Mechanical Bending Moment and Rod Deformation Calculations

[Apple&Orange]

Homework Statement

A) A circular rod is subjected to a bending moment of 315Nm. What is the minimum diameter of the rod required so that the maximum stress does not exceed 200N/mm²?

B) If the Modulus of Elasticity for the material from which the rod is made of is 100kN/mm², what radius will the rod be deformed to when stressed to the maximum allowable?

Homework Equations

A) $\frac{M}{I}$ = σ

B) $\frac{M}{σ]}$ = $\frac{ρ]}{E]}$

Second moment of Area for a circle of diameter d, about its Neutral Axis is $\frac{∏d}{64}$

The Attempt at a Solution

A) M=314N/m
σ=200MN/m²
I=$\frac{∏d]}{64}$

[itex]\frac{314}{$\frac{∏d}{64}$}[/itex]=200×10⁶

d=0.0752m

B) M=413Nm
E=100GN/m²
σ=200MN/m²

[itex]\frac{314}{200×10⁶}[/itex]=[itex]\frac{ρ}{100×10⁹}[/itex]

ρ=157,000m (Ridiculous answer, I know)

I haven't done mechanics in a while, so I was wondering if someone could double check that I'm on the right track.

Chuur Chuur

 

[LawrenceC]

Bending stress is Mc/I not M/I. Check your units.

 

[Apple&Orange]

Bending stress is Mc/I not M/I. Check your units.

But how would that formula work though?

I wouldn't be able to solve for c, since I'm asked to find the diameter. If I substitute that in, then I'd have $\frac{M\frac{d}{2}}{\frac{∏d}{64}}$=σ

where c = $\frac{d}{2}$

 

[LawrenceC]

But how would that formula work though?

I wouldn't be able to solve for c, since I'm asked to find the diameter. If I substitute that in, then I'd have $\frac{M\frac{d}{2}}{\frac{∏d}{64}}$=σ

where c = $\frac{d}{2}$

You are provided with the bending moment and the maximum stress. You can easily determine diameter from that information. The formula you have above is missing an exponent.

 

[DeeNos]

I would first like to commend you for attempting to solve these problems and showing your work. Your approach to solving the problems seems correct, but there are a few things that could be improved upon.

For part A, you correctly used the formula for calculating stress, but your value for the bending moment (M) is incorrect. It should be 315 Nm, not 314 N/m. Additionally, when converting the units for the second moment of area (I), you should be using meters instead of millimeters. This will give you a diameter of 0.0752 meters, which is equivalent to 75.2 mm. This seems like a reasonable answer for the minimum diameter of the rod.

For part B, your approach is correct, but there is a mistake in your calculation. The bending moment (M) should still be 315 Nm, not 413 Nm. This will give you a value for ρ (radius) of 0.00157 meters, which is equivalent to 1.57 mm. This is a reasonable answer for the deformation of the rod when stressed to the maximum allowable.

Overall, your approach is sound, but it's important to pay attention to units and use the correct values in your calculations. Keep up the good work!