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BOUGHT WITH THE INCOME OF THE
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' THE GIFT OF
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1891
Cornell University Library
TA 770.P12
Retaining walls; their design and constru
3 1924 015 698 016
PI Cornell University
P Library
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RETAINING WALLS
THEIR DESIGN AND CONSTRUCTION
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PUDIISHERS OF BOOKS FO B^
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EETAINING WALLS
THEIR DESIGN
AND CONSTRUCTION
BY
GEORGE PAASWELL, C. E.
First Edition
McGRAWHILL BOOK COMPANY, Inc.
NEW YORK: 239 WEST 39TH STREET
LONDON: 6 & 8 BOUVERIE ST., E. C. 4
1920
\
lij.lvl I \M ■ U ': I I Y
^^SfoSS
coptkight, 1920, bt the
McGrawHill Book Company, Inc.
THE MAPLE PRESS TORE: PA
PREFACE
The presentation of another book on retaining walls is made
with the plea that it is essentially a text on the design and con
struction of retaining walls. The usual text on this subject
places much emphasis upon the determination of the lateral
thrust of the retained earth; the design and construction of the
wall itself is subordinated to this analysis. Without gainsaying
the importance of the proper analysis of the action of earth
masses, it is felt that such is properly of secondary importance in
comparison with the design of the wall itself and the study of
the practical problems involved in its construction.
It is the purpose of the first chapter to present the existing
theories of lateral earth pressure and then to attempt to codify
such theories evolving a simple, yet wellfounded expression for
the thrust.
An attempt is made to continue this codification throughout
the theories of retaining wall design so that a direct and continu
ous analysis may be made of a wall from the preliminary selection
of the type to the finished section. Such mathematical work as
is presented is given with this essential object in view.
Under Construction advantage is taken of a classic pamphlet
on Plant issued by the Ransome Concrete Plant Co. (which pam
phlet should be in the possession of every construction engineer)
to illustrate the principles of proper plant selection.
A retaining wall is a structure exposed to public scrutiny and
must, therefore, present a pleasing, but not necessarily ornate
appearance. Since, in the case of concrete walls, the appearance
of the wall is dependent upon the character of the concrete work,
it is essential that the edicts of good construction be observed.
For this reason the modern development of concreting is pre
sented fully with frequent extracts from some of the recent im
portant reports of laboratory investigators.
It is hoped that proper credit has been given to the authors of
all such quoted passages, as weU as to other references used.
A vast amount of literature exists on the subject of retaining walls
vi PREFACE
and earth pressure (see bibliography at the end of the book),
and in view of the absence of a proper collation of all this material
there is, of course, much duplication of the analysis. It is hoped
that before future studies are made of earth pressure phenomena,
an attempt will be made to examine existing literature and that
a due appreciation will be had of the subordinated importance
of the determination of lateral pressure.
I must take this opportunity to thank Mr. Arthur E. Clark,
Member, Am. Soc. C. E., for his patient reading of the text and
his many helpful hints.
To Mr. F. E. Schmitt, Associate Editor of the Engineering
NewsRecord, I am deeply grateful for encouragement and aid
in preparing the book and in arranging the subject matter in a
logical and clear manner.
The Author.
New York,
Feb, 1919.
CONTENTS
Preface. v
List OP Plates ... vii
PART I
DESIGN
CHAPTER I
Earth Pressures . . . . 1
History of the various theories of earth pressure — Exact analysis
of the action of earth masses — The ideal earth and the fill of
actual practice — The two theories — Rankine's Theory — Cou
lomb's method of the wedge of shding — Various other methods of
thrust calculations — Experimental data — Wall friction — Cohesion
— Surcharge — Pressure on cofferdams — Pressure of saturated soils
— Sea walls — Problems.
CHAPTER II
Gravity Walls 42
Location and height of wall — General outline of wall — Two classes
of walls — Fundamental principles of design — Concrete or stone
walls — Thrust and stabiUty moments — Foundations — Distribution
of base pressures — Factor of safety — Footing — Direct method of de
signing the wall proper — Revetment walls — Problems.
CHAPTER III
Reinporceb Concrete Walls 79
General principles — Preliminary section — Distribution of base pres
sures — Tables and their use — Theory of the action of reinforced
concrete — Bending and anchoring rods — Vertical arm — Footing —
Toe extension — Counterfort walls — Face slab — Footing — Counter
fort — Rod system — Problems.
CHAPTER IV
MiSCELLANBOXTS WaLL SECTIONS 122
Cellular walls — Hollow cellular walls — Timber cribbing — Concrete
viii CONTENTS
cribbing — Walls with land ties — Walls with relieving arches —
Parallel walls enclosing embankments — Abutments — Box sections
subject to earth pressures — Advantage of the various types of walls
— Problems.
CHAPTER V
Tempebatttbe and Shkinkagb. Genebal Notes .151
General theory of the flow of heat and the range of temperature in
concrete masses — Shrinkage — Settlement — Expansion joints —
Construction joints — Wall failures.
PART II
CONSTRUCTION
CHAPTER VI
Plant 165
Relation between plant and character of works — Standard plant
layouts — Subdivision of field operations — Mixers — Distribution
systems — Examples of plant layouts.
CHAPTER VII
FOBM WOBK 181
Subdivision of forms — Concrete pressures — Major Shunk's experi
ments — Robinson's experiments — ^Lagging, joists and rangers — Tie
rods — Bracing — Stripping forms — OUing and wetting forms — Pat
ent Forms — Hydraulic, Blaw — Supporting the rod reinforcement
— Examples of form work — Problems.
CHAPTER VIII
CONCEETE CONSTBTJCTIGN 197
Modern developments — Prop. Talbot's notes on concrete —
Conclusions of Bureau of Standards — Pbof. Abram's analysis of
concrete action — ^Importance of the water content — Prof. Abbam's
conclusions — ^Apphcation of theory to practice — Concrete methods
distributing concrete — Keying lifts — ^Use of Cyclopean masonry —
Winter concreting — Acceleration of concrete hardening — ^Concrete
materials — Cements — Sand — Crushed stone and gravel — Fineness
modulus — Method of surface areas (Capt. E. N. Edwards) —
Crum's method of proportioning aggregates.
CHAPTER IX
Walls other than Concrete 227
Plant required — Mortar — Construction of wall — Coping — Face
finish — Special stone — Plaster coats — Cost data.
CONTENTS ix
CHAPTER X
Architectural Teeatmbnt; Dkainagb; Waterproofing 232
Architectural treatment — Face treatment — Rubbing — Tooling —
Special finishes — Colored aggregates — Artistic treatment in general
— Hand rails and parapet walls — Drainage — Examples in practice
— Waterproofing.
CHAPTER XI
Field and Office Work. Cost Data 242
Surveys necessary — Construction lines — Walls on curves — ^Lines
for concrete forms — Computation of volumes — Isometric repre
sentation of wall details — Cost data — ^Labor costs — Examples of
cost of work.
APPENDIX
Specifications; Bibliography; General Index 254
Index 271
LIST OF PLATES
Facing Page
Plate I . .... 46
Fig. A. — Dry rubble wgll along highway.
Fig. B. — Characteristic appearance of cement rubble wall.
Plate II 150
Fig. a! — Crack in reinforced concrete wall at junction of wing wall
and abutment.
Fig. B.^Structural steel supports for special type of retaining wall.
Plate III 150
Crack in sharp corner of wall due to tension component of thrust.
Plate IV 191
Fig. A. — ^Unsatisfactory rod detail for concrete pouring.
Fig. B. — Holding vertical rods in place before concrete is poured.
Plate V 227
Fig. A. — Method of laying stone wall by series of derricks.
Plate VI 227
Fig. A. — Uncoursed rubble wall with coursed effect given by false
pointing.
Fig. B. — Rubble wall (Los Angeles) with face formed by niggerheads. '
Plate VII 236
Fig. A. — Showing effects of poor concrete work.
Fig. B. — Ornamental parapet wall. Tooled with rubbed border.
Plate VIII 236
Fig. A. — Ornamental handrail — approach to viaduct.
Fig. B. — Picket fence wall lining open cut approach to depressed
street crossing. ^
Plate IX . . . 236
Fig. A. — Ornamental concrete handrail approach to concrete arch.
RETAINING WALLS
THEIR DESIGN AND
CONSTRUCTION
PART I
DESIGN
CHAPTER 1
THEORY OF EARTH PRESSURE
The Development of the Theory of Earth Pressure.' — A search
through engineering and other scientific archives fails to yield
any evidence that prior to 1687 an attempt had been made to
analyze the action of earth
pressure upon a retaining
wall. Undoubtedly, rough
methods of computing wall
dimensions existed back in
prehistoric times, since the
art of constructing retaining \ '''©
walls is as old as building art
itself. In 1687 General
Vaxiban,^ a French military
engineer gave some rules for
figuring walls, but presented
no theoretical basis for these
rules. It is questionable whether such existed. In 1691 Bullet'
advanced a rather primitive method, assuming that the angle
of shding (see Fig. 1) is 45°. The weight of this sUding
1 The facts in the historical outline are taken from "Neue Theorie des
Erddruckes," E. Winkler, Wien, 1872.
" Traite de la defense des places.
3 Traite d'architecture practique.
1
Fig. 1.— Method of Bullet.
IS
2 RETAINING WALLS
wedge ABC is resolved into components parallel and normal
respectively to the plane of slip. The former component was
the only one considered, and by taking moments about A,
proper wall dimensions are found to resist this thrust. Couplet
in 1727 makes the plane of cleavage pass through the outer edge
of the wall (see Fig. 2) at D. , The prism ACFE is resisted by
AED, the remaining portion of the wall EBID supportmg the
wedge EFB. As before, the weight of this latter wedge EFB is
resolved into parallel and normal components and the former
applied directly to the portion of the wall concerned. To get
the angle that the plane of
^ P B I cleavage makes with the
vertical, he followed the
method of Mayniel,* tak
ing this angle equal to that
of the slope of a uniformly
built pile of shot, the
tangent of which angle is
Vs.
Sallonmeteb and
RONDELET (1767) follow
the method of Couplet,
save that the plane of cleavage starts from the back of the
wall. Belidor,^ an architect formulated a method in which
the action of friction is considered. Proceeding as in the above
methods, he arbitrarily assumes that onehalf of the wedge weight
is consumed in overcoming friction, the balance, properly re
solved into parallel and normal components, acting upon the wall.
Coulomb in 1774, presented the first rational theory making
proper allowance for friction and then determining the wedge
of maximum thrust. Following him, Navier and finally
PoNCELET developed the theory into its present form, the ele
gant graphical method of determining the amount of thrust be
ing due to the latter.
It was to be expected that the brilliant school of the English
and French mathematical physicists of the middle of the last
century would attempt to analyze the action of earth pressure.
Levy, Boussinesq and Resal of France and Rankine of England,
' Traite de la pousee des terres. Memoire publiee dans I'histoire de
Taoademie des sciences, 1728.
'La Science des Ingenieurs L. I., 1729.
Fig. 2. — Method of Couplet.
THEORY OF EARTH PRESSURE 3
applied the methods of the theory of elasticity of solids to granu
lar masses with varying degrees of success. Rankine's results
are best known. Utihzing the socalled ellipse of stress (the
stress quadric of elastic theory) he developed his theory of con
jugate pressures. His results are probably the most universally
applied of all the varied methods.
Later analysts of earth pressure have attempted to include
in the theory the elements of friction between the earth and the
back of the wall and that of cohesion in the mass. Such attempts
leave intricate expressions of decidedly questionable practical
value.
The want of agreement between theory and experiment has
led to many attempts to estabUsh empiric relations between the
width of the wall base and the height without determining the
earth thrust. Sir Benjamin Baker, the illustrious English
engineer, under whose supervision the London tubes and outlying
extensions were built, advocated a value of this ratio of about
0.4, one which Trautwine warmly seconds in his handbook.
Such empiric constants were of value when walls were of the
rectangular section, or verging upon the revetment type. With
the modern development of the concrete walls, both gravity and
reinforced sections, the use of such empiric relations is decidedly
questionable and good engineering practice requires that a
rational method of ascertaining the wall pressures be used in
determining the proper dimensions of a retaining wall.
Exact Analysis of the Action of Earth Masses. — The correct
interpretation of the character, distribution and amount of pres
sures throughout an earth mass typical of ordinary engineer
ing construction, cannot be expressed by exact mathematical
analysis. The usual earth mass retained by a wall contains so
many uncertain elements (see page 4) that can neither be
anticipated nor determined by typical tests, that it becomes very
hard to assemble sufficient data for a premise upon which to
found any satisfactory conclusion. To analyze an earth mass
an ideal material must first be assumed. The divergence in
properties between that of the actual material and the ideal
material determines, in a more or less exact degree, the
approximation of the results found theoretically.
Under such uncertain circumstances and with a consequent
skepticism of mathematical results, the natural query is — why
attempt a refined mathematical analysis? There are several
4 RETAINING WALLS
praiseworthy reasons. The general action of earth pressures
may be indicated and reasonable theories may be advanced as
to the probable character of pressures to be anticipated. A good
framework may be built upon which to hang modifications
experimentally determined. The several mathematical modes of
treatment may indicate a common and possibly a simple expres
sion for the pressures, of easy and safe apphcatipn to most of
the conditions occurring in actual practice. Finally, the analysis
of the ideal earth mass may show the maximum pressures that
can exist in the usual fills, which pressures the actual ones may
approach as the character of the fill approaches that of the ideal
one assumed. Thus the probable maximum value of earth
pressures may be established; an important function and an
indication of the probable factor of safety so far as the amount
of the earth thrust is concerned.
The Ideal Earth and the Fill of Actual Practice. — The mathe
matical discussions of the action of earth masses premise a granu
lar, homogeneous mass, devoid of any cohesion (see page 20) a,nd
possessing f rictional resistance between its particles . In addition,
■the surface along which sliding is impending is assumed to be a
plane. Such a fill is rarely found in practice. Fills, ordinarily,
are made either from balanced cuts for street or railroad grading,
or depend upon local excavations. In the usual city work,
materials for fill may be expected from other local improvements,
public or private, which may be prosecuted simultaneously, or
which may be induced to be prosecuted because of the expected
place of disposal for spoil, In out of town improvements special
steps, such as the employment of borrow pits, may beconie
necessary to provide the needed material. It becomes evident
that the character of the fill may vary greatly, containing any one
or several types of earth, and including, usually, a large propor
tion of excavated rock.
The construction of the embankment itself may be carried out
in widely different manners. It may be built up from a tem
porary railroad trestle, the materials dumped from cars and
against the wall, if it be already built. Ordinary teams, or motor
trucks may dump materials upon the ground, riding over the fill,
or may dump over the slope of the fill already formed. Little
homogeneity can be expected from either of these methods.
Attempts to puddle a fill to give it eventual compactness and in
creased density make it difficult to team over the puddled portion
THEORY OP EARTH PRESSURE 5
and are usually abandoned on this account. While specifications
often require the construction of an embankment in thin well
rammed layers, this requirement is observed more often in
the breach than in the observance. It is a costly timecon
suming expedient and unless required by special types of design
(see page 21) may safely be ignored.
Rarely then, in either the type of the earth, or in the mode
of utihzing it to make a fill, can the engineer make any definite
assumptions as predicated for the ideal earth, nor would he be
justified, from the standpoint of economy, in limiting the selection
of materials for fill to such as approach the character of the ideal
material, especially in view of the uncertainty of local geologic
conditions. Obviously, refinements in the theory of earth pres
sures and attempts to predict with any degree of exactness the
angles of repose become matters of more or less academic interest
only.
Bearing in mind these limitations placed upon the ideal
material assumed in the following analysis and that the mathe
matical work is developed solely as a means toward an end, as was
pointed out in a previous page, a proper appreciation will be had
of the relative value of the discussion in the next sections.
The Two Theories. — The theoretical treatment of the action of
earth pressures follows along two fairly distinct lines. The
Rankine method is an analytic one, starting with an infinitesimal
prism of earth and leading to expressions for the thrust of the
entire earth mass upon a given surface. The Couloaib method,
or the method of the maximum wedge of sliding is essentially a
graphical one, as finally shaped by Poncelet and treats the
mass of earth in its entirety, finding by the principle of the
sliding wedge, the maximum thrust upon a given surface. It will
be noticed that the final algebraic expressions for the thrust, as
determined by either method, are similar in form, and, when
certain reasonable modifications (introduced by Prof. William
Cain) are placed upon the Coulomb method^ are approximately
alike in value also.
The Rankine Theory. — The angle of internal friction (approxi
mately equal to the angle of repose) of an ideal earth as defined
above, is the angle 4>, (see Fig. 3) which the resultant force R
makes with the normal to the plane when sliding along this plane
is just about to start.
In a mass of earth unlimited in extent, select a minute triangular
6 RETAINING WALLS
prism, whose section parallel to the page, is a right angle tri
angle, as shown in Fig. 3. In addition, let the prism be so
selected that only normal stresses exist upon its arms. These
stresses are then termed principal stresses, and the planes to
which these stresses are normal, are termed principal planes. The
existence and location of such planes are found by simple methods
given in the text bboks on applied mechanics. For earth masses,
whose upper bounding surfaces are planes, Rankine has shown
that the principal planes are parallel and normal, respectively to
the upper boundary plane.
Fig. 3.
TP and q are, respectively, the normal stress intensities upon the
principal planes shown in Fig. 3.
Since there is a limiting value of the angle <i>, which limiting
value is the angle of repose, or better termed, the angle of internal
friction, and since the angle i of the triangular prism may vary,
it is possible to determine a maximum value of for some value
of the angle i. The ratio between the principal stress intensi
ties p and q may be shown to be independent of the angle i^
and can be denoted by some constant. With the value of the
angle thus defined, it is possible to express it in terms of the
ratio f/q, since the angle i may be eliminated after its value
rendering <j> a maximum is found. Knowing the maximum value
of 0, from the physical properties of the earth in question, it is
thus possible to express the stress intensity ratio in terms of the
'See Howe's "Retaming Walls," 6th Ed.
THEORY OF EARTH PRESSURE 7
angle <j). This work may be carried out by utilizing the statics
of the force system as given in Figure 3.
From the statics of Fig. 3
tan {i  <l>) = Q/P
and, since Q = qb, and P = pa,
Q/P = q tan i/p; since b/a = tan i.
Place the ratio of the intensities q/p = n.
The above equation then becomes,
tan a — (j}) = n tan i (1)
Denote tan i and tan <^ by a; and y respectively, and expand tan
{i — (t>) by the formula
,. ,, tan I — tan <i
tan {i — <^) = ^ , , TT 7
1 + tan I tan <p
Equation (1) becomes
_ xjl  n) ,
y~ 1 + nx' ^'
By the principle of the theory of maxima and minima, this ex
pression is found to have a maximum value when x = l/\/n.
The expression for y, or rather, tan 0, for this value of x is
tan (j) = 1= (3)
2\/n
To reduce this to the form as finally given byRankine,note that
tan (^
sin (^ = ,, „ ,
V(l + tanV)
which trigonometric relation reduces (3) to
and similarly
^ q ^ 1  sin <^ ^,
'^ p l + sin cj} ^ ^
This gives the fixed relation between the principal intensities
of stress when the maximum angle of friction is given, and the
upper surface is a horizontal plane. The value of the principal
intensity p upon the horizontal plane, is easily seen to be the
weight of the earth mass above this plane. If the depth to this
8
RETAINING WALLS
plane is h and the unit weight of the material is w then p = wh
and (4) becomes
1 — sin <^
q ='wh
1 + sin 4>
(5)
which is the classic relation between the vertical and horizontal
pressures as first given by Rankine.
This is the fundamental equation of the Rankine method and
the following theorems are deduced directly from it:^
(a) The direction of the resultant earth pressure against a
vertical plane is parallel to the free upper bounding surface and
is independent of the interposed wall.
(&) For an earth mass whose upper bounding plane makes an
angle a with the horizontal (see Fig. 4), the intensity of pressure
parallel to CA is
cos a — vcos^ a — cos^ <t> ,„,
(6j
I = wh cos a
cos
a + \/cos^ a — cos^
This expression may be simplified by placing cos <j> / cos a =
sin u whence
t = wh cos a tan^ (m/2) (7)
Note that, in this expression, < is a linear function of the depth
of earth h, so that the value of the entire thrust upon a plane
.45 of depth /lis
T = thy 2. (8)
and the point of application of this thrust is at one third the dis
tance h above B.
(c) Thefinal resultant thrust upon the back of the wall BC is
compounded of the above
thrust and the vertical weight
GoftheprismA£C(seeFig.4).
It is to be noted that no
allowance is made for any
frictional resistance that may
exist between the back of the
wall and the earth mass im
mediately adjacent to it.
The upper surface must be
free, i.e., the mathematical
, ,. , treatment excludes external
loading upon the upper bounding surface. J. Boussinesq has
' Howe, "Retaining Walls," 5th Ed., p. n et seq.
Fig. 4.
The Rankine method of determin
ing the thrust.
THEORY OF EARTH PRESSURE
9
attempted to extend the theory of Rankine to include frictional
action between the earth and wall.^ The complexity of his
analysis and the arbitrary premises although of the utmost
elegance, preclude its acceptance by engineers. In fact, it is
quite doubtful whether the Rankine method can be extended
much beyond that set forth above.
The average earth fill has an angle of repose approximately
equal to 30°. As pointed out on page 4, no refinements in the
selection of this angle are justified by practical conditions. The
expression for the thrust upon a vertical plane with this value of <^
becomes with t = wh/3
T = w^ (9)
D
Taking the value of w as 100 pounds per cubic foot, this becomes
r = 16?i2 (10)
For a wall with sloping back (the usual form of wall), as shown
in Fig. 5 the thrust is found by combining the thrust upon the
vertical plane AB with the
weight of the earth over
the batter of the back.
The upper bounding sur
face shown in Fig. 5 is
that typical of the usual
composite fill and sur
charge equivalent loads
(see later pages in the
chapter for a full discus
sion on surcharges) . Most
retaining walls support an embankment of this type. For
upper surfaces of varying types, a detailed analysis is given on
pages 25 to 31.
The angle of friction is taken at 30°, with the consequent
simpKfication of the Rankine formula. The ratio of the height
h' to h is denoted by c, whence the total depth of fill acting upon
the plane AB, Fig. 5, is h{l + c). The thrust acting upon this
plane is then
P = wh' (1 + c)V6.
1 See an admirable resume of his work in this direction in a series of articles
by him in the Annales Scientifiqws de UEcole Normale SuperUre, 1917 and
reprinted in pamphlet form by GauthierVUlars, Pans, 1917.
t>j(l+c)hJ
Fig. 5. — Typical loading Rankine method.
10
RETAINING WALLS
The ratio c is small, generally less than onethird, whence it
is permissible to substitute 1 + 2c for (1 + c)^. The expression
for P takes the form
P = ^,.L+2£l (11)
Note here, that if a trapezoid be drawn as shown in Fig. 5 with
ordinates at the top and bottom of the wall the earth pressure
intensities at these points, the area of this trapezoid becomes
„. 1 + 2c 1
and the center of gravity hes at a point Bh above the base, where
B has the value
B =
11 +3c
(12)
3 1 + 2c
From (11), the area of this trapezoid may be taken equivalent
to the thrust upon the plane, and consequently, equivalent to
the horizontal component of the resultant thrust upon the back
of the wall AB. The thrust is located at the center of gravity of
this trapezoid as found aboye.
The weight of the earth mass superimposed upon the back of
the wall is
G = w (h'h tan b M ^ ) ^ w/i^tan b — ^ — (1^)
This is the vertical component of the resultant thrust upon the
back of the wall and the value of the thrust T is
T = V(P^ + G')
where J is equal to ^ \/(l + 9 tan^b)
Table 1
(14)
(15)
^6"
J
e°
6°
J
fl"
0.33
14
0.42
23
2
0.34
4
16
0.44
25
4
0.34
8
18
0.47
26
6
0.35
11
20
0.49
27
8
0.36
15
22
0.62
28
10
0.38
17
24
0.56
29
12
0.40
21
THEORY OF EARTH PRESSURE
11
To aid in the computation of the thrust when the height of
wall and the amount of surcharge is given, as well as the slope of
the back of the wall, Table 1 has been prepared covering a number
of values of J for the varying values of the angle b.
The angle which the thrust T makes with the normal to the
back of the wall is (see Fig. 5)
0= tani (G/P) b = tani (3 tan b)  b
(16)
from equations (.11) and (13) above.
For a basis of comparison with the formulas developed later,
a table of values of d for the several values of the angle b is given
in Table 1.
To summarize briefly the results above, it may be said that
equation (14) is the Rankine expression for the thrust of an earth
with an angle of repose of 30° whose upper surface is a horizontal
plane. The former remarks upon the usual nature of embank
ments as found in actual practice justify a blanket assumption of
30° for this angle of repose and the resulting simplification of the
thrust expression strengthens the reasons for the selection of that
particular value of the angle of repose. For a wall with sloping
back retaining a fill of shape shown in Fig. 5 equation (14) gives
the expression for the thrust. The computation of this thrust is
to be aided by the use of Table 1.
Coulomb Method of Maximum Wedge of Sliding. — The same
assumptions as to the properties of the ideal earth mass are made
as were made in the preceding theory. Referring to Fig. 6 any
Fig. 6. — Method of maximum wedge of sliding.
prism of earth AFC, where AG makes an angle a with the hori
zontal, which is greater than the angle of repose <t>, will tend to
slough away from the remaining ^earth bank and will therefore
require a retaining wall with back AF to hold it. In this prism of
12 RETAINING WALLS
earth the forces acting upon it are its weight G, the reaction of
the thrust T upon the wall and the reaction of its pressure Q
upon the remaining bank. As different wedges of possible sliding
are selected, some one wedge will produce the maximum thrust
upon the wall AF, which is the actual thrust sought.
From the equilibrium of the figure, the forces T, G and Q,
are concurrent, i.e., must meet in a common point. From the
law of concurrent forces
r/sin t = G/sin g = Q/sin q. t, g and q are the angles as
shown in the figure.
G is the weight of the irregular prism AFEC and is resolved by the
methods of equivalent figures (any elementary text in plane
geometry) into the triangular prism ABC. If a sUce of earth
of unit thickness is taken and its unit weight denoted by w, the
value of G is
= w 2 (17^
AT is normal to BC
From the sine relation above shown
T = '^ATXBC^^ (18)
^ sin g
To obtain the maximum value of this expression, it is neces
sary to separate its factors into those which remain constant as
various planes of sliding are selected, and those which vary with
the different planes of sliding. This is effected as follows :
Draw, in Fi^. 6, what may be termed a base line AZ making
an angle <^ + 9 with the normal to the back of the wall. (The
explanation of the angle <i>' will be given later.) Parallel to
this line draw BO and CI. In ACI, from the law of sines
CI/AI = sin t/ sin g.
(Note in the figure that the angles g, t and q and their supple
ments are denoted by the same letters.
In similar triangles CID and BOD
CI/ID = BO/OB and BC/BD = 01 /OD.
Inserting these values in (18)
rp _w BD CI W/ATXBDXB0\IDX0I ,,^.
In this expression all factors are invariant for the figure except
the factor —  — and to obtain the maximum value of the
THEORY OF EARTH PRESSURE 13
thrust T, it is sufficient to find the maximum value of this variable
factor. Upon placing AI = x, AD = a and AO = h, introducing
these values in this factor and then proceeding to find the maxi
mum value by the differential calculus, this maximum value is
found to occur when
a; = VJaF) (20)
In other words the maximum thrust exists upon the back of the
wall when AI is a mean proportional between AO and AD. Fig.
7 shows a simple method of finding a mean
proportional by geometric construction. S
The value of T as given in (19), with this y\
new value of the term — — may further y\ \ 
be simplified by noting that triangles DTA / \\ /
and CHD are similar, whence AT/CH = Ay^ _ ___m/
AD/CD; BO/OD = CI/ID; BD/OD = ^;^^^^^^^^^^^^
CD/ID. Substituting these values in that
J. ,1 .. , ,1 . ,, Fig. 7. — Geometric
expression for the thrust, there is the construction for mean
simple form proportional.
(CH X CI) (21)
If, with 7 as a center, an arc CC is described, the area of tri
angle CC'I, multiplied by the unit weight of the earth is equiva
lent to the maximum thrust T.
The direction of the thrust is assumed, in the original method,
to be normal to the back of the wall, but Prof. Cain has modified
this so that the direction of the thrust makes an angle 0' with the
normal to the back of the wall. The angle (f>' is the angle of
friction between the earth back of the wall and the wall masonry.
(See page 19 for a discussion of this frictional action between
earth and wall.)
The above method as outlined is essentially a graphical one and
in order to make a comparison between the results of this method
and the results of the Eankine method, it will be necessary to
obtain an algebraic expression for the thrust. To avoid needless
complications, the profile of the earth surface will be assumed to
have the shape shown in Fig. 8. Without entering into the
tedious but quite simple steps in reducing the geometric substi
14 RETAINING WALLS
tutions above to algebraic ones, the thrust is finally found to
have the form
(22)
where
cos(j)' + b)
cos" {(!>' + 4> + by
p = \/m sin <j>
vd
n  ^—,
sin<^
/ u + V tan b , j. j, i „+ „•
/ = n/m m = — —. — ; — a = tan & + cot z
■> ' sin<3!»
u =
sin (<^^ + <^ + h)
cos {<i>' + 6)
V = —
cos (<^' + «j> + b)
cos (<^' + 6)
c D
Fig. 8. — Typical loading CoulombCain Method.
Table 2
6°
K
6°
K
«' = 0°
«' = 15°
«' = 30°
4.' = 0°
.<.' = 15°
*' = 30°
0.33
0.30
0.29
16
0.45
0.42
0.43
3
0.36
0.32
0.32
18
0.4S
0.45
0.47
6
0.38
0.34
0.34
21
0.51
0.48
0.50
9
0.40
0.37
0.37
24
0.54
0.52
0.57
12
0.43
0.40
0.39
When the back of the wall is vertical, i.e., 5=0, and the uppe,
surface is horizontal and at the level of the top of the wall, i.e.
c = i = 0, the expression for the thrust reduces to
T = w ,^ 1  sm<l>
2 1 + sin (^
(23)
which agrees with the expression obtained on page 7 using the
Rankine method, and there is the important note that the Thrust
THEORY OF EARTH PRESSURE 15
upon a Wall with Vertical Back Due to a Fill Whose Upper Sur
face is Horizontal and Level 'with the top of the Wall is found to
Have the Same Expression in Both Rankine and Coulomb Methods.
In the equation for the thrust (22), the term c^/may be neglected
and as before the term (1 + c)^ may be replaced by 1 + 2c,
whence the expression takes the form
T = wh^K^^ (24)
K = L {1 — pY. K is finally reduced by substituting the above
values of m and p in it and, without introducing the trigonometric
steps, is given by
^ _ cos icl>' + b) r / sin (j> sin {<!>' + <^) ] ' .or.)
cos^ (4,' + ^ + b)[ yjcos b cos (0' + b)\ ^ '
To compare the values of this constant K with the constant
of parallel meaning J found on page 10, Table 2 has been pre
pared covering a range of values of b and <^'. As before the value
of the angle of repose has been taken as 30°.
Note that if in Fig. 8, the trapezoid ABCD be drawn with base
Kwh ( 1 + c) and ordinate at A Kgch, its area is
wh^K'±^
Ji
which is equivalent to the value of the thrust as found in equation
(24). A comparison of these two expressions for the thrust,
found by the Rankine and by the Coulomb method and a study
of the tabular values of J (Table 1) and K (Table 2) shows the
following points:
The form of the expression giving the thrust is the same by either
method.
For values of the angle b less than 5°, K with 4>' equal zero is
the same, approximately, as J.
For values of the angle b greater than 5°, K with <i>' equal to 30°
is the same, approximately, as J.
For the values of <t>' as noted in the preceding the directions of
the thrusts are approximately alike using either theory.
From the above comparative study (also see examples at the end
of this chapter giving numerical comparisons of thrust computa
tion by either method) it is seen that, with the limitations as shown
above (see pages 1 9 and 20 for a discussion of the proper values of the
16
RETAINING WALLS
angle of friction to be assumed between the back of the wall and
the earth) eitherof equations (14) or (24) maybe used to obtain the
value of the thrust. As a matter of fact the expression as deduced
from the Rankine equation (14) will be used to obtain the thrust,
and the Coulomb form of the thrust given in (24) will only be
used where its form lends itself more readily to the analysis
of the special problem at hand.
To recapitulate: The thrust upon any wall with sloping
back, and earth profile as shown in Fig. 5, is to be found from
T = Jivh^
1 + 2c
where / is the earth pressure constant to be taken from the
values of J found in Table 1, c is the surcharge ratio, and w is
the unit weight of the earth. The point of application of the
thrust is located at a distance Bh above the base of the wall,
where the values of the ratio B, is to be found from Table 3.
