Nonlinear Finite Element Analysis of Concrete Structures

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affecting the descending curve in the post-failure region. Eq.
(3) is a four-parameter expression determined by the parameters
a , e , E., and D, and it infers that the initial slope is E.,
and that there is a zero slope at failure, where (a,e) = (- c ,
-e ) satisfies the equation. The parameter D determines the postfailure
behaviour, and even though there are some indications of
this behaviour, e.g. Karsan and Jirsa U969),the precise form of
this part of the curve is unknown and is in fact, not obtained
by a standard uniaxial compressive test. Therefore, the actual
value of D is simply chosen so that a convenient post-failure
curve results. However, there are certain limitations to D, if
eq. (3) is to reflect: (1) an increasing function without inflexion
points before failure; (2) a decreasing function with at
most one inflexion point after failure; (3) a residual strength
equal to zero after sufficiently large strain. To achieve these
features A > 4/3 must hold, and the parameter D is subject to
the following restrictions
(1-35A) 2 < D _< l+A(A-2) when A < 2;
0 _< D 4/3 is in practice not a restriction, and, in
fact, eq. (3) provides a very flexible procedure to simulate the
uniaxial stress-strain curve. For instance, the proposal of Saenz
(1964) follows when D = 1, the Hognestad parabola (1951) follows
when A = 2 and D = 0, and the suggestion of Desayi and Krishnan
(1964) follows when A = 2 and D = 1. In addition, different postfailure
behaviours can be simulated by means of the parameter D
and this affects only the behaviour before failure insignificantly.
This is shown in fig. 3, where A = 2 is assumed and where
the limits of n are given by zero and unity.
Using simple algebra, eq. (3) can be solved to obtain the actual
secant value E
s
of Young's modulus. The expression for E
s
con-
tains the actual stress in terms of the ratio - 0/0 . For unic
axial compressive loading £ = - a/a
for E
holds, and the expression
can therefore be generalized to triaxial compressive load-