Environmental Engineering Reference
In-Depth Information
Conditional equations for damage variable
k
fat
c
fa
c
on the compression meridian are described in
Equations 4.20 to 4.25. Based on the S-N curves for uniaxial repeated compressive
loads, we get the following ranges:
When 0
The courses of the damage variable
k
<
S
cd
;
min
<
0
:
8, then
logN
1
¼ð
12
þ
16
S
cd
;
min
þ
8
S
cd
;
min
Þð
1
k
fat
c
Þ
(4.20)
log
N
2
¼ 0
:
2 log
N
1
ðlog
N
1
1Þ
(4.21)
log
N
3
¼ log
N
2
ð0
:
3 3
S
cd;min
=
8Þ=D
(4.22)
S
fat
Specifying the number of fatigue cycles log N associated with
k
c
:
If logN
1
6
then
logN
¼
logN
1
;
(4.23)
If log
N
1
>
6 and
D
S
0
:
3 3
S
cd;min
=
8
;
then
(4.24)
log
N
¼ log
N
2
If log
N
1
>
6 and
D
<
0
:
3 3
S
cd;min
=
8
;
then
S
(4.25)
log
N
¼ log
N
3
s
cd;min
f
c
ðj; r; uÞ
where
S
cd;min
¼
(4.26)
fa
c
S
cd;min
s
cd;min
¼ minimum stress
f
c
ðj; r; uÞ¼multi-axial concrete strength according to ½93;
D
S
¼ k
see also Section 3
:
6
:
2
The damage variable
k
fa
c
can be determined iteratively from Equations 4.20 to 4.25. It
is presumed here that we know the number of load cycles log N for which the damage
variable is to be determined. First of all, the damage parameter for the actual minimum
stress S
cd,min
is estimated from Figure 4.32 depending on the number of load cycles
log N. The recommendation here is to read off the damage variable
k
fa
c
for a marginally
higher number of load cycles. The value read off is entered into Equations 4.20 to 4.25
and the number of fatigue cycles log N thus calculated then checked against the initial
value. If the number of fatigue cycles calculated is too small, the damage variable
fat
c
must be increased, and if the value calculated is too high, the variable must be reduced.
The number of fatigue cycles must be recalculated for the corrected damage variable
and again compared with the initial value. This iterative procedure should be repeated
until the values coincide with sufficient accuracy.
k
fat
t
Conditional equations for damage variable
k
fat
The courses of the damage variable
t
on the tension meridian are described in Equations
4.27 and 4.28. The equations were developed from the S-N curve for a uniaxial repeated
tensile load. The damage variable
k
k
fat
t
can be calculated directly from the equations
depending on the number of load cycles log N and the effective minimum stress S
cd,min
.
