Nonlinear Finite Element Analysis of Concrete Structures

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ment and for unembedded reinforcement:
a
•i
ii
a
*- e ^ _ E
a) usual reinforcement b) unembedded reinforcement
Fig. 3-1: Uniaxial stress-strain curves for usual reinforcement
and for unembedded reinforcement.
I.e. a trilinear stress-strain curve applies to usual reinforcement
while a quadrilinear stress-strain curve applies to unembedded
reinforcement. The slope of the lines is arbitrary except
that it is non-negative. In the present section emphasis is given
not to the constitutive modelling as such, as it is quite trivial,
but rather to a formulation that is computationally convenient
in the finite element program.
As outlined in section 4.3 usual reinforcement may consist of
bars or of membranes while unembedded reinforcement, i.e., springs,
obviously are treated as uniaxial loading. In accordance with
the formulation of the constitutive equations for concrete in
terms of secant values of Young's modulus and Poisson's ratio,
the constitutive equations for usual reinforcement are based on
a total formulation instead of the generally more accurate incremental
formulation. For usual bars carrying stresses in one
direction only the two formulations coincide when loading alone
is considered, but for membrane reinforcement differences exist.
The incorrect response to unloading inherent in the total formulation
employed is considered to be of minor importance as only
structures subject to increasing external loading are dealt with.
For unembedded reinforcement, i.e., springs, a different approach
is followed that considers both loading and unloading in a correct
way. This numerical approach is outlined in section 4.4 and
no more attention will be given here to unembedded reinforcement