Zachary R. answered • 02/24/22
B.S. in Mechanical Engineering / M.S. in Materials Science & Eng.
Hello Mark!
Here is some help with your Materials Science problem! Let me know if you have any questions afterwards, I'd be happy to help!
Given Parameters:
Initial Length: L0 = 4.0 m
Initial Width: w0 = 80 mm
Initial Thickness: t0 = 40 mm
Tensile Force Loaded Along Length: F = 320 kN
Young's Modulus of the Steel: E = 400 x 103 Pa (?)
Poisson's Ratio of the Steel: ν = 0.32
You didn't report the units of the Young Modulus, and if I assumed that it was in Pascals, it seems a little bit low of a value for steel -- so I will show how to work this problem out in terms of variables. You could always plug in the numbers later!
So we are stretching the bar along its length-axis, and want to calculate the induced change in the width- and thickness axes. We should expect that this bar's width and thickness shrink slightly due to Poisson's ratio effects. Let's start with the definition of Poisson's ratio and work from there...
Let's define the Z-direction to be along the length of the bar, the X-direction to be along the width axis, and the Y-direction to be along the thickness axis....
Poisson's Ratio Definition:
v = - (εxx) / (εzz) = - (εyy) / (εzz)
... now rearrange to be in terms of εxx and εyy...
εxx = εyy = - vεzz
So, to learn things about how much strain occurs in the X- and Y- directions (width- and thickness- directions), we must first calculate how much strain is occuring along the Z-direction (length-direction)...
To find an expression for εzz we can use Hooke's Law...
Hooke's Law:
σzz = E * εzz
εzz = σzz / E
... where σzz is the axial stress along the Z-direction (along the length-direction)...
Now how can we calculate that axial stress? We just need to use the definition of an axial stress...
σzz = Fz / A0
where Fz is the force applied along the Z-axis and A0 is the cross-sectional area of the bar perpendicular to the Z-direction... we can express the area in terms of the relevant dimensional parameters...
σzz = Fz / (w0 * t0)
All of these parameters on the right side are given to us in this problem. Now plug this back into our Hooke's Law equation...
εzz = σzz / E
εzz = Fz / [E * w0 * t0]
...now plug this Z-axis strain expression back into our Poisson's ratio formula...
εxx = εyy = - vεzz
εxx = εyy = - [v*Fz] / [E * w0 * t0]
Now we have an expression for the strains in the X- and Y- direction in terms of only known parameters.
But we want to take it one step further, we don't just want STRAINS, we want the change in the dimensions in millimeters... Let's look at the definition of strain to do so...
Eng. Strain Definition:
εxx = Δw / w0
εyy = Δt / t0
... lets substitute this expression in for our strains in our earlier expression and solve for Δw and Δt...
εxx = - [v*Fz] / [E * w0 * t0]
[Δw / w0] = - [v*Fz] / [E * w0 * t0]
Δw = - [v * Fz * w0] / [E * w0 * t0] (w0 terms cancel...)
Δw = - [v * Fz] / [E * t0]
εyy = - [v*Fz] / [E * w0 * t0]
[Δt / t0] = - [v*Fz] / [E * w0 * t0]
Δt = - [v * Fz * t0] / [E * w0 * t0] (t0 terms cancel...)
Δt = - [v * Fz] / [E * w0]
All you'd need to do would be to plug in the relevant numbers to the above equations and you'd know the change in width and thickness of the steel bar.
Hope that helps! Let me know if you have any questions!
--Zach
