Environmental Engineering Reference
In-Depth Information
concentrations of force transfers. Such situations are also found in towers and become
significant when fatigue limit states can occur, as in the structural members of wind
turbines (see Section 4.9). Three-dimensional mechanical models for concrete will
therefore be looked at below; they are dealt with in more detail in [37,8]. Such models
can provide a realistic description of the non-linear material behaviour, the crack
formation, or rather the progressive damage, and the potential failure conditions, and
can be fed into a numerical analysis with FEM programs.
The literature contains many uniaxial models which can satisfactorily describe non-
linear material properties such as strain hardening, softening or crack formation.
However, the three-dimensional models available are still unsatisfactory and thus
require further development. Their parameters should be chosen in such a way that they
can be calibrated with the help of tests.
3.6.1 Failure envelopes and stress invariants
The failure envelopes are frequently described geometrically by the stress invariants I
1
,
J
2
and J
3
, see [38], for instance. Here, I
1
represents the hydrostatic stress state, whereas
J
2
and J
3
are expressed by the components of the stress deviator. Formulating with the
help of the Haigh-Westergaard coordinates
j
,
r
and
u
is expedient. Figure 3.13 shows
the intersection curves between the failure envelopes and a principal meridian plane
and the deviatoric cross-sections.
The intersection between the failure envelopes and the stress plane
s
11
s
22
(biaxial
stress) results in a closed curve which can be compared with the results of tests
according to Kupfer [39].
Fig. 3.13 Failure envelopes illustrated by means of Haigh-Westergaard coordinates
