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 · F · S) (t2· w), 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.
|
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
|
(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 ism: the Weibull modulus. Find the characteristic strength (s0) (63.2% probability of failure).