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53_HWKTTE 4004c

53_HWKTTE 4004c - ln =-=(5-20 For both Equation 5-19 and...

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43 Figure 5-2. Probability Density Functions for Normally Distributed Load and Resistance And the probability of failure, p f may be represented as ) ( 1 Q R P p p s f < = - = (5-18) where the right hand of Equation 5-18 represents the probability, P, that R is less than Q. It should be noted that the probability of failure can not be calculated directly from the shaded area in Figure 5-2. That area represents a mixture of areas from the load and resistance distribution curves that have different ratios of standard deviation to mean values. To evaluate the probability of failure, a single combined probability density curve function of the resistance and load may developed based on a normal distribution, i.e. ( 29 Q R Q R g - = , (5-19) If a lognormal distribution is used the limit state function g(R,Q) can be written as

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44 ( 29 Q R Q R Q R g ln ) ln(
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Unformatted text preview: ) ln( ) , ( =-= (5-20) For both Equation 5-19 and 5-20 the limit state is reached when R=Q and failure will occurs when g(R,Q)<0. Reliability index. The reliability index is a simple method of expressing the probability of failure using function g(R, Q) (Eq. 5-20). The frequency distribution of g(R,Q) would look similar to the curve shown in Figure 5-3. Figure 5-3. Definition of Reliability Index, β for lognormal Distributions of R and Q Evident from the curve is that if the standard deviation is small or the mean value is located further to the right, the probability of failure will be smaller. The reliability index β , is defined as the number of standard deviations, ξ g , between the mean value, g (average), and the origin, or: g g ξ β = (5-21)...
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