Consider a rigid retaining wall of a massive type, generally not vertical, its contact surface AB is described by a certain angle β in Fig.1. Assume what follows.

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THE COULOMB-PONCELET ultimate earth-pressure theory

1. Assumptions

Consider a rigid retaining wall of a massive type, generally not vertical, its contact surface AB is described by a certain angle β in Fig.1. Assume what follows.

1) There is a plane state of displacements typical for long objects; only a representative 1m section is considered; the wall can move outwards the soil (to the left in Fig.1, f < 0) or towards the soil (to the right in Fig.1, f > 0),

2) the backfill parameters are c = 0 kPa, ϕ > 0, γ > 0; the backfill top boundary is generally not horizontal, usually ε ≥ 0,

3) ultimate shearing (soil sliding) happens if

• τ = σ⋅ tg ϕ , for a sliding inside the soil, ϕ = internal friction angle,

• τ = σ⋅ tg δ , for a sliding on the contact wall-soil, δ = external friction angle at the contact, 4) when the wall moves infinitesimally, a rigid wedge of soil is observed (triangle ABC of the

weight G

ABC

[kN/m]) which a little bit slides downwards in limit equilibrium along the wall AB and along a narrow shearing zone BC in the soil interior; three separate parts - the wall, the backfill triangle ABC and the rest of the backfill - are considered as rigid bodies,

5) there are used general symbols δ

ⁱ

, because the signs (orientations) are very important, but of course the values are different δ

1

= -ϕ, δ

2

= δ,

6) by assumption, the BC curve is linear, so determined by an angle χ .

The details are presented in Fig.1, also the angles β, ε, χ, δ

2

which are positive here (but δ

1

< 0).

Fig.1. Denotations.

The force polygon is presented on the right:

vertical vector G

ABC

is known (if a value is assigned to the angle χ), directions of both vectors E

a

[kN/m] and R [kN/m] are known, too.

The triangle can be solved; the value of E

a

is of interest (the value of R is not).

+

χ A’

H A

B

C

β

ε E

a

R G

ABC

δ

¹

δ

2

L

∠ = χ+δ

¹

∠ = 90

^o

-β-δ

2

f < 0

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2. The Coulomb-Poncelet concept

The Coulomb idea is to decompose the vertical vector G

ABC

into two vectorial components E

a

and R, along two given action lines, Fig.1. The value E

a

= E

a

 is to be found, the reaction R

= Ris less important. Mathematically, this is a simple geometrical solving of the force triangle in which all angles are known, as well as one side G =  G

ABC

 ;

the theorem called law of sines can be used, which yields E

a

= G ⋅ sin( χ + δ

¹

) / sin(90

^o

- β - δ

²

).

For the allowed horizontal translation f < 0 (Fig.1), a virtual downward sliding of the triangle ABC is expected. This simply remark has very important consequences – it means that shearing components of both forces E

a

and R have a – more or less – downward orientation, along AB and BC respectively; both forces happen above the normal line perpendicular to the surface.

Therefore, the signs δ

2

≥ 0 and δ

1

< 0 are justified.

The BC line is entirely included in the soil mass interior, so for the limit equilibrium there is δ

¹

= - ϕ as usually for shearing in soils; the situation on AB depends on roughness of the soil- concrete contact, 0 ≤ δ

²

≤ ϕ . Some comments on realistic δ

²

values are discussed in Lecture 5.

In particular:

• min δ

2

= 0

^o

corresponds to a perfectly smooth surface (no shearing component, no friction),

• max δ

2

= +ϕ (still greater value at the contact is impossible, because this is the maximal value in the soil interior, maybe only one milimeter next to the contact).

Note that hypothetically opposite δ

2

sign is also possible in such a sense that the Poncelet approach is still effective. In practice, this could happen for large settlements of the wall (hitherto only horizontal translation f < 0 was assumed causing a dropping of the ABC triangle); this case looks much less probably and is not considered here.

Following conclusion is of the special importance:

the right selection of the angle signs depends

on the relative kinematics of the sliding block ABC and the wall itself (Lecture 5).

The convention of δ signs should be used only in conjunction with the normal vectors (dashed lines) outer to the wall-surface AB. This can be a source of misunderstandings, because two walls in Fig.2 are physically the same in the sense of the soil-pressure value E

a

; one should not think that δ

²

angles to the normal are opposite here.

