Environmental Engineering Reference
In-Depth Information
means for the failure envelope is that at the points of the uniaxial compressive strength, both
the value on the hydrostatic axis and the distance
r
increase over the course of the fatigue
loading (see Figure 4.33). If these changes are known for significant loading conditions and
an increasing number of fatigue cycles, then the modified formof the failure envelope can
be described mechanically. It therefore becomes possible to calculate the fatigue strengths
for other loading relationships depending on the numbers of fatigue cycles.
Introducing damage variables
The failure model of [41], introduced in Section 3.6.2, is used for the mechanical
description of the multi-axial concrete strength. The model describes the shape of the
failure envelope with the help of five parameters. Knowledge about the changes to the
five parameters depending on the numbers of fatigue cycles enables the multi-axial
fatigue strength to be determined by way of the changes to the failure envelope.
In order to describe the changes to the principal meridian for fatigue loads, the damage
variable
k
fat
t
for the tension meridians. These damage variables enable the decrease in strength under
fatigue loads to be described with the damage model of [41]. For simplicity we shall
assume that the fatigue behaviour along the principal meridians can be determined
approximately by one damage variable for each case.
fat
c
is introduced for the compression meridians and the damage variable
k
Boundary conditions for principal meridians subjected to fatigue loads
The curvatures of the principal meridians are described by Equation 4.18 for the tension
meridian and Equation 4.19 for the compression meridian. Furthermore, compliance
with convexity conditions according to [41] is essential. The resulting boundary
conditions for formulating the principal meridians for fatigue loads are described in
Section 3.6.2 and listed in Table 4.10 together with the damage variables introduced.
Parameters for principal meridian equations
By including the parameters for fatigue loads, the parabolic equations for the principal
meridians according to [41] (see Section 3.6.2) can continue to be used. These parameters
can be determined by taking into account the boundary conditions given in Table 4.10.
The parameters of the parabolic equation for the tension meridian are in accordance
with Equation 4.18:
r
2
15
2
2
p a
c2
a
1
4
3
k
fat
t
fat
t
(4.18)
a
0
¼
a
c2
a
2
þ
k
a
c
;
2
r
6
15
a
2
þ
a
ct
;
1
a
c
;
2
1
p
k
fat
t
a
1
¼
2
a
c2
a
ct
;
1
2
a
c2
þ a
ct
;
1
r
6
15
r
6
5
a
c
1
p
d
Z
a
ct
;
1
a
c
;
2
a
ct
;
1
a
c2
þ
2
a
c2
þ a
ct
;
1
a
2
¼
a
2
p a
c2
a
c
þ
1
p a
ct
;
1
a
c
2
3
a
ct
;
1
a
c2
fat
t
2
a
c2
þ a
ct
;
1
c
k
p
a
1
4
a
0
a
2
a
0
¼
a
1
where
2
a
2
k
fat
c
