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Adapting Graphical Coulomb Methods to Numerical Solutions

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Chapter 10: Adapting Graphical Coulomb Methods to Numerical Solutions By Karl Hanson, S.E., P.E.*

August 2012

10.1 Introduction:

Many of the design approaches that we use today to design walls,

foundations and excavation bracing originated from Coulomb’s

stability methods developed in 1796. One of the earliest known

physicists, Coulomb is also famously responsible for his discoveries

in the field of electricity.

Today, we understand that Coulomb’s soil pressure theories are

based on energy principles. In Coulomb’s day and age, the concept

of energy was not quite articulated as well as today.

Charles Coulomb

(1736-1806)

What became known as “Coulomb’s Theory” was soon adapted into graphically based

methods. Using a pencil and paper and drafting board, a solution for either active or

passive wall force can be determined. The “answer” for a force against a wall is measured

from the drawing. These methods are tedious, yet they clearly demonstrate failure planes.

The appeal of these methods has been lost, since the accuracy of the answer literally

depends on how well one draws and how sharp a pencil is used!

Notably, the father of soil mechanics, Karl Terzaghi, routinely used

these graphical methods to develop his geotechnical theories on many

subjects. In Terzaghi’s 1943 book, “Theoretical Soil Mechanics” (Ref

1), he derived a theory for soil bearing failure, by treating the soil mass

under a footing as similar to the problem of passive pressure against a

wall. In connection with the bearing capacity problem, Terzaghi wrote:

“…the computation of the critical load…requires the determination of

the component of… the passive earth pressure which requires several

hours of work.” (meaning hours of drawing and re-drawing failure

planes)

Karl Terzaghi

(1883-1963)

To expedite the use of his theories, Terzaghi opted for a “simplified” bearing pressure

equation, which did not require graphical methods. This trend has continued, perpetuating

various forms of bearing pressure equations developed by researchers, typically derived

from the Theory of Plasticity (Ref 4).

Why not “dust off” these old methods, and use our computers to construct the

graphics? Today, we have an opportunity to resurrect these old graphical methods,

adapting them to numerically based solutions. Perhaps, with these methods, engineers can

better visualize soil failure mechanisms for the structures which we design.

(* DesignCalcs, Inc., http://www.designcalcs.com)


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10.2 The Physics Behind Energy Principles:

We begin this discussion by asking, “What is energy?” The term “energy” is now so

common place in our language, it’s something that we take for granted. Putting aside

modern day concepts represented in advertising, we need to define energy using the

language of physics.

Leibniz was the first to note the importance of the kinetic

energy term, “mv2”. Also, about the same time, Newton, in his

“Third Law of Motion”, implicitly stated the whole doctrine of

energy. It wasn’t until 1807 that the term “energy” was first

coined by Thomas Young.

Maxwell explained energy principles as follows:

“(Energy)..has no individual existence…The transactions of

the material universe appear to be conducted, as it were, on a

system of credit. Each transaction consists of the transfer of so

much credit or energy from one body to another. This act of

transfer or payment is called work.” (Ref 2)

James Clerk Maxwell

(1831-1879)

Various Forms of Energy

Kinetic Energy The most obvious type of energy is

kinetic energy:

A mass, “m”, moving at a velocity, “v”,

has kinetic energy,

K.E. = ½*mv2

Potential

Energy

A more complex form of energy is the

potential energy stored in a spring:

A spring with force, “F”, compressed

“D” has potential energy,

P.E.=1/2*F*D

Other Energies Even less obvious forms of potential energy are chemical energies,

electrical-mechanical energies, nuclear energies, thermal energies –

basically, things that we can’t see that are locked into a system.

When Albert Einstein came along, physicists were faced with the fascinating idea that

some systems may not have constant mass!

An isolated system (a system that is not acted on by another system), will have a total

energy comprised of the above forms of energy. For a system that has constant mass,

the total energy of an isolated system will remain constant.


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For practical engineering purposes, we are usually only interested in the kinetic energy

and the potential energy of a system:

Kinetic Energy

+

Potential Energy

= Total Energy

(A constant for a

system with

constant mass)

We are interested in physical predictions. Given a system in state “A”, there may be an

infinite number of physically viable next states, “B1”, “B2”,… How can we predict

what will be the next state? By using the following law:

The tendency of all physical events is to maximize kinetic energy and minimize

potential energy.

