161
C H A P T E R 4
Reliability of a Component
under Static Load
4.1 INTRODUCTION
According to the applied loading on a component, component design typically can be grouped
into component design under static loading and component design under cyclic loading. is
chapter will discuss component design under static loading. e component design under cyclic
loading will be discussed in the book of Reliability-Based Mechanical Design, Volume 2: Component
under Cyclic Load and Dimension Design with Required Reliability.
According to the task of component design, component design can be grouped into com-
ponent design check and component dimension design. Component design check means that
we check the actual reliability of a component when loading, geometric dimensions, and material
properties are all specified. If the actual reliability of the component is larger than the specified
reliability, the component is classified as a qualified design. Otherwise, the component needs to
be redesigned. Component dimension design refers that we design component dimensions with
the required reliability under the specified loading and material. is chapter will discuss com-
ponent design check for a component under static loading. e component dimension design
will be discussed in the book of Reliability-based Mechanical Design, Volume 2: Component under
Cyclic Load and Dimension Design with Required Reliability.
In this chapter, first, we will discuss how to describe all design parameters: dimension,
loading, and material properties as random variables. en we will discuss the reliability of
component under some typical static loadings such as rod under axial loading, pin under direct
shearing, shaft under torsion, beam under bending. Finally, we will discuss component under
combined loadings.
4.2 GEOMETRIC DIMENSION AS A RANDOM VARIABLE
Any geometric dimension is always associated with a corresponding dimension tolerance, which
is controlled by a manufacturing process. For example, the possible dimension tolerance of a
component dimension between 0.4
00
–1.97
00
vs. its different manufacturing processes [1] are listed
in Table 4.1.
e geometric dimensions of a component can be classified as a free dimension and a
mating dimension. A free dimension refers that the variation of the dimension will not affect
162 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
Table 4.1: Dimension tolerance vs. its different manufacturing processes
Dimension Range
0.4˝– 0.71˝ 0.71˝ – 1.19˝ 1.19˝ – 1.97˝
Manufacturing Process Tolerances in ousandths of an Inch
Lapping and honing 0.2–0.3 0.25–0.4 0.3–0.4
Cylindrical grinding 0.3–0.7 0.4–0.8 0.4–1.0
Surface grinding 0.3–1.0 0.4–1.2 0.4–1.6
Diamond turing 0.3–0.7 0.4–0.8 0.4–1.0
Diamond boring 0.3–0.7 0.4–0.8 0.4–1.0
Broaching 0.3–1.0 0.4–1.2 0.4–1.6
Reaming 0.4–2.8 0.5–3.5 0.6–4.0
Turing 0.7–10 0.8–10 1.0–10
Boring 1.0–10 1.2–10 1.6–10
Milling 2.8–10 2.5–12 4.0–16
Planing and shaping 2.8–10 2.5–12 4.0–16
Drilling 2.8–10 2.5–12 4.0–16
component functions and other adjacent components. e dimension tolerance of a free dimen-
sion will typically be the default tolerance, which is the most economical way to manufacture this
dimension in a company. A schematic of a simple base-shaft assembly is shown in Figure 4.1,
where a base is a rectangular place with a hole in the center and the shaft is a cylindrical round
bar. e shaft is fitted through the hole of the base. e dimension d1 in Figure 4.1 is a free
dimension.
A mating-dimension refers that the variation of this dimension will affect component
functions and other adjacent components. e mating dimension is the dimension on an in-
terface. e tolerance of it will be determined by the required function, such as a clearance fit,
interference fit, or transient fit. e diameter d 2 of the shaft and the diameter d 2 of the hole
will be a mating dimension.
e dimension tolerance can be expressed with symmetrical tolerance such as
1:000
00
˙0:005, or a unilateral tolerance such as or a bilateral tolerance such as 1:000
00
C0:008
0:002
.
e general form of a dimension with tolerance can be expressed as d
t
U
t
L
, where d is the nom-
inal dimension, t
U
is the upper limit of tolerance and t
L
is the lower limit of tolerance. If the
final value of the dimension of a manufactured component is in the range .d C t
L
; d C t
U
/, the
component will be classified as a qualified component. If the final value of the dimension of a
manufactured component is out of the range .d C t
L
; d C t
U
/, the component will be classified
as an unqualified component and will be discarded.
4.2. GEOMETRIC DIMENSION AS A RANDOM VARIABLE 163
d1
d2
Shaft
Base
Figure 4.1: e schematic of a base-shaft assembly.
Before a component has been manufactured, the actual dimension of a component is un-
known due to the associated dimension tolerance. erefore, component geometric dimension
is a random variable. It is typically treated as a normally distributed random variable d . Accord-
ing to the definition of dimension tolerance, the components’ dimension inside the dimension
tolerance range
.d
C
t
L
; d C t
U
/ will be accepted. For a normal distribution, the probability of
event .
d
4
d
d
d
C 4
d
/ will be 99.9968%. is event can be used to represent the
dimension tolerance range with a very small error (0.0032%). erefore, the mean and stan-
dard deviation of a normally distributed dimension random variable d can be determined per
Equation (4.1):
d
D
.
d C t
L
/
C .d C t
U
/
2
D d C
t
L
C t
U
2
d
D
.
d C t
U
/
.d C t
L
/
8
D
t
U
t
L
8
:
(4.1)
Example 4.1
A shaft diameter with a dimension tolerance is 1:250
00
˙ 0:005. Determine the mean and stan-
dard deviation of the shaft diameter if it is treated as a normally distributed random variable.
Solution:
Per Equation (4.1), the mean and standard deviation of the shaft diameter are
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