Elastic curve of cantilever beam

[chris54]

Homework Statement

Using method of integration, determine the elastic curve for the beam and calculate the total deflection at a point A (end) of the beam. Use EI as a constant.
See word document attached with diagram.

Homework Equations

EIV''(x) = - M(x)

The Attempt at a Solution

I know how to integrate to get the deflection formula but I'm not sure how to calculate M(x) with the moment at the free end.
Is it M(x) = -w/2(L-x)^2 + M(a)(x/L)
or M(x) = -w/2(L-x)^2 + M(a)

Why do some problems in my textbook (Mechanics of materials, Gere) include the support reactions in M(x) and others don't? Do I need to include the support the reactions?

Also will my answer be in meters e.g. v = 100m/EI ?

Any help is much appreciated.
Thanks.

 

[radou]

If there's a moment on the free end of the beam, the moment diagram M(x) is constant, right? You can set your coordinate system with a positive direction from the free end to the fixed end, and then use integration to determine w(x). You need a boundary condition to find the constant of integration, which actually equals the deflection at the free end.

 

[chris54]

Thanks for the reply.

So it will be M(x) = -w/2(L-x)^2 + M(a) ?
It's for an assignment and the question is given with the origin of the coordinate system at the fixed end, so I can't change it. The boundary conditions are v'(0) = 0 and v(0) = 0, so both constants will be zero and I should get:

EIv''(x) = (wL^2)/2 - Lwx + (wx^2)/2 - M(a)
EIv'(x) = (wLx)/2 - (wlx^2)/2 + (wx^3)/6 - M(a)x
EIv(x) = (wlx^2)/4 - (wlx^3)/6 + (wx^4)/24 - (M(a)x^2)/2

The last one will be the elastic curve and then I just sub all the values in, with x=L and I will get the deflection at the free end. Does that sound right?