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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 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 V, H
and M at the point A, where V is the sum of all the vertical forces, i.e., all gravity
forces and Pv, H, the sum of all horizontal forces (here Ph only since we ignore earth
pressure from the toe soil) and M, 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 = stabilising moments
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 V and H and to obtain the point of intersection C, we isolate M and V at
A. Now V and M together are equivalent to V at a distance L (L = AC), where
L =
In the above M includes the moments due to all the forces including Pv and Ph and
V, the sum of all vertical forces including Pv. (The above result is obvious since in the
reverse order V at C is equivalent to V and M = V.L at A.) The system is complete
when H is also transferred to the point of intersection C, even though V and M are
sufficient to locate the point of intersection since both the points A and C are on the
line of action of H, H has the same effect at both A and C; in other words, transfer
of H from A to C does not involve any change in the net effect.
Now, e =
2
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 =
(1 +
6
) (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 V and H at point C, it is obvious that H 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 V 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 =
(13.3)
The minimum value of Fs normally specified is 1.5. For coarse-grained soils free from
silt, 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 M at A and H would have
been favourably influenced by the passive pressure, V 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 sat is assumed as equal to (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 and
consequently on Ka is negligible.) Based on Rankines 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 Rankines 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 whKa and height h. (Note that this is a mathematical
result, as one cannot ascribe any physical meaning to multiplying water pressure wh
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 sat and sub
2
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Now, sub = sat - w - w, it follows that w = -
2=
2
We can therefore superimpose forces Ph and Pv 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 Pv 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 V is more affected than H. This means
a lesser V is called upon to resist a less H, 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 Rankines theory for limiting active and
passive earth pressures. Accordingly the active pressure at depth z from the ground
surface is:
pa =
where is the unit weight of the soil, and Ka = [1
1+]= tan2 (45 -
2). Ka is the coefficient
of active earth pressure, which is dimensionless. We shall now rewrite,
pa = KAz, such that KA = Ka
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 , to be expressed in
kN/m3. In the same way, we shall define the passive pressure increment KP, as Kp,
where KP is the coefficient of passive earth pressure, using which pp = KPz.
We know,
Kp = 1
= [
1+
1] = 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., H = 0 and M = 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 H = 0, we get,
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H - 1
2( )
2 + ( ) = 0 (13.4)
From which,
f = ()
22
2() (13.5)
Eq.(13.5) expresses the unknown f as a function of the unknown d.
Taking M about B gives:
( + ) 1
2( )
2.
3+ ( ).
3= 0 (13.6)
Substituting for f from Eq.(13.5) in Eq.(13.6) and simplifying, we get,
4 8
()2
12
() [
2
]
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 M 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 H = 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 M = 0 and H =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 , but also the tension T and the maximum bending
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moment Mmax have also decreased. If we attempt to increase the depth of penetration
further, we will find that the quantities , T and Mmax register a further decrease. This
will continue until we reach the depth at which = 0, indicating fixity at the base. At
this depth it has been found that T and Mmax reach their minimum values. Since 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 ( = 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. Blums 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,
M about B gives,
R2.d = 1
2( ). . .
3
simplifying which R2 = ( )2
6
From the above we get,
d = 6.2
() (13.8)
In the above equation for M = 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 + 6.2
()] (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 =
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 = 2
2 Kp, and Pa =
2
2 Ka, from which the anchoring capacity,
A = Pp Pa = 2
2 (Kp Ka)
Since factor of safety is to apply only on the passive trust, we can state,
T =
- 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.
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