PRACTICAL DESIGNING
OF
RETAINING WALLS.
BY
ARTHUR JACOB, A. B.,
ASSOCIATE OF THE INSTITUTE OF CIVIL ENGINEERS; LATE EXECUTIVE ENGINEER, H..L BOMBAY SERVICE.
NEW YORK:
D. VAN NOSTRAND, PUBLISHER,
23 M3URRAY AND 27 WARREN STREET.
1873.








PRACTICAL DESIGNING
OF
RETAINING WALLS.*
IRTRODUCTORY.
In designing masonry works there is
hardly any subject that presents itself more
frequently than the retaining or revetment wall; and in some form or other it is
found to enter into almost every design.
To the military engineer no less than to
his civil brother is the subject one of importance and interest, forming as the revetment wall does, for the most part, a
component element in works of defence.
To military engineers in truth is due some
of the most valuable  information  that
v Practical Designing of Retaining Walls, by Arthur Jacob,
A. -B.




4
civil engineers possess regarding the theory
of earth pressure, and although further
considerations are involved in designing
revetments for military works than the
mere support of earthwork, there is still to
be derived from the experiments and researches of military men information of
much value to civil engineers. The subject is one that has received the fullest and
most able treatment at the hands of mathematicians, and solutions for every case that
could possibly occur in practice are to be
found in our text-books. But the mathematical investigations of this and many
other questions of common occurrence in
practice, unquestionably valuable as they
are, in determining the principle involved,
and establishing final rules applicable to
practice, are, it is believed, but rarely resorted to by practical engineers. Even
when such examples have to be dealt with
by those sufficiently acquainted with the
mathematical mode of proceeding, they are
generally solved without hesitation by some
empirical rule, derived from experience.
Such a method may, and doubtless occa



5
sionally does, lead to accident from weakness, and not unfrequently to clumsy waste
of material and consequent expense. But
it is not clear that less of failure or clumsiness would result if every retaining wall
were calculated with mathematical precision, for in truth the data involved are so
variable and imperfect, and the disturbing
causes are of such a character as to neutralize to a great extent the accuracy of the
investigation. With certain specific data
theoretical accuracy can always be attained;
but the engineer as a rule knows nothing
with absolute certainty either of the weight
of the earth he has to sustain in position,
or of the masonry that he intends to adopt
in doing so. These and other data he
must assume before he enters on his calculations; and though there is not in these,
as in many other investigations, any
necessity to attempt an extreme degree of
refinement, which would be inapplicable
for every-day practice, yet there can be no
good excuse for dealing with the matter by
hap-hazard and guess-work.
It is not proposed now to regard with




6
more than a cursory glance the principles
involved in determining the strength of
walls to support earthwork. Such simple
rules will be given, as it is hoped will
serve —due regard being had to the peculiarities of each particular case-to guide
the less experienced in designing works
of this class. The empirical mode of dealing with the question is clumsy and unscientific, whilst the formulae usually given
are so complicated as to render their application to practice out of the question.
SPECIFIC CAUSES OF FAILURE.
It must not be presumed that the failure
and destruction of a retaining wall is necessarily due to the wall being of itself insufficiently strong. It may be quite heavy
enough to resist the pressure of a bank, if
due regard be had to the mode of forming
the earthwork, and to drainage; but if
these points be not fairly considered and
observed at first, a retaining wall of quite
sufficient thickness will probably give way
sooner or later. As much care should in
fact be devoted to the method of backing




up and draining a wall, as to the calculation of its section; for indeed if these matters be disregarded, no retaining wall, properly so called, can be implicitly relied
upon to stand. With the exception of one
particular case, which will be noticed hereafter, walls are designed on the assumption that they are to support a dry
material-or one, at any rate, not permeated by water-and that the material is
to be deposited in such a manner as to
have no predisposition to slide against the
wall. It is, of course, also presumed that
the wall shall be of fair workmanship and
materials, and where these points cannot
be relied upon, as is sometimes the case,
especially in foreign works, some allowance should be made in the dimensions of
the wall. It has not unfrequently happened that a retaining wall will have stood
for a considerable number of years without
showing any appearance of yielding, and
yet will give way suddenly and completely,
without apparent cause. Such failures can
generally be accounted for by the fact of
the wall not being designed to resist a




8
maximum pressure, and never having been
tried fully till the time of its destruction.
Much apparent anomaly is observed in the
way that retaining walls are found to fulfil
the purpose for which they are designed:
for whilst some will yield, others of less
dimensions will continue to stand; such
apparent inconsistency giving occasion for
ingenious theories, most of them entirely
unsupported by fact or experience. The
truth is, that imperfect drainage, defective
foundations, or rotten work will account for
almost every conceivable case of failure.
FIRST CASE-HYDROSTATIC PRESSURE.
The first and simplest case of a retaining
wall to be considered is that in which the
pressure of water has to be counteracted;
not indeed that the question in such a form
belongs strictly to the subject under notice;
but it nevertheless becomes absolutely the
method of determining the strength of walls
for certain positions. It not unfrequently
happens, as in some hydraulic works or
with the wing-walls of aqueducts, that the
infiltration from behind, which is not al



