Boussinesq - at rest pressure

[jeanb88]

I face a strange problem when calculation the loads on a retaining wall.

For example, if we have a soil with φ = 30° behind a wall, we would have an a rest pressure coefficient of K[sub]0[/sub] = 1 - sin φ = 0.5. If we have an infinitely extended surcharge of 50 kN/m² we would thus have 25 kN/m² lateral earth pressure. I suppose this is logic for everyone.

If we have a line load behind the wall on a given distance, I can use the Boussinesq equations. Attached I have a comparison showing the NAVFAC formula and the classic Boussinesq ones. This shows the famous factor 2 for the "mirror effect" (note: this factor is under discussion in Bowles book)

Now; I can integrate the Navfac and Boussinesq formula to calculate the effect of an infinitely extended surcharge (area load). This gives a rather stranger result. I would except 25 kN/m² for the case described above. This is correct for the classic Boussinesq equation but not with the NAVFAC equation. The latter is clearly twice the Boussinesq value.

So, is there something wrong with doubling the load for the mirror effect?

 

[FixedEarth]

Everything else being equal (poission's ratio, phi, Gamma, Ka or K0), a vertical uniform surcharge will yield the same thrust as a strip load only when the strip load is taken as far back as the active failure plane. So for example (phi=30, Gamma=120 pcf and H=10 ft) a 0.5 ksf uniform vertical surcharge will yield 1.67 kips/ft thust. To get that same thrust from a strip load, using the 2x equation, you will need a strip load of 0.5 ksf from 0 ft to 6.9 ft setback. However, still, the uniform surcharge thrust is acting at 5 ft below top of wall, while the strip load surcharge thrust is at acting 3.5 ft below top of wall. So the strip load thrust in this case will induce larger moment on your retaining wall stem.

So you have to use judgement on which surcharge is the most approprate to use. You would not want to use uniform surcharge when a strip load is actually the correct model and vice-versa. That is 250 psf uniform surcharge is incorrect for use when you have a fence wall with a 2 ft wide footing, at a setback of 5 ft to 7 ft with intensity of 1.2 ksf.

http://www.soilstructure.com/

 

[RFreund]

I think FE has a nice explanation.

Now that you are familiar with Boussinesq try to set up a calculation as Bowles suggests in his 4th edition (I believe) where you use a series of point loads to replicate a strip load. If you compare the results of this to the strip load equation, I get drastically different numbers. I will try to find my post on this.

Have question for you:
What did you use to compose that calculation? PYLab? Could you tell me more about this?
Thanks!

EIT

http://www.HowToEngineer.com

 

[Doctormo]

Keep in mind that Boussinesq does not use use the phi angle as it is an elastic soil analysis based on Poisson's ratio based on a lot of assumptions.

Usually the strip load equation will equal Ka = 0.5 for zero offset and an infinite load width as I recall. You can then double that if you believe that theory.

 

[jeanb88]

@FixedEarth : I follow your idea but I still does not explain the difference. The formula for the line load may be integrated to form an infinity extend area load. I did this for the navfac formula and for the classic Boussinesq formula. For 50 kN/m² (or the units you prefer) infinity extend area load I expect to find for normal soils (say φ= 30°) the at rest pressure to be about 25 kN/m². This is clearly the fact for the Boussinesq approach. Navfac formula leads to an at rest pressure of 50 kN/m² and thus an at rest pressure coefficient of 1,0 which you will have for φ = 0° or OCR > 1
Navfac formula uses thus the 2x factor for the mirror effect or is way to conservative.

@ RFreund: If I have some time this evening, I will test the summation of point loads. This must have the same results as the Boussinesq results I produced above.

About the software:
I have used the IPython web based notebook. (See

http://ipython.org/notebook.html)

I have full access to the python programming language and can use scipy, numpy and pylab to provide something similar to matlab. The notebooks can be converted to html and latex (and to pdf) afterwards.

 

[RFreund]

See here for my problem of point loads verse strip equation:

http://www.eng-tips.com/viewthread.cfm?qid=334036

also when you say you have full access to python. Is this through ipython or by other means?
Is it free/opensource?
Just out of curiosity have you ever tried Smath Studio?

