Understanding Maximum Moment for Moving Loads: Strength of Materials Reviewer

[J000e]

Hey guys, I've been reading my strength of materials textbook and its pretty much like a reviewer that contain several problems and short descriptions of their concepts.

There's this formula that computes the maximum moment for two moving loads and its:

[PL-(Psmall)(d)]^2/4PL
where P= Psmall + Pbig, L= length of beam, d=distance of two loads, Psmall= small load, Pbig= larger load

Can anybody tell me how was this derived?

 

[pistolpete]

is that equation for a specific type of supported beam? Supported at both ends, cantilever,ect?

 

[JBA]

It would be best if you could upload a copy of the entire example with any included diagram of the beam with loads.

 

[J000e]

here is the figure

 

[Nidum]

Question could have more than one interpretation but if we take the simplest which is that the two wheels just apply simple static point loads to the beam then problem can be solved by writing down a general equation for the bending moment which takes into account the variable positions of the loads and then finding the maximum value .

Before doing any actual analysis just think about the problem - what does your intuition tell you about roughly where the wheels need to be located ?

 

[J000e]

Im pretty sure moments with the highest value are located when the resultant force is near the midspan of the beam.
So the position of the wheels would look like this where
Pb=Bigger load
Ps=Smaller load
R=Resultant Force

 

[JBA]

Actually the highest moment will be at largest loading point, second highest at the lower load point and decrease at a linear rate to the lower load point. The maximum deflection point will be at some point between the two load points.

For verification of this see any technical reference giving the analysis and the moment diagram for "a simply supported beam with two equal loads". For that case, the maximum bending moment is equal at both load points and remains the same value between those two points.

 

[J000e]

I just saw a pdf file that has simplified everything. It's just a matter of analyzation in order to come up with the needed values.
I've been trying to find a textbook that will explain this subject before, its a shame that google and the right keywords are the only things required to have an understanding.
Anyway, thank you so much!