9
ways avoidable, may produce.such a pressure as no retaining wall properly so called
could be expected to bear. With this view
the engineer's limit of safety will be attained when the structure is designed to sustain the full hydrostatic pressure. The
pressure of water upon any plane surface
immersed is known to be equal to the area
of that surface, multiplied by the depth of
its centre of gravity below the level of the
water, and by the weight of a unit of water.
Generally speaking the unit adopted in calculation is a foot; and the unit of water
being taken at a cubic foot, weighing 62.5
lbs., the resulting product, from the multiplication of the three quantities, will give the
pressure in pounds on the surface immersed.
Let it be supposed for simplicity that water
to the depth of 10 ft. has to be sustained
by a vertical rectangular wall. It is usual
to take but I ft. length of the wall for the
calculation, though it will not affect the result whether 1 ft. or 100 ft. be the length
assumed. We then have the surface under
pressure= 10 sq. ft., the depth of the
centre of gravity =  5 ft., and the weight of




10
a cubic foot of water = 62.5 lbs.; the product of which quantities give us 3,125 lbs.,
thepressure on 1 ft. length of the wall. But
this pressure is not the whole of the force
that the wall has to resist; the leverage
that it exerts must also be taken into account. In the example under consideration-namely, that of a vertical plane, with
one of its sides coinciding with the surface
of the water, as in Fig. 1.-the whole of
the pressure is so distributed as to be equal
to a single force acting at a point one-third
of the depth from the bottom. Thus the
total force to be resisted by the wall is
3,125 X 3.33 =  10,416, which is the moment tending to overturn the wall.
MOMENT OF RESISTINCCE TO OvERTURNING.
It is evident that a certain weight of
wall must be opposed to this overturning
forc6; and as the height of the wall and
the length are determined quantities, the
thickness alone remains for adjustment.
But as a rectangular wall in upsetting is
considered to turn upon a single point, F,
Fig. 1. —namely, the outer line of the foot




11
of the wall, there will be a certain amount
of leverage to assist the wall in resisting
the pressure of the water. This leverage is
the horizontal distance of the centre of
gravity of the wall from the turning point
F, and when the structure is rectangular
and vertical, it is equal to half the thickness.  The amount of the wall's resistance
will then be equal to the number of cubic
feet in one foot of its length, multiplied by
the weight ofa single cubic foot of masonry,
and by half the thickness of the wall.
Taking w = the weight of a cubic foot of
water = 62.5 lbs., w l = the weight of a
cubic foot of masonry, say 112 lbs.; x =
thickness of the wall, and h = the height;
the condition of simple stability will be fulfilled when
w1 xh   h
W1 X h X  X2 —wXhX- X-2   (1)
wa h    w w h3
2  -  6
and solving for x we get
-- ='''W.        (2)
The thickness of the wall. 4 ft. 4 in.




12
EXAMPLE.
A simple example has been selected for
illustration, but of course a rectangular
section of wall would not be found generally applicable in practice, nor would it be
expedient to limit the dimensions of a retaining wall of whatever kind to the minimum that would sustain the pressure;
some margin of safety must therefore be
allowed, to cover inferior work and materials. It is true that no account has been
taken of cohesion, which, if the wall be
founded on rock or concrete, may be assumed to add to its stability about 7,000 lbs.
for every square foot of base. In addition
to this, practice seems to indicate an increase
on the calculated thickness, and in the,example the mean width might be augmented to 5 ft., the stability being further
increased by altering the section from a
rectangle to a battering wall with offsets at
the back.
A good general rule for the dimensions
of a wall designed to support water or earth
in a semi-fluid condition will be



13
Top breadth = 0.3
Middle do. - 0.5
Bottomdo. -0.7
The height being represented by unity.
Proceeding to the consideration of walls
for the support of dry earth, it will be found
that the question is one that will in general
require the engineer to exercise his judgment, to determine what angle of repose he
will base his calculation upon. The natural
slopes assumed by earths of different tenacity are so various, that an average figure
cannot be adopted with safety; the calculation of pressure from earth, in fact, depends
essentially on this point, and a disregard of
it will lead to very doubtful results. The
following are a few of the slopes assumed
by different materials, but it is probable
that the engineer's judgment will be of
more service than any table in deciding the
angle of repose. The examination of a district in which works are intended to be
carried out will always suffice to satisfy the
designer of the nature of the material that
he is dealing with, and may enable him to




