Turbochargers Are Us, a new start-up company, hires you to design their new turbocharger. They explain that they want to replace their metallic superalloy turbocharger with a high-tech ceramic that is much lighter for the same configuration. Silicon nitride may be a good choice, and you ask Ceramic Turbochargers, Inc. to supply you with bars made from a certain grade of Si3N4. They send you 25 bars that you break in three-point bending, using a 40-mm outer support span and a 20-mm inner loading span, obtaining the data in the following table.
(a) Calculate the bend strength using σf = (1.5 FS)/(t2w), where F = load, S = support span (a = 40 mm loading span and b = 20 mm in this case), t = thickness, and w = width, and give the mean strength (50% probability of failure) and the standard deviation.
(b) Make a Weibull plot by ranking the strength data from lowest strength to highest strength. The lowest strength becomes n = 1, next n = 2, etc., until n = N = 25. Use F = (n – 0.5)/N where n = rank of strength going from lowest strength n = 1 to highest strength n = 25 and N = 25. Note that F is the probability of failure. Plot ln [1/(1 – F)] as a function of ln sf and use a linear regression to find the slope of the line, which is m: the Weibull modulus. Find the characteristic strength (σ0) (63.2% probability of failure). [Hint: The characteristic strength is calculated easily by setting ln [1/(1 – F)] = 0 once you know the equation of the line. A spreadsheet program (such as Excel) greatly facilitates the calculations.]
Bar | Width (mm) | Thickness (mm) | Load (N) |
1 | 6.02 | 3.99 | 2510 |
2 | 6.00 | 4.00 | 2615 |
3 | 5.98 | 3.99 | 2375 |
4 | 5.99 | 4.04 | 2865 |
5 | 6.00 | 4.05 | 2575 |
6 | 6.01 | 4.00 | 2605 |
7 | 6.01 | 4.01 | 2810 |
8 | 5.95 | 4.02 | 2595 |
9 | 5.99 | 3.97 | 2490 |
10 | 5.98 | 3.96 | 2650 |
11 | 6.05 | 3.97 | 2705 |
12 | 6.00 | 4.05 | 2765 |
13 | 6.02 | 4.00 | 2680 |
14 | 5.98 | 4.01 | 2725 |
15 | 6.03 | 3.99 | 2830 |
16 | 5.95 | 3.98 | 2730 |
17 | 6.04 | 4.03 | 2565 |
18 | 5.96 | 4.01 | 2650 |
19 | 5.97 | 4.05 | 2650 |
20 | 6.02 | 4.00 | 2745 |
21 | 6.01 | 4.00 | 2895 |
22 | 6.00 | 3.99 | 2525 |
23 | 6.00 | 3.98 | 2660 |
24 | 6.04 | 3.95 | 2680 |
25 | 6.02 | 4.05 | 2640 |
* This problem was contributed by Dr. Raymond Cutler of Ceramatek, Inc.
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