Nonlinear Finite Element Analysis of Concrete Structures

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E =
8E WXT E_ E„
MN A M
E»F„ + B«E MXT (E„ - Ej
A M f MN V M
(2.2-6)
in which E
, depending on B, is the secant value along the original
post-failure curve MN obtained by means of eq. (4), using
the negative sign. Likewise, the constants E, and E,„ are secant
A M
values at failure also determined by eq. (4) using the positive
and negative sign, respectively, and the nonlinearity index value
at failure, i.e. B = B f - The preceding moduli are shown in fig.
4. Eq. (6) implies a gradual change of the post-failure behaviour,
both when the stress state is changed towards purely compressive
states, or towards stress states where cracking occurs.
1.0
Pi
0.6
P
0.2
" A/
— Æ ^
-/E 4
//
.•••.
• — •
'•.
1 'V
M
-*.
/ :
VE M '••
/
• •
••.
'"-•,
> •
"...N
—
-
-
Fig. 2.2-4: Post-failure behaviour for intermediate stress states
that do not result in cracking or compressive crushing
of concrete.
2.2.3. Change of the secant value of Poisson's ratio
Let us now turn to the determination of the secant v.lue u of
s
Poisson's ratio. Both for uniaxial and triaxial compressive loading
we note that the volumetric behaviour is a compaction followed
by a dilatation. The expression of M
for uniaxial compressive
loading is therefore generalized to triaxial compressive
loading by use of the nonlinearity index B. Hereby we obtain
u = u. when B < B_
S I —a
u s -
u f - (u f - v A - (i^) : (2.2-7)
when B > B
" a,
in which u ± = the initial Poisson ratio; and u f » the secant
value of Poisson's ratio at failure. Eq. (7) is shown in fig. 5