14
proportion his works very nearly to the requirements of safety and economy:Angle of repose.   Slope.
Dry sand, clay and mixed fFrom 379.. 1.33 to 1*
earth.................   to 21.. 2.62 to 1
Damp clay..............     45Q.. 1   to 1
Wet clay............. Fr to  1     43.23 to 1
From 48  0.9 4   to 1
Shingles and gravel...... F o 450.. 0.9 to 1
to 350    1.43 to 1.From 450.. 1 to 1
Peat..................................
to 14     4   to 1
To which might be added as a special feature London clay; it appears under the influence of weather to be exceedingly unstable, slipping away to almost any angle
of repose.
THEORY OF IARTH PRESSURE.
It has been ascertained by M. Prony that
when a vertical wall sustains the pressure
of a bank of earth the top of which is
horizontal, the maximum horizontal pressure to which the wall can be subjected will
be reached when the plane of fracture of
the earth bisects the angle that would be
formed were the earth to slope from the
* Rankine's " Manual of Civil Engineering."




15
foot of the wall backwards at the natural
inclination. This fact is somewhat striking,
for it would appear at first sight, and was
for long assumed, that the angle of fracture
ought to coincide with the natural slope of
the earth; such is, however, not really the
case. If we suppose the angle made between the sloping plane and the vertical to
be bisected, the prism of earth enclosed between the bisecting plane and the wall will
represent the mass, the pressure of which
has to be resisted; and this being the maximum pressure that a horizontal topped
bank is capable of exerting, it is usually
the point to be determined.
Referring to Fig. 1, the principle of
earth pressure will readily be understood.
Supposing the plane of rupture to bisect
the angle c-which will be the case when
the pressure is a maximum-the prism cut
off will be the whole weight that the wall
will have to sustain. Taking this prism
for a single unit of length or thickness, the
superficial area will represent the cubic
contents. But the area of the triangle,
taking h as the height of the wall, will be




16
h2 tan. i-c
2
c being the angle contained between the natural slope of the earth and the back of the
wall. It is only necessary to multiply this
value by w, the weight of a cubic foot of the
bank, to get the total weight of the prism.
FIG. 1.
D    E
//~_~   //
F    B
This prism of earth is then like any other
body-resting upon an inclined plane; which
in this case is the plane of rupture. It is
sustained in position by the wall on one
side and by the fixed portion of the bank




17
on the other; and may be regarded as a
solid mass of material without motion
amongst its parts. The line K M represents the direction of the force of gravity,
and the lines K L and K 0 the pressures
exercised against the wall, and the force of
the bank respectively.  These pressures
produce a certain amount of friction against
the wall and the bank, but, as the friction
against the wall does not materially affect
the question, the friction of the bank alone
is considered, and taken into account in
arriving at the following formula which
applies to the case of a vertical wall supporting a bank with  a horizontal-topped
bank:
to hz
P      =   tan..  * (3)
Having calculated the pressure of the earth,
the next step will be to determine its moment to overturn the wall, and this can be
ascertained, as in the case of water, by
multiplying the pressure by one-third of
the wall's height.  This having been determined the next consideration will be,
what weight of wall will suffice to sustain




18
it; and the method of arriving at this is
similar for the most part to that adopted
for water. Taking, as above, the moment
of the wall to resist the pressure, the following equation will represent the conditions of stability: —
wh' h xt  w tw h    h
tan. i c2L     z          3
And solving for x, the thickness of the wall,
we have/to wh tan. 2 i c
3-.'. o               (4)
If the weight of a cubic foot of earth be
taken equal to a cubic foot of the wall, the
value will be/hi tan. 2 C
"  1/  3.       (5)
wiich would give a thickness of 2.69 ft. for
a rectangular wall of 10 ft. high supporting
a bank of earth, the angle of repose being
taken at 40 deg. The average weight of
brickwork and ordinary clay will generally
be nearly the same; but if great accuracy
be desired, and the respective weights of
the materials be known, the general formula No. 4 must be used.




