Nonlinear Finite Element Analysis of Concrete Structures

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the "embedded" concept where reinforcement can be located arbitrarily
within the concrete elements. Assuming consistent displacements
for reinforcement elements and concrete elements the
displacements and thereby the response of the reinforcement can
be described by the nodal displacements of the concrete element.
The advantage is that reinforcement can be located at will but
assumption of perfect bond is inhersnt in the approach. In the
present report this latter consideration for reinforcement is
employed. A quite similar formulation was given by Zienkiewicz
et al. (1972), but the present concept "n its original form and
the corresponding procedures in the finite element program were
given by Tingleff (1969, 1973).
Figure 1 illustrates how the discrete reinforcement bars are
smeared out to equivalent "shells" possessing volumina and stiffness
characteristics identical to those of the bars. This means
that all three elements can be treated in the same way except
for their different stiffness characteristics. In general all
three types of reinforcement will therefore be referred to as
bars.
In addition to the in-plane forces of the bar, transverse forces,
i.e., dowel forces, may develop as a result of cracking. Relative
displacements parallel to a crack plane result mainly in local
bending of the bar as well as local crushing of the concrete in
the vicinity of the bar. Such crushing of the concrete might be
simulated by suitable link elements which, however, is beyond
the possibilities of the present approach. Local bending of a bar
could in principle be considered, but this would require knowledge
of the displacement fields of two subsequent triangular
elements. As a result nodal points are coupled that generally do
not interact. This coupling would in general almost double the
band width of the equation system thereby increasing the computer
time inadmissibly, at least for the equation solver used here.
The only possibility to deal with dowel action, therefore, is to
consider the shear stiffness of the bar. The corresponding shear
forces that might be present in RZ- and membrane reinforcement
are shown in fig. 1. This approach to consider dowel action is
evaluated in sections 5.1 and 5.3 and it is shown there that