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Modeling of reinforced concrete (RC) structures is a problem having
complex character. It involves both modeling of concrete and reinforcing
steel and also the interaction of these materials. As it is widely known,
concrete belongs among so called quasi-brittle materials.
Its behaviour is strongly influenced by a stress state to which it is
exposed. The behaviour of concrete varies from brittle to very ductile
according to the lateral confinement which can be provided either by
outside constraints or by transversal reinforcement, which is the common
case. All types of transversal reinforcement such as different ties,
stirrups or steel sections can be used to provide this lateral confinement.
The change of concrete behaviour according to lateral confinement yields
the need of triaxial modeling of concrete. Without respecting the triaxial
behaviour of concrete one cannot describe the reality of RC structures.
There is also one very important feature of concrete. General loading
cases lead to the phenomenon called softening. Softening of concrete
is characterized by a progressive loss of material integrity, which
yields the descending load-deflection diagram.
Modeling of reinforcing steel must include elastoplastic behaviour of the
material. Longitudinal reinforcement is usually placed near the surface
of concrete and fixed laterally by transversal reinforcement. If the RC
structure is subjected to the compressive load the longitudinal reinforcement
can buckle. Thus, the model of longitudinal reinforcement must include also
the possibility of buckling of steel.
What are the material models capable to describe triaxial nonlinear
behaviour of concrete? Well, there is not much choices. Classical models
are based mainly on the theory of plasticity. Theory of plasticity was firstly
developed for modeling of metals, but it was also enlarged for modeling of
concrete (see e.g. [1,2,3]). Theory of plasticity provides nonlinear description of concrete
including loading, unloading and path dependence. The key point of these
models is the definition of yield condition which is usually formulated
in the stress space. This formulation can be very complicated and moreover
multidimensional formulation can be hardly imagined and physically interpreted.
Another type of models are based on continuum damage mechanics. These models
are able to describe material stiffness degradation according to the certain
damage parameter, which can be defined as a single scalar parameter or a tensor
of higher orders. A popular damage model was proposed by Mazars in [4]. Some
models combine the attitude of plasticity and damage mechanics (see [5]).
All these types of models suffer from some kind of insufficiency. One of the
major errors can be caused by not respecting of anisotropy development
within the material microstructure. Deficiency of these models is that they
are usually derived in principal strain space. They do not respect the rotation
of principal axes during loading process that can lead to big errors. Especially
description of softening is very sensitive to it, because damage development
changes the material from isotropic to highly anisotropic. The right way to
solve this problem is to link damage with its orientation in the material
and compute the material response directly for this concrete orientation.
This is done so in concept of microplane model [6,7,8]). Final
response is then given by combination of the responses from different orientations.
The classical approach to the constitutive modeling is based on a direct relationship
between strain and stress tensors and their invariants. In contrary to it, constitutive
relations of microplane model are formulated in terms of strain and stress components
on planes of arbitrary spatial orientations, so called microplanes. This attitude
excels in conceptual simplicity and allows straightforward modeling of anisotropy
and other processes connected with planes with different orientations. The penalty
to be paid is a great increase in computational effort. The relationship between micro
and macro level is obtained by projecting strain tensor to the particular microplanes
(so called kinematic constraint) or by projecting stress tensor (static constraint).
Then constitutive relations between microstrains and corresponding microstresses are
evaluated. The missing link (between microstresses and macrostress for kinematic
constraint and between micro and macro strain for static constraint, respectively)
is obtained by application of principle of virtual work. That kind of material
model is capable to describe triaxial nonlinear behaviour of concrete including
tensional and compressive softening, damage of the material, different types of
loading, unloading or cyclic loading.

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*Daniel Rypl *

2005-12-03