By Dr. Naveed Anwar
I wrote this article on moment-curvature curve as being a well kept secret in structural engineering, a few years ago and gauging from the interest it had raised, I thought of updating it and complementing it with a video to demonstrate some of the ideas discussed there.
You can find the original article here.
One of the interesting developments since the publications of the previous article is the generation of the “curvature capacity surface” (shown in the background image above), which shows the variation of the maximum curvature capacity of a cross-section along various bending directions and its relationship with axial load. This surface is not widely available or discussed, but can provide a good understanding of the ductility of the cross sections sections, subjected to axial load and bi-axial bending, such as columns and shear walls. It also shows how rapidly a beam behavior changes to column behavior as the axial load crosses about 10% of the capacity
Another valuable insight that the moment curvature curve can help to provide is the understanding of various types of stiffness (initial, tangent and secant) and its variation with moment and curvature, which can reduce rapidly (even becoming negative!) after reaching the yielding capacity. This effective stiffness can then be used for computing rotations and deflections, and for nonlinear modeling and analysis.
In addition to looking at the relationship of curvature to moment, the same procedure and process can also be used to plot the variation of maximum and minimum strain, axial force in compression and tension, corresponding stresses and neutral axis depth with change in curvature. Such plots can provide significant insights into the cross-section response and help improve the design and evaluation of reinforced concrete members.
With increased focus on displacement based and performance based design and evaluation of structures, the role of moment curvature curve and its applications are becoming even more important. So, let us keep exploring this versatile tool, and keep finding more treasures hidden beneath it!