Superposition to a beam to calculate or show beam deflection

In summary, the conversation discusses applying the principle of superposition to calculate the deflection of a beam. The equation for maximum deflection is given as (WL^3) / (48EI). The person is having trouble figuring out the equation for the second position, when it is 3/4 or 1/4 of the total beam length. They mention needing to calculate the deflection at a given point and using tables to arrive at the result. They also mention needing to compare the theory of superposition to real-life results in a lab. The conversation concludes with a discussion about calculating the weight in Newtons.
  • #1
EngNoob
38
0

Homework Statement



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


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)



The Attempt at a Solution




I think i need to sum the two figure, however, can't get a figure for point two.

Think i need an equation to calculate deflection at a given point?
 
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  • #2
EngNoob said:

Homework Statement



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


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)



The Attempt at a Solution




I think i need to sum the two figure, however, can't 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.
 
  • #3
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
 
  • #4
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?
 
  • #5
This question is directly related to your other questions on the reciprocal beam theorem. While you won't 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
[tex]

\delta_a = f_{aa} \cdot P_a + f_{ab} \cdot P_b

[/tex]
 
  • #6
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?
 
  • #7
[tex] 3 \times 9.81 [/tex]

the units for Newtons are
[tex]

\frac{kg \cdot m}{s^2}

[/tex]
 

1. What is superposition and how does it relate to beam deflection?

Superposition is a principle used in structural analysis that states the total response of a structure is equal to the sum of its individual responses. In the case of beam deflection, this means that the deflection caused by multiple loads acting on a beam can be calculated by adding the deflections caused by each individual load separately.

2. How is beam deflection calculated using superposition?

To calculate beam deflection using superposition, the deflection caused by each individual load is first calculated using the relevant equations and then added together to determine the total deflection. This method is most commonly used for beams with multiple point loads or distributed loads.

3. Can superposition be used for all types of beams?

Superposition can be used for most types of beams, including simply supported, cantilever, and continuous beams. However, it may not be applicable for more complex structural systems such as curved or nonlinear beams.

4. Are there any limitations to using superposition for beam deflection calculations?

While superposition is a useful tool for calculating beam deflection, it does have limitations. It assumes that the beam is linearly elastic, meaning that it will return to its original shape after the loads are removed. It also does not take into account any deformations caused by shear or rotational forces.

5. Are there any alternative methods for calculating beam deflection?

Yes, there are other methods for calculating beam deflection such as the moment-area method and the conjugate beam method. These methods may be more accurate for certain types of beams or loading conditions but may also be more complex to use. Ultimately, the most appropriate method will depend on the specific structural system and loadings being analyzed.

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