EIT

http://www.HowToEngineer.com

 

[jeanb88]

@RFreund: IPython is indeed free/open source. IPython just replace the python shell. You can have it in command line, as a Qtconsole or as an interactive web based notebook.
I never tried Smath. We have Mathcad at our office but I mostly use IPython. (With the Unum module for unit aware calculation if needed)

 

[jeanb88]

I made the integration for surface load based upon the Boussinesq point load. Integration was done for a width of 50 x H and for -50 to 50 m. Again the same conclusions.

 

[RFreund]

Thanks for the information, I may have to look into that. I have been want to get to know python. Do you use if for most of your structural calcs?


Anyway, here's what I really meant (also I will look into explaining your question hopefully this evening)
First lets define "area load" as defined area so the surcharge has a defined width perpendicular to the wall and a defined length parallel to the wall. "strip load" = defined width perpendicular to the wall and runs continuously parallel to the wall. "line load" no width and extends continuously parallel to the wall. And "point load" which is a single point load.
So try the following (if you so care to do so):
1.) Use the strip load equation of navfac without the factor of 2 and also make sure it's not the equation that has "H" as opposed to pi in it.
2.) "Discretize" your strip load in to a series of point loads (i.e. select some width and divide the strip load in to point loads). Then use the boussinesq basic equation for a point load use summation to find the total pressure at each elevation. Bowles 4th edition gives an example of this.

You would think that the summation would like to the same result but I find that it does not.

I will try to give the doubling load some thought, but I've always thought the 2x was conservative. If you try the discretized point load (again see Bowles chapter 11 4th edition) you will get much less lateral pressure. Unless I am experiencing some sort of rounding error, which could be possible as I created the spread sheet in excel and there is a lot of small summation.... not sure.

EIT

http://www.HowToEngineer.com

 

[jeanb88]

Off topic: I use Excel, IPython and GNU octave as numerical packages for writing custom worksheets.

Concerning your comparison request. I'm not sure I got the whole part but I used the data of your other topic: 10' wall height, a 10' width strip load of 200 psf, at 5' from the wall. (I'm not familiar with the units but they seem consistent for me)

I used navfac DM7.01 strip load and the Boussinesq equation from Bowles which I integrate from -inf to inf and from x1 to x2 to form the strip load.

In total I found:
total Navfac = 330.482 psf (isn't this close to your value?)
total Boussinesq = 411.920 psf (I could not find your value for this one)

Can you check this against your result? I think I used the same values ...

 

[RFreund]

Sorry for carrying on this side conversation, however you have sparked my interest. Regarding excel vs Ipython vs GNU - Do you use these together or different calcs are in different programs? Meaning why not just use 1? I'd like to learn and use 1 software that can handle all (most) hand calcs that I would write. I'm sure excel w/VBA could do the trick but it can very difficult (or time consuming) for others to follow calcs in excel). I may put some time into Ipython it seems as this is gaining a lot of attention lately (maybe I'm wrong here).

Back to the calculation at hand. Let me go through your results as your answers are much closer than what I found.
Trying to work with units you're are not familiar with...now that's hard work!!

EIT

http://www.HowToEngineer.com

 

[RFreund]

OK so my results are a little different.

NAVFAC: I find the 330 plf (pounds per foot as this is the force on the wall as opposed to pressure)
Boussinesq: 227 plf

If I make the wall 20' tall the results get more distorted
NAVFAC: 505plf
Boussinesq: 297plf

Here is the link to where I discuss this:

http://howtoengineer.com/surcharge-analysis-elastic-methods-strip-load/

The comparison spreadsheet near the bottom is the spread sheet I use.
Although another spreadsheet can be found here:

http://files.engineering.com/getfil..._Boussinesq_Yakpol_update_multiple_loads.xlsm

I will look at this closer to see if I can find a difference.

Someone also posted this at one point in time but I've never been smart enough to read past the first sentence:

http://files.engineering.com/getfil...rce_at_a_point_in_the_semi-infinite_solid.pdf

EIT

http://www.HowToEngineer.com

 

[RFreund]

One other thing I was wondering if you could try ->
Currently your setting nu=0.5 what result do you get if you set this equal to 0.33 and/or 1.0?
Thanks.

