Radius of Curvature Derivation for Bending of Beam w/Overhangs

  • Thread starter Thread starter Freyster98
  • Start date Start date
  • Tags Tags
    Curvature Radius
Click For Summary

Discussion Overview

The discussion revolves around deriving an expression for the radius of curvature of a simply supported beam with overhangs, focusing on geometric methods and the application of the Pythagorean theorem. Participants explore different approaches to this derivation as part of a lab assignment.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents a derivation using the Pythagorean theorem, leading to an expression for the radius of curvature.
  • Another participant confirms that using the Pythagorean theorem is acceptable but suggests that a factor of 2 may have been overlooked in the derivation.
  • A different method involving similar triangles is proposed as potentially quicker and less prone to error, though one participant expresses confusion about this approach.
  • One participant admits that their attempt using similar triangles resulted in an expression for the diameter rather than the radius, indicating a misunderstanding in the application of the method.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for deriving the radius of curvature, with multiple approaches discussed and some uncertainty expressed regarding the similar triangles method.

Contextual Notes

There are indications of missing assumptions or steps in the derivations, particularly regarding the factor of 2 mentioned by participants. The discussion also highlights the potential for confusion between diameter and radius in the context of similar triangles.

Freyster98
Messages
49
Reaction score
0

Homework Statement



Derive an expression geometrically for the radius of curvature of the following beam. This is part of a lab assignment for the bending of a simply supported beam with overhangs.

** I did this crappy diagram with AutoCAD, so I couldn't ( or didn't know how to ) include greek letters. Let's let r= \rho, and d= \delta for my derivation.

Homework Equations



a2+b2=c2

The Attempt at a Solution



I just used the pythagorean theorem to solve for \rho.

Starting with: \rho2= (\rho-\delta)2+(L/2)2.

Factoring out (\rho-\delta)2 , solving for \rho and simplifying , I end up with the following expression:

\rho=(\delta/2)+(L2/8\delta)I guess I have this question...is this the proper way to derive the radius of curvature geometrically? Is it ok to do it this way?
 

Attachments

  • radius of curvature.jpg
    radius of curvature.jpg
    11.6 KB · Views: 458
Last edited:
Physics news on Phys.org
Freyster98 said:
Derive an expression geometrically for the radius of curvature of the following beam.
…
I just used the pythagorean theorem to solve for \rho.

Starting with: \rho2= (\rho-\delta)2+(L/2)2.

Factoring out (\rho-\delta)2 , solving for \rho and simplifying , I end up with the following expression:

\rho=(\delta/2)+(L2/8\delta)

I guess I have this question...is this the proper way to derive the radius of curvature geometrically? Is it ok to do it this way?

Hi Freyster98! :smile:

(have a rho: ρ and a delta: δ :wink:)

Yes, Pythagoras is fine :smile: (though you seem to have lost a factor of 2 somewhere :confused:).

But there is quicker method (with less likelihood of a mistake):

Hint: similar triangles :wink:​
 
tiny-tim said:
Hi Freyster98! :smile:

(have a rho: ρ and a delta: δ :wink:)

Yes, Pythagoras is fine :smile: (though you seem to have lost a factor of 2 somewhere :confused:).

But there is quicker method (with less likelihood of a mistake):

Hint: similar triangles :wink:​


I ran through it a few times...I don't see where I'm losing a factor of 2.
 
sorry … my similar triangles method (have you tried that yet?) gave me the diameter, not the radius :rolleyes: …

so i got an extra 2 :redface:
 
tiny-tim said:
sorry … my similar triangles method (have you tried that yet?) gave me the diameter, not the radius :rolleyes: …

so i got an extra 2 :redface:

Ok, thanks. No, I haven't tried the similar triangles because, well, I don't get it :rolleyes:
 
Freyster98 said:
Ok, thanks. No, I haven't tried the similar triangles because, well, I don't get it :rolleyes:

ok … the triangle with sides d and L/2 is similar to the triangle with sides L/2 and … ? :smile:
 

Similar threads

Engineering Bending Moment for Simply Supported, Overhanging Beam with two Overhangs
  • · Replies 4 ·
Replies
4
Views
3K
Beam deflection and curvature radius formula doubts
  • · Replies 13 ·
Replies
13
Views
6K
Undergrad Reframing Light Bending Using Density Instead of Radius
  • · Replies 20 ·
Replies
20
Views
1K
High School Curvature radius question for my muon beam experiment
  • · Replies 7 ·
Replies
7
Views
2K
What Equations Are Needed for Calculating Beam Stress and Curvature?
  • · Replies 65 ·
3
Replies
65
Views
7K
Why bending moment is constant on an end loaded beam
  • · Replies 2 ·
Replies
2
Views
3K
Acceleration and radius of curvature of a particle
  • · Replies 5 ·
Replies
5
Views
4K
Calculating the radius of curvature given acceleration and velocity
  • · Replies 14 ·
Replies
14
Views
4K
Moments of deflection/ curvature of radius of a supported beam
  • · Replies 4 ·
Replies
4
Views
6K
How to Derive Radial Geodesics in Kerr Metric?
  • · Replies 11 ·
Replies
11
Views
4K