Table 3
c
B
c
B
c
B
0.0
0.33
0.5
0.42
1.0
0.44
0.1
0.36
0.6
0.42
1.5
0.46
0.2
0.38
0.7
0.43
2.0
0.47
0.3
0.40
0.8
0.44
Infinite
0.50
0.4
0.41
0.9
0.44
Admittedly, neither theory meets rigorously the application of
actual conditions, nor are they confirmed, experimentally (see
page 18 for some experimental data on earth pressures) to
any great degree of exactness. It follows, then, since refinements
are not only unnecessary but superfluous in earth pressure
theories, that such assumptions and approximations as have been
noted and applied above, should suffice for all retaining wall
design.
It is essential that simplicity of thrust calculation be kept in
mind, as it is by far more important that a standard method
of such thrust determination be had, than that the refinements of
such analysis be noted. The emphasis upon retaining wall
design must be placed upon the actual design of the wall itself
and not merely upon the derivation of the thrust.
THEORY OF EARTH PRESSURE 17
As a matter of interest, several of the other methods of thrust
determination are given in the following section.
Various Methods of Thrust Calculation.— Most of the empir
ical expressions for the thrust have the form
T = ch^ (26)
with various assumptions as to the value of c. On page 9
above, the value of c, from Rankine and from Coulomb, when
the angle of repose <^ is taken as 30°, was found to be 16.
In an interesting series of discussions of earth pressures ^
this value of c, namely 16, met with considerable approval.
The analogy between lateral and hydrostatic pressures has
been utiUzed in some formulas by assuming the earth to be a
fluid with unit weight varying from 25 to 62 pounds per cubic
foot, the latter amount supposedly used to insure a satisfactory
factor of safety. These assumed weights would give to c in the
above empiric equations a value varying from 12.5 to 31.
C. K. Mohler, in the Journal of the Western Society of
Engineers, Vol. 15, gives a modified form of hydrostatic pressure
in the compromise formula
T = wh^{l  sin <t>)/2 (27)
where w is the unit weight of the material and <^ is the socalled
"angle of flow." He states that the lateral earth pressures due
to earth surcharges is probably insignificant and illustrates this
by an ingenious arrangement of cylinders. Considerable skep
ticism, however, is shown in regard to this latter statement in the
discussions on his paper, and doubtlessly, the author of the
paper has not credited a correct effect to such surcharges.
In Vol. 19 of the same Journal, a modified form of the Rankine
formula is given and is urged as a true expression for both lateral
and vertical pressures.
To summarize the various comments upon the methods of
deriving an expression for the earth thrust, it may be stated that
although objections are raised to practically every suggested mode
of treating such pressures, it is generally conceded that retaining
wall failures are not due to weaknesses in the theory of pressures,
but are primarily due to faulty design and construction. This is
a vital conclusion and is a further justification for the adoption
of the simple, and mathematically sound, expressions given in the
1 Western Society of Engineers, Vol. 16, 1911.
2
18 RETAINING WALLS
preceding pages. Examples at the end of the chapter will
illustrate the application of the various formulas and will show
the simplicity of application as well as the approximate cor
rectness of these concise expressions.
It may be stated that rule of thumb methods, both for the
computation of the earth thrust and for the relations between the
wall dimensions are undesirable, are of questionable profes
sional practice and, in the case of reinforced concrete walls, are
not only inapplicable, but even dangerous.
Experimental Data. — The various attempts to determine earth
pressure values experimentally, have been quite disappointing,
so far as definite results are concerned; but they have led to
several important conclusions. The results of two such series
of experiments are given here, and are of value, not only for
the conclusions reached in the papers themselves, but also
because of the summary of previous experiments given therein.
In a paper by E. P. Goodrich, "Lateral Earth Pressures and
Related Phenomena," Trans. A.S.C.E., Vol. liii, p. 272, the
following may be quoted as of some bearing:
Sir Benjamin Baker has pointed out that the coarser the
materials the less the lateral pressure.
A. A. Steel. 1 For dry and moist earth the lateral pressure
is from H to }i the vertical and, in saturated materials is
practically equal to it.
Some of Mr. Goodrich's important conclusions are as follows :
(a) The point of application of the resultant thrust is above the
}i point, usually about 0.4 of the height of the wall.
(6) Rankine's theory of conjugate pressures is correct when
the proper angle of friction is found (the italics are mine), and
probable adaptations of his formulas will be of most practical
value.
(c) Angles of internal friction and not of surface slope must be
used in all formulas which involve the shding of earth over earth.
(Such tables are to be found in the author's paper.)
It must be emphasized that the experiments mentioned above
were performed upon a more or less homogeneous material. The
actual composition of fills has been described on page 4.
In a papers by WilUam Cain, the conclusions, after analyzing
'Engineering News, Oct. 19, 1899.
2 ''Experiments of Retaining Wails and Pressures on Tunnels," Trans
A. S. C. E. Vol. Ixxii, p. 403.
THEORY OF EARTH PRESSURE 19
some experiments performed by the author and analyzing also the
extensive experiments carried on in the past, are :
"1. When wall friction and cohesion are included, the sliding wedge
theory is a reliable one, when the filling is a loosely aggregated granular
material, for any height of wall.
"2. For experimental walls, from 6 to 10 feet high, and greater,
backed by sand or any granular material possessing 1 ttle cohesion,
the influence of cohesion can be neglected in the analysis. Hence
further experiments should be made only on walls 6 feet and preferably
10 feet high.
"3. The many experiments that have been made on retaining walls
less than one foot high have been analyzed by their authors on the
assumption that cohesion could be neglected. This hypothesis is so
far from the truth that the deductions are very misleading.
"4. As it is difficult to ascertain accurately the coefficient of cohesion,
and as it varies with the amount of moisture in the material, small
models should be discarded altogether, in the future experiments
and attention should be confined to large ones. Such walls should be
made as light, and with as wide a base as possible. A triangular frame
of wood on an unyielding foundation seems to meet the conditions for
precise measurements.
"5. The sliding wedge theory, omitting cohesion, but including wall
friction, is a good practical one for the design of retaining walls backed
by fresh earth, when a proper factor of safety is used."
Clearly, experimental data verifies neither of the above theories
with any degree of exactness, yet does indicate that either of the
two theories may form a rational basis for a working formula.
Equation (14) may again be brought forward as the practical
formula to be used in obtaining the thrust upon a wall, due to
the usual type of embankment loading.
The above work has frequently discussed the items of wall
friction and cohesion and these two factors will be taken up in
the following sections.
Wall Friction. — The question, whether frictional resistance
between the back of a retaining wall and the adjacent earth is,
or is not, a permissible factor to be included in the computation
of the thrust and in the determination of its direction, plays an
important role in various theories of earth pressure. Since the
earth backing exerts a pressure upon the wall, then by the ele
mentary theories of physics, there must be friction between the
two surfaces in contact. The angle of friction cannot be assumed
larger than the angle of friction of the earth material, since if it is
20 RETAINING WALLS
larger, and this is quite possible, the effect is that a layer of earth
will adhere to the waU and sUpping will take place between this
layer and the remainder of the earth bank. If allowance is made
ior such frictional resistance, it is customary to take the angle
of such friction (<^') the same as the angle of repose. This
angle has been taken as 30°, and <^' may therefore be given the
same value.
The question of lubrication between the earth and wall due
to the presence of water, must be taken into account and gener
ally the more vertical the wall is, the greater will be the effect
of this lubrication upon the angle of wall friction. The use of
equation (14) founded upon the Rankine method, automatically
provides for this condition, as was pointed out in the comparison
between the Rankine and Coulomb method on page 15.
It will be seen later, in analyzing the various types of walls,
that in finding the proper dimensions of a gravity wall to safely
withstand a given thrust, quite an economy in the necessary
section of the wall is effected by a favorable consideration of wall
friction. It is to good advantage, then, that the back of the wall
be stepped or roughened so as to fully develop such wall friction.
It seems better engineering practice to make allowance for
such a force than to ignore it and assume that a factor of safety
of unknown value is thereby added to the wall. Such uncertain
conditions as exist in wall design may more properly be allowed for
in a final factor of safety of some assumed value, than to merely
add blind factors by ignoring forces which must surely exist.
The question of wall friction plays an unimportant role in
the design of reinforced walls (whose backs are usually nearly
vertical) and as its neglect simplifies the calculation of the wall, it
is permissible to ignore it — not on the basis that it does not exist,
but because it has no effect upon the attendant analysis.
Cohesion. — Cohesion, as it exists in an earth mass, is rather a
loosely appUed term, which had better be called cohesional
friction. Prof. WilHam Cain, has defined its action r^
"The term 'cohesive resistance' of earth may properly apply either
to its tensile resistance or to its resistance to sliding along a plane in
the earth, dependent on the viewpoint. However, as the tensUe resist
ance of the earth is rarely called for, the term 'cohesive resistance of
earth' from Coulomb's time to the present, has been generally restricted
to mean the resistance to sliding as affected by cohesion * * *."
'■ Proc. A. S. C. E. Vol. xUi, August, 1916, p. 969.
THEORY OF EARTH PRESSURE 21
To properly appreciate the effect of this cohesional friction,
it must be borne in mind that it exists to some extent, varying
from a slight amount to a very large amount, in all earth masses.
It is the one element that probably accounts for the large diver
gence between theoretically determined and experimentally
determined thrusts. It is least for dry granular masses, and
reaches a maximum value in the plastic clays.
In the ordinary fills as found in engineering practice (and
over 90 per cent, of walls retain embankments of fresh fill) its
presence is a highly uncertain one and in view of the mixed char
acter of such a fill containing boulders, cinders and other miscella
neous material, its existence as a definite resisting force to sliding
must be ignored. General practice while admitting that cohe
sion does exist in earth masses, has taken the very wise step, to
ignore its action. While this may increase the amount of thrust
upon a wall, it is very possible that, due to vibrations, or other
disturbances, the cohesive action in the earth is destroyed,
temporarily at least making the actual thrust approach very
closely, in value, the theoretical thrust. I he conclusions of
Prof. Cain, quoted on page 19 may again be noted, where the
method of the sHding wedge, ignoring cohesion, is recommended
as one properly determining the thrust.
Under certain conditions, where a direct effort is made to obtain
and preserve a cohesive effect in the earth mass, it is within rea
sonable practice to take advantage of the force. When a wall
retains an old embankment, where only a thin wedge of new fill is
placed between the old fill and the back of the wall, there is good
justification for assuming that cohesion will be a permanent force.
Again, by carefully placing and ramming in thin layers a specially
selected fill, cohesion is practically assured and the design of the
wall, may safely include this factor. The retaining walls of
the approach to the Hell Gate Arch^ over the East River, New
York contain a fill placed with extreme care and the determina
tion of the thrust included the factor of cohesion, permitting the
construction of a fairly thin wall, where, under ordinary granular
theory, a wall of prohibitive section would have been required.
The effect of cohesion may be interpreted in two manners.
It has been noticed that the bank of a freshly cut trench will keep
its vertical slope for quite a period, and then as it sloughs away
will gradually approach a parabolic shape, with the upper portion
' Engineering News, Vol. 73, p. 886.
22 RETAINING WALLS Jl^
more or less vertical. It will be remembered that the granular
theories above discussed have assumed that the surface of rupture
is a plane. To allow for the cohesive action as described, a much
steeper angle of slope for the material may be assumed than its
ordinary angle of repose would warrant, in that way approaching
the parabohc curve or it may be assumed that for a certain dis
tance below the surface of the ground there is no lateral pressure,
the surface of rupture being a vertical plane, and below this
critical point the material observes the ordinary laws of the granu
lar materials.
The first method is an empiric one and seems a rather perilous
one to adopt, in view of the uncertainty of cohesive action. The
above mentioned retaining walls of the Hell Gate Arch Approach
were designed on this basis, the fill taking a very steep slope. '
Table 4
Material
c in lbs.
sq. ft.
per
1
5
8
3
6
4
23
1
18
5
39
5
Dry sand
Wet sand
Very wet sand
Clayey earth '
Damp fresh earth
Clay of httle consistency.
A theoretical discussion of cohesion^ indicates that the latter
method is founded on more logical a basis. The effect of cohe
sion is to lower the "head" of earth pressure so that a soil pos
sessing cohesion exerts no lateral pressure until a certain vertical
pressure has been reached, corresponding to a depth x in the earth.
The value of x is given by the expression
a; = ^San(45° 4 I) (28)
c is the coefficient of cohesion for the material and may be taken
from Table 4. w is the unit weight of the material and <t> is
the usual angle of repose of the material. Below this depth x, the
earth pressures follow the ordinary laws of noncoherent earths
(see Fig. 9). An appUcation of the above formula to ordinary
^ See previously quoted article in Engineering News.
2 Cain, "Earth Pressure, Walls and Bins," p. 182 et seq.
THEORY OF EARTH PRESSURE
23
earth with some cohesion shows that this lowering of the head is
but a slight one and for all practical purposes may be ignored.
For a densely compacted material, approaching a plastic clay this
lowering of the head reaches a value that has a marked effect
upon reducing the amount of the thrust.
In an interesting paper on the lateral and vertical pressure of
clay^ a set of formulas for the stress system in a coherent earth
mass was given, after a careful experimental study of the neces
sary coefficients. While of hmited apphcation (they are prima
rily for the clayey materials) they are worthy of quotation and
may prove of service in interpreting the action of materials
Fig. 9. — Coherent earth.
of that nature. Before presenting these equations it may be
well to note the character of some of the stresses. In a material
more or less plastic there is a tendency for the surface adjacent
to an applied loading to heave and raise. This may be shown by
a mathematical discussion of the stress distribution in a material
of that character^ and is clearly demonstrated by experiment.
Under a retaining wall the pressure is generally nonuniformly
distributed, having a maximum value at the toe and a minimum
value at the heel. From the foregoing note it is clear that
when the wall bears on a plastic coherent soil, there must be a
certain minimum downward pressure at the heel to compensate
for the upward heaving pressure caused by the soil loading. This
is given below. The loading which a soil can stand without
excessive yielding is usually termed its passive stress, as distin
guished from the stress which it exerts (the lateral stress) and
which is termed its active stress. The passive stress is frequently
called the ultimate bearing value of the soil.
1 Bell, "Minutes of the Proceedings of the Institute of Civil Engineers,"
Vol. cxcix, p. 233.
2 See Howe, 5th Ed., "Retaining Walls."
24 RETAINING WALLS
Table 5
Character of clay
k tons, sq. ft.
at
Verv soft Duddle clav
0.2
0.3
0.5
0.7
1.6
0°
Soft puddle clay
3°
Moderately firin clav
5°
Stiff clay
7°
Very stiff boulder clay
16°
The retaining wall is subjected to a lateral pressure from the
coherent material of intensity pi, which is given by the equation
Pi = wh tan^ (^  I)  2A; tan i^  ^j
(See Fig. 9.) a and k are the constants of the coherent material,
and may be taken from Table 5. From the above expression
it is to be noted that within a given distance x below the surface,
there is no intensity of pressure. This value of x,
cot (^  2) (29)
w
may be compared to the similar value of x given in equation (28)
on page 22.
If Pi is the minimum permissible intensity of downward pres
sure on the foundation at the heel of the wall, where the depth
isH
Pi = wH tan* (ir/4  a/2)  2k tan^ (7r/4  a/2)  2k tan
(7r/4  a/2) (30)
The retaining wall rests in a trench and its footing butts
against the forward part of the trench when the earth pressure
acts upon the wall. The maximum intensity of horizontal
resistance in front of a wall at any depth d (note that tl^is is a
passive stress) is
ri = wd tan^ (7r/4 + a/2) + 2k tan (7r/4 + a/2) (31)
The maximum permissible intensity of downward pressure on
the foundation at the toe of the wall, where the depth is D (note
that this is a passive stress, usually termed the safe bearing value
of the soil) is
ra = wD tan* (7r/4 + a/2) + 2k tan^ (x/4 + a/2) + 2k tan
(7r/4 + a/2) ■ (32)
THEORY OF EARTH PRESSURE 25
While the above series of equations are intended primarily
for the clays, they are applicable to all materials upon proper
adjustment of the values of the coefficients. Thus for non
coherent or ordinary granular masses, the cohesion coefficient k
is zero and the angle replaces the angle a.
In a discussion upon the results given by Bell, Prof. Cain
has noted, that if A is the value of a unit area, then the relation
between the k given here and the c of his material is fc = cA .
In the analysis of the walls in the following chapters and
in the application of the results of the text to specific problems
the action of cohesion will be entirely ignored, the formulas given
in equations (14) and (24) being used to obtain the thrust upon
the wall.
In determining the strength of an existing wall retaining a
wellsettled and aged embankment, there is Httle doubt of the
existence of cohesion, and with the aid of the preceding equations
a proper determination of the load carrying capacity of the wall
may be obtained. Whether to increase the load upon the wall,
by addition of a surcharge, because of thelowered lateral pressure,
is a matter of judgment and in view of the uncertain character
of cohesion and the possibility of its absence for some unforeseen
reason, a careful engineer may sacrifice apparent economy to an
easier conscience.
Surcharge. — While a surcharge denotes an earth mass above
the level of the top of the wall, it is customary to reduce applied
loadings on the upper surface to equivalent surcharges. In the
theory of the distribution of stress through elastic solids, it
has been proven' that such distributions are independent of the
manner of the local loading except for points fairly close to such
loads and it is permissible to substitute the resultant load for
this distribution, or conversely a distributed loading for a series
of concentrated loads.
It seems quite justifiable to extend this law to granular masses
and, in fact, it is generally accepted that applied loadings may
be reduced to a distributed earth surcharge equivalent. The
reduction of dynamic loadings is, possibly more involved than
that of the reduction of still loadings.' Nevertheless, it would
seem that in view of the comparative inelastic properties of a
granular mass and of the large amounts of voids in the material,
1 See for example, J. Boussinesq, "On the Applications of the Potential,"
etc.
26 RETAINING WALLS
the vibrations are completely "dampened" before they reach the
wall. If this is conceded, no distinction need be made between
static and dynamic loads. In any event, impact coefficients
of as great value as are applied to elastic solids should not be
applied to the earth mass.
While there may be some question as to whether a surcharge
loading produces a lateral pressure of intensity proportionate to
the fill proper, below the level of the top of the wall a theoretical
analysis gives no foundation for such doubt, and there is as
tangible a basis for assuming the full proportionate effect of
the surcharge upon the wall as there is for the other theoretical
assumptions of earth pressures.
When the surcharge is uniformly distributed over the top
of the embankment and extends to the back of the wall, equations
(14) and (24) give the amount and Table 3 gives the location of
the resultant thrust. When the surcharge is not of uniform
distribution, or does not extend to the back of the wall, the con
ditions require special analysis. The following treatment of
such surcharges is given primarily for the same reasons as in
the treatment of earth pressures in general and is to be used in the
same sense.
When an external loading upon an embankment has been
reduced to a uniformly distributed loading equivalent to the
same weight of earth, a new profile has been given to the top of
the embankment. It must be noted here, however, that when a
wedge of earth is about to sUde along some plane in the fill proper,
this plane cannot extend at the same slope throughout the sur
charge, but must be directed vertically upwards after reaching the
surface of the ground upon which the surcharge rests (see Fig.
10). The method of the maximum wedge of sHding is most
easily apphed to the discussion of this case and a simple graphical
analysis follows."
Let the equivalent surcharge extend to v. Draw a line parallel
to the upper surface and at a distance 2h' above it. Draw bn
parallel to ov. Connect o and n. The intersection s of this line
with the ground surface is the usual base point to construct
the equivalent thrust triangle. Thus through s, let sa be parallel
to the base line oz. Locate d as the mean proportional between
oA and oD, and locate c by drawing through d a line parallel
'Taken from MEHETENs"Vorlesungen ***** Baukonstmctionen" as
translated by G. M. Ptjbveb, Engineering <fc Contracting, Nov. 2, 1910.
THEORY OF EARTH PRESSURE
27
to the base line. Through c draw uk parallel to no. With d
as a center describe an arc cm. The thrust on the wall due to
earth and the surcharge is the area of the triangle udm multi
phed by the unit weight of the earth. It is shown* that this
triangle is equivalent to the area of cdm multiplied by the ratio
{h+2h')/h = l+2c where c is the usual surcharge ratio.
The triangle cdm is the measure of the thrust upon a wall, with
Fig. 10. — Surcharge not extending to back of wall.
no surcharge, whose back is the line so, making the angle a with
the vertical. The thrust may then be expressed algebraically
T = £^!0_+2£) K (33)
with K as given in (25) with the value of 6 = a. When the
surcharge extends to the back of the wall, then the b of the wall is
equal to a and the form for the thrust in this case is the same
as that given in (24), which is a measure of the approximation
of that formula.
To determine a denote the distance vb by r and let this be
equal to yh. Let the angle voN he 0. h tan ^ = r  h tan b
or tan P = y  tan b. bm = 2h' tan j3 = 2ch tan /3. Nv =
h tan /3. mN = r  bm  nv = h[y  (l+2c) tan p]. tan a =
ml^ _ ya±2c)(^tanb) ^ ^anb  r^^ y. (34)
h(l + 2c) ~ l+2c l+2c
It is to be noted that a may be negative. For K then see
Table 13.
The application of the wedge of maximum thrust to the case
1 lUd.
28
RETAINING WALLS
of isolated loads on the surface, is quite lengthy and involves
considerable geometric construction. It is discussed fully in
the lectures mentioned previously. For ordinary practice it
seems quite sufficient to replace it by its equivalent uniform
spread over the surface and then to apply the wedge theory to
a surface of broken contour, as shown in Fig. 10. <
An effective and simple manner of treating this case has been
devised by the Design Bureau, PubHc Service Commission, 1st
district N. Y. and is as follows:
In Fig. 11 there is a concentration oiL/a as shown, a surcharge
of h', and the earth back of the wall. For some plane of rupture
Fig. 11. — Surcharge concentrations.
BN all three exert a maximum thrust upon the wall. A few
trials are ample to determine this plane with sufficient accuracy.^
Let the plane of maximum thrust make an angle m with the
horizontal. The thrust Ti due to the concentrated load is —
a
tan (m — <^). The thrust T2 due to earth and surcharge is
g/j2(X 4 c)^ L
5 cot m tan (m — 0) and the total thrust is —tan
i a
(m — 0) + ^ (1 + c)2 tan (m — (f) cot m the maximum value
of this is found either graphically as noted above or by equat
ing the derivative of this last expression to zero, whence, upon
placing the ratio of L/a to ^ — ^ I'" '^' = r
r =
sin (m — (j)) cos (m — (j>)
sin^m
— cot m
(35)
'See Cain, "Earth Pressure, Walls and Bins,'' p. 43 for an excellent
graphical solution of this case.
THEORY OF EARTH PRESSURE
29
Assuming = 30° and simplifying the expression
sin (2m  120°)
2 sin^m
(36)
The relation between m and r is shown on Curve Plate 1. When
the value of m brings the wedge of thrust inside the distribu
tion of the loading L, it is reasonably certain, unless L is small,
that the maximum thrust upon the wall occurs when the plane of
/
/
80
/
/
y
/
/
z'
>
y
^
y
70
y
y
^
y^
y'
y
^
^
^■^
y^
^
■^
O.l O.E
0.3
0.4
Ratio = 5 TO '^— '2 •!
Curve Plate No. 1.
sUp just encloses the spread of the load L. Where the back of
wall is battered, the above method may be applied to the ver
tical plane through the heel of the wall, and this thrust may be
combined with the superimposed weight of the wall over the
back.
The apphcation of the earth and surcharge thrust, if, as before,
(1 \ cY'is replaced by 1 + 2c, (see page 15) is at the center of
gravity of the trapezoid of loading, or at a distance Bh above the
bottom of wall, with B as given in Table 3. The thrust due to
the isolated load may be assumed to be distributed uniformly
along the back of the wall, from the base of such load to
30 RETAINING WALLS
the bottom of wall. As shown in Fig. 10 its lever arm is
then C/2.
A simple method of reducing isolated concentrated loads to a
uniformly distributed surcharge, making the standard thrust
equations (14) and (24) apphcable is as follows. The concentrated
load is assumed to be transmitted along slope lines making an
angle of 30° with the vertical. (See the following pages of this
chapter for the experimental justification of this assumption.)
At the point I, where this distribution strikes the Une AB, see
Fig. 11, determine the intensity of vertical pressure. With this
as the new surcharge equivalent, employ the above equations
to determine the thrust. This method is, of course, quite ap
proximate, and should be used more as a method of confirming
the results obtained in the more exact construction above, than
as a primary method of getting the thrust. An example at the
end of this chapter will illustrate the two methods.
The preceding discussion of surcharge loadings has confined
itself to the lateral effect of such loadings upon a retaining wall.
It may be of interest to determine the vertical intensity of such
loadings at distances below the upper bounding surface. The
intensity diminishes as the distance from the upper surface in
creases and its spread may be said to be confined, roughly,
within the surface of a cone. Several expressions are given for
the intensity at any plane below the upper surface.
In Vol. 20, Journal of the Western Society of Engineers, Mr.
Lacher has given the following expression for the vertical live
load intensity at any depth h below the surface (due to locomotive
wheel loads)
11000
8 + 2hx
where x is the incUnation of the spread planes in fractions of a
foot per foot of depth.
The distribution of pressure through soil has been experimen
tally determined' and for depths of over 3 feet there is a spread of
fairly uniform intensity extending within slope planes making an
angle of 30° with the vertical.
An empiric expression given by Prof. Melvin L. Enger in the
Engineering Record Jan. 22, 1916, p. 107, for the intensity of
' Proc Am. Soc. Testing Materials, Vol. 17, part 2, 1917.
THEORY OF EARTH PRESSURE 31
vertical pressure at any depth as experimentally determined is as
follows :
A = pB
where A is the intensity of pressure at a depth h in inches, B
is the surface intensity of pressure and p is the percentage of
the surface intensity given by the following
p = 91 di«V^'"
The authors of the paper doubt whether the above expression has
general application. It would show, roughly, however, that such
transmitted pressure varies as the inverse square of the distance
below the loaded surface. A. E. H. Love has shown^ that the
transmitted pressure through an isotropic soHd, at a distance
h below the loaded surface and directly below the loaded point is
D 3T7 1
so that there is a striking agreement in the variation of trans
mitted pressure in soHd and granular masses. For an interesting
treatise on the distribution of pressure through soUds for any
character of surface loading, See "Application des Potentials"
by J. Boussinesq, pp. 276 et seq.
Pressure on Cofferdams. — A cofferdam retaining earth is in a
sense, a retaining wall subject to the ordinary theory of lateral
pressures. The cofferdam itself is an assembly of sheeting, wal
ing pieces, or rangers and braces, the design of which follows the
ordinary theory of the design of timber structures. Mr. F. R.
Sweeny^ has presented a thorough investigation of the loadings
upon such a structure together with a study of the economics of
its design.
His design has been predicated upon the assumption that the
ratio of the unit lateral pressure to the unit vertical pressure
is given by a constant c (corresponding to the earth pressure
coefficients K and J of the preceding pages) . The unit weight of
the material outside the sheeting is denoted by w. To quote the
author:
"The values of w and c are not easily determined being largely matters
of mature judgment. In any event, it is important to look into the
1 "A Treatise on the Mathematical Theory of Elasticity," 1st Ed., p. 270.
^ Engineering NewsRecord, April 10, 1919, pp. 708 et seq.
32 RETAINING WALLS
matter of possible saturation of the soil to the point where hydrostatic
pressure will be developed and superimposed upon the earth pressure."
The economic proportions and the best dimensioning of the
timbers and sheeting (wood and steel) are given in the article and
the entire design is exhaustively treated.
Pressures of Saturated Soils. — With the presence of water in
a soil, an additional lateral pressure is exerted from the plane of
the water surface to the bottom of the wall. An interesting
paper by A. G. Husted^ discusses in detail this important ques
tion. The following quotations from the paper cover the salient
features of the treatment.
"Formulas giving the lateral pressure of earth against vertical
walls may be found in many text books and hand books. These for
mulas, however, usually refer to dry earth and not to earth which is
saturated with water. The writer has had occasion when designing
structures, wholly or in part below water level to calculate the lateral
pressure of saturated earth, and being unable to find a satisfactory
method for computing these pressures has worked out the method
herein set forth."
The writer of the paper states that he will apply the Rankine
relation between the lateral and vertical intensities as given by
equation (14).
"As has been noted before, the formula assumes that the lateral pres
sure at any point bears a definite relation to the vertical pressure, this
relation depending entirely upon the angle of repose. It will thus be
seen that the second part of the equation can be divided into two parts,
wh representing the unit vertical pressure and (1 — sin 4>)/{ 1+ sin 0)
representing the relation between lateral and vertical pressures.
"Two methods of applying this formula to cases involving saturated
earths have been and are still in quite general use. One of these
methods consists in computing the total lateral pressure in the usual
way using for w the weight of dry earth and for <^ the angle of repose
of dry earth. To this pressure, then, is added full hydrostatic pressure
below the plane of saturation. This method may quite often give
results close enough to actual conditions for ordinary purposes of design,
but it appears to the writer to be at variance with the fundamental
formula. In the first place, no allowance is made for the fact that satu
rated earth has a smaller angle of repose than dry earth, and in the
1 Engineering NewsRecord, Vol. 81, p. 441 et seq.
THEORY OF EARTH PRESSURE
33
second place it is assumed that earth weighs the same in water as it
does out of water.
"Another method of calculating lateral earth pressures consists in
computing the total lateral pressure in the ordinary way and adding
to this, partial hydrostatic pressure below the plane of saturation. The
amount of the partial hydrostatic pressure is determined by taking the
difference between full hydrostatic pressure and lateral earth pressure
for an equivalent depth. This method, however, can easily be proved
erroneous by applying it to a fill of completely saturated earth. In
this case the partial hydrostatic pressure to be added will be the difference
between full hydrostatic pressure and lateral earth pressure for the total
depth of earth. It can thus be seen that the total lateral pressure at
the bottom would be exactly equal to full hydrostatic pressure. This is
absurd.
"In order to correct the errors in the above mentioned methods, a
method has been worked out which the writer believes to be theoretic
ally correct. In this method the following assumptions are made:
Lateral earth pressure varies directly with the vertical earth pressure for
earth with any given angle of repose and is equal to the vertical pressure
multiplied by (1 — sia <^)/(l + sin (j)).
Water exerts full hydrostatic pressure
laterally as well as vertically regard
less of the amount of the space oc
cupied by earth.
"It is a well known fact that the
angle of repose of earth in water is
less than the angle of repose of
dry earth. Therefore the ratio of
lateral pressure to vertical pressure
is greater below the plane of satu
ration than above. On page 580
of Merriman's "American Civil
Engineers' Pocket Book" the angle
of repose of dry earth is given as
36°53' while that of soil under water is given as 15°57'.
"Above the plane of saturation the lateral pressure is computed in
the usual manner. Below the plane of saturation the lateral pressure
is obtained by multiplying the total vertical pressure less the buoyant
effect of water by (1  sin 0)/(l + sin <^) and adding to this the full
hydrostatic pressure. For example, in Fig. 12 the unit lateral pressure
p„ at point a which is above the plane of saturation is Wih{l — sin <t>)/
(1 I sin 0). wi is the weight of the dry earth per cubic foot, h is the
distance of the point a below the surface and (^ is the angle of repose
of dry earth. Likewise the unit lateral pressure ps at point 6 below the
plane of saturation is (wi/ii + wji^) (1  sin 4>)/{\ + sin 4>) + 62.5^2.
3
j^™ Plane of
^ Safurafion
I
i'^i'.
FiQ. 12.
34 RETAINING WALLS
Ml as above is the weight of the dry earth per cubic foot, hi is the distance
from the ground surface to the plane of saturation, W2 is the weight per
cubic foot of earth under water, hi is the distance of the point & below
the plane of saturation and <t>2 is the angle of repose of earth under water.
"It will be noticed that in this method, for points below the plane
of saturation, hydrostatic pressure and earth pressure are separated;
that full hydrostatic pressure is allowed; that the vertical pressure is
obtained by adding the total weight of earth above the plane of satura
■ tion to the net weight (weight under water) of earth below the plane
of saturation; that the lateral earth pressure is obtained by multiplying
the vertical pressure by (1 — sin 4>^)/il + sin ^2) ; that the total lateral
pressure is obtained by adding the hydrostatic pressure to this lateral
earth pressure.
"It can be readily seen that if a smaller angle of repose is assumed
for saturated earth than for dry earth, there will be a decided increase
in the unit lateral pressure at the plane of satura
tion. In other words, the unit lateral pressure an
kXMisaevu'vi'iii infinitesimal distance below the plane of saturation
^ y'Mm''mm/ will be much greater than that at an infinitesimal
^ distance above the plane of saturation.
