- a Standard deviation.
- <(>.
- It is paradoxical that one of the great contributions to physical science, namely the search for consistency and reproducibility in nature, grew out of an idea that was only partially valid.
- As another example, the variability in tensile strength in bolts is shown in the histogram of the ultimate tensile strength of 539 bolts in Fig.
- For example, stresses permitted by AISC code for weld metal in fillet welds in shear are 40 per- cent of the tensile yield strength of the welding rod.
- where n is the design factor..
- (2.2) does not serve well as it stands, and so designers fit statistical distributions to histograms and estimate the risk of failure from interference of the distributions..
- and the load-induced stress is normally distributed, a ~ N(^i 0 , cr a.
- 2.4, then the z variable of the standardized normal N(0,1) can be given by.
- and the reliability R is given by.
- If the strength is lognormally distributed, S ~ LN([Ls, ^s), and the load-induced stress is lognormally distributed, a ~ LTV(Ji 0 , a a.
- R = I- <J>(-2.34.
- The coefficient of variation of the design factor n can be approximated for the quotient S/aas.
- The mean and standard deviation of the companion normal to n ~ ZJV are shown in Fig.
- F(Za R(ZO TABLE 2.1 Cumulative Distribution Function of Normal (Gaussian) Distribution.
- (2.7) Equation (2.7) is useful in that it relates the mean design factor to problem variabil- ity through C n and the reliability goal through z.
- Note that the design factor n is independent of the mean value of S or a.
- DESIGN FACTOR n.
- If the coefficient of variation of the design factor Cl is small compared to unity, then Eq.
- The role of the mean design factor n is to separate the mean strength S and the mean load-induced stress a sufficiently to achieve the reliability goal.
- They are useful because they came to the attention of the statisti- cal community as a result of a pressing practical problem.
- Table 2.2 identifies seven useful distributions and expressions for the probability density function, the expected value (mean), and the variance (standard deviation squared)..
- TABLE 2.2 Useful Continuous Distributions Distribution.
- If the data reduction gives estimates of the distributional parameters, say mean and standard deviation, then a plot of the den- sity function superposed on the histogram will give an indication of fit.
- The CDF is just the probability (the chance) of a failure at or below a specified value of the variate x.
- If the CDF is plotted against the variate on a coordinate system which rectifies the CDF-A: locus, then the straightness of the data string is an indication of the quality of fit.
- TABLE 2.3 Transformations which Rectify CDF Data Strings.
- When the machine is set up to produce a diameter at the low end of the tolerance range, each successive part will be slightly larger than the last as a result of tool wear and the attendant increase in tool force due to dulling wear.
- If the part sequence number is n and the sequence number is n f when the high end of the tolerance is reached, a is the initial diameter produced, and b is the final diameter produced, one can expect the following relation:.
- From Table 2.2, take the probability density function for uniform random distribu- tion.
- Third, apply transformations from Table 2.3 and look for straightness..
- When a pair of dice is rolled, the most likely sum of the top faces is 7, which occurs in 1/6 of the outcomes, but 5/6 of the outcomes are other than 7..
- Histographic data of the ultimate tensile strength of a 1020 steel with class intervals of 1 kpsi are as follows:.
- From Table 2.2, the mean and standard deviation of the companion normal to a log- normal are (Ref.
- FIGURE 2.6 Histographic report of the results of 1000 ultimate tensile strength tests on a 1020 steel..
- the mean and standard deviation (vari- ance) are preferred.
- A plot of the histogram and the density is shown in Fig.
- The mean and standard deviation of a function (|)(jci, J t 2.
- Equations (2.12) and (2.13) for simple functions can be used to form Table 2.4 to dis- play the dominant first terms of the series.
- For a more complete listing including the first two terms of the Taylor series, see Charles R.
- is of the form a XI x 2b x 3c.
- The estimate of the mean in a functional relationship comes from substituting mean values of the variates.
- Equation (2.15) says that the variance of § is simply the sum of the weighted vari- ances of the parameters, with the weighting factors depending on the functional rela- tionship involved.
- In terms of the standard deviation, it is a weighted Pythagorean combination..
- If 12 random selections are made from the uniform random distribu- tion U[0,1] and the real number 6 is subtracted from the sum of the 12, what are the mean, the standard deviation, and the distribution of the result?.
- From Table 2.2,.
- (a + b)n = (O + l)/2 = 1/2 V 2x = (b- a) 2 H From Table 2.4, the mean is the sum of the means:.
- From Table 2.4, the standard deviation of the sum of independent random variables is the square root of the sum of the variances:.
- From the central limit theorem, the sum of random variables asymptotically approaches normality.The sum of 12 variates cannot be rejected using a null hypoth- esis of normality.
- If RANDU is the subprogram name of the uniform random number generator in the interval [0,1], and IX and IY are seed integers, then.
