Beam that is simply supported in the middle

[vettett15]

Hey Guys,

I am working on a situation where I have a beam that is simply supported in the middle, and the two ends of the beam are .005" higher than the middle where the support sits. I am trying to figure out with a linearly increasing load what the force is to make the two ends ,which are sitting .005" higher, the same heigh as the middle. Basically I want to flatten the piece.

EI = 731 lbf*in^2
length = 3.522in
width = .404in

So this is what I have:

I broke it down into two beams that are cantilevered

So for each cantilever:

load = (displacement * 81 * EI ) / (7*(Length/2)^4) Length/2 because I broken the simply supported beam into 2 cantilevered beams

load = 4.399 lbf/in

pressure = load/width of beam = 10.9 psi

My problem is assuming the above is correct, why does it not work out the same when I break that linearly increasing load into the resultant force:

resultant force for one of the cantilevered = .5 * L/2 * load = 3.872lbf

So since we need 2 of those forces, one on each end 2/3rds of the way up the linearly increasing triangle, that would equal 7.744 lbf

My problem is now if I say that force (7.744 lbf) / (L*width) = 5.442 psi

5.442 psi doesn't equal the 10.9 psi?

I know it is different but I did the same thing using a uniform pressure, where I broke the uniform pressure into two resultant forces and then divided those forces by the area and got the same pressure as the uniform pressure I had originally calculated.

Any help would be appreciated, thanks.

 

[dirk_mec1]

Are both ends inclined? (Engineers love drawings so make one)

 

[vettett15]

Yes both ends are inclined. They are 0.005" higher than where the support it

 

[FredGarvin]

You are confusing me with what you term a "linear increasing load." It looks like in one case you mean a simple distributed load and the other you are doing a distributed load that is a function of the distance down the length of the beam, i.e. a triangular shaped distributed load. I understand what you are doing and as a rough first guess I would have done what you did. What I don't understand is the second part you are comparing to.

 

[vettett15]

Yeah by linearly increasing load I mean a triangular distributed load that is increasing the further away from the support you get.

So in the first part I have the load required for the triangular distributed load

and then I found the resultant force of that triangular distributed load by using the formula
.5*L*load. Load being in the form of (lbf/in)

When I compare the pressure from the triangular distributed load and the pressure from the resultant forces, I get two different pressures.

 

[mathmate]

If you have triangular loads which increase towards the free ends on both sides of the middle support, then you have the equivalent of a cantilever with half the length.
(Ref: http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_bc_cantilever.cfm)
Let the Triangular load be 0 at the support, and p at the far end, then it is equivalent to adding a uniform load of p to the cantilever, and subtracting the inverse triangular load, which is p at the support and 0 at the far end.
The deflections at the "free" end of the cantilever for the first and second cases are
w1=pL⁴/8EI, and
w2=pL⁴/30EI
Subtracting
w1-w2=pl⁴/EI*(1/8-1/30)=11pL⁴/120EI
now
L=1.761
EI=731
w1-w2=0.005
we have
11*p*L⁴/(120*EI)=0.005
Solving for p,
p=160039/38600=4.1461
pressure
=4.1461/.404
=10.263 psi (at the far end where the load intensity is maximum)
Very close to what you had at the beginning.