19
The following table gives the weight per
cubic foot in pounds avoirdupois of such materials as come under our consideration in
solving questions relative to retaining walls:
Weight of a
cubic foot
in pounds.
Sand-damp.......................    120
Do.   dry........................  90
Marl............................    100
Clay..............................    120
Gravel...........................  125
Brick.........................     130
Brickwork........................    112
Masonry...........................    130
Mortar............................    110
PARTIAL RETAINING WALL.
Having so far considered the first two
cases, namely, those of a wall supporting a
horizontal-topped bank of earth in a semifluid condition, and also in a state of comparative dryness, the next example that
suggests itself to our notice for examination
is that of a partial retaining wall, or a wall
from the top of which the bank slopes away
for a certain height-called the surcharge
— either at the natural slope of the earth or
at a less inclination. Such mode of con



20
struction is of very common occurrence,
dwarf walls being frequently adopted on
railway works where the cuttings or embankments are of considerable height, and
when carefully designed are found to effect
a saving of expense, both in construction
and in the item of land. In cuttings the
walls are carried up to such a height as
economy dictates, and the slope is then
trimmed back at the proper angle. Similarly with embankments, the walls are so
disposed as to cut off the foot of the slope.
In either case a little consideration will suffice to show whether the saving of earth
and land area will cover the cost of the retaining walls. In military works, as well
as civil, the partial revetment is very commonly used, being, indeed, a component
part of almost every system of fortification.
The first particular case belonging to this
class, though not of the commonest occurrence in civil practice, is when a partial retaining wall supports a bank, the face of
which slopes back at an angle less than the
natural slope of the earth. As M. Prony's
rule, that the plane of rupture bisects the




21
angle between the natural slope of the earth
and the back of the wall, only holds good
when the surface of the bank is at right angles to the plane of the wall, another mode
of determining the angle for the maximum
pressure must be resorted to. The simple
construction given in the note enables us to
arrive at the maximum pressure for a wall
at any given batter, with the surcharge
above sloping at any inclination.  The
equation arrived at is the expression for the
maximum horizontal pressure:
p  w h2  tan. 0 tan. 2 ( )  (6)
-41    tan. 0 - tan. c
the angle c being that between the back of
the wall and the natural slope; 0 = the angle made by the face of the bank with the
plane of the wall; and p = the angle between the plane of rupture and the back of
the wall. The value for c - b will be found
in the note. Taking, for example, a vertical wall of 10 ft. high, supporting a bank
that slopes back at an inclination of 20 deg.
with the horizon, the natural slope being
40 deg., the value of tan. (c. -- /) will be




22.4610; inserting this value and working
out the equation, we arrive at a pressure of
2,100 lbs. against the back of the wall.
For the case of a revetment sustaining a
surcharge the centre of pressure will be, as
in the former case, at one-third of the height
of the wall, giving a leverage of 3.33 feet.
This gives 2,100 lbs. X 3.33 =-6,993, the
moment of the earth tending to overturn
the revetment. Equating this value to the
moment of the wall, taking the cube foot of
brickwork at 112 lbs., the same weight
as the earth, and solving for x the thickness, we find it to be 3.53 ft.
DEFINITE SURCHARGE.
The next case to be considered is one of
much more frequent occurrence in practice
than that just mentioned; it is a partial retaining wall supporting a surcharge of earth,
sloping away at the natural inclination,
and terminating in a horizontal plane above.
Cuttings and embankments partly supported
by masonry works furnish familiar examples of this, which is denominated the
" definite surcharge." The most convenient




23
method of determing the thickness of wall
in -this instance will be to consider, first, the
conditions of stability for an infinitely long
siope, which, however, can only.[have a
FIG. 2.
theoretical existence; and having arrived
at the thickness of wall necessary to support
such a bank, a simple reduction will give




24
the thickness required when the length of
slope is limited.
It has been mentioned that when a vertical wall sustains a bank with a horizontal
top, the plane of rupture for the maximum
pressure is found to bisect the angle between the natural slope and the vertical.
It is also an ascertained fact, that as the
angle of the surcharge increases, the angle
4, or that between the plane of rupture and
the back of the wall, also increases; until
the face of the bank slopes at the natural
inclination of the earth, and then the plane
of rupture becomes parallel to it. From
this it would appear that when the slope is
infinitely long-a condition that could not
exist in practice —the pressure will also be
infinitely great; but such is not really the
case. The ratio of the pressure of a bank,
whatever its inclination, to the pressure
exerted by an embankment level with the
top of the wall can never exceed 4: 1. The
formula, then, for finding the maximum
horizontal pressure exerted by an infinitely
long slope against a vertical wall will
be



25
to h'(
P=  2-sin. 2 C..  (7)
the notation being exactly the same as in
the other cases. If we work this pressure
out, using the same values for w, h, and c,
as taken above, we shall find P=3,281
lbs.
Now for the leverage: we have, as in
every other case, simply to divide the height
of the wall by 3, which in our example
gives 3.33 and the moment to overturn the
wall = 3,281 X 3.33=10,925. Proceeding
in the same manner as before, the width of
a wall of brick to counterbalance an infinitely high bank sloping at the natural
inclination, will be found to be 4.43 ft.
When the surcharge is very high as
compared to the height of the wall, no reduction of the thickness will be necessary,
for practically the slope may be considered
infinite; but when the bank does not overtop the wall by a great height it will be
Well to apply the following formula to ascertain the corrected thickness. Let h —
height of wall-10 ft., h'=height of surcharge above the wall, which we shall take




