What is the Maximum Deflection of a Beam with a Point Load and Supports?

[skaboy607]

Homework Statement

A beam of I cross section rests on supports 8.00m apart, and carries a
load of 70kN at a distance of 3.00m from one end. If the second
moment of area is 2.8 * 104 cm4 and E = 210 kN/mm2, find the deflection
at the load, and the position and magnitude of the maximum deflection.
Neglect the self weight of the beam.

Ans: 11.16mm and 11.68mm, 3.72m from end nearest load

Homework Equations

I used Mcauley's method.

The Attempt at a Solution

I have calculated the deflection at the load and obtained the correct answer. Where I am stuck is in the calculation of the position and magnitute of maximum deflection. I thought at first that max deflection was where the load was but I guess not...

What I tried is reverting back to the equation for slope. I know that where the slope is 0, the deflection is max (I think-have it written down in lecture notes). From this, I rearranged to find x (distance from end) however the answer I get is 7.28, which as you can see is wrong. Don't know what else to do now. The answer divided by 2 is not that different although not close, but I have rearranged the equation over and over again in case I had done it wrong but always get the same answer.

Any help would be most appreciated.

Thanks

 

[mathmate]

If you show your workings, we can better help you find your problem.
To confirm your observations: unless the load is extremely skew, the maximum deflection is usually around mid-span.

 

[skaboy607]

Ok I will try and write them, here goes:

Using Mcauley's method I have, EIdv/dx=R1x^2/2+W/2*(x-a)^2+C (think this is right-doing it from memory-dont have workings with me)
Where dv/dx is the slope and max deflection is when this is 0. Equation is now 0=R1x^2/2+W/2*(x-a)^2+C. I rearrange for x and get I get the answer that I said in the above post 7.28.

Thanks

 

[mathmate]

It looks like the slope expression is missing some terms. In any case, the slope for a simply supported beam with a non-central point load has two expressions, one on each side of the load.
The maximum deflection is generally on the side further away from the support.
If you have equated the slope expression on the short side, the results will unfortunately be not correct.

If you can, I suggest you redo the derivation and post it here if the answer still does not correspond to the given answers.

As a confirmation, the answer of 3.72 (or 3.7183) from the end closer to the load is correct.
11.68 (11.6813) mm is also correct for the maximum deflection.

 

[skaboy607]

Hi

Thanks mathmate! I got it in the end! One day i'll understand this statics stuff. Out of interest to solve x, I used the quadratic equation which yields two answers, bar the fact that one of the values is longer than the beam itself, is there any other reason why you disregard it or is it just as simple as that?

Thanks

 

[mathmate]

I would discard it simply based on the fact that it is not a physically possible solution.
I'm glad you've got it in the end. What you have derived, will stay with you for a long time, and that is the reward of hard work!
Keep it up!