1.2. CYCLIC LOADING SPECTRUM 3
is chapter describes how to establish the limit state function of a component under a
cyclic load, and then how to determine the reliability of a component under such cyclic
load. e book presents two fatigue theories to calculate the reliability of a component
under cyclic load. e first theory is the P-S-N (Probabilidtic-Stress-Number of cycles)
curve approach. e second theory is the probabilistic fatigue damage model (the K-D
model). Five typical component cases under cyclic load presented in this chapter include
bar under cyclic axial load, pin under cyclic direct shearing, shaft under cyclic torsion,
beam under cyclic bending, and a rotating shaft under cyclic combined loads.
• Chapter 3: e Dimension of a Component with the Required Reliability
is chapter presents how to design the dimension of a component with required reliability
under static load or cyclic load. For the dimension of a component under cyclic load, the
second fatigue theory, that is, the K-D model is mainly used. Five typical component
dimension design with required reliability presented in this chapter include bar under static
axial load or cyclic axial load, pin under static direct shearing or cyclic direct shearing, shaft
under static torsion or cyclic torsion, beam under static bending or cyclic bending, and a
component under combined static loads or cyclic combined loads.
• Appendix A: Computational Methods for Calculating the Reliability of a Component
is appendix concisely describes the procedure of the H-L, R-F, and Monte Carlo meth-
ods for calculating the reliability of a component, which has been presented in detail in
Volume 1 [1].
• Appendix B: Samples of Six MATLAB Programs
is appendix provides six MATLAB programs as a reference, including three programs
for the calculation of reliability and another three programs for dimension design with
required reliability.
1.2 CYCLIC LOADING SPECTRUM
Mechanical devices or systems always have at least one moving component. Due to the repeated
functions or stop-start process or mechanical vibration, mechanical components are typically
subjected to a cyclic load. A schematic of a cyclic load is depicted in Figure 1.1. e maxi-
mum stress
max
and the minimum stress
min
of cyclic stress (loading) are the maximum and
minimum values of the cyclic stresses, as shown in Figure 1.1. e mean stress
m
, the stress
amplitude
a
, and the range of stress
r
of the cyclic stress (loading) are defined as:
m
D
max
C
min
2
(1.4)
a
D
max
min
2
(1.5)
4 1. INTRODUCTION AND CYCLIC LOADING SPECTRUM
r
D
max
min
D 2
a
: (1.6)
e stress ratio S
r
is defined as the ratio of the minimum stress
min
to the maximum stress
max
, that is,
S
r
D
min
max
: (1.7)
A cyclic stress curve can be treated as a wave. One complete of the wave such as the minimum
point to the adjacent minimum point, or the maximum point to the adjacent maximum point is
one cycle of the cyclic stress, as shown in Figure 1.1.
Stress
Time
One
Cycle
σ
a
σ
r
σ
max
σ
m
σ
min
Figure 1.1: A schematic of cyclic stress with a constant stress amplitude.
e magnitude of cyclic stress can be fully defined by any two out of these six variables
max
,
min
,
m
,
a
,
r
, and S
r
. e duration of the cyclic stress will be defined by the number
of cycles of the cyclic loading. One special cyclic stress that has a zero mean stress is called the
fully (completely) reversed cyclic stress, as shown in Figure 1.2. For a fully reversed cyclic stress,
it has:
max
D
min
, and the stress ratio S
r
D 1. is type of cyclic stress is a special case
because lots of fatigue strength data are based on fatigue tests under a fully reversed cyclic stress.
Example 1.1
Cyclic stress has a stress amplitude
a
D 10 ksi and the stress ratio S
r
D 0:5. Calculate the mean
stress
m
, the maximum stresses
max
, the minimum stresses
min
, and the range of stress
r
of
this cyclic stress.
Solution:
Based on Equations (1.5) and (1.7), we have:
a
D 10 D
max
min
2
(a)
S
r
D 0:5 D
min
max
: (b)
1.2. CYCLIC LOADING SPECTRUM 5
Time
Stress
Figure 1.2: A fully reversed cyclic stress.
