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footing with biaxial moment
4

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.

RE: footing with biaxial moment

My foundation design textbook has a method for calculating effective widths and lengths for biaxial footings, but the solutions result in charts that you have to read.

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

There are various solutions but a trial and error approach as outlined in Peck/Hanson/Thornburg "Foundation Engineering" (Mine is 2nd ed. -1974 p391) is rather straight-forward.    

RE: footing with biaxial moment

StructuralEIT:

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

i have worked out eccentricity for both directions. If one will mean pad is non bearing for a length (say due mx) I have used that bearing pressure in my P/A +- Mx/Zy +- My/Zx formula then for the other way applied the reduced bearing area to work out my Z in other direction.

Sorry if the symbols are different from USA but you get the idea.  

RE: footing with biaxial moment

Here's a PDF of my standard mathcad file.  The approach is simple.  Determine the zone from the eccentricity and then determine the pressure from the formula for that zone.  Unfortunately, I didn't record where this was published (duh), and the engineer who gave it to me has passed away.  If anyone knows where this comes from, I would appreciate that info.

RE: footing with biaxial moment

As a first approximation, calculate the resultant moment acting on the footing, i.e. the vector sum of the two moments.  Assume the line of zero pressure occurs parallel to the resultant moment (using the right hand rule).  Take an educated guess where the zero pressure line is and determine whether the resulting stress block satisfies both load and moment.  Modify the position of the zero pressure line until load and moment are approximately satisfied.  

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

Last one of these I did I modeled in RISA3D on compression-only springs. Took me all of 20 minutes.

Once you get outside the kern in one or both directions things go haywire for bi-directional bending.  

RE: footing with biaxial moment

From personal experience, here are my views:
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

(OP)
Well, I'm just evaluating software, so I think it important to check by hand.

RE: footing with biaxial moment

StructuralEIT...good point...most states require hand check on the software, or at least same model run through two or more software programs.

RE: footing with biaxial moment

Are we talking about a square footing?  Why not give us the load, P also Mx and My and see how we might deal with it?

Best regards,

BA

RE: footing with biaxial moment

Do the moments happen at the same time or independently?  

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

(OP)
The moments are concurrent.  These are all service loads using the appropriate load combinations.
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

(OP)
I can get within 100 psf (of the max bearing pressure spit out by the program) using the method I stated above, but I'd really like to use a more rational approach and see where I get.

RE: footing with biaxial moment

can someone convert k to kg or KN for me and tell you what I get using spreadsheet

RE: footing with biaxial moment

That size seems about right.  qmax = 2.36ksf is what I'm getting with that size footing.  

RE: footing with biaxial moment

(OP)
Willis, can you try it with P = 20.818k, Mx=87.842 k-ft, and My=4.226k-ft?

RE: footing with biaxial moment

When I add in the weight of the footing, P = 50 kips.  The My moment is so small it is basically negligible.  And so q = 2P/3aB = 1.03 ksf.  I think you can make the footing smaller.

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

(OP)
my P had the footing weight added in already.  I think I noted that in the post.

RE: footing with biaxial moment

1.89ksf

RE: footing with biaxial moment

I get about 1.9ksf also.

RE: footing with biaxial moment

(OP)
Interesting.  I come up with 1.88ksf and the program is spitting out 1.93 ksf.  That seems close enough to me, it's just that my method involves a very "non-engineering" type estimation, which I really hate to use.

RE: footing with biaxial moment

I think you are using the wrong shape of footing.  Why not elongate it in the direction of the larger moment?  That should permit a reduction in footing area.  Even better, if possible, use a combined footing so that the loads are balancing each other.

herewegothen,

You've got to be kidding, right?

Best regards,

BA

RE: footing with biaxial moment

(OP)
BA-
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

StructuralEIT,

Okay...no offense intended.

Best regards,

BA

RE: footing with biaxial moment

(OP)
none taken.

RE: footing with biaxial moment

There still seems to be something wrong with the results you guys are getting.  First of all, a 10'x10'x2' footing weighs approximately 30 kips.  I don't know what you mean by "an appropriate DL factor" but, for the sake of the problem, let us say that:

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

(OP)
If the controlling load combination is 0.6DL-W, then you need to multiply the ftg weight by 0.6 - that's what I meant by "the appropriate DL factor".
I did tweak the original loads I gave a little to be exact, see the later post.  

RE: footing with biaxial moment

So, are we all in agreement that the maximum soil pressure with the given loading is in the order of 2.35 ksf, not 1.9 ksf?  If not, then I still do not understand how you arrive at the maximum pressure.

Best regards,

BA

RE: footing with biaxial moment

I agree with myself.  =)

RE: footing with biaxial moment

The 10' X 10' X 2' footing weighs 30 k
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

(OP)
chicuck,
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

For the loads that were specified, the formulas I attached in my earlier post match WillisV's results exactly.  Although the PDF is a printout of a Mathcad file, all the formulas are shown, and can easily be followed in a hand calc.

RE: footing with biaxial moment

Using the revised values of 20.818k and 87.842k-ft, I would agree that the maximum pressure is in the order of 1.9 ksf.  

