Nonlinear Finite Element Analysis of Concrete Structures

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within the element are still given by eqs. (4.2-3) and (4.2-4).
Therefore, the condition e = 0 can be obtained simply by replacing
all elements in the third row of the B-matrix given by
eq. (4.2-7) with zeros. Then correct strain values follow and
as the constitutive matrix 5 given by eq. (4-2-10^ also applies
for plane strain the correct stiffness matrix is obtained directly.
Correct expressions for strains, stresses and nodal
forces hold even when temperature loading is present. Therefore,
the plane strain concrete element is formulated completely identical
to the axisymmetric element just by modifying the B-matrix
as stated above.
Turning to the plane stress concrete element located with its
plane in the RZ-plane we have a a = 0 according to the plane
u
stress assumption. As before, the displacements within the element
are given by eqs. (4.2-3) and (4.2-4). Using the standard
transformation formula, cf. for instance Timoshenko and Goodier
(1951) p. 34, and replacing E with E(l+2v)/(l+v) 2 and v with
v/(l+v) then the D-matrix for plane strain transforms to the D-
matrix for plane stress except for the third rov» and column that
correspond to c. and e , respectively. If the B-matrix is modified
as for plane strain then e„ = 0 follows, but it is easily
w
shown that the true plane stress stiffness matrix is obtained
and if the third row of the initial strain vector e given by eq.
(4.2-11) is set to zero then correct nodal forces due to temperature
loading also result. Except for ø fi and e fi the true stresses
and strains are obtained as well and finally a« is therefore
simply set to zero while e fl is made directly equal to - v(e +e )
/(1-v) + (l+v)a AT/(l-v). When cracking is involved, and obviously
no radial cracks can be present, no temperature loading
is considered and e fi is then made equal to - v e where c is
the strain parallel to the crack direction. This is just to say
that e„ is independent of the strain normal to the crack plane
and that isotropic properties exist along the crack plane. A
similar expression was suggested by Phillips and Zienkiewicz
(1976)- Summarizing, the plane stress concrete element is obtained
from the axisymmetric element by modifying the B-matrix
as for plane strain. Moreover, the above-mentioned transformations
for E and v are applied and the initial strain vector due