Environmental Engineering Reference
In-Depth Information
l
c3
(N, r)
modification factor for compression meridian stresss (Section
4.9.5.2)
l
c2
(N,
a
)
modification factor for biaxial action effects (Section 4.9.5.3)
N
actual number of fatigue cycles
r
¼
(
s
11
¼s
22
)/
s
33
loading relationship on compression meridian
a¼s
11
/
s
22
loading relationship for biaxial loading
All other designations are as described in Section 4.9.2.
The stresses present in the principal direction of the loading are related to the uniaxial
compressive strength. The influence of the multi-axial loading state is taken into
account solely by the modification factors. The introduction of the modification factors
means it is now possible to avoid a comparatively more involved, direct calculation of
the multi-axial concrete strength.
4.9.5.2 Derivation of modi
cation factor
l
c3
(N, r) for fatigue loads on
compression meridian
Comparison of concrete strengths for uniaxial and multi-axial loads
Assuming that fatigue loads on the compression meridian enable the qualitative S-N
curves to be used for uniaxial repeated compressive loads allows the modification
factor
l
c3
(N, r) to be determined directly from a comparison of the concrete strengths
for uniaxial concrete strength f
c1
and multi-axial concrete strength f
c
(
j
,
r
,
u
). This is
shown for the effective maximum stress S
c33,max
and the number of fatigue cycles
N
¼
1; it is also equally valid for higher numbers of fatigue cycles.
Accordingly, the initial equation for the values of the effective maximum stress in the
axial direction S
c33,max
on the compression meridian is
fat
fat
S
c33
;
max
¼
s
c33
;
max
ð
r
¼
0
Þ
f
c1
¼
s
c33
;
max
ð
r
6¼
0
Þ
f
c
ðj; r; uÞ
(4.30)
We can use Equation 4.30 to obtain
f
c
ðj; r; uÞ
f
c1
fat
c33
max
ð
r
6¼
0
Þ¼s
fat
s
max
ð
r
¼
0
Þ
(4.31)
;
c33
;
The modification factor
l
c3
is defined as
f
c1
f
c
ðj; r; uÞ
l
c3
¼
(4.32)
The modification factor
l
c3
therefore depends on the triaxial loading state
j
,
r
and
u
,
which in the case of compression meridian stresss can be expressed by the loading
ratio r
¼
(
s
11
¼s
22
)/
s
33
. The stresses
s
11
and
s
22
describe the radial stress,
s
33
describes the axial stress. In the end, substituting Equation 4.32 in Equation 4.31
leads to
fat
fat
s
c33
;
max
ð
r
¼
0
Þ¼l
c3
ð
r
Þs
c33
;
max
ð
r
Þ
(4.33)