From Equations (a) and (b), we have:
max
D 40 .ksi/;
min
D 20 .ksi/: (c)
Based on Equations (1.4) and (1.6) by using information from Equation (c), we have:
m
D
max
C
min
2
D
40 C 20
2
D 30 .ksi/
r
D
max
min
D 40 20 D 20 .ksi/:
ere is a lot of different cyclic loads. e cyclic loading spectrum refers to a description
of cyclic stress (loading) levels vs. corresponding cycle numbers. Generally, three parameters,
including stress amplitude, mean stress, and the number of cycles of the cyclic loading, are
used to describe a cyclic loading spectrum fully. However, for fatigue design, a non-zero mean
cyclic loading is typically converted into a fully reversed cyclic loading with an equivalent stress
amplitude, which will be discussed in detail in Chapter 2. So, two parameters including the
fully reversed stress amplitude
a
and the cycle number n
L
of cyclic loading are typically used
to describe a cyclic loading spectrum for the reliability calculation of a component under cyclic
load. Both stress amplitude
a
and the cycle number n
L
can be a constant (deterministic value)
or several constant or random variable. With the reasonable combinations of variations of stress
amplitude
a
and the cycle number n
L
, the systematic description of cyclic loading spectrum
will include the following six models. ese six cyclic loading spectrum models [2] can describe
any cyclic loading spectrum.
Model #1: A constant stress amplitude of cyclic loading with a constant cycle number.
Model #1 is the simplest cyclic loading spectrum for fatigue design. For example, the component
under design is subjected to a constant cyclic stress amplitude
a
D 15 (ksi) with a given cycle
number n
L
D 5 10
4
(cycles).
6 1. INTRODUCTION AND CYCLIC LOADING SPECTRUM
Model #2: A constant stress amplitude of cyclic loading with a distributed cycle number.
Model #2 is a typical cyclic loading spectrum for a component with a single steady function,
that is, a constant cyclic stress amplitude. However, the cycle number n
L
of the cyclic loading is
treated as a random variable and is described by a probabilistic distribution function. How can
this be? It is because, in the reliability-based mechanical design, the components under design
are a batch of “identical” components in service. Each component during its service life had one
value of the number of cycles. All of those can be used to determine the distribution function
of the cycle number. For example, the component under design is subjected to a constant stress
amplitude of a fully reversed cyclic stress
a
D 15 (ksi) with a normally distributed cycle number
n
L
, which has a mean
n
L
D 3 10
5
(cycles), and a standard deviation
n
L
D 3500 (cycles), that
is, n
L
D N.3 10
5
; 3500/. Here, the expression X D N.
x
;
x
/ means that the random variable
X is a normal distribution with a mean
x
; and a standard deviation
x
.
Model #3: A given fatigue life (cycle number) with a distributed amplitude of a cyclic loading.
Model #3 is a typical cyclic loading spectrum for the component with specified service life,
but the fully reversed stress amplitude
a
varies and can be treated as a random variable, and
is described by a probabilistic distribution function. For example, the component under design
with a cycle number n
L
D 8 10
4
(cycles) is subjected to a fully reversed cyclic stress. e stress
amplitude
a
of this cyclic loading follows the uniform distribution between 25 (ksi) and 35 (ksi).
Model #4: Multiple constant amplitudes of cyclic loadings with multiple constant cycle num-
bers.
Typically, model #4 could be used to describe the cyclic loading spectrum of a machine with
several distinguished functions or actions. For example, the cyclic loading spectrum of the com-
ponent under design is:
Cyclic stress level 1 W
a1
D 20 .ksi/; n
L1
D 260;000 .cycles/
Cyclic stress level 2 W
a2
D 30 .ksi/; n
L2
D 50;000 .cycles/:
Model #5: Multiple constant stress amplitudes of cyclic loadings with multiple distributed cycle
numbers.
Model #5 is a common cyclic loading spectrum and can be used to describe many loading condi-
tions for machines with several distinguished functions. For example, the cyclic loading spectrum
of the component under design is:
Cyclic stress level 1 W
a1
D 20 .ksi/I n
L1
D N.260;000; 10;000/
Cyclic stress level 2 W
a2
D 25 .ksi/I ln.n
L2
/ D N.8:425; 0:136/:
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