present case by definition the .