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Modules 24,25/Topic 13

GEOTECHNICAL DESIGN (STABILITY ANALYSIS) OF RETAINING

STRUCTURES

Retaining structures constitute the most important class of substructures after

foundation structures.

‘Retain’ means ‘hold’ or ‘support’. The function of a retaining structure is to provide

lateral support for a mass of soil. In other words, it retains the soil on its side and hence

the name. The major difference between a foundation structure and a retaining

structure is that, while the former is mainly subjected to vertical loads, the latter is

predominantly subjected to lateral load, which is invariably the pressure exerted by the

backfill soil which it retains.

13.1 Situations requiring retaining walls

Fig.13.1a shows the example of a highway embankment retained by retaining walls

on either side, the provision of which is necessary within the limits of busy cities where

constraint on space prevents sloping it down as in rural stretches. Fig.13.1b shows the

example of a playground, built on a mound, with the fill retained by the retaining wall.

In addition to the lateral earth pressure from the backfill, the wall can be subjected to

the effect of dead and live loads, if any, such as when the backfill supports a loading

platform, or moving loads such as from a roadway or railway on top of it (Fig.13.2),

transmitting their effects through the backfill to the retaining structure. For the purpose

of analysis, these loads are normally treated as equivalent static ‘surcharges’ on the

backfill.

We had a detailed presentation of the subject of “earth pressures” under Topic 4.

13.2 Types of retaining structures: gravity and non-gravity types

From the point of view of the source from which the retaining structure derives its

stability, we have basically two types of retaining structures, viz., the gravity and the

non-gravity types. The gravity type structures are normally ‘rigid’ and the non-gravity

type, ‘flexible’. The gravity type of retaining structure derives its stability mainly from

the self weight of its components, while in the case of the non-gravity type, the factors

contributing to stability are other than gravity or self-weight forces. Old masonry type

of retaining walls (Fig.13.3) and the comparatively new reinforced concrete retaining

walls – both of the cantilever (Fig.13.4) and the counterfort (Fig.13.5) types – are

examples of gravity type of retaining structures, but with the difference that, whereas

in the case of the stout masonry wall, the self weight of the wall alone is the main

source of stability, what contributes to stability is not only the weight of the thin R.C.

structural elements, but also that of the soil on the base slab, in the case of the latter.

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There is a host of types of retaining structures which derive their stability from

sources other than gravity. The foremost example of this category is the sheet pile wall

which is too thin, whether in steel, reinforced concrete or timber, for any stability to be

derived from its self weight. While in the case of the cantilever wall of this type

(Fig.13.6), the only source of stability is penetration into the soil below, penetration

and anchorage together contribute to the stability of anchored bulkheads (Fig.13.7).

Diaphragm walls (Fig.13.8) and bored pile walls (‘contiguous’ and ‘secant’ types –

Fig.13.9) are thin structures which are invariably anchored into the side soil using

‘prestressed ground anchors (Topic 22), when they are called upon to function as

retaining structures. Hence in this state, their stability comes mainly from anchorage.

The most modern type of flexible retaining structure is the reinforced earth, where

a thin ‘facing skin’ is held in position by a large number of thin ‘reinforcing strips’ tied

to it and running through the backfill (Fig.13.10). This type of wall owes its retentive

action to the mechanical friction between the reinforcing strips and the backfill soil.

This in a sense one may look upon it as the facing skin anchored into the backfill, even

though the facing skin has a very minor role to play in this system. A major difference,

however, between the anchoring action in the case of bulkheads and diaphragm walls

on the one hand, and reinforced earth on the other, is that whereas the former two can

be described as examples of ‘terminal anchorage’, the latter represents a case of

‘continuous friction anchorage’. The subject of reinforced earth is covered in greater

detail in Topic 25.

Among the non-conventional types of retaining structures must be mentioned crib

walls and gabions (Topic 28) both of which are predominantly gravity structures, but

to a degree flexible, in nature. Even ‘tetrapods’, laid along coast lines as a protective

measure against sea erosion, can be considered as falling under the broad category

of retaining structures.

13.3 Drainage of retaining walls

The retaining wall, except when it serves as a water-front structure (Fig.13.11),such

as a quay wall, where the water table in the backfill eventually attains the same level

as the water in front of the wall, is not called upon to resist water pressure from the

backfill. We have already noted in Topic 4 that water pressure is of the order of twice

the active earth pressure in the dry state, and thrice the submerged earth pressure. It

will therefore be highly uneconomical to design the wall for such a high extra load.