^ "At first thought this appears absurd, but it can
be seen that it should be so. It can perhaps be
best illustrated by an exaggerated example. Take
the case of a retaining wall supporting a bank of
earth loaded with timbers (Fig. 13), the lateral pressure of the timbers
against the wall is zero, but at an infinitesimal distance below the
surface of the earth the pressure is a considerable amount due to the
load that is superimposed.
"The difference is plainly due to a difference in the angle of repose."
While the preceding analysis is a correct mathematicalinterpre
tation of the action of saturated, homogeneous material, devoid
of cohesion, and may be used with the same degree of freedom as
any of the carefully worked out theories of earth pressure, it is
open to the same vital objections as were stated on the pages
preceding. However, as long as a proper appreciation is had of
the limitations of theory in general and if the lateral pressures
are computed as suggested on page 16 and as given by the
equations there shown the method presented by Mr. Husted is a
practical one and should be followed provided a safe lateral thrust
of saturated soils is sought.
Sea Walls.— A sea wall is essentially a retaining wall with a
fill of varied character behind it, composed, usually of riprap,
THEORY OF EARTH PRESSURE 35
earth, cinders and the like, and subject to a hydrostatic pressure
varying with the tide. An analysis of the pressure to which sea
walls are subjected is given in an article byD. C. Berber, Engineer
ing News, August 23, 1906, excerpts of which are quoted below.
Walls with vertical backs are the only type treated. The Rank
ine method, as applied in the previous pages, is used in this
treatment, the thrust intensity being given by equation (5).
It is assumed in the paper that the fill varies by strata, a hori
zontal plane separating the fills of different character. If the
fill back of the wall is assumed to be composed of two such mate
rials, of weights wi and W2, respectively and separated from each
other by a horizontal plane, /la above the bottom of the wall and
hi below the top of the wall Mr. Serber notes the following im
portant conclusion (theoretically deduced) :
" The total pressure on the lower section of the wall (i.e., below the
plane of separation) is entirely independent of the angle of natural
repose of the material above the plane of separation."
If the angle of repose of the upper material, of weight lOi is <t>i
and that of the lower material, of weight W2, is <j>2 and if, for the
sake of simplifying the resulting expression there is put
m = hi/h2; n = Wi/Wi andai =0(90° — 0i) 02 = s (90° — ^2)
the total pressure P2 on the back of the wall is
P = ^ ' [m^n tan^ cti + (2mn + 1) tan^ 02]
An ingenious graphical method of obtaining the total pressure
of two or more layers of different fill is presented in the paper
founded upon the reduction of the different weights in terms of
one of the weights.
The effect of surcharge upon a sea wall is discussed as follows :
"Merchandise, cranes and other loads of considerable weight are apt
to be stored temporarily or permanently on the sea wall and the backing
immediately behind it. The Department of Docks and Ferries of
New York City assumes a uniform vertical load of 1000 pounds per
square foot, * * *. When the bottom is very soft mud of consider
able depth and a pile foundation is to be resorted to, the normal dif
ficulties of sustaining a retaining wall are so great that it becomes
highly desirable to avoid the additional thrust due to the surcharge.
In such cases a platform may be built * * * supported on an in
dependent foundation sufficient to carry the surcharge, thus' relieving
the wall of the thrust * * *."
36 RETAINING WALLS
The inclusion of hydrostatic pressure upon this wall may be
dealt with in the manner outlined in the preceding section, the
formulas of Mr. Serber being readily adaptable to the principles
given in that section.
It must be emphasized that a sea wall is a structure of peculiar
importance in the design of which the paramount question is not
one of ascertaining how great the thrust upon its back is, but
how can its foundation carry the loads brought upon it. Accord
ingly due appreciation to this question must be given before
attempting refinements in the calculation of the thrusts that may
be induced in the wall by the fills deposited behind it.
A number of problems have been prepared at the end of this
and the succeeding chapters to illustrate the application of the
several tables, curves and equations given in the text immediately
preceding. They will also serve to demonstrate, numerically,
the points discussed in the chapter, bringing home more forcibly
the truths quoted than the literal equations.
Problems
1. A wall with a back sloped to a batter of one on four and 30 feet high
supports a level fill subject to a surcharge loading of 600 pounds per square
foot. What are the thrusts, by both Rankine's and Coulomb's methods
(a) when there is no surcharge; (6) when the surcharge extends to the wall
a (see Pig. 5); (c) when the surcharge extends up to the point 6, directly
over the heel of the wall.
The angle that the back makes with the vertical is tan>(M) = l*" For
the condition of no surcharge, from (14) and Table 1 with J = 0.42 for 6 =
14°.
100 ^ ^0^
T 9 ^ 042 = 18,900 pounds.
From Table 1, 9 = 23° and the angle that the thrust makes with the hori
zontal is 23° + 14° = 37°.
From (25) and Table 2 for 0' = 0°, 15° and 30°, K = 0.44, 0.41 and 0.42
respectively and the values of the thrusts are accordingly, 19,800 18 500
and 18,900 pounds.
For the condition of the surcharge extending to the back of the wall,
the constants remain as above and since c = %q = 0.2, the thrusts are
each increased by (1 + 2c) or by 1.4. The thrust, using Rankine's method
is then 1.4 X 18,900 = 26,500 pounds. The three thrusts, employing the
method of the sliding wedge method become, respectively 27,800, 25 900
and 26,500 pounds as the angle of friction between wall and earth is taken
as 0° 15° or 30°.
When the surcharge extends to 6 the condition under which the method
THEORY OF EARTH PRESSURE
37
of Rankine is used must receive special investigation, since equation (14)
no longer applies. From (11) with c = 0.2, the thrust is
The weight of the triangle G is, since ab = 30 X H = 7.5, 11,250 pounds
and the resultant thrust upon the wall is
^'o = V(21,000)2 + (11,250)2 = 23,700 pounds.
The angle which this final thrust makes with the horizontal is
tan' (11,250/21,000) = 28°.
With the expression given in (33), the method of the sliding wedge may
be employed, after the proper value of a has been found. The value of the
ratio y is 7.5/30 = 0.25. From (34) tan a = 0.25  j^ 0.25 = 0.18, from
which a = 10° and the coiresponding values of K for the angles of friction
0', 15° or 30"= are 0.42, 0.39 or 0.39 giving for T the corresponding values
23,500, 24,500 or 24,500 pounds.
K, C P
Fig. 14.
Allowing for friction between the back of the wall and the retained earth,
a close agreement is again to be noted between the two methods of computing
the thrust.
2. A wall with vertical back 20 feet high supports an embankment as
shown in Fig. 14 subject to a surcharge of 800 pounds per square foot.
Determine the thrust for the two conditions of no surcharge and surcharge.
For the condition of no surcharge, equation (22) may be used. Here
h' = 6 feet approximately and c is then 6/20 = 0.3. The angle 6=0° and
f the friction between wall and earth is ignored (which is advisable when
38 RETAINING WALLS
the back of the wall is vertical, as it is in this problem) <^' is also zero.
Again the angle of repose and the angle i are both equal to 30". The various
factors in the expression then take the following values:
L = l/oos^ <t> = H d = cot »' = cot (f>. u = sin </> and v = —cos <t>.
cos cot ■» ^ _ ^^^3 ^. „ = sin ^/sin .^ = ], and / = cot^ <t>
sm <^
= —3. p = sin = }>i.
T = ^' X I (l.3  ^V].32 + 3 X 0.09) '
= 9,600 pounds.
If the expression in (24) had been used with K = H and with the same value
of c = 0.3, the value of the thrust thus found would be
100 X 400 X1.6 _^, 0,00
^ X o
The latter method, or rather, equation (24) is apparently sufficiently exact
for the conditions under which the problem was analyzed.
For the surcharge of 800 pounds per square foot, as shown in the figure,
the graphical construction of Poncelet is employed to determine the thrust.
Draw aoh, making the triangles aof and coh of equivalent area. (A few
trials will determine the location of this line. In fact the accuracy of the
problem is easily satisfied by locating the line aoh by inspection.) Draw
Ah, then ah parallel to it and proceed as before with this method. The
thrust is then the area of the thrust triangle inm, multiplied by the unit
weight of the earth 100 pounds per cubic foot and is then equal to
16.7' X 100 .„„„ ,
^ = 13,900 pounds.
As a check upon this method, note that the line aoh makes an angle of 41°
with the horizontal. The method, using equation (22) may be employed
with the new surface ahi. . . With the same scheme of substitution as
employed in the first part of the problem, with i = 41°, n = cot 4> cot i =
2.0 and c = 1^0 = 0.7. The thrust is then found from the expression
_ 100X20^X4/, „ 1 , \2
T = 2X3 i ^ ~ 2^^'^' + 2 X 0.49 j = 13,700
affording a satisfactory check upon the graphical calculation.
3. A material is so densely compacted and well drained upon being
placed behind a retaining wall that it is safe to take its angle of slope as 46°
Derive an expression for the thrust against a vertical wall and also against
a wall with a batter of one in four.
With the surface horizontal and against a vertical wall the expression
for K in both the Rankine and Coulomb method is
1 — sin <t>
1 + sin <t>
which becomes for a value of <^ = 45°, closely onesixth. The thrust for
this material is then onehalf of the normal thrust against a vertical wall,
the normal thrust being that produced by a material with a slope angle of
30°.
^%"
1
* 1
^
v
/ _\ ll
l»
1
1
'"'"■' 1
i Y
THEORY OF EARTH PRESSURE 39
The value of the slope angle is 14°. From (14) the expression for the
thrust becomes, using the above value of and 34 for tan 6
T = 0.3 g^'(^ + 2c)
the value of J now being 0.3, which may be compared to the value 0.42 for
<t> = 30°
The corresponding values for the thrust as determined by the method of
the sUding wedge are easily found by proper
substitution of the value oi (j> = 45° in the
constant K, in the expression as given in (25).
This arithmetic work need not be given here.
4. A building wall running parallel to a re
taining wall, as shown in Fig. 15 carries a load
of one ton per square foot and has a spread of
four feet at a base four feet below the top of tt i n
the retaining wall. The retaining wall is
subject to no surcharge load other than that produced by the bearing wall.
What is the total thrust upon the wall and where is it located?
Referring to Fig. 15, the value of L/a is four tons or 8000 pounds per lineal
foot of wall. There is no surcharge and with A = 25 feet
gh^l + c )' 100 X 625 „, „.„ ,
" — ^~ — — = ^ = 31,250 pounds.
The ratio L/a to gh^(l + c)V2 is 0.256. This is the value of the ratio r.
With this value entering curve plate No. 1, the value of m for a maximum
wedge of sliding is 74°. It is observed that this plane will intersect the foot
ing and accordingly the maximum plane of slip is made to pass through the
inner edge of the base. This gives a value of 69° for m.
The thrust due to the concentrated load is
8000 tan(69°  30°) = 6480 pounds.
That due to the earth wedge is
100 X 625 ^^^ ggo ^^^(ggo _ 3Q0) ^ 9700 pounds.
The point of application of the thrust due to the concentrated load is 10.5
feet above the base of the vertical wall. That of the earth wedge is one
third of the distance up or 8.33 feet. The total thrust is then 6500 
9700 = 16,200 pounds and is located
6500 XKX5+ 9700X8.33 ^ ^ ^ ^^^^ ^^^^^ ^^^ ^^^^ ^j ^j^^ ^^1,
Assuming that the transmitted pressure of the bearing wall is contained
within planes making an angle of 30° with the vertical, at a point approxi
mately 11 feet below the surface the distribution of the load would strike
the back of the retaining wall. With a uniform distribution of the load at
this plane, the intensity of the transmitted pressure is soo^g =670
pounds per square foot. If this is treated as a surcharge at the surface and
40 RETAINING WALLS
equation (24) is employed to obtain the thrust, c is then ^J^s = 0.27.
With K taken as H
_ 100 X 6 25 X 1.54 ,„..„ ,
T = Trm = 16,050 pounds.
Z X o
1 81 X 25
From Table 3 the point of application of this thrust is located ' ^  ,
= 9.8 feet above the base of the wall. See page 30 for a discussion of
the use of this method of analysis as a check upon the prev ous method.
As a problem illustrative of _,the action of saturated earth the
author of the paper on page 32 has given the following example:'
"Take for example a wall supporting ten feet of earth the lower 6 ft.
of which are below water level and hence saturated. Assume dry
earth at 100 pounds per cubic foot and earth under water at 70 pounds
per cubic foot. Assume a natural slope for dry earth of 1.5 to 1
(<^i = 33°41') and for earth under the water of 2.5 to 1 (^2 = 21°48').
"Lateral pressure at the plane of saturation due to dry earth = 100
X 4 X (1 — sin 4>i)/0 + sin 0i) = 114.4 lbs. per square foot.
"Lateral pressure at the plane of saturation due to saturated earth =
100 X 4 X \ ~ g^ ^^' = 183.2 lbs. per square foot.
"Lateral earth pressure at the bottom
(100 + 4 + 70 X 6) j ~ ^1° ^^ = 374.6 lbs. per sq. ft.
"Hydrostatic pressure at the bottom = 62.5 X 6 = 375 lbs. per
square foot.
"Total lateral pressure at the bottom = 374.6 + 375 = 749.6 lb. per
sq. ft.
"Total resultant lateral pressure above the plane of saturation per
foot length of wall is 114.4 X 0.5 X 4 = 228.8 lb. This is applied at a
point 13^ ft. from the plane of saturation or 73^ ft. from the bottom of
the wall.
"Total resultant lateral pressure below the plane of saturation is 0.5
(183.2 + 749.6) X 6 = 2798.4 lb. This is applied at a distance of
6(749.6 + 2 X 1 83.2) „,, ,,
— 3(749 g _L 103 2) — °^ ^^ '■^®'' irom the bottom.
"The resultant lateral pressure against the wall per foot of length
is then 228.8 + 2798.4 = 3027.2 lb. This is applied at a distance of
228.8 X 7.3 + 2 798.4 X 2.4 „ „„ , ,
2027 2 ~ 2.77 feet from the bottom."
BiBLIOGKAPHT
For an exhaustive bibliography on the various theories and experiments
upon earth pressures, both active and passive see Howe, " Retaining Walls "
5th Ed. (see also Appendix) .
1 A. G. HusTED, Engineering NewsRecord, Vol. 81, p. 442.
THEORY OF EARTH PRESSURE 41
The following is a list of interesting papers upon the subject matter of
the chapter.
Earth Pressures: A practical comparison of theory and experiments,
Cornish, Trans. A. S. C. E., Ixxxi, p. 191.
Cohesion in Earth: Cain, Trans. A. S. C. E., Ixxx, p. 1315.
Earth Pressure Lateral: Cornell Civil Engineer, April, 1913.
Lateral Pressure of Clay : W. L. Coombs, Journal Western Society of Engi
neers, Vol. 17, p. 746.
Retaining Wall Theories: Pebby, Journal Western Society of Engineers,
Vol. 19, p. 113.
Retaining Walls: Based entirely upon the theory of friction, P. Dozal,
Buenos Aires. Translated.
CHAPTER II
DESIGN OF GRAVITY WALLS
Location and Height of Wall. — The need for a retaining wall
arises from the construction of a cut or an embankment, whose
side banks are not permitted to take their natural slopes. Where
the amount of land necessary for the construction of such a fill
or cut is, to all intents, un
limited, the wall may be
located at any point where
economy dictates that a wall
of the necessary height and
section is cheaper than the
additional cut or fill which it
replaces. Thus in Fig. 16 the
wall replaces all fill shown
crosshatched. A comparative estimate, taking into considera
tion the cost of masonry, of embankment, or excavation for the
wall footing, will show, after a few trials as to location, at what
point the wall should be placed to obtain the minimum cost.
If the wall, however, is to run along a highway or other fixed
property line, then, this at once determines its location. Again,
RoadSurface
Easemerrf Jp\ t^"'^
Line .
Fig. 16.
Fig. 17.
Fig. 18
in railroad work through cities, especially grade eUmination and
track elevation work, easements are costly and are generally re
stricted by the municipalities which grant them, so that it is
necessary to get the wall as close to the tracks as possible, whence
a wall is placed as shown in Fig. 17. Even in the case where ease
42
DESIGN OF GRAVITY WALLS 43
ments are cheap and unlimited, an eye to future development and
consequent increased trackage may make it desirable to so con
struct a wall, that the additional fill necessary for the future tracks
may easily be placed. In Fig. 18 the wall may be so built, that,
with placing a new top above A, the section will be ample to take
care of the new fill and live load, or the wall nday be built to the
future required height at once. This latter may, however, prove
unsightly.
General Outlines of the Wall.— The section of a wall should be
so chosen that, at a minimum cost, it yields a maximum area for
the improvement work. When this work runs through valuable
property acquired at high cost, so that every square foot possible
must be made available for the roadway or tracks, the front
face, on the property line, should be made vertical as shown in
Fig. 17 and placed as close to the line as the details of the coping
and footing will permit. To insure no possible encroachment at
a future date, due to settlement of the wall, surveying or con
struction errors and the like, it is better to place the coping a
few inches back from the line. The coping usually projects a
few inches beyond the face of the wall.
Before entering into a discussion of the relative merits of walls
with various outlines, it is necessary that the principles upon
which the walls are designed, be first explained. This will be
done in the following pages. The section of the wall may be
controlled not only by these general principles, but also by specific
limitations foreign to the actual stress system existing in the wall.
Architectural treatment may determine the shape of the wall,
when the wall is part of some general landscape scheme. The
selection of a type of wall that will suit peculiar foundation condi
tions is discussed in detail in later chapters. Generally speaking,
however, that section of wall is chosen which can be most econom
ically and expeditiously built.
The Two Classes of Retaining Walls. — 'Retaining walls fall into
two broad classes. The walls which retain an earth bank wholly
by their own weight are termed gravity walls. This type is dis
cussed in the present chapter. Those which utilize the weight
of the earth bank in sustaining the pressures of the bank form the
reinforced concrete type of walls. This latter class, because of the
mobile character of reinforced concrete has an infinite variety
of shapes. The following chapters will take up in detail the
analysis of the shapes occurring in ordinary construction work.
44
RETAINING WALLS
Since the active element of support in the gravity wall is the
material out of which it is composed, the wall may be made of
other materials besides concrete. The reinforced walls are made
of concrete and steel.
Fundamental Principles of Design. — 'A retaining wall, in sup
porting an earth bank must successfully withstand the following
possible modes of failure :
(a) The overturning moment caused by the earth thrust may
exceed the stability moment of the weight of the wall, or in the
case of the cantilever type, of the combined weight of the wall
and relieving earth weights. Thus in Fig. 19 the thrust moment
Tt is greater than the stability moment Gg, and the wall will
Fig, 19. — Criterion of overturning. Fio. 20. — Criterion of sliding.
rotate about its toe. To remedy this, the weight G or the lever
arm g is increased by adding to the dimensions of the wall, usually
by widening the base.
(6) The pressure on the toe caused by the resultant forces of
the thrust and weight of wall and earth may exceed the bearing
power of the soil at that point, crushing the ground and causing
the wall to tilt forward and, in the extreme case, topple over.
The remedy hes in a wall properly shaped and dimensioned to
insure safe soil pressures, or where dimensions alone will not
suffice the preparation of a proper foundation either by further
excavation to a better bottom or by the use of timber or pile
foundations.
(c) The frictional resistance between the wall base and the
foundation may be insufficient to overcome the horizontal com
ponent of the thrust and the wall will slide forward along the base.
In Fig. 20 fG is less than T^. f is the coefficient of friction, a
table of which for various materials, is shown here (Table 6).
Th is the horizontal component of the thrust. With a wall pro
perly proportioned against failure through overturning ot exces
DESIGN OF GRAVITY WALLS 45
sive bearing on the foundation, this condition rarely exists. It
is most likely to occur on a clay bottom, if water is present, since
the wet clay acts as a lubricant. To remedy a condition of this
kind, the base may either be widened, increasing the weight on
the wall, or a bottom may be prepared offering mechanical as well
as frictional resistance to sliding. If narrow trenches are dug in
the foundation, projections will be formed which will materially
increase the resistance. Again, the bottom may be tilted up
wards towards the toe, giving a horizontal component of resis
rnmrnm.
Fig. 21. — Types of bottoms to increase resistance against sliding,
tance in addition to the frictional (see Fig. 21 for both cases).
Filling the foundation trench completely with masonry, so that
the front of the wall butts against the original earth of the trench
(not any backfill) may also prove efficacious.
Table 6
Character of foundation
Coefficient
Dry clay
Wet or moist clay . . . .
Sand
Gravel
Wood (with grain) . . .
Wood (against grain) .
.50
.33
.40
.60
,60
.60
These are, then, the potential modes of failure of a retaining
wall, and the wall satisfying most economically these criteria
against failure has been properly designed.
To recapitulate, the following equations must be satisfied:
{a) Gg must be greater than Tl.
(b) Si must be less than S (where »Si is the toe pressure actually
induced and S is the permissible soil pressure.)
(c) fG must be greater than Th.
Concrete or Stone Walls.— In spite of the wellnigh universal
adoption of concrete as a retaining wall material, many yards of
46 RETAINING WALLS
stone wall are still being built. Under certain conditions, this
type of wall is the more economical one. The cut stone walls,
however, with their ashlar or coursed masonry faces are much
more costly than the concrete walls and are only used when
necessitated by architectural treatment. With the development
of the artistic treatment of concrete' faces and with the ability to
duplicate practically every cutstone effect in concrete, the need
of stone walls for even this purpose is rapidly diminishing. The
rubble walls, both mortar and dry, do have an important applica
tion and where local stone cuts are available, are far the cheapest
material out of which to build the wall.
When a wall is to be built adjacent to property, to which no
access is permissible, even during construction, thus preventing
the placing of the bracing and concrete forms, a stone wall be
comes a very convenient type of wall to build. Rubble walls
were so used in the track elevation of the Philadelphia, German
town, and Norristown Railroad through Philadelphia.^
The dry rubble wall is frankly a temporary expedient, awaiting
further local improvements, upon the arrival of which, the need
for the wall itself is either removed or else the walls are replaced
by those of more permanent and jileasing effect. The word
"temporary" should be used most qualifiedly, for many dry
rubble walls have existed for long periods of time, exceeding, by
far their expected duration of life. In municipal improvements,
as for, example the grading of a highway, leaving surrounding
unimproved property below the future grade, it is customary to
place a dry rubble wall along the highway with the expectation
that when the adjacent property is improved or graded, the wall
will either be removed or buried (see Plate 1, Fig. la).
The cement rubble wall is of as permanent a nature as the
concrete wall. Its face, unless more or less screened is not as
pleasing as a concrete face when viewed at close range. At com
paratively small distances away, however, it presents quite a
pleasing effect, the variegated coloring of the local stone showing
to advantage (see Plate 1, Fig. 16).
The stone walls require a distinct class of labor, familiar with
the work. Stone masons are not always available and because
of the diminishing amounts of stone walls built, are becoming
fewer in number. The universal adaptability of concrete, its
independence of local material conditions and the large amount
1 See S. T. Wagner, Trans. A.S.C.E., Vol. Ixxvi.
Plate I
Fig. a. — Dry rubble wall along highway.
...^ E«*.,«=»,^' ^^#i?S:' j:^«,.,ir/ f.— ^i.^^:f'*.,i:i< :■=■
^:^^i *«.'*.=i^^**'^r!*X'
Fio. B. — Characteristic appearance of ceirjcrit rubble wall.
(Facing page 40)
DESIGN OF GRAVITY WALLS 47
of concrete laborers and foremen all tend to explain the waning
popularity of stone masonry, i
Where the selection of the material out of which the wall is to
be built is governed solely by economic reasons, then, with labor
and material conditions of equal weight the costs of the dry
rubble wall, the cement rubble wall and the concrete wall stand
in the order one, two and three, i.e., the cost of the cement rubble
wall is twice that of the dry rubble wall and the concrete wall
three times that of the dry rubble wall. It is understood that
there are available local stone quarries for the rubble wall.
A very long haul for the stone makes the cost of the wall far too
high to permit a serious consideration of its construction.
When using a dry wall, care must be taken to allow for the
voids in assuming the weight of the masonry. The voids may
vary from 15 to 40 per cent, of the section. A problem at the
end of this chapter brings out this in some detail.
Thrust and Stability Moments. — The method of determining
the thrust upon the back of a gravity wall follows the recom
mended form of procedure given on page 16. The thrust T upon
the back of the wall is located at a point Bh above the bottom of
the wall, where the value of B is found from Table 3. The stand
ard type of surcharge loading of height h' is used (see Fig. 5) and
the ratio h'/h is denoted by c. The amount of the thrust is
T = Jgk^ "^
where / is the adopted earth pressure coefficient to be taken from
equation (14) or from Table 1. The unit weight of earth is g
(replacing w in the original equation to avoid confusion with a
more natural form of lettering used in the following algebraic
work).
If, under special conditions (see problems at the end of this
chapter) it is decided to use the method of the maximum wedge
of sliding, with the equation 24 on page 15, the thrust is
T = Kgh^^^
where K is the earth pressure coefficient of this method corre
sponding to / above and is to be taken from equation (24) or
' See Engineering NewsRecord, Vol. 81, p. 890 for a description of the
iise of dry rubble walls to retain the HetchHetchy Railroad. The cuts
for the highway afforded large amounts of stone.
48
RETAINING WALLS
from Table 2. Unless the back of the wall has a small batter
(less than 5°) it is recommended that a value of 0' = 30° be used
in finding the value of K.
Following are some general relations between the wall factors
and the thrust, covering all shapes of
gravity walls and all varieties of earth
pressures.
Let Fig. 22 represent a general sec
tion of gravity wall. Assume that
the thrust has been found, in value T
and located at a point Bh vertically
above the base. The weight of the
wall G is ■ usually found by breaking
up the figure as shown into triangles
and rectangles. Algebraically then,
by taking moments about some con
fer example, at the toe A, both the thrust
the stability moment Gi^i + 62^2 + GzQb
are •'  ^ 
FiQ. 22. — Stress system in
gravity wall.
vement point, as,
moment Tt and
easily found. Graphically by means of an equilibrium
polygon it is a simple matter to locate the resultant of the forces
both in amount and in point of application. In the above alge
braic method it is necessary to proceed further to obtain the
resultant in both location and in amount. Fig. 23 shows the
^v , Intersechbn
of Rays 1*5
7
K Drawn Parallel
I '1x1 R in Polygon
Fig. 23. — Graphical analysis of gravity wall stresses.
method of applying the thrust polygon to the determination of
the stability of the wall.
The wall is on the verge of overturning when the stability
moment is equal to the thrust moment or what is the same thing
when the resultant just intersects the toe of the wall. For this
condition the factor of safety is one.
DESIGN OF GRAVITY WALLS 49
As long as the stability moment exceeds the thrust moment,
or as long as the point of application of the resultant falls within
the base, the wall is safe against overturning. The proper
location of the resultant depends not only upon the factor of
safety thought desirable but also upon the question of a satis
factory foundation pressure. Before entering upon a discussion
of a safety factor against overturning, it may be well to discuss
the matter of foundations.
Foundations, those most vexing problems of engineering
practice, are of paramount importance in both wall design
and construction. Generally a correct foundation design de
mands a uniform distribution of load as its most important
premise. Unfortunately, the economics of retaining walls
usually forbid the fulfillment of this premise. The wall is
considered satisfactorily designed so long as the resultant of the
pressure on the base falls within the middle third of the base,
and more often at the outer edge of this middle third, so that
the pressure intensity on the base varies from nothing at the
heel to the maximum at the toe.
For foundations varying from rock to hard soils, such as
coarse sands and gravels or loamy soils, i.e., a mixture of gravelly
sand and clay, the relative settlements due to the varying loads
is small and a nonuniformly distributed load may safely be
placed upon them. For the finer sands, wet soils, reaching down
to the plastic bottoms, it is imperative to have a uniform dis
tribution of pressure and foundations must be designed to
secure this or recourse must be had to special types of walls,
such as the cellular and similar types (see later pages).
There is no intention of entering into a detailed analysis of
the proper selection and preparation of a foundation. ^ A brief
description only of the various types of bottoms will be given.
Various phases of this subject, however, will be taken up under
the headings of "Varied Types of Walls," "Settlement," etc.
Rock is an elastic term, embracing all the types from a dis
integrated product, that can easily be picked and shovelled
to the hard gneiss, trap and granite which prove so costly to
drill bits. The poor rocks, when stripped of a one or two foot
layer usually present a bottom sufficiently strong to take as heavy
a load as the safe crushing strength of the wall material will
permit, and this is, of course, the maximum pressure that can
1 See texts by Jacoby & Davis; Patton; Folwell, etc.
4
50 RETAINING WALLS
be allowed on any masonry foundation. Under these conditions,
the resultant may intersect the outer edge of the middle third
with a triangular distribution of base loading. Occasionally the
resultant is permitted to fall outside the middle third, so that the
wall bears on only part of the foundation. While, theoretically,
tension must then exist between the base and the foundation to
wards the heel of the wall, the rock is unyielding, so that there
can be no opening at the heel while the criteria of overturning
and safe bearing loads are satisfied. In the gravity walls, when
this type of foundation is adopted, care must be taken that the
tension then developed in the back of the wall at the base does
not exceed the tensile strength of the masonry. If it does, it is
necessary to reinforce the back with rods.
With a rock bottom well cleaned, left in the usual rough
condition, and, with a good bond secured between it and the base
of the wall, there is ample resistance to shding.
Shales, cementations gravels, coarse sand and gravel, in similar
fashion present but httle difficulty and it is customary, here also,
to permit a triangular distribution of soil pressure. Shading
off into the finer sands, dry clays and bottoms of Hke type with
moderately yielding propensities, a theoretical discussion' of
passive earth pressures seems to indicate that in yielding soils
there is an upward heaving of the soil adjacent to the down
ward loads, so that, to counteract this tendency, there must
be a minimum downward pressure on the base. For this
reason, the resultant of the pressures should strike the base
within the middle third, giving a trapezoidal distribution of
pressure.
Coming down to the plastic bottoms, there must be a uniform
distribution along the base not to exceed the safe bearing value
of the soil in question. If this is not possible it is necessary to
place piles. It is highly desirable that the piles carry equal
loads. If the base pressure is not uniform a uniform pile
loading may, nevertheless, be secured, by proper spacing of
the piles.
Distribution of Base Pressures.— The analysis of the loadings
upon the wall determines, finally, the location and amount of the
resultant pressure upon the base of the wall. Since this re
sultant force is eccentrically placed upon the base, it is necessary
to obtain the manner of the distribution of the pressure due to
1 Howe, "Retaining Walls, Earth Pressures and Foundations."
DESIGN OF GRAVITY WALLS
51
this resultant. The vertical component of the resultant is ana
lyzed here; the horizontal component affecting only the frictional
resistance between the wall and the earth.
Referring to Fig. 24, let R be the vertical component of the
resultant of all the pressures upon the base. Si and S2 are the
extreme pressure at the toe and heel
respectively. With these Hmiting in
tensities found all' the necessary data
for the footing is had.
Take moments about (the heel) g^
SiW^ . {Si  S2)io''
kwR =
and
6
Si + 2^2 = 6kR/w
Fig. 24. — Foundation
pressures.
(37)
Again, since the area of the trapezoid is equivalent to the value
of the resultant R
Si + 82 = 2R/w
Solving these simultaneous equations, there is
w
(38)
(39)
^.=^(3fcl)
(40)
When fc = }i, i.e., when the resultant intersects at the outer
edge of the middle third — a very common condition, Si = 2R/w
and S2 = 0. When fc = H, i.e., when there is a uniform
distribution of pressure along the base Si = S2 = R/w.
Note that when, fc is less than onethird, there is pressure along
only a portion of the base. The point of zero intensity is given by
X =
w 1  3fc
3 1  2fc
(41)
where x is the distance from the heel to the point of zero in
tensity.
Table 7 gives the permissible intensities of soil pressures as
allowed by the various codes.
52 RETAINING WALLS
Table 7. — Pehmissible Soil Pressures in Tons per Square Foot
Soil
Quicksand, silt
Clay, soft
Clay and sand
Sand, clean, dry
Sand compacted, well cemented
Gravel and coarse sand
Gravel and coarse sand well com
pacted
Clay, hard, moderately dry
Clay, hard, dry
Rock, soft to hard
M1
M2
24
24
46
68
810
4r6
68
5200
1
2
4
6
10
4
75*
1
2
3
6
10
4
840
1
t2
4
1220
A. Prof. Cain.
B. Public Service Commission, 1st District, New York City.
C. Building Code, New York City.
D. Building Code, Dist. of Washington.
E. Building Code, Baltimore.
* Sound ledge rock.
t Clay or clay mixed with sand, firm and dry. 3 tons.
