Environmental Engineering Reference
In-Depth Information
We can see from Equation 4.33 that the maximum axial stress for compression
meridian stresss can be simplified to the uniaxial compression load by the modification
factor l c3 (r). This means that only the axial stresses (maximum and minimum) plus a
modification factor l c3 (r) dependent on the loading ratio r have to be known when
calculating the number of fatigue cycles to failure for triaxial compression meridian
stresss. And this means that within the scope of a fatigue analysis based on the linear
accumulation hypothesis, Equation 4.29 can be used to determine the triaxial fatigue
strength directly from the uniaxial S-N curves for repeated compressive loads
according to Section 4.9.4.2.
Determining the modification factor
l c3 (N, r) from the failure hypothesis
according to [41]
The calculation of the modification factors l c3 (N, r) is shown below. The yield
condition according to [41] gives us (see Section 3.6.2)
p
5
f c1 ¼
r ðj; u INT Þ
(4.34)
Here, the failure curve on the compression meridian ( u INT ¼ 60 ) is described by a
parabola. Substituting the parabolic equation in Equation 4.34 results in
2
p f c1 ¼ b 0 þ b 1 f c1 þ b 2
1
f c1
(4.35)
In the case of compression meridian stresss, Equation 4.36 enables the effective
deviatoric stress
/f c1 described by Haigh-Westergaard coordinates to be expressed by
the effective principal stress
r
fat
s
c33 ; max (r)/f c1 and r:
r
2
3
s fat
c33 ; max ð r Þ
f c1
f c1 ¼
ð 1 r Þ
(4.36)
Equating Equations 4.35 and 4.36 leads to the following quadratic equation:
r
2
15
b 1
p ð 2 r þ 1 Þþ 3
ð 1 r Þ
3 b 0
b 2 ð 2 r þ 1 Þ
0 ¼
2 þ
2
b 2
ð
2 r þ 1
Þ
(4.37)
! 2
s fat
fat
c33 ; max ð r Þ
f c1
s
c33 ; max ð r Þ
f c1
þ
The values specified in, for example, [74] enable the parameters of the parabolic
equation b o ,b 1 and b 2 to be determined for N ¼ 1 and subsequently the calculation of
the stress
fat
c33
max (r) from Equation 4.37 depending on r.
Equation 4.33 should be used to determine the parameters for the parabolic equations
for numbers of fatigue cycles N
s
;
>
1 and to solve Equation 4.37. The modification factor
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