Put another way: Given 10 different scenarios, the one with the larger fraction of

kinetic energy and smaller fraction of potential energy will be the likely next event.

Illustrating this by an engineering example, two walls are shown below:

The wall on the left is expanding outward. In a sense, it is like a spring that is expanding.

It is a system that is gaining kinetic energy and losing potential energy. The larger “Pa”

is, the faster will be the rate that the system gains kinetic energy.

The wall on the right is a being pushed against. It is like a spring that is being condensed.

It is a system that is losing kinetic energy and gaining potential energy. The smaller “Pp”

is, the slower will be the rate that the system gains potential energy.

For both of the above systems, the soil mass behaves like a spring. The next likely event

will be the configuration that minimizes the soil mass’s potential energy.


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10.3 A Practical Application: The Trial Wedge Method

The “Trial Wedge Method” is a very useful graphical method that can easily be adapted

to a spreadsheet solution. The below sketch shows a simple, generic wall showing three

trial failure planes, amongst an infinite number of possible failure planes. (Terazaghi

determined that the true failure plane actually has a slight bit of curvature following a log

spiral)

By a trial an error process we compute “Pa” until it is a maximum. We do this as follows:

1. Try several failure planes. Compute weights of each trial wedge.

2. Compute the total weight, “W”, on the failure plane (W=W1+W2+W2)

3. Compute the cohesion, “Cs”, along the failure plane.

4. Compute the wall adhesion, “Cw”.

5. Construct a vector force diagram:


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6. Compute point (X1,Y1):

X1=Cw*cos(aaaa) , Y1=-Cw*sin(aaaa)

7. Compute point (X2, Y2):

X2=X1 – Cs*cos(rrrr), Y2=Y1 – Cs*sin(rrrr)

8. We know only the direction of “R” and that it intersects at (X2,Y2). Compute the

equation of the line Y=m1*X + b1:

m1 = tan-1

(r + 90 –f)

b1=Y2-m2*X2

9. Also, we know the direction of “Pa” and that it intersects at (0,-W). Compute the

equation of line Y=m2*X + b2:

m2 = tan-1

(180 - a + d)

b2=-W-m2*0 = -W

10. Solve for the intersection of the two lines at (X3,Y3):

m1*X3 + b1 = m2*X3 + b2

X3=(b2-b1)/(m1-m2)

Y3=m1*X3 + b1

11. Solve for the active force, Pa:

Pa = [(X3-0)2 + (Y3+W)

2]1/2

The preceding algorithm can be easily put into a spreadsheet. This spreadsheet can be

downloaded at http://dcalc.us/Downloads/Trial wedge.xls (note the underscore

“…l_w…”)

DCALC’s Trial Wedge Application: TRIALWEG

The TRIALWEG application was written using steps similar to the above procedure, but

with some variation.

• This application can handle multi-layered soil, with or without water tables.

• Reads soil boring data from the BORING program

• Provides a graphical presentation of the failure plane and the force vectors.

• Uses 100 iterations to determine the failure plane.

In addition, TRIALWEG provides a pressure diagram, with separate diagrams for the

basic force components (earth, surcharge, seismic and water pressures).

Similar to other DCALC applications, TRIALWEG produces a calculation file that is

designed to be used by other DCALC applications.


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10.3.1 Trial Wedge Analysis Example 1

The below trial wedge example was calculated by J. E. Bowles using graphical methods

as example E11-6 on p. 504 of Reference 5:

The above sketch shows the failure plane computed using the TRIALWEG program. The

sketch was drawn in DesignCAD. Areas and lengths were measured directly in

DesignCAD.

Bowles also applied a wall adhesion force, which appears to account for the slight

difference in results.

Comparison of Results:

Calculation Source Maximum Active Force “Fa”

Computed in Bowles’ Example 11.90 k/ft

Computed by TRIALWEG 13.15 k/ft

Computed graphically (above) 13.00 k/ft


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The below trial wedge is similar to the previous example, except the soil cohesion has

been set to zero.

Again, the above sketch shows the failure plane computed using the TRIALWEG

program.

For this verification, the Excel spreadsheet solution has also been made, shown on the

next sheet.



Computed by TRIALWEG 22.60 k/ft

Computed graphically (above) 22.42 k/ft

Computed by spreadsheet (next page) 22.70 k/ft


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Note: This spreadsheet does not include a provision for cohesion. Note also that the two 8

k concentrated surcharge loads were input as 4 ksf over a width of 2 feet each.