Proper Centering for Piles. — Since the retaining wall brings
a nonuniform distribution of loading upon the base, a uniform
spacing of piles would produce unequal loading upon them.
This is not a desirable type of loading
for piles. The following is a method
of so spacing the piles as to secure a
uniform loading.
The piles may be spaced either in
rows parallel to the face of the wall, or
in^ rows perpendicular to the face of
the wall. A graphic and an analytic
method are outlined below for either
of these two methods of spacing the piles.
Let P be the safe bearing value per pile. In Fig. 25 divide
the base into a series of strips of equal width v. From the
eccentric position of R determine the extreme bearings, Si and S^
and lay these off to scale. The soil pressure in any strip v, Sy
is readily obtained by scaUng the figure. vS^ then gives the total
load on the v strip taken for a unit width of wall. Dividing
P by this product determines the spacing necessary in that strip.
The^minimum spacing of piles is about three feet, so that, when
Pig. 25. — Pile spacing.
Case I.
DESIGN OF GRAVITY WALLS
53
the spacing in a strip is found to be less than this minimum, it
is necessary to take the strips closer together. When this fails
the base must be widened by placing a toe extension.
The piles may be spaced perpendicularly to the paper at equal
intervals, but at varying distances along the base of the wall
(see Fig. 26) . Assume that a width
of wall is taken (perpendicular to U.
the sheet) equal to the permissible 'd^
or desirable spacing of piles. The
values of R, Si and S2, as found
above are increased accordingly.
Making a scale layout as above,
trial irregular widths are taken
decreasing in width towards the
toe, each being equivalent to the
safe bearing of one pile. The following is an analytic discussion
of the two cases.
Case I. — From the geometry of Fig. 25 the total pressure in
any width v of the base (a unit's thickness of wall is assumed) is
Casein.
+ {i  l)v
vSvi = S2V + V '■
w
(Si  S2)
i is the number of the division, counting from the back of the
wall.
Replacing ^Si and S2 by their values in terms of R and A;
,S.*=^(3fcl)+3^?^(l2/c) (42)
Since the pile can take P as a safe load, the required spacing of
piles in the "i" the row is, then
P
vSvi
a =
(43)
Case //. — ^Let it be assumed that the rows of piles, parallel to
the page, are spaced m feet apart. The total vertical load on the
foundation is then mR and if, as above, P is the safe load per pile,
the number of piles required in each thickness m of the wall is
mR/P = n and this is the required number of spaces of equal area
into which it is required to divide the trapezoid, in Fig. 26. Com
plete the triangle as shown and let the area of BCO be Po. The
area of any other triangle, bound, say, by the vertical side b.,
54 kETAlNlNG WALLS
as base, is Po + iP, where i is the number of divisions, or of
piles, from the back of the wall. Since the areas of similar
triangles are to each other as the square of their homologous
sides
bi^ Po + iP ^r, 61' ^0 + P a 1 1 J, 2 hi
W=, = Po + a  DP ' *^'" w = ^T' ^^"^"'^^ ^' = ^'
P0 + 2P _ , 2P0 + 2P
TT+P'" "pT"
Extending this result to the general case
,, = ^0^^^ (44)
Let k be the distance from B to the corresponding i line
then k= hbo = bo [sp^  l) Since Po =^', 60 =
and if, finally, Si and S2 are replaced by their values in terms of
R and k
. = .p(^iHi);P = 3f^;H = (^;l =
^ (45)
(3A;  1)P
That the distance between the two piles adjacent to the toe
shall not be less than a specified amount a (usually about three
feet) it may be necessary to extend the base by means of a toe
With sufficient exactness the distance a may be taken as onehalf
the distance between the toe and the point Z„_2. Then
Wo — ln2 = 2a
Replacing L2 by its value from (45), simplifying the resulting
equation and ehminating the radical and putting 2a/wo = X
^ ^l^)+2(lX), N=^^
3fc  1 F
and solving for k
_ iV(lX)(l3X)
" 6X(1  X) ^46)
If the width including the toe extension is Wo, and the width with
out the toe extension is w, letting 2a/w = X' and noting that
Wq = w {1 + i) and X' = X (1  i) also k = y^. (see Fig. 24).
DESIGN OF GRAVITY WALLS
55
A = 3X' [X' +
Equation (46) becomes a cubic in (1 + i) or u
2(le)],B=6\'MlM);
2
u' + 2X'm2  Au + B = 0. (47)
In view of the fact that i is small in comparison with unity, (it
cannot exceed }i for a valid solution), it is permissible to replace
M^ by 1 + 3i, and m^ by 1 + 2i, which makes (47) hnear in i
and gives the relation
^  2X'  2/r,
6/n + 4X'  A ^^^)
This apparently comphcated analysis together with the entire
mathematical treatment of pile loading is given with the idea
of affording a direct solution of pile spacing problems for ec
centric distributions of loading. The problems at the end of
the chapter will bring to bear the arithmetic appHcation of the
hteral equations just developed. The work just shown of
determining the proper offset to maintain the minimum pile
spacing replaces a rather tedious method of trial and error. In
all the above work it is understood that a uniform loading of the
several piles used is the result sought.
For the special case of fc = ^, i.e. the resultant intersects the
base at the outer edge of the middle third, and (45) becomes
(49)
Table 8 gives values of F and H.
Since either method, theoretically,
must give the same density of piles, it
is immaterial, from the standpoint of
the number of piles required, which
method is adopted. Practically, how
ever, it seems simpler to use the latter
method of distribution since the piles
are lined up in both directions. In the
former, they are in line longitudinally,
only, i.e. parallel to the face of the wall
making the work in the field a little
more cumbersome than in the latter
method.
Occasionally eccentric bearing is
allowed on piles, the piles then being
Table 8
*
F
H
.36
.10
131.0
.37
.14
65.0
.38
.19
36.8
.39
.26
23.0
.40
.33
15.0
.41
.43
10.0
.42
.54
7.11
.43
.69
5.00
.44
.89
3.61
.45
1.20
2,50
.46
1.58
1.66
.47
2.30
1.10
.48
3.67
.62
56
RETAINING WALLS
unequally loaded. This practice is far from coramendable,
since, a pile is, by its very nature, a yielding support (unless
driven to absolute refusal) and unequal settlement is unavoid
able. Pile foundations, and, in fact, all foundations, demand
most mature engineering judgment in their planning and con
struction and time and money spent in consulting experienced
men on this part of the work is an ideal assurance towards a
safe and weUappearing wall.
A problem at the end of this chapter illustrates the application
of the above analysis to a concrete case.
Factor of Safety. — It has been seen that, as long as the resultant
intersects the base inside the toe, there is no danger that the wall
will overturn. Since the thrust is computed from the maximum
load possible or anticipated upon the wall, a factor of safety
but little greater than one seems ample. However, to insure
that there will be no tension in the back of the wall, the resultant
should intersect within the middle third.
Fig. 27. — The retaining wall and
its foundation.
The wall may be divided into two parts; that portion (see
Fig. 27) above the ground surface, retaining the fill; and the
foundation course. At the junction of these two parts, that is, at
the surface of the ground, the resultant should intersect at the
outer edge of the middle third. This insures the most economi
cal wall above the surface and at the same time prevents any
tension in the wall. The dimensions of the footing are then solely
governed by the permissible soil pressures.
The ratio between the moment tending to resist the over
turning of the wall and the moment tending to overturn the wall
has been termed the factor of safety against overturning'.
Referring to Fig. 28 the overturning moment is Tut and the
DESIGN OF GRAVITY WALLS 57
resisting moment is Gx + ^[(l + i)w  Bh tan b]. Denoting
the factor of safety by n
Gx + n[(l + i)w  Bh tan b] = nT^t
Taking moments about the point where the resultant intersects
the base G(x  zw) = m  r„[(l + i  z)w  Bh tan b]
Placing A = T,[il + i)w  Bh tan b] the two equations become
Gx + A = n Tht; Gx  Gzw = T/^t + T, zw  A. Combining
these two equations and solving for n
^ Td ■^zw(ff+ r.) _ zw{G + n)
Tnt ~ ^ + fd (^^^
and conversely
_ {n \) Tht
^ ~ (G + T.)w' (51)
Prof. Cain' advocates designing a wall for a definite factor of
safety and recommends the following values of n for walls sub
jected to vibratory loadings, such as walls adjacent to passing
trains :
Walls less than 10 feet high n = 3.5
Walls from 10 to 20 feet high n = 3
Walls around 50 feet high n = 2.5
Prof. HooP recommends a factor of safety of 2 for the average
retaining wall.
To assign a definite, integral factor of safety against overturn
ing locates the position of the resultant upon the base without
regard to the character of the distribution of the pressure upon
the soil that seems most desirable. Walls fail because of founda
tion weakness (see pages 160163) rarely because the overturn
ing moment exceeds the stabihty moment. An integral factor
of safety reverses this order of importance and makes the less
usual potential mode of failure the more important criterion. It is
better procedure to decide upon the location of the resultant of
the pressures and then to learn what factor of safety is to be had
following the method given on page 56. It is assumed, in figur
ing the factor of safety against overturning, that the wall will
revolve about its toe as a fulcrum. This is possible only upon an
unyielding soil; for the other soils, as the wall tends to turn on
1 Trans. A. S. C. E., Vol. Ixxii.
2 "Reinforced Concrete Construction," Vol. 2.
58
RETAINING WALLS
its toe, the ground in the immediate vicinity of the toe will
crush so that the conditions under which the factor of safety was
computed will no longer be vaHd.
It is doubtful whether, in actual practice this factor against
overturning is ever predetermined or subsequently ascertained.
It is well, however, as an additional precautionary measure, to
find its value in the manner outlined before.
Footing. — The retaining wall proper may be considered to end
at the bottom of the fill retained, or at the natural ground
surface (see Fig. 27). It is then necessary to design a footing
that will properly distribute upon the soil the pressures brought
to it from the retaining wall. If the base of this wall proper is
projected vertically downwards, and if the
values of Si and ^2 as found on page 51 in
equations 39, 40 are within the allowable
pressures as shown in Table 7 no extension
of the base is necessary. When these values
exceed the permissible ones a toe extension
becomes necessary. This may be found as
follows: In Fig. 29 let ew locate the position
of the resultant pressure and let S be the
permissible soil pressvue. The offset iw is that necessary to
make the value of Sx approach as nearly as possible the allowable
value S. Referring to equation (39), the value of fc is now
H't
Fig. 29.— Toe
extension.
k
The value of S^ is
_ {i + e)w _i + e
~ (1 + z)wJ ~ i + 1
Si =
2R
w(l + i)
(^'m)
Place
w Si/2R = r
and the above equation becomes
_ 2  3e  z
*" (1 + iy
which is a quadratic in i, which when solved gives
V 12 7(l  e) + 1  (2r + 1)
t =
2r
(52)
(53)
(54)
(55)
(56)
DESIGN OF GRAVITY WALLS
59
The usual value, and the one most properly taken for e is 3^
This makes (56)
. _ V{8r + 1)  (2r + 1)
2r
(57)
which determines the necessary offset for the base when the
resultant is given in amount and location and the value of the
soil pressure intensity has been assigned. To aid in the deter
mination of the offset when the value of r is given, Table 9 has
been prepared giving the values of i for a range of values of r.
Some examples at the end of the chapter illustrate the application
of Table 9 to specific problems.
A less frequent requirement, but one which may possibly
exist (see problems at end of chapter) is the determination of a
toe offset to give a minimum intensity *S2 at the heel. With the
value of k as in equation (52) and from (40) after placing
& = WS2/2R
2i + 3e
(1 + iy
There is obtained a value of i
t =
For e = 3^, this becomes
1 
fi
V 1  s(2
 3e + 1)
JCOl
nes
1 
s
i
s V(l 
2s)
(58)
(59)
(60)
(61)
Table 10 has been prepared giving a range of values of i for the
possible variations in the ratio s.
Table 9 Table 10
l.OOj
.00
■9..
.04
•8 J
.08
•7,1
.12
.6
.18
•Si
.24
A
.31
.375
.33
s
i
.00
.00
.05
.02
.10
.05
.15
.09
.20
.13
.25
.17
.30
.22
.35
.29
.375
.33
60 RETAINING WALLS
The toe extension is a cantilever beam and must be so dimen
sioned as to satisfy the shear and bending moment requirements
of such a beam. Let the thickness of the toe be d. Since the
extension is usually small in comparison with the rest of the
footing, the distribution of soil pressure may be taken as uni
formly spread over the toe and equal in intensity to Si, per unit
of length. If /„ is the concrete stress allowed in compression,
the external moment equated to the resis ting mom ent gives
8ri'wy2 = /,dV6 and d = Uw, with k = VWUW
It is necessary here to locate the principal planes to determine
along what plane there exists a maximum tension, i.e., the plane
of weakness of the step. The stresses on the principal planes
are given by the expression / = c/2 + \/(cV4 + p^). c is the
unit compressive stress and p the unit shearing stress found in
the body with the axes corresponding to the axes of loading of
the body, i.e., as in the sketch, vertical and horizontal. In
* C CI 47}^
shghtly altered form, this may be written j = ^ — ^■\ 1 \ — ^ •
For concrete c is large in comparison with p and in developing
the radical by the binomial theorem it will be permissible to stop
with the second term, whence/ = p^/c, or p = ■\/{fc) The unit
shear is then a geometric mean^ between the tension and com
pression as exerted along the vertical and horizontal planes
of the body. In the first expression for the principal
stresses, the minus sign was taken since the principal tension
was sought.
The angle between the principal tension plane and the
vertical plane is given by tan~^ ( — 2p/c), or using the approxi
mate relation between p and c is equal to tan "■'2 ^/" Upon the
recommendation.of the special concrete committee of the A.S.C.E.
(a summary of which is given later in a section on "Reinforced
Concrete") the ratio //c is to be taken as Ke, and this angle be
comes tan "^( — H) or the ratio of the extension to the depth is
onehalf.
The maximum tension then exists along a plane making a slope
of one to two with the vertical. Again, it has been demonstrated
that the transmission of loading through a solid is contained with
' In "Reinforced Concrete" by Mobsch, as translated by B. P. Good
rich, this theorem is established by somewhat different an analysis.
DESIGN OF GRAVITY WALLS 61
in planes making an angle of about 30° with the vertical. For
both these reasons, good practice would demand that, wherever
possible the ratio of step to depth for a foundation offset be one to
two.
The maximum pressure that can be brought to bear upon
a foundation is Kmited by the permissible bearing on the
masonry, usually taken at about thirty tons per square foot or
about 400 pounds per square inch. From the preceding formula
for the depth of step as required because of the bending moment,
k is then less than 2, so that a step of 1 to 2 will always satisfy
the bending moment requirements with the above maximum
loading. Ihe shear on the plane where the toe joins the footing
is Sjiw/d = Si/k. If the shearing stress is taken as 75 pounds
per square inch, then as long as ^i does not exceed 150 pounds
per square inch or about ten tons per square foot, a value of
fc = 2, is good. When the soil pressure does exceed this amount,
it will be necessary to reinforce the base.
For all ordinary soil pressures, then, a step of one to two is
satisfactory and should be adopted for the toe extension.
A Direct Method of Designing the Wall Proper. — In the ordi
nary course of design of a gravity wall, a tentative section,
governed by the judgment and experience of the designer, is
selected. This is analyzed in accordance with the methods out
lined in the preceding pages. It has been pointed out that the
usual goal of the designer is to select such a section of wall that
the resultant intersects exactly at the outer edge of the middle
third. As the tentative section does not, at first choice, fulfill
this condition, one or more succeeding sections are chosen until
the final one does meet this criterion. By using the criterion that
the resultant must intersect at the outer edge of the middle third
and by giving the thrust the standard form of expression on page
16, it is possible to effect a direct solution of the required dimen
sions of the wall. The analysis following develops an equation,
predicated upon these assumptions, from which Table 12 has
been prepared. This table covers the usual range of the factors
controlling the wall section and is to be used in place of the
method of trial and error as stated above. The numerical ap
phcation of the table and of the equations upon which it is based
is to be found in the problems at the end of the chapter.
The general gravity type of wall is shown in Fig. 30. The rec
tangular wall, the wall with a vertical front face and the wall
62
RETAINING WALLS
with a vertical rear face are, of course, but special cases of this
general type.
In taking moments about the outer edge of the middle third,
i.e., about the point 7, the moment of the thrust must be equal
to the wall moment. These
moments are found as follows :
Extend the sides of the wall
to their intersection at A
roject the point A vertically
down upon the base, meeting
the base at the point D. The
vertical distance that A is
above the top of the wall is t.
Let the ratio t/h be put equal
to p. The front face of the
wall makes an angle a with
the vertical; the rear face (the
face adjacent to the earth
embankment) an angle b.
Place tan a and tan b equal to
M and A'' respectively. Tak
FlG
30. — Design of gravity wall.
ing moments about the point D, the location of the point of ap
plication of the weight of the wall with respect to D is x, where
x = (N  M)
hi +Bp + dp^
l + 2p
(62)
The distance of G from the point 0, i.e., from the toe of the wall
IS
(1 + p)Mh + X
(63)
and from 62 this becomes
h/ 2 + Qp + Zp^ 1 + 3p + 3p' „
3V l + 2p ^ 1 + 2p ^^
(64)
This expression, locating the center of gravity of a general type
of gravity wall with respect to the toe may be further simplified
by putting the ratio of the upper to the lower base equal to u.
Then
u = p/(l + p).
(65)
DESION OF GRAVITY WALLS
63
Calling the distance of the center of gravity of the wall from the
toe, q, from (63) and (64)
q = ^iU,M+ U^N)
(66)
where
C/i =
2 + 2u  u^
U.
1 +u + u^
1  u^
Table 11
u
Ui .
Ut
.0
2.00
1.00
.1
2.21
1.12
.2
2.46
1.29
.3
2.76
1.53
.4
3.14
1.86
.5
3.67
2.33
.6
4.44
3.06
.7
5.70
4.30
.8
8.22
6.78
.9
15.73
14.27
Table 11 has been prepared giving the values of these coeffi
cients for the range of values of u. The table, and the above
formulas for the center of gravity with respect to the toe are
applicable to any method of analyzing the wall, not only the
special method now being followed.
The distance from the outer third point I to the point of
application of the force G is x, where
x= {l + v)Mh + x\{l{v){M\N)h (67)
When simplified this value becomes
hll\ 3p + 3p'
_hn
~ 3\
M +
pi
N
l + 2p  ' 1 + 2v' I (^^)
If the unit weight of the masonry is m pounds per cubic foot,
then the value of G is
G = ^,. (1 + 2P)(M + JV)
and its moment about the outer third point I is Gx, or
Gx = ^V(l + 3p) + (M + N)v'} {M + N) (70)
(69)
64
RETAINING WALLS
To determine the thrust moment resolve the thrust into its
horizontal and vertical components as shown on page 10. The
horizontal component is Tn and its value is
Th = ghKl + 2c)/6 (71)
The vertical component is Ty and its value is
n = gh'il + 2c)N/2 (72)
Taking moments about the outer edge of the middle third /, and
letting the thrust moment be Mo.
Mo = TkBH  r.[ (1 + p)(M N)h  BhN'\
gJl
6
(1 + 2c){J5  iV[2(l + p)(M + iV)  WN]} (73)
Equating this thrust moment to the stability moment of the
wall, putting the ratio of the unit weight of the earth g to the
unit weight of the masonry m equal to s, and writing the equation
in the form of a quadratic in p(M + N),
(M + NYv^ + Iv{M + N) + H = Q (74)
J = 3M + 2sN{l + 2c); ff = M{M + N)
6MN  3m + 3c(l  4MiV  N')]
'[1
3^
It will be noticed that the quantity p(M + N) is the ratio of
the width of the top of the wall to the height of the wall. Table
12 has been prepared based upon equation 74, giving the ratios
Table
12
iV = 0.0 AT = 0.1 1 AT = 0.2 JV = 0.3 1 JV = 0.4 j JV = 0.5
M
\ \ c \ c \ c \ .
.2
.4
.2
.4
.2
.4
.2
.4
.2
.4
.2
.4
.47
.60
.70
.40
.50
.58
.33
.41
.47
.25
.33
.37
.17
.23
.28
.07
.17
.23
.47
.60
.70
.50
.60
.68
.53
.61
.67
.56
.63
.69
.57
.63
.68
.57
.67
.73
.33
.46
.56
.26
.36
.44
.19
.27
.34
.10
.18
.24
.02
.09
.14
.05
.08
.1
.43
.56
.66
.46
.56
.64
.49
.57
.64
.50
.58
.64
.52
.59
.64
.65
.68
.22
.34
.44
.15
.24
.32
.07
.15
.22
.06
.11
.02
.02
,2
.42
.13
.54
.24
.64
.33
.45
.05
.54
.14
.62
.22
.47
.55
.05
.62
.11
.56
.61
.01
.62
.72
.3
.43
.05
.54
.15
.63
.23
.45
.54
.05
.62
.12
.55
.61
.01
.61
.4
.45
.55
.07
.63
.15
.55
.62
.03
.61
.5
.57
.05
.63
DESIGN OF GRAVITY WALLS 65
of the top and bottom widths of the wall to the height of the wall for
a sufficient range of values to determine very closely the required
dimensions of any gravity type of wall, assuming that the ratio
of the weight of the earth to masonry is % {i.e., s = %) and
that the resultant intersects the base of the wall proper at the
outer edge of the middle third.
With both M and N zero, the wall is the rectangular type.
With M zero, the wall is the vertical front and battered back type,
a very popular type forming a large percentage of all gravity
types built and very efficient where maximum trackage and
minimum easements are wanted (see page 42). With N zero
there is the less usual type, but a most economical one with
vertical back and battered face. A slight face batter and a
larger back batter make a wall of economical section and pleasing
appearance. It is understood in selecting the dimensions of the
wall that a proper footing is to be developed as shown on the
preceding pages, to give the correct distribution of pressure upon
the foundation.
The converse problem, given the section of a retaining wall,
to locate the position of the resultant pressure upon the base
may be solved as follows : Referring to Fig. 30, with the weight
of the wall G a distance q from the toe and the point of applica
tion of the resultant pressure a distance zw from the toe where
zw = 01, as in Fig. 23, take moments about I
G(q  zw) + T^{w  BhN  zw) = ThBh
and solving this expression for z,
z =
Gq + T.iw  BhN)  TkBh .„..
((? + T.)w ^^^^
The value of q and of the thrust components may be taken
from the appropriate equations and tables given in the preceding
work. — — I
Revetment Walls. — The wall leaning toward the earth bank ;
which it supports, as shown in Fig. 31, is termed a revetmenUvall^
It is more of historic than of present interest. pPf^TCam Has
showni that when the angle b is less than 10°, the ordinary theory
of earth pressure as given by the method of the wedge of maxi
mum thrust (see pages 1115), may safely be applied to deter
mine the thrust.
' " Earth Pressure, Walls and Bins," pp. 96, 97.
5
66
RETAINING WALLS
That the wall be selfsustaining while under construction, it
is necessary that its center of gravity projected down, always
falls within the base. To effect this, denote the ratio of the
width of base to height of wall (a parallelogram is the only type
of section discussed in detail here) by k. That the wall be
selfsustaining, it is necessary
that h be greater than tan h.
As in the former pages, a
direct method of determining
upon the ratio k for any
character of loading, predi
cated upon the resultant in
tersecting at the outer edge
of the middle third may be
found for this type of wall.
In the following work the
earth pressure coefficient is
K, defined by equation (25).
In view of the fact that the
angle h is now negative. Table 13 has been prepared giving the
values of this coefficient K for negative values of the angle h.
The thrust moment is
TXAO (76)
From (24)
Flo. 31. — Design of revetment wall,
gh'K
1 + 2c
AO = EF = ED  FD.
ED = Bh cos{<t>'  b)
FD = {Bh tan 6 f 1 kh) sin (</>'  6)
and (76) becomes
g Kh^il + 2c)lW cos (<^'  6)  (35 tan b + 2k) sin(,>'
b)]
The stability moment of the wall (both of the moments are
taken about the outer edge of the middle third, i.e. 0) is
mkh' ( tan b + kh/2  kh/Z) = m ^ (3 tan 6 + fc)
Equating these two moments, and writing the resulting equa
tion as a quadratic in k
k^ + Rk = S
(77)
DESIGN OF GRAVITY WALLS
67
where
72 = 3 tan 6 + 2s(l + 2c) sin (0'  6)
cos <l>'
S = s(l + 3c)
s is the ratio
gK
cos b
(78)
(79)
Table 13
!>
^' = 0°
*' = 15'>
*' = 30''
0°
.33
.30
.29
5°
.30
.27
.26
10°
.27
.24
.23
Table
14
«' =
«'=15°
<*.' = 30°
6 = 5°
5=10°
t = 5°
6 = 10°
6 = 5° 6 = 10°
.35
.25
.30
.21
.23
.17
.2
.47
.37
.40
.29
.31
.23
.4
.57
.46
.48
.37
.37
.29
Table 14 gives a series of values of the ratio, k, based on the
above equation for several values of b and <^'. Revetment walls
are usually built of stone masonry, presenting quite a rough
surface adjacent to the earth bank, and it is therefore safe to
allow the usual value of ^' (about 30°). Revetment walls,
because of construction difficulties are rarely built of concrete.
If concrete should be used, its smooth surface, together with the
possibility of lubrication due to water, makes it inexpedient to
allow for any frictional resistance between the wall and the adja
cent earth.
Problems : Gravity Walls and Foundations
Note. — A comparative study of various sections of walls, with illustra
tive plates, is given in a pamphlet published by the Engineering News, 1913,
entitled "Comparative Sections of Thirty Retaining Walls and Some Notes
on Design," by E. H. Carter.
1. A wall with a slight face batter and battered back, 25 feet high, sup
ports a fill level with its top and subject to a uniformly distributed load of
600 pounds per square foot. What is the necessary width of the base as
suming that the top width is taken as 2' 6" wide? Determine the offset of
its footing that the toe pressure shall not exceed 6000 pounds per square foot.
What is the factor of safety of the wall? If the method of the maximum
wedge ol sliding is used where is the point of application of the resultant
located and what is the factor of safety (a) when the angle of friction is
assumed as 30° (6) and when it is assumed as 0° between earth and back
of wall?
The equivalent surcharge to a load of 600 pounds per square foot is six
feet, whence the value of c is ^5, or 0.24. The ratio of top width to height
is 2.5/25 or 0.10. By interpolation in Table 12 the values Af = 0.067 and
68 RETAINING WALLS
N = 0.5 satisfy the given avguments and the resulting width of base is
^(0.1 + 0.5 + 0.067) = 16.7 feet. The face batter is J^" to the foot and
the rear 6" to the toot.
To obtain the proper soil distribution, the weight of the wall (taking the
masonry unit weight 150 pounds per cubic foot) is 35.9 kips (i.e. a kip is a
1000 pound unit). The vertical component of the thrust is (Eq. 72) T„ =
23.1. The vertical component of the resultant pressure upon the base is
the sum of these two forces or is equal to 35.9 + 23.1 = 59.0 kips. From
(54) r = 0.85 and from Table 9, i = 0.057, whence the necessary projection
is iw or 1' 0". Since the wall foundations are carried down about four
feet to prevent fiost action and surface water erosion, the step of one foot
to four feet is a satisfactory one.
Fiom (50) referring to Fig. 28, B from Table 3, is 0.39 whence « = 0.39 X
25 + 4.0 = 13.75. zw = ),i of 16.7 + iw = 6.56 and the horizontal com
ponent of the thrust from (71) is 15.4 whence the factor of safety = 1 +
1.8 = 2.8, a satisfactory one from Prof. Cain's recommendations, page 57,
but clearly without significance, unless taken in conjunction with the loca
tion of the resultant and with the manner of the distribution of pressure
upon the soil.
By the sliding wedge method the horizontal component of the thrust is T cos
(b + 0'), with T as given in (24). For N = 0.5, 6 = 26° 34' and from
(25) K = 0.60. Tk and Tv are then 15.4 and 23.3 respectively. (Cf. cor
responding values by other method.) The location of the weight of the
wall G is obtained from (66) and Table 11 with u = 0.10/0.567 = 0.18.
3 = ~ (2.41 X 0.067 + 1.25 X 0.5) = 6.55, whence from (75) z = 0.364,
not at large variance with the value of i + e in the Rankine's method.
The toe pressure is from (53) 6.4 kips, approximating with sufficient exact
ness the result obtained in the suggested standard method of obtaining
the thrust.
If the frictional resistance between earth and masonry is ignored, K =
0.64 and Tj,, T„ are respectively 26.5 and 13.2. With the revised values,
z = 0.163, a very unsatisfactory result. If the section of wall is changed
to give a value oi z = 0.333 by the last method, a much heavier section of
wall results, showing the costly effect of omitting the consideration of fric
tional action of the earth upon the back of wall. All the standard sections
exhibited in the abovementioned pamphlet would develop high tension at
the heel of the wall and a high bearing at the toe leading to the disfiguration,
if not destruction of the wall were they designed in accordance with the
maximum wedge of sliding, ignoring frictional action between the earth
and wall. The sections are all extensively used in actual practice with
excellent results.
Allowing for frictional resistance between earth and wall the. factor of
safety is 3; ignoring such action the factor becomes 1.5, i.e., such favorable
consideration doubles the factor of safety.
2. A standard wall for highways, is to be built, with a face batter of lyi"
to the foot and a back batter of 4" to the foot. Give a section with the
proper tabular dimensions. Also prepare plans for the proper foundation
dimensions for (a) coarse sand and clay, well compacted, permissible bearing
DESIGN OF GRAVITY WALLS
69
4 tons per square foot, (6) coarse sand, permissible bearing 3 tons per square
foot (c) fine sand, where a maximum intensity of toe pressure is 2 ton per
square foot and a minimum intensity of heel pressure is 0.6 tons per square
foot. Also give a pile foundation section, allowing twenty tons per pile.
With the batters as given, M = 0.125, and AT = 0.333. For highways,
an average uniformly distributed load of 500 pounds per square foot will
safely provide for the heavy surface loadings. Then for h = 15, c = 0.33;
for 7i = 20, c = 0.25; for h = 25, c = 0.20; lor h = 30, c = 0.17. From
data obtained from Table 12, the following table of top and bottom widths
of wall has been prepared, (d is the top width, b the base width.)
h
d
6
15
2' 5"
9' 3"
20
2' 8"
11' 10"
25
3'0'
14' 5"
30
3' 4"
17' 0"
Following the preparation of this table, a similar one may be prepared, giving
the data necessary to compile the required toe extensions for the several
allowable pressuies.
Si
= 4 tons
Si
= 3 tons Si = 2 tons
52 = 0.5 tonl
h
G
n
T,
li
r
i iw
r
i \ iw \
r i
s i
iw
1
13
13.1
6.2
6.2
19.3
1 *
*
•
.96
.02
.24
.17°
l'6"
20
21.7
10.0
10.0
31.7
! •
*
*
.75
.10
.19
.13°
l'6"
25
32.7
14.6
14.6
47.3
*
.90
.035
0'6"
.61
.18°
.15
.09
2'6"
30
45.7
20.1
20.1
65.8
*
.78
.08
l'4"
.52
.23°
.13
.08
4'0"
* No offset necessary. "This value governs.
For the coarse sand and clay bottom (4 tons per sq>iare foot) no toe extension
is necessary.
In preparing typical pile foundation plans, it is assumed that the piles
win be in line both transversely and longitudinally (Case II).
h = 15'. Assume two piles to a section. If the rows are m feet apart,
and with a beaiing value of 40 kips each, the necessary spacing of the rows
is 80 /R = 4.15; therefore space these rows on four foot centers. The total
load on each row is then 4i2 = 4 X 19.3 = 77K. With a value of k = }i,
Eq. 49 is applicable and h = w\/o.5 = 6.55. The location of the pile is
at the center of gravity of this triangle or at a distance H X 6.55 from the
heel. The pile is, accordingly 4' 4" from the heel. The other pile is at
the center of gravity of the trapezoid bounded by the toe and the line li.
The center of gravity of the trapezoid may be found in a manner similar to
the location of the center of gravity of the earth pressure triangle Fig. 5. The
, t • I, 6.55
value of c is here ;
= 2.4 and the value of B from Table 3 is 0.47.
■ 9.25  6.55
The center of gravity is then *Koo of the distance 2.70 from the toe, or
approximately 1' 3" from the toe. It is safe, generally to take the pile at
the center of the trapezoid, the erroi being one of a few inshes only.