26
at 20 ft., t=thickness of wall to support a
horizontal bank, as found in the first case
_2.69 ft., T=the thickness of a wall for a
20 ft. surcharge, t'=thickness for indefinite
slope as found=4.43. Working this out
the thickness is found to be 408 ft.
T -h t + 2 h' t'     (8)
h + 2 h.
So far we have considered the cases of
more usual occurrence in practice, namely
those in which the back of the wall is vertical or stepped, which is practically the same
thing. For the calculation of leaning walls
the reader is referred to the general formulas (A) and (B) given in the note; from the
latter formula the horizontal resistance of
any bank, supported by a wall at any angle.
of inclination, can be ascertained, and the,
leverage being in every case taken at one —
third of the height of the wall, there will be
no difficulty in designing a wall of such a
section as will resist the pressure of the
bank effectually.
The point to be kept in view is the moment of the wall, and this must be made:




27
to exceed the maximum overturning force
of the embankment. It will not suffice to
equalize the moment of the earth's force to
the resistance of the wall, as has been done
FIG. 3.
_            ~1in the examples above; a certain excess of
resistance will be necessary, and this can
easily be attained by giving the wall a
batter, or else sloping it back so as to throw




28
the centre of gravity of the mass as far
back as possible, in a horizontal direction
from the outer line of the foot of the wall.
The line of the centre of gravity must not,
however, be allowed to fall inside the base
of the wall, otherwise the stability of the
structure will become dependent on the
support of the bank, and will have a tendency to slide away from its position.
It has been stated, and taken for granted,
that banks of earth, when they destroy retaining walls, do so by turning them over;
this is, however, not invariably the case.
It has occasionally happened that walls
have been moved bodily forward, sliding on
their base. Subh an occurrence is certainly
accidental, and is probably the result of the
wall having been founded on an unstable
material, perhaps on an inclined bed of
moist and uncertain soil. Walls have also
given way in rare instances by the upper
courses of the structure yielding to pressure,
breaking off and falling over; a contingency that is probably due to the upper part
of the bank becoming suddenly charged
with water, and exercising an undue pres



29
sure on the wall before there is time for the
water to drain away. These must be regarded as rare contingencies, arising out of
some defect of the foundations, or backing;
and cannot affect the consideration of the
wall's stability generally. The theory of
the wall being turned over on its base provides for the greatest trial to which the
structure can be subjected, or, in other words,
the wall would as a general rule give way
under a much less pressure by falling over,
than would be required to overcome friction, and move the wall forward in its entire state; if therefore the structure is considered as having to withstand the overturning force, it will always be strong
enough to resist being pushed forward.
RETAINING WALL WITH CURVED BATTER.
A form of retaining wall commonly met
with in practice, especially in brickwork
structures, is that with a curved batter,
stepped in offsets at the back. The curve
usually adopted is the arc of a circle, the
radius of which is from 21 to 3 times the
wall's height; and the centre of the curve




30
is as a rule in the same horizontal plane as
the top of the wall. In such structures the
courses are made to radiate from the centre,
and the result is that the joints of the
brickwork at the back are thicker than is
either necessary or advisable. When the
radius of curvature is large, the increase of
thickness is inconsiderable, but it becomes
decidedly an objection when the curve is a
short one; for the thickness of the wall will
not become reduced in the same proportion
as the height or as the radius of curvature.
The dimensions of a wall of this kind may
be determined with sufficient accuracy, by
first considering it as a leaning wall at a
given slope, and using the general formula
(6), and in this manner a very close approximation to the thickness may be arrived
at. There are, it is true, specific formula
given by some authors for determining the
thickness of curved walls, but they are too
complex for application in practice. The
effect of the curvature will be to add to the
wall's stability by bringing the centre of
gravity farther in towards the bank, and
this, indeed, is the only advantage that the




31
curved form possesses; it is difficult to construct, and consequently expensive; for the
saving of material, if any, is very trifling.
In architectural effect it certainly has no
advantage over the wall with a straight
batter, for the simple reason that it does
not convey the same idea of strength. If
the curved wall is supposed to derive any
additional stability from its curvature, on
the principle of the arch, as some have
fancied, it must be recollected that an arch
with but one abutment is a very unstable
kind of structure, and such kind is simply
what the curved retaining wall is. Quays
and river walls may, indeed, be designed of
a curved form with advantage, for such will
allow of ships coming closer to the brink,
than they could were the wall a straight
one. And sea walls, also, are not unfrequently built of a curved section on the face,
this form being under certain circumstances
better adapted than a straight wall to resist
the force of waves.
In situations where a retaining wall has
but one purpose to fulfil —that of supporting a bank of eartli-.L lb  usual to give




