## Beam deflection problem with multiple Uniformly Distributed Loads

## Beam deflection problem with multiple Uniformly Distributed Loads

(OP)

Hi all,

I am trying to solve this beam problem to find the deflection at different points along the length of the beam. However I am having a hard time writing the equation for bending moment using Macaulay's theorem since there are 2 uniformly distributed loads of different magnitudes.

Can someone help solve this problem?

Thank you in advance!

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

_{A}and R_{B}directly. From there, you can find the moment at any point distant 'x' from Point A using simple statics.Macaulay's Theorem is just another name for the Double Integration Method. I was going to refer you to the following for a simple example of a beam with a point load at midspan. Unfortunately, the man in the video gets the expression for dy/dx (slope) and y (deflection) wrong. Can you find his error?

https://www.youtube.com/watch?v=cSzzTbA267I

Personally, I would not use the Macaulay Theorem to solve this problem, unless you are required to use it in your assignment. There is nothing wrong with the method, but with that many loads, it is very easy to drop a term or make some arithmetic error. There are a number of easier ways to solve it. One method would be to treat each load separately, then use superposition to find the combined answer for slope and deflection at any point distant 'x' from Point A.

We can't solve your problem for you, but it really is pretty straightforward, even though it has a lot of terms. Start by solving for the two reactions, then express the moment at any point distant 'x' from Point A and show us what you've got.

BA

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

Cheers

Greg Locock

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## RE: Beam deflection problem with multiple Uniformly Distributed Loads

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

Although he came up with the correct answer for deflection at midspan, his expression for deflection was not valid for the left half of the beam.

BA

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

_{x}_{}= R_{A}.x - P(x-a)(x>a) - Q(x-b)(x>b) - R(x-c)(x>c)is the moment at any point of a simple span beam, distant x from Point A, with point loads P, Q and R at a, b and c respectively from Point A. The boolean expressions, shown in red have the value of 1 or 0 depending on whether the bracketed expression is true or false.

BA

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

I was actually able to write moment equations for individual sections and integrated them twice(which ended up with a lot of constants of integration - 8 in total). Then I applied multiple boundary conditions to solve for all the constants. boundary conditions were:

1. deflection = 0 at supports - (solved for 3 constants).

2. deflection is the same at a particular point, meaning I could equate the deflections given by 2 different equations (points C and D) - (solved for 2 constants).

3. slope is the same at a particular point, meaning I could equate the slopes given by 2 different equations (points C, D and B) - (solved for 3 constants).

Thus I now have different bending moment, slope and deflection equations for different sections of the beams!

thank you all for the great inputs :)

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

BA

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

I was taught to write equations like that as...

M=_{x}R_{A}x-P<x-a> -Q<x-b> -R<x-c>,with the pointy brackets being zero if the contained value was equal to or less than.

--

JHG

## RE: Beam deflection problem with multiple Uniformly Distributed Loads

BA

## RE: Beam deflection problem with multiple Uniformly Distributed Loads