70
RETAINING WAI^^LS
h = 20'. Assuming two piles in a row here, with the value of i? = 31.7
gives a spacing between rows of 2.5' which is too close to space the piles;
therefore three piles are taken. With this value m = 3.8 and may be taken
as 4'. To ascertain whether a toe extension is necessary to permit a mini
mum spacing of 3' between the piles adjacent to the toe, the value of i from
equation (48) with X' = 6/11.83 = 0.606; e = H; and n = 3, is found to
be 0.073. The required toe extension is thus 0.073 X 11.83 = 0.86 or 10".
The corresponding value of k is ;; — ; — , = 0.37.
From Table 8, F and H are
l+i
respectively 0.14 and 65.0. Applying equation (45)
li = (11.83 + 0.83)0. 14(\/ l +22  1) = 6.75
I2 = 12.66 X 0.14(Vl +441) = 10.2
The pile adjacent to the heel is 4' 0" from the heel, and bearing in mind
the remarks previously made, the other two piles are 8' 6" from heel and
1' 2" from toe respectively.
h = 25'. Here B = 47.3. With an assumed number of piles, 4 to a
row, the required spacing between rows is found to be 3' 6" To get the
toe extension, X' = 6/14.42 = 0.414. Accordingly i = 0.16 and the toe
extension is 0.16 X 14.42 = 2' 4" For simplicity make this 2' 0" A; is
0.14 +0.33
then :r7. = 0.41 and F and ff are 0.43 and 10.00 respectively.
1.14
From (45)
li = (14.42 + 2.0) X 0.43 X 0.87 = 6.11
I2 = 16.42 X 0.43 X 1.45 = 10.2
Is = 16.42 X 0.43 X 1.92 = 13.5
Coarse Sand and Clay ^x"\::
Coarse Sand I^ty;^
Fine Sand — t^y^. .
Piles  A4^.'"'''
men A IS over tff,Sfep Base t
as shown.
ffei'nforce Base when A is
overZft.
h
d
b
At
A?
A3
A4
is
E5"
9'3'
16'
20
ZS"
ni(f
16"
■10'
25
30'
145
06"
2fi"
ZQ~
30
3^4"
ir0'
14
4'0'
30
Fig. 32.
The pile adjacent to the heel is 4' 0" from the heel; the next is 8' 0"; the
third 12 0" and the face pile is ) ' 6" from the toe, this spacing closely ap
proximating the centers of the several pressure trapezoids.
h, = 30'. R = 65.8. With 5 piles in a row, the required spacing between
rows is found to be 3'. e is 0.353 and i = 0.2. The toe extension may be
taken as 3' 0" The value of k is 0.42 and F and H are 0.54 and 7.11 respec
tively. Then li = 6.0, I2 = 10.4; I3 = 13.9; U = 17.2. The piles are
spaced 3' 0"; 8' 0"; 12' 0"; and 15' 6" from the heel and the face pile 1' 6"
from the toe.
DESIGN OF GRAVITY WALLS
71
Figs. 32 and 33 show the wall proper and its foundations. It is under
stood, of course, that in preparing actual plans for construction that the
plans will cover a much closer variation in the heights.
3. A wall of "quaker" section, 25 feet high is to rest upon a rock bottom.
A surcharge of 500 pounds per square foot extends to the back of the wall.
It will be permissible to let the resultant intersect at the outer J^ point.
Any tension developed in the wall because of this location of the resultant
must be carried by steel reinforcement.
M 4' >j< 4' 4
V
Hi
%
"ft
l_.
f ' ^' 't
h=15'
1
.i__l
h=eo'
4^
h=25'
n=30
Fig. 33.— Pile layout.
In order to effect a direct design of a wall of this section, when the position
of this resultant is at the outer quarter point, it will be necessary to proceed
as in the present chapter. Referring to Fig. 30 and equation (62) with
M = 0,
_ h 1 + 3p + 3p'
1 +2p
N
From the quarterpoint to I> is
and G from (69) is
(1 + p)hN
4
mhHl + 2p)Ar
2
The lever arm of G about the quarterpoint is then
h 1 + 3p + 3p' J. hN _ hN 1 + 3p + 6p^
And the moment of G about this point becomes
TO ^ (1 + 3p + 6p2)
The horizontal and vertical components of the thrust are respectively from
C71 72)
gfeHl + 2c) ghHX + 2c)N
6 ' 2
72
RETAINING WALLS
The lever arm of the horizontal component is simply Bh and that of the
vertical component is
hN
y^a + v)hN  BhN = ^ [3(1 +p)  4B]
The overturning moment due to the thrust is
^[3(1+,
S(l + 2c)h
^+2c) ^^ _ ghHl+2c)N _ hN ^^^^ _^ ^^ _ ^^j
2^ AB  3N'[3(1 + p) m
Equating the stability and overturning moments
miVHl + 3p + 6p2) = g{l + 2c) {4B,  3M3(1 + p)  4B]}
and replacing, as before g/m by s
4s(l + 3c)
1
6pm^ + 3pN^ + N^
9s(l + 2c)m  9s(l + 2c)pN' +
4s(l + 3c)Ar2
" 6pW2 + ZpNI +J=0
where I = Nil + 3s(l + 2c)]
/ = N^[l + s(5 + 6c)]  4s(l + 3c) /3
Solving the quadratic
pN = }{2{VQP  24/  3/}
Table 15
0.1
0.2
0.3
0.4
.31
.23
.16
.08
.39
.41
.43
.46
.48
.38
.30
.21
.11
,2
.49
.48
.50
.51
.51
.46
.35
.25
.14
,4
.57
.56
.55
.55
.54
To establish a table (see Table 15) take the ratio s at its usual value %.
To apply the results of the above to the problem at hand note that c =
5/25 = 0.2. Let the coping width be placed at 2 feet. From the variation
of the top and base ratios as seen in the table the base width may be taken
as 0.5 X 25 or 12.5 feet.
To determine the character of the stresses in the wall it becomes neces
sary to locate the line of resultant pressures, or thrusts in the wall. This
is best done graphically. The wall is divided up into sections five feet high.
The weight and thrust upon each section is determined as shown in Pig. 34.
The points of application of each of these forces are found as follows; the
center of gravity of the masonry trapezoids is taken from equation (66)
and table 11, where q = hUiN/S, or, since N = 0.42 and h is constant and
equal to 5 for each section,
3 = 0.7 [/a
DESIGN OF GRAVITY WALLS
73
For the five sections starting from the top the ratios of the upper to lower
base (m) are respectively 0.50; 0.66; 0.74; 0.79 and 0.83 and the correspond
ing values of q are then 1.64; 2.60; 3.60; 4.45 and 5.7. The weights of these
sections are respectively 2.3; 3.8; 5.4; 7.0 and 8.6. The centers of gravity
of the thrust triangles are found most easily from table 3, using the proper
value of»c. Since the surcharge is 5 feet, the respective values of c to be
used in determining the values of B to locate the point of application of the
thrusts are 1; 2; 3; 4 and 5 and the point of application above the base of
Fig. 34.
each trapezoid is 2.2; 2.35; 2.38; 2.41 and 2.42. For the sake of simplicity
and to reduce the number of lines to be drawn the resultant of each of these
two forces will be used. To determine the line of thrusts it is most easy to
apply the principles of the funicular polygon. The load polygon, at the
right of the figure is first drawn. The direction of each of the resultants
is found to be the same and parallel to the total resultant at the base of the
wall. The pole of the polygon is taken at convenience and the rays are
drawn to the individual resultants. The
funicular polygon is drawn in the usual
manner and the location of the resultant
thrust upon each section is determined by
the intersection of the corresponding ray
with ray 1, extended when necessary. A
line through this intersection parallel to
the direction of the resultant shown in the
load polygon determines this location of the
resultant thrust. The vertical components
of the resultant pressure upon the base of each section is scaled from the
load polygon.
Whenever the point of application of the resultant thrust lies within the
outer third there is tension developed at the rear of the wall and it is neces
sary to determine this amount and supply sufficient steel rod reinforcement
to take care of this tension, it being assumed that the wall shall take no
tension whatsoever. From an inspection of the figure it is seen that above
Fig. 35.
 Amount of tension
in wall.
74
RETAINING WALLS
the line a the resultant pressure lies within the middle third and there is
consequently no tension in the concrete above this point.
From Fig. 35, the steel area necessary to take the tensile stresses developed
is that required by the shaded portion. The area of this portion is x Si/2.
From (41),
w \ 3k
and Si from (40) is
3 1  2A;
27?
Si = — (1 — 3A;), disregarding the negative sign.
The area is then
ii; (1  3kY
3 1  2fc ^ ^
where
y 1 (1  3kY
3 \ 2k
Table 16 gives a list of the values of V for several values of fc less than J^.
Table 16
k
V
.33
.00
.30
.01
.25
.04
.20
.09
.15
.14
.10
.20
.05
.27
The values of R as determined from the load polygon for each of the sec
tions 6, c, d and e are, respectively 10.3; 19.2; 31 and 45.5. The correspond
ing values of k (by scaling) are %o; i%i; 1^2 and i^o (Note that
this last value of k affords a check upon the algebraic method of obtaining
the dimensions of the wall; having assumed the location of the resultant at
the outer quarter point) or 0.26; 0.25; 0.23 and 0.25 for which the values
of V are 0.04, 0.04 ; 0.06 and 0.04. The total area of the sections, or rather
the total tension that must be taken by the steel are respectively 410 77o'
1860 and 1820. Assuming that the steel rods can take 16,000 pounds per
square inch, a % inch square rod every 12' will afford sufficient section to
take the maximum stress. Since it is not necessary to have this amount
of metal to the plane 6, the rods will be spaced at 12" centers to the plane c
and at 24" centers to the plane a. The rods will be placed 3" from the back
of the wall. Figure 36 shows the final wall section.
4. A dry rubble wall, 35 feet high with front face battered oneinch to
the foot and rear face battered 4^ inches to the foot weighs 125 pounds
per cubic foot. The earth surface is horizontal and is subject to a live load
of 500 pounds per square foot. The soil pressure must not exceed 3 tons
per square foot. Determine the proper wall and footing dimensions
DESIGN OF GRAVITY WALLS
75
For this problem M = ]ri2 and N = %. Referring to (74), s
i''%25 = 0.8 and c = 5^s = H
/ = %2 + 2 X 0.8(1 + M)H = 1.02
H
J. / J_ 3\ _ OS
12 \12 '''8/ 3
1
exf^xf
d X g2 + ^
U * A 12 ^ 8 8^
1 =
0.15
The quadratic now becomes, after putting p(ikf + i^) = a;
x2 + 1.02a;  0.15 =0
From which
0.13
The base ratio is 0.13 {■ {M + N) = 0.59. The top and base width of
the wall are then 0.13 X 35 = 4 feet 6 inches and 0.59 X 35 = 20 feet.
Note that for a wall of concrete or rubble masonry weighing 150 pounds
per cubic foot the top and base ratios for the same conditions as the wall
in the problem are, from table 12, 0.07 and 0.53 or the widths become 2'6"
and 18'6" respectively. The area of this latter wall is 85 per cent, of the
area required of the dry rubble wall. That is, 15 per cent, more area is
required when the unit weight of the masonry is decreased 15 per cent. — a
result quite obviously expected.
The vertical component of the thrust is from (72)
(1 + 2c)fe'
N
and in the variables of this problem
. .z'e'
100 X 1.286 X 352 X 0.375/2 = 29.5
The weight of the wall is
35 X 125 X ^'^ ^ ^° =53.5 kips.
The total vertical component of the resultant pressure upon the base is
83 kips. The permissible soil pressure intensity is 6 kips per square foot.
From (54) r = wSi/2R = 0.723 and from Table
9 with this value of r, the necessary value of i
is 0.11. The toe extension is 0.11 X 20 = 2'3".
As indicated on page 61 the depth of footing
will be 4'6" The complete section of the wall
is shown in Fig. 37. In conformity with the
usual practice the coping is made of concrete
and carried back 2'6"
6. A rectangular wall is to line a rock cut
twenty feet high and may be subjected to
hydrostatic pressure up to onehaK of the full
water pressure. Determine the necessary wall
thickness. To avoid the necessity of placing
steel in the wall the point of application of the resultant should lie at the
outer third point. .
Onehalf fluid pressure is 31 pounds per cubic foot. For a wall witn
vertical back the lateral earth pressure has an intensity of H of the vertical
Fig. 37
■20 ■■■■^
Dry rubble wall.
76
RETAINING WALLS
Fig. 38. — Footing
for uniformly distrib
uted base load.
and with earth at 100 pounds per cubic foot (the usual value) this intensity
is 33 pounds per cubic foot. The problem is then merely to find a wall
satisfying an earth pressure thrust as given in (14) with c = and K = }i.
From Table 12 with N = M = and c = 0, the required ratio of base to
height is 0.47. The necessary thickness of the wf>ll for the conditions of
the problem is 9'6".
6. A wall, whose resultant brings a vertical component of 35 kips per
linear foot of wall located at the outer third point must have a uniform dis
tribution of loading. The base of the wall proper is 12 feet wide. Design
the foundation.
For a uniform distribution the resultant must be at the center of the foot
ing. Since, under the conditions of the problem the location of the point
of application of the resultant is 8 feet away from the
heel, the footing must be 16 feet wide, necessitating
a fourfoot toe extension. The uniform intensity of
pressure is then 3^g or 2.2 kips per linear foot.
The shear at the cantilever support, see Fig. 38, is
4 X 2.2 = 8.8 kips. Since the usual depth of footing
is four feet, to bring the base of the wall below the
fiost line, the step will be made 4 feet high as shown
in figure. The unit shear is 8800/(48 X 12) = 15 pounds per square inch.
The cantilever moment at the same point is 8800 X 24 = 211,000 inch
pounds. The section modulus is hd'/6 = 8600. The tension at the
lower edge of the base is then 211,000/8600 or 24 pounds per square
inch. Clearly no reinforcement is necessary. For the economy of the
material the step will be made in two sections oi like dimensions. The
shear is now 4400/(24 X 12) =15 pounds per square inch and the moment
is 220 X 24 = 53,000. The section modulus is 12 X 24 V6 = 1160. The
unit tension is 63,000/1150 = 46 pounds per square inch. The safe value
is slightly less than this (40 pounds per square inch) but this variation
from the safe stress is a permissible one and no reinforcement will be added.
7. In the wall of problem 3 an opening is to be placed as shown in Fig. 39.
Determine whether it is necessary to rein
force the section of the wall to make it
span safely the opening.
The resultant load per lineal foot of the
wall was found to be 46.5 kips per foot.
The span in the clear is 20 feet.
The wall is on a rock footing, so that set
tlement is improbable and it seems reason
ably safe to take the wall as a fixed beam,
with moment wl'/12 at the support. How
ever, since the wall may crack near the supports for some reason unfore
seen, it is better to investigate the stresses at the center of the span on
the assumption that the beam is a simple one, and to make provision for
stresses at the support in accordance with the assumption of a fixed
beam. As a simple beam the moment is 45.5 X 400/8 = 2275 kip feet.
As a fixed beam the moment is 45.5 X 400/12 = 1520 kip feet. The
total moment is then 45.5 X 400/12 = 1520 kip feet. The shear is 455
10
Fig. 39.
DESIGN OF GRAVITY WALLS
77
Anchor Fods
kips. The area of the wall is 26,100 square inches giving a unit shear o
455,000/26,100 = 17 pounds per square inch.
The apex of the section (produced) is about 5 feet above the top of the
wall. Analogous to the location of the center of gravity of the thrust
triangle the center of gravity of the beam section is located a distance Bh
above the base, where with c = 0.2, B =0.38 from Table 3 and the center
of gravity of the section is located 0.38 X 25 or 9.5 feet above the base.
From the "Carnegie Handbook" (p. 137) the moment of inertia of the
section about its center of gravity axis is given by an expression
J ^ d'(b' + 4bbi + 61")
36(6 + 61)
where d is the depth corresponding to h here and 6 and 61 are respectively
the lower and upper bases.
Using foot* units, the moment of inertia of the given section is
/ = 25^(2^ + 4 X 12.5 X 2 + 12.5')
36(2 + 12.5) ~ '**""■
To the extreme fibre in tension at the center of the span, the distance
is 9.5 feet, and the section modulus becomes 7800/9.5 or 820.
The unit tension per square foot is then
2,275,000/820 = 2780 or 19 pounds per
square inch. No reinforcement is then
necessary. Over the supports the maximum
tension occurs at the top of the wall. The
distance of the extreme fibre is now 25 —
9.5 = 15.5 and the coiresponding section
modulus is 7800/15.5 = 500. The unit
tension per square foot is 1,520,000/500 =
3040 and the unit tension per square inch
is 21 pounds.
While no steel is necessary theoretically, a
prudent engineer may specify light reinforcement over the supports, at the
top of the wall and along the bottom of the wall from support to support
(see Fig. 39;.
Some Examples in Recent Practice
1. Wall, Reinforced on Bottom on Account of Threatened Settlement,
Engineering Record, Vol. 64, p. 715.
2. Wall Across Marsh, on Piles, Engineering Record, Vol. 61, p. 242.
3. Wall on Piles, Engineering Record, Vol. 66, p. 132.
4. Heavy Gravity Section, Railway Improvement, Engineering Record,
Vol. 66, p. 720.
5. Wall, 33 Feet High, on Piles, Journal Western Society of Eng., Vol.
16 (1911), p. 970.
Walls to Meet Special Conditions
1. Retaining Wall as Beam over Arch, Engineering Record, Vol. 64, p. 715.
2. Raising Existing Wall (see Fig. 40), Journal W. S. E., Vol. 16, p. 970.
Original
Wall
Fig.
40. — Reconstruction of
gravity wall.
78 RETAINING WALLS
Section avoided the necessity of deep excavation, with consequent heavy
shoring of adjacent tracks. The abutting piivate property made it impos
sible to place face forms for a concrete wall, and a rubble masonry wall
was built instead, backed by concrete. The author adds an interesting
note: "It has occurred to the writer, that there is one feature of this type
of wall, that might frequently be employed as a measure of economy. That
is the saving in excavation and masonry effected by setting the foundation
of the heel higher than the foundation of the toe. There are usually but
two reasons for carrying the foundations of a retaining wall lower than the
surface of the ground. The first is to reach a material that will sustain a
greater pressure and the second, to get the foundation below the action of
frost. The first is usually only necessary at the toe of the wall, for almost
any good soil will sustain the heel pressure. The second, also, is only neces
sary under the toe for the heel is protected from frost by the embankment."
CHAPTER III
DESIGN OF REINFORCED CONCRETE WALLS
General Principles. — Reinforced concrete retaining walls form
a class of walls in which the weight of the earth sustained is the
principal force in the stability moment. Typical sections of this
class of wall are shown in Fig. 41. The same fundamental prin
ciples governing the general outlines of the gravity wall, as given
in the preceding chapter, likewise govern the outlines of this type
of wall and the same criteria against impending failure must be
satisfied. The actual section of the wall, once the forces upon
it are known, is determined from the principles of design of
reinforced concrete, a brief outline of which principles is given
in this chapter.
As in the case of gravity walls, the stress system, soil pressures
and other wall functions are known only when the final section
'iT Cantilever
1
ilr
3
.JilJil.nl'
"t'Canti lever Courrterfbrt
Fig. 41. — Typical reinforced concrete sections.
of wall is known. This, of course, necessitates a process of
trial and error until a wall section has been found satisfying
most economically all the necessary requirements of the data at
hand. On the other hand, assuming the standard type of loading
as shown in Fig. 5 and using the standard thrust equation as
given in (14), and adding a few approximate conditions, a ten
tative section may be chosen from appropriate tables, varying
but little from the final section of wall.
79
Ihch
' T :
iw
Bhi ;
i y
:3
Lj
80 RETAINING WALLS
Preliminary Section. — The masonry composing the wall proper
of a reinforced concrete section plays but a minor role in control
ling the final wall section. The difference in its weight and the
weight of the earth retained may thus be ignored and a skeleton
section of wall treated as shown in Fig. 42. The thickness of
the vertical arm of the wall is that demanded by the stresses
existing within it (for a certain minimum thickness because of
construction limitations, see the following pages) and whatever
batter is given to the back of the arm is that necessary to take
care of the increasing moments and shears in going toward the
base of the wall. This is comparatively a small batter, and for
a tentative design may be ignored. The
back of the wall is then taken vertical and
the thrust upon it is assumed to have a
horizontal direction. The value of the
earth pressure coefficient J is, for this
condition }i (see Table 1).
The required outline of the wall is satis
factorily determined when the ratio be
tween the width of base and height of wall
Fig. 42.— Skeleton wall, is known. This ratio is denoted in the
following work by k. Controlling the de
termination of this ratio are the location of the point of
application of the resultant pressure, the toe extension, if any
is assumed, the maximum permissible intensity of pressure upon
the soil at the toe and the factor of safety. The value of the
determination of this factor has been discussed on page 57.
The approximate assumption as to a skeleton outline of wall
in addition to the adoption of the standard forms of loading
and thrust makes it possible to determine directly the value of
the ratio k depending upon the various functions enumerated
above. While this section is not to be taken as the final one,
it is sufficiently correct a section upon which to base estimates
of cost and to determine the limitations of the various types of
the walls to the peculiar conditions at hand.
Based upon the above assumptions the following relations
between the various criteria affecting' the wall section are found.
Refer to Fig. 42. This is known as the "T" type cantilever
wall and is together with its modified "L" shape wall, the type
of most frequent occurrence. The thrust T is found from equation
(14) and is located at a distance Bh above the base, where B has
REINFORCED CONCRETE WALLS 81
been defined by equation (12) and may be found from Table 3.
The moment of the thrust about the toe P is then
TBh
and if these quantities are replaced by their values as taken from
the equation mentioned, the thrust moment is
„ _ 1 + 2c 1 l + 3c , 3
= J Jg{l + Zc)¥ (80)
as before g is the unit weight of the retained earth, and is
ordinarily taken as 100 pounds per cubic foot.
The stability moment of the wall, Ms is
Ms = Gy (81)
Since, as per the adopted approximation, the difference in weight
between the masonry comprising the wall and the weight of the
retained fill is ignored, the value of G is
G= gh (1 + c) w (1  i) (82)
i is the ratio between the length of the toe extension and the
entire width of the base. The value of the lever arm is
y = ^ L(Lpl = iL+^ (83)
Let the ratio between the width of base, w, and the height of
wall, h, be denoted by k. If the factor of safety of the wall is
taken to mean the ratio between stability moment and the
overturning moment, and is denoted by n,
Ms = nMo (84)
From (81), (82) and (83)
Ms=lgk'{l + c)il i^)h' (85)
From equations 80, 84, 85,
^J(l + 3c)/i= = 2 gk'a + c)(l  i'W
and finally
4.
J(l + 3c)w
3(1 + c)(l  P)
(86)
82 RETAINING WALLS
expressing the ratio between the width of the base and the height
of the wall in terms of factor of safety assumed and the width of
toe extension. The surcharge ratio c and the earth pressure
coefficient J are, for the purposes of the problem, independent
of the functions of the wall outlines.
To establish the base ratio k in terms of the location of the
point of application of the resultant and the toe extension (and
these are the two functions generally known, or easily found
in advance), take moments about the point of apphcation of the
resultant, if „, the thrust moment remains the same as before
and is given by equation (80). The newy. stability moment M,
is related to that found in equation (81) in the ratio of the respec
tive lever arms of the force G, or if M'^ denotes the new stability
moment
M'JM. = p_ = 1  if, (87)
2
Taking moments about the point 0, M's = M^ and from (.87) and
since Ms = nM„
A relation between the factor of safety, the location of the point
of application of the resultant and the toe extension ratio.
Inserting this value of n in (86)
i= / J(l + 3c)
A/3(l + c)(li)(14
(89)
/3(1 + c)(l i)(l + i 2e)
which may be written
/ J(r+3^) \ 1 ,qn^
''Vsd+c) \Jl^eY  ii  ey ^^^^
Inspecting this last expression, it is seen that A; is a minimum
when the factor {i — e) in the denominator vanishes, or for i = e.
For a given location of the resultant pressure the most economical
width of base is had when the vertical arm is placed over the assumed
point of application of the resultant pressure.
When the back of the wall is vertical, as is assumed in the
present analysis, J has the value }i, which should be inserted
in expressions (86) and (90). Again, introducing this value of
REINFORCED CONCRETE WALLS 83
J and also, the economical criterion established above (90)
becomes
k = I j^^\/(l+3c)/a + c) (91)
The application of these equations to specific problems is
shown at the end of the chapter.
Distribution of Base Pressures. — The manner of the distribu
tion of pressure on the base is again controlled by the type of soil
upon which the wall will rest, with an advantage over the gravity
type of wall in that, any tension developed in the wall may be
taken care of by proper reinforcement. Continuing the approxi
mations given above, further guidance may be had in shaping
the wall to meet the anticipated soil conditions.
The total load upon the base of the wall is G. From (39) of
Chapter 11 and from (82) above
S, = ?^ (2  3e) = 2gMl + c)(li) _ 3
Place H = h {1 + c); that is, H is the total depth of fill plus the
depth of surcharge. Solve the equation for e, taking the unit
weight g of the earth as 100 pounds per cubic foot and expressing
both this weight and the soil pressure intensity Si in tons. There
is
2 lOSi ,.„,
' = 3  3^0."^) ^^^^
When the maximum soil pressure intensity S is given as well as
the toe extension ratio i, this equation may be used to locate the
point of application of the resultant pressure upon the base.
When this value of e has been found, equation (90) is then ap
plied to find the value of the base width ratio k.
Conversely when the point of apphcation of the resultant is
assigned (and with a foundation known in advance, the location
of the point of application of the resultant is usually indicated)
the toe extension necessary to give this resultant location is
found from
If, in equation (93), i is put equal to e (the economy criterion),
and the resulting equation is solve d for e
84
RETAINING WALLS
Under the above conditions, given H and *Si, the toe extension
ratio i is determined at once. ' The conditions under which the
location of the stem is governed solely by the economy of the
wall have been previously touched upon (see pages 42^4)
and will be discussed in more detail further on. Clearly, if no
limitation is placed upon the location of the vertical arm, it
should be placed where the economy criterion dictates : directly
over the indicated position of the point of application of the
resultant upon the base.
Tables and Their Use. — Tables are readily founded upon the
preceding equations and simplify the necessary calculation of
the wall outlines. From the relation existing between the loca
tion of the point of application of the resultant, the factor of
Table
17. — Values of e
2
3
4
5
.25
,33
.38
.40
.1
.27
.37
.41
.44
.2
.30
.40
.45
.48
.3
.33
.43
.49
.4
.35
.46
.5
.38
, .50
safety and the amount of toe projection, equation (88), Table
No. 17 has been prepared. With a given location of the resultant
and an assigned factor of safety, the required toe projection is
taken from the table. Again, for an assigned location of the
point of application of the resultant and a given toe projection,
the factor of safety may be taken from the same table. For
the criterion of economy i.e. i — e, this relation becomes
1 1 +e
 n=
1 1 — e
e =
(96)
Table
18.
— Values
OF
7c
i =
=
i=M j
i = K
i = K
c
i
(N
CO
■^

w
«
^
■o
"
^, j « j ^,
l"
«
cq
CO
Tf
ic
II
II
II
II
II
II
II
II
II
II
II 1 II 1 II 1 II
II
II
II
II
II
»
^
w
w
w
ID
lU
•u
<»
u
<a 1 (B
1 ^
1 »
»
<»
w
.37
.43
.63
.74
.37
.42
.48
.57
.77
.38
.42
.48
.56
.70
,41
.45
.50
..5fi
,67
.1
.42
.49
.60
.85
.43
.48
.54
.66,
.88
.44
.48
.54
.64
.81
,47
.51
.57
.64
76
.4
.47
.54
.66
.94
.47
.53
.60
.72
.97
.48
.53
.60
,70
,69
.52
.57
.63
,71
.84
REINFORCED CONCRETE WALLS 85
A general table, Table 18 has been prepared, giving the value
of k, as found from equation 90, for a range of values of c, e and i.
The earth pressure constant J, has been taken as 3^.
With the general outlines of the wall approximately established
by aid of the foregoing, it is possible to proceed with the actual
design of the several members composing the reinforced concrete
retaining wall. While it is not the purpose of the preceding
analysis to replace a careful, exact analysis of the wall, its prime
intent is to permit an intelligent selection of a wall without a
tedious process of trial and error. It should be pointed out, that
the approximations consist in ignoring factors which have proven
negligible in controlling the wall dimensions, so that even though
the selection of the wall outlines are finally determined by these
approximations, no serious error has been committed. However,
a careful and painstaking designer will analyze the completed
wall, to see whether the stress system in it checks with the one
first determined.
Theory of the Action of Reinforced Concrete.— The assump
tions in the design of reinforced concrete beams are those of the
ordinary beam theory, namely: the Bernoulli — Euler theory of
flexure. The fundamental premise is that a plane section before
bending, remains a plane section after bending, with the further
assumption that Hooke's Law, i.e. the stress is proportional to
the strain, is true.
Although the brilliant researches of Barre de St. Venant,
have shown that plane sections do not remain plane during bend
ing, the error becomes appreciable when the ratio of depth of
beam to span exceeds onefifth. Since for such ratios, stresses,
other than those induced by bending moment, usually govern
the required reinforcement and depth of beam e.g. the unit shear
and adhesion, these assumptions of plane sections may be taken
as valid, so long as the stresses induced by the bending moment
govern the required depths and amounts of steel reinforcement.
The concrete is assumed to take no tension.
The excellent report of the Special Committee on Concrete
of the A.S.C.E., has set the seal of approval on this mode of
figuring the action of reinforced concrete after most thorough
investigation, both from a theoretical and experimental stand
point, and the engineer may accept this method, with no fear of
beam' failure ensuing, so long as care has been taken of all the
stress criteria.
86
RETAINING WALLS
Under load, the distribution of stress across a section normal
to the axis of the beam is shown in Fig. 43. Adopting the recom
mended nomenclature as suggested in the above report, Es is
the steel modulus, Ee the concrete modulus, and n the ratio of
the steel modulus to the
■^c ^^ concrete modulus. A, and
»«o\ fe/ Ac are the areas of the steel
' i'"* / T and concrete in the section
respectively, fc and /, are,
respectively, the unit con
crete and steel stresses.
Let fio be the displacement
of the section at a and e, that at b. From the assumption that
a plane section remains a plane section after bending, and from
Hooke's Law
Fig. 43. — Theory of reinforced concrete.
kd
e, (1  k)d
/„= CcEo]/, = e,E,As; As,
(97)
pAc = pbd
and, by summation of all the horizontal forces
fc kbd _ . EcBckbd „ , ,
'—jz — = e.E,A„ or — ^ — = e.E.pbd
kecEc
whence
and equating this to (.97)
2pn
= e,E,p
2pn
^ ~k~
k
and finally
Solving this for fc
1 fc
fc2 I 2kpn  2pn = 0.
(98)
(99)
fc = VpV + 2pn 
pn
(100)
which locates the position of the neutral axis, once the ratio of
the two moduU are adopted and the percentage of steel assumed.
It is to be noticed that it is a function of these two quantities only.
The resisting moment of the section may be expressed with
bd^
REINFORCED CONCRETE WALLS 87
either the steel force or the concrete force as the force factor in
the couple. If M„ and M, are the concrete and steel resisting
moments respectively,
A; (l  )
Mo = /c  ^ 2^^  bd^; M, = fAs {^l)d= fsP (l  I)
1 — fc/3 = j and is the effective lever arm of the couple, corre
sponding to the effective depth of homogeneous beams. The
moments may be expressed as
M, = kM^;M, = ksbd^ (101)
where
K = fckj/2; K = fspj (102)
Ordinarily, the most economical section is that one in which
the concrete and the steel are each stressed to their permissible
limits. The percentage of steel to satisfy this condition may be
f Qund as follows :
Since, from the summation of horizontal components of stress
intensities across the right section of the beam, the total concrete
stress must be equal to the total steel stress
A J, = pbdfs = kbdfJ2
from which equality
k = 2p{' (103)
Equating this value of k to that found in equation (100) and
replacing the ratio 2f,/fc by a, and solving the equation for p
^ 2n ^ 2n_
If in the ratio a, the unit stresses are those allowed for the
material at hand, than this value of p proves to be the most
economical one to use.
The above analysis is of course, predicated upon the assump
tion that the section is controlled by the bending moment. Other
stresses may determine the percentage of steel or the depth
of the section. When the percentage of steel is above that neces
sary for the economical steel ratio as given by (104), then the
concrete stress in the section will determine the resisting mo
ment to be used and the section constant is found from fc^, as
88 RETAINING WALLS
defined above. With this value of k^, the proper percentage
of steel is to be taken from Table 19. Again the depth of the
section may be greater than required by the bending moment, and
accordingly the percentage of steel to satisfy the bending moment
will be less than that required by equation (104). The steel
stress will be the governing stress in the section and the section
constant to be used will be k, as defined in equation (102). The
proper percentage is found from Table 19 with this value of k^.
