Civil Engineering Reference
In-Depth Information
Figure 2.4
Moment-curvature diagram for reinforced concrete beam with tensile reinforcing only.
To further illustrate the three stages of beam behavior which have just been de-
scribed, a moment-curvature diagram is shown in Figure 2.4.
1
is de-
fined as the angle change of the beam section over a certain length and is computed by the
following expression in which
For this diagram,
is the strain in a beam fiber at some distance
y
from the
neutral axis of the beam:
y
The first stage of the diagram is for small moments less than the cracking moment
M
cr
where the entire beam cross section is available to resist bending. In this range the
strains are small, and the diagram is nearly vertical and very close to a straight line.
When the moment is increased beyond the cracking moment, the slope of the
curve will decrease a little because the beam is not quite as stiff as it was in the initial
stage before the concrete cracked. The diagram will follow almost a straight line from
M
cr
to the point where the reinforcing is stressed to its yield point. Until the steel
yields, a fairly large additional load is required to appreciably increase the beam's
deflection.
After the steel yields, the beam has very little additional moment capacity, and only a
small additional load is required to substantially increase rotations as well as deflections.
The slope of the diagram is now very flat.
1
MacGregor, J. G., 1997,
Reinforced Concrete Mechanics and Design
, 3rd ed. (Upper Saddle River, NJ:
Prentice-Hall), p. 90.