Therefore any water that seeps into the backfill must be expeditiously drained away

(Fig.13.12) before allowing it to build up and exert a hydrostatic head on the wall. A

system of drainage is therefore an essential component of a retaining structure. More

details on drainage will be found under Topic 21. To prevent the drainage system from

becoming impaired due to clogging, the system must be designed to satisfy the

requirements of an inverted hydraulic filter (Topic 6).

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13.4 Backfill material

As regards backfill material, where we have a choice, cohesionless soils such as

sand and gravel must be preferred on account of the lower active pressure due to

higher shear strength, and at the same time, and the higher permeability which

facilitates drainage. Clayey soils are clearly inferior in both these respects.

In this Section, our discussion on geotechnical design will be confined to 1)

reinforced concrete cantilever retaining wall (gravity type), and 2) sheet pile walls or

‘bulkheads’ (non-gravity type).

13.5 Stability analysis

The two phases of design of foundations, namely geotechnical design and

structural design that we noted at the beginning, apply equally to retaining structures,

with the only difference that ‘stability analysis’ takes the place of geotechnical design

in respect of the latter. And just as in the case of geotechnical design, the product of

stability analysis is the cross sectional dimensions of the retaining structure such as a

retaining wall, which are arrived at satisfying all the requirements of stability

independently whereupon its structural design takes over.

In the case of a two-dimensional structure such as a cantilever retaining wall (its

counterpart in foundations is the continuous footing), the dimensions to be so fixed are

the cross sectional dimensions of the retaining wall. And just as we had two aspects,

viz., bearing capacity and settlement to be satisfied in the case of geotechnical design

of foundations, there are several aspects of stability, which are to be satisfied

independently in the stability analysis of the retaining wall. Therefore, when the critical

factor relating to stability, which may vary with the parameters of the problem at hand,

is just satisfied, all the other factors of stability will be oversatisfied. This is a situation

which cannot be helped and which does not therefore imply overdesign. This is indeed

similar to the situation with respect to satisfying bearing capacity and settlement in the

case of geotechnical design of foundations. And as in the case of geotechnical design,

stability analysis turns out to be an indirect exercise in that one has to initially assume

tentative dimensions, check for stability, which if not satisfied in full, revise the

dimensions, in stages, till all aspects of stability are fully satisfied, with one aspect just

satisfying. This implies that, in the initial trial, if the requirement with regard to all the

aspects are undersatisfied, we have to increase the dimensions. On the other hand, if

all the requirements are oversatisfied, we have to decrease the dimensions until the

results converge to the correct value from either the higher or lower side. At that value,

one requirement – which we shall call the critical aspect - will be just satisfied, the rest

of the requirements being oversatisfied. Thus the end product of stability analysis is

a properly dimensioned retaining wall, at which stage its structural design takes over,

which completes the design process. It is interesting to note that while structural

design in the case of a foundation, such as a footing, fixes its thickness and amount

of steel, in the case of an R.C. cantilever retaining wall, since the cross section is

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finalised at the stage of stability analysis itself, the role of structural design reduces to

simply fixing the quantity of steel. And in this respect it has a close parallel to the

structural design of deep foundations, such as a pile, as already mentioned under

Topic 11.

13.6 Earth pressure

We have already come through classical earth pressure theories due to Coulomb

and Rankine under Topic 4. However, if we can resort to standardisation in the matter

of typifying the backfill material, it is possible to have a semi-empirical approach to the

problem of determining earth pressures, which would be more expedient in dealing

with practical problems of design. Accordingly four types of backfills have been

identified in the order of decreasing quality and charts have been prepared which

enable the expeditious determination of the horizontal and vertical components of the

total active earth pressure on the vertical face passing through the rear end of the

base slab (not the stem, which is used only in its structural design). This vertical face

extending from the base of the base slab up to the top of the backfill is called the

‘virtual back’. These charts are presented in Fig.13.13.

It is to be noted that the coefficients kh and kv used here are not the dimensionless

earth pressure coefficients K(Topic 4), but coefficients obtained empirically, and

multiplied by assumed values of the unit weight of the soil. Hence they have the

dimensions of unit weight [kN/m3]. It is seen that these coefficients and hence the

corresponding earth pressures increase with decreasing quality of the backfill, except

in the case of backfill of Type 4 which gives a zero vertical pressure. A further point

that is noticed is that the total pressures are rather insensitive to the angle β at the

lower ranges of values for the same.

