Civil Engineering Reference
In-Depth Information
8110 1985. However, it has been pointed out that the plastic range of the
curve should be given a slight inclination to facilitate computation
(Kotsovos, 1984) (
Figure 9.10
)
.
9.13
Cracking in concrete and yielding in steel
Concrete has a very limited tensile strength. If the principal tensile stress in
the concrete exceeds its tensile capacity, cracks will develop in the direction
perpendicular to the appropriate principal stress. The tensile stress that
fictitiously existed just prior to cracking has to be removed and transferred
to other parts of the structure. This is done by working out the equivalent
nodal forces in the element due to the excess stress and treating them as
additional external loads in the next cycle of iteration in non-linear analysis
(Zienkiewicz, 1971; Valliappan and Doolan, 1972).
(9.47)
Similarly, if the stress value exceeds the yield strength of the material, the
excess stress is also to be removed and transferred in the same manner. In
this case only the portion of the stress exceeding the yield stress is removed;
whereas in the case of cracking, all of the normal stress perpendicular to the
crack becomes excess stress.
9.14
Stiffness of cracked element
Once yielding or cracking has started to form and develop in an element, the
material elasticity matrix will be different and its stiffness will decrease. The
element which has cracked should have its stiffness reduced before going
further with the analysis.
For a quadrilateral element, direct integration cannot be performed on the
cracked element because the stiffness function is no longer continuous. Only
approximate integrations are possible with cracked elements.
In the smeared cracking model, it is assumed that the concrete
becomes anisotropic with one of the material axis
x
Ó being oriented along
the direction of the crack. The modulus of elasticity along the direction
perpendicular to the crack will be reduced to zero whereas the modulus
of elasticity along the direction of the crack may remain to take its
appropriate value under uncracked condition. Hence, the elasticity
matrix [
D
xÓ
] should be modified to