32
the base of the wall a certain amount of
inclination to the horizontal, the slope being perpendicular to the batter of the face;
or if the wall have a curved batter, the
plane of the base will simply radiate from
the centre of curvature. Such mode of
construction is calculated to increase the
frictional stability, for it brings the thrust
of the earth from behind more nearly perpendicular to the bearing surface.
COUNTERFORTS.
Counterforts are frequently constructed
at the back of retaining walls, and, although generally approved of, appear to be
a somewhat doubtful mode of distributing
material. Mr. iosking, in a paper read
before the Institute of Civil Engineers,
deprecates their use and, with some reason,
advocates the use of ribs or arches from
wall to wall. These ribs seem to have been
suggested by the cast-iron beams used to
support the falling walls on the London
and North Western Railway between
Euston Station and Primrose Hill. Mr.
Hosking proposes that his arches of brick




33
should pass completely over the road, and
that they should consist on plan of a pair
of flat arches placed back to back. Such
an  arrangement would doubtless prove
effective, and the expediency of adopting it
would evidently be determined by the cost
of the work and the value of land adjoining-a mode of construction in common
use in metropolitan works, and in other
situations where land is very valuable, is
that shown in Fig. 4. It consists of a series
FIG. 4.
BUTITRESS.    [PLAN.]       BMREMS.
of buttresses and inverts, the convexity of
which latter is opposed to the thrust of the
backing. Such a distribution of material
is most suitable in situations where the
projection of the buttresses is not found
inconvenient. In quay and river walls it




34'would not answer of course to have any
such projection, as the near approach of
ships and boats is an essential consideration.
The distribution of the material in the
form of counterforts is attended with a
slight saving, and where buttresses would
be inadmissible on account of their encroaching on the roadway, counterforts
may be adopted. They have at least one
use, that they oppose more friction to the
earth than a plain wall, and, being easy of
construction, are productive of but little
additional expense. In order to ascertain
what additional mean thickness a wall derives from the counterforts, it is only necessary to multiply the length of the counterJort by its mean width, and divide the
product by the distance from  centre to
centre of two counterforts. The form and
dimensions of counterforts vary with circumstances, the narrow and deep disposition of the material being probably the best
as a general rule. The late Lieutenant
Hope, of the Royal Engineers, conducted
some interesting experiments on the sta



35
bility of retaining walls generally, and arrived at the conclusion that a thin wall,
with frequent thin counterforts, was the best
arrangement of the material.
Two points of importance relative to counterforts demand particular attention-the
first, that they should be built simultaneously with the wall; and the second,
that the wall should be well bonded into
the counterforts, otherwise they detract from
the wall's strength, instead of augmenting
it. It is evident that without some special
system of bond, counterforts reducing the
thickness of the wall, as they are generally
understood to do, must prove detrimental
rather than advantageous; but if plenty of
hoop iron be used, which is not usually the
case, counterforts may be made to contribute
in a very considerable degree to the stability
of the wall. In fact, quite as much as buttresses.
MODE OF BACKING AND DRAINAGE.
That accidents frequently occur from due
care not being exercised in the mode of
backing-up retaining walls is undoubted,




36
and indeed to this cause alone the majority
of failures is attributable; not, as is frequently supposed, to the insufficient section
of the wall. The drainage of masses of
earth sustained by walls, is a matter that
can only be disregarded with risk of ill
consequences. It is a difficult thing to prevent surface water from finding its way
into earth-work, and therefore the simplest
method of dealing with it will be to provide
efficient means for its escape. To this end
holes or weepers should be left in the wall
at different levels, to relieve it from pressure
from behind; and in order to admit the
surface water to these points of escape, it
will be advisable to back up the wall with
dry stone, quarry shivers, or whatever else
will admit the free passage of water. If a
wall be backed up in this way by a rough
angular material, it will be relieved of almost all pressure from the earth. Economy
will, however, generally preclude such an
expedient in works of considerable extent,
and then it will be necessary to form the
embankment with great care, adopting every
precaution to prevent the tendency of the




37
earth to slip in the direction of the wall.
It will be evident from the calculation of
the pressure exerted by earth, that the less
the angle of repose is, the greater will be
the pressure on the wall; and, as a matter
of course, any means that will tend to increase the angle of repose, will relieve the
wall of a certain amount of pressure. Effectual drainage will do much towards this
end; but the mode of depositing the earth
will also affect the angle of natural slope in
a considerable degree. The same earth
under different treatment will assume different slopes; if dry, it will fall when
tipped-at a low angle, but if damped, and
well rammed, will adapt itself to a much
higher one. It has even bees found that a
bank when constructed in such a manner has
stood for a considerable time perfectly vertical. The best mode of backing a wall up with
earth will then be, to commence depositing at
the foot of the wall, and to lay the earth in
layers inclining against the wall, as shown
by the dotted lines in Fig. 3, each layer
being well rammed before another is commenced. This will not only consolidate the.




