While teaching my graduate students in the Advanced Concrete Structures class, last year, I asked them, “How many of you know what a moment-curvature curve is, and what it can be used for?” Just one out of forty students raised hand. Considering that these students had already obtained the bachelor’s degree in civil engineering from good universities, and were therefore, qualified professionals, I expected that most of them would be aware of it. I thought maybe they are shy or not comfortable raising hands; I then asked the same question individually to each one of them. This time, just two more seemed to know something about this curve but were unable to describe how can it be generated or used. To further explore, I posed the same question to a group of practicing structural engineers, participating in one of our seminars, and was surprised to find that only very few of them had good knowledge about it. Among those who knew, hardly few were using it in their day-to-day practice of structural design.
This led me to think that the moment-curvature curve of a structural member’s cross-section must to be an intentionally well-kept secret in structural engineering. Considering its significance, I thought someone should “leak” this secret to the structural engineering community and it might as well be me. I am sure many of you, reading this article, will already be familiar with it (but are probably keeping it secret from others), but for the benefit of those who are not, I am going ahead and talk about it.
To begin with, we can say that for any deformable physical object subjected to some applied force actions, the resulting deformational effects can be related to those applied actions through the corresponding action-deformation relationships/curves. These curves provide very useful information about the overall behavior of that object. From structural engineering point-of-view, the entire response of a structure can be described in an integrated manner using these curves. The overall response of a structure is derived from its members depending upon the structural configuration, geometry and member behavior. The response exhibited by an individual member is derived from the cross-sectional behavior, which ultimately is dependent on its constituent material behavior. Therefore, the action-deformation curves can be obtained at several levels, for example,
Structural Level: Load – Deflection Curve
Member Level: Moment – Rotation Curve
Cross-section Level: Moment – Curvature Curve
Material Level: Stress – Strain Curve
The load-deformation curves can also be plotted between the axial load and axial shortening of a member, the shear force and shear deformation, and the torsion and corresponding twist angle.
However, the moment-curvature curve is probably the most interesting, important and useful action-deformation curve especially for the design of axial-flexural members such as beams, columns and shear walls. And as I have realized, it is probably also the least-understood or least-utilized tool in the normal structural design practice. Many of the text books, design codes and handbooks do not provide sufficient information for the computation and use of these relationships.
The moment-curvature curve of a cross-section is dependent on several parameters including the cross-sectional stiffness (which itself is comprised of material and geometric stifnesses) and the level of axial load on the cross-section. The term “curvature” can be defined in several contexts. In geometry, it is the rate of change of rotation. In structural behavior, the curvature is related to the moment through stiffness. For a cross-section undergoing flexural deformation, it can be computed as the ratio of the extreme fiber strain to the depth of neutral axis and is measured in radians/length units. For the reinforced concrete members, the direct solution to determine the moment-curvature curve is not possible because the determination of neutral axis depth for a given extreme fiber strain and for a given set of axial load is an iterative process. Those who are interested in the detailed discussion about the complete process of generating the moment curvature curve, can refer to the book “Structural Cross Sections: Analysis and Design”.
Let’s talk about the application of this curve. The most interesting information contained in these curves is the effective flexural stiffness of the cross-section for any given moment, or for any given curvature. It can be determined as the slope of moment-curvature curve at the corresponding given point. The curvature of a cross-section at a given moment can be converted to other deformation-related responses (e.g. strains, rotations and deflections) at any point in the member. The strain value (corresponding to a certain moment) at the bottom of a reinforced concrete beam can be used to determine the crack width for an assumed crack spacing or pattern. The moment-curvature curves are useful in determining the ductility of the cross-section (defined as the ratio of the curvature at any given point to the curvature at the yield of first rebar). In fact, several ductility ratios can be computed for various required or specified performance levels during the nonlinear analysis of structures.
And if that is not enough, the information provided by the moment-curvature curve is also very useful for the non-linear analysis of a structure including the post-elastic behavior. Being an important component of a nonlinear structural model, these relationships are also the basis for the capacity-based or the performance-based design methods. They are useful in determining the rotational capacity of plastic hinges assigned in inelastic models for the static pushover analysis and the detailed nonlinear time history analysis of structures.
So, if I may conclude, the moment-curvature curve is a valuable tool which can provide an insight of structural response at an individual cross-section level. It can help in quickly understanding the nonlinear behavior of a member or any assembly of members. The information provided by this curve should be considered by the practicing structural engineers for an effective design structures, not only for the earthquake effects and extreme events, but also for the normal loads. This should not be a secret, rather disclosed and discussed extensively for the benefit of structural engineering and for designing the structures with better performance.
The secret is out! Let’s make use of it.
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