Nonlinear Finite Element Analysis of Concrete Structures

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to that of nonlinear elasticity and the model indeed corresponds
to the bei aviour of a fracturing solid, Dougill (1976). Moreover,
the described model is able to represent in a simple way most
of the characteristics of concrete behaviour, even for general
stress states. These features include: (1) the effect of all three
stress invariants; (2) consideration of dilatation; (3) the obtaining
of completely smooth stress-strain curves; (4) prediction
of realistic failure stresses; (5) simulation of different
post-fajlure behaviours and (6) the model applies to all stress
states including those where tensile stresses occur. In addition,
the model is simple to use, and calibration to a particular type
of concrete requires only experimental data obtained by standard
uniaxial tests. The construction of the model can conveniently
be divided into four steps: (1) failure and cracking criteria;
(2) nonlinearity index; (3) change of the secant value of Young's
modulus and (4) change of the secant value of Poisson's ratio.
The failure and cracking criteria utilized in this section are
the ones proposed by the writer and dealt with in sections 2.1.3
and 2.1.4. In the finite element program the modified Coulomb
criterion described previously is also used together with the
following stresr-strain model and it should be emphasized that
any failure criterion can be employed in connection with the described
constitutive model, and no change as such is necessary
because the criterion in question i s involved only througn the
determination of the nonlinearity index, as defined.
2.2.1. Nonlinearity index
Let us now define a convenient ir ;asure for the given loading in
relation to the failure surface. First of all, we have to determine
to which failure state the actual stress state should relate.
Although there is an infinite number of possibilities,
four essentially different types can be identified. To achieve
a simple illustration, we will at present adopt as failure criterion
the Mohr criterion shown in fig. 1 a), which also shows
the actual stress state given by a, and o~. Failure can be obtained
by increasing the a,-value as shown by circle I, or alternatively
by fixing the (o, + o,)/2-value as shown by circle II.
However, tensile stresses may then be involved in the failure