Help with a BMD and deflection of beam problem

[kostoglotov]

Homework Statement

I am given this beam:

imgur link: http://i.imgur.com/tw6EEJz.jpg

and asked to express the slope at A and the max deflection in terms of a, E, I, and Mo.

I am being told that my answer has the wrong multiplier...so I assume that my overall approach is ok (??) but I've missed dividing something by 2 somewhere or something like that.

I transform that above diagram into an equivalent:

imgur link: http://i.imgur.com/uevlWxU.jpg

(I whited out the bending moments, pretend they aren't there anymore)

I figure those same bending moments could be produced by a load P like the one I've drawn.

Doing that leads to the shear force and bending moment diagrams that look something like this:

The lines on the BMD are $y = \frac{P}{2}x$ and $]y = \frac{-P}{2}x+\frac{3Pa}{2}$

Thus I get $P = \frac{2Mo}{a}$

and I have the equations $v_{max}=\frac{-PL^3}{48EI}$ and $\theta_{max}=\frac{-PL^2}{16EI}$

Now that I have P in terms of Mo and a, and L = 3a, through basic substitution I get $v_{max}=\frac{-9M_0a^2}{8EI}$ and $\theta_{max}=\frac{-9M_0a}{8EI}$

It does seem strange to me that the constant is the same out the front of both, and I am being told that my multiplier out the front is wrong. What have I missed?

Homework Equations

The Attempt at a Solution

 

[KataKoniK]

Hello,

Thank you for sharing your approach and calculations. Overall, your approach seems to be correct, but there are a few things that you may have missed that could have led to the wrong multiplier.

First, when you transformed the original beam into an equivalent one, you assumed that the bending moments at A and B were equal, which is not necessarily the case. It is possible that the bending moment at A is different from the bending moment at B, which could affect your calculations.

Second, in your shear force and bending moment diagrams, you have assumed that the load P is acting at the midpoint of the beam, which may not be the case. The load could be acting at a different point along the beam, which would affect the equations for the shear force and bending moment.

Finally, when calculating the maximum deflection and slope, you have assumed that the beam is simply supported at both ends, but this may not be the case. The beam could be fixed at one end and simply supported at the other, which would result in different equations for the maximum deflection and slope.

I would recommend double-checking your assumptions and considering the possibility that the beam may not be simply supported at both ends. Additionally, it may be helpful to draw the shear force and bending moment diagrams for the original beam and compare them to the diagrams you have drawn for the equivalent beam to see if there are any discrepancies.

I hope this helps and good luck with your calculations!