Nonlinear Finite Element Analysis of Concrete Structures

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total formulation as opposed to an incremental formulation is
employed. However, for each loading level iterations are carried
out until the constitutive equations for concrete, reinforcement
and prestressing are in accordance with the total loading in
question. This procedure is sketched in fig. 2.
loading
level 3
level 2
level 1
displacement
Fig. 4.6-2: Numerical solution of the nonlinear equation system
All nonlinearities can in principle be treated by the initial
stress method where, as described in section 4.3.2, the stiffness
matrix K is maintained constant while the force vector F is
modified appropriately. This approach enables a quick solution
of the nonlinear equation system (1), but when nonlinear material
behaviour becomes dominant it is known that convergence
proceeds rather slowly. This is illustrated in fig. 3 where
identical plastic strains exist at points A and B and where the
predictions C and D are shown after 4 iterations. Therefore, to
improve convergence an occasional updating of the total stiffness
matrix is appropriate, cf. for instance Phillips and Zienkiewics
(1976). A general acceptable criterion for determining
when updating should occur is apparently not available. However,
as cracking of concrete as opposed to the gradual development
of plastic strains is an irreversible distinct phenomenon, it
seems convenient to update the stiffness matrix every time
cracking occurs. Moreover, plastic strains in concrete as well
as cracking in concrete in general develop simultaneously in
most structures when the loading is increased. Therefore, the
extreme choice to treat all concrete nonlinearities by directly
changing the total stiffness matrix K and to treat all nonlinearities
present in reinforcement and prestressing by appropriate