footing with biaxial moment
footing with biaxial moment
(OP)
I've been looking for a reference as to how to determine the max soil pressure for a footing that has moments in both directions, but only has partial bearing. The moment in one direction would the load in the kern (if it were the only moment), and the moment in the other direction would put the load outside the kern. I can't find a reference on this in my foundations book, and I could go through the math of it (but that would take a REALLY long time, and I honestly don't want to spend an entire day to figure it out), but I figured someone else has to have done this before.
My first inclination was to take the max pressure of the moment causing partial bearing and adding that to the max pressure caused by the full bearing (M/S), then I realized that I couldn't us the full S of the footing (for the smaller moment) because the whole footing isn't in bearing anymore. I tried estimating the amount of the footing that would be in bearing and using that S. That would get me close, but I'm really trying to be exact because I'm evaluating a program. The line of zero bearing stress is not perpendicular to either edge of the footing because of the moments in both directions, but again, I don't know how to address this without a day-long geometry session.
My first inclination was to take the max pressure of the moment causing partial bearing and adding that to the max pressure caused by the full bearing (M/S), then I realized that I couldn't us the full S of the footing (for the smaller moment) because the whole footing isn't in bearing anymore. I tried estimating the amount of the footing that would be in bearing and using that S. That would get me close, but I'm really trying to be exact because I'm evaluating a program. The line of zero bearing stress is not perpendicular to either edge of the footing because of the moments in both directions, but again, I don't know how to address this without a day-long geometry session.






RE: footing with biaxial moment
I have attached a link to a website that has an excel spreadsheet with sited references that will do a biaxial footing
http://www.seaofsc.org/Alex's%20Corner.htm
RE: footing with biaxial moment
RE: footing with biaxial moment
Any chance of using grade beams to avoid the problem? I always like to KISS the situation, if you know what I mean. Not that the solution you are looking for is hard, but it could save you concrete in the footing.
Mike McCann
MMC Engineering
RE: footing with biaxial moment
Sorry if the symbols are different from USA but you get the idea.
RE: footing with biaxial moment
RE: footing with biaxial moment
Then check the moment normal to the resultant vector. If it is not zero, modify the direction of the zero pressure line until it is close enough.
Best regards,
BA
RE: footing with biaxial moment
Once you get outside the kern in one or both directions things go haywire for bi-directional bending.
RE: footing with biaxial moment
1. As suggested by MSQUARED48, avoid the situation to the best you can.
2. As JLNJ pointed out, use FEM with compression only spring.
3. For personal satisfaction, you can write your own spreadsheet program assuming the footing and soil both are linear elastic (linear stresses). The difficult part is once a corner has developed negative pressure (uplift), the bearing area is reduced, and the new bearing area/neutral axes need to be found, its properties re-calculated (quite mathematically challenging), and load redistributed. The iteration process stops when the footing is fully in bearing.
Have fun.
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
Best regards,
BA
RE: footing with biaxial moment
If they happen independently, then use the worst case of two conditions, P + Mx and P + My.
If they happen concurrently, then resolve the moments to a Mxy and rotate the square footing to the axis of the combined moment, allowing a solution similar to a simple P + M solution for inside or outside of the kern.
Mike McCann
MMC Engineering
RE: footing with biaxial moment
P=20k
Mx=88k-ft
My=4k-ft
The shears are already included in the moments (that's why they're not included above), and the footing weight is included in P (at the appropriate DL factor). The progam is spitting out a 10'x10'x2'thick footing.
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
For those cases where the biaxial moments are both large, I use the graph on page 133 of Foundation Design by Teng.
DaveAtkins
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
herewegothen,
You've got to be kidding, right?
Best regards,
BA
RE: footing with biaxial moment
I'm just trying to verify what the program is spitting out, nothing more. There may be a time when this comes up in a job and I want to trust the program when it does.
RE: footing with biaxial moment
Okay...no offense intended.