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

(OP)
I don't think that is such a bad safety factor for OT when you consider that it is the 0.6DL-W load combination.
That's actually better than 1.5 for 1.0DL and wind.

RE: footing with biaxial moment

Well, I agree with you on that point.  The 0.6 factor is certainly not in my code (Alberta Building Code) and from your latest comments, it would appear that you don't take it very seriously either.  

Best regards,

BA

RE: footing with biaxial moment

I never design a footing to land outside the kern distance. Several geotech books I have seen recommend not doing this either.

RE: footing with biaxial moment

(OP)
I would generally agree with not having a footing with partial bearing, but sometimes you could end up with footing with huge plan dimensions and/or much thicker than they need to be.

RE: footing with biaxial moment

haynewp, what you say makes sense if it is a permanent load, but it is a bit conservative if you are looking at overturning from wind load.

RE: footing with biaxial moment

Are you implying that because by a short term nature of wind thought?   

RE: footing with biaxial moment

Yes,

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

I can see that but I could also see compaction ocurring at the toe from a footing rocking back and forth under wind to create increasing deflections. But this is better answered by a geotech in my opinion, you may be right because it is not really my cup of tea.

RE: footing with biaxial moment

(OP)
miecz-

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

Risafoot will solve this problem using FEM.

RE: footing with biaxial moment

The AASHTO Code has a method for coming up with the qmax as well.  See Figure 4.4.7.1.1.1C in the 17th Edition.

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

StructuralEIT-

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

(OP)
miecz- Thanks.

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

With only P and Mx, the eccentricity is 3.84', or 0.66' from the edge of the footing.  With a triangular distribution of pressure, the effective width of the footing is 3 * 0.66 = 1.98' and the maximum pressure = 2.09 ksf.

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

Just a reminder, by reading your loads, the resultant force is outside the kern w/r to x-axis (ex = Mx/p > middle third), the footing is not fully engage in bearing, and its geometric properties have changed.

RE: footing with biaxial moment

(OP)
kslee- No kidding?!  That's been the whole point of this post.  It's a piece of cake if there's full bearing.

RE: footing with biaxial moment

(OP)
miecz-
I'd give you a second star if I could. Thanks!

RE: footing with biaxial moment

StructuralEIT:

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

(OP)
What?

RE: footing with biaxial moment

2.31

RE: footing with biaxial moment

(OP)
Thanks Willis.  I am trusting the formulas for miecz's sheet, but does anyone have an idea where they come from?

RE: footing with biaxial moment

If you have located the principle nutral axis, the area remains in compression is trapezoidal, all forces are shifted as well as the stresses. Check the P/A term, it should reveal something.

RE: footing with biaxial moment

The AASHTO Standard Specs have Figure 4.4.7.1.1.1C to solve this kind of problem.  It comes from the old AREA Railway Manual.  It gives the same results, but, because it is a graphical solution, it gives information that formulas do not.  Specifically, with the loads and dimensions you have, the bearing pressure is very sensitive to small changes in vertical load or eccentricity.  Just reducing P 6% to 17.56 kips raises the bearing pressure to 3380 psf.  Could your program be fiddling with your input?

RE: footing with biaxial moment

The graph and formulii look like the exact same ones that i found in an old foundations text.

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

(OP)
csd-

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

The book was 'Foundation Design' By Wayne C Teng, Prentice Hall 1962.

RE: footing with biaxial moment

(OP)
That book is going for $325.00 online.  Is there any chance you have those pages scanned?

RE: footing with biaxial moment

It must be out of print as it is selling as a collectors item. I see no issue with providing copies of 46 year old out of print books so I will scan and post it on Monday.

RE: footing with biaxial moment

(OP)
Awesome!  Thanks!

RE: footing with biaxial moment

Hey!  I mentioned the Teng text way back at the beginning of this thread (on January 29).

No one gave me a star sadeyes

DaveAtkins

RE: footing with biaxial moment

(OP)
Dave-
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

Here is one way to do it.  

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

Oops, I meant to say:
until Cx = A/2 - Mx/P and Cy = A/2 - My/P.

Best regards,

BA

RE: footing with biaxial moment

(OP)
Is there anyone that has the Teng text who can scan and post the relevant pages?

RE: footing with biaxial moment

I have a copy of Teng at home.  I'll bring it in and scan it in the morning.

RE: footing with biaxial moment

If anyone is interested, the RISAFoot program (even the free Demo version) provides information that can be used to verify a bi-axial soil bearing profile.

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

In the attached diagram, the pressure at corner b and d is zero.  The pressure at c is k times the pressure at a, namely h.  The volume abcd is actually a truncated triangular pyramid, so can be expressed as abe - cde.  The volume and centroid of a regular pyramid is well known, so no need to integrate.

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

Bendog,

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

Viewing the soil pressure as a truncated pyramid makes it seem so easy.... At least in comparison to all the integration that I've been doing.

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

JoshPlum,

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

StructuralEIT, I don't know if this would influence your decision, but I found the book by Teng for about $150 here:

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.

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