Nonlinear Finite Element Analysis of Concrete Structures

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Finite element formulation using eq. (23) results in a stiffness
m&crix that depends on the unknown displacements and as solution
of the resulting equation system therefore necessarily involves
iterations, the inverse of the coefficient matrix of the equation
system has in principle to be determined in each iteration.
This is very time-consuming so instead the finite element formulation
can use eq. (24) resulting in a constant coefficient matrix
while only the nodal forces due to the initial stresses are
changed in the iteration process. The contribution to the nodal
forces due to the initial stresses is, cf. eqs. (4.1-19) and
(4.1-25)
F a = - | 1 T o Q dV (4.3-25)
° V
This initial stress method was proposed by Zienkiewicz et al.
(1969) and as shown by Zienkiewicz (1977) p. 459 it corresponds
to the modified Newton-Raphson method.
Let us now determine the nodal force contribution due to plastic
deformation of the reinforcement using this initial stress
method. It should be noted that the corresponding implementation
in the finite element program was performed mainly by Herrmann
(1975). Moreover, as the primary reinforcement forces are those
located in the reinforcement plane, and to facilitate the calculations
the influence of shear tresses due to dowel action
on the plastic deformation of the reinforcement is ignored.
Firstly, the forces at the nodes of the reinforcement element
are detsrmined in the local R'Z*-coordinate system, cf. fig. 2.
Secondly, these forces are transformed to the global RZ-coordinate
system and then they are transferred to the nodes of the
involved triangular element. The forces at the nodes of the reinforcement
element are determined by means of eq. (25), i.e.,
f 0 o b * ~ j § b T 5 ob dV * f b T 5 ob 2 * r * dt (4 ' 3 " 26)
bar vol.