1.7. RELIABILITY VS. FACTOR OF SAFETY 15
• In 1965, the DOD (Department of Defense) military standard MIL-STD-785 “Reliabil-
ity Program for Systems and Equipment,” which was revised in 2008.
Reliability in mechanical engineering design is founded based on similar concepts and
principles which were established through the extensive research on military electronic equip-
ment. e following are several notable events for reliability in mechanical engineering design.
• In 1951, Weibull, W. of the Swedish Royal Institute of Technology published a statistical
distribution for material strength [11]. is distribution is called Weibull distribution and
has played an important role in the development of reliability in mechanical engineering.
• In 1968, Professor Edward B. Haugen published the book
Probability Approach to De-
sign [12], which was directedly focused on mechanical design with reliability.
• In 1972, Professor A. D. S. Carter published a book Mechanical Reliability [13].
• In 1977, Dr. D. Kececioglu published a paper “Probabilistic design methods for reliabil-
ity and their data and research requirements” to present the approach for dealing with
reliability in fatigue failure [14].
Nowadays, reliability in engineering design has become an important concept and tool
for mechanical engineering design.
1.7 RELIABILITY VS. FACTOR OF SAFETY
e differences between reliability and factor of safety will be very clear after Chapter 4 has been
read. Before that, we will use some simple examples to explain their differences.
In the traditional mechanical design approach, the factor of safety is typically defined as
the ratio of component’ strength to the maximum component stress induced by the operational
load. For example, if the failure mode of the component is a static failure, the factor of safety is:
n D
S
Q
> 1; (1.4)
where S is the average of the component’s strength index such as tensile yield strength or ul-
timate tensile strength for static failure design. Q is the average of the component’s maximum
stress, which can be determined by the component’s geometrical dimensions and operational
loading. n is the factor of safety, which links together component’s material strength, compo-
nent geometrical dimensions, and operational loading for component design.
Reliability of a component is defined as the probability of a component, a device or a
system performing its intended functions without failure over a specified service life and under
specified operation environments and loading conditions. e mathematical equation for the
reliability is shown in Equation (1.1) and is repeated here:
R D P .S Q/; (1.1)
16 1. INTRODUCTION TO RELIABILITY IN MECHANICAL DESIGN
where S is the component’s strength index such as tensile yield strength or ultimate tensile
strength for static failure design. Q is the component’s maximum stress, which can be deter-
mined by the component’s geometrical dimensions and operational loading. R is reliability.
Both the reliability and the factor of safety serve the same purpose and are a measure for
creating the design equations. However, the key differences between them are:
• the factor of safety is a deterministic approach in which all design parameters are treated
as deterministic values. e factor of safety intends partially to consider the uncertainty of
the design parameters; and
• the reliability is a probabilistic approach in which all design parameters are treated as ran-
dom values. e uncertainties of the design parameters are assessed by reliability.
e following example can explain in detail the similarities and differences between the
reliability and the factor of safety.
Example 1.3
In the traditional design approach with a factor of safety, the design parameters for three design
cases with different materials and different operation loading are treated as deterministic values
and are listed in Table 1.4. When the uncertainties of the strengths and stresses of the three
design cases are considered, the strengths and stresses of the components of the same design
cases are treated as normal distribution random variables (note: the normal distribution will
be discussed in Section 2.12.4). e corresponding distribution parameters are also listed in
Table 1.2:
1. conduct the design check, that is, calculate the factor of safety and the reliability of com-
ponents; and
2. discuss the results.
Table 1.4: e design parameters of three design cases
Traditional Approach
with a Factor of Safety
e Probabilistic Approach with a Reliability
Strength S
(normal distribution)
Stress Q
(normal distribution)
Case #
Strength Stress Mean μ
S
Standard
Deviation σ
S
Mean μ
Q
Standard
Deviation σ
Q
1 100 (ksi) 80 (ksi) 100 (ksi) 5 (ksi) 80 (ksi) 5 (ksi)
2 100 (ksi) 80 (ksi) 100 (ksi) 5 (ksi) 80 (ksi) 20 (ksi)
3 100 (ksi) 50 (ksi) 100 (ksi) 30 (ksi) 50 (ksi) 30 (ksi)
1.7. RELIABILITY VS. FACTOR OF SAFETY 17
Solution:
1. e factor of safety and the reliability.
e factor of safety of three design cases can be calculated per Equation (1.3) and are
listed in Table 1.5. e reliability can be calculated according to Equation (1.1), which will
be discussed in detail in Section 2.12.4. e random event .S Q/ is the same random
event
.
