Nonlinear Finite Element Analysis of Concrete Structures

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where the B-matrix is given by eq. (7) and evaluated at the centra
of the triangle while D and é are given by eqs. (10) and
troid
(11).
4.2.2. Cracking in the concrete element
Suppose now that tensile cracks according to the clacking criteria
of section 2.1.4. initiate within the element. The present
section deals with the corresponding modifications in the finite
element approach of the concrete triangular axisymmetric element.
Due to rotational symmetry only two types of cracks can exist,
namely radial cracks where the crack plane follows a radial
plane and circumferential cracks where the crack plane forms a
rotational symmetric surface. These two types of cracks are il-
Circumferential cracks
Fig. 4.2-2: Type of cracks in an axisymmetric structure.
lustrated in fig. 2. In addition, combinations of these cracks
are possible namely: a radial crack together with a circumferential
crack, two circumferential cracks with different directions
of the crack planes and finally these last named two circumferential
cracks together with a radial crack.
When a crack forms then in principle a discontinuous displacement
field results. However, this can be represented only in the
finite element approach either by forcing the cracks to follow
the boundary of the elements and then introducing new nodal
points along these boundaries so that separation can occur, or
by allowing the cracks to propagate through the elements and then
define new elements and nodal points so that representation of