In the empirical approach based on the above charts, even when the natural soil at

site is used as the backfill material, one can still make use of the chart pertaining to

the standard backfill which the actual backfill resembles most, with necessary

adjustments for design, by way of interpolations.

13.7 Initial proportioning of the retaining wall

Since stability analysis is essentially iterative, one needs to start with initial trial

dimensions. Instead of starting with random dimensions, guidelines have been

evolved from experience, which when followed, have been found to give satisfactory

results calling for the least number of revisions.

These guidelines are essentially based on the total height of the wall (H) which

must be fixed taking into consideration the height of the soil retained. (In fact, if the

stem of the retaining wall can be considered as the superstructure, the base slab can

be looked upon as its structural foundation. Accordingly the depth of foundation, Df,

can be fixed based on the considerations of determining the same in the case of

shallow foundations.) These guidelines are stated below with reference to Fig.13.14.

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1) Width of base slab (B): 0.4 to 0.65 of H, the smaller ratio applying when the base

is supported by firm soil and when the backfill is horizontal and is of silt, clean sand or

gravel. The ratio increases with decreasing quality of the subsoil and increasing slope

of the backfill. Live loads on the backfill will, however, need increased widths.

2) Thickness of the base slab: 1/12 to 1/8 of H

3) Width of stem at top: 300 mm

4) Rate of increase of the width of stem: 20-60 mm per m height

5) Projection of toe from the stem: 1/3 of B

13.8 Reduction of the system of forces (Fig.13.15)

As earlier, we shall take a unit length of the retaining wall into consideration. Before

analysing the retaining wall for stability, it shall be convenient to reduce the effect of

all the forces about the tip of the toe, marked as point A in the figure.

This is done on the basis of the fact that the effect of any force such as V (see inset

of Fig.13.15) about a point such as A, is the same as the force V plus a couple = V x

v, where v is the lever arm of the force about the point A. The same applies to a

horizontal force such as H, also shown in the inset.

Thus the system of forces shown in the block under consideration, consisting of the

gravity forces and the two components of earth pressure, is equivalent to ΣV, ΣH

and ΣM at the point A, where ΣV is the sum of all the vertical forces, i.e., all gravity

forces and Pv, ΣH, the sum of all horizontal forces (here Ph only since we ignore earth

pressure from the toe soil) and ΣM, the algebraic sum of all the moments, i.e., due to

all gravity forces, as also Pv and Ph.

13.9 Aspects of stability

The following are the aspects of stability which are to be independently and

simultaneously satisfied in the stability analysis.

1.Overturning about A

Strictly speaking it is not necessary to consider this aspect as it is automatically

satisfied when the Aspect 2. below is satisfied. We shall, however, state the

requirement in terms of a factor of safety against overturning but without specifying

any minimum design value for it.

Fo = Σ stabilising moments

Σ overturning moments (13.1)

In the above the numerator includes all moments including that due to Pv, all of which

contribute to stability. The denominator is the moment due to Ph which causes

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overturning. Conservative designers, however, will be reluctant to include Pv as

contributing to stability being a non-gravity force.

2.Tension at the base

The vertical soil pressure acting upwards on the base slab will remain compressive

throughout when the resultant of the system of forces intersects the base within the

mid-third of the width B (the ‘mid-third rule’ – see Kurian, 2005: Sec.8.8). The vertical

and horizontal components of the resultant force R, acting on the base slab are the

same as ΣV and ΣH and to obtain the point of intersection C, we isolate ΣM and ΣV at

A. Now ΣV and ΣM together are equivalent to ΣV at a distance L (L = AC), where

L = 𝛴𝑀

𝛴𝑉

In the above ΣM includes the moments due to all the forces including Pv and Ph and

ΣV, the sum of all vertical forces including Pv. (The above result is obvious since in the

reverse order ΣV at C is equivalent to ΣV and ΣM = ΣV.L at A.) The system is complete

when ΣH is also transferred to the point of intersection C, even though ΣV and ΣM are

sufficient to locate the point of intersection since both the points A and C are on the

line of action of ΣH, ΣH has the same effect at both A and C; in other words, transfer

of ΣH from A to C does not involve any change in the net effect.

Now, e = 𝐵

2− 𝐿

Which must be ≤ B/6, for no tension, i.e., for no tensile soil pressure to develop at the

base.

