Continuous Beams - A Structural Analysis

[DangleSnipe267]

I recently ran a series of simple structural analysis for girder in the following center post bent arrangement. 



 

I have arbitrarily utilized a uniform floor load of 1000 lb/ft in each case so as to exaggerate the results for clarity. I would like to note that this is not a finite element model and as a result, assumes that both the posts and braces provide the exact same level of support. The stiffness of each element has not been properly accounted for. 

1. SIMPLE BEAM

First, lets look at a simple beam. A simple beam spans from the outside post to the center post (ignoring any knee braces) and is generally the de-facto standard for sizing a beam. In this case we are looking at a girder that spans 14-feet.





As expected, we find that the maximum bending stress occurs at the center of the span and that the maximum shear occurs at the supports

2. "Simple Beam" with support at Braces

For this case, I ran the same 14-foot long beam but assumed that it was actually also supported by the knee braces. In a way, this creates a 3-span continuous beam.



 

What is interesting here is that now the maximum moment and maximum shear have occurred over the top of the braces. Having said that, the overall bending moment has significantly reduced from 25.1 to only 8.9 kip-ft. The shear and moment at the support post becomes almost negligible. This is a good example of why it is conservative to size for a simple beam. 

3. Continuous Beam
Prior to continuing, it is important to note that the continuous beam model is representative of a one singular member that spans the entire bent. 

Now we will look at a continuous beam that is only supported at the posts. You will notice that this is a similar diagram to what you may find floating around the site.



 

As expected, we notice that the inflection point for the moment occurs at approximately the 1/4 point from the center support. This means that the bending stress is near zero at this location. It is important to note that we have completely ignored all of the braces in the system at this point though. Overall, the maximum moment remains the same but its position has shifted to the center support post.

4. Continuous Beam with support at Braces

Now things get interesting, I have modeled the continuous beam to be appropriately supported by each brace within the system.



 

We now observe that the maximum moment and shear has shifted to the brace supports. Again, the actually applied moment is significantly reduced compared to that of a typical continuous beam supported only at the post. We see a reduction in bending from 25.1 to 4.8 kip-ft. This indicated that if a scarf truly acts as a continuous beam, it is actually seeing the highest stresses in the bent. However, having said that, it is seeing seriously reduced stress that it is capable of handling in my opinion.

5. Continuous Beam with support at only 1 Brace

The last model I ran was a continuous beam supported at each post and at only 1 brace. I did this because generally when positioning a scarf, it is over a brace despite every other brace in the structure being ignored.



 

Now we see that the maximum moment has shifted back to the post but we also find that the the moment over the brace is actually relatively high. A little less than 1/3 the total moment observed from the simple beam model.

Closing

I am not suggesting that the frame acts one way or the other but was simply looking to illustrate a couple of different ways that a bent might be analyzed. Draw from this what you will but there are several factors that need to be considered. I am happy to further discuss these results and my interpretation of them with anyone.

-Tim

[Don P]

Thanks for that Tim.
There is a simplified diagram that pops into my head to when the not so "continuous" beam discussion comes up.



 
On the left, 2 simple spans of 14', 1000 plf beam load
Max bending moment will be at midspan, WL/8... 14000lbs x 14/ 8=24,500 ft-lbs

On the right sketch. The connection is a pin, free to rotate. There is an identical uniform load. But because the right beam is being supported on the overhanging end of the left beam, there is a point load on the end of the overhang. 3' overhang, 5.5' (5500lbs) of the right beam loading the overhang end. This is a simple cantilever equation Pa, 5500 lbs x 3'= 16500 ft lbs over the center post.
Add to that the moment over the post from the uniform load wa²/2 ...1000plf x 3'²/2= 4500 ft-lbs

Add em up 16500 ft-lbs + 4500 ft-lbs= Max moment over the mid post is 21000 ft-lbs.

Max moment between posts is going to be about the same.

[Don P]

This is very close to a fixed end condition, when you rigidly hold both ends of the beam and don't allow them to rotate it reduces the bending moment in span to 1/3 of a simple beam and at those rigidly held ends it produces a moment about 2/3 of the center moment in a simple beam. The span and load also reduced by 6'.



 
Max moment in span WL/24, 6700lbs x 6.66'/24=1859 ft-lbs
Max moment at ends WL/12 ...3719 ft-lbs

I use the simple beam calc for a bolstered post just showing why I said it is stronger than advertised.