Boussinesq Equation for Surcharge and Retaining Wall Design

[PatsSuperfan]

I have searched the web for an example and am unsuccessful in finding a retaining wall design using boussinesq's equation for surcharge. Does the surcharge have to be calculated twice - once applied to the stem and once applied to the virtual back behind the heel - to design the stem structurally and then check the wall for overturning/sliding? Doing so gives different offsets and values for the surcharge. Thanks in advance for the replies.

 

[FixedEarth]

See attached.

http://www.soilstructure.com/

 

[RFreund]

Nice example FE!

I would say that yes you could do two different calculations for an offset surcharge. One considering the offset distance from the back of the stem (for the design of the stem) and the other from the back of the heel for external stability design. Or conservatively use the back of the heel.

EIT

http://www.HowToEngineer.com

 

[Doctormo]

Boussinesq is a little screwy for loads that are some distance behind a wall. This is based on the assumption that a soils is homogeneous, isotopic, and perfectly elastic which it is not. Add to this the 2X factor and the results can be quite odd.

To simulate a uniform surcharge, the offset can be set at zero and the load width at infinity (1000 feet) and you will get an earth pressure coefficient of 0.50 or 2X that amount which is much closer to Ko or 2X Ko which would not be appropriate for a yielding wall system.

You can also put the offset in at 50 ft. and you will still get a lateral load which is silly nut the result of a perfect elastic analysis.

I like Bousinessq for strip loads directly behind a wall but that is about it. The 2X factor is fine for rigid structures but not really appropriate for yielding structures nor at the back of the footing since there is soil on both sides of the analysis.

 

[PatsSuperfan]

Thanks for the replies everyone.

 

[FixedEarth]

To me the backfill soil is more important than the rigidity of the wall. Think of that strip load as inducing a vertical stress component & horizontal stress component. The "radiating stress" is a function of the load intensity, strip width, backfill soil type and setback distance. Whether the "receiving end" of this stress is sheet pile wall or slurry wall is insignificant. The strip load could give hoots about this wall.

If you look at my attachment, it is implied in these equations that poisson's ration is 0.5 and my resultant, which is not shown, is 1.94 kip/ft applied at 4.5 below top of wall (assuming the footing strip is at the ground surface).

According to Bowles, he disagrees with the 2X and says just Boussinesq equation is sufficient. So the resultant now should be 0.97 kip/ft and location of thrust is still @ 4.5 ft below top of wall.

If you look at Terzaghi & Peck 1967, p 367-368, if we assume sandy soil, we get a thrust of 0.51 kip/ft, silty soil backfill, we get a thrust of 0.74 kip/ft & if it is Clay backfill we get 1.9 kip/ft (exactly same as 2X factor). So backfill soil type does make a difference.

If you look at Newman's,"Standard Cantilever Retaining Walls" book, 1976, p22, his method which is based on the City of Los Angeles Building Code, Circa 1970's, I get 0.07 kip/ft @ 4.2 ft thrust location. This is only 7% of the unmodified Boussinesq. It is just too small a value and I would dismiss this approach.

So we get a low value of 0.5 kip/ft & a high of 1.9 kip/ft. You decide. We didn't even get into trial wedge method & approaches by several other texts.

http://www.soilstructure.com/

 

[PatsSuperfan]

To be a little more specific with my question, the surcharge on the wall is from a Cooper E 80 live load. I have attached a sketch of the offsets and a reference I found from Union Pacific RR.

I have read Bowles' discussion regarding the Boussinesq equation, and am using the version provided by AREMA. Where I am having trouble, is applying the equation to the retaining wall for stem design and overturning/sliding. The figures provided in Bowles section 12-5 show the earth pressure force applied to the stem, and to the virtual back behind the heel. Also, every example I find has the equation applied to a cantilever wall.

The more I think about it, the equation should be applied against the wall, with different off-sets for change in stem thickness and location of footing. I also attached a excel file I created to find the resultant. Does that logic make sense?

Thanks for the replies everyone.

 

[PatsSuperfan]

Sorry its my first time with uploading, but here is the pdf.

 

[PatsSuperfan]

One more time lol.

 

[Doctormo]

Fixed Earth - If you are a true believer in Bousinesseq, your position is correct that the wall type does not matter. However, you will then acknowledge that a RR loading that is 50' back from a 10' wall will apply a lateral load of 30 psf since the calculation says it is so. There is clearly something wrong with the application of Bousinesseq in real life since the results can be senseless due to purely mathematical considerations of an elastic analysis.

I think you have to consider trial wedge as another way of looking at things. The pressure distribution of a trial wedge analysis is the opposite of a Bousinesseq analysis yet they are in the same text books just a few pages apart. They both can not be right at the same time. Bowles does a good job of discussing the differences although there is a time and place for each method in my opinion. Trial wedge is solely a function of geometry, soil strength and weight whereas Bousinesseq is solely a function of Poisson's ratio which explains part of the problem.

Trial wedge generally indicates that things that happen outside a 1:1 influence line from the bottom of a wall (or heel of the footing) do not have an effect on the wall design. This can be wrong in the case of really poor soils and/or back slope conditions or really large strip loads that can make the influence line a little greater. If the backfill is high strength or the loads are lower, the influence line zone can be shorter but 1:1 is good rule of thumb.

