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## Monday, February 12, 2018

Earlier this year, I made a submission on how to calculate column axial loads by considering beam support reactions. Just in case you missed it, you can read it by following the link below. But this post highlights the methods of loading columns in a building in order to obtain the maximum design moment.

How to Calculate Column Axial Load By Considering Beam Support Reactions

Columns in buildings may be axially, uniaxially, or biaxially loaded. An axially loaded column is column that is subjected to axial load only. This type of columns usually exist in centres of building where bending moments from floor beams will normally neutralise each other. When a column is subjected to moment in one plane, then it is uniaxially loaded, and when it is subjected to moment in both planes, it is biaxially loaded.

Consider the general arrangement of a building that is as shown above;

Column A1 is a biaxially loaded column because the load from beam 1:A-B is generating bending moment in the x-direction, while the load from beam A:1-2 is generating moment on the column in the z-direction. This moment is usually in the form of fixed end moment, which is finally distributed to the other members meeting at that node based on their stiffness.

Column B1 is a typical example of a uniaxially loaded column. The bending moments from beams 1:A-B and 1:B-C will typically neutralise each other. While the load from beam B:1-2 will exert bending moment in the z-direction.

A typical example of axially loaded column in the building are columns B2 and C2.

When a design is to be carried out manually, the design engineer has to determine the bending moment on columns as appropriate, alongside the axial loads.

However, while I pointed out above that column B1 is uniaxially loaded due to the beams on gridline 1-1 neutralising each other, have you considered a situation whereby PANEL 1 is fully loaded with stored materials, while PANEL 2 empty? This situation is very possible during the service life of the structure, and as a result the column at that instance will not behave as a uniaxially loaded, but as biaxial. This is a just to give us a little idea of what may influence our loading while carrying out such analysis. Our consideration is usually the worst possible load regime.

Reynolds and Steedman (2005) gives us a good idea of how to approach some aspect of this issue. Loading all the spans at ultimate limit state (say 1.35gk + 1.5qk) will seldom give us the maximum design moment. However, by alternating the loads (say one span 1.35gk + 1.5qk and the other span 1.0gk) will give us a heavily unbalanced moment, which is a possible scenario in practice.

So I am going to give us a run down of the methods that we can use to load structures (sub-frames) in order to determine the maximum bending moment on columns.

(1) Method 1

Let us consider the image above showing the sub-frame of a building with the intent of determining the maximum design moment of the lower column at joint C. Now, what we have to do is to load 1.0gk and (1.35gk + 1.5qk) on the spans adjoining span C such that the unbalanced moment is maximised. In this case, it is preferable to apply 1.35gk + 1.5qk on the longer span which is CD, and 1.0gk on the shorter span BC. While this is more representative, the set back is that you will have to solve 5 x 5 simultaneous equation before arriving at your answer, and hence may not be very handy for simple scientific calculator process.

(2) Method 2

In this case, the sub-frame is simplified as shown above, but it works best if span BC is longer than the adjoining spans. If span BC is shorter than the adjoining span, you can switch to the use of method 3. Here, we apply a load of 1.35gk + 1.5qk on span BC, and 1.0gk on span AB and CD. We only solve 2 x 2 simultaneous equation here, and it is very convenient for classroom and simple design purposes. Realise also that we have to reduce the the stiffness of the adjoining beams by half because we are actually overestimating the actual stiffness of the beam by considering all ends to be fully fixed.

(3) Method 3
In this case, we are going to load (1.35gk + 1.5qk) on the longer span, and 1.0gk on the shorter span such that the unbalanced moment at C is maximum. We are still going to reduce the stiffness of the beam by half for the same reasons stated in method 2 (see the link below for how to calculate the stiffness of beams). The advantage of this method is that we are going to solve for just one unknown (rotation at point C) in order to obtain the design moment in question. This is the most popular approach that is taught in most classrooms and utilised for simple designs or checks. I have written some MATLAB programs for carrying out such analysis.

After loading the, we can now analyse the structure completely by using the stiffness method (my recommendation). An example of this has been done in the next post, and you can take a look by clicking the link below;