Page 62 - Shigley's Mechanical Engineering Design

bud29281_ch02_031-070.qxd 11/12/2009 7:59 pm Page 37 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas:

Materials 37
description of strength, a material property, is distributional and thus is statistical in
nature. Chapter 20 provides more detail on statistical considerations in design. Here we
will simply describe the results of one example, Ex. 20–4. Consider the following table,
which is a histographic report containing the maximum stresses of 1000 tensile tests on
a 1020 steel from a single heat. Here we are seeking the ultimate tensile strength S ut . The
class frequency is the number of occurrences within a 1 kpsi range given by the class
midpoint. For example, 18 maximum stress values occurred in the range of 57 to 58 kpsi.

Class Frequency f i 2 18 23 31 83 109 138 151 139 130 82 49 28 11 4 2

Class Midpoint 56.5 57.5 58.5 59.5 60.5 61.5 62.5 63.5 64.5 65.5 66.5 67.5 68.5 69.5 70.5 71.5
x i , kpsi

The probability density is defined as the number of occurrences divided by the total
sample number. The bar chart in Fig. 2–5 depicts the histogram of the probability den-
sity. If the data is in the form of a Gaussian or normal distribution, the probability
density function determined in Ex. 20–4 is

2
1 1 x − 63.62
f (x) = √ exp −
2.594 2π 2 2.594

where the mean stress is 63.62 kpsi and the standard deviation is 2.594 kpsi. A plot
of f (x) is also included in Fig. 2–5. The description of the strength S ut is then
expressed in terms of its statistical parameters and its distribution type. In this case
S ut = N(63.62, 2.594) kpsi, indicating a normal distribution with a mean stress of
63.62 kpsi and a standard deviation of 2.594 kpsi.
Note that the test program has described 1020 property S ut , for only one heat of
one supplier. Testing is an involved and expensive process. Tables of properties are
often prepared to be helpful to other persons. A statistical quantity is described by its

Figure 2–5 0.2

Histogram for 1000 tensile
tests on a 1020 steel from a
single heat.
f(x)

Probability density 0.1

0
50 60 70
Ultimate tensile strength, kpsi