Beam deflections due to moments created by casters

[kjoiner]

Hello,

It's been a while since I've worked through beam calculations that are more than the standard conditions so please go easy.

I have a shelf with 4 leveling feet and a 500 pound load sitting on top of two rails that have 2 swiveling casters each. For simplicity I'm looking at one side of the assembly so each leveling foot has 125 pounds and sits on top of one rail with the two swiveling casters. I do have a cabinet frame made from 8020 and its total weight is 125lb so there is a load of 31.25lb at each end of each caster rail but I'll focus on just the shelf load for now and assume the rail is simply supported.

I've run through the reactions and shear moment diagrams, but in determining the deflection of the beam I'm running into some issues. The casters create vertical reactions to the loads from the shelf feet, but since the caster wheels are not directly under the plates that fasten them (their axles are 1.75 out from the pivot point) they also create moments and these moments will also vary depending on which way the casters are turned.

In Roarks I see some equations for concentrated intermediate moments (p.106, 6th edition) and then there are the standard ones for point loads. Is there a way to add up the deflections due to the point loads and the moment loads to come up with a max deflection value?

Thanks,

Kyle

 

[Cockroach]

Why not introduce these localized moments at the position of the casters? Would these not be inline with your beam when aligned parallel? That would be the worst case loading, which is what you need to design for.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[kjoiner]

Kenneth,

Thank you for responding. Your suggestion is exactly what I need to do. In the time from my initial post, I revised things and placed the casters directly under the feet of the shelf and this simplifies things somewhat since the loads and reactions are symmetrical along the beam. I also increased the load of the scale to 1000lb so each foot sees 250lb. Finally, to complicate things a litte, I did go back and place 31.25lb at each end of the beam so now my beam has some overhanging loads.

Based on that, my maximum bending moment appears to be 375 in-lb . With that value and the S of the extrusion, the stress calculates to 714 psi which is a very moderate value.

An "eyeball" evaluation of the beam seems to indicate that the defelction will be low, but I would like to be able to calculate it.

The area where I am unclear is in determining the deflection of the beam due to the concentrated moments from the casters is in a few areas:

1. I have two concentrated 500 in-lb moments, both 4" from their respective ends. Can I simply run the calculation for one moment and then double it since the other is the same value and for cases where the casters are opposing each other they will create an overall positive bending moment on the beam and therefore deflect it either up or down.

2. Where the casters are pointing in the same direction, their direction of "rotation" will be the same direction, but one will be trying to deflect the beam down and the other up - creating an "s" curve. The node will be in the middle of the beam where deflection curve crosses the neutral axis. Because the node is in the middle of the length of the beam (14" in my case) , does each concentrated moment effectively "see" a beam that is half as long which then reduces the deflection?

3. Can I take the deflection values that are created by the point loads and then add them to the deflections of the concentrated moment loads? That may require a graphical solution.


If my post is becoming to wordy, I'll gladly post a PDF of a sketch of what I'm trying to do.

Thanks again,

Kyle

 

[msquared48]

Regarding #3, sure - that is the principle of Superposition, and works well. Should be able to solve it via the numbers routs simply without any graphics.

I'll have to see your diagram to answer #1 and #2.

Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask

 

[kjoiner]

Hello,

Thanks again for the responses. I'm linking a PDF of the diagram for reference. I've run the calculations for deflection due to the concentrated moment and came up with .0058 in. Adding the other moment load should give me .0116.

The diagram shows the vertical reaction loads for the casters and also the moments created by the casters.

Thanks,

Kyle

 

[kjoiner]

Hello,

My link did not work the first time. I'll try again.

http://files.engineering.com/getfil...9895ec19fda&file=Caster_rail_load_diagram.pdf

Kyle

 

[desertfox]

Hi kjoiner

If your pdf you have uploaded is correct then you can use Macaulay's method for beam deflection and only need consider half the beam because of symmetry.

try this link:-

http://www.freestudy.co.uk/strength of materials/tutorial 2 1.pdf

I started to work out the problem you posted but couldn't finish it because you haven't provided the "I" of the beam.

desertfox

 

[kjoiner]

Desertfox,

I had found Mcaulay's method in some of the research I was doing so it's good to have confirmation I am using the right procedure.

Sorry about not posting the I values for the beam. It's a piece of 8020 brand 1530 lite extrusion (6061 T5 aluminum). The Ix is .3935 in^4 and the Iy is 1.3847 in^4. In this appilcation I am using the profile in the flat orientation so .3935 will the be used.

Later I'll be running a calculation for the span across the width of the entire frame and for that application the profile will be turned up which will be important since that piece will be 72" long.

Kyle

 

[desertfox]

Hi kjoiner

Thanks for the information, I have completed my calculation based on one half of the beam you posted with you latest information and at the beam centre I obtain a deflection of 0.03771".
Your figure is different but I don't know at which point on the beam you're calculating the deflection for.

desertfox

 

[kjoiner]

Desertfox,

Thanks for calculating the deflection. The main area I am interested in is at the load cell feet. The slope of the deflection is probably most important since I am using load cells and want to avoid a slop that would affect their accuracy. From your calculation, it appears that the slope is very low (.11°) so that should not be an issue.

Kyle

 

[desertfox]

Hi kjoiner

Where did the load cells come from? you never mentioned them before.
I expected my deflection to correspond with your calculation but it doesn't and we need to understand why.
I took the beam as you drew it ie simply supported at the castors and therefore using standard theory there would be no deflection at these points, so any deflection would be generated by the fixed moments applied to the beam, however I don't understand from your description or sketches what actually generates these moments.

desertfox

 

[kjoiner]

Desertfox,

I apologize for confusion. Here is some additional detail.

In the application, there is a platform sitting on load cells. The load cell leveling feet of the load cells sit on the beam shown in the diagram. The total weight of loaded platform can be up to 1,000 lbs so each load cell sees 250lb of the total. The leveling feet are the leveling feet of the load cells which are sitting on the the 28" beam which is supported by the casters. The 492.75 in-lb moments shown in the diagram are due to the 281.25lb reaction of the casters multiplied by their offset of 1.75" (so they will swivel around). The casters create the 281.25 lb reactions but in doing so also induce the concentrated 492.75 in-lb moments at their mounting points.

Kyle

 

[boo1]

How does a caster introduce a moment vs point load?

 

[msquared48]

boo1:

If the position if the axle of the caster wheel is not directly below the geometric center of the four (?) mounting bolts, then a moment is introduced to the connection.

Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask

 

[boo1]

How can a caster point load transfer a moment?

 

[Cockroach]

Simple, offset load pinned at a distance, moment = load X arm length.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada