# Help to find force given a diameter reduction (Poissons Ratio)

1. Jul 31, 2013

### LDC1972

1. The problem statement, all variables and given/known data

Mild steel bar 40 mm diameter and 100 mm long, subjected to a tensile force along its axis (through central length)
Youngs Modulus = 200GN m^-2
Poisson's ratio = 0.3

Calculate the force required to reduce the diameter to 39.99mm

(there is also a picture saying Use the x- y co-ordinate system above). I shall upload that momentarily if needed?

2. Relevant equations

Poissons ratio = Transverse strain = - εt / εl

Diameter = 0.04m
Lo = 0.1m
E = 200GN m^-2 = 200x10^9 Pascals (Pa)
V = 0.3

Transverse strain εt = (Reduction in Diameter - Original Diameter) / Original Diameter

Axial strain = Transverse strain / V

Δd = εtd

ε = εt / -V

ε = σ / E

σ = F/A

3. The attempt at a solution

So here's what I did and hoping it is right?

Transverse strain εt = (Reduction in Diameter - Original Diameter) / Original Diameter
(0.3999 - 0.4000) / 0.4000 = -0.00025 Transverse strain (εt)

Axial strain = Transverse strain / V
-0.00025 / 0.3 = .00083333333 Axial strain

I confirmed these by dividing Transverse strain by Axial strain = -0.3

Since decrease in diameter = transverse strain x Original Diameter then:
Δd = εtd = -0.00025 x .40 = 0.1mm which is correct.

As εt = -Vε Then:
-0.00025 = -0.3 x ε

ε = εt / -V
ε = 0.00083333333

ε = σ / E
σ is unknown ∴
σ = ε x E
σ = 16666666 Pa (stress value)

σ = F / A

Force = σ x A (Area)
= 16666666 Pa x 0.00125663706
= 20943.95019 Nm

I think here I can bring the figure down a thousand to 20.944 KNm?

Thank guys for ANY help! This took me ages!!

2. Jul 31, 2013

### SteamKing

Staff Emeritus
1. What are the derived units for a Pascal?
2. Since when did force in SI become newton-meters?

3. Jul 31, 2013

### LDC1972

Oh, every time one of us Brit's ask a question regarding GNm, we get told to use the proper term GPa (I think that's the equivalent). So I swapped the GNm to Gpa. Yes just checked online and 1 Newtom metre = 1 Pascal.

Force SI should be Newtons? As in N or KN or MN but not NM?

Thanks!

I'm worried my force looks too low? .01mm should surely take more than my final figure?

Thanks

Lloyd

4. Jul 31, 2013

### LDC1972

Just went through double checking figures as "felt" force too low to make a .01mm impact on diameter.....

Found σ was missing a unit, one 6 short!

New final figure: 209439.51N
Or simplified to 209.43951KN
Or to 0.20943951 GN?

5. Jul 31, 2013

### SteamKing

Staff Emeritus
Not even close.

The pascal is the SI unit for pressure or stress, which implies a force divided by an area.

1 Pascal = 1 Newton / m^2

6. Jul 31, 2013

### SteamKing

Staff Emeritus
F = 209.44 kN looks like a good result.

7. Sep 23, 2013

### oxon88

ignore this.

Last edited: Sep 23, 2013