Biaxial Overturning Factor of Safety 1

[RyUIUC]

I am looking at a footing that has biaxial overturning. The footing is not symmetrical and I have found that the factor of safety about each direction to be adequate but of course if the moments occur simultaneously I do not want to double count the resisting moments due to gravity loads, intertia of the footing. Thus I was thinking the formula below may be appropriate, thoughts please...

(1/FS)lateral + (1/FS)longitudinal = (1/FS)total

The idea here is that the resisting moment relative to the overturning moments is effectively the common denominator and thus not double counted in the total FS.

 

[JStephen]

I think my first approach would be to combine the moments and evaluate the overturning stability in the plane of the total moments.

 

[RyUIUC]

I have tried several examples and short of just finding the resultant overturning forces and resisting moment I am actually finding if I square all the fields above it is more accurate.

(1/FSlat)^2+(1/FSlong)^2=(1/FStot)^2

 

[ron9876]

The safety factor comes from 0.6D when you are using dead load to resist overturning.

 

[jeff91]

Assuming that the maximum overturning moments occur at the same time about each axis, an alternate approach may be to compute the FS against overturning about each axis individually but only take credit for 50% of the resisting moment within each individual calculation.

A more important check may be the bearing stress that occurs at each corner of the footing and verification if uplift is occuring at any of the corners.

 

[paddingtongreen]

The only exact way is to resolve the moments into a single moment in an oblique direction and then calculate the FS. This is particularly true if the resultant load is outside the kern.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[ToadJones]

I have run into this before.
I only checked overturning in each direction independently.

The more difficult task is determining the bearing pressures and % bearing area for a footing with biaxial moments. It can turn into a bit of a science project.
Checking overturning in the resultant direction may be even more difficult. About which point does one sum moments for the resultant direction?

 

[RyUIUC]

If you have a square footing and equal moments about each axis you would assume the resultant moment was in the direction of the corner at 45 degrees correct? Now your resisting moment arm just increased by the same factor your resultant overturning moment did (sqrt 2)and your total factor of safety is the same.

 

[JoshPlumSE]

I have always only checked the overturning in the two directions independently.

Though, I agree with PaddingtonGreen that the exact way would be to resolve this into a single moment along an inclined plane. Then calculate the overturning and resistance relative to that inclined axis.

 

[ToadJones]

My intuition tells me that for overturning on a rectangular footing summing moments about an inclined plane may be good for a check, but in reality I suspect the footing itself would not rotate about that plane. It would seem as though the footing may initially rotate about the plane, but then would lose some initial stability and tend to rotate in the horizontal plane as well.

 

[JStephen]

"If you have a square footing and equal moments about each axis you would assume the resultant moment was in the direction of the corner at 45 degrees correct? Now your resisting moment arm just increased by the same factor your resultant overturning moment did (sqrt 2)and your total factor of safety is the same."

I had to pull up a spreadsheet and play with the numbers a bit.
It looks like for a rectangular footing, the combined safety factor will always be in between the safety factors in the two principle directions, so if they're both okay, the combination is okay. This is for checking rigid-body overturning about a point, only.

Note that in the first case above, the soil bearing due to moment is doubled at that corner, so if that's the limiting factor, it's different.

For a round footing, the moment arm doesn't change, so checking in two directions is unconservative.