Two circular beams where one is solid of diameter,D and other a hollow of outer diameter Do and inner diameter Di are of same length,same material and of same weight.Find the ratio section modulus of?

The section modulus of a beam is a measure of its ability to resist bending stress. It is given by the formula:

Z = (π/32) * (D^3 - Di^3)

For the solid beam, the diameter is D and the inner diameter is 0, so the section modulus is:

Zsolid = (π/32) * D^3

For the hollow beam, the outer diameter is Do and the inner diameter is Di, so the section modulus is:

Zhollow = (π/32) * (Do^3 - Di^3)

Given that the two beams have the same weight, we can equate their volumes:

π/4 * D^2 * L = π/4 * (Do^2 - Di^2) * L

Simplifying, we get:

D^2 = Do^2 - Di^2

Now we can substitute this into the section modulus formula for the hollow beam:

Zhollow = (π/32) * (Do^3 - (Do^2 - D^2)^3)

Zhollow = (π/32) * (Do^3 - (Do^3 - 3*Do^2*D + 3*Do*D^2 - D^3))

Zhollow = (π/32) * (3*Do^2*D - 3*Do*D^2 + D^3)

Now we can find the ratio of the section modulus of the hollow beam to the solid beam:

Zhollow / Zsolid = (3*Do^2*D - 3*Do*D^2 + D^3) / D^3

Zhollow / Zsolid = 3*Do^2/D^2 - 3*Do + 1

Therefore, the ratio of the section modulus of the hollow beam to the solid beam is 3*Do^2/D^2 - 3*Do + 1.

Section Modulus for Solid Beam:

$$\frac{I}{c}=\frac{\pi D^4}{64}$$

where:

- I is the moment of inertia
- c is the distance from the center of the beam to the extreme fiber

Section Modulus for Hollow Beam:

$$\frac{I}{c}=\frac{\pi (Do^4-Di^4)}{64(Do-Di)}=\frac{\pi (Do^2+Di^2)(Do-Di)}{64(Do-Di)}$$

Ratio of Section Moduli:

$$\frac{Section\ Modulus\ of\ Solid\ Beam}{Section\ Modulus\ of\ Hollow\ Beam}=\frac{\pi D^4/64}{\pi (Do^2+Di^2)(Do-Di)/64(Do-Di)}$$

Simplifying the expression:

$$\frac{Section\ Modulus\ of\ Solid\ Beam}{Section\ Modulus\ of\ Hollow\ Beam}=\frac{D^4}{(Do^2+Di^2)(Do-Di)}$$

Given that the beams have the same weight:

$$W=V\gamma=\frac{\pi D^2}{4}L\gamma=\frac{\pi (Do^2-Di^2)}{4}L\gamma$$

where:

- V is the volume of the beam
- γ is the specific weight of the material
- L is the length of the beam

Cross-multiplying:

$$\pi D^2L\gamma=\pi (Do^2-Di^2)L\gamma$$

Dividing both sides by πLγ:

$$D^2=Do^2-Di^2$$

Rearranging:

$$Di^2=Do^2-D^2=Do^2-D^4/Do^2=(Do^2-D^4)/(Do^2)=(Do-D)(Do+D)/(Do^2)$$

$$Di=\sqrt{\frac{(Do-D)(Do+D)}{Do^2}}=\sqrt{\frac{Do^2+D^2-2DDo}{Do^2}}=\sqrt{1-\left(\frac{2D}{Do}\right)^2}$$

Substituting this value of Di into the ratio of section moduli expression:

$$\frac{Section\ Modulus\ of\ Solid\ Beam}{Section\ Modulus\ of\ Hollow\ Beam}=\frac{D^4}{(Do^2+(1-\left(\frac{2D}{Do}\right)^2)Do^2)(Do-\sqrt{1-\left(\frac{2D}{Do}\right)^2})}$$

Simplifying further:

$$\frac{Section\ Modulus\ of\ Solid\ Beam}{Section\ Modulus\ of\ Hollow\ Beam}=\frac{D^4}{Do^2\left(1-\left(\frac{2D}{Do}\right)^2\right)}=--\boxed{\frac{1}{\left(1-\left(\frac{2D}{Do}\right)^2\right)}}$$