When the mid-third rule is satisfied, the requirement regarding overturning can be

deemed to have been automatically satisfied, which makes a check on overturning -

based on a minimum factor of safety – redundant.

When e > B/6, there will be resultant tension at the heel and consequently a

redistribution of soil pressure takes place to keep it compressive throughout (see

Kurian, 2005: Sec.8.8).

When the base slab is supported on rock, mid-half is substituted for mid-third in the

above rule.

3.Maximum toe pressure

Referring to Fig. 13.15,

pmax = Σ𝑉

𝐵(1 +

6𝑒

𝐵) (13.2)

pmax should not exceed the design value of ‘allowable soil pressure’. The latter should

be obtained from bearing capacity and settlement, considering the eccentricity of load.

Further, if pmax were a net quantity, both bearing capacity and settlement should have

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been based on net considerations. On the other hand, if pmax is a gross quantity, as

obtained in the present case, both bearing capacity and settlement should also be

based on gross values. What is, however, invariably done is considering a

presumptive value of the allowable soil pressure for comparison with with pmax

obtained as above.

4.Sliding at the base

Considering ΣV and ΣH at point C, it is obvious that ΣH will cause a tendency for

the retaining wall to slide at the base, which is prevented by the friction that is

mobilised, the maximum value of which is ΣV multiplied by the coefficient of friction

between the base slab and the soil below. Therefore the factor of safety against sliding

can be stated as :

Fs = 𝛴𝑉𝑥𝜇

𝛴𝐻 (13.3)

The minimum value of Fs normally specified is 1.5. For coarse-grained soils free from

silt, µ may be taken as 0.55, while for coarse-grained soils with silt the same may be

taken as 0.45. For pure silt the value goes down to 0.35. If the base rests on soft clay,

on the other hand, in place of friction what gets mobilised is the ‘adhesion’, the

absolute maximum value of which may be taken as half the unconfined compressive

strength.

Designers are aware that such a seemingly simple requirement as the above is

practically the most difficult to satisfy among all the aspects of stability discussed

above, in retaining wall design. A useful result of this situation is that the value of B

corresponding to this critical aspect can be obtained by setting up Eq.(13.3) which will

reduce as a linear or quadratic equation. The B so obtained can be advantageously

taken as the initial trial dimension, as done in the corresponding design software

(Sec.13.13).

If the initial dimensioning fails to satisfy the requirements 2., 3. and 4., the solution

lies in increasing B, or if oversatisfied, by decreasing B. Such a change in B, however,

modifies the data itself, since practically all the self weight terms, as also Pv and Ph

get modified on account of it. Indeed the essence of the solution is the differential

variation of the vertical and horizontal forces with change of B. A further point to be

noted is that modification of B at any stage results not only in the change of magnitude

of forces and moments, but necessitates checking for stability through all stages prior

to it including that stage (Fig.13.16). This results in increasing looping of operations

with the advancement of the stage at which revision of dimensions is called for. The

repetitive nature of the work can turn out to be highly tedious depending upon the

combination of parameters involved in a specific problem, which makes it ideal for

programming the work on the computer. This task has been happily attempted in the

design software referred to above.

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A further point to be noted is that, had we considered the passive pressure at the

toe either in full or to a factor of safety, it would have considerably added to stability

thereby leading to a more economic design. (Note that ΣM at A and ΣH would have

been favourably influenced by the passive pressure, ΣV remaining the same.) We

have, however, chosen to ignore it, to be on the safer side, and also on account of the

uncertainty regarding the mobilisation of passive pressure at the toe. This fact,

however, need not discourage us from considering the beneficial influence of the

weight of the soil on the toe slab which is fully dependable on account of its simple

physical presence.

13.10 Submerged backfill

If the retaining wall serves as a water-front structure (see Fig.13.11), such as a

quay wall, water from the front will seep into the backfill and eventually the water table

in the backfill will rise to the same level as the free water level in the front (see

Fig.13.18). The water in the backfill running through the pore spaces is continuous

and will exert the same hydrostatic thrust as the water in the front, cancelling each

other, leaving the retaining wall being subjected to the active thrust from the

submerged soil in the backfill.

Submergence of the base slab and the part of the stem as also the soil on the toe

and heel slabs affects the magnitude of the self weight forces. Referring to Fig. 13.18,

if γsat is assumed as equal to γ(Sec.10.4.1), the net effect of submergence is obtained

by superimposing an upward force equal to the weight of the body of water ‘abcd’

(Fig.13.18) acting at its centre.

