Strip Load Surcharge Analysis using Elastic Methods

UPDATED – Problem Solved…

Here we will dig deeper into analyzing strip loads with elastic methods. First to avoid confusion between seeing similar equations which use different reference angles, we will set our own nomenclature.

Strip Load – Elastic Methods RSF Nomenclature

Nomenclature

K = Constant typically between 1 and 2, depending on the stiffness of the wall. It may be appropriate to use K=1 for flexible walls such as cantilevered sheet pile, segmental walls. It may be appropriate to use K=2 for rigid walls. Refer to CivilTech Software manual (http://www.civiltechsoftware.com/downloads/sh_manu.pdf).

Below are (2) common equations which result in the same solution but are shown differently. I will refer to these equations as the integrated method or equations. Most codes and design guides refer to these equations but it seems too differ on who is credited. Some refer to them as Boussinesq as modified by Tang or Spangler, others Mindlin or Terzaghi (although Terzaghi is mostly responsible for modifying point and line load elastic equations).

Note that β is located at half of θ and not half the width of the strip load.

The other

In Joseph Bowles’ “Foundation Analysis and Design” he discuses different methods of analyzing offset surcharges using elastic methods. He discuss the work of Spangler and the factor of two suggested by Mindlin. In summary he suggests that Spangler’s experiments were not accurate and possibly flawed due to the setup geometry. Also the equations derived by Spangler use a Poisson’s ratio of 0.5 which may not be correct for all soils. However Spangler’s equation for the strip load results in the same equation as shown above. Bowles also states that Mindlins reasoning for the factor 2 (Mindlin says that a rigid wall produced a mirror effect) is wrong.
Bowles suggests to only use the original Boussinesq equation.

[EQN – 1]

With these equations one can “discretize” the strip load. Meaning you divide the strip load into a series of concentrated loads. Then apply the Boussinesq equation to each point load. Then the sum the results to find the resulting pressure at a certain elevation on the wall. You then find these resulting pressures at a certain number of elevations on the wall and sum these to find the resulting total force. It should be noted that Poisson’s ratio should be modified so that it represents a plan strain condition.

where represents a triaxial condition
Refer to Excavating Systems, Planning, Design and Safety, 2009 for the following suggested values of
Moist clay soils: 0.4-0.5
Saturated clay soils: 0.45-0.5
Cohesionless, medium dense: 0.3-0.4
Cohesionless, loose to medium: 0.2-0.35

Here are some notes on the discretized method (semi-large file – about 10MB, I will try to shrink later):
Boussinesq Strip Load Discritization Method 2

Now my problem is this – when I compare the “discretized” approach to the integrated I do not get similar values, not even close. I have to think that I am in error somewhere, but I cannot find where. I have compared spreadsheets to other programs to hand calcs. I will continue to search, but for now, I have attached a couple hand calcs showing the discrepancy. I will also attach a spreadsheet comparing techniques.

UPDATE:

I believe I have now solved this problem. Where I went wrong – When integrating for a strip load you would use boundary conditions from +infinity to -infinity. Which is NOT what I was doing. I was taking the load as if it was an AREA surcharge over a one foot length (along the wall). This is not right. I must say though that Bowles does a poor job explaining that part (in my opinion). Therefore if you want to simulate a stip load you MUST enter a large “Y” dimension so that the program realizes the effect of a STRIP load and not an AREA load. This should really be built into the worksheet but is not yet.

Attached is a hand calc comparison between the “integrated” and discretized approach.
Boussinesq Integrated vs Discretized
Elastic Method – Compare Integrated to Discretized

Elastic Methods Nomenclature Notes

Elastic Methods Spangler Derives Integrated Method
Offset Surcharge Strip Load Comparison Spreadsheet

Another problem when using elastic methods is that all equations assume that the load is applied at the same elevation as the top of the wall. How can we handle a slope atop the wall with a load applied at the top of the slope? Well the railroad design manuals (See UPRR Shoring Manual) distribute the load down through the soil at a certain angle (they use a 2V:1H) and find a new uniform load applied at this elevation.

Elastic Methods – Slope Diagram

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Analysis of Offset Surcharges on Retaining Walls Using Elastic Methods

Offset surcharges are always up for some debate. Which method to use? What is the line of influence? Is the wall flexible or rigid? Well in this segment we will cover the analysis of surcharge loads on retaining walls using elastic methods. You may commonly here this as a Boussinesq analysis or a Boussinesq as modified by Terzaghi or Tang. Also other equations have been produced by Spangler and Mindlin. Trying to wrap you head around all the different equations and who did what can be confusing. We will try to straighten some of this out and present some of the different approaches and how most of the equations are similar but the equations are presented slightly differently.

