Yes, I guess there could be some confusion. I get soil bearing allowable in terms of allowable net bearing pressure. Say 4000psf. But geotech guy does not know the depth of various foundations I might select. If soil unit weight is 100pcf and my fdtn is at 5ft, then gross allowable is 4000 + 5ft x 100psf = 4500psf. If fdtn depth is 3ft, then gross is 4400psf. Gross is net + overburden stress. Overburden stress changes with depth, but net never does. Its a quote from my soil mechanics class professor.
I was trying to find a URL with the same, but could not. I did find something similar, equation same, at Uni Bristol, but didn't like how the page was written. I offered them a suggested revision.
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Following are some comments I have concerning the referenced URL.
"When excavating for a foundation, the stress at founding level is relieved by the removal of the weight of soil. The net bearing pressure (qn) is the increase in stress on the soil.
qn = q - qo
qo = g D
where D is the founding depth and g is the unit weight of the soil removed."
The above statement implies that removal of soil overburden increases the net bearing pressure, when it is actually the gross bearing pressure at depth D that can be increased by the weight of soil removed. "As net never changes", it is easy to remember the difference
qo = overburden stress
Therefore q (what I call the gross allowable stress) = qn + qo
Your equations are correct, as
qn = q - qo does equal q = qn + qo
however it is far easier to immediately see that when the equation is presented in the later form.
I also do not think this following paragraph is correct.
"The allowable bearing capacity (qa) is the maximum bearing stress that can be applied to the foundation such that it is safe against instability due to shear failure and the maximum tolerable settlement is not exceeded. "
Allowable soil bearing capacity is not the maximum stress that can be applied to the foundation. It is the maximum stress that can be applied by the combined loads of structure and foundation TO THE SOIL.