As regards the horizontal pressure, since thy hydrostatic pressure diagrams on

either side cancel each other, one is left with the net active pressure diagram due to

the dry unit weight of the soil above the water table and the submerged unit weight of

the soil below the water table. (Note that the influence of submergence on Ø and

consequently on Ka is negligible.) Based on Rankine’s theory, it is seen from Fig.

13.17 that the soil pressure at any depth can be written as the sum of the pressures

at individual heights making up the total height. Considering Rankine’s theory for earth

pressure, following Fig.13.18, one notices that the net pressure diagram due to

submergence is obtained from the original pressure diagram by subtracting a triangle

the base of which is equal to γwhKa and height h. (Note that this is a mathematical

result, as one cannot ascribe any physical meaning to multiplying water pressure γwh

by Ka.) In other words, the effect of submergence is obtained by superimposing on

the original pressure diagram the above triangular diagram with its resultant acting in

the opposite direction.

If the backfill is sloping and if we propose to use the charts (Fig.13.13), the above

superimposition can be effected in the following manner.

We assume γsat ≅ γ and γsub ≅𝑟

2

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Now, γsub = γsat - γw ≅ γ - γw, it follows that γw = γ - γ

2=

γ

2

We can therefore superimpose forces P’h and P’v each of which is half the chart values

obtained for the depth h, and acting at a height h/3 from the base, in directions opposite

to the original thrust components Ph and Pv (Fig.13.19). (Note that, even though there

may be no P’v due to water, it must be included since we are using a soil chart to

consider the effect of water indirectly.

The above approach of superimposing vertical and horizontal forces due to

submergence on the original system of forces will be found to be very convenient in a

systematic approach to stability analysis without and with water table above the base.

Magnitudewise, however, the stability of the retaining wall will indeed be affected to

the extent of the differential influence of submergence on the original vertical and

horizontal forces.

As a matter of fact, due to submergence ΣV is more affected than ΣH. This means

a lesser ΣV is called upon to resist a less ΣH, clearly indicating the need for an increase

in B.

13.11 Bulkheads

Bulkheads constitute the most important type of retaining structures after retaining

walls. Bulkheads are sheet pile walls which are made up of a number of thin vertical

elements, called, ‘sheet piles’, driven in such a way as to form a continuous and

reasonably tight wall (see Figs.13.6,7). Bulkheads are mostly used as water-front

structures in marine constructions. Since the sheet piles are thin, the stability of the

wall comes not from self weight, but from the depth of penetration in the case of the

cantilever wall (see Fig.13.6). Anchoring is a necessary step for anything bur short

heights of soil to be retained. The term ‘bulkhead’ is normally reserved for the

anchored type of wall, the cantilever type being normally referred to as the cantilever

sheet pile wall. Even though there is no bar on calling an anchored bulkhead an

anchored sheet pile wall, one shall not call a cantilever sheet pile wall a cantilever

bulkhead.

The geotechnical design, or rather the stability analysis, of a bulkhead involves the

determination of the depth of penetration and the forces in the anchoring system which

together ensure stability of the wall. Hence these are the aspects we shall be

addressing ourselves to in the following.

13.11.1 Stability analysis of a cantilever sheet pile wall

Before analysing a regular cantilever sheet pile wall, we shall consider the rather

hypothetical case of a sheet pile wall subjected to a concentrated load H at a height h

from the ground (Fig.13.20). The latter is actually meant to facilitate the analysis of the

former. Our aim is the determination of the depth of penetration d which will provide

the requisite stability. Being a starting case, we shall determine H at the stage of

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incipient or imminent failure, i.e., corresponding to a factor safety of unity. We shall

consider a case where ground water is nor encountered within the depth of

penetration. It shall be convenient for us to use Rankine’s theory for limiting active and

passive earth pressures. Accordingly the active pressure at depth z from the ground

surface is:

pa = 𝛾𝑧𝐾𝑎

where 𝛾 is the unit weight of the soil, and Ka = [1−𝑠𝑖𝑛∅

1+𝑠𝑖𝑛∅]= tan2 (45 -

∅

2). Ka is the ‘coefficient

of active earth pressure’, which is dimensionless. We shall now rewrite,

pa = KAz, such that KA = 𝛾Ka

calling KA the active pressure increment, since it gives the rate of increase of active

pressure with depth. KA therefore has the same dimension as 𝛾, to be expressed in

kN/m3. In the same way, we shall define the passive pressure increment KP, as 𝛾Kp,

where KP is the ‘coefficient of passive earth pressure’, using which pp = KPz.