Constraints and Assumptions

There are generally 4 types of surcharges considered – Point load, Line load, Strip load and Area load. Point loads and area loads have a finite length. Line and strip loads are assumed to be of infinite length parallel to the wall. The back slope is generally considered to be flat. Elastic methods do not consider any soil parameters such as effective shear strength or wall friction. The only way to try and account for different soil types is by adjusting Poisson’s ratio or adjusting the factor of safety. However most of the theories assume a Poisson’s ratio (u) of 0.5 which simplifies the analysis, but more on this latter.

The basis of the elastic methods discussed below is Boussinesq’s equation for a surcharge on a semi infinite mass

Equation 1:

Boussinesq Equation

Boussinesq Equation Diagram

Where:

P = point load

v = Poisson’s ratio

History

Most of this is from Bowles 4th Edition Foundation Analysis and Design.

1936 Spangler – Performed experiments to measure the lateral pressure on a wall due to point loads form a truck behind a rigid retaining wall. Spangler used Boussinesq’s equation with u=0.5 and found that the actual lateral pressure was approx 2x the pressure found by equation 1 with u=0.5. Bowles points out some flaws of this experiment – loose soils with a wall of ‘finite’ length and ‘old’ technology used in the load cells.

1936 Mindlin – Discusses the results of Spangler. He explains that the 2x factor could be due to the rigid wall producing a mirror load effect. This reasoning is disputed by Bowles.

1954 Terzaghi – suggests 0.4*H (wall height) is a critical distance behind the wall where two different equations should be used for point loads and line loads. Terzaghi proposed (2) different modified boussinesq equations for point and line loads both using v=0.5.

1962 Teng – Similar modification claiming that the pressure on rigid walls for strip loads should be 2x the results obtained by the ‘integrated’ Boussinesq equation. Some also credit Teng the strip load equation shown below as a plastic’solution as opposed to the Boussinesq solution which is an elastic solution (See Civiltech Software Design Manual and Foundation Design by Wayne C. Teng 1962).

1972 – Rehnman and Broms showed that when the soil behind the wall was dense the lateral pressure from point loads was much less than with loose soils. Also gravelly backfill’s produced larger lateral pressures than finer-grained soils. This observation would mean soil state and Poisson’s ratio are significant.

 When and How to Apply Elastic Methods

First the how – The equations presented below give pressures at a certain height on the wall. Therefore it is suggested that a program or spreadsheet is made that uses these equations to find pressures at set intervals or segments of the wall (sigmah). Then the force is found by multiplying the height of this segment by the pressure (Fi=force at elevation ‘i’ = sigmah * hseg). Then you solve for the height above the bottom of the wall to this segment (yi = height to force, Fi). Then you solve for the resultant force and it elevation. Fh=sum(Fi’s) Then find the elevation Ybar = Sum(yi’s * Fi’s)/Sum(Fi’s). This is similar to finding the centroid of an area by parts. This pressure/force is then superimposed on the soil pressure.

Zone of Influence

Second the when – This is a much more difficult question to answer. There are many different ways that an offset surcharge maybe handled see here for a more indepth look. So if we are specifically looking at when to apply elastic methods the answer would be – anytime. We can superimpose the results of the elastic method results on to the soil pressure as would be found using Rankine or Coulomb equations (see here for refresher). However elastic methods will yield a horizontal force for any surcharge applied at any distance behind the retaining wall. This may be overly conservative however it is up to the engineer to determine what distance beyond the wall that the surcharge would have no affect on the retaining wall. Usually this information is given in the governing code. Some examples – many use the Rankine or Coulomb failure plane as the determining distance however a trial wedge may give different results when finding the failure plane angle. Edward White has suggested that pressures may distribute down at 1V:1H (See Foundation Engineering Handbook by Hans F. Winterkorn and Fang). Sometimes a 2V: 1H is used as in a Meyerhoff or Boussinesq bearing pressure distribution. While a 2V:1H has been found accurate at short depths (See Bowles 3rd Edition p172) and is usually conservative when finding vertical stress at a certain depth but can be unconservation when finding horizontal stresses. Most rail road design manuals suggest at 1.5H:1V influence line. The NCMA recognizes a 2H:1V as a conservative estimate to find the horizontal distance to which you may disregard surcharge loads.

