Boussinesq - at rest pressure
Boussinesq - at rest pressure
(OP)
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 K0 = 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?
For example, if we have a soil with φ = 30° behind a wall, we would have an a rest pressure coefficient of K0 = 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?





RE: Boussinesq - at rest pressure
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/
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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.
RE: Boussinesq - at rest pressure
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.
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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)
RE: Boussinesq - at rest pressure
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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 ...
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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-elasti...
The comparison spreadsheet near the bottom is the spread sheet I use.
Although another spreadsheet can be found here:
http://files.engineering.com/getfile.aspx?folder=4...
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/getfile.aspx?folder=c...
EIT
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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
RE: Boussinesq - at rest pressure
http://files.engineering.com/getfile.aspx?folder=b...
EIT
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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
www.HowToEngineer.com
RE: Boussinesq - at rest pressure
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.