Radius of curvature usng: M/I= σ/Y = E/R

In summary, the formula for finding the radius of curvature is M/I = σ/y = E/R, where M is the bending moment, I is the second moment of area, σ is the stress, y is the distance from the neutral axis, E is the modulus of elasticity, and R is the radius of curvature. To use this formula, you can rearrange it and use any pair of the equations to solve for the radius of curvature.
  • #1
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Question:

How do I use this formula to find the Radius of curvature?

Formula:

M/I= σ/Y = E/R (M = bending moment, I = second moment of aria, σ = stress, y = distance from nutral axia, E = modulus of elasticity & R = radius of curvature)

Attempt:

In this question, I have all of the values except the radius of curvature, but I don't understand how to use this formula. If someone could rearrange the formula for me and post it, it would be much apriciated.

Thanx
 
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  • #2
They are all equal to one another. But if you have to use them to solve anything, you need to use any pair of the equations. For example, I could use

M/I = σ/y OR M/I=E/R
 

Related to Radius of curvature usng: M/I= σ/Y = E/R

1. What is the significance of the equation M/I= σ/Y = E/R in determining the radius of curvature?

The equation M/I= σ/Y = E/R is commonly used in structural engineering to calculate the radius of curvature of a beam. It represents the relationship between the bending moment (M), the second moment of area (I), the stress (σ), the yield strength (Y), the elastic modulus (E), and the radius of curvature (R). By rearranging the equation, R can be solved for and used to determine the shape of the beam.

2. How is the radius of curvature affected by changes in the bending moment and stress?

The radius of curvature is inversely proportional to the bending moment and directly proportional to the stress. This means that as the bending moment increases, the radius of curvature decreases, and as the stress increases, the radius of curvature also decreases. This relationship is important in designing beams that can withstand different loads without failing.

3. Can the equation M/I= σ/Y = E/R be used for all types of beams?

Yes, this equation can be used for all types of beams, as long as the material properties and the load distribution are known. However, it is most commonly used for beams with a rectangular or circular cross-section.

4. How does the elastic modulus (E) affect the radius of curvature?

The elastic modulus, also known as Young's modulus, is a measure of a material's stiffness. A higher elastic modulus means that the material is more resistant to deformation under stress. As a result, a higher E value will result in a larger radius of curvature, and a lower E value will result in a smaller radius of curvature.

5. What are some limitations of using the equation M/I= σ/Y = E/R to determine the radius of curvature?

One limitation is that the equation assumes a linear relationship between stress and strain, which is not always the case for certain materials. Additionally, it does not take into account other factors that can affect the beam's shape, such as shear stress and imperfections in the material. As with any equation, it is important to consider its limitations and use it in conjunction with other methods for accuracy.

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