First moment of area (S) for different shapes (over the height)

[jdeboon]

I made a mathcad file, that calculate the area, first moment of area (whole), second moment of area, centroidal axis etc.

All only depending on the coördinates that you choose.

It is usefull to quickly see the moment of inertia of each shape and apply this in further calculations.

But for futher calculation, now i want to know the First moment of area  (S), some call this the static moment, over the height.

This to easy see the critical points of the beam. In case of shear.

Hope that you are able to help me.

[AlvaroDíaz]

Hi Johan.

Maybe the spanish article from wikipedia could help: Primer momento de área - Wikipedia, la enciclopedia libre

There: S.CM = 0, S.axis = A*c, where c=dist(CM,axis), where CM = mass center.

For partial moments: For rectangles 2h*b (see the figure in the wiki) it is: Q.y = b*(h^2-y^2)/2.

Best regards.

Alvaro.

[jdeboon]

Hi Alvaro,

Thanx for your awnser, I already knew the theory behind this.

I have 2 problems nemely that:

- I want to calculate many different beams (symmetric over the x-axis)

- And I want to apply this theory over the height of this beams, so that will make visible (and proof) what the critical points here are.

I added a figure, to make things more clear:

In the upper figure i take a random beam, and do the basic calculation for the static moment (the way you said it)

But now i want to make this calculation automatially, no matter what kind of figure u use.

By myself i am thinking about using a intergral, something like Sy= Intergral.A(Y*DA)

But i am not sure that this will work and how to apply this in mathcad.

Hope this make my question more clear

Greets Johan

[Fred_Kohlhepp]

You can integrate to find what you want, see the attached file (Prime 3.0)

I beams and strange shapes may require some ingenuity to define functions.  (I normally do this for closed solid sections rather than flanged beams.)

[jdeboon]

Thank you for the responce.

Is there a possibility to safe the file as a mathcad 15 file?

I do not have a mathcad prime account.

[Fred_Kohlhepp]

Version 15 per your request.

[jdeboon]

Thankyou,

I still have 2 questions:

- The s in the file is the distance in x direction?

- Is there a possibility to plot in the x-direction (or s), this to make visual what the first moment of area does over the height of the beam?

[Fred_Kohlhepp · Accepted solution]

s is a variable of integration, yes in the x direction

Attached is what you want, I think.

[jdeboon]

Thank you, this is exactly what i was looking for!