The conditions under which these constants control are best
illustrated by specific problems as given at the end of the chapter.
no
100
90
80
'iSl.S
70
60
e 50
40
30
20
^
\,
\
V
\
\,
V
N
\
Sr
■s
^
!:5bn^
.^
"^
t:;^
—t^'
n<
'
~~
"^
zz
n'li
—
nlO
rl
t
0.OOZ 0.003 0.004 0.005 0.006 0.007
As
j>=5+eel Ro+io'T^
Curve Plate No. 2.
Economical steel percentage.
0.00& 0.009 0.010
To simplify the use of equation (104), Curve Plate No. 2 has
been drawn from which the proper value of p may be taken once
the value of n and of the ratio a are known.
In the report of the Special Concrete Committee, mentioned
above, the foUowing values of n are suggested, depending upon
the ultimate strengths of concrete :
n = 15. Ultimate strength equal to or less than 2200 lbs. per sq. in.
n = 12. Ultimate strength between 2200 and 2900 lbs. per sq. in.
n = 10. Ultimate strength greater than 2900 lbs. per sq. in.
Table No. 19 is a compilation of the values of the several
functions entering into the computation of a concretestee
section It IS noticed that the terms are not carried out to the
usual degree of refinement. In view of the approximation in
REINFORCED CONCRETE WALLS
89
both the theory and in the experimental determination of the
concrete constants, it does not seem good practice to carry the
work out to any greater degree of exactness than shown here.
Table 19.— Reinforced Concbete Constants
,.
= 10
k
n =
i J
12
iik
Pi
,. = 15
p
Ic
3
Hik
Pi
k
;■ y^jk
pi
.002
.18
.94
.09
.002
.20
.93
09
.002
.22
.93
.10
.002
.004
.25
.92
.12
.004
.26
.91
12
.004
.29
.90
.13
.004
.006
.29
.90
.13
.005
.31
.90
14
.005
.34
.89
.15
.005
.008
.33
.89
.14
.007
.35
.88
15
.007
.38
.87
.17
.007
.010
.36
.88
.16
.009
.38
.87
16
.009
.42
.86
.18
.009
.012
.38
.87
.17
.010
.41
.86
18
.010
.45
.85
.19
.010
.014
.40
.87
.17
.012
.44
.85
19
.012
.47
.84
.20
.012
.016
.43
.86
.19
,014
.46
.85
20
.014
.49
.84
.21
.013
.018
.45
.85
.19
.015
.48
.84
20
.015
.51
.83
.21
.015
.020
.47
.85
.20
.f)17
.49
.84
21
.017
.53
.82
.22
.016
.025
.50
.83
.21
.021
.53
.82
22
.020
.57
.81
.23
.020
.030
.53
.82
.22
.025
.56
.81
23
.025
.60
.80
.24
.024
In addition to determining the resisting moment of a section,
it is necessary to find the unit shear and the unit adhesion, each
of which stresses may demand more resisting material than that
required by the moment.
Analagous to a steel or other section of homogeneous material
the shear over any section is assumed distributed over the effec
tive depth {jd) of the section, so that, if s is this unit shear, and
V is the total shear over the section
s = .,
jhd
(105)
The unit adhesion corresponds again, to the horizontal shear,
and since the unit vertical shear is equal to the unit horizontal
shear, the periphery of the steel embedded in the concrete per
unit length must carry the unit horizontal shear (or its equiva
lent, the unit vertical shear.)
If r is the periphery of the rods per unit length, and q is the
permissible adhesion stress,
q = ^ (106)
^ jar
90
RETAINING WALLS
Table 20. — Standard Ultimate Strengths op Aggregates as Suggested
BT THE Special Committee on Concrete A. S. C. E.
Aggregate
1:1:2
1;1M:3 1:2:4 ' 1:2^:5 1:3:6
Granite, trap
3300
3000
2200
800
2800
2500
1800
700
2200
2000
1500
600
1800
1600
1200
500
1400
Gravel, limestone
Soft limestone
1300
1000
Cinders
400
The following are the percentages of the above ultimate stresses that may
be allowed:
Bearing. — Compression applied to surface twice the loaded area, 32.5
per cent.
Axial Compression. — Where the length is not greater than twelve diame
ters, 22.5 per cent.
Compression Extreme Fibre. — 32.5 per cent.
Shear and Diagonal Tension. — Beams, with horizontal bars {i.e., bars
parallel to the longitudinal axis of the beam only) no web reinforcement,
2 per cent. .
Bond. — 4 per cent. In case of wires 2 per cent.
Upon the recommendation of the above Committee, Table 20
was compiled, giving the standard ultimate strengths for the
several combinations of the different aggregates, and then the
percentages of these ultimate loads to be used for the different
type of stresses.
Bending and Anchoring Rods. — Rods are anchored in the con
crete by (1) carrying them beyond the theoretical end, a distance
sufficient to develop, in bond, its tensile stress; (2) hooking the
end of the rod around a rod at right angles to it; (3) threading
the end of the rod and bolting it to a steel washer or other steel
device buried in the concrete (4) making a U turn in the rod.
The first and last methods are the usual ones because of cheap
ness of these details. The second and third are used only where
lack of room makes such details necessary. Bending rods around
another rod, and threading and bolting rods are expensive details
to be avoided as far as possible.
If the unit adhesion is q and /, is the steel stress, then, if L is
the length necessary to carry the rod beyond its theoretical end
4g<L = /,«2 and L = 1^ t. (107)
The value of this fraction varies from 40 to 50 (the unit stresses
taken from Table 20) and the rod is carried passed the theoretical
end, this number of thicknesses.
REINFORCED CONCRETE WALLS 91
If a rod is twisted about another rod then the twist should be
at least one complete turn (360°) and carried beyond about
six inches, not only to satisfy the theoretical requirements, but
to aid the work in the field. In bending a rod care must be taken
that the radius to which the rod is bent is sufficiently large that
the bearing induced on the concrete will be within the allowable
limits. For a rod bent to a circular arc, carrying a tension of T
at either end, the condition is similar to that of a hoop (see any text
on applied mechanics) and the compressive stress upon the concrete
per linear unit of the rod is
C = T/R
Where R is the radius of the bend. If c is the permissible unit
bearing on concrete and/s is the permissible steel unit stress, then
introducing these factors in this last equation
R =^t
c
t is the thickness of the rod. The ratio fjc has a value of about
30 and in the work that follows this proportion will be used in
determining the proper radius to turn the rod.
To get the area of a washer necessary to hold the bar, with A
the area of the washer and c the unit concrete bearing, let d
be the side of the square (if a square washer be used) and with the
same units as before, the total bearing is Ac. Since Ac =
c{d'' r)
d = tVif./c + 1) (108)
With the usual unit stresses, d is about six thicknesses of the bar.
If d is the diameter of a round, washer
With the usual values, the diameter of a round washer should be
about seven and onehalf thicknesses of the bar. '
Vertical Arm. — The vertical arm of a reinforced concrete wallas
shown in Fig. 42 and as tentatively analyzed on pages 80 and 82 is
a cantilever beam, subjected to a horizontal load of T, located at
a point Bh above the base. In the skeleton wall, the basis for the
approximate analysis, h is measured from the bottom of the wall.
In the actual final section, the correct value of h must be used.
92 RETAINING WALLS
namely the height of the vertical wall above the top of the foot
ing. The discrepancy in the assumed and correct h may be
ignored in the tentative selection of the thicknesses of the arm
and footing.
As above shown the cantilever moment in the arm is TBh, and
if T is replaced by its value in (14), and B by its value in (12)
then
M = ijsf(l + 3c)¥ (110)
The value of J is taken as onethird (see page 80). g is
the unit weight of earth and c is the ratio of the surcharge height
to the actual height of wall assumed. The standard type of
loading as shown in Fig. 5 is to be used.
While the shear and the unit adhesion may, and frequently
do, control the depth of beam required, this depth will not vary
much from that required by the bending moment depth and
it is safe in this preliminary analysis to work with the depth
required by the bending moment. The resisting moment has
been given by (101) and equating this to the external moment
given in (110), and solving for d
d = h^^'^^i^ (111)
J may be given the value I'i as above, g is taken at the usual
weight 100 lb. per cubic foot. If the economy criterion of
(104) is used, and if in accordance with general practice a 1:2:4
concrete is specified with the resulting permissible stresses as
given in Table 20, from Curve Plate No. 2 with n = 15, the steel
ratio p is 0.0075. From Table 19, O.bkj is, for this value of p,
0.17 and since /„, in conformity with the other terms of (111) is
to be expressed in units of pounds per square foot, the bending
moment constant h from equation (102) is about 16,000.
With these values equation (111) becomes
d = 0.0185;i^'V(l } 3c) (112)
The depth d necessary to satisfy the bending moment due to
the earth thrust may be closely approximated from this equation
and the same expression may be used to find the required depth
at any point on the cantilever arm, by using the proper values of
c and h.
Table 21
REINFORCED CONCRETE WALLS 93
To determine the depth to satisfy the shear requirements, ap
ply equation (105). V is the thrust T and j may be safely
taken, for the purposes at hand, at %. With the same concrete
constants as assumed above, the shearing value for a simply rein
forced beam is s = 40 pounds per square inch or 5760 pounds per
square foot. The required value of d is
d = r/5040 = Jgh^il + 2c)/10080 = O.Q03dh^(_l + 2c) (113)
Comparing this equation with (112), the shearing stress will
control the required depth of the arm, whenever
the value of d as found from (113) is greater than
that value as found from (112). Solving this
inequality, the shearing stress will determine the
necessary depth when
31(1 + 3c)
"^ (1 + 2cy ^^^^^
This may be termed the "critical" value of h
and Table 21 gives the values of the "critical"
value of h for several values of surcharge ratio c.
Its use is explained in the problems at the end of
the chapter. The above equations suffice to de
termine, approximately, the thickness of the arm
to satisfy the stresses induced by the earth thrusts.
While such thicknesses are fairly accurate (the problems at the
end of the chapter are illustrative of this) it is
better practice to take the wall thus approxi
mately outHned as the tentative section and
design finally by the more exact methods the
required dimensions of the wall.
Footing. — ^The footing, see Fig. 44, is again a
cantilever, with its maximum moment at the 44 _t
foot of the vertical arm B. Its loading is the ing on footing,
net difference between the downward weight of
the retained fill and the upward thrust of the soil pressures. The
soil pressure intensity at B is
Sb = S, + (Si  S2)
Taking moments at B
Mb = Gp/2  >S2pV2  ('
pV6
.0
31
.1
28
.2
25
.3
23
.4
21
.5
19
.6
18
.7
17
.8
16
.9
15
1.0
14
(1
— i)w
w
= (1
t)Si
+ iS,
(115)
?B
 Sdp
76 =
Gp/2
 (2S2
+ Sb)
(116)
94
RETAINING WALLS
From (115)
2^2 + Sb = (I i) S, + (2 + i) S2
and from (39) and (40) of chapter 2
2S2 + & = 6^[e  i{l  2e)]
The expression (116) for the bending moment now becomes
M.^^[l^^[eiil2e)]] (117)
Note that p = w{l — i) and that
G = gh{l + c) (1 — i)w; and w = kh
Using the value of h as found in equation (90), the expression
for the bending moment (117) is finally
Mb = Ig{l + 3c)/iV18 (118)
^ _ {I  i)\l  2{\  i)[e  i{l  2e)\\
where,
\\i 2e
(119)
Comparing the value of this moment as given in equation with
that of the vertical arm, as given in equation (110), it is seen that
the footing moment is / times the arm moment with I varying
from one to onehalf. Table 22 gives a series of values of I.
Table 22
e =
e = 0.4
e = 0.5
I
Q
I
e
I
Q
.0
1.00
.00
1.00
.00
1.00
.00
.1
.96
.03
.95
.05
.90
.10
.2
.88
.11
.85
.14
.80
.20
.3
.76
.19
.72
.25
.70
.30
.33
.72
.22
.69
.28
.60
.33
.4
.64
.27
.62
.34
.60
.40
.5
.50
.33
.50
.43
.50
.50
As before, the shearing stresses and the adhesion stresses must
be found. The compUcated type of loading upon the footing
makes it impossible to find an easily apphed expression for these
stresses and resort must be had to specific problems to illustrate
REINFORCED CONCRETE WALLS
95
the effects of these stresses. Some problems at the end of this
chapter bring out in detail these points.
Toe Extension. — The approximate design of the toe extension
of the footing, if such an extension is used, follows along lines
similar to those of the preceding paragraphs. Referring again
to Fig. 44 with the value of the soil intensities as previously found
Sb is taken the same as in the design of the heel extension. For
the exact analysis, the moments for the heel and the toe are taken
at the intersection of the rear and face planes of the vertical arm
respectively. For the approximate solutions now sought this
refinement is unnecessary and taking moments about B
M'b = Sb i%W/2 + ^' ~ ^" PkW = (Sb + 2Si) ^ (120)
and again replacing the soil intensities and k by their values,
QghU{l + 3c)
6
where
2t"2[2  3e  2i(l  2e)]
M'b =
(121)
Q =
(122)
1 +i  2e
The toe footing moment is thus Q times the arm moment,
with Q varying from zero to onehalf. Table 22 gives a set of
values for Q.
Fig. 45. — Graphical analysis of reinforced concrete wall.
It is again necessary to emphasize the fact that the shearing
and adhesion stresses must be ascertained.
The dimensions of the wall are thus approximately determined,
and with the outlines of the wall previously found, it is possible to
proceed with the definite final design. Laying out the wall in
96
RETAINING WALLS
accordance with these dimensions, the thrust may be found by
the graphical methods or may, once more, be taken with J =
onethird as urged in Chapter I and then combined with the ver
tical weight of the earth on the projection of the back of the arm
(if the arm be battered from the minimum practical width at the
top to the required width at the base). With the thrust deter
mined, the location of the resultant and the soil pressure intensi
ties are found and checked with the location and intensities of
pressure assumed originally. This is best found graphically as
shown in Fig. 45, where the properties of the funicular polygon
are utilized. Several problems at the end of this chapter develop
in greater detail the methods sketched here.
Counterfort Walls. — 'A study of the expressions determining
the thicknesses of the members of the cantilever walls discussed
in the preceding sections, will show, that as the walls increase
in height, the required thicknesses of these members become
very largej* To reduce the sizes of the arm and of the footing,
~suppbr"ting walls are introduced between these members, termed
loosely, counterforts. See Fig. 46. These serve a function similar
to that performed by the gusset
plate on a through girder, an
choring the wall and base slab
to each other.  —
This combination of counter
fort, wall and footing, forms a
structure quite difficult to
analyze exactly and, generally,
no such exact analysis is at
tempted. The usual modes of
treating the wall and base slabs
of the counterfort wall are as
follows :
(a) The wall and the footing
slabs are treated as composed of a series of independent
longitudinal strips, freely supported at the ends, i.e., at the
counterforts. The bending moment is then WL/S. W is the
total weight acting upon the strip in question.
(b) The wall and footing are treated in strips as above, but the
supports are taken as fixed at the counterforts. Although, ex
actly speaking, for this condition, the moment at the support is
WL/12, and that at the center of the beam is TFL/24, the moment
, Jx(i+Cx)g
Fig. 46. — Stresaes in a counter
forted wall.
REINFORCED CONCRETE WALLS 97
is assumed alike at the center and at the support and of value
WL/12.
Method (b) is the one generally used in the design of the slabs
forming the counterfort wall and will be used in the present text.
The design of the counterfort itself is a matter of much con
troversy and practice is far from uniform here. ^ It may be taken
as a tension brace, simply anchoring, by means of the rods con
tained in it, the base slab and the wall slab to each other, the
concrete merely acting as a protection to the steel; as a cantilever
beam, anchored at the base and receiving its load from the wall
slab, or as the stem of a "T" beam. In the following work the
counterfort will be treated as a cantilever beam. Prof. Cain has
made an exact analysis of a beam of this wedge shape (see his
"Earth Pressures," etc.) but the theory of retaining walls and of
earth pressures does not seem to justify such refinements of
design.
Not only are all of the methods of stress computation above
discussed approximate, but it is difficult to make an estimate as to
their degree of exactness. If the slabs are designed as outlined
under (o) and (6) the relieving action of the portion of the slab
adjacent to the strip under question is ignored. That is, no
account is taken of the plate action that may exist in the slab.
Toward the junction of the base and the arm, the two members
tend to mutually stay each other, reducing the possible deflec
tion and thus the resulting stress. It is clear that there is con
siderable latitude permissible in making stress assjimptions and
here again, simplicity of design should dictate the formulas to
be used rather than an intricate analysis of questionable accuracy.
While attention has been paid only to bending moments in
discussing stresses, it is understood that the other stresses, such
as shear and adhesion are likewise to be ascertained, and, in
fact, it will be seen that these latter stresses may more often con
trol the required dimensions than the bending moment stress.
Face Slab. — The same assumptions as to standard character
of loading, of amount of earth thrust etc. will obtain here as have
obtained in the former work on the design of the walls. The
intensity of earth pressure upon any horizontal strip (see Fig. 46)
at a depth x below the top of the wall is
Jxil + c,)g (123)
'See E. Godfrey, Trans. A.S.C.E., "Vol. Ixx, p. 57, and accompanying
discussion.
7
98 RETAINING WALLS
where J is to be taken at its usual value 3^; d is the ratio of the
surcharge height h' to x and g is the unit weight of the earth.
If m is the counterfort spacing, and if the moment is as above
defined WL/12, then
M  a;(l + OsmVSe (124)
Placing X = vh, so that v is the ratio between the distance from
the top of the wall to the point in question and the total height of
the wall, then Cx — h' /x = c/v; where c is the standard ratio
between the surcharge height h' and the total height h. The mo
ment may now be placed
M = ^0 + v) gmySQ (125)
As before (see page 92), the resisting moment of the slab, for a
condition of balanced reinforcement may be placed equal to kcd^.
Equating this to the external moment (125), and solving for d
Ordinarily this depth is less than a certain minimum necessary
for good construction and a minimum depth of from 12 to 18
inches is usually specified to make the working conditions fa
vorable for good concrete work (see later sections).
The shear (see equation 123), is found to be
V = ^J X m{l + Cx)g = g mh{c + v)g
From (105) the necessary value of d is
d = ^M^) (127)
Since the beams are comparatively short (the counterforts are
generally spaced about 8 to 10 feet apart) it is quite likely that
the unit adhesion stress will be high, and may, in fact, control
the thickness of the concrete and the spacing of the reinforce
ment.
The use of the preceding formulas, and the relative value of
the several stresses and their effect upon the dimensions of the
member are illustrated in some problems at the end of the chapter.
Footing.— The loading upon the base slab is the net difference
between the downward weight of the retained fill and the upward
REINFORCED CONCRETE WALLS 99
soil pressure. (In this work the weight of the slab itself is neg
lected, since Its downward weight is practically reflected in the
upward soil pressure intensity caused by this weight.) The load
distribution upon the slab is quite problematical, and the net
difference as stated above does not exactly give the actual loads.
The distribution of soil pressures is of course conditioned upon
the deflection of the base slab,i so that at those portions, where
there is a maximum stiffness of base there will be less pressure
(other things being equal) . Accordingly for the counterfort walls,
the maximum deflection of the base slab will occur midway be
tween the counterforts and toward the heel and the minimum at
the counterforts and toward the junction of the arm and footing
slabs. These niceties of pressure distribution will not enter into
the following treatment of the design of the base slab but they
should be borne in mind, and it is permissible to let the true
state of affairs color, more or less, the computations involved in
the design of this slab. Essentially, however, the following
analysis, gives a simple method of design, with probably a stronger
section of base than is actually required, but not enough stronger
to justify a highly refined analysis. It may again be emphasized,
that a little excess section may be sacrificed to simplicity of
analysis.
So long as there is not a uniform distribution of soil pressure,
the minimum upward pressure occurs at the heel. Since the
downward load is, to all intents, uniformly distributed, the maxi
mum net intensity of load occurs at the heel. Again, the
maximum soil pressure occurs at the toe, and since its intensity
will be larger than the downward intensity of pressure, there will
likely be a net difference of pressure upon the base of consider
able magnitude and directed in an opposite direction to the net
pressure at the heel. This may be brought out algebraically as
follows (see Fig. 46) :
The unit downward load is gh{l + c)
where the variables have the usual meaning as defined in the
preceding pages. The soil pressure intensity, Sx, at a point x
from the heel is, from (39) after makng the proper substitutiens,
S. = 2ghil + c)(l  i) [Se  1 + 3(1  2e) ^] (128)
^A discussion of this point is given in Cain "Earth Pressures, Walls
and Bins," p. 157.
100 RETAINING WALLS
making the net downward load at the point x, Px
Px = gh{l + c)Jx (129)
where
/, = 1  2(1  z)[3e  1 + 3(1  2e)] (130)
The maximum net downward pressure, at the heel, Pi, is, with
X =
Pi = ffA (1 + c)/i (131)
where
Ji = 1  2(1  i)(3e  1) (132)
and the maximum upward net pressure, at the toe, P^, with x = w
' P2 = ghil + c) J2 (133)
with
Ji = 1  2(1  z)(2  3e) (134)
When the point of application of the resultant falls within
the outer third of the base, the soil intensity at the end of the
heel is zero and
Pi = ghil + c) . . (135)
The above equations determine the loads to be used in de
signing the longitudinal strips of the base slab and with m the
distance between the counterforts, the moment is
M = PtoV12 (136)
where the proper value of P from the preceding equations is to
be used. The shear is P/2.
Similarly to the design of the face slab, the required depth of
the slab, due to the bending moraent is
d = my/j^gh{l + c)/]2 (137)
A theoretical comparison, based upon the bending moment
requirements ofily, may be had between the depths of the base and
of the arm slabs. The depth of the face slab is governed by the
thickness required at the base of the arm; that of the base slab
by the thickness required by the maximum value of J. When the
resultant falls at the outer third point, or within the outer third
the value of J^ is 1 . D enoting the respective required thicknesses
of face and base slabs by rf„ and 4 respectively, comparing equa
tions (126) and (137), after placing v = 1, there is
d, = 4V(l73/5 (138)
REINFORCED CONCRETE WALLS 101
and with J^ = 1, this relation becomes
d, = 0.584 ■ (139)
This relation, however, is more of academic than practical
interest, since it will be found that the thicknesses of these slabs
are controlled by factors other than the bending moments.
Later on this relation will serve a fairly useful purpose in obtain
ing relative economy of the several wall types, for which purpose
it is of some practical application.
Counterfort. — The counterfort is designed as a simple canti
lever beam, with effective depth e as shown in Fig. 46. For the
reasons given on the preceding pages no other refinement is de
sirable in treating this member. To anchor the slabs to the coun
terfort, rods are placed as shown in Fig. 51 of a section sufficient
to hold the stresses induced by the loadings. For the face slab
the necessary rod area to hold a strip of face bounded by the
two horizontal lines xi and Xa from the top of the wall, with m
the distance between the counterforts, and taking the earth pres
sure coefficient J as }^i is (see Fig. 46).
A. = mg (^x + 2h' + x.) {x,  x,) ^^^^^
fs is the permissible unit steel stress, and g the weight of the earth
per cubic foot. Using a value of 16,000 pounds per square inch
for fs and 100 pounds per cubic foot for g, this last equation takes
the form
yoU
To anchor the base slab to the counterfort it is noticed (see Fig.
46) that beyond the point A the slab and the counterfort are in
compression. It is therefore necessary to provide anchorage for
the portion of the base between A and B, only. The point A is
located as follows: The soil intensity at A is found from (128).
At the point A this intensity is equal to the downward intensity
gh{l + c). Forming this equality, and solving for x
1  2(1  t)(3e  1)
^ ~ '^ 6(1  i){l  2e)
or
X = Dw (142)
102
RETAINING WALLS
To facilitate the computation of D, Table 23 has been pre
pared covering a range of values of i and e. The total rod area
necessary to hold the portion of the slab AB to the counterfort
is then that area required to
Table 23.— Values op "D" hold the net difference in the
upward and downward load
ings between these two points.
Two conditions exist (see Fig.
47) : when the point of appli
cation of the resultant force
lies within the outer third,
and when it lies without the
oirter third. For the former
case, the point of zero soil
intensity has been given by equation (41) of Chapter 2, and the
net difference in loading is
1 1  3e
(143)
^\
.1
.2
.3
.4
.5
.50
.52
.54
.57
.61
.67
.1
.50
.52
.55
.59
.63
.71
.2
.50
.53
.57
.62
.68
.78
.3
.50
.55
.60
.68
.77
.92
.33
.50
.56
.62
.72
.83
1.00
.4
.50
.59
.71
.86
.95
1.00
D +
3 1  2e
T = mghw (1 + c) ~
which may be written, simply
T = mghw(l + c)E,
where E represents the fraction in the above equation.
(144)
< Dw ?
^0m^
<.
.. Dw —
— >
A
1
Sa
IP^
Fig. 47
Again, when the point of application of the resultant pressure
is without the outer third, i.e., when the soil distribution is a
trapezoidal one, the value of T may be given by
T = mghw(l + c)E'
where
E' = D[l{1 i) [2(3e  1) + 3(1  2e)D]] (145)
Table No. 24 gives the values of E for a range of values of
i and e.
REINFORCED CONCRETE WALLS 103
The application of the above expressions to specific problems
IS given at the end of the chapter.
The required rod area to hold the load T is
As = TIU
where/, is the permissible unit steel stress.
(146)
Table 24. — Valtjes op "E"
.1
.2
..3
.5
.42
.43
.44
.45
.47
.50
.40
.41
.42
.44
.46
.50
.36
.38
.40
.42
.45
.50
.29
.31
.34
.38
.43
.50
.25
.28
.31
.36
.42
.50
.15
.19
.24
.31
.40
.50
The preceding analysis, involving as it does a series of mathe
matical expressions, is not to be taken as interpreting with
exactness the stress system in the
counterfort wall. The difficulty of
attaining such exact statement
has been pointed out above. The
work as given is to be used as a
logical stepbystep process of
taking care in as simple a way as
possible the stresses that are in
dicated by a general study of the
wall. The equations together with
the tables based upon them are
readily applied to numerical problems (as given at the end of
this chapter) and cover in sufficient detail the necessary work
in determining the wall dimensions and th^ size and distribution
of the rod system.
Rod System. — The anchorage of the rod system into the wall
members forms the vital part of the design of the counterforted
wall. While it may seem a simple feat to anchor such rods to the
face slab (note that, in what follows, particular stress is laid
upon the face slab ; the thickness of the base slab is such that ample
room is had for anchorage of the tierods by simple extension
of their length and no further treatment is thus required) by
threading their ends and bolting through steel plates or washers
or even to assembled steel sections; or by bending around rods
at right angles to the anchoring rods, such details involve costly
field work, the use of an expensive class of labor and slow up to a
considerable extent the progress of the work. Simple details are
essential. In a problem, discussed in some detail at the end
of the current chapter, a detail is given showing such anchor
rods bent into Us of a radius large enough to prevent crushing
of the concrete and lying in a vertical plane. Rods of small
thickness are usually used because of the greater total surface
presented for adhesion.
104 RETAINING WALLS
Problems
1. A wall, of height 25 feet, retains an ordinaiy railroad fill subject to a
surcharge of 600 pounds pei square foot. It is placed along the easement
line, beyond which no encroachment is permissible. The soil is a sandy
loam on which four tons per square foot is allowable (see Table 7). A
design as a "L" shaped cantilever, and as a counterforted reinforced con
crete wall is desired.
With the above data c = 6/25 or 0.24; i = 0. From (93), page 83, with
H = 31 and i = 0, the location of the resultant is
e = 2/3  40/93 = 0.24
With this value of e and with i = 0, the factoi of safety against overtarn
ing is 2 (Table 17), a satisfactory one according to Hool, but less than the
2.5 suggested by Cain. See page 57. Adopting this value of e, from Table
18 the required value of k is 0.57 and accordingly the base will be made
14 ft. wide.
From Table 21 the shear and the bending moment require about the
same depth. Using the shear equation (113)
d  0.0033 X 252 X 1.48 = 3.05
and the thickness of the base will be taken as three feet. At a point haU
way up the wall for which c = %25 = 0.48, the moment determines the
depth at this point, as can be seen from Table 21, and from (112)
d = 0.0185 "X 44.2 X \/(l + 144) = 1.28
For the sake of simplicity of forms, bracing and rods, the wall will be
given an unbroken batter from the coping to the base, with the top width
a minimum practical width of one foot.
At the midpoint just investigated, the thickness will then be two feet, in
place of the required 1.28 feet. In the final design of the wall, the rod sec
tion will be diminished to allow for the decreased moment.
Footing. — Prom (111,118) the required depth will be y/l times the depth
necessary for the arm (siBce the arm depth here is that practically demanded
by the moment). From Table 22, since i = 0, / = 1, and the depth will
be the same as that required of the arm at its base, namely 3' 0" This
thickness will be maintained to the end of the heel.
Enough data has now been gathered to prepare an exact and final design.
From table 3, for c = 0.27, B = 0.40; whence Bh = 0.40 X 22 = 8.8.
Note here that the exact length of the arm is now considered, proper allow
ance having been made for the thickness of the footing. The batter of the
back is two feet in twentytwo feet, or 6 = tan"' (Ki) = 5° 40' = 6°.
From Table 1, / = 0.345 and B = 9°. The value of the thrust T is, from
(14), 16 kips, and is inclined at an angle of 15° (9  6) to the horizontal.
The weight G of the supei imposed earth on the footing is 22 ) 6 = 28 X
11 X 0.1 = 30.8 kips. The weights of the footing and of the rectangular
and triangular portions of the arm are respectively, 6.3, 2.2 and 2.2 kips
(see Fig. 48). Graphically, the resultant is found to intersect the base 3.5
feet from the toe or exactly }4 of the distance from the toe, checking the
REINFORCED CONCRETE WALLS 105
first assumption. The horizontal and vertical components of the resultant
found graphically ate respectively 15.8 and 46.5 kips. With the latter
value and using equation (39) S, = ^^ (2  0.75) = 8.3 kips, a permissible
variation from the 4 tons or 8 kips assumed.
..(.*^^'^
i5
*^^^A
miitjttn
^
Fio. 48.
Resistance to Sliding. — The coefficient of friction between sandy loam
and concrete is about 0.5 (an average between sand and gravel), see Table 6.
The sliding resistance is then 0.5 X 46.5 = 23.2 kips. The horizontal com
ponent is 15.8 kips, giving a factor of safety against sliding of 23.2/15.8 =
1.5 which is ample.
Design of the Vertical Arm. — The actual loading on the arm extends to
the top of the footing and for the arm, h is 22' and c = 0.27. EVom (110)
with J = }i, M = 106.5 kip ft. and the shear is 12.4 kips. Taking, as
before, the steel ratio for balanced reinforcement, or p = 0.75 per cent.,
0.5kj = 0.16 and h = 105. With b = 12", the required depth d in inches
is
^ 106,500 X 12
12 X 105
whence d = 32". From (113) the depth required on account of shear is
_ 12,400 ^ r„
0.89 X 12 X 40
(From Table 20 with c = 0.27, the shearing stress governs, when h is greater
than 27'.)
106 RETAINING WALLS
The steel area required at B is 0.0075 X 32 X 12 = 2.88 square inches.
This is a rather heavy reinforcement, necessitating great expense in handling
and placing bars. If a thicker wall is assumed, e.g., d = 40", then, from
.• T 106,500X12 . Ranannn 
the properties of the section fc. = — 12 X 40" — "^ ^^ ^ bb/lb,UUU 
0.004 and the required percentage of steel is 0.4 per cent. The steel area
is, then, 0.004 X 40 X 12 = 1.92 square inches and 1 inch square bars on
6" centers will give the necessary area. The unit adhesion is
12,400 ^
.89 X 40 X 8
The permissible stress is 80 pounds per square inch. To determine at what
point it is possible to stop one half of these rods, i.e. to space the rods 12"
apart, note that the external moment is given by the expression (110) or
M = 66.7(1 + Sc)x'
Since the coping width is taken as 12", the effective thickness at any point
of the wall x is
d = 9" + ~ (40  9) = 9" + 1.5x.
The resisting moment is given by Af = k,bd', where k, = f,pj. For small
values of p, pj may be taken equal to p, and with /, = 16,000 pounds per
square inch, and noting that since the area of steel is to be one square inch,
p = 1/bd, the resisting moment becomes
M = 16,000d = 16,000 (9 + 1.5.i;)
Equating the resisting moment to the external bending moment and replac
ing c by its value 6/x, there results a cubic in x
a;3 + 1SC2 _ 360j;  2160=
which is satisfied by a; = 15. Accordingly, at a point 15' below the top of
the arm the rods will be spaced 12" apart. Since a further reduction in
the spacing would place the rods 24" apart, which is not good practice, the
12" spacing will be continued to the top of the arm.
