Nonlinear Finite Element Analysis of Concrete Structures

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of the resulting predictions with the experimental data of
Kupfer (1973) has previously been demonstrated, cf. fig. 2.2-6.
In this figure, the value D = 0 was used instead of D = 0.6,
but this affects the post-failure behaviour, only. In general,
the weaker the concrete the more ductile is its post-failure
behaviour, cf. for instance, Hognestad et al. (1955). This
suggests the use of D = 0.6 instead of D = 0 as is apparent
from fig. 4, where the normalized stress-strain curves using
these two D-values are shown. The predicted failure load using
the above concrete parameters underestimates the actual failure
load by only 3%, and is plotted in fig. 8. Therefore, the calculations
are in agreement with the experimental evidence showing
that within the considered variation of the a -values, a
linear relation exists between pull-out force and compressive
strength.
It is remarkable that the prolongation of the experimental line
in fig. 8 intersects the ordinate axis at some distance from
the origin. However, two aspects of concrete behaviour are
dependent on compressive strength namely the ductility and the
ratio of tensile strength to compressive strength. As has
already been touched upon, the post-failure behaviour is more
ductile the weaker the concrete. To investigate the influence
of minor variations in the post-failure behaviour of the concrete,
a calculation was performed using again the concrete having a
strength of 18.7 MPa, but now having lesser ductility. Therefore
the value D = 0 was used instead of the more realistic one D =
0.6, cf. fig. 4. This in fact decreases the predicted failure
load by 5% as shown in fig. 8. That the failure load depends
on the particular softening behaviour of the concrete is indeed
to be expected considering previous remarks in relation to fig.
7. However, comparison in fig. 4 of the concrete having a =
18.7 MPa and D = 0 with the concrete having a = 31.8 MPa and
D = 0.2 shows an almost similar normalized behaviour. Moreover,
the a /a -ratios are identical for these concretes. Using
dimensional analysis, the failure loads should therefore be
almost proportional to the a -value and this is in fact also
observed for the two predicted failure loads, cf. fig. 8.