Environmental Engineering Reference
In-Depth Information
3.6.2 Common failure models for concrete
The following one-parameter models are among those described in the literature [40] as
the simplest approaches:
- the Rankine failure theory,
- the Tresca yield condition, and
- the Von Mises yield condition.
These models can be used to describe brittle (Rankine) or ductile (Tresca, Von Mises)
material behaviour, but not the full complexity of concrete. Therefore, the following
two-parameter models [40] are frequently used for concrete (and related mineral
materials):
- the Mohr-Coulomb yield condition,or
- the Drucker-Prager yield condition, which describes a cone whose axis of rotation is
the
j
axis:
f
j; ðÞ¼
6
p
p
2
a j þ r
k
¼
0
In order to adapt this yield condition to the biaxial failure condition [39], the angle of
the surface of the cone
must be small. Distinguishing between ductile and brittle
concrete failures is then only possible if it is postulated that both the Drucker-Prager
yield condition and the principal stress criterion of Rankine are satisfied - as is
illustrated in Figure 3.14 for biaxial stress.
The three-dimensional edges that result from the intersection between the Drucker-
Prager cone and the three Rankine planes cannot be understood in physical terms and
represent a problem for the numerical analysis. On the other hand, fracture surfaces
evolve continuously when higher-value models are used, for example:
- the Willam-Warnke three-parameter model,or
- the Willam-Warnke five-parameter model [41].
The three-parameter model develops the failure envelope from the Drucker-Prager
cone by introducing the tension and compression meridians with different angles and
generating the intervening cone surfaces by way of elliptical interpolation. The
compression meridians describe stress states
s
11
<s
22
¼s
33
; the tension meridians,
on the other hand, describe stress states
s
11
>s
22
¼s
33
. The three free parameters are
determined from the uniaxial compressive strength f
c,1
, the uniaxial tensile strength f
ct,1
and the biaxial compressive strength f
c,2
[41].
The difference between the five-parameter and the three-parameter models is that the
principal meridians are not assumed to be straight lines, but rather second-order
parabolas:
f
j; r; u
INT
a
r
ð
Þ ¼
p
5
Þ
1
¼
0
f
c
;
1
r
j; u
INT
ð
Compression meridian:
r
1
ðÞ¼
a
0
þ
a
1
þ
a
2
2
j=
f
c
;
1
j=
f
c
;
1
