Nonlinear Finite Element Analysis of Concrete Structures

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dealt with until now. In principle., such considerations could be
performed using spring elements that could be formulated similarly
to usual reinforcement elements. However, as concrete and
tendon deform independently except at the anchor regions such a
formulation would couple nodal point* that in general are far
from each other. As a result a very large bandwidth of the equation
system would exist making such a formulation prohibitive.
Instead, after specification of the initial prestressing forces,
attention is focussed directly to the additional tendon forces
caused by deformations. In the program the dependence between
these forces and the relative deformations of the ends of the
tendons is specified as the quatrolinear dependence shown in
fig. 3-1 b) where as before consideration has to be taken to the
initial prestressing force. The corresponding nodal forces then
depend on the unknown displacements and this infers that an iterative
process is necessary even when material behaviour is linear.
As we are dealing here mainly with short-term loadings this
is considered to be only a minor disadvantage as changes in tendon
forces caused by deformations are usually of interest only
for loadings where material nonlinearities are involved and
where iterations therefore necessarily must be performed. It is
to be noted that unloading is treated correctly.
4.5. Plane stress and strain versus axisymmetric formulation
Until now only the axisymmetric finite elements have been dealt
with. However, the formulation both of plane stress and plane
strain elements follows very much the same lines and they will
therefore be treated only schematically in this section. Naturally
the objective for the derivation of plane elements is to
achieve formulations that are, as far as possible, analogous to
the axisymmetric case so that, except for certain modifications,
identical subroutines can be utilized in the computer program.
Let us first consider the plane strain concrete element and let
the tangential stress and strain correspond to quantities in the
longitudinal direction of the structure, i.e., according to the
plane strain assumption we have e 0 = 0. Now, the displacements