Elastic Method Equations

Below are the ‘typical’ equations used in most design manuals for using elastic methods. There are links to other posts which will give further discussion, examples and spreadsheet calculations. The equations assume rigid walls (pressures maybe less for flexible walls, Civiltech Software recommends 0.5 for flexible, 0.75 for semi-flexible and 1.0 for rigid – see further discussion), a Poisson’s ratio of 0.5, and the pressure maybe combined by method of superposition.
Nomenclature:
σh= horizontal pressure
H = Height of wall (excavation)
x1 = m*H = distance to surcharge (point load, line load, strip load)
zi = n*H = distance from top of wall to elevation under consideration
Q = point load or line load surcharge
q = uniform surcharge

Point Loads

Point Load Section

Point Load Plan View

Boussinesq equation as modified by Terzaghi.

To evaluate a point at an angle to the point load along the wall:

See here for a thorough discussion

Line Loads

Line Load Section

See here for a thorough discussion

Strip Loads

Strip Load Section

See here for further discussion.

Area Loads

Will update shortly

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Trial Wedge Method

Update: I will be updating this post or possibly creating a new post that demonstrates a simpler, clearer way to implement the trial wedge method. It should be completed by 2013/2/17.

Update: I realized a couple of algebra errors were made and now the spreadsheet should be functioning correctly. I have verified with Keystone’s Retaining Wall Software “Keywall” (its free upon request). I still have the “CT” equations as well which could be deleted as they are there for comparison. I believe the results are different because I derived my equations with a different geometry. I will also try to post hand calc example as well.

Trial Wedge Method

Trial Wedge Spreadsheet (Use RSF Equations)

Cali T&S Manual 2011

Trial Wedge Method Video (coming soon)

There are many graphical solutions to the trial wedge analysis shown in textbooks however I was not pleased with them. They usually use ‘force polygons’ where you have to find the weight of each soil wedge for all the different failure angles. So given that this is an iterative process I thought excel would be a good tool. It turns out the trig/algebra/geometry is time consuming but I think I have the basics pretty close. Please check carefully for errors as there are many places for them to occur, please let me know what you find.

The calculation is limited to the geometry of which I selected – a flat section behind the wall, and a broken back slope.

However the procedure is applicable to any geometry.  I hope to add the option of a water table elevation in the future.

Trial Wedge Forces

Nomenclature:

Geometry:

H = Height of the retaining wall

h0 = Height from the bottom of the wall to h2

h1= Height of the brocken backslope

h2= Height from the top of wall to where the extension of the brocken backslope intersects the wall

h3 = Height from h0 to the intersection of the failure plane and the ground in section 2.

h4 = Height above the top of wall to the top of the broken back slope.

alpha = α = angle of failure plane measured CCW from the horizontal.

beta = β = angle of the broken back slope.

omega = ω = wall battered measured CW from the vertical.

 Forces

Ca = Adhesion

Co = Coheasion

La = Length of adhesion

Lo = Length of cohesion

Ka, Kah, Kav = Active pressure coefficient, horizontal component and vertical component

Pa, Pah, Pav = Active force on retaining wall, horizontal component and vertical component

R, Rh, Rv = Resultant force of soil wedge weight and friction force direction is determined by effective friction angle and magnitude by the weight of the soil, horizontal component and vertical component

delta = δ = wall friction angle (effective friction angle between soil and wall)

phi = φ = effective friction angle of soil

kappa = κ = δ – ω

lambda = λ = α – φ

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This post is an extension of a previous post https://howtoengineer.com/retaining-wall-lateral-earth-pressure/

The spreadsheet will use the nomeclature found in NCMA’s Design Manual for Segmental Retaining Walls and Coulomb Theory.

See here:
Lateral Earth Pressure – Soil Basic 1 Geometry Sketch

Also here is a ‘fun’ spreadsheet where you can enter values in green columns. There are a bunch of graphs to show you how ka (the horizontal earth pressure coefficient will change with different values for backslope, effective friction angle, wall batter, and friction between wall and soil.

Earth pressure spreadsheet: Lateral Earth Pressure – Coulomb

Attached are a couple of TEDDS calcs that show the analysis of active and passive pressures based on Coulomb Theory. I am working on incorporating a berm distance into the passive equations and will probably present this in a seperate post. Using Coulomb equations for Toe slopes and backslopes should be used with caution and these conditions may warrant a Global (or slope) Stability Analysis!

Soil Evaluation NAVDAC DM – 7.2

Soil Evaluation – NCMA

Equivalent Slope

Another note: When there is a toe slope that passive pressure will be reduced a good reference for this condition is CALTRAN Trenching and Shoring Manual 2011 and NAVDAC DM 7.2 page 7.2-65 Figure 4 .

The post Lateral Earth Pressure II appeared first on How To Engineer.

