Environmental Engineering Reference
In-Depth Information
N
f
I
1
n
i
m
=
(12-9b)
where
i
= index of layers in the spectrum
I
= number of layers in one spectrum
m
= number of repetitions of the spectrum required to cause fatigue failure
n
i
= number of cycles applied at stress level
S
i
N
i
= fatigue lifetime at a constant stress level
S
i
(cycles)
S
i
= stress parameter in the cycle upon which fatigue damage is primarily de-
pendent (
e.g.
, maximum stress, cyclic stress,
etc.
) (kN/m
2
)
N
f
= fatigue lifetime under spectrum loading (cycles)
Inherent in Equation (12-12) is the assumption that the order of applying the layers of stress
to the material is not important. While this may not be true for a single spectrum with only a
few layers, it has been found to be acceptable for design when high- and low-stress layers are
randomly mixed and a large number of spectra is applied.
The dependence of
N
on repeated cycles at a constant level
S
, illustrated by the
S-N curve
in Figure 12-2, is typically expressed by a power-law equation [Basquin 1910, Manson 1965
and 1966], as
S
=
S
1
N
a
(12-10a)
(12-10b)
S
³
S
e
where
S
1
= empirical stress coefficient (kN/m
2
)
a = empirical exponent
S
e
=
endurance limit
, below which no fatigue damage occurs (kN/m
2
)
On log-log coordinates Equation (12-10) is plotted as a straight line with an intercept of
S
1
at
N
equal to one cycle and a slope of
α
, down to a stress of
S
e
. Typically, a is less than about
- 1/8. Solving Equation (12-10a) for
N
with
S
=
S
e
, we obtain the fatigue life at the so-called
“knee” of the S-N curve. To the right of this point the curve becomes a horizontal line.
The next step in defining an allowable stress for the spectrum is to express the layer
stresses as fractions of the largest stress in the spectrum,
S
maxmax
. This produces a set of nor-
malized stresses analogous to the normalized loads in Table 12-1. In this way
S
maxmax
can be
used as a scale factor to proportionately increase or decrease all stress levels in the spectrum
until the required fatigue lifetime is achieved. Let
S
i
=
s
i
S
maxmax
(12-11)
where
s
i
= stress ratio for the
i
th layer
S
maxmax
= largest stress in the spectrum (kN/m
2
)
Combining Equations (12-9), (12-10), and (12-11) produces the following equation for cal-
culating allowable stresses by the S-N linear damage method:
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