Single/Double Curvature Bending of Columns

So let’s say you have a structural analysis program and you forgot how to figure whether the bending moment diagram produces a single or double curvature. What now?

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It is important to know whether a column bends in a single or double curvature depending on the end moments and end forces. Both ACI and AISC have similar provisions regarding curvature to compute the magnification factor d for end moments for both sway and non-sway conditions.

Let’s say you have to investigate manually whether you can do anything to make that particular column pass. By experience, ETABS (I use ETABS mostly. I don’t know about other software) has some fat stored in its design (i.e. slenderness factor k equals 1.0 when you can reduce it depending on the end condition of the column, Cm is automatically 1.0, and others).

But before that let’s have a brief code recap and brief background on the application of moment curvatures.

ACI 318-11 section states that for nonsway moment magnification,

 

Cm
Cm computation. Screen grab from ACI 318M-08

 

And in turn, the term Cm will be used to compute d which is the required magnification factor.

delta
Sway magnification computation. Screen grab from ACI318M-08

 

Now, is it single or double curvature?

So, let’s have some mental math without putting in some digits in the equations. If the column is bent in a single curvature, the ratio of M1 over M2 will be added to 0.60. Now in equation (10-12), the bigger the value of Cm, and the greater the magnification factor d becomes. So how does d and in turn the design moment be equal to one (remember, d should always be ≥ 1.0 as the code stipulates!)? The column should be in double curvature where the ratio of M1 over M2 will be deducted from 0.60

But then again, you can’t always force a column to bend in double curvature unless you’re at liberty to alter the framing system which is usually you don’t.

So let’s say you have a structural analysis program and you forgot how to figure whether the bending moment diagram produces a single or double curvature. What now? Simple:

Recall that in theory of structures, the forces, deflected shape, column shear, and bending moments should look like this:

single double curvature
Single or double curvature based on loadings and bending moment diagrams. Drawn by yours truly.

So for example in ETABS, you will know that the column is bent in single curvature if the bending moment diagram only have one color and double curvature if the bending moment diagram has two colors.

With that known, you can now go figure whether that column do pass or whether you have to do something else to make it work.

Author: The Romantic Alpha

Aside from physical and intellectual strength, a man is also given a heart to feel and share love and value. Dive into the mind of the alpha, feel his heart beat, and let me carry you with my strong yet gentle hands. I'm continuously innovating myself to be a better man, husband, father, friend, and structural engineer. Love's not overrated. The world needs the true essence of which.

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