We know,

Kp = 1

𝐾𝑎= [

1+𝑠𝑖𝑛∅

1−𝑠𝑖𝑛∅] = tan2 (45+

∅

2)

For the purpose of analysis we consider the sheet pile as a rigid body, and further that

its rotation and the corresponding translation mobilises the limiting active or passive

pressure, as the case may be. As usual, being a 2-dimensional situation, the analysis

is per unit length of the wall.

If the pile rotates about the base (point B), passive earth pressure develops to the

left and active earth pressure to the right. Since the former is higher than the latter,

the resultant earth pressure is the triangular figure on the L.H.S. with a base intensity

of (KP – KA)d. (Note that (KP – KA) is the slope of the line FA.) If, on the other hand,

the rotation takes place about the ground point A, we have passive pressure on the

right and active pressure on the left with a resultant pressure diagram with the same

base intensity as earlier, but on the R.H.S. Therefore, for a point of rotation between

A and B, it shall be logical for us to consider the resultant pressure diagram as

beginning to show a reverse trend at some depth and going over to the right. Point

C corresponding to this change is defined by the height f from the base, in the figure.

As a result of the above, we are now left with a second unknown f in addition to the

primary unknown d. We shall now proceed to determine their values by invoking two

conditions of equilibrium, viz., ΣH = 0 and ΣM = 0.

In writing the equilibrium equations, it shall be convenient for us to add the area

BDCF to the pressure diagrams both on the L.H.S. and R.H.S., since the same area

gets cancelled when the algebraic effect is taken.

Accordingly, taking ΣH = 0, we get,

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H - 1

2(𝐾𝑃 − 𝐾𝐴)𝑑2 + (𝐾𝑃 − 𝐾𝑎)𝑑𝑓 = 0 (13.4)

From which,

f = (𝐾𝑃−𝐾𝐴)𝑑2−2𝐻

2(𝐾𝑃−𝐾𝐴)𝑑 (13.5)

Eq.(13.5) expresses the unknown f as a function of the unknown d.

Taking ΣM about B gives:

𝐻(ℎ + 𝑑) −1

2(𝐾𝑃 − 𝐾𝐴)𝑑2.

𝑑

3+ (𝐾𝑃 − 𝐾𝐴)𝑑𝑓.

𝑓

3= 0 (13.6)

Substituting for f from Eq.(13.5) in Eq.(13.6) and simplifying, we get,

𝑑4 −8𝐻

(𝐾𝑃−𝐾𝐴)𝑑2 −

12𝐻ℎ𝑑

(𝐾𝑃−𝐾𝐴)− [

2𝐻

𝐾𝑃−𝐾𝐴]

2= 0 (13.7)

Eq.(13.7) is of the fourth degree in d, solving which we get the value of d, which is the

depth of penetration for equilibrium.

(Instead of solving a higher degree equation such as the above by trial and error, one

may use the Newton-Raphson method, which enables the quick determination of the

root starting from a random trial value. This is fully explained by Kurian

(2005:Sec.6.3.1).

Even though the stability analysis, or geotechnical design, is over with the

determination of the depth of penetration d, we are in a position to draw the resultant

earth pressure diagram by substituting d in Eq.(13.5) and obtaining f.

As stated at the beginning, the above analysis was carried out to assist the stability

analysis of an actual cantilever sheet pile wall. Referring to Fig.13.21, under condition

of retention of the soil, it is obvious that active pressure will develop behind the wall

over the full height of the soil retained. The active pressure will start decreasing below

this height reaching 0 at A beyond which we shall assume a pressure condition

depicted by Fig.13.21. It therefore follows that if we are in a position to determine H

and h, we can continue with the earlier analysis and complete it on the lines described

above. For this we first determine the depth n to point A, which is obtained by invoking

the condition that GF is a straight line having a slope of (KP – KA). H is nothing but the

area of the triangle JGA and in order to determine h, one only has to take moments of

the two triangles constituting JGA about A.

After determining d and f, we have to replace H by the actual pressure diagram

represented by the triangle JGA. Actually we have no use of this pressure diagram

until we come to structural design for which this is the loading diagram on the sheet

pile wall using which we draw the B.M. and S.F. diagrams needed in the structural

design.

