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 σ_f = (1.5 FS)/(t²w), where F = load, S = support span (a = 40 mm loading span and b = 20 mm in this case), t 5 thickness, and w 5 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 5 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 σ_f and use a linear regression to find the slope of the line, which is m: the Weibull modulus. Find the characteristic strength (s₀) (63.2% probability of failure).

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Bar 1 23 4567 9 10 11 12 13 14 15 16 17 18 19 20 22222 24 Width (mm) 6.02 6.00 5.98 5.99 6.00 6.01 6.01 5.95 5.99 5.98 6.05 6.00 6.02 5.98 6.03 5.95 6.04 5.96 5.97 6.02 6.01 6.00 6.00 6.04 6.02 Thickness (mm) 3.99 4.00 3.99 4.04 4.05 4.00 4.01 4.02 3.97 3.96 3.97 4.05 4.00 4.01 3.99 3.98 4.03 4.01 4.05 4.00 4.00 3.99 3.98 3.95 4.05 Load (N) 2510 2615 2375 2865 2575 2605 2810 2595 2490 2650 2705 2765 2680 2725 2830 2730 2565 2650 2650 2745 2895 2525 2660 2680 2640