Civil Engineering Reference
In-Depth Information
Figure 3.1
Cracked rectangular sections (singly reinforced)
those between the cracked sections. Therefore, the application of Bernoulli's compatibility
hypothesis to cracked reinforced concrete members should theoretically be based on the
smeared stresses
and
smeared strains
of mild steel bars embedded in concrete. An accurate
relationship between the
smeared stress
and the
smeared strain
is presented in Section 6.1.9
(Chapter 6).
In the linear bending theory of reinforced concrete, however, two assumptions are made
to simplify the bending analysis and design. First, the stress-strain relationship of steel bars
is based on the bare bars, not on the steel bars embedded in concrete. In other words, the
local
stress-strain relationship of steel at the cracks is assumed, rather than the
smeared
stress-strain relationship. Second, the tensile stress of concrete is neglected in the bending
analysis. Since these two assumptions are physically correct at the cracks, the linear bending
theory is accurate in the prediction of yield moment. However, the omission of tensile stress of
concrete will cause the bending rigidity and stiffness to be significantly under-predicted, and
the yield curvature to be significantly over-predicted. For the calculation of bending deflection
in Section 3.1.5, this weakness is remedied by using an 'effective bending rigidity' which has
been adjusted to fit the deflection test data.
3.1.1.2 Moment and Curvature
The characteristic of a bending member is represented by its moment-curvature relationship.
Consequently, the crux of the flexural analysis is how to find the moment and the curvature
using Navier's three principles of stress equilibrium, strain compatibility and stress-strain
relationships of materials. For a bending member with materials that satisfy Hooke's law,
