Civil Engineering Reference
In-Depth Information
n
i
Δ
S
i
FIGURE 5.35
Frequency distribution histogram of stress ranges.
principal importance. The Palmgren-Miner linear damage accumulation rule is
n
i
N
i
=
1.0,
(5.47)
where
n
i
is the number of cycles at stress range level,
S
i
, developed from an appro-
priate cycle counting method. There are also many techniques for counting the cycles
of variable amplitude load or stress traces. However, for many structures, the rainflow
method appears to provide the best results (Dowling, 1999). Instead of counting only
the tensile portion of stress range cycles, AREMA (2008) recommends counting all
live load stress range cycles as a complete tensile cycle (even those with a compres-
sive component due to stress reversal). This is appropriate because near flaws and
details the member will be subjected to a fully effective tensile stress range cycle
due to superposition of tensile residual stress. This is analogous to raising the mean
stress such that the entire stress range cycle is in tension. A frequency distribution
histogram for the numbers of stress range cycles,
n
i
, can be developed from rainflow
cycle counting, as shown schematically in
Figure 5.35.
N
i
is the number of cycles to failure at stress range level,
Δ
Δ
S
i
. A log-log straight
line relationship exists between
N
i
and
S
i
(Basquin, 1910). This relationship is also
observed in constant amplitude fatigue testing of members and details (Kulak and
Smith, 1995).
Crack growth behavior, as defined by the Paris-Erdogan power law,
∗
can be used to
establisharelationshipbetweenstressrangeandnumberofcyclestofailure.Thecrack
growth rate is
Δ
d
a
d
N
=
K
m
,
C
Δ
(5.48)
where
a
is the crack length,
N
is the total number of constant amplitude stress range
cycles causing the same fatigue damage as the total number of variable amplitude
∗
This is a log-log linear relationship. At crack growth rates below log-log linear behavior, a threshold
exists below which cracks will not propagate. At crack growth rates above log-log linear behavior,
fracture occurs at critical stress intensity equal to the fracture toughness of steel (Barsom and Rolfe,
1987; Anderson, 2005).
