Nonlinear Finite Element Analysis of Concrete Structures

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From this equation and the relation
3 = 6 I (4.2-18)
we derive the transformation formula
6 = T T B' f (4.2-19)
Suitable expressions for the D'-matrix will now be investigated
for different types of cracks. Before cracking isotropic behaviour
exists determined by the two parameters E and v. After
cracking a stratified material results where it is reasonable to
assume that a plane of isotropy parallel to the crack plana is
present. Following Lekhnitskii (1963) such a material is termed
transverse-isotropic and it is characterized by five parameters.
It is obviously not trivial how to determine these parameters
knowing in advance the isotropic parameters E and v alone, and
various procedures may be found in the literature.
Consider first a circumferentia 1
crack where the crack plane
forms a rotational symmetric surface, compare fig. 2. Just before
cracking we have an isotropic material, i.e., the D'-matrix in
eq. (17) relating the stress vector o'
and the strain vector e'
is given by eq. (10). Moreover, the principal stresses in the
RZ-plane are assumed to fellow the directions of the R'-and Z'-
axis. The a R ,-stress is then a principal stress and it is
assumed that it is the largest principal stress. As the crack
plane is assumed to be normal to this stress it follows the
direction of the Z'-axis as shown in fig. 3. After cracking, the
inability of the material to carry tensile load in the R'-direction
is obtained by replacing all coefficients in the corresponding
row of the 6*-matrix with zeros. As the constitutive matrix
is symmetric the corresponding column consists also of zeros. In
the plane of isotropy now created, it is assumed that plane
stress conditions exists. Before cracking the shear stiffness in
the RZ-plane along the direction of the crack plane is given by
the shear modulus G = E/2(l+v). After cracking, it is assumed
*
that only the shear stiffness nG, where 0 < n < 1 applies, is