38
earth, and prevent any shock that might
occur from sudden settlement, but will increase the angle of repose, and give the
earth a tendency to slip away from the wall,
rather than towards it.
TRE LAND TIE.
An expedien' for securing retaining walls
that is simple and not expensive, is the land
tie; it consists of an iron plate, with a rod
passing through its centre, the plate being
placed vertically in the bank behind the
wall, and the end of the rod passed through
the wall and secured. The holding power
will depend on the area of the plate, and
the depth at which it is sunk beneath the
surface. But it is evident that, in order to
act most effectually, land ties should be attached to the wall at the height of the centre
of pressure.
NOTE I.
The following construction, given by
Mr. Neville in the "Transactions of the
Institute of Civil Engineers, Ireland," vol.
i., shows the method of determining the




39
pressure exerted by a bank, whatever may
be its inclination:
FIG. 5.
/ _,,_,.  t n-c —-., /
~~/..
Let C D represent the wall; D E the
face of the bank sloping at any angle; and
C H the line of natural slope. Draw any
line perpendicular to the line C HI, cutting
the line of the wall produced at A, and also
a line drawn parallel to the face slope at
0. On A 0 describe a semicircle. From




40
A0, as a centre with the radius 0 HI, describe an arc cutting A 0 in I: draw I C.
The triangle C D F represents the maximum to be resisted. The angle - 0 - c.
The complement of the angle of repose
=p; and the face C D = h
tan. (c —I)=(tan. 2 3+tan. c tan. d) i —tan. d. (A)
Putting R for the maximum horizontal resistance, and w for the weight of a cubic
unit of the bank, the resistance of pressure
will be
R1  n h2 tan. 0 tan. 2 (c-).  (B)
Z    tan. 0 —tan. c.
in which the value (c-b) found above
must be substituted.  When C D E is a
right angle we shall have
R=-' tan. 2 J  c       (C)
the equation given in the first part of this
article; and that which holds good when
the slope of the bank is at right angles to
the face of the wall.
The following tables calculated by Mr.
J. H. E. Hart, Executive Engineer of the




41
Bombay Department of Public Works, are,
by his kind permission,' appended to this
pamphlet, and will be found very convenient for the calculation of Retaining
Walls.
Knowing the angle of repose of the earth
to be supported, and the relative weights of
the masonry of the wall and of the earth
per cubic unit, a simple reference to the
table will give a coefficient, which multiplied by the height will give the requisite
thickness. For example, supposing a horizontal topped bank has to be supported by
a masonry wall of 10 ft. high, and of twice
the specific gravity of the earth, the angle
of repose of the latter being 35 deg.
Under the fraction i, and opposite to 35
deg., will be found in Table A the fraction.212, which multiplied by 10, the height of
the wall, gives 2.12 ft., the required mean
thickness.




TABLE A.
Table of coeficients of h for finding the Thickness of Ctandard Rectangular WaVls,
when the top of the bank is horizontal.
Ratios of1.            I        I  i| I                                  2
W  toW'. +  TI    - TfI 1-  T.    T5 T.6 T.Y  T'.       T. I  A    V..   Y.4...
Angle of
Repose.                              K   K    K
0.577 550.527.506.488.472.456 443.430.419.408.399 389.381.373.365
309.334.317.304.292.282.272.263.256.248.242.235.230.224.220.215.211
31~.327 311 298.286.276.267.258.251.243.237.231.225.220.216.211.207
32~0.320) 305.292.281.270.261.253.245.238.232.226.221 216.211.206.2()2    L
330.314 299.286.275.265.256.248.240.234.228 222.217.212.207.203.198
34Q.3t7. 293.9 280.269.260.251.2433.236.229.223.217.212.207.203.198.194
35Q   3.28(;6.274.263.254.245.237.230.224.218.212.207 202.198  194.190
360.295.280.268.258.248.240.232.225.219 213.208.203.198.194.190.186
370.288.274.262.252.243.235.227.221.214.209.203.199.194.190.186.182
380.282.268.257.246.2381.230.222.216.209.204.199.194.189.185.182.178
390.276.262.251.241.233.225.217.211.205 199.194.190.185.180.177.174
40Q   269.256.245.235.227.219.212.206.200.195 190.185.180.177.173.170
41~   263 250.240.230.222.215.207.202.196.192.186.181.177.173  170.166
42~.257.244.234.225.216.210 202.197.191.186.181.177.173.169.165.162
430.251.239 229.220.212.205.198.192.187.182.177.173.169.164.162.158
449.245.233.223.214.206 200.193.188.184.177.172.I69.164  161.158.154
450   239.227.21&. 209.202.2951.188.183.178.173.168.165.161.157  154   151




