How can I use integration to find the maximum deflection in a beam?

In summary, the conversation discusses finding the maximum deflection in a beam with given parameters, including the use of the method of superposition and integration for accuracy. The formula for deflection is also mentioned, with a suggestion to check the calculation for the moment of inertia.
  • #1
Colnago
9
0

Homework Statement



Find the maximum deflection in the beam.

E=205Gpa
Beam breadth = 195mm
UDL on beam: 40 Bar

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Homework Equations



I have useed the method of superposition to try and solve it. Is this the best way?

It has been suggested that finding the displacement by integration would give me a more accurate answer but I'm unsure how to apply it in this case.

I am using Mechanics of Materials by R.C. Hibbler for reference.

The Attempt at a Solution



I = (bd3) / 12


I = (0.195 x 0.053) / 12 = 2.03125x10-6


Formula: y= (-5 x w L4) / (384 E I )

y = (-5 x 4x106 x 0.1054) / (384 x 205x109 x 2.03125x10-6)


y = 0.015mm

Thanks for any help.
 
Last edited:
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  • #2


Hello! It looks like you have made a good attempt at solving this problem. The method of superposition is a valid approach for finding the maximum deflection in a beam, but as you mentioned, using integration can provide a more accurate answer.

To apply integration in this case, you would need to find the equation for the deflection curve of the beam. This can be done by considering the bending moment at any point along the beam and using the differential equation for beam deflection. From there, you can integrate the equation to find the deflection at any point along the beam.

I would recommend checking your calculation for the moment of inertia (I) as it seems to be missing a factor of 10. It should be (0.195 x 0.0533) / 12 = 2.03125x10^-5. This will change your final answer to 1.5 mm, which is still within the same order of magnitude as your initial estimate.

I hope this helps! Let me know if you have any further questions.
 

1. What is beam deflection?

Beam deflection is the amount of bending or deformation that occurs in a beam when a load is applied to it. It is a measure of how much the beam has moved from its original position.

2. How is beam deflection calculated?

Beam deflection can be calculated using various mathematical formulas, depending on the type of beam and the type of load applied. Some common methods include the moment-area method, the method of virtual work, and the conjugate-beam method.

3. What factors affect beam deflection?

The amount of beam deflection is influenced by several factors, including the type of material the beam is made of, its cross-sectional shape and size, the type and amount of load applied, and the distance between supports.

4. How does beam deflection impact the structural integrity of a beam?

Excessive beam deflection can lead to structural failure if the beam is not able to support the load it is carrying. It can also cause damage to the beam itself, such as cracking or permanent deformation.

5. What are some common methods for reducing beam deflection?

There are several ways to reduce beam deflection, including using a stiffer material, increasing the size or cross-sectional shape of the beam, adding additional supports, and redistributing the load by using multiple beams instead of a single one.

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