Boussinesq equation under a rectangular loaded area

[canwesteng]

My old college textbooks are giving me a bit of heartburn. I'm looking to program in a function to determine stress below any point for a rectangular foundation, but the way the formula is written, it looks like x/y/z or some combination thereof are variables being integrated in the equation, when in fact I don't believe any variables should be integrated (i.e. it is a constant function being integrated over the base of footing). I think x and y should be cartesian coordinates of the point where stresses are being determined, z is depth below the foundation, and dx/dy are just an infinitesimal square of the foundation to represent a point load. Is this correct? Some snapshots from Das and from my calc below.

Thanks

 

[Celt83]

The foundation applies pressure to the soil surface and the point in the soil below is influenced by every dxdy of pressure so you need to integrate over the bounds of the applied pressure area. X,Y end up being the distance from the point you want stress for to the dxdy load differential area.

Here is the formula for stress below the corner of the flexible foundation:

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[canwesteng]

Isn't that what my formula is doing though? I'd prefer to avoid making a mess with all the influence factors if possible, as that is very tedious for multiple points and creating all these fictional rectangles.

 

[Celt83]

your integral is hard coded with the integral bounds being 0,L and 0,B and you've also subbed a,b in for x,y.
The integral range will vary based on the point you want stress for and a,b should in fact be x,y so that the get integrated they are not constants.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[canwesteng]

Why do x and y get integrated? I read the text as x and y being cartesian coordinates of the point where stress is being calculated.

I guess the other alternative is that the formula should look like this, although I'm not sure how to rationalize it.

 

[Celt83]

x and y are the distance from the point you want stress at to each dxdy of pressure

Edit: the integral ranges are only 0->L and 0->B if your looking at a point on the corner of the foundation area.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[Celt83]

1-D:
As dx approaches 0 the initial x1 approaches 0 and the final xi approaches the length of the foundation

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[Celt83]

2-D:
as dx and dy approach 0 x1 and y1 approach 0 and the last xi approaches B and the last yi approaches L.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[canwesteng]

Good feedback, I'm not following why the integral only works for the corner if integrated from x 0 to B and y 0 to L though...This is where I'm at now, for any one point A at some point x.i, y.i, z in space.

 

[Celt83]

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[canwesteng]

Good sketches, I follow how this needs to be integrated from -B/2 to B/2 for the case at the centre of the slab, but for the case of a point outside the slab, my equation would hold true, right?

 

[Celt83]

That seems correct to me, so instances of x in the original formula would be replaced with (a+x).

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[bdbd]

Thanks guys, see the calculation below.

https://i.imgur.com/CV8fVFC.png

 

[BigH]

Find a copy of Poulos and Davis' Elastic Solutions for Soil and Rock Mechanics - it gives all the equations to use. You can find it on the internet - I think Poulos has downloaded it as has Paul Mayne

 

[geotechguy1]

I like to ignore all the fuss about Bousinesq and just use a 1H:2V projection.