Nonlinear Finite Element Analysis of Concrete Structures

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1.0
i i 1 r
0.6
0.2
J I I L
0.2 0.4
v s
Fig. 2.2-5: Variation of secant value of Poisson's ratio.
The second of these equations, which represents one-quarter of
an ellipse, is valid only until failure. Very little is known of
the increase of u in the post-failure region, but it is an experimental
fact that dilatation continues here. Now, for a given
change of the secant value E , there corresponds a secant value
u*, so that the corresponding secant bulk modulus is unchanged.
In this report, we decrease the E value by steps of 5% in the
post-failure region, and to ensure dilatation in this region
also we then simply put u = 1.005 u* in each step, although
other values may also be convenient. A similar approach is usea
for the intermediate stress states where tensile stresses are
present but no cracking occurs. In the model, u < 0.5 must always
hold, but this limit is achieved only far inside the postfailure
region. In eq. (7), a fair approximation is obtained
when the following paraireter values are applied for all types of
loading and concrete
0 a = C.8; u f = 0.36 (2.2-8)
As before, the 8 value to be applied in eq. (7) is determined
by eq. (1) when only compressive stresses occur, and by eq. (2)
when tensile stresses are present.
In summary, the constitutive model is based on nonlinear elasticity,
where the secant values of Young's modulus E , and
Poisson's ratio u , are changed appropriately. We select a failure
criterion, and on this basis calculate the nonlinearity index