Not sure what beam deflection formula to use...

[NCCaryME]

I've been through my Machinery's and Marks', but can't find a beam bending diagram for a beam that is loaded at 2 points, one on the cantilever end and one between two other supports, sort of like this:

W W
----------------------------------
R R

Where "W" are the loads, and "R" are the supports. I realize that I may not have the most complete set of beam diagrams with these two books. How would you go about determining the maximum deflection in this beam?

 

[csd72]

I know what you mean, this is a common situation but this exact formula is hard to find.

2 ways you can do this:

1. 3 separate calculations, 1 for the cantilever, 1 for the u.d.l on the span, and one for the span with a moment on the end then add the relevant deflection/rotations up to get the total.

2. formulas are

cantilever deflection = (wa/(24EI))x(3a^3 +4a^2Xl - l^3)
span deflection = (wl^2/(384EI))x(5l^2-12a^2)
where a = cantileveer (an not more than l/2),l = span.

Be aware that this is for uniform load on the whole beam only and if you have only partial load then the deflection will be more.

 

[UcfSE]

Find a diagram for each load location, one at a time. If you are remaining leanearly elastic, you can superimpose the response.

If it's structurally determinant, I would instead find the reactions and draw the shear force (V) and bending moment (M) diagrams. Bending moment at a given location is the area under the shear force diagram up to that point (M=[∫]V and V=[∫][loading diagram].

 

[oneintheeye]

sometimes when formula are not easily found or complex you can convert your bending moment obtained back into an equivilent UDL and apply the standard UDL deflection formula. Will be conservative answer.

 

[Tomtation]

Do an internet search for beam2D software (Orand Systems). This software comes with a 30 day free trial and is an excellent program. It will solve for any number of supports and loads.