We focus on the left case.

3. The Coulomb-Poncelet solution for the active pressure

Neither the wall height H nor the vertical axis z but the AB-interval length L in Fig.1 is the leading parameter of the model; all intervals are going to be expressed using L along AB, not the height H nor z for example.

The ABC-triangle area equals S = h⋅AC/2:

• for the triangle height h = A’B = L ⋅ ^sin( ∠ ^CAB) ⁼ ^L ⋅ ^cos( ε ^- β ^),

• for the triangle base AC = L ⋅ sin( β - χ +90

^o

) / sin( χ - ε ) using again the law of sines.

The triangle weight is equal to G = S ⋅γ [kN/m] therefore from the force G decomposition:

Fig.2. Two equivalent design situations of δ

²

> 0.

+ +

E

a

E

a

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( χ _{) = ½⋅γ⋅} ⋅ sin (χ − ϕ)

sin (χ − ϕ + 90 − β − δ ) ⋅ cos (ε − β) ⋅ sin (β − χ + 90 ) sin (χ − ε)

Value of one parameter is missing in this solution and this is the sliding wedge angle χ in Fig.1.

Coulomb proposed – as every good engineer would had to do – that the unknown variable χ takes the worst value as possible, maximal in this case E

a

= max{E

a

(χ)} = E

a

(χ

a

), for which dE

a

/dχ = 0 if d

²

E

a

/d

²

χ > 0. This way, the trigonometric equation dE

a

/dχ = 0 yields an “ultimate”

angle χ

^a

and finally E

a

= E

a

( χ = χ

^a

).

This is J.V.Poncelet who first solved this general situation.

The solution is usually presented in the following form:

=

^{( )}γ

⋅ _γ ⋅ _/2 for

( )

= ( ϕ ₋ β ₎

cos( β _{+ ) ∙}

1 1 + !sin( ϕ + ) ∙ sin(" − #) cos($ + ) ∙ cos(# − $)%

Note again that these formulae can be different sometimes if another sign convention is used.

4. Comments

1. The general concept comes from Ch.Coulomb (1773) but he solved in detail only the

simplest case of “The Coulomb wall”, i.e. with β = 0, ε = 0, δ

2

= 0 for which χ

a

= 45

^o

+ ϕ /2 and next K

aγ

= (1-sin ϕ )/(1+sin ϕ ) = tg

²

(45

^o

- ϕ /2).

2. The Coulomb concept is a prototype of a much more general variational approach, where generally the shape, of the “ultimate” sliding line BC is of interest, i.e. a certain functional equation z(x) for BC has to be found; in the family of linear functions, the problem is

algebraized, because such lines are fully described by the only one numerical parameter χ.

3. Note that the simply assumption of the linear sliding surface (triangular sliding wedge ABC) finds a strong justification in experiments; for Coulomb walls, this assumption and the theory are exact, worse agreement happens for situations being far from Coulomb walls.

4. The reader should appreciate a great skilfulness of J.V.Poncelet (mathematician in fact), who managed to overcome this toilsome derivation as early as just before 1840 (moreover, in pre-computer times he invented a useful graphical method supporting the earth-pressure calculations).

5. About one hundred years ago, Műller-Breslau preferred H coordinate instead of the length L and he got the similar solution =

(&'()γ

⋅ _γ ⋅ _{) /2 .}

Simply substitution of L = H/cos( β ) from Fig.1 expresses

^(&'()_γ

=

( )γ

⋅ /cos (β) , so this is nothing original.

Just

^(&'()_γ

values are presented in the Polish National Code PN-83/B-03010; nevertheless,

the use of the Poncelet original notation and his coefficients

^{( )}_γ

is strongly recommended

because this relates earth pressure to the physical contact surface AB, not to the abstract

variable H.

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5. Uniform load q = const on the boundary

The idea for q > 0 is similar but the vertical weight force G

ABC

is completed by q-action

integrated over AC, i.e. Q

AC

= q⋅AC [kN/m]. The weight G

ABC

is always the vertical vector but Q

AC

is not necessarily so; the vectorial sum G

ABC

+ Q

AC

must be decomposed into E

a

and R – as previously.

F ig.3. Case of q > 0.

