Deflection cantilever beam

[raymanmusic]

Homework Statement

How do I find an expression for the deflection at midspan for a cantilever beam loaded with a uniformly distributed load?

Illustration of beam: http://goo.gl/3SxeVZ

Homework Equations

$\delta = \frac{Qx^2}{24EI}(6L^2-4Lx+x^2)$

The Attempt at a Solution

Attempt at solution: http://goo.gl/umUBkU

My current answer is: $\frac{49QL^4}{1280EI}$, I think this is wrong. Putting $x = \frac{L}{2}$ in the elastic curve formula I get: $\frac{17QL^4}{384EI}$, I think this is the correct answer.

 

[SteamKing]

You can always check your formula for the deflection by setting x = L and seeing if your result is the same as that from a beam table.

 

[nvn]

raymanmusic: qx = q, not qx = q*x/L. Try again.

 

[raymanmusic]

Yes, that was the mistake. I got the correct answer now: $\frac{17QL^4}{384EI}$. Thank you nvn.

 

[rezeew]

I would first clarify the problem statement and make sure that the beam is indeed a cantilever beam and not a simply supported beam. This is because the deflection at midspan for a simply supported beam with uniformly distributed load is different from that of a cantilever beam.

Assuming that the beam is indeed a cantilever beam, the expression for the deflection at midspan can be derived using the Euler-Bernoulli beam theory. This theory states that the deflection of a beam is directly proportional to the bending moment and inversely proportional to the flexural rigidity (EI) of the beam.

In this case, the deflection at midspan can be found by integrating the bending moment equation over the length of the beam and then applying boundary conditions. The bending moment equation for a cantilever beam with a uniformly distributed load is given by:

M(x) = \frac{wL}{2}(x-L)

where w is the uniformly distributed load and L is the length of the beam.

Integrating this equation and applying boundary conditions (at x = 0, M = 0 and at x = L, M = \frac{wL^2}{2}), we get the following expression for the deflection at midspan:

\delta = \frac{wL^4}{8EI}

This is the correct expression for the deflection at midspan for a cantilever beam loaded with a uniformly distributed load.

It is important to note that the elastic curve formula provided in the problem statement is only applicable to simply supported beams and cannot be used for cantilever beams. As a scientist, it is important to understand the assumptions and limitations of different theories and equations in order to accurately solve problems and make informed decisions.