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
P = 20k
Mx = 88k-ft
Assume that My = 0 for now.
The eccentricity in the x direction is 88/20 = 4.4', so for a 10'x10' footing, the effective length of bearing on the soil is (5 - 4.4)3 = 1.8' and the effective width is 10'.
The soil pressure has a triangular distribution. Average pressure = 20/(10*1.8) = 1.11 ksf. Maximum pressure is 2.22 ksf. Minimum pressure is zero. So far, we have said that My = 0.
If My = 4k-ft, surely the maximum pressure will increase slightly, so the maximum pressure by my calculation is approximately 2.22 + 0.13 = 2.35 ksf.
Am I misinterpreting the problem?
Best regards,
BA
RE: footing with biaxial moment
I did tweak the original loads I gave a little to be exact, see the later post.
RE: footing with biaxial moment
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
the soil above it weighs about 16.28 k for bearing depth at 3.5' below grade.
If total P = 20 k, then is there an uplift load on the footing?
chichuck
RE: footing with biaxial moment
No, there is no uplift on the footing. The axial load from the column (for the controlling load combination of 0.6DL-W is 2.818k. While the weight of the footing is 30k, you can only use 18k (because the load combination is 0.6DL-W), hence the TOTAL P=18k+2.818=20.818K (as noted in an earlier post). I am not considering any soil overburden on the footing (whether that is right or wrong is irrelevant to what I am trying to do), I can specifiy overburden in the program, but all I want to do is verify that any situation that the program is faced with will be done correctly.
BA,
No, the max pressure is around 1.9ksf (for P=20.818, Mx=87.842k-ft, and My=4.226k-ft).
RE: footing with biaxial moment
RE: footing with biaxial moment
If the original values of 20k and 88k-ft are used, the maximum pressure is in the order of 2.35 ksf.
This indicates that the footing is extremely sensitive to minor changes in load which suggests to me that this would be a very bad design, no matter what your computer program tells you.
One point to note is that the factor of safety against overturning is only 20.818*5/87.842 = 1.185 which is completely inadequate by any standard.
Best regards,
BA
RE: footing with biaxial moment
That's actually better than 1.5 for 1.0DL and wind.
RE: footing with biaxial moment
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
The only reason why it may not be a good idea for partial loading is because of settlement. For settlement you need more time than a gust of wind takes.
RE: footing with biaxial moment
RE: footing with biaxial moment
I'm looking a little more closely at the sheet you posted. I'm having a hard time following some things. For example, in the bearing pressure calcs, what is "S"? Also, for the Zone II calcs, it has ey in the denominator (which is 0), but the f2 value is the same as I would get by hand (but the equations shown don't work out because of the 0 in the denominator). What is the px in the Zone II calcs?
RE: footing with biaxial moment
RE: footing with biaxial moment
I am not sure if the same figure appears in the new one (LRFD only). I think it is strange that the code includes this figure because elsewhere in the code they do not permit bridge foundations to be loaded outside the kern...
On another note, we design sign and billboard foundations often enough, and it is entirely uneconomical to design a spread foundation with the resultant within the kern for wind load. It would irresponsible of an engineer to require the load to remain within the kern in such a circumstance.
RE: footing with biaxial moment
The "S" used in the bearing pressure calculations, has no name, it's simply a variable defined by the formula.
You're right, ey is 0. To avoid dividing by zero, I've adjusted ey (and ex) by ε, as shown in the formulas on sheet 1. ε is defined as .001 feet, but that definition is not shown anywhere on the printout.
px is the soil pressure at the center of the footing, if Mx=0k'. I use px to calculate the bending moment at the center of the footing, where I believe it is maximum.
Again, I don't know where the soil pressure formulation comes from. I got it at least 20 years ago and have verified it against many other sources over those 20 years. It is always dead on.
