Environmental Engineering Reference
In-Depth Information
Table 4.10 Boundary conditions for principal meridian equations
Boundary condition
j=
f
c
;
1
r=
f
c
;
1
u
INT
r
ð
j=
f
c1
; u
INT
Þ
q
k
fat
t
1
fat
t
fat
t
a
ct
;
1
0
s
11
¼ k
f
ct
;
1
k
a
ct
;
1
3
p
1
fat
t
r
1
k
a
ct
;
1
p
q
k
fat
t
2
fat
t
fat
t
0
s
11
¼ s
22
¼k
f
c
;
2
k
a
c
;
2
a
c
;
2
3
p
2
fat
t
r
1
k
a
c
;
2
p
fat
t
fat
t
0
—
k
a
c
k
d
Z
fat
t
r
1
k
a
c
q
k
fat
c
1
fat
c
60
s
11
¼k
f
c
;
1
k
3
p
fat
c
1
fat
c
r
2
k
p
—
fat
c
fat
c
60
fat
c
k
a
c
k
d
D
r
2
k
a
c
—
fat
c
60
fat
c
k
a
0
0
r
2
k
a
0
The parameters of the parabolic equation for the compression meridian are in
accordance with Equation 4.19:
2
(4.19)
fat
c
fat
c
b
0
¼k
a
0
b
1
k
a
0
b
2
r
2
15
1
1
fat
c
b
1
¼ k
p a
0
b
2
1
p
þ a
0
r
2
15
1
1
3
p
a
0
þ
p
d
D
ð
a
0
þ a
c
Þ
b
2
¼
1
3
1
3
ð
a
0
þ a
c
Þ a
c
p
a
0
þ
p
k
fat
c
The damage variables introduced now have to be determined depending on the numbers
of fatigue cycles in order to describe the failure envelope of the concrete when
subjected to fatigue loads.
fat
t
The damage variables are derived from the qualitative S-N curves for uniaxial and
multi-axial fatigue loads because these describe the decrease in the concrete strength as
the number of fatigue cycles increases for various loading situations.
Determining the damage variable
k
fat
4.9.4.2 Derivation of damage variables k
c
and k
fa
c
, which is intended to describe the changes to the
compression meridian for fatigue loads, requires the S-N curve for a uniaxial repeated
compressive load. On the other hand, the description of the damage parameter
k
fa
t
for
the tension meridian is based on the strength development for a biaxial repeated
compressive load.
The equations for the S-N curves given in Model Code 90 are used for the mathematical
description of both damage parameters (see Section 4.9.2.2). Whereas Model Code 90
