# Superposition to a beam to calculate or show beam deflection

1. Homework Statement

Apply the principle of superposition to a beam to calculate or show beam deflection.

2. Homework Equations

Maximum Deflection.

(WL^3) / (48EI)

Equation for possition two, when possition two is 3/4 or 1/4 of the total beam lengh

(Cant figure out the above)

3. The Attempt at a Solution

I think i need to sum the two figure, however, cant get a figure for point two.

Think i need an equation to calculate deflection at a given point?

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PhanthomJay
Homework Helper
Gold Member
1. Homework Statement

Apply the principle of superposition to a beam to calculate or show beam deflection.

2. Homework Equations

Maximum Deflection.

(WL^3) / (48EI)

Equation for possition two, when possition two is 3/4 or 1/4 of the total beam lengh

(Cant figure out the above)

3. The Attempt at a Solution

I think i need to sum the two figure, however, cant get a figure for point two.

Think i need an equation to calculate deflection at a given point?
The deflection you have shown is for a beam on simple supports with a concentrated load W at mid-point. Is that your first loading case? Did you use tables to arrive at that result? For the
2nd case, with the load at a quarter point, you'll need to calculate or look up in a table the deflection along the length of the beam, and add (superimpose) the results to the first case for the total deflected shape. But the point of max deflection for each case is not the same, so I'm not sure what you are being asked to do, and whether you can use tables or if you need to use some other analytical method.

I have for formula for load at point 1, i need to calculate the deflection at point 2 due to a load at point 1, point 2, and both together.

Formula i posted can be applied to point 1, dead center, as its maxium deflection?

I am been asked to compare the theory of superpossition, to a real life result that has been done in a lab, and compare results.

So i need to calculate theory for superpossition first.

I am googleing to find the answer, but so far, none of the equations give me a result anywhere near the tests.

The only formula that i can get to work is the one i postsed above...

Any ideas, very appreciated

PhanthomJay
Homework Helper
Gold Member
I notice you've posted this same question in another forum. Stewart gave a good set of charts, like the 4th one down. Use it for each load case, and add them up. It's a bit of math for sure. What material are you using? Did you calculate E and I correctly?

This question is directly related to your other questions on the reciprocal beam theorem. While you wont need the theorem here to solve the problem, the flexibility coeficients you derrived for loading the beam at a L/4 and finding the deflection at L/2 in your previous question will provide you with everything you need to solve this case.

From super positioning we get
$$\delta_a = f_{aa} \cdot P_a + f_{ab} \cdot P_b$$

The material i am using is steel.

I have calculated E correctly and I.

However, i am unsure i have calculated the weight correctly?

I have 3kg of weight, its need in newtons, i have been using 3000, but is this correct? or is it 3 * 9.81?

$$3 \times 9.81$$

the units for Newtons are
$$\frac{kg \cdot m}{s^2}$$