I might suggest that Bousinesseq is appropriate for higher loads within the 0.5H zone behind a wall but after that I think trial wedge may be more appropriate which is basic active earth pressure theory that depends on a yielding wall structure for its equilibrium analysis. Non yielding walls have to be more careful since there is no way to dissipate load through mobilization of the soil shear strength thus Ko designs and Bousinesseq pressures.

Carusso24 - If you are doing a project for UPRR then you follow their manual regardless of what is right or wrong. Railroads tend to be like this.

I am not trying to be difficult but I have wrestled with this problem for years and there still is no solution, just good engineering judgement for the particular situation at hand. I tend to favor trial wedge primarily for its ability to deal with back slope configurations and soil strengths for most retaining walls. I have done UPRR walls per their manual as well. I hope to solve it one day.

 

[RFreund]

FE and Dr. Mo good comments.
I hope to someday sort through all the different methods I have seen and used and write a short post about them (when can we start using FEM for this stuff ?)

I think what Curusso is asking though is, can he set up two different cases? Meaning a case for pressure against the stem to design the reinforcement for the stem and a case for stability so the wall can be checked for bearing, sliding and OT.

I would say yes. Meaning if you have 2ft toe and a 10ft heel and your offset surcharge is 20ft from the toe. Then for your stem case you would have an 18ft minus the stem thickness offset and for stability you would have an 8ft offset.

Maybe I'm missing the question.

EIT

http://www.HowToEngineer.com

 

[PatsSuperfan]

RFreund is right in that was the question I was asking. I understand the load case for overturning/sliding, in that projecting the earth pressure force against the virtual back behind the heel neglects contribution of its vertical component to resisting overturning/sliding. The more I think about it, I agree with Doctormo that its inappropiate to apply it behind the footing since there is soil on both sides of the analysis. Also, Boussinesq doesn't yield a vertical component to the surcharge earth pressure force, so maybe one load case is all that is needed. Applied on the retaining wall with different offsets for the stem and heel?

Also Doctormo, after reading Bowles' section on surcharge my interpretation is that there are three ways to calculate the surcharge earth pressure force: (1) equaivalent infinite strip, (2) trial wedge, and (3) boussinesq. I believe he says that almost every surcharge can be converted into an infinite strip load if its within the rankine soil wedge. Trial wedge is accurate for surcharges within the rankine wedge, however inaccurate but conservative for surcharges outside. Boussinesq probably gives a more accurate value for surcharges outside the wedge, and he then goes into discussion of the 2X thing (didn't read extensively into that section).

I am only checking a calculation, so I want to make sure it is being applied correctly. If I was calculating it myself, I would probably calculate the surcharge all three ways and use judgement to determine a value/method.

Also, the most interesting thing I found in the chapter was his discussion on the case when there is limited backfill, and the need to account for backfill compaction. I wish there was more written on that.

 

[Doctormo]

Carusso24 - Loads within the Rankine wedge (approx 50%H) are obviously the most critical. However, the trial wedge analysis with either Rankine or Coulomb assumptions will get flatter than 50%H if there is a back slope or the strip load requires it for maximum load.

I do not always agree with Bowles as there are lot of different situations so sweeping statements tend to be wrong at times. The fundamental assumption of trial wedge is active earth pressure where as Bousinesseq is much higher. If you use Bousinesseq for a uniform live load, the thrust is a K = 0.50 (or 1.0 if using 2X) vs. a Ka of 0.33 for active EP. The resultant forces can be in different positions also which affects overturning.

I would not use the words more accurate or more conservative as they are different methods with different results for different conditions. Just have to learn when to use them and for what. I use them both.

 

[FixedEarth]

RF - I agree Finite Element method is another tool to use.

DM - Yes I like Boussinesq method. The 2:1 V:H footing stress distribution by Boussinesq will miss the stem+footing, yet we still get calculated lateral stress per Boussinesq method. More research is needed.

http://www.soilstructure.com/

 

[BigH]

Just an observation . . . for a surcharge as far away as you say, you might have more influence on the earth pressure on the wall for the soil neglecting the surcharge just due to variations in the unit weight of the material behind the wall!

 

[Doctormo]

BigH

Valid point as well as what is the strength of the heavier material.

I quote from Terzaghi "For example, pressure cell measurements on the back of a reinforced concrete retaining wall 34 ft. high indicated that within one year the pressure varied from the average value by ±30% (McNary 1925). The maximum value of the earth pressure of backfills subject to seasonal change is greater than the Coulomb or Rankine value."

The exercise is actually hopeless when you consider real world effects, actual strengths and weights, non-homogeneous materials, etc and thus we use classical earth pressure theories with a bit of conservatism in the design assumptions since the process seems to work ok based on the number of retaining walls that perform just fine historically. On the other hand as Terzaghi notes, many walls may be "safer" than necessary due to actual design situation and materials used.

In my opinion, the main difference now is there are more people pushing the envelope with higher soil strengths and more liberal analysis than there used to be in order to be "more competitive" and the contractors try harder to reduce their costs as well. Thus the overall conservatism of retaining walls is probably less now than in the past so history may not be a good guide of future performance.