# Homework Help: Beam deflection problem

1. Apr 12, 2010

### IIan

1. The problem statement, all variables and given/known data

Determine the slope and deflection at the point x on the beam (attached image)

2. Relevant equations

Bending stiffness equation: EIV'' = M
where E = young's modulus, I = second moment of area, V = deflection, M = moment

3. The attempt at a solution

By taking moments about both supports I have determined the reactions at the left and right supports to be b/(a+b) and a/(a+b) respectively.

Cutting the beam between point x and the right hand support:

EIV'' = M = bx/(a+b) - <x-a> (where <> are macaulay brackets)

EIV' = bx^2/2(a+b) - (<x-a>^2)/2 + A

EIV = bx^3/6(a+b) - (<x-a>^3)/6 + Ax + B

Here's where I get confused:

Using V=0 at the boundary conditions (x=0 and x=a+b) to find the integration constants, B is found to be 0 but A always comes out as a complicated fraction that I can't seem to simplify to get anything sensible.

I know the final answers are supposed to be: V' = (ab/3)((a-b)/(a+b))
and V =-a^2b^2/3(a+b)

I have worked through the question several times and I can't figure out where I'm going wrong so any help would be much appreciated.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

#### Attached Files:

• ###### beamQ.GIF
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2. Apr 12, 2010

### pongo38

Have you checked your complicated expression for "A" by putting a=b and setting the gradient to zero at the centre? It is no guarantee if it is correct, but, if it is incorrect in this special case, then it will be incorrect more generally.