Units of measurement for structural analysis

Question

FEA engine

I'm using CCX finite elements analysis engine along with its pre- and post-processor CGX:

http://www.dhondt.de/index.html

https://github.com/calculix

Example

I'm looking at one example here that is using steel material:

https://github.com/calculix/CalculiX-Examples/blob/a3ef0b86de47ad3d8f7e4bb39a3200c839a55f2c/Elements/Solid/solid.inp#L7

Elastic specifications

In the example above, the declaration is done for Young’s modulus, Poisson’s ratio, and Temperature:

*ELASTIC,TYPE=ISO
210000,0.333333333,0

Density

Mass density is initialized by:

*DENSITY
7.85e-9

Gravity

Gravity, i.e. acceleration vector, is defined like this:

*DLOAD
Eall,GRAV,9810.,0.,-2,0.

Input values

Therefore, the input values are:

• Young’s modulus = 210000
• Mass density = 7.85e-9
• Gravity acceleration = 9810.

Question

What are the units of measurement for the above example?

My guess

I know the gravity acceleration is 9.8 m/s2. So, 9810. means mm/s2.

But I'm confused about Young’s modulus and Mass density. What are their units? They should be consistent with the mm. But I cannot figure them out.

Solved

This table from the latest CalculiX solver documentation helps with unit conversion. The units of the above example are mm,N,s,K.

Answer 1

DISCAIMER: I haven't used calculix.

From this page, I found out that there are 3 predominant unit systems that are being used

• m,kg,s,K : meter, kilogram, second, Kelvin
• mm,N,s,K: millimeter, N, second, Kelvin
• cm,g,s,K: centimeter, g, second, Kelvin

You are right in that the length units seems to be consistent with mm, so most likely the second is the one being used. If you notice though the second is the only one that used Force (N) instead of mass (kg, or g).

This is consistent with the Modulus is 210000 MPa (Mega Pascal) i.e. $\frac{N}{mm^2}$ (modulus of steel is 201000 - 210000MPa).

However it gets a bit confusing with the density. More specifically:

• Mass density of steel should be equal to 7800 $\frac{kg}{m^3}$.

In the page mentioned above you can see that density is given by: $\frac{N\cdot s^2}{mm^4}$ in the 2nd system of measurement. The reason is that because $mass = \frac{Force}{\left(\frac{Length}{time^2}\right)}\rightarrow mass = \frac{Force\cdot time^2}{Length}$, and volume is $Length^3$, therefore the units for density are:

$$[density] = \left[\frac{Force \cdot time^2}{Length^4}\right]$$

The actual value of steel density is 7800 $\frac{kg}{m^3}\equiv \frac{N\cdot s^2}{m^4}$. So when you convert m to mm you get

$$7800\cdot \frac{N\cdot s^2}{m^4}=7800\cdot \frac{N\cdot s^2}{(1000mm)^4}=7800\cdot \frac{N\cdot s^2}{10^{12}mm^4}=7800\cdot 10^{-12}\frac{N\cdot s^2}{mm^4}$$

and finally: $$7800\cdot 10^{-12}\frac{N\cdot s^2}{mm^4}\Rightarrow 7.8\cdot 10^-9\frac{N\cdot s^2}{mm^4}$$