Environmental Engineering Reference
In-Depth Information
log N
¼
12 just satisfies the convexity condition of the failure model used, which is
described in [41].
Furthermore, it is clear in Figure 4.35 that the presence of even just a small transverse
tensile action effect leads to a significant decrease in strength with a fatigue load.
Failure curves for other minimum stresses are shown in Figures 4.36 to 4.39.
4.9.5 Design proposal for multi-axial fatigue
The use of the extended, energy-based damage model described in [74] is recom-
mended for more accurate fatigue investigations involving multi-stage and multi-axial
loads. The non-linear damage process can be ascertained very well with this damage
model.
However, its use presumes the iterative calculation of the energy component dissipated
in damage in the fatigue process from the monotonic curve. And that requires the use of
a computer program.
In addition, the material parameters, for example the volume-specific crushing energy,
must be known.
Where the influence of multi-axial fatigue loads is to be estimated with the help of the
linear accumulation hypothesis, it is sufficient to determine the numbers of fatigue
cycles to failure according to Section 4.9.4.2 and evaluate the individual damage
components according to the Palmgren-Miner hypothesis (see Section 4.9.2.2).
4.9.5.1 Procedure for designing on the basis of the linear
accumulation hypothesis
If as an approximation the S-N curves for uniaxial repeated compressive loads are
presumed for fatigue design in the principal direction of the loading, then the
numbers of fatigue cycles to failure can be calculated with Equations 4.7 to 4.13
given in Section 4.9.2.2. However, in doing so, the stresses present must be related to
the multi-axial concrete strength. This has to be calculated in each case for the actual
loading relationship and can, for example, be determined according to [41].
The alternative for practical applications is to express the multi-axial strength in the
form of a modified uniaxial concrete compressive strength. To do this, modification
factors
l
c2
and
l
c3
are calculated in Sections 4.9.5.2 and 4.9.5.3 respectively, and
presented in the form of charts. The modification factors are incorporated directly in the
equations for evaluating the S-N curves. The numbers of fatigue cycles to failure can
then be determined with the initial values according to Equation 4.29:
S
cd
;
min
¼ g
sd
l
c2orc3
ð
N
; a
or r
Þs
c
;
min
h
c
=
f
cd
;
fat
(4.29)
S
cd
;
max
¼ g
sd
l
c2orc3
ð
N
; a
or r
Þs
c
;
max
h
c
=
f
cd
;
fat
where
S
cd,min
;S
cd,max
minimum and maximum repeated compressive loads in the prin-
cipal direction of the loading related to the uniaxial compressive
strength