Footing. — To analyze the footing stresses, a moment diagram has been
drawn in Fig. 48. Note that the moment at B is very nearly equal to the
arm moment at B, affording another check upon the approximate method.
With Mb = 110 foot kips, and for balanced reinforcement, the required
depth is 34" The necessary amount of steel is 0.0075 X 34 X 12 = 2.73.
This again demands too heavy a reinforcement for efficient handling, and a
thicker concrete will be assumed. With d = 40, fc, = ^^°'OQOX 12 ^ ^^^
and pj = 69/16,000 = 0.0043. The steel area is then 2 square inches and
one inch bars spaced 6" apart will be used. To determine, again, at what
point it will be possible to reduce the steel section to oneinch bars at 12"
spacing, the resisting moment of such a steel section, since the thickness of
the base is kept constant, is found to be, with p = 1/(12 X 40) = 0.0021.
M = 0.0021 X 16,000 X 12 X 40" = 635 inch kips or 53 foot kips!
Plotting this value upon the moment diagram of the footing, it is found that
at a point 6 feet from the heel it is possible to reduce the rod section to one
REINFORCED CONCRETE WALLS 107
inch bars 12" apart. For the reasons outlined above, there will be no further
increase in this spacing.
To develop the adhesion in the vertical and horizontal rods, which must
be carried out 50 thicknesses or 4' beyond the point of maximum moment,
it is necessary to place a 6" projection at the toe and into the footing as
shown in Kg. 50.
The spacing of the secondary rod system for shrinkage, settlement and
temperature will be discussed in a later chapter.
Counterforted Wall. — Adopting the economical' spacing of ten feet for
the counterforts; from (126), with v = I, h = 16,000, h = 25,
d = 0.7 feet.
It is impractical to pour concrete in a wall this thickfor the height as given
and a minimum thickness of 12" will be adopted.
From (139) the required thickness of the footing slab is Vs times that
required of the vertical slab. It will be seen later that this thickness will
be controlled by a thickness necessary to get a practical spacing of rods for
adhesion. The dimensions of the separate members as now found are less
than those of the cantilevered wall, and since that wall as finally designed
agreed with the approximate dimensions it is clear that the counterforted
wall, will likewise agree and it will not be necessary to recheck the outline
dimensions of the section.
In selecting rod systems, both spacings and sizes, and wall thicknesses,
it must be borne in mind that there must be sufficient working space to
pour the concrete; that small sizes of rods are relatively more expensive than
the larger sizes; that many variations in both length and spacing tend to
cause confusion in construction. This limitation of the economical section
on paper by field conditions, is discussed more in detail in the foUowmg
chapter.
The moment at the base of the vertical slab (here h = 23.5 feet), with
c = .265, and P =  ^°° ^ ^^"^ = 0.98 kips, is from (125) 8.3 kip feet. As
before the depth for this moment, with balanced reinforcement is 0.73 feet,
but, for reasons, outlined above the thickness will be taken as 12". With a
wall of this thickness the utmost care must be exercised in pouring concrete
into it. See Chapter VIII for the precautions to be used to insure a well
mixed and rammed concrete.
With a depth to steel of 10", h = 8300 X 12/12 X 100 = 83 and p =
0.004, which gives a required area of 0.48 square inches, which H" rods on
The total shear is 980 X 5 = 4900 pounds, and the unit shear from (105) is
i?52 = 45 pounds per square inch
0.89 X 10 X 12
which is so slightly in excess of the permissible stress of 40 pounds that the
section wUl be maintained as assumed. The area required for adhesion is
from (106)
„ i?52 = 6.9 square inches
" 0.89 X 10 X 80
1 See problem. Chapter IV, p. 150.
108 RETAINING WALLS
The adhesion stress thus governs the spacing of the rods and %" rods spaced
5" apart will give the required periphery of section.
At h = 15, the moment is 5.8 kip feet and the shear is 700 X 5 = 3.5
kips. The area required for the bending moment is accordingly 0.18 square
inches, while that required for adhesion is found to be 5 inches.
At A = 10 feet, the periphery required for adhesion is 3.8 and at ft = 5
feet, the required periphery for adhesion is 2.6 inches.
It is seen that the adhesion stress will determine the spacing of the rods
throughout the arm. At h = 15, since 5 inches are required for adhesion
the spacing at the base will be maintained beyond this point. At fe = 10
feet, since r = 3.8 the rods may be spaced on 10" centers. At ^ = 5' the
value of r required wUl not permit a further reduction in the spacing of the
rods. There will thus be J^" square rods spaced on 5" centers from
the base to A = 10 feet and ten inch spacing from there to the top of the
wall. To take care of the equal but negative moment at the counterfort,
with the corresponding adhesion stresses, the same spacing will be main
tained on the inner face of the vertical slab. Since the rods must be
carried beyond the point of zero moment (approximately the quarter point)
the rods on the inner face will be made five feet long centered at the
counterforts.
Footing. — The net weight on the footing excluding the excess weight of
the masonry over the earth, is 3100 pounds. As before a depth to satisfy
the bending moment, is from (138) 18". For adhesion
3100 X 5
.88 X 16 X 80
= 13.7
which cannot be readily and practically provided. Conversely since it is
desirable to use a rod not exceeding the section of %" rod whose minimum
spacing is 6" on center, d is found
d = 3100X5 _
0.88 X 7.2 X 80
and the total depth of the footing slab is thus 30 + 2" = 32".
The point where the upward and downward intensities balance each other
is, from (142) and Table No. 23 with i = and e = J^, at the midpoint' or
seven feet from the end of the heel. To avoid many changes in the spacing
of the rods, the %inch square rods will be spaced on 5inch centers to a
point 3.5 feet from the heel and thence, to the midpoint on 10 inch spacing.
For the portion between the midpoint and the vertical arm it is reasonable
to assume that the slab is supported on three edges — ^the counterfort edges
and the vertical arm — and that such support is uniformly distributed along
such edges. 1 From (134) /j = 1  2(2  %) = 1.5.  P, = 1.5 X 100
X 28 = 4200 lb. The total net load between the counterforts reacting
upward upon the slab is then, since the intensity is zero at the midpoint,
4.2 X 7 X 9/2 = 132 kips. The total length of supporting edge is 2 X 7 +
1 For an interesting discussion of this modification of plate theory it may
be well to consult Prof. Eddy's brilliant little book on the "Theory of Rec
tangular Plates."
REINFORCED CONCRETE WALLS 109
9 = 23 feet and the shear per linear foot is 5.7 kips. For a 30" slab the unit
u ■ .V 5700
shear is then gg x 12 X 30 " ^^ '^' °°iisiderably below the allowable
and the periphery of rod required for adhesion is ^^ =27
.89 X 80 X 30
square inches. It is then sufficient to carry the H," rods on 10" spacing
to the toe of the base. The rod spacing will be duplicated on the opposite
face to take care of the negative moment and reversed stresses. Thus
from the midpoint out to the heel the rods on the lower face will be carried
full length and those on the upper face five feet beyond the counterfort.
From the midpoint to the heel the rods on the lower face will be carried
for the full length and those on the upper face will be extended five feet on
either side of the counterfort.
, Counterfort. — Designed as a cantilever beam, the moment at the base
is then TBh, with the thrust taken for a length m of the wall. T = 10 X
13 = 130 kips. With c = %2 = 0.27; B = 0.39 and h = 22,
M = 130 X 0.39 X 22 = 1,140 kip feet
The depth e of the cantilever is 14'. Assume, tentatively, its thickness
as 1.0 feet.
kc = 41, making p = 0.001 and the required area of steel in square inches
is 2 square inches. Therefore two inch square bars are ample to take care
of the moment in this counterfort. Investigating the unit adhesion, it is
found that, with a value of F = 130,000 pounds, the unit adhesion is 110
pounds. If two 1}4:" bars are used, the unit adhesion is found to be 85
pounds per square inch, a permissible variation from the allowable 80 pounds.
To anchor these rods into the base it is necessary to carry them fifty thick
nesses or about five feet into the foundation. For this reeison an extension
will be buUt into the foundation two feet below the slab and carried six
inches on either side of the counterfort. The radius to which the rods must
be bent in going into the base slab is 30 X IM = 3' 0".
To anchor the face slab to the counterfort, since the thickness of the face
slab does not permit a straight extension of the rods into it, it will be neces
sary to adopt the expedient of bending the rods into a U, with the radius
of the curve 30t.
From (141) for the top five feet of the wall
A. = ^ (5 h 12) X 5 = 0.89
Therefore two H" Us give sufficient bond for this length. The bars must
be bent to a radius of 15". For the next five feet the required amount of
steel is 1.4 and two ^" rods bent to a i7 with radius of 18" provide the requisite
bond. For the five feet below this section A, = 1.9, and three %" Us as
shown in Fig. 51 satisfy the requu'ements of this portion. The remaining
space from 15' to 22' is divided into two parts, the area of the first part is
found to be 1.6, of the lower part 1.9. Therefore three %" Us as previously
detailed will provide the remaining bond rods.
To gee the necessary rod area to anchor the heel portion of footing to the
counterfort (the portion from the midpoint to the heel) from (144) and
no
RETAINING WALLS
Table 24 with £? = 0.33 the total load to be held by these rods is 10 X 100 X
28 X 0.33 X 14 = 129 kips. The steel area is then, from (146) 129/16 =
8.1 square inches. Using )4 square rods, one on either side of the counter
fort 32 are required. With 6" spacing 15 spaces will carry the rods beyond
the midpoint. The depth of the footing is ample to develop these rods in
adhesion without any special detail and they will be carried to two inches
from the bottom, of the footing. Theoretically they need be carried into
the counterfort the same distance, but it seems better practice to carry
the rods for the full height of the portion of the counterfort affected (see
Fig. 51).
2. Modify the preceding problem to carry a railroad track system with
wall track 8 feet away from the face of the wall and the other tracks on 12.5
foot spacing. Assume that all tracks but the wall track are loaded; then
assume no tracks loaded. In what way is the pressure upon the footing
affected, and do any of the stresses exceed those for the case of all tracks
loaded (the former case)?
Doffed Li'msshov
Force Polygon for
No Surface Loading.
T=I0.6
Te.O
Fig. 49.
For this case, see Fig. 10, the surcharge extends to 14 feet fronj the wall
face. As above 6 = 6° and from (32) of Chapter I
The proper value of a to use in determining the coefficient K is from (34)
with y = 14/22 = 0.64
54
tan a = tan 6°  ~^ 0.64 =  0.119
1.04
whence
a
7°
From table No. 13, allowing no friction uponthe back of the wall, K = 0.286
and the thrust is then 10.6 kips. Fig. 49 shows the force system on the
wall for this case.
REINFORCED CONCRETE WALLS
111
For the second condition, no loading upon the surface, the thrust becomes,
with K = 0.33 and c = 0, r = 8 kips. Pig. 49 shows the force system
for this case.
I no
ZI4 Bars
140
Fig. 50. — Cantilever wall.
Fig. 51. — Counterfort wall.
Lower Face
Footing
Upper Face
Footing
Fig. 52. — Rod layout counterfort wall.
From Fig. 49, e for the former condition is 0.28 and B = 37.5. For the
second condition e = 0.35 and R = 37. From (38) Si for the former is
6750 pounds per square foot and for the latter is 5000 pounds per square
112
RETAINING WALLS
301
7g"
foot. It is obvious that the analysis of the first problem will require no
modification of stress distribution because of these latter conditions.
Fig. 50 gives the detailed layout of the "L" shaped cantilever. Fig.
51 gives the rod layout of the counterfort and Fig. 52 of the vertical and
base slabs. In neither of the sketches are the temperature and check rods
shown. A later chapter will indicate such distributions.
3. A "T" shaped cantilever wall is to be built, retain
ing an embankment as shown in Fig. 53. The em
bankment is subject to a surcharge live load of 750
pounds per square foot. The foundation pressure
must not exceed 5000 pounds per square foot. Deter
mine the proper wall dimensions and details.
For the condition of no surcharge, both the exact
and the approximate expressions for the thrust, as given
on page 14 may be employed. Exactly, ,with the
6 = 0' =■ and <^ = 30°, L = l/cos" (^ = 4/3; m = sin (^; « =
cot (t>; m = 1; n =
20'
Fig. 53.
angle i = 30°
—cos <i>; d
and/ = 3.
cot^ (j> = —3; c ■
The expression for the thrust is then
0.5; p = sin <#> = H
T =
2
10.7
xx
[1.5  0.5 ■v/1.52 + 0.25 X 3]'
The approximate method, which since c
15), gives a value
0.5, is not to be used (see page
T =■
(1 I 2c) = 13.3.
A variation from the true value too excessive to permit of its use.
For the condition of a live load surcharge,
in place of the graphical method of obtain
ing the thrust, the compromise, algebraic
geometric method outlined in the problem
at the end of Chapter 1, may be used. The
value of i is determined graphically, the
line forming the equivalent triangles as
shown in Fig. 54. With aoe making an
angle of 35°, the triangles afo and obe are
equivalent. With this value the thrust may
be determined as above. From Eq. 22 L =
l/cos^ <t>; u = sin <t>;v= —cos 0; n = —cot 35° cot (^ = — 2.43; p = sin
4> = 0.5; TO = 1 and/ = —2.43.
J, ^ 100X400 ^ 4 ^ g^g _ 1^1.875^ + 0.761 X2.43J ^
= 13.6
Refer to Figs. 42 and 53 assuming, as the condition of economy, that i=e.
In addition, assume that the resultant intersects the base at the outer third
point, I.e. i = Y^. Noting that g = 100; h = 20 and tan 35° = 0.7 the
weight G has the value
g(l  iywHan 35°
Fig. 54.
G = g{\  i)wh + '
= 0.67 w(2 + 0.023«))
(A)
REINFORCED CONCRETE WALLS 113
Taking moments about 0, and noting that without serious error the point
of application of the weight may be taken at the middle of the base
G(l  i)w/2 = Th/3.
Introducing the values above, this equation becomes
(1 — i)w
and with i = }i
G = 273/u) (B)
Equating (A) and (B), there results a cubic in w
410 = 2w^ + 0.023U)'
which is satisfied by the root, w = 13.5. With this value oi w, G = 20.2
and from (39)
Si = IGlw = 40.4/13.5 = 3 kips.
The projection of the toe beyond the face of the wall is 4' 6" Assume
tentatively that the thickness of the base and of the vertical ar i at its base
is two feet. The thiust, for the purposes at hand may be assumed to vary
as the square of h. Since the effective height of the wall, so far as the arm
is concerned is 18 feet,
18^
T =~X 13.6 = 11.
and its point of application is onethird of ^ or 6 feet above the top of the
footing. The bending moment is then 11 X 6 = 66 and with k = 16,000
for balanced reinforcement, the required depth on account of moment is
d = \/(66/l6) = 2.03
The shear is 11,000 pounds and the depth to satisfy this amount is
d = 11,000/5040 = 2.18
The thickness of the vertical arm at its base may be taken as 2' 6" The
back will be battered to a top thickness of one foot.
Footing. — The face of the vertical arm is, on the assumptions previously
made at the third point or 4' 6" from the end of the toe. The moment of
the heel cantilever is then taken at a point 4' 6" + 2' 6" from the toe or
6' 6" from the end of the heel. At this point, since Si is 3000 pounds, the
6.5
soil intensity is — — X 3 = 1.44.
13.5
Taking the approximate value of G as 20.2 and again assuming that it is
directed over the center of the heel cantilever, the bending moment becomes
20.2 X 3.25  h^^^^ x 2.2 = 55.4
The shear is
20.2  1.44 X 6.5/2 = 15.5.
Evidently the shear will control the depth required and
d = 15,500/5040 = 3.08
114
RETAINING WALLS
Whence take 3' as the required thickness of base.
It is now possible to proceed with the exact design. (See Fig. 55.)
thrust is found from equation (22), with c = 17.5/17 = 1.03 and
The
^ 100 X 17^ A,
T = X3I2.O3
[2.03^^;
2.03' + 1.06 X 3
This will be applied at a point 17/3 or 6.65 feet above the top of footing.
The weights of the earth has been divided up into the triangles dbc = Gi)
ade = Gs and the lectangle dcfe = Gi. The weight of the masonry has
been divided into the triangle Gi and the rectangles Gi and Ge. The weights
are: . '
Gi = 9 X 4.75 X 100/2 = 2.14 kips.
G2 = 6 X 17 X 100 = 10.2 kips.
6,t6,
Fig. 55.
Note that the two above act in practically the same vertical line, so that
the two may be added and treated as one force
Gi + G2 = 12.3
G3 = 2 X 17 X 100/2 = 1.7
G4 = 1 X 17 X 160 = 2.55
Gs = 1 X 17 X 150 = 2.55
Ge = 3 X 13.5 X 160 = 6.07
With the forces as above found the polygon is drawn in the usual manner,
see Fig. 65, and the location and amount of the resultant pressure is found.
The actual value of k is 5.6/13.6 = 0.4 and R = 26.5.
S,
j^ (2  0.12) = 3.00 and ^2 = ~ (1.2  1.0)
Vertical Arm. — The moment of the thrust is 8.9 X 5.65
depth to satisfy this moment is
d = V(50.4/16) = 1.78
The shear is 8900 and the corresponding depth required is
d = 8900/5040 = 1.77
= 0.75
60.4 and the
REINFORCED CONCRETE WALLS ' 115
The required depths are thus identical and the total thickness of slab at
the base of the arm will be 2' 0", allowing 3" for a protective concrete coat.
Since, for balanced reinforcement the steel ratio is 0.0075, the amount
steel required is
A, = 0.0075 X 21 X 12 = 1.89.
Spacing 1 inch square bars (deformed) 6" apart will furnish the necessary
section. Assuming that there is a triangular distribution of pressure, ' the
moment diagram is shown in Pig. 55. To obtain the thrusts for the
moment, note that at the points 15', 10' and 5' from the top of the wall the
corresponding values of the surcharge ratio are 1.17; 1.75 and 3.5. The
values of the thrust are then
100 X 15' 4
ii5 2 ^3
_, 100 X 10' .. 4
1 10  2 ^ 3
100 X 5' 4
^'  2 ^3
2.17  ^'\/ 2.17' + 3 X 1.17' =6.9
2.75  ^"\ 2.75' + 3 X 1.75' ' = 3.3
["4.5  2^ 4.5' + 3 X 3.5'1 ' = 0.9
The moments are, assuming again that the thrusts are }i of the distance
above the point in question,
Mii = 6.9 X 5 = 34.5
Mio = 3.3 X 3.3 = 10.9
Ms = 0.9 X 1.67 = 1.5
At some intermediate point along this arm, it will be found that one half
of the rods are sufficient to carry the stress; i.e., the rods from this point
on may be carried on 12 inch spacing. 7^ before the width of the wall at
the coping will be taken as 12 inches. With a spacing of 12 inches for the
oneinch rods at
;i = 15; d = 19" and p = 1/(19 X 12) = 0.0044
h = 10' d = 16" p = 1/(16 X 12) = 0.0052
The corresponding values of pj are 0.0042 and 0.0047, and the resisting
moments are then, expressed in footpound units,
Mu = 144 X 0.0042 X 16,000 X 1.59' = 24.3
Kio = 144 X 0.0047 X 16,000 X 1.33' = 19.2
Plotting these two values on the moment diagram. Fig. 55, it is seen that
the resisting moment of oneinch rods on twelveinch centers, is equal to
the external bending moment at a point approximately 4.5 feet above the
footing. The sixinch spacing will then be stopped at a point 5' above the
top of the base slab. As previously explained, this spacing will be continued
to the top of the arm.
1 While this is, strictly speaking, incorrect, since the thrust is not a linear
function of h, which condition is the necessary one that there be a triangular
distribution of pressure, the ease of handling the problem with that assump
tion counterbalances the slightly excessive pressures thus found.
116 ■ RETAINING WALLS
Footing. — The force acting upon the base slab over the heel is (?i + G2
or 12.3 kips. The weight of the base slab (maintaining the thickness first
found) is 6.5 X 3 X 150 = 2.9 kips. The total downward load upon the
1 3 X 6.5
heel is 15.2 kips. The upward soil pressure is — —  — '— = 4.22 kips. The
moment for the heel is thus
15.2 X 3.25  4.22 X 4.33 = 31.2 kip feet
The shear is 15.2 — 4.2 = 11 kips. The required depth, for shear is 2.18,
which clearly, is greater than that required for the moment. With a pro
tective concrete over the rods the thickness of the heel slab will be taken
as 30". With the net depth (eilective) of 27", fc, = 31.2/2.252 = 42.5 and
Vi = 42.5/16,000 = 0.003. The steel ratio is then 0.003 and the necessary
section of rods becomes 0.003 X 27 X 12 = 0.97 square inches. Oneinch
rods spaced twelve inches apart will provide the requisite steel area and
this spacing will be carried out to the end of the heel.
Toe. — At the toe the cantilever moment is
M = ^^ t t^i^^^' = 26.8 kip feet
and the shear is
(3 + 2.3)4.5/2 = 11.9 kips
As before the shear requirement will control the depth of the section
d = 11.9/5040 = 2.33
The same thickness of both heel and toe will be used, which in view of the
usual manner of pouring the wall is practically mandatory.
_ 26,80 X 12 _
"' 12 X 272 ~ ^^
and vj = 37/16,000 =0.0023. The steel ratio is then 0.0023 and the area
required is 0.0023 X 27 X 12 = 0.83. 1inch bars spaced twelve inches
apart will provide the steel reinforcement.
Since a 1" bar requires four feet to develop its tension by adhesion, the
heel rods will be carried four feet beyond the rear face of the vertical arm
and the toe rods four feet beyond the front face of the vertical arm. For
the reinforcement of the vertical arm, an extension 1' 0" wide and 1' 6"
deep will be built into the foundation to provide the required length.
Fig. 56 shows the complete section of wall. The rods necessary for shrink
age and temperature stresses have not been shown.
4. In the wall of problem 1, it will be necessary, for a given stretch to
provide a footwalk as shown in Fig. 57. Without changing the outlines
or the design of the wall proper, design the bracket to carry this walk, sub
ject to a live load of 100 pounds per square foot.
Assuming that the concrete bracket will be 6" thick, the dead load will
be 75 pounds per square foot, making the entire load upon the bracket 175
pounds per square foot. For balanced reinforcement
d = v/(790/16,000) =0.22, or 3" thick.
With 2" protective concrete over the rods the total thickness of slab is 5".
REINFORCED CONCRETE WALLS
117
The required steel area is 0.0075 X 3 X 12 = 0.27, and }i inch square
rods, 12" apart will provide the required steel section. The unit shear is
' = .89X3X12 = 1^ P°""'*'
The unit adhesion, with r = 2, is
= 98 pounds per square inch.
3 =
525
X 3 X 2
This latter value is excessive and the depth of section must be increased at
this point. If at the cantilever junction between the wall and bracket a
fillet is placed as shown in Fig. 57, the unit adhesion at the point D is % of
that above found or 70 pounds per square inch. To provide the necessary
bond the J^" rods will be bent as shown and carried into the vertical arm.
/feinforcemen^
■y'of YirHccU^rm
Fig. 56.
Fig. 57.
B. A counterforted wall, resting upon a rock bottom, is to take a surcharge
of 500 pounds per square foot. The easement does not permit a toe exten
sion. Determine the general wall outlines from the approximate formulas
given and design a counterfort made up of a steel truss.
With i = 0, and the ' foundation rock e may be taken equal to K,
giving a value of fc from Table 18 of 0.51. The width of base is thus 0.51 X
50 = 25' 6". From Table 17 the factor of safety is found to be two. As
sume that the counterforts will be spaced ten feet apart. The pressure at
the base of the vertical slab is JghiX  c) = 0.33 X 50 X 1.1 X 0.1 = 1.83
kips per square foot. From (126)
= in /l 100 X 50 X 1.1
\3 12 X 16,000
1.0
The depth for shear is
d =
1.83 X 5
5.04
= 1.83
118
RETAINING WALLS
It will be found, later that the thickness of the face slab at the base will
be controlled by the necessary dimensions of the member composing the
vertical arm of the truss. The thickness of the base slab is controlled by
the depth necessary for the adhesion stresses. If 1" square bars, spaced
6" apart are to be used, then the depth necessary to satisfy the limiting
adhesion stress of 80 pounds per square inch is
d =
5500 X 5
80 X 0.89 X 8
= 49"
To avoid the use of so heavy a slab throughout the base, a fillet of con
crete will be placed at the junction of the base and counterfort, dimensioned
as shown in Fig. 58. The main body of the slab will then be taken as 2' 9"
thick.
Fig. 58. — Counterfort wall.
The design of the counterfort proper (note that a final check of the dimen
sions just found is omitted — in actual practice such omission is poor design)
is most conveniently made by graphical methods. The skeleton outline
of the truss is shown in Fig. 58. The loads at the panel points A, B, C are,
allowing for the ten foot spacing of counterforts ;
„ 1.83X16,5X16 „
n  ^ + g   28
„ 7 X 16 , 5.5 X 16 , 1.83 X 16 , 2 X 5 X 16 ,,„
P" = ~^r~ + — 6""" + — 2 — + — 6 — = ^12
p 12.5 X 15 . 5.5 X 15 , 7 X 16 , 2 X 5.5 X 16 ,„,
The stress polygon is drawn as shown and the stresses are denoted plus
or minus as they are, respectively tension or compression. The vertical
REINFORCED CONCRETE WALLS 119
members of the face and the horizontal member of the base, must carry
the moment induced by the slab reactions. These moments are
4 3 V 1fi2
Mat = „ = 138 ft. kips
O
Afj. = lO^je" ^ 320 ft. kips
Mci = llAJA^ = 424 ft. kips
The unit stress in tension will be assumed to be 16,000 pounds per square
inch. That ia compression, long column, 12,000 pounds per square inch.
The vertical arm and the base arm are buried in concrete. It is the practice,
for members thus stressed, to let the concrete take the load from the steel
member by adhesion so that the member carries only the bending load.
Such practice will be adopted here.
Where deductions from gross section are necessary because of rivet holes,
ij^6 inch open holes will be assumed. The actual work of the design is
not shown here.
2 Ls 3.5 X 3.5 X %
2 is 6 X 3.5 X H
2 Ls 6 X 6 X Ke
2 Ls 6 X 6 X Kg
2 Channels 15" 40#
Since the member AB is subject to bending only,
AB, M = 138 Sect. Modulus 138 X ^Ke = 103
Web plate 15 X %; 4 Ls 6 X 3.5 X ^
EB M = 320. S. M. = 320 X ^Ke = 240
Web plate 18X%;4Ls6X6X ^He
FD M = 424. S. M. = 424 X iJie = 318
Web plate 24 X M; 4 Ls 6 X 6 X%.
The details are not given of the connections, etc.
It will be assumed that the truss work is either encased, member by mem
ber in concrete, or is coated with gunite, or other preparation of similar
nature.
6. A counterforted wall, 24 feet high, subject to a surcharge of 6 feet, is
to rest upon a soil capable of holding not more than 6000 pounds per square
foot. Determine the general wall outlines and design the toe extension.
From (95), with Si = 2.5 tons and H = 30 feet, and i = e
5 ^ ^ M , """ _ r, 28
ag.
S = 61.
A = 6J^6 = 3.8
ah.
S = 112.
A = ii^e = 7.
he
S 125.
A = 12^2 = 10.4
eg
S = 183
A = 18^6 = 114
cd
S = 315
A = 3i^g = 19.7
6
W'+w = »=
From Table 17, for e = 0.25, k = 0.56, and the width of the base is 0.56 X
24 = 13' 6". The toe projection is 0.28 X 13.5 = 3.8 or 4' 0". Without
attempting to design the separate sections of the wall and then redetermining
these general outlines from the more exact data, let it be assumed that
these preliminary outlines will remain in the final analysis.
120 RETAINING WALLS
The loading upon the toe extension is shown in Fig. 59. 22 = 30 X 9.5 X
0.1 = 28.5 kips. From (39) Si = 4.9 kips checking the first assumption.
From (41), the location of the point of zero intensity of soil pressure is
found at a; = I j^ = 4.5(0.16/0.44) = 1.63 feet from the heel. The
center of gravity of this loading may be found by aid of Table 3, noting that
the value of c is 7.8/4.0 = 2 approx., whence B = 0.47 and the location of
the force is 1.88 from the toe, little error would
/3ig« ...^ have resulted in taking the center of gravity
I I at the center of the load. The total load is
/'^>i/.fi3k ^tMx4=16.6. For shear d= 16,600/5040
= 3.3. The moment requirement is less and
the depth chosen will be that required by the
shear. The total thickness of the toe, includ
ing the protective concrete over the steel rods
\^4'0'4< 40'> ^iu be 3' 6".
_ 16,600X2.12 X 12 ^
^""/P^.m, ' ' 12X39^
and pj = 23/16,000 = 0.0014. This is sub
FiG. 59. stantially the steel ratio p. The area of steel
required becomes 0.0014 X 39 X 12 = 0.66
square inches. Taking j again as 0.89, the periphery of steel necessary
for the proper adhesion stress, namely 80 pounds per square inch, is
16,600 „ . ,
— 6 sq. mches.
0.89 X 39 X 80
This latter requirement controls the selection of the reinforcement and
J^ inch square bars spaced on 6" centers will be used. Since, to develop
the stress (and in accordance with the principle of the proper detailing of
structures, the section as used is developed and not merely the stress exist
ing in it) the bars will be carried by the face of the vertical arm for 50 X
M = 4 feet.
The toe as finally laid out is shown in Fig. 59.
It must be again emphasized that in none of the preceding problems have
the secondary rod systems, for temperature, etc., been shown. In a later
chapter these rod systems will be completely detailed, with reference to
these problems.
Bibliography
The following is a list of articles on reinforced concrete walls :
Standard Design of 5516 Linear Feet of Wall, 9 to 24 Feet in Height,
Steptoe Smelter, Engineering Record, Vol. 61, p. 209.
Recent Retaining Wall Practice, Journal Western Society of Engineers,
Vol. 26.
Tables for Reinforced Concrete Walls, Based on Fluid Pressures of 20 and
26.6 Pounds per Cubic Foot, Engineering & Contracting, Vol. xlii, p. 146.
Reinforced Briclfwork, The Engineer (London, England), July 2, 1915.
REINFORCED CONCRETE WALLS 121
Design of Retainiag Walls, Engineering and Maintenance of Way, March,
1912.
Reinforced Concrete Retaining Walls, Cornell Civil Engineer, March, 1913.
Some Economical Types of Retaining Walls, Railway Age Gazette, April 6,
1917.
Counterforted Walls, Lining a Stream Channel, Engineering News, Vol. 72,
p. 1258.
Walls for Yale Bowl, Maximum Height 42 Feet, Engineering News, Vol. 72,
p. 997.
Counterforted Walls with Structural Steel Frame, Enginemng News, Vol.
73, p. 776.
The Design of Counterforted Walls, E. Godfrey, Engineering & Contracting,
Vol. xxxiv, Dec. 21, 1910.
(See Also Bibliography in Appendix.)
CHAPTER IV
VARIOUS TYPES OF WALLS
The types of walls discussed in the previous chapters are
those generally used in engineering practice. Occasionally, condi
tions are such that these general types are inapplicable and it
becomes necessary to devise special types to meet the peculiari
ties of the given environment. Such walls_are described briefly
below.
Cellular Walls. — A type of wall insuring a light foundation
pressure approaching a uniform distribution is shown in Fig. 60.
It is essentially a gravity type, the interior concrete replaced by
V//////1
\ I
Tt
a
aJ
Section aa
Plan
Fig. 60.— Cellular wall.
an earth fill. The principles governing its outlines are thus iden
tical with those governing the outlines of the rectangular gravity
walls, with the correct allowance made for the reduced stability
moment. In a finished wall, complete with the fill outside and
inside, the rear wall is under no pressure. To insure no possi
bihty of failure during construction or at some later date in con
sequence of an adjacent excavation, it is well to make the rear
wall like the face wall. Theoretically the wall may be built
without a base. Practically, to insure an even distribution of
pressure upon the bottom, and to avoid unsightly settlement, a
base is generally used.
The design of the separaite members is identical with the
method used in the design of the several members composing
the counterf orted wall. For the base, when such is used, the slab
should be designed for the net difference between the upward
122
VARIOUS TYPES OF WALLS
123
and downward loads. A description of a wall of this type is
given in Engineering & Contracting, Vol. 35, p. 530, by J. H. Prior.
