## Continuous Beams With Cantilevers by Macaulay’s Method

I have updated the Macaulay spreadsheet (described here) to allow the analysis of continuous beams with cantilevers at one or both ends.  The updated spreadsheet (including full open-source code) can be downloaded from Macaulay.zip.

My first attempt was to write a Cantilever user defined function (UDF), then apply the end moment and reaction from this function to a continuous beam with simple supports at the beam ends.  I ended up with a different approach, but the Cantilever UDF was completed, and is included in the download spreadsheet.

Cantilever Function Output

It turned out to be simpler to just amend the code to allow the input of the position of the end supports, with some fairly minor adjustments to the Macaulay analysis code.  Input and results for a three span beam with cantilevers both ends are shown in the screenshots below (click on any image for full size view).  As before, the results have been compared with an analysis in Strand7, and show near exact agreement.  See earlier posts in this series for more details of the background to Macauley’s method, and details of how to use the functions.

Conbeam Input with Cantilever Ends

Conbeam output compared with Strand7 results

Conbeam and Strand7 Results Compared

This entry was posted in Beam Bending, Excel, Frame Analysis, Newton, UDFs, VBA and tagged , , , , , . Bookmark the permalink.

### 12 Responses to Continuous Beams With Cantilevers by Macaulay’s Method

1. George says:

Hi there,
The formulas are very impressive, and allow the user to input a variety of loads.
Is it possible to choose the support you want (fixed or not) in the ends of the continuous beam?

Thank you

George

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• dougaj4 says:

Geotge – You can have supports at the ends, but in the current version all supports are simple. The next version will allow moment restraint and translational or rotational springs at any support.

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2. metroxx says:

Waiting for new version.

Thank you,
Andrej

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3. John M Bowers PE says:

got the impression that there were only very few uniform loads and concentrated loads that could be input. Real-life designs could have at least 4 different uniform loads per span, and at least 3 different concentrated loads per span, but not necessarily so many on every span.

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• dougaj4 says:

John – you can have up to a total of 1,048,576 distributed loads, and the same number of point loads! The examples are set up to allow over 30 separate loads, which should be more than enough for most purposes, but if you need more, just add more data to the bottom of the list, and adjust the Dloads and/or Ploads ranges in the function arguments. You can also have the output wherever you like, with as many rows as you need.

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• dougaj4 says:

for the latest version, which is now called ConBeamU, and is units aware, and will analyse moving groups of point loads.

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4. John M Bowers PE says:

Doug –
I understand now. BTW, I looked at my old college textbooks (ca. 1973), and I could find no mention of Macaulay.

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• dougaj4 says:

I hadn’t heard of it until quite recently either. The Colin Caprani link is a good source for background information:

Click to access Macaulay%27s%20Method%201011.pdf

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• John M Bowers PE says:

Doug,
It will take me a while to comprehend all that because I am not academic-oriented, having been design-oriented for so many decades.
I do find Caprani’s approach (with Macaulay) to have a significant advantage over the way I was taught, because he presents a method to analyze any continuous beam, with or without cantilever at either/both ends, and any complex loading, in a logical sequence of calculate reactions, calculate moment diagram (no magic there), then calculate exact deflections. In other words, all the information we need to design any continuous beam design, from start to finish.

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5. Kyle says:

Doug, could you please explain a bit more on how you dealt with cantilever ends? What did you end up doing instead of analyzing a cantilever beam and putting the moment and reactions on a simply supported one?

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• dougaj4 says:

The Macaulay function treats applied loads and reaction loads in exactly the same way, so the procedure is:
1) Find the shear force and moment at the support end of the cantilever(s).
2) Find the reaction loads on all the supports by analysing a continuous beam without cantilevers, but with the cantilever actions applied to the beam ends.
3) Analyse the entire beam (including cantilevers) applying both the applied loads and the support reaction loads.

I am currently in the process of re-writing all these functions so I can use a common set of routines for both the continuous beam and frame analysis programs. It’s taking longer than I had hoped, but I should be publishing something quite soon.

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