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13.11.2 Stability analysis of anchored bulkheads

The depth of penetration needed for static equilibrium in the case of a cantilever

sheet pile wall normally exceeds the height of the soil retained by it. It follows from the

above that a cantilever sheet pile wall is not an economically feasible proposition for

anything but short heights of soils to be retained. Hence one must seek additional

sources of stability when there is need for retaining higher depths of soil. Anchoring

the sheet pile wall at the top to a stable terminal anchor established in the backfill at a

distance sufficiently away from the sheet pile wall constitutes a viable solution to the

problem which can bring down the depth of penetration dramatically thanks to the

additional stability of the system provided by the anchorage. Such sheet pile walls are

called anchored bulkheads. The stability analysis of anchored bulkheads involves the

determination of the depth of penetration and the tension in the anchor rod per unit

length of the wall.

We shall now proceed with the analysis of an anchored bulkhead. For the sake of

convenience, we shall assume that the sheet pile wall is anchored at the ground level

(Fig.13.22).

We shall start by assuming a one-sided pressure distribution below point A as

shown in Fig.13.22. We shall soon see that it is possible to satisfy equilibrium by such

a pressure diagram.

The unknowns in the problem to be determined are the depth of penetration d below

the point A, and T, which is the tension in the anchor rod/cable per unit length of the

wall. This is possible after determining n in the same manner as described in

Sec.13.11.1

In order to determine d we shall take ΣM about the anchor rod. This will result in a

cubic equation which on solving gives d. We can now complete the pressure diagram,

using which we express ΣH = 0. The latter gives the second design parameter T.

13.11.3 Anchored bulkheads with ‘fixed earth support’

The case analysed above pertains to an anchored bulkhead with the minimum

depth of penetration for static equilibrium. If we want to increase the depth of

penetration beyond the minimum depth, the pressure diagram will change by

developing a zone to the R.H.S.as in Fig.13.21. Since d is known in the present case,

the unknowns are f and T which can be determined by taking ΣM = 0 and ΣH =0,

respectively.

If we take the slope of the sheet pile wall at the toe in the present and previous

cases, it will be found that the same has decreased with increased penetration. (Slope

is actually relevant, if at all, only in the part on structural design, but we need it here

for developing the present topic.) We would find that at the increased depth of

penetration, not only the slope θ, but also the tension T and the maximum bending

13

moment Mmax have also decreased. If we attempt to increase the depth of penetration

further, we will find that the quantities θ, T and Mmax register a further decrease. This

will continue until we reach the depth at which θ = 0, indicating fixity at the base. At

this depth it has been found that T and Mmax reach their minimum values. Since θ has

already become zero, any further increase in depth is not going to affect the results

any further.

The above results are schematically assembled in Fig. 13.23. Case 2 in which the

bulkhead is driven to the minimum depth of penetration consistent with static

equilibrium, is referred to as a bulkhead with free earth support. Case 4 which is the

limiting case (θ = 0) where the bulkhead has been driven to a depth consistent with

fixity at the base is referred to as a bulkhead with fixed earth support. The depth of

penetration corresponding to fixity is also referred to as, (1) favourable depth of

penetration, and (2) economic depth of penetration. The term favourable is used on

account of fixity at the toe. The term economic is used since the lowest values of T

and Mmax associated with this depth indicate economy in the section of the anchor

cable and the sheet pile. As regards the latter, it has been found that economy is not

confined to the section of the sheet pile alone, but there is overall economy in material

even after offsetting for the extra depth, when compared to the minimum depth of

penetration.

13.11.4 Analysis of anchored bulkheads with fixed earth support

Since the criterion for fixity is the no-slope condition at the base, in order to

determine the depth of penetration for fixity, one has to try increasing values of depths

from the minimum depth for static equilibrium until one reaches the depth at which the

slope of the wall at the base is zero. Being a trial-and-error approach, this method is

cumbersome and one has to think in terms of a direct method, however approximate

it may be, for determining the depth of penetration for fixity. Blum’s equivalent beam

method is one such method, the features of which we shall examine in the following.

13.11.4.1 The equivalent beam method

The basis of this method lies in the structural behaviour of the sheet pile wall which

is necessary for us to examine to develop the subject. It has been found from the

bending moment diagram of an anchored sheet pile wall driven to fixed earth support

that the point of contraflexure almost coincides with the point A at which the resultant

earth pressure is zero. A further finding that follows from the above is that the portion

of the sheet pile wall above point A can be treated as a simply supported beam,

supported at the point of the anchor cable and the point of zero pressure A, and loaded

by the triangular pressure diagram (Fig.13.24). This beam is called the equivalent

beam. A further finding has been that the maximum bending moment obtained by

analysing this beam is practically the same as the maximum bending moment obtained

in the actual analysis of the entire sheet pile height.

