Nonlinear Finite Element Analysis of Concrete Structures

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the discontinuity of the displacement field can be node lied.
These approaches to cracking are often termed discrete crack
modelling. The first method was proposed by Nilson (1968) as an
extension of the model of Ngo and Scordelis (1967) who considered
only predefined cracks. Obviously this first method places severe
restrictions on the possible crack directions and it is almost
abandoned today. The second method is physically attractive, but
like the first method it implies considerable computational
effort as a complete redefinition of the structure is necessary.
However, very recent progress in the latter approach to cracking
has been given by Grootenboer (1979).
Apart from the discontinuity in the displacement field another
crucial feature of cracking is that the material loses its ability
to carry tensile load normal to the crack plane. This very
important aspect may easily be incorporated in the finite element
formulation as it can be accomplished simply by appropriately
changing the constitutive matrix D when determining stresses from
strains and when evaluating the element stiffness matrix compare
eqs. (9) and (12). This procedure was proposed by Zienkiewics and
Cheung (1967) and Rashid (1968) and constitutes the most often
applied consideration to cracking.
In the present report we also adopt this cracking model that is
often termed the smeared or continuous cracking approach as the
discontinuity in the displacement field is ignored while the
inability of concrete to carry tensile load normal to a crack
plane is considered by changing the B-matrix from an expression
corresponding to isotropic material behaviour to ar appropriate
anisotropic formulation. Moreover, it is assumed that when a
crack forms in an element it intersects the complete element.
Following Mohraz and Schnobrich (1970) we consider the strain
state at an arbitrary point. The strain vector £ referred to
the original RZ-coordinate system is related to the strain vector
e ' referred to the rotated R'Z'-coordinate system, cf. fig.
3, through
£' = 1 c (4.2-14)