The situation becomes very simple if q > 0 is vertical. Indeed, Q

AC

algebraically increases the weight of the triangle ABC. Instead of the real triangle loaded by q, it suffices to consider the unloaded triangle ABC increasing its unit weight from the real γ to a virtual γ > γ; note that the*

“ultimate” angle χ

a

does not depend on the unit weight γ thus the solution will be correct.

To keep the equivalence of forces (weights), there must be G = G +* Q

AC

or G* = S ⋅γ* = γ*⋅ h ⋅AC/2 = G + Q

AC

= γ⋅ h ⋅AC/2 + q⋅AC.

Finally, γ * = γ + 2q/h = γ + 2 ⋅ q/(L ⋅ cos( ε - β )) = const( χ ).

The general formula =

^*

∙ +

^∗

∙ ∙ is true for every unit weight, thus making the substitution of γ * results in:

= 1

2 ∙ γ _{∙ ∙} _{+ - ∙ ∙}

_.

where

_.

=

_{4 5(6'7)}^/⁰¹²³

.

6. Earth pressure as a continuous loading

Since the resultant earth pressure = 8 9 ( ):

_<^;

, so there is 9 ( ) = : ( ) : ⁄ and finally:

9 ( ) = γ _{∙ ∙} _{+ - ∙}

_.

This is a trapezoidal shape. For a homogeneous cohesionless backfill, the earth pressure e

a

[kPa] increases linearly along the wall and has constant angle δ

2

to the normal vector.

7. Layered soils

Layered backfills are not in use (exception for local drainage layers) but such a case can happen for slurry walls or sheet-pile walls embedded in natural soils, in their upper, not anchored part. Simple engineering approach is as follows.

+

χ q A

B

C

β

ε E

a

R Q

AC

δ

1

δ

2

L

∠ = χ+δ

1

∠ = 90

^o

-β-δ

²

f < 0

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Simplify (replace) the boundary of the lower soil “2” in Fig.4 by a straight line at B

1

which is parallel to the ground surface. For the two layers the angles ε and β are the same but the angles δ

2

generally are not, being dependent on different ϕ

i

. On the segment AB

1

the solution is the same as previously (starting with L = 0 at A), on the segment B

1

B

2

the solutions starts anew for q

2

= const instead of q

1

for “1” (starting with L = 0 at B

1

). Therefore, the earth pressure e

a

is bilinear, having discontinuities in both the angle δ

²

and the value e

a

at B

1

. In Fig.4 there must be ϕ

2

> ϕ

1

(explain why – two reasons). The same idea is effective for 3, 4 and more layers.

Note that the method is approximate and it overestimates a little bit the earth pressure, because q

2

is locally overestimated on the separation level near the point B

1

; indeed, a part of vertical loadings from q

1

and γ

1

is locally transmitted along AB

1

due to friction.

8. Multi-linear wall profiles

For multi-linear wall profiles, like the one in Fig.5, the situation from Fig.4 can be adopted.

The backfill is the same, here only the angles differ β

1

≠ β

2

.

The earth pressure e

a

is bilinear due to discontinuity of the angle β and therefore also

discontinuity of the value e

a

at B

1

occurs. In Fig.5 there is β

¹

> β

²

. The same idea is effective for 2, 3 and more corners on the wall; the analogous overestimation of e

a

takes place.

9. Passive earth pressure

Ch.Coulomb did not analyse walls pushed into the soil but J.V.Poncelet did. Since f > 0 in Fig.1, so the relative kinematics is different – the rigid triangular wedge ABC moves virtually upwards;

this way, the ultimate shearing stresses τ have signs opposite to the ones in the active case (δ

1

= +ϕ and -ϕ ≤ δ

2

≤ 0; as previously, there can be some exceptionsJ).

q

1

A

B

1

B

2

„2”

„1” q

2

h

1

Fig.4. Two-layer system.

The upper layer has an assumed design thickness h

1

;

for “1”: q

1 ≥ 0, γ¹

, ϕ

¹

, c

1

= 0, h

1

for “2”: q

2

= q

1

+ γ

1⋅h1⋅cos(ε), γ2

, ϕ

2

, c

2

= 0 and next calculate as if the wall started at B

1

q

1

A

B

1

q

2

h

1

Fig.5. Two-layer system.

The upper virtual layer has an assumed thickness h

1

(see two parallel lines, it is a vertical distance).

