Calculation of brace Young's Modulus (eq. 4.4-4)

[robanomoly]

Hello everyone, I've been trying to calculate the Young's Modulus of some bracing material (Spruce) using equation 4.4-4, and am having a bit of a fit with it. Using equation 4.4-4, the resulting units for "E" end up being g/mm instead of g/mm^2. So, I’m a bit confused how to get that into GigaPascals (N/m^2). Here are the values I have:
Length: 508.5 mm
Width: 51.0 mm
Height: 19.0 mm
Mass: 172 gram
Frequency: 414.6
E: 10,517,855,577 (g/mm)
I: 29,150.75 (mm^4)

If the units for E were g/mm^2, that converts to 103,145 GPa...which does not sound right at all.

Any help would be greatly appreciated!

[Trevor Gore]

So, I’m a bit confused how to get that into GigaPascals (N/m^2).

Pascals are a measure of pressure, units of newtons per square meter. A newton is defined here. Young's modulus, E, is stress/strain. Stress has units of pressure as above, strain is meters/meter, so the units cancel and strain is dimensionless. So E has units of pressure also. Giga means 10^9, so GPa is 10^9 Pascals = 10^9 N/m^2.

By Newton's Law, Force = Mass * Acceleration. The acceleration due to gravity is ~9.81m/s^2, therefore the reaction force required to prevent a mass of 1kg from falling towards the center of the earth is ~9.81 newtons (or colloquially, 1 kg = ~9.81 newtons force)

You're likely confusing mass and force. Working in SI units (meters, kilograms, seconds) rather than cgs metric (centimeters, grams, seconds) or other versions of metric, eliminates most of the problems.

[robanomoly]

Sooo.....when I weigh the piece and the scale says 0.172 Kg, that equals 1.687 Newtons of force? Thus:

[((414.6 * 0.5085^2) / (1.0279 * 0.019)) ^2] * [1.687 / (0.5085 * 0.051 * 0.019)] = 103,144,928,394 N/m (103 GN/m)....which is not a unit of pressure....(scratching head).....umm....where am I going wrong? I appreciate your patience and understanding in responding to a simple American trying to grasp SI units.

Using imperial units:
[((414.6 * 20.02^2) / (1.0279 * 0.748)) ^2] * [0.379 / (20.02 * 2.008 * 0.748)] = 588,972,719 lb/in....still not a unit of pressure...?

[Trevor Gore]

Density is mass/unit volume, so you need what the scales say, namely 0.172 Kg. If I change 1.687 to 0.172 in your formula I get 10.5 GPa, which is about right.

Straight off the spreadsheet: =(((414.6 * 0.5085^2) / (1.0279 * 0.019)) ^2) * (0.172 / (0.5085 * 0.051 * 0.019))

I can't see where you're going wrong with the units. I did a dimensional check on Equ, 4.4-3 and 4.4-4 and they came out as expected.

Going through Equ. 4.4-3 (4.4-4 is just an inversion of that) and checking dimensions (Kg, meters, seconds) = M, L, T (Mass, Length, Time):

={ [M.L/T^2]/L^2 . (L^4) }/ { (L^4) . (M/L^3) . (L^2)}

=(M.L^8)/(M.L^8 .T^2)

= 1/T^2 which is frequency squared, as expected.

Doing Equ 4.4-4 just for completeness:

= {(1/T^2) . (L^4) / (L^2)} . {M/L^3}

=(M.L/T^2) / (L^2) = Force/Area = Stress, same as pressure.

[robanomoly]

Trevor, thank you very much for your time in writing thorough responses. I think the [M.L/T^2] portion was tripping me up.

[John Greenlaw]

I'm just figuring this out as well. I just made this little chart and I think it looks ok though I'm not sure if 14 Nm2 is too high for spruce of these dimensions. Any thoughts?

[Trevor Gore]

You have too many zero in your first row of data, column E.

Flexural rigidity is a pretty straightforward calc. You need to develop some confidence in your results, best done by "triangulation". i.e. calculate three different ways and if you get the same answer three times it might be right!

[John Greenlaw]

Ah yes! I forgot about triangulation. Thanks for the advise Trevor!