- The mean of <)>.
- Table 2.5 shows the mean and standard deviation of <J>.
- TABLE 2.5 Stochastic Parameters of Fatigue Ratio (J>*.
- Transactions of the American Society of Mechanical Engineers, Journal of Vibration, Acoustics, Stress and Reliability in Design, vol.
- Multiplying the 0.506 by the mean and standard deviation, one can write <j>o.
- Table 2.6 shows approximate mean values of $ b for several material classes..
- The results of an ultimate tensile test on a heat-treated 4340 steel (382 Brinell) consisting of 10 specimens gave an estimate of the ultimate tensile strength of S M.
- <j>.
- kpsi The estimated mean of the endurance limit S' e is given by.
- 6 is lognormal.The 99th-percentile endurance limit is found from the companion normal to the endurance limit distribution as follows:.
- without fatigue testing from the history of the 133 steel materials ensemble embod- ied in <|>.
- One can expect 99 percent of the instances of endurance limit to exceed 69.2 kpsi given that the mean tensile strength is 190 kpsi..
- Moore testing of the 4340 gave MS.
- Figure 2.8 graphically depicts the two and one-half times dispersion resulting from use of the correlation rather than R.
- Defining the design factor as the quotient of the response potential divided by the stimulus is more general and useful.
- 2.9a the probability density of the response potential S is/i(S), and in Fig.
- 2.9b the density function of the stimulus a is/ 2 ( CF).
- which is the differential reliability dR, or.
- FIGURE 2.9 (a), (b), and (c) Development of the general reliability equation JlRiJR 2 by interference.
- Define RI as a function of the cursor position x:.
- 70 Define R 2 as a function of the cursor position x:.
- Geometrically, the area of the triangle in Fig.
- 2.10 is which equals 0.09, and the ones complement is the reliability R = I .
- 2.9/shows that the largest contribution to the area under the curve is near R 1 = I', consequently, the tabular method will begin with R 1 = 1 at the top of the table.
- Column 2 contains the values of the cursor location x corresponding to R 1 .
- Column 3 consists of the values of R 2 corresponding to the cursor location x, namely R t)/5.The ordinates to the curve are in the R 2.
- 0.09 and the reliability is.
- 1 x<A (B-X)I(B-A) A<x<B O x>B and the survival function R 2 is given by.
- l -(b-a)(B-A)l (b - x)dX = l -2(b-a)(B-A) TABLE 2.7 Reliability by Simpson's Rule.
- Note that the reliability declines from unity as the square of the overlap (b -A).
- A very useful three-parameter distribution is the Weibull, which is expressed in terms of the parameters, the lower bound Jt 0 , the characteristic parameter 0, and the shape parameter b, displayed as x ~ W[x 0 ,0, b].
- The mean and standard deviation are found from the parameters as.
- For interference of a Weibull strength S ~ W[x Q1 , 0i, bi] with a Weibull stress a ~ W[Jt 02 , 02, b 2 ] 9 use a numerical evaluation of the integral in Eq.
- Write the strength distribution survival equation in terms of the cursor location jc as.
- Noting that the survival equation for the stress distribution in terms of the cursor location jc is.
- If S ~ W[AQ 9 50,3.3] kpsi and a ~ W[30,40,2] kpsi, then Table 2.8 follows.
- The means of the strength S and the stress a are.
- Since the distribution of the design factor as a quotient of two Weibull variates is not known, discovering the design factor corresponding to a reliability goal of (say) 0.999 becomes an iterative process, with the previous tabular integration becoming part of a root-finding process, quite tractable using a computer..
- Machine parts often exhibit geometries with coefficients of variation that are very small compared with that of the load.
- The set of all integers is the set of the counting numbers 1,2,3.
- The set of all rational numbers mln is constructed from the integers (dividing by zero excepted).The set of all real numbers is constructed by adding limits of all bounded monotone increasing sequences (irrational numbers) to the set of rational numbers.
- Another form of number is the incomplete number.
- It can be formed from the true value of the number simply by truncating after a prescribed number of digits..
- When one encounters random variables, certain properties of the distribution will be useful, such as the mean, standard deviation, etc.
- Now Ji is the incomplete number repre- senting an unbiased estimate of the population mean, and a is the incomplete num- ber representing an unbiased estimate of the population standard deviation.
- Moore specimen diameter gives <(>o.3o ~ LN The companion normal has a mean of y and a standard deviation s y of.
- It would be misleading to consider the mean of the fatigue ratio as 0.506 to three digits or even to call it 0.5 because it is not known to three significant digits.
- Dispersion in the mean is addressed by the standard deviation of 0.070.
- The unbiased estimator of the mean is 0.506, a rounded incomplete number, and the meaningful qualifica- tion is 0.4860 <.
- Transac- tions of the A.S.M.E., Journal of Mechanical Design, vol