TABLE B.
Tab,'e of coe~fciernts of h for finding the Thickness of Standard Rectangular Walts,
when the top of the bank s.opes away at the Ang~e of Repose.
Ratios of    l!~        I tL                            ~ -s 1 ~ I; 
WtoW'. I  I..-2.3.4 T.-  1.6 1.7  1 -T.   2  2-. 2'.*  2.*   24   2
Angle f                              K   K  | K
Repose.                                                         ___..
30n.500.476.456.438.423.409.395.384.372.363.353.345.337.330.323.316
31Q   494.471.452.434.418.404.391.80.368.359.3 01.342.333.326.320.313
32~   489.466.447.429.414. 400.387.376.365.355.346. 338 330.323.316.309
33~   483.461.442.424.409.395.382.371.360.351.342.334 326.319.312   306    c
340~.478.456.437.419.404.391.378.367.356.347.338.331.322.316.309.303
350~.472.450.4311.414.399.386.373.362.352.343.334.327 318.312.305.299
36Q.467.445.426.409. 95.382.369.358.348.339.330.323 315.308.302.295
37Q.461. 439.4211.404:389.377.364.3451.343.334.326.319.311.304.298.291
380.4551 433 4151.399.34.372.359.349.339.330 39.330321.314.306.300.294..288
390   448.427 409i 393.379.367.354.344.334.325.317.310.302.296.290.284
400.442.421 404.388.374.361.349.339 329.321.312.306.298.292 286   279
410.435.415.397.382.368 356.344.334.324.316.307.301 293.287.281.275
42~.429.409.391.376.363.351.339.329.319.311 303.296 289.283.277.2'71
430.422[ 402.385.370 357.345.333.324.314.306.298.292.284.278.273.267
440.415 395 379.364 351.339.32r.' 18.309.301.293.287.280.274.268.262
450.408.389.3731.358.345 334.4322.313.304.296.288.282 1.275.269.264.258




44
NOTE II.
The following graphic method for determining the pressure of earth against
a retaining wall, we take from " Engineering:"
Referring to Fig. 1, let us determine,
first, the pressure exerted by the wedge,
A, B, 0, the angle, B, A, E, being greater
than 0, the limited angle of resistance of
the wedge against A, B, which is also
identical with the natural slope of the
earth. The friction against the back of
the wall is neglected. We have now  3
forces to deal with-namely, the weight
of the triangle, A, B, 0, acting vertically
through its centre of gravity, and therefore passing through the point c, where
A, c =  I A, B; next, the resistance of the
plane A, B, the direction of which is inclined at an angle, 0, to the normal to A,
B; and, lastly, the thrust against the back
of the wall acting horizontally through
c, and cutting A, C in a point, g, where
A  — A, C.




45
Now, since the weight of the wedge, A
B C, is proportional to C B, the height of
the wall remaining constant, and the slope
varying, if we set off c b = C B, and complete the triangle of forces, a b will represent the thrust against the back of the
wall.
Let us now observe the effect produced
by altering the slope A B, to A D.
Construct the triangle of forces d e f, as
before, making I f=  C D, to represent
Fro. 1.
E
the weight of A C D. The angle, e d f, is
now greater than the angle a c b, by the




46
same amount that we have increased the
slope of A B to A D; that is to say, the
angle e df= — a c b + angle D A B.
Also, the length of b c has decreased to
fd in the ratio of C B to C D.
Supposing now we divide up the angle
C A F into any number of equal angles by
the radial lines Al A2 A3, etc. (see Fig. 2),
and imagine the slope of A B to be altered
FIG. 2.
I I       3  4  5   6    7     F
C
A. 
to each of these positions successively, we
shall then for each alteration have a new
triangle of -forces; for instance, in moving
from the position Al to A2 the angle b A p,
of the triangle of forces will increase to c A
q in the same ratio that the slope of the




47
plane varies, and the side A b will decrease
to A c in the same ratio that C, decreases
to C2.
We are thus enabled to make a diagram
illustrating the successive changes by a
curve, e d c b, A b, A c, A d, etc., being
respectively equal to C1, 02, C3, etc. The
lines b p, c q, d r, e s, and c drawn at right
angles to A b, A c, A d, etc., will now represent the thrusts against the back of the
wall at the different slopes, and it will be
observed on examining the diagram that
the position which gives the greatest magnitude to the line representing the thrust is
the slope A4, which bisects the angle C
A F.








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12