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

[LDC1972]

Homework Statement

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?

Homework 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

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!

 

[SteamKing]

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

 

[LDC1972]

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

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

 

[LDC1972]

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

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?

 

[SteamKing]

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

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

 

[SteamKing]

F = 209.44 kN looks like a good result.

 

[oxon88]

ignore this.