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As regards design it has been further found that the simply supported reaction R1

at the point of anchorage, obtained by taking moment about A, is very close to the

actual tension T in the anchor cable. In order to find the depth of penetration beyond

A, we make use of the simply supported reaction R2 acting at point A.

As stated in Sec.13.11.3, the resultant soil pressure diagram at the depth

corresponding to fixity has a zone on the R.H.S. from A up to the base. Reverting to

the equivalent beam, the sheet pile being continuous, the simply supported reaction

at A can be looked upon as the action of the portion of the sheet pile wall below A on

the portion above A. Hence the action (or rather reaction) of the sheet pile wall above

A on the portion below A is a force at A equal in magnitude but opposite in direction to

R2 (Fig.13.24). Considering the forces on the section of the4 sheet pile wall below A,

ΣM about B gives,

R2.d = 1

2(𝐾𝑃 − 𝐾𝐴). 𝑑. 𝑑.

𝑑

3

simplifying which R2 = (𝐾𝑃 − 𝐾𝐴)𝑑2

6

From the above we get,

d = √6.𝑅2

(𝐾𝑃−𝐾𝐴) (13.8)

In the above equation for ΣM = 0, there is an error due to including area 1 and

excluding area 2. The effect is cumulative and not compensatory. In order to account

for the same, it has been recommended that 10 % may be added to the above value

of penetration giving the total penetration as,

d = 1.1[n + √6.𝑅2

(𝐾𝑃−𝐾𝐴)] (13.9)

and that completes the stability analysis of an anchored bulkhead with fixed earth

support by the celebrated equivalent beam method.

13.12 Anchorages for bulkheads

Anchors for bulkheads are classified as continuous anchors and individual anchors.

Continuous anchors run parallel to the sheet pile wall at some distance away

(Fig.13.25a).Individual anchors are those to which the anchor cables are individually

connected (Fig.1 3.25b).

We shall only take up anchor wall which is an example of a continuous anchor and

examine how it derives its anchoring capacity.

As regards ‘anchoring capacity’, we can state,

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The allowable tension in the anchor cable/tie rod = 𝐴𝑛𝑐ℎ𝑜𝑟𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑐ℎ𝑜𝑟

𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑠𝑎𝑓𝑒𝑡𝑦

In the case of the anchor wall, since the anchor cables terminate at continuous

structures on either end, the allowable anchor pull on the continuous anchor per unit

length is the same as T obtained in the stability analysis of the bulkhead.

13.12.1 Anchor walls

These are rigid continuous vertical walls. Their anchoring capacity depends upon

the resultant of the passive and active thrusts on either side of the wall. While the

passive pressure contributes to stability, the active pressure acts against it.

Considering a unit length of the wall which starts from the ground level down

(Fig.13.26), we can write,

Pp = 𝑟𝐻2

2 Kp, and Pa =

𝑟𝐻2

2 Ka, from which the anchoring capacity,

A = Pp – Pa = 𝑟𝐻2

2 (Kp – Ka)

Since factor of safety is to apply only on the passive trust, we can state,

T = 𝑃𝑃

𝐹 - Pa (13.10)

From Eq.(13.10) we can determine H, the height of the wall, which constitutes the

geotechnical design in this case.

Kurian (2005: Ch.6) gives detailed Design Plates covering R.C. cantilever retaining

wall and cantilever sheet pile wall and bulkheads of various types, which the student

must attentively learn to understand the nuances of the analysis.

13.13 Design software for retaining structures

The software carries out the geotechnical design (by stability analysis) of R.C.

cantilever retaining walls, cantilever sheet pile walls, anchored bulkheads, and many

other retention structures such as cellular cofferdams and cut supports, not covered

here.

In the case of R.C. cantilever retaining wall, the initial trial dimension (base width)

is determined on the basis of base friction, which is found to reduce the number of

iterations to zero in most cases. The software is general, which also accounts for the

position of the water table, which may be either below or above the toe soil. The scope

of the design of sheet pile walls and bulkheads has been greatly enhanced by the

software by enabling the consideration of layered soils above and below the dredge

line, besides a variable factor of safety on passive soil resistance.