EIT

http://www.HowToEngineer.com

 

[jeanb88]

Just out off curiosity, I checked my Boussinesq value with a double integration of point loads against a simple integration of line loads. The latter gives the same as the navfac approach but is however smaller than the value from the point loads. Your value from the point loads is very low. Can the reason be that I can calculate from -infinity to +infinity. (quad uses an integral transform to change the boundaries [0,1] )

Varying the poisson coefficient yields negative lateral pressure on the wall which is not possible.

About the side conversation: I use the tool which fits best for my needs. Calculations with lots of tables are best fit for Excel. For heavy simulations which lot's of "if statements", optimization, etc. I make my choice between Octave and IPython. Capabilities are about the same. I prefer IPython because it has a cleaner style and the nice web notebook interface.
Example of calculations:
Bearing capacity of foundation, torsion on concrete section, simple retaining wall -> Excel
Lateral load of pile, FROM analysis of structures, etc -> IPython because it requires some non linear equation solving.
Most economical solution of rebar -> Octave because it has much easier optimization routines than python

 

[RFreund]

That is strange. Attached is a segment from Bowels which shows where the idea comes from. I will try to devote some more thought to this on the weekend.

http://files.engineering.com/getfil...e=Bowles_5th_Ed_Ch11_-_Lateral_Pressure_2.pdf

EIT

http://www.HowToEngineer.com

 

[RFreund]

You know, it seems clear to me now. I believe you are correct. When using the "discretization method" (where you simulate a strip load with a series of point loads) is really more like a line load that runs perpendicular to the wall. The only adjustment for the fact that this load is continuous parallel to the wall is that it uses the plane strain nu. However it basically assumes there is no load parallel to the wall, i.e. it is not from -infinity to +infinity how your integration is done.

If you have time read page 365 from the above attachment. Maybe I'm misreading this but I think Bowels method of setting up that program is in error. He states "For a line load with the plane strain assumption, use a single point load perpendicular to the wall location where the pressure profile is wanted and use u' (plane strain). For a strip use a unit width opposite eh the wall, divided into as many unit areas as necessary to define the strip width and again use u'. For a finite-loaded area divide the load into as many unit areas as necessary and use u (not u')....."

Thanks for integrating the equation both from a point load (double integration) and from the line load (single integration). Interesting results.

Back to the original question:
I think you are basically asking can we say that the factor of 2x is too conservative? Is this the original question?
Well it probably is conservative in some cases and usually I will take CivilTechs approach -> 1 for yielding structures (MSE (actually even less for MSE) or steel cantilever/embedded), 1.5 for semirigid (cantilever concrete) and 2 for rigid (braced). Personally there are other techneques as well if the load is near the wall. You could do a trial wedge or Terzaghi gives some factors based on soil type.
Basically I think what happened is that you had the boussinesq equation and you had some tests (Spangler, Gerber) that showed found the stress was higher than that predicted by Boussinesq. So they said 2x would cover these test results. However must of the testing was done with old equipment and with point (wheel) loads. There may have been some scale effects going on. However I can possibly see justifying the 2x factor in that if you have a non-yielding wall you create more of a passive pressure situation when you apply a surcharge. Meaning that passive pressure is cause by an object moving or compressing the soil. The surcharge is compressing the soil on to the wall (well sorta) and thus creates this "passive" condition that you could argue (maybe) is higher than an at rest pressure.

Side conversation:
Does the program do symbolic integration? Can it integrate with units? Can it solve non-linear equations w/or w/out units?


EIT

http://www.HowToEngineer.com

 

[jeanb88]

Ok about the original question. I think I will just calculate conservative unless this yields very over dimensioned structures.

Side discussion: sympy can be used to do symbolic calculations (including integrals etc.) It also contains a unit module but it is not really useful. Every SI unit is set to its base unit. Consider the following example:

>>> from sympy.physics.units import *
>>> a = 1.0*m
>>> b = 3.0*m
>>> Q = 20.0*kilo*N
>>> Q/(a*b)
6666.66666666667*kg/(m*s**2)
>>> _ / (kilo*N/m**2) #convert to kN/m² but the unit will be lost
6.66666666666667
>>>

I use the Unum package for units.