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The below example appears on page 84 of Wayne Teng’s book (Reference 6), as a

numerical example of “wedge theory”.



Computed by Wayne Teng 8.1 k/ft

Computed by TRAILWEG program 8.48 k/ft

Computed by graphically (above) 8.343 k/ft


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We can easily check the trial wedge method results for a wall with a simple slope, for

various soil internal angles and slopes. For the design of walls on slopes, the theoretical

exact Rankine coefficient for the horizontal active pressure is given as,

Ka = cosb *(cos b – (cos2b – cos

2f).5)/ (cos b + (cos

2b – cos2f)

.5) (Eq. 11-7a Ref.5)

Fa (k)

f b Cos f Cos b Cos2 f Cos

2 b Ka Exact TRIALWEG Graphical

(deg) (deg) Solution solution Solution

20 10 0.940 0.985 0.883 0.970 0.531 11.687 12.520

20 15 0.940 0.966 0.883 0.933 0.603 13.262 14.050

20 18 0.940 0.951 0.883 0.905 0.697 15.335 15.710 15.718

30 10 0.866 0.985 0.750 0.970 0.350 7.689 8.220

30 15 0.866 0.966 0.750 0.933 0.373 8.205 8.840

30 18 0.866 0.951 0.750 0.905 0.395 8.686 9.320

38 10 0.788 0.985 0.621 0.970 0.246 5.420 5.740

38 15 0.788 0.966 0.621 0.933 0.258 5.677 6.080

38 18 0.788 0.951 0.621 0.905 0.268 5.903 6.320

We observe that the trial wedge solution computed using both TRIALWEG and

graphically agree, but the results are slightly higher forces than the exact solution.


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The below sketch shows a sheet piling wall located in three different layers of soil, with

water to a depth of 21 feet. In this calculation, the TRIALWEG application has

determined the effective pressure with depth. Although the pressures from TRIALWEG

do not exactly match the hand calculations, in this particular case there is a close fit

Both Terzaghi and Bowles have discussed the difficulty of determining the pressure

distribution using the trial wedge method. We also acknowledge the difficulty of this

particular aspect of the use of this method. In some cases, especially involving multi-

layered soil, the pressure distribution may not be realistic, although the total force is

reasonably accurate. Please consult References 1 and 5 for additional guidance.


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10.4 Solving for the Ultimate Bearing Pressure Under a Foundation or Shaft

Using a technique similar to the trial wedge method, we can solve for the ultimate

bearing pressure under a footing. In this case, a portion of the failure plane must follow

the shape of a log spiral, which is a property of soil parameters.

By a trial an error process, we will try different points of tangency. The passive force,

“P”, under the wedge surface will vary, depending on the angle, u. We will go through

a process of iteration, trying various angles, u, to determine the minimum P.

Why does the soil fail in a log spiral shape? Exactly what is a log spiral curve? To

understand this, we need to review how slip lines form in the soil failure element.

Mohr’s Circle

Principle Stresses

In the diagram, “s1” is the passive stress and “s2” is the active stress. In Mohr’s circle

diagram, we see that the two failure planes are separated by an angle “180+2f”.

Therefore, in the actual soil specimen, the failure planes are separated by an angle of


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“90+f”. These failure lines will be oriented at an angle “90+f” everywhere within the

plastic failure zone.

In this case, the failure lines must “turn the corner”, maintaining a failure angle of

“90+f”. This is accomplished by a spiral function, as follows:

Using simple trigonometry,

dr = r*du * tan(f)

therefore,

dr/du = r*tan(f)

The following function satisfies this

differential equation:

r=r0*e(u*tan(f))

(r0=beginning radius under footing)

We construct the trial wedge model using 10 slices in the tangent portion, and 10 slices in

the spiral wedge portion, as shown below:

Slices 1-10 in the tangent portion are spaced equally. For slices 11-20 in the spiral

portion, it is easiest to divide the angle “u” by 10, and compute (x,y) coordinates along

the bottom.

We will work from right to left.

• The first segment will have no horizontal forces acting from the right. We will

solve for the horizontal force on the left of slice 1.

• Then we will apply the horizontal force from slice 1 to slice 2. We will solve for

the horizontal force on the left of slice 2.

