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Lateral pressure due to strip load surcharge

In Principles of Geotechnical Engineering by Das, the lateral stress due to a strip laod is given by:
Sigma(x)=2q[Beta{Sin(Beta)(Cos(Beta))}]/H
Where H is the height of the wall. Intutively, it seems that the lateral pressure/stress from a given loading, a given distance from the wall should not be dependent on the height of the wall. Some other references, such as Poulus and Davis' Elastic Solutions book give the same formula but with pi instead of H. Can someone shed light on this?
Thanks 

GTeng (Civil/Environmental) 
21 Sep 04 15:15 
The NAVFAC Design Manual and Elastic Solutions for Soil and Rock Mechanics by Poulos and Davis give similar forms for calculating horizontal stress, i.e.,
Sigmax = (q/pi)[alpha  sin(alpha)Cos(alpha +2delta)]
where alpha is the angle encompassed between the left and right edge of the strip loading, and delta is the angle encompassed between vertical line through point of interest, and the near edge of the strip load.
However, some sources, like the American Railway Engineering and Maintenance of Right of Way Association (AREMA) design specifications, and the sheet pile design manual, present similar equations with varying definitions of angles) but with a factor of 2 added to the numerator. So depending on the source used, the calculated stress can vary by a factor of 2. Anyone know the real formula? 

BigH (Geotechnical) 
21 Sep 04 23:36 
The "real" formula is without the factor of two; this has to do with Spangler's experiments back in 1930s and the "mirror image" theory. Bowles (5th Edition) gives a good discussion of the background of surcharge pressures  Section 1113. While some (sorry, mon ami Focht3) do not like the late Dr. Bowles' book), it does give good background that you can pursue as you wish. One point is that poisson's ratio does have a significant effect. Hope this points you in a direction of study. 

kkoloj (Civil/Environmental) 
3 Oct 04 13:19 
earth321:
I know exactly what you mean about how intuitively one would think that the horizontal stress from a surcharge load should be indepedent of the wall height. I have spent a lot of time gathering information (articles, sections of books, etc.) about surcharge loads (infinite strip, finite strip, line, point, and sloped backfill) behind retaining walls in order to write spreadsheets for a structural standard pool plan. I have also taken the time to graph how the horizontal stress varies with depth from a line load surcharge. I have done this for several wall heights (e.g., 2', 4', 6', and 8'). Then I used Mathcad to integrate the areas to determine the force behind the wall. What happens is that the force is the same regardless of the wall height. MY question is if one is designing say an 8ft retaining wall with two stems (one from 0' to 4' and the other from 4' to 8'), what does one use for the upper stem. Should one use the stress distribution for an 8ft wall then integrate from finish grade to the location of the upper stem (upper 4 ft), or when designing the upper stem simply use the stress distribution for a 4ft wall??? I think the latter is overly conservative.
SORRY for the long message, but I think the discussion above is more for line and point loads behind retaining walls where the stress is inversely proportional to H.
I will refer you to a great article that solves strip load surcharge analytically. The equations in this article do not have the H in the denominator. Please see...
Total Lateral Surcharge Pressure Due to Strip Load by Ramon Jarquio, J. of the Geotechnical Engineering Division, Proceedings of the American Society of Civil Engineers, Vol. 107, 1981, pp.14241428. 

GTeng (Civil/Environmental) 
6 Oct 04 15:57 
Thanks for your reply. I look forward to another refernce on the issue. Could you describe the geometry of the wall in more detail? I don't understand what you mean by two "stems". If I understand the geometry correctly, in my opinion, the design of the upper 4 ft portion of the wall is the most straightforward aspect of the problem. The evaluation of bending and shear above this depth (4 ft)isn't really affected by the geometry of the portion of the wall below, and the earth forces would be calculated using the correct form of the Boussinesq or empirically modified formula, if anyone can figure out the true form of the equation. If there is a horizontal stem at 4 ft depth then the earth pressures below that point would presumably be less than the "full" value, depending on the strength and rigidity of the stem. The minimum value would be the earth pressure from the weight of the material below the upper stem, plus any other surcharge loads that are not "shadowed" by the upper stem. Am I totally misunderstanding what you mean by two stems? Incidentally, Bowles presents a discussion of the "mirror" load which one of the earlier postings refers to. Mathcad can be used to integrate that formula. Height of the wall has no influence but poissons ratio has a big effect. 

kkoloj (Civil/Environmental) 
7 Oct 04 23:59 
GTeng:
When you said "I look forward to another reference on the issue", were you interested in another article?
When we design tall retaining walls, we use multiple stems to try to engineer a slightly more efficient retaining system. In order to do this, the Retain Pro software we use allows us to design multiple vertical segments of the wall. Say for a 15ft wall, we might have a bottom stem (measured from the top of footing) that is 3 ft tall and consits of 16" concrete with #7 at 6" o.c., a middle stem that is 6 ft tall (measured from the top of the bottom stem) that is 12" concrete with #6 at 12" o.c., and a top stem that is 6 ft tall with #4 at 16" o.c. You can see the efficient "step down" of concrete and steel.
However, my question, which I think you addressed, was how would you design say the middle stem when you have a line load surcharge. I agree, that all loading below where the middle stem begins will not impart any force on the middle stem. And with typical triangular loading for a cantilever wall the solution is quite easy. Yet I see a problem when you consider a line load surcharge (I admit I may be making this more difficult than it is).
With line load surcharge, H is in the denominator of the horizontal pressure equation (modified Boussinesq). With a given line load Q, if you plot sigmaH versus depth for two different H's and integrate you will get the same horizontal resultant force. Applying this to my multiple stem wall: if you plot sigmaH first for the entire 15 ft and integrate you will get a horizontal force used to design a 15 ft wall. But what about the middle stem? Do you (a) simply integrate your previous sigmaH versus depth, from finish backfill to a depth of 15'3' = 12', or (b) recalculate sigmaH with the same Q for a wall that is 12'. If you do this, you will get the same force you found in (a). Again, (b) seems incorrect?
Thanks for the response. 



