164 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
d
D d C
t
L
C t
U
2
D 1:250 C
0:005 C 0:005
2
D 1:250
00
d
D
t
U
t
L
8
D
0:005 .0:005/
8
D 0:00125
00
:
Example 4.2
e diameter of a hole with a dimension tolerance is 1:500
00
0:009
0:001
. Determine its mean and stan-
dard deviation of the hole diameter if it is treated as a normally distributed random variable.
Solution:
Per Equation (4.1), the mean and the standard deviation of the shaft diameter are
D
D d C
t
L
C t
U
2
D 1:500 C
0:001 C 0:009
2
D 1:505
00
D
D
t
U
t
L
8
D
0:009 .0:001/
8
D 0:001
00
:
4.3 STATIC LOADING AS A RANDOM VARIABLE
In traditional mechanical component design under static loading, the maximum static load-
ing will be one unique value. However, in reliability-based mechanical component design, the
loading for component design under static loading is a random variable. In reliability-based
mechanical design, a component under consideration is not a single specific component, but a
batch of the same component. Each of this batch of the component might be used by different
customers in different service conditions. One value of the maximum loading for each compo-
nent can be obtained. is value can be treated as one sample data of the static loading. Suppose
that this batch of the component under consideration is 50,000. We could collect one sample
data per each component. erefore, we will have 50,000 sample data of the static loading for
this batch of the same component. Based on these sample data, we can determine the type of
distribution and calculate the corresponding distribution parameters of the static loading.
If a loading P such as concentrated force, concentrated moment or torque, is expressed as
a range of value such as .P
low
; P
up
/, it could be treated as a normally distributed random variable.
We can use the same reasoning and similar equation as Equation (4.1) to determine its mean
4.3. STATIC LOADING AS A RANDOM VARIABLE 165
and standard deviation:
P
D
P
low
C P
up
2
P
D
P
up
P
low
8
:
(4.2)
For a component design under static loading, loading could also be presented by other three
types: (1) one single loading, such as 2000 lb; (2) a discrete distributed random variable such
as the loading in Example 4.4; and (3) a continuously distributed random variable such as the
loading in Example 4.5.
Example 4.3
A component is subjected to a concentrated force P D 2025 ˙ 125 (lb). If it is treated as a
normal distribution, determine its mean and standard deviation.
Solution:
Per Equation (4.2), the mean and standard deviation of this concentrated force are:
P
D
P
low
C P
up
2
D
Œ
.2025 125/ C .2025 C 125/
2
D 2025 (lb)
P
D
P
up
P
low
8
D
Œ.
2025 C125
/
.2025 125/
8
D 31:25 (lb):
Example 4.4
A company will design a component. e company sent out a survey to potential customers
for identifying the maximum static loading for the component. e collected data of maximum
static loading for the batch of the same component are listed in Table 4.2. In this table, the first
row is the maximum possible loading for the component. e second row shows the number of
selected loading by potential customers. For example, in the fourth column, “1100” in the first
row refers that the maximum static loading is 1100 lb. e “40” in the second row refers that 40
customers in the survey selected 1100 lb as the maximum static loading. Build the distribution
function for the loading.
Table 4.2: e collected data for the maximum static loading
Maximum static loading L (lb.) 1000 1050 1100 1150 1200
Number of selected loading 10 20 40 32 5
166 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
Solution:
In this example, the maximum static loading, according to the survey, is a discrete random vari-
able. e total number of sample data will be 10 C 20 C 40 C 32 C 5 D 107. en the probabil-
ity of the loading equal to 1100 will be 40=107. So, we can use the following PMF to describe
the loading:
p
.
l
/
D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
10=107 D 0:0935 l D 1000 (lb)
20=107 D 0:1869 l D 1050 (lb)
40=107 D 0:3738 l D 1100 (lb)
32=107 D 0:2991 l D 1150 (lb)
5=107 D 0:0467 l D 1200 (lb):
Example 4.5
A company collected maximum static loading for a component. One hundred components at
different services were selected for collecting loading data. For each component, the maximum
loading on the component was collected during one-month regular service. e collected data
are listed in Table 4.3. Create the histogram and then determine the type of distribution and
corresponding distribution parameters.
Table 4.3: One hundred sample data of the maximum static loading on the component
e Maximum Static Loading X on Components (lb)
2532, 1987, 2877, 2588, 2511, 2682, 2568, 2026, 2377, 2384, 2211, 3207, 2056, 2232, 2130, 1922, 2363, 2327,
3139, 2338, 1971, 3172, 3006, 2347, 1772, 2250, 2380, 2574, 2332, 2650, 2626, 1887, 2023, 2117, 2221, 2306,
2456, 1087, 2244 ,3009, 1970, 2870, 2608, 2437, 2532, 1746, 2412, 3172, 2494, 2469, 2120, 2436, 2555, 2642,
2282, 2344, 3361, 1434, 3453, 2602, 2900, 1701, 2184, 2325, 2640, 1698, 2662, 1904, 2480, 2744, 2597, 2937,
2903, 2157, 2566, 2025, 1855, 2866, 2450, 2425, 2860, 2718, 2608, 3013, 2868, 2558, 2139, 2157, 2986, 1725,
2439, 1573, 2909, 2838, 2451, 2418, 1331, 2712, 1463, 1406,
Solution:
1. Use MATLAB to create a histogram.
We can follow Section 2.8 and use the MATLAB program to create the histogram. We
need to input the data of from the Table 4.3 in an Excel file, where the data will be inputted
in the first column. e file name will be “Example 4.5.” en we can use the following
MATLAB program to import the data and create the histogram. e histogram is dis-
played in Figure 4.2. From the histogram, we could assume that the loading follows a
normal distribution.
4.3. STATIC LOADING AS A RANDOM VARIABLE 167
%Import the data from an Excel file
X=xlsread(`Example 4.5}'),
% Create a 10-bin histogram
histogram(X,10)
xlabel (`Bins')
ylabel(`Frequency')
25
20
15
10
5
0
1000 1500 2000 2500 3000 3500
Bins
Frequency
Figure 4.2: Histogram of the maximum static loading.
2. Run the goodness-of-fit test and calculate the distribution parameters.
We can follow Section 2.13 to check whether it is a normal distribution. We can use the
following MATLAB program to run the goodness-of-fit test. After running the MAT-
LAB program, we have the value of “h” is zero. So the load can be assumed as a normal
distribution. e MATLAB program can also calculate the mean and standard deviation:
X
D 2400:2 (lb);
X
D 457:4 (lb):
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