S Q 0
/
. Let us use Z to represent the new random variable S Q , that is,
Z D S Q. Equation (1.1) can be rearranged as:
R D P
.
S Q
/
D P
.
S Q 0
/
D P .Z 0/:
Since both strength S and stress Q are normal distributions, the random variable Z will
also be a normal distribution. e mean
Z
and standard deviation
Z
of Z can be calcu-
lated by the means and the standard deviations of S and Q. ey are:
Z
D
S
Q
I
Z
D
q
.
S
/
2
C
Q
2
:
After the mean and standard deviation of the normally distributed random variable Z are
determined, the reliability can be directly calculated based on R D P .Z 0/ and are listed
in Table 1.5. (Note: e calculation procedure will be discussed in detail in Section 2.12.4.)
Table 1.5: e factor of safety and the reliability of three design cases
Case #
e Factor of Safety Approach e Reliability Approach
Strength Stress
Factor of
Safety
Mean μ
Z
Standard
Deviation σ
Z
Reliability
1 100 (ksi) 80 (ksi) 1.25 20 (ksi) 7.1 (ksi) 0.9977
2 100 (ksi) 80 (ksi) 1.25 20 (ksi) 20.6 (ksi) 0.8340
3 100 (ksi) 50 (ksi) 2 50 (ksi) 42.4 (ksi) 0.8807
2. Discuss the results.
• Both the factor of safety and the reliability are the measure of the status of safety
of the components. However, the reliability R not only predicts the status of safety
of the component but also indicates failure probability F , which is equal to 1 R.
For example, from Table 1.5, the reliability of components in Case #1 is 0.9977, and
the failure probability of a component is 1 0:9977 D 0:0023 D 0:23%. e factor of
safety cannot provide any information about the possible failure of a component.
18 1. INTRODUCTION TO RELIABILITY IN MECHANICAL DESIGN
• From Table 1.5, both Cases #1 and #2 have the same factor of safety: 1.25. e same
value of the factor of safety implies that both Case #1 and Case #2 will have the
same measure of the status of safety. However, when the reliability approach is used
to check the status of safety of Case #1 and Case #2, the reliabilities of these two
designs are quite different, as shown in Table 1.5. Case #1 has a reliability 0.9977,
and Case #2 has a reliability 0.8340 only. e cause for this inconsistent result is
due to the uncertainty of design parameters. e traditional design approach with
the factor of safety cannot quantitively consider the effect of uncertainty. e design
approach with reliability does consider the effects of uncertainty.
• From Table 1.5, the factor of safety in the design Case #3 is 2 and is larger than the
factor of safety 1.25 of the design Case #1 from the table. According to the meaning of
factor of safety, this indicates that the components from the design Case #3 should be
relatively safer than the components from the design Case #1. However, the reliability
of the components for the design Case #3 is much less than the reliability of the
component for the design Case #1 from Table 1.5. e cause for these contradictory
conclusions is mainly due to the uncertainty of design parameters. So, a higher factor
of safety does not guarantee a much safer component. However, higher reliability will
certainly guarantee just that.
• From Table 1.4, the simple information about the design parameters, that is, deter-
ministic values, are required when the design approach with a factor of safety is used
for component design. However, when the design approach with reliability is used
for component design, a large amount of information about design parameters are
needed because the type of distributions and corresponding distribution parameters
are required.
In summary, both the factor of safety and the reliability are the measure of the status of
safety of a component. Both are successfully used for mechanical component design. e ad-
vantages of the factor of safety are simple and do not require much information about design
parameters. e disadvantages are: (1) it cannot be used to explain possible component failure;
(2) the higher the factor of safety of components does not guarantee that it will be much safer;
and (3) it cannot include the effects of uncertainty of the design parameters. e advantages
of reliability are: (1) it not only indicates the probability of safe components, but also indicates
the probability of component failure; (2) the higher reliability of a component certainly guar-
antees that it is much safer; and (3) the approach with reliability can fully consider the effects
of uncertainties of design parameters. e main disadvantage of the design approach with reli-
ability is that much more information or a large amount of data about uncertainties of design
parameters are required. Without reliable descriptions of uncertainties of design parameters, the
implementation of the design approach with reliability is meaningless.
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