Retaining Wall – Lateral Earth Pressure

Update: For spreadsheets and more examples of calculating active and passive pressures see Lateral Earth Pressure II

We will briefly discuss lateral earth pressure caused by soil weight and ground water effects. I’m not going to go through all the derivations just the results and how they are typically used in practice. More of the ‘I don’t want to hear about the labor just show me the baby’ technique.

See this post for a broader overview of earth retention design:

General Earth Retention Design

Rankine and Coulomb Methods

The most common theories for determining lateral pressure due to soil are Rankine and Coulomb methods. Both methods use an idealized failure plane where the soil ‘shears’ itself and causes the soil mass to move toward the wall. The Rankine method assumes that the soil is cohesionless, the wall is frictionless, the soil-wall interface is vertical, the failure surface on which the soil moves is planar, and the resultant force is angled parallel to the backfill surface. The Coulomb method accounts for friction between the wall and the soil and also a a non-vertical soil-wall interface (battered wall). Earth pressures may also be found in geotechnical reports as Equivalent Fluid (or Lateral) Pressures (EFP or ELP). Which are given in units of lbs per sq ft. per ft of depth or pcf. All this represents is a lateral earth coefficient already multiplied by the soil density. So if you find your active pressure coefficient using one of the formulas below say Ka=0.33 and multiply this by the soil density say 120 pcf you get about 40pcf. Because earth loads are applied as uniformly increasing loads (triangular distribution against the back of wall). The equivalent lateral pressure is 40psf / ft of depth.

Basic Geometry Sketches

Lateral Earth Pressure – Soil Basic 1

Lateral Earth Pressure – Soil Basic 1 Geometry Sketch

Rankine equations for Active and Passive pressure (more on that below):

φ = (phi, html format looks slightly different than image) effective friction angle of the soil

β = Angle of backslope from the horizontal

For the case where β is 0, the above equations simplify to

 Coulomb equations:

φ = (phi, html format looks slightly different than image) effective friction angle of the soil

β = Angle of backslope from the horizontal

δ = effective friction angle between the two planes being evaluated. Usually between wall and soil with typical values being 2/3*φ or between two soil surfaces (i.e. for segmental retaining walls – reinforced zone soil and retained soil)

θ  = batter or angle of wall from the horizontal (you may see some coulomb eqns which use values from the horizontal so don’t be confused)

To account for wall batter the hoizonatal and vertical component of the active pressure are:

Kah=cos(δ+θ)

Kav=sin(δ+θ)

The Active state referes to pressures where the soil is sliding toward the wall or the wall is giving. The Passive state refers to soil pressures where the soil is being compressed such as soil at the low side of a sheet pile wall. Passive pressures will be higher than active as you can imagine that the soil will ‘push back’ when it is being pushed. The soil may also be ‘at-rest’. You may wish to use at rest pressures when designing concrete basement walls which do not allow much movement or other type retaining walls where minimal movement is wanted.

At rest pressure coeffcient:

K0= 1 − sin
(φ)

References

Reference for wall movement under to ‘engage’ active pressure:

In Winterkorn and Fang, “Foundation Engineering Handbook” Table 12.1
Sand:
Active Pressure: Parallel to Wall .001H
Active Pressure: Rotation About Base .001H
Passive Pressure: Parallel to Wall .05H
Passive Pressure: Rotation About Base >.1H
Clay:
Active Pressure: Parallel to Wall .004H
Active Pressure: Rotation About Base .001H
Passive (No values given) however NAVDAC DM2.2 states that the required strain or wall movement required to mobilize the passive soil is about 2x the movement required for active pressure.
A great reference, but I’m sure it’s been out of print for a while.  My copy is dated 1975.

Water Pressure

Water pressure can be greatly reduced by providing drainage aggregate and drain pipe directly behind the wall. The density of water is much less than most soils (64 pcf) however its lateral pressure coeffcient is = 1.0 so the Equivalent Fluid or Lateral pressure is 64.5 psf/ft which is higher than most soils in an active pressure case. Therefore water pressure can have a serious impact on the lateral load applied to wall and should be given proper attention if the water is not given a relief source as mentioned above.

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Offset surcharges have provided me with some of my largest learning hurdles as a young engineer. We will spend some time discussing different design options when faced with an offset surcharge. We will not spend much time on uniform surcharges as these are relatively straight forward. They are also covered extensively in texts and most importantly there is a a general consensus on how to evaluate the resulting lateral pressure. We will also provide some spreadsheets and written examples. We will also try a few video tutorials as well.

Upcoming Topics:

1. We will first cover the basics of a uniform surcharge. Generally Fq=Ka*q*H.
2. Elastic Methods – Point Load, Line Load, Strip Load, Area Load, with spreadsheet.
3. Other approximate methods
4. What is appropriate and when

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