Hollow Cellular Walls. — To insure even lighter soil pressures
than given by the type previously discussed, a hollow cellular
wall may be used, as described in Fig. 61. Its stability is
_,
~rr_ """ 7  
(^
=^i
Section aa Plan
Fig. 61. — Hollow cellular wall.
furnished by the small amount of earth fill resting immediately
upon "it and by the weight of the track ballast, in addition to the
weight of the separate members composing the cells. It is
essential, because of the light weight of the wall that adequate
attention should be paid to its tendency to sHde forward. The
face of the lower part of the wall should abut against the firm
ground, and, if possible, extensions should be built into the bot
tom to add to the sliding resistance. Two interesting types
of the wall are described here. The former, as shown in Fig. 61,
■ Openings in
ParHfion Walls
IB>
z'o"
ySeam Sfruis
_. Slope Line of
Pressure
"CT
Fig. 62. — Cellular wall on timber wibbing.
termed the "Lacher" wall is described in detail in an article by
J. H. Prior.i While this was the most expensive type of five
types analyzed for the track elevation work of the Chicago,
Milwaukee and St. Paul (gravity, "L" shape, counterfort "L,"
cellular as described previously and the hollow cellular) it was
the only type insuring a safe permissible pressure on the soil
encountered in the work. The maximum soil intensity was two
^Engineering & Contracting, Vol. 35, p. 530.
124
RETAINING WALLS
tons per square foot. This type also permitted a full use of the
easement for tracks. It was not feasible to use piles.
The second type, shown in Fig. 62, was used in supporting the
Speedway, a highway along the west bank of the Harlem River,
New York City. It is described in the Engineering Record, Vol.
66, p. 22. A good foimdation could be had upon a timber
cribbing already in place, below mean high water, giving promise
of little future settlement. The wall is about square in section
and the sidewalk forms the upper slab of the cell. The walls
are thinned down towards the top and a circular segment is cut
out of the transverse walls, to diminish the load upon the base.
The distribution of the pressure is practically a uniform one.
To quote from the article :
"The transverse walls are so spaced that their weight is evenly dis
tributed upon the foundation cribs by the 3 foot concrete flooring.
It was assumed that the line of thrust at the base of these walls due to
their weight and the weight of the sidewalks which they carry, would
be at an angle of 45°. Upon this basis, the lines of thrust from the
bottoms of successive transverse walls intersect just at the base of the
3 foot concrete floor, causing a uniform application of the loads upon
the foundation cribs." See Fig. 62.
Timber Cribbing. — Walls have been constructed of old ties,
forming practically cellular walls. The transverse ties are
spiked to the stretcher ties forming the rear and front faces. See
Fig. 63. Such a wall was used in Chicago by the Chicago, Rock
Fig. 63. — Timber orib.
Island and Pacific Railroad for heights varying from four to
twenty feet. There is an interesting discussion on the use of this
type of wall in the Joural of the Western Society of Engineers,
Vol. 20, 232 et seq.
Concrete Cribbing. — In exactly identical fashion with the use
of timber cribs, concrete cribbing may be used, the members
constructed in units of a shape similar to a tie and reinforced at
the four corners. A description of the use of such cribbing in
Oregon along a highway is given in the Engineering News
Record, Vol. 81, p. 763. It is pointed out in this article that the
VARIOUS TYPES OF WALLS
125
life of timber cribs is so short that their use is not economical.
Concrete cribs, would not be open to this objection.
Walls with Land Ties (or Backstays). — This is a practically
obsolete type of wall, but is occasionally used for small light
walls usually along the water front . A typical wall of such charac
ter is described in Engineering and Contracting, Vol. 37, p. 328.
It is shown in Fig. 64. Its design follows from the ordinary
Fig. 64.— Wall with land ties.
principles of statics and the force system is shown in Fig. 64.
If the tie is a metal one, there is danger of its gradual destruction
by rust. It should be encased in concrete, which adds consider
ably to the expense of the wall. On a fair foundation and for a
small wall, this type may prove economical. The theory of such
walls is given by Rankine 23rd Ed., 1907, pp. 410, 411.
Walls with Relieving Arches. — This is another type of his
torical interest rarely used now. As constructed of brick with
Fig. 65. — Wall with relieving arches.
cheap labor it afforded an economical type of substantial con
struction. The theory of such a wall is given by Rankine7in
his 23rd Ed., p. 412. Fig. 65 shows a typical view of such a
wall.
An interesting example of a wall of this kind is given on p.
353 Handbuch Fur Eisenbetonbau III Band. The relieving arches
126
RETAINING WALLS
are of cast iron and the wall masonry of brick. The section ofj
,the,wall is shown in Fig. 66.
A novel type of wall is shown in Fig. 67, and is a compromise
between a cellular and cantilever type. It is taken from the
handbook on concrete quoted above.
rn
rm
L
Fig. 66. — Briek wall with oast iron relieving arches.
'^ — n
4' ^
4'
r 1
i
^
\^ 120 A
Fig. 67. — Special shape wall.
Euorpean Practice. — Some very interesting types of walls,
mostly of European origin are given in the Handbuch Fiir Eisen
betonbau III Band, pp. 369 to 402. The intricate rod systems
and complicated form details necessary in the construction of
these walls would preclu,de their use in America. It is notable
to see the latitude allowed individual engineering talent in the
adoption of the various designs and such freedom of thought
should prove, in the long run, very fruitful in useful wall sections.
Embankments Bounded by Two Walls.^ — ^The construction of
embankments through narrow easements, requiring retaining
walls on either side of the fill makes it possible to utilize the
mutual action of the two walls to effect quite a reduction in the
section of each wall required. The wall thus built is in effect
a modification of the counterforted wall and so far as the actual
design of the wall itself, the theory as previously given is sufficient
VARIOUS TYPES OF WALLS
127
rjf.
■S7S
to design this wall. Two interesting examples of this type of
construction are given here.
Retaining Walls, New York Connecting Railroad,
Hell Gate Arch
Approach. — 'The embankment to be retained was practically
of square section, 60 feet wide and high. The ordinary theory
of earth pressure would have necessitated enormous sections. A
carefully specified embankment
well drained and compacted
made it possible to reduce the
thrusts (see page 21). The
walls were divided into ten foot
square panels, at each corner of
which a tie rod 23^^ inch diameter
extended between the walls and
was anchored to a steel channel
embedded in the face walls (see
Fig. 68). Every fiftyfeet, a
partition wall ran between the face walls giving additional stabil
ity to the section, and especially stiffness against wind stresses
prior to the placing of the fill within the wall. A most careful
system of drainage was placed at every row of tie rods to prevent
the accumulation of water with a consequent increased pressure.
tHU
\z!i'°Rods
Fig. 68
Walls of Hell Gate arch
approach.
Interboro Rapid Transit Railroad, Eastern Parkway
Improvement
The walls here were about 25 feet high and tied to each other
at intervals of 20 feet by reinforced concrete partition walls
(see Fig. 69).
In both examples it is to be noticed that no bottom slab is
used, forming the true cellular wall as described by Lacher in
the previously mentioned issue of the Journal of the Western
Society of Engineers. The interesting details in connection
with the use and nonuse of expansion joints are discussed in the
following chapter.
The widening of an existing right of way prior to its final com
pletion (White Plains Rd. Extension, Interboro Rapid Transit
Co.) made it possible to adopt an unusual expedient of anchoring
the new wall directly to the existing wall. Structural steel
128
RETAINING WALLS
frames were anchored through the existing wall as shown in Fig.
70 (See Plate II, Fig. 26). The new face wall consisted of
slabs supported by upright channels. To insure the permanence
of the anchors they were embedded in concrete partition walls.
In placing the fill care was observed to carry up the fill levels at
the same rate on either side of these partition walls to prevent
aJ
Plan
Section aa
Fig. 69. — Walls Eastern Parkway Extension Interboro Rapid Transit R. R.
placing an earth pressure upon them. The thickness of the
face slabs was the minimum width it was found practicable to
construct in the field with the equipment at hand.
Abutments. — The design of the abutment differs from that of
the ordinary retaining wall, merely in that an extra dead or dead
and alive load, is superimposed upon the wall and serves to
counteract the overturning moment of the earth pressure. This
r—
?n M
t~i C3
\ ^
T
\
( i
)
Plan
Section aa
Fig. 7G. — Anchoring new'wall to old wall.
additional load, resting upon the abutment is assumed to be
uniformly distributed along the abutment and is, thus, treated,
mathematically, as an additional masonary surcharge. The
variable conditions of loading make it necessary to investigate
all possible states of loading, in order to ascertain the maximum
forces upon the wall.
The following combinations of dead and live loads are all
possible ones and each is worthy of investigation. The ac
VARIOUS TYPES OF WALLS
129
companying Fig. 71 may serve to give a better idea of these
combinations as listed below.
(a) The earth backing in place, but no span construction set.
The abutment is a plain retaining wall.
(6) The crane to be used in erecting the span is in place behind
the abutment. Here the abutment is a retaining wall with a
surcharge load due to the erecting crane.
(c) The construction complete. Live load approaching the
span. The abutment has the full earth and surcharge load, but
only the dead load of the span as a relieving load.
(c) (d)
Fig. 71 . — Conditions of Abutment loading.
(rf) The live load is on both the span and back of the abutment.
There is here the maximum earth pressure and maximum relief.
This latter case gives the greatest total loading upon the base.
The others, however, may give a greater toe intensity.
In connection with the conditions of loading subsequent to
the completion of the structure, the span construction, in ad
dition to the relief afforded by its weight upon the wall also
exerts a horizontal relieving action, forming a beam out of the
abutment with both a top and bottom support. Such relief,
however, is most difficult to compute, due to the uncertainty
of the action of the roller bearings and had better be neglected
in the design of the wall.
The designer should, of course, govern the design of the wall
by the above four conditions and not attempt to control
the field conditions, such as the sequence of operations in the
placing of embankment and erection of the bridge, by his design.
It is, of course, within the' province of the experienced engineer
to determine how best to adapt the design to take care of the
130
RETAINING WALLS
construction loadings. The factor of safety against sliding and
overturning may be temporarily lowered to take into account the
conditions prior to final completion, but it does not seem advis
able to permit the soil intensity under any combination of
loading, temporary or otherwise, to exceed the safe allowable
pressure.
Fig. 72.
FiQ. 73.
Abutment types.
Fig. 74.
The location of an abutment is usually transverse to the right
of way, permitting the footing to encroach upon the crossing,
whether pubHc or private. It is thus possible to secure the best
type of soil pressure distribution, keeping, at the same time, an
economical section of wall. Since the abutment is a combination
of a retaining wall and an ordinary pier subject to vertical loads
only, it is customary to extend both the heel
and toe (see Figs. 73, 74, 75).
Abutments may be either composed of plain
masonry or ot reinforcedconcrete, as economy
or other factors dictate. The flexibihty of
reinforcedconcrete in permitting slender walls
with projecting heel and toe indicates that for
practically every condition a reinforcedconcrete
type of wall may be found that will prove more
economical than the gravity masonry walls.
The counterforted retaining walls may readily be adapted to
form an abutment, by placing a cap over the top to form the
girder seat (see Fig. 72). Several of the usual types of abut
jnents are shown in Figs. 73, 74 and 75.
r Wingwalls. — The wing walls attached to the abutments are
' ordinary retaining walls and are so designed. Their location
\ is governed by the conditions of the intersection and may either
' be in line with the abutment, following the slope of the fill, or
Fig. 75.— Re
inforcedconcrete
abutment.
VARIOUS TYPES OF WALLS
131
r"
'•1
Fig. 76.— Plan of
"T" abutment.
if the condition of the easement does not permit may make
an angle with the abutment determined by the economical
hmitations. The combmation of wing wall and abutment, makes
it possible to devise ingenious schemes to effect an economy of
material used. The walls and abutment may form a Uband of
constant crosssection as described in Engineering News, Mar. 8,
1917, p. 393, the walls partially buried in the fill and holding, by
friction, the abutment portion of the U. Cellular abutments
have also been used.
Occasionally an abutment is supported by a
stem buried in the retained embankment, forming
a T (see Fig. 76).
An exhaustive analysis of abutments and wing
walls, with a wealth of practical hints, is given by
J. H. Prior in the American Railway Engineering
Association, Vol. 13, p. 1085.
C. K. Mohler,^ Consulting Engineer, has
pointed out the economy effected by turning
back the wing wall in place of merely extending it in the line of
the abutment to follow the slope of the retained embankment.
E. F. Kelly has pointed out^ that for minimum wing length,
the face of the wing should bisect the angle between the
shoulder of the fill (sometimes termed the berm) and the
face of the abutment produced. This assumes that the
end of the wing wall becomes a line, in place of, as in
actual practice, the wall being cut off at a convenient height.
Since the end of the wall has no serious effect upon the Entire
amount in question, such approximation has but negligible
effect. To take into account such practical factors, the author
of the paper has prepared curves giving the actual angle required
when the character of the end detail is taken into account to
gether with the character of the junction of the wing with abut
ment at the shoulder. It is emphasized^ that where minimum
volume, rather than minimum length is sought, the above rule
and curves do not hold. For minimum volume the wing wall
carried out directly in the plane of the abutment face gives the
least volume until the angle between the wing and the axis of
the retained embankment exceeds a right angle.
' Engineering NewsRecord, Vol. 80, p. 168.
«Ibid, p. 785.
3 Ibid, p. 1243.
132 RETAINING WALLS
For track elevation, where full trackage on a limited easement
is essential, the abutment frames into the two parallel retaining
walls on either side of the embankment forming a boxlike
structure. Other details are made to fit into the special cir
cumstances of the given location.
A number of examples of the varied types of gravity and re
inforced concrete abutments is given in the Handbuch fur
Eisenbetonbau iii Band, pp. 415 to 422.
For ordinary highway abutments it is possible to compile
standard sections to cover practically all the cases expected.
Thus H. E. Bilger in a paper read before the Illinois Society
of Engineers and Surveyors' states: For walls up to 25 feet in
height:
(a) For ordinary earth bottoms, the base is }^ the height;
(6) For rock or shale bottoms the base is ^'i the height.
The footing is 18 inches thick and is offset 9 inches at the heel
and toe. The back of the wall is vertical. Gravity walls
are generally used because the character of local labor does
not permit the use of the reinforced concrete sections.
Box Sections Subject to Earth Pressures. — The section, shown
in Fig. 77, subjected to earth pressure, both horizontal and ver
tical requires an intricate analysis, if de
,^"S!^" signed as a monolith. Since such struc
tures, though otherwise designed, are
actually rigid frames, it is quite desirable to
'ci„^r ^^^^^ t^® t^^® stresses existing in them.
'■Line The principles of the theory of least
work apphcable to the problem in question
I t ijH,mi,mww, may be stated as follows :
(a) The work performed by the shear
FiQ. 77. — Subsurface j xu j. • i i i • • ■,,
structures. ^°" ttirust IS neghgible m comparison with
the work done by the moment.
(b) The work performed by the moment between any two
points Si and Si is given by the expression :
(c) The derivative of this expression with respect to a force
that does no work i.e., a force whose point of apphcation is at a
fixed point, is zero.
^ Given in Engineering Record, Vol. 63, p, 205.
VARIOUS TYPES OF WALLS
133
Corollary : It is permissible to differentiate the expression under
the integral sign with respect to a variable other than the variable
of the integrand, thus
5£V(i^,«)<^.p^'^ (148)
Finally, it shall be arbitrarily taken that a moment which causes
compression in the outside of the member is positive.
In Fig. 78 the moments between the following points are :
C to a: M = Mi + Hx
a to A: =  Afi  W{x  a) + Hx
AtoB: = Ml  wlh  a) + Hh
B to D: same as c to A
The total work is, with 7i and /a the moments of inertia of the
roof and sidewalls respectively, and E the modulus of elasticity
Eh\jo
2EI
BZijo
Ja
{Mr + Hxydx+l [Ml W(x a)+ Hx^dx \
[Ml W(h a)+ Hh^dx (149)
w
w
H Ml
Fig. 78. Fig. 79.
Loads on subsurface frame.
Vd
The forces Mi and H shall be taken as the forces with respect
to which the partial derivatives of the work are zero. The points
C and D are taken as fixed. From the corollary and since
dw/dH = dw/dMi =
^\ r2(Mi + Hx)xdx +
EliXJo
/:
[Ml  W{x  a) + Hx\xdx\ +
1
EI
2EIi
L\ r  2i Ml + Hx)dx +
1I2 [Jo
r 2[Mi  Wih  a) + Hh]hdx\ =
134 RETAINING WALLS
i
 2[ Ml  W{x a) + Hx]dx] +
r
Solving these two simultaneous equations for H and ilf i
,^ Fa(fe  a)[/i/i(fe  g) + 6/2(2/}  a)] .. .„.
^' = mhh + 2bh) ^^^^^
„ WQi  ay[h{h + 2a)7i + h{2h + a)/;,] ^i^.x
^  /i'(Mi + 26/2) ^^^^^
In similar fashion,' referring to Fig. 79, the base moment and
horizontal thrust due to concentrated load upon the roof is found
to be
M.  ^„(6  «)r{jj^^,  5(^f5Jj (153)
Using these four equations as a foundation, it is possible to
establish some general conditions of loading on either roof, side
walls or upon both simultaneously. For a uniformly distributed
load on the roof of w per foot, replace in (152) a by a;, H^ by w,
multiply the expression by dx and integrate between the hmits
and h. The expressions for the thrust H and the mo ent Mx
are
For a uniformly distributed loading p on the side walls, in similar
manner integrate the expressions given in (150) and (151) be
tween the hmits and h. The thrust and moment are then
Mi=^ ^^^ + ^^^^ (157)
Again for a triangular distribution of loading on the side wall,
with maximum base intensity q, the expressions become
20 hh + 2bh ^^^^^
^^' 60 hh + 2bh ^^^^^
'See HiBoi, "Statically Indeterminate Structures."
VARIOUS TYPES OF WALLS
135
Denote the ratio ^ by e and let 1/(1 + 2e) = Zi; (2 + 5e)/
(1 + 2e) = Zi. Then (1 + 3e)/(l + 2e) = Za  1; (7 + 16e)/
(1 + 2e) = 3 + 2Z2; (3 + 8e)/(l + 2e) = 1 + 2Z2. Table 25
gives the values of Zi and Z2 for several values of the ratio e.
With the above substitutions the expressions given in (154 to
159) become
For uniform loading on roof.
4A
H =
Z^,
M. = ^Z.
(160)
For uniform loading on side wall.
Ml =
H = ?^Zi,
For triangular loading on side wall.
H = ^(3+ Z,),
Ml =
I2
60
To apply these expressions to a sub
surface structure subject to earth pres
sure upon roof and sidewalk, let the
loading above the roof Hne be treated as
a surcharge, with the usual terminology
that c is the ratio of this surcharge height
to the full wall height h. The roof load
ing w is then gch and the side wall
pressure is compounded of a uniform
intensity p = Jgch at the top of the side
wall, and a triangular loading with base
intensity q = Jgh. For a loading upon
the roof alone the respective thrust and moment are
H = ^fZi
{Z2  1) (161)
(1 + 2Z2) (162)
Table 25
e
Zi
Zi
1.00
2.00
.2
.72
2.14
.4
.56
2.22
.6
.45
2.27
.8
.38
2.31
1.0
.33
2.33
1.5
.25
2.37
2.0
.20
2.40
Infin.
.0
2.60
Ml
gch^h
12
Zx
(163)
(164)
For a loading upon the side wall alone the thrust and moment are
Jg\^
H =
20
[3 + (2 F 5c)Z2]
Ml = 
60
[1  5c I (2 f 5c)Z2]
(165)
(166)
136 RETAINING WALLS
For a simultaneous load upon roof and sidewall the two above
expressions are added to give the total thrust and moment. It
is possible, of course to have a different surcharge for the roof
than for the sidewall, since there may be no surface load over the
roof and a surface load whose weight will affect the sidewall pres
sure. This is taken care of by giving the proper values to the
surcharge ratio c in the above expressions.
With the thrust H and the base moment Mi known the moment
at any other point of the frame can easily be found by the ordi
nary principles of statics.
Fig. 89 is a typical section of such a structure analyzed by
the above method. A radically different distribution of stress
exists in this structure when analyzed exactly as above than
when it is treated as an assembly of independent units. It is
the very essence of the design of such structures, usually subsur
face, that they be waterproof. Any cracks developed in the
structure due to ignored stresses are fatal to the integrity of the
structure. It is patent that regardless of what method is em
ployed in designing such structures, provision must be mad ; for
stresses as found above.
The theory as above outlined and the formulas as given are
ample to analyze any subsurface structure subject to lateral and
vertical pressures.
The mutual effect of the members upon each other makes it
essential that such conditions be combined as will produce the
maximum stresses at the separate points of the structure.
It may be interesting to note, while treating subsurface struc
tures that avery thorough analysis, both theoretical and practical,
of stresses in large sewer pipe is given in Bulletin No. 31, issued
by the Engineering Experimental Station of the Iowa State
College of Agriculture and Mechanic Arts. See also for a com
parison between theoretical and actual stresses "Analysis and
Tests of Rigidly Connected ReinforcedConcrete Frames" by
Mikishi Abe, Bulletin No. 107. Engineering Experiment
Station, University of Illinois.
Economy of the Various Types.— Broadly speaking, the selec
tion of a given type of wall is governed by one, or more of the
following reasons: economy of section; character of foundation;
demands of the environment, in which latter may be included
the relation between walls and property line; architectural
treatment, the wall entering into a part of some general landscape
VARIOUS TYPES OF WALLS 137
scheme ; the availability of materials necessary for its construction
and the character of the labor to be had in the vicinity of the
work.
So far as the economy of the section is involved, it must be
noted that the relative economy of gravity and reinforced con
crete walls is not that given merely by a parallel comparison of
materials required for the finished wall. The reinforced concrete
wall has thinner members, requiring more form work per cubic
yard of concrete. The slenderness of this wall, together with
the network of rods within it, makes it more difficult to properly
place and distribute the concrete, necessitating more skillful
labor and more competent foremanship. The gravity walls
are more capacious within the forms, the laborers have, conse
quently, more room to move about and can thoroughly spade
and turn over the mix, giving better assurance of a flawless wall.
This is a very important item and one too frequently overlooked.
A concrete gang of the average type, i.e., a class of men just a
shade above the common excavators, will tackle a gravity section
of wall and turn out a good looking section. Upon attempting
to pour a reinforced concrete wall, a very inferior piece of work
is constructed. Before preparing plans for a thin reinforced
concrete wall, it is essential to insist upon a capable contractor,
equipped with the proper labor gangs to do such work. With
a policy of awarding the work to the lowest bidder where competi
tive bids are asked, it is necessary that the engineer adapt the
type of wall to one that can safely be built by the general run of
low bidders.
Unsuspected variations in the character of foundations, may
demand an abrupt change in the section of wall. For a rein
forced concrete wall the rods are usually ordered some time in
advance of the actual construction of the wall. It is necessary
that the section of the wall be determined at the time of ordering
the rods^. Despite careful boring made at the site of the work,
the soil encountered at the proposed bottom of the wall may prove
to be different from that assumed and it may thus become neces
sary to excavate deeper to obtain the desired character of bottom,
or even to change the type of wall. Since the rods have been
ordered, the wall design is inflexible and if a new section is
' While it is possible to get shipments from local markets at short notice,
quite a premium must be paid for this material and such orders are given
only when economy must be sacrificed to urgency.
138 RETAINING WALLS
ordered, it may mean delay awaiting mill shipments of the new
lengths needed, costly orders of rods from stock supplies, the un
desirable splicing of rods or the placing of a plain concrete base
to bring the actual bottom level up to the theoretical one — all
expensive and undesirable expedients. For this condition the
gravity wall is the more flexible type and the section may be
changed without any additional trouble should soils at
variance with the originally assumed ones, be encountered.
On the other hand, where the character of the soil is assured,
the reinforced concrete type of wall may be molded to adapt
themselves to any distribution of soil pressure desirable. This
has been shown in the previous work.
It has been pointed out^ ... for walls of the height re
quired for track elevation and track depression a gravity wall,
will under ordinary conditions be cheaper than the reinforced
concrete types.
Again, in the same issue of the Journal in discussing the relative
demerits and merits of the cellular types it was pointed out^ in
connection with track elevation work, that such a wall, with the
bottom left out offers great resistance to sliding and overturning
and "occupies the right of way so as to afford little opportunity
for encroachment. It permits of ready driving of a pile trestle
right over it." On the other hand "it occupies considerable
space before filling and may thus interfere with the use of the
tracks. Settlement may also give an unpleasing appearance."
So far as the actual amounts of materials involved, both
during construction (forms, etc.) and in the permament structure
it is possible to determine the more economical wall by com
parison of two types or by mathematical and tabular methods as
given at the end of this chapter. It is understood that the proper
weight is given to the indeterminate factors of cost as above
mentioned i.e. the construction limitations of the several types.
It must be emphasized that wall details should be simple.
Shapes that apparently make for economy may prove exceedingly
difficult to pour in the field. Thus for example, a section of a
cantilever wall as shown in Pig. 80 (see also Photo Plate No. 4a)
with a net work of obstructing rods at A makes it very hard to
get a good concrete at and below that point. The break in the
form work is also objectionable because of the added labor and
1 Journal of Western Society of Engineers, Vol. 20, p. 653.
^ P. 232, et seq.
VARIOUS TYPES OF WALLS 139
difficulty of pouring the concrete. When a shape, such as just
shown IS much more economical than the straight battered back,
It will be found that the counterforted wall will prove even more
economical, and should therefore be adopted.
Fig. 80.
Fig. 81.
Sloping the footing as shown in Fig. 81 may prove troublesome
and more costly in the end than the plain rectangular section.
Much, of course, depends upon the ability of the contractor to
carry out the niceties of the design and it is thus incumbent
upon the engineer planning an intricate section of wall to see
that its execution is placed in the proper hands.
One is tempted, in designing counterforted walls to mold cor
ners and make steel details as shown in Fig. 82, in order to effect
a thorough bond between the slab and the counterfort. These
Fig. 82.
details, again, demand extra form work, steel work and labor
and should therefore be employed with due appreciation of
the possibility of their added expense.
On the whole, that wall is most effectively and economically
designed which is most compactly and simply shaped.
With the rapid development of thin slab construction as
markedly shown in the construction of concrete ships and barges,
there is excellent promise of the extension of such work to re
taining walls. If the construction of thin slabs and intricate
140
RETAINING WALLS
details becomes commercially applicable, then a vast field is
opened to economic wall design, permitting the shape to follow
every peculiarity of the environment and to take advantage of
whatever economies the site may. offer. At present the prac
tical limitations of construction have restricted retaining walls to
but few types which in turn are limited in economic thickness by
field conditions.
Problems
1. An abutment is to carry two tracks as shown in Fig. 83. Each of the
stringers, under full load brings a reaction of 50 tons upon the abutment.
Determine the necessary dimensions of both a gravity and a reinforced
concrete "T" wall.
An abutment is ia combination of a retaining wall and a pier. Its eco
nomical design is affected not only by the type adopted, but also by the as
sumed location of the girder reaction. In the case of a gravity wall, the
vertical girder reaction, while assisting in the stability of the wall, may by
the location of its point of application, induce tensile stresses in the back of
the wall. Thus in Fig. 73, the girder load falls within the outer third,
violating an essential requirement of gravity walls. The selection of a type
as shown in Fig. 74 brings the girder reaction towards the center of the wall
and assists quite materially in the stability moment of the wall.
The distribution of these girder loads may be assumed to foUow within
planes making an angle of 30° with the vertical as shown in Fig. 83. The
abutment should be made long enough to permit the distribution to follow
along these planes. In addition, it is assumed that (for reasons given in
the following chapter) the abutment is independent of the adjacent struc
tures, so that the span loads will be confined within the abutment proper
as shown in Fig. 83.
Since the reaction from each girder is 100 kips, the area for bearing upon
the concrete, allowing 0.5 kip per square inch, is 200 sq. in. A plate 12" X
18" provides this bearing area. The plate will be placed as shown in Fig.
VARIOUS TYPES OF WALLS 141
84, where the remaining details of the girder seat are shown. As shown in
Fig. 83, the distribution of the loads spreads between a distance of 48',
niaking the load per linear foot at the foot of the abutment ^o%8 = 8.3
kips. As a retaining wall, prior to the setting of the steel, the height is
30' (above the footing) without any surcharge. From Table 12, a face
batter of 5" to the foot will give the necessary dimensions for stabiUty, and
will also satisfy the details of the girder seat.
The crane load is taken equivalent to 500 pounds per square foot. The
cases are lettered and discussed in the same order as on page 129. The
graphical analysis is shown in Fig. 85.
Fig. 84.
FiQ. 85. — Graphical analysis of abutment.
(o) The resultant intersects the third point (Checking the tabular value)
and B = 28.2 + 4.5 = 32.7
<Si = 65/13.5 = 4.8 kips per square foot. The permissible soil intensity
in this and the following work is taken as 4 tons per square foot.
(b) The resultant intersects at the ^ij point, and from (39)
fifi
Si = 70^(2 — 3 X 0.185) = 7 kips per square foot; which is within the
permissible value,
fifi
S2 = T^( — 0.44) = 2120 pounds per square foot, or 15 pounds per
square inch. This tensile stress in the concrete, developed under a crane
load prior to the setting of the span, is a permissible stress.
(c) This condition is quite similar to the preceding one, with the excep
tion that the indetenhinate factor of the frictional resistance between the
girder bearing and the abutment, together with the dead weight of the span
add to the wall stabiUty.
(d) For this case (that of full loading) the resultant is found to intersect
exactly at the third point. /J = 42 kips
Si = 84/13.5 = 6.2 kips per square foot.
The section, then, satisfies all the necessary conditions of design and
construction.
Reinforced concrete section. Assume, as in the case of the ordinary re
inforced concrete retaining wall, the criterion of economy, i = e. Let the
total toe pressure not exceed 7 kips per square foot, leaving a margin for the
142
RETAINING WALLS
toe pressure caused by the girder load. Note here, that since a skeleton
section of wall is assumed, with the point of application of the resultant
located at the vertical stem of the wall, the girder load, which is at the same
point, can have no effect upon the wall dimensions, and merely increases
the intensity of the soil distribution. From (95), with Si = 3.5 tons per
square foot, H = 38' (taking the thickness of footing 3 feet) allowing for a
five foot surcharge:
5
'=1
IV
1 +
120 X 3.6
= 0.26
6\ ' 38
Take the point of application of the resultant, and the location of the face
of the abutment at the quarter point of the base. Prom Table 18 with
this value of e and i, k = 0.50 and the base width w is, accordingly 16.5
feet. With a girder load of 8.3 at the quarter point, from (39)
2 X 8.3 „
0.75) = 1.25
and the total toe pressure is 8.25 kips, a permissible excess over the allowp.ble
4 tons per square foot.
The height of the vertical stem is 30', and from Table 21 the critical height,
above which the shear controls the thickness of the stem is less than 30'.
The thrust for the given surcharge is 20 kips, located 11.4 feet above the
top of the footing. From (113), the thickness of wall because of shear is
d = 20/5.04 = 3.95
A thickness of 4' will be used at the base.
The footing moment is found to be 119 ft. kips and the depth for balanced
reinforcement is, from (101)
d = V(iiM6) = 2.75
requiring a thickness of 3 feet. If no special stirrup reinforcement is placed
to take care of the diagonal tension, an excessive depth will be required for
Fig. 86. — Graphical analysis of abutment.
the shear (24.5 kips). For this reason it will be assumed that such rein
forcement is employed here and the depth of the slab adopted will be that
required by the bending moment. The thickness of the toe extension will
also be taken as 3 feet, bearing in mind that the thickness of the footing.
VARIOUS TYPES OF WALLS 143
both heel and toe, must, for construction reasons, be kept the same
The introduction of concrete fillets at the junction of the footing and arm
would obviate the need for web rods and a comparative estimate may prove
that the fiUets, with the extra work involved, are cheaper than the compli
cated rod details of web reinforcement.
Discussing the separate cases of loading, treated graphically in Fig 86
for the case of total loading (Case d) the point of appHcation of the resultant
is at e = 4.75/16.5 = 0.288; whence from (39), with J? = 59 kips. Si = 8100
pounds per square foot.
V^
oV">tV
a4"
i_r
Fig. 87.
eroundSurface ^Surcharge of SO'
T'iiiiii)iii)>iimi)imimmmi wuui'Kiimmii i iminiiiimmmmr
^
2B0
FiG. 88.
Omitting the span load (Cases 6 and c) the point of application of the
resultant is at e = 4.5/16.5 = 0.273 and with R = 51 Si = 7.3 kips per
square foot.
The section as shown therefore satisