Deflection of beam with multiple cross sectional areas 3

[bbarter]

Hi,

How would one go about calculating the deflection of a beam with multiple I-values.

The beam is simply supported, and symmetrical. It is made up of 3 sections, with the two ends near the supports being I1 and the middle section being I2. There are two equal point loads being applied to the beam at the points where the beams cross-section changes from I1 to I2, and I am trying to calculate the deflection at the points where the load is applied.

I have tried calculating the deflection of equivalent beams with cross sections I1 and I2 at these points and summing the deflection together but FEA analysis has given completely different results, meaning this method is completly wrong and I need to somehow calculate the beam as one.

Any help would be much apprciated.

Cheers

 

[FlashSet]

Look at page two for an example that you could follow:

https://engineering.purdue.edu/~ce474/Docs/Beam_Examples01.pdf

 

[Qshake]

If you have FEA software, then you should discretize the beam into 3 sections. There will be 4 joints total, 2 end supports and two interior joints which should reflect the continuity of the conditions, that is that all releases are non-zero. As such the beam will be continuous from one support to the other.

Since this is a simple problem you can check the behavior of this non-prismatic beam by inputing the same value for I1 and I2 and checking against well known formulas for deflection or moment or shear. Then you can change I2 incrementally noting how it changes the results until you have I2 at the value you want.

Regards,
Qshake

Eng-Tips Forums:Real Solutions for Real Problems Really Quick.

 

[IDS]

The spreadsheet linked here:

http://newtonexcelbach.wordpress.com/2013/11/22/unit-aware-continuous-beam-spreadsheet-update/

Will calculate shears, bending moments, rotations and deflections for simply supported spans or continuous beams with any number of changes of section.

Since this is a symmetrical problem you could also analyse it as a cantilever fixed at mid span with a fairly simple calculation.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[GregLocock]

First you draw your shear force diagram
then you draw the bending moment diagram
then you use M/I=E/R to get the radius of curvature at each point

then you integrate the curvature once for gradient, and again for deflection.

This is what we used to call engineering.



Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[CELinOttawa]

So mean Greg! Correct, accurate, and even reasonably helpful.... But MEAN! lol.

 

[dik]

We used to use numeric integration for that type of stuff... don't remember how we used to do it... almost like 2 cycle moment distribution... also forgotten...

Dik

 

[JStephen]

You should be able to calculate deflections in each section using standard deflection formulas for simple beams and then combine them. Don't forget to include rotation of the outer beams due to deflection in the middle beams. This may or may not be simpler than integrating from the start.

 

[CELinOttawa]

JStephen: That approach is quite non-conservative... Or do you mean something more complicated than a superposition solution?

 

[CELinOttawa]

Dang it: CONSERVATIVE. Very conservative to add all deflections...

 

[rb1957]

I think JS's approach is very conservative since the beam has varying cross-sections ... i don't think you can use typical deflection curves (based on constant cross-sections); certainly you can't substitute the local I and hope for an accurate number. of course you could always use the minimum section, and if the results are ok then you're ok; possibly some sort of average section.

Quando Omni Flunkus Moritati

 

[JStephen]

No, I don't mean superposition. Break the beam up into three sections, find the end moments and end shears of each section, and you can then apply beam deflections formulas from Roark or other sources to each section to find local deflections in those sections. If it's symmetrical and loads are applied at the points where the section changes, it would work out pretty easily, otherwise it could get involved. In either case, you do have to keep in mind the translation and rotation of the whole sections as well as the local deflection.

 

[paddingtongreen]

Greg Locock is almost there. Draw the load diagram, from that draw the shear diagram.
Draw the bending moment from one end by adding the areas under the shear diagram from the point at hand and the support, work your way across (no big deal with this case). in this case, you go from the support to the first load point, then the second.
Now, you load up the beam with the bending moment diagram as though it were a load, now calculate a "Secondary shear" caused by this load. It is convenient to divide the answers by EI for this diagram. this diagram shows the slope of the beam. From this "Secondary shear" diagram, calculate a "secondary moment" diagram. This secondary moment diagram shows the deflection of the beam.

Understand that you keep track of units as you perform this method. It takes less time to do it than for me to write this.

It is strange to say, but this was a method of choice when I started work right up until reliable computer programs appeared on the scene.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin