Nonlinear Finite Element Analysis of Concrete Structures

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If displacements are prescribed in other directions than the R-
or Z-axis, i.e., if skewed kinematic constraints are present
then eq. (1) has to be transformed to the R'Z*-coordinate shown
in fig. 1. After that a modification corresponding to eq. (2)
is performed and a retransformation back to the original RZ-
Z
V
_R
Fig. 4.6-1: Skewed geometrical constraint
coordinate system is then carried out. The result of these trivial
matrix multiplications may be found in appendix B.
Having then introduced the prescribed displacements for a fixed
stiffness matrix K and a fixed force vector F standard routines
are available for solution of the equation system (1). In the
present case, the equation solver is termed BANDSYMEQ and is
available at Risø's computer, Sørensen (1968). As the name indicates,
this solver takes advantage of the symmetry and banded
structure of the stiffness matrix. A direct solution is applied
that uses the square-root method, i.e., a Cholesky decomposition
of the stiffness matrix into triangular matrices. Special care
is taken to minimize rounding-off errors.
Different strategies exist for determining the structural response
when material behaviour becomes nonlinear. In the present
case as a nonlinear elastic model is utilized for the concrete
and as a secantial formulation has been employed so that dilatation
and softening of concrete can be considered, the equation
system (1) is set up and solved when the force vector F includes
the total loadings applied to the structure. This means that a