First solve along AB

1

ignoring B

1

B

2

for “1”: q

1

≥ 0, γ

¹

, ϕ

¹

, c

1

= 0, h

1

is known.

Next solve along B

1

B

2

ignoring AB

1

for “2”: q

2

= q

1

+ γ

1

⋅ h

1

⋅cos(ε), γ

1

, ϕ

1

, c

1

= 0.

And so on, if there are still further, deeper intervals:

always focus only on the lower one.

B

2

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For q = 0 [kPa] the following result can be derived basing on E

p

= min{E

p

(χ)} = E

p

(χ

p

):

>

= 1

2 ∙ + ∙ ∙

^>^{( )}

where

>( )

= ( ϕ ₊ β ₎ cos( β _{+ ) ∙}

1 1 − !sin( ϕ − ) ∙ sin(" + #) cos($ + ) ∙ cos(# − $)%

Note that

^{( )}_γ

≠ 1 /

^{( )}_γ

, generally.

For the Coulomb wall ( ε = 0, β = 0, δ

2

= 0) this is the exact solution,

_>^{( )}

=

>

=

^*?5@AB_*'5@AB

= tg (45

+ φ/2) and the curve of sliding BC is linear, indeed.

The case of q > o can be solved in the analogous way as in the active case.

Important example Let:

ϕ = 45

^o

(large value but possible for crushed basalt, granite or dense gravel, etc.), c = 0, δ

²

= - ϕ = -45

^o

(possible for perfectly rough surface), β = ε = - ϕ = -45

^o

.

The data means a right-angle 2D pyramid which is in limit equilibrium, Fig.6.

The situation in Fig.6 is realistic – though unstable; in particular, no additional loading can be applied to the boundary because the slope angle equals the internal friction angle. In contrast, one obtains here

_>γ^{( )}

=

_>.^{( )}

= +∞ (!), thus a great vertical ultimate loading 9

>

( ) =

_>γ^{( )}

⋅γ⋅ L could be applied, even the infinite one. This is nonsense, but what is wrong?

There are no objections to the decomposition of the weight vector G, the extreme principle E

p

= min{E

p

(χ)} = E

p

(χ

p

) is also reasonable, so the assumption about linear sliding-curve BC is not correct here. The Poncelet analysis of this example reveals that the ultimate sliding line BC becomes parallel to the soil boundary (χ

p

= ε) thus its length equals infinity. This finding explains singularity of the Poncelet solution for the above specially selected set of parameters: even small shearing stress along the infinite line cause infinite forces;

therefore, no linear line BC can be accepted for such unusual situation.

Conclusion:

In principle, the Polish National Code PN-83/B-03010 is based on the Poncelet solution, though in the Műller-Breslau formulation

_>γ

=

_>γ^(&'()

=

_>γ^{( )}

/ cos (β) , therefore the code had to introduce a correction coefficient η ≤ 1 and recommends using of η⋅

_>γ

in place of

_>γ

to avoid overestimation of the values yielding from more adequate, better confirmed models.

The Eurocode EC7-1 introduces

_>γ

values in some charts which already do not need any reduction; they yield from some more complex numerical procedures in which BC line is not assumed as linear, of course.

Fig.6. Unstable slopes in the ultimate equilibrium (slope angle equals the internal friction angle).

e

p

A

B ϕ ϕ

β

ε

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There is no problem with

_γ

values, because BC is almost linear here, so many different methods are convergent.

10. Ultimate earth pressures of cohesive soils

The backfill is usually non-cohesive but the earth pressure theories are also very important for cohesive soils. For example, the bearing capacity of shallow foundations is a consequence of differences between the passive earth pressure and the active one.

The solutions for c > 0 can be easily derived from the same situations for c = 0. Details are presented elsewhere on this WWW as the so-called equivalent states principle.

11. Concluding remaks

1) The Coulomb-Poncelet method belongs to a group of limit equilibrium methods applied to a rigid (triangular) sliding wedge thus it differs by the used tools from the Coulomb-Mohr method, which operates locally on stresses.

2) The Coulomb-Poncelet method is very effective for the ultimate active earth-pressure, because the fundamental assumption about the linear sliding curve BC is realistic.

3) The Coulomb-Poncelet method can be misleading for the ultimate passive earth-pressure in situations which are “far” from the Coulomb wall. Usually, it could be a dangerous overesti- mation of soil response because passive earth-pressure is a stabilizing factor.