• We will iterate this process until we get to the footing edge at slice 20


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The above diagrams show the forces and vector diagrams. The objective is to compute

“PLn” for each slice. The mathematics is similar to what was described for the trial

wedge method in Section 10.3. Equations are contained in the spreadsheet.

For slices 1 to 20, we compute the following:

1. Compute Wn

2. Compute the angle r

3. Compute the cohesion along the bottom, Csn.

4. Apply the force PRn from the previously computed slice (=PLn-1)

5. Compute (X1,Y1)

6. Compute (X2,Y2)

7. Knowing the direction of “Rn”, compute the equation of the line y=m1*x + b1

8. Knowing the direction of “PLn”, compute the equation of the line y=m2*x+b2

9. Solve for intersection of the two lines at (X3,Y3)

10. Solve for force “PLn”.

11. Go to Step 1 for the next slice


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For slice 21, which is directly under the footing, we will need to resolve forces as shown

below:

For this particular slice, if there is cohesion, we will also include a cohesion force, “C”,

along the failure plane.

The angle “a”, depends on the roughness of the footing base. Terzaghi used the

following assumptions:

• For a smooth base, a = 45 + f/2

• For a rough base, a = f

“Qult” is equal to the sum of the vertical forces “P” and “C” acting on two symmetrical

failure planes, divided by the unit width, “B”.

The above algorithm has been setup in a spreadsheet that can be downloaded at

http://dcalc.us/Downloads/Bearing_pressure.xls (note the underscore “…g_s…”)

DCALC’s Soil Bearing Pressure Application: BEARPRES

The BEARPRES application was written using the steps similar to the above procedure,

but with more complexity.

• The application reads soil boring information from the BORING program

• It can handle multi-layered soil, with or without water tables.

• The ground can be flat or sloped.

• Footings (strip, square or rectangular) or round shafts

• The pressure failure shape is shown graphically to scale.

Also, the user has the option of viewing detailed calculations, showing the force

equilibrium of each slice, for checking purposes.


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10.4.1 Ultimate Bearing Pressure Examples

The below sketch shows the typical output from the BEARPRES program, depicting the

log spiral failure plane for the square footing in uniform soil:

BEARPRES outputs both a numerical solution and a conventional solution for ultimate

and net bearing pressure, therefore it is easy to check. The following table shows a

comparison of several verification problems:

Description g g g g

(kcf)

F F F F

(deg)

C

(ksf)

Ultimate bearing pressure, Qult

By Conventional

Formula

By

“BEARPRES”

4’x4’ footing, 4’ deep 0.120 30 0 17.16 ksf 15.63 ksf

4’ strip, 4’ deep 0.120 30 1 44.31 ksf 41.69 ksf

2’ diameter, 20’ deep 0.120 30 0 77.27 ksf 58.82 ksf

2’ diameter, 20’ deep 0.120 0 1 8.50 ksf 12.90 ksf

A particular advantage of this program is that it shows the shape of the failure surface.

This feature alone provides significant meaning about the mechanics of bearing failure.

A Shaft in Granular Soil

A Footing Next To A Slope


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List of References:

Reference 1: “Theoretical Soil Mechanics” by Karl Terzaghi, published by John Wiley

and Sons, 1943

Reference 2: “Matter and Motion” by James Clerk Maxwell, originally published 1877,

re-printed 2007 by Cosimo, Inc.

Reference 3: “The Ultimate Bearing Capacity of Foundations” by G. G. Meyerhof,

Geotechnique V2 no. 4, 1951

Reference 4: “Plasticity and Geomechanics” by R.O. Davis and A.D.S. Selvadurai,

published by Cambridge University Press, 2002

Reference 5: “Foundation Analysis and Design” by Joseph E. Bowles, 4th

Edition,

published by McGraw Hill, 1988

Reference 6: “Foundation Design” by Wayne C. Teng, published by Prentice Hall, 1962

Website Credits: For an excellent paper on how to compute bearing capacity in soils, with many examples,

search for “Bearing Capacity v1.00 September 2010”:

http://data.bolton.ac.uk/staff/phm2/files/Semester1/J3%20PJ4%20Geotechnics/Bearing%

20Capacity%20v1.00%20Sept2010.pdf

(This paper, titled “Bearing Capacity – Shallow and Deep Foundations”, does not

contain a copyright or the names of its authors. It is apparently from a class at Bolton

College in the UK)

Photo Credits: Photographs/Images of Coulomb, Terzaghi and Maxwell are from Wikipedia.

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