RE: footing with biaxial moment
miecz and willis-
Would it be possible for both of you to plug in the following loads in your respective sheets and tell me what you get?
footing is 9'x9'x2' (but the self weight of the footing is already figured into P - using the 0.6DL factor)
P=18.632K
Mx=71.52k-ft
My=5.646k-ft
Using miecz's formulas I'm coming up with 2306psf (which is roughly what I would expect), but this program I'm checking is showing 3380psf (almost 50% higher). I found another method that I was trying out, but that's only giving me 1330psf (which is way too low, so I'm throwing that out).
RE: footing with biaxial moment
Now, considering My = 5.64k' and a section modulus of S of 1.98*9^2/6 = 26.7 for the effective portion of the footing, the additional stress is M/S = 0.21 ksf.
Adding the two together gives 2.30 ksf.
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
I'd give you a second star if I could. Thanks!
RE: footing with biaxial moment
If you have done hand calculation by defining principle axes, you will know you haven't found/confirmed the maximum stress yet.
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
They can be back checked when you realise that the resulant is a triangular area with a stress increasing from zero to the maximum in the corner of the footing.
It is really just a geometrical problem with the same principles as the 2d case. e.g. centre of reaction = centre of loads e.t.c.
RE: footing with biaxial moment
Can you tell me what text that was?
I know it is a geometric exercise (and not difficult to work through), but is very time consuming to work through from scratch.
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
No one gave me a star
DaveAtkins
RE: footing with biaxial moment
You're right. Here you go.
The AASHTO spec does give equations that are less straightforward, but will get you the answer. I'm anxious to see the equations out of the Teng text.
RE: footing with biaxial moment
On the attached sketch, Vol. abcd is the volume under the stress block.
Cx and Cy are measured from the edge of footing to the centroid of the stress block.
h is the height at point a.
kh is height at point c.
0 is the height at b, c and e.
Using Excel, one can experiment with different x and k values until Cx = A/2 - P/Mx and Cy = A/2 - P/My.
Finally, calculate "h" to satisfy the equation that the volume under the stress block is equal to P, the applied load.
Best regards,
BA
RE: footing with biaxial moment
until Cx = A/2 - Mx/P and Cy = A/2 - My/P.
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
RE: footing with biaxial moment
That program will show you the soil bearing pressure at each of the 4 corners and will also display the distace from the point of maximum compression to the neutral axis location.
If you know that information it is not all that difficult to verify that the total pressure load equals the applied vertical load. I did this for one of our users yesterday. It takes me awhile to remember how to do the integration, but once I remember then it's not that bad.
You can even use this same information to verify that the centroid of the soil pressure corresponds to the load eccentricity location. But, the integration gets more complicated, and I tend to "guesstimate" it for my hand calculations instead.
RE: footing with biaxial moment
The procedure is to guess at x and k and iterate to a correct solution such that Cx and Cy correspond to the known location of the load P. When this has been found, the highest pressure, h (at point a) can be found by equating the volume of abcd to applied load P.
A bit messy, but it can be done.
Best regards,
BA
RE: footing with biaxial moment
RE: footing with biaxial moment
Thanks for sharing this. It is is a very comprehensive treatment of the subject of eccentric loads on rectangular footings.
By the way, my solution given in my last post agrees precisely with Case III from Teng.
Best regards,
BA
RE: footing with biaxial moment
Now I'm embarassed that I never made that connection. But, am very thankful to BAretired (and EngTips in general) for showing me the light. :)
Thanks!
RE: footing with biaxial moment
There is one slight disadvantage to the truncated pyramid model. That is, when My = 0, k is 1.0 and the expressions for Cx and Cy become indeterminate (0/0).
With the Teng model, D and A may be evaluated when R is 1.0.
Of course, when My is zero, it is not biaxial moment?
Best regards,
BA
RE: footing with biaxial moment
http://www.alibris.com/
Its used of course, but I have usually found their prices to be somewhat lower than others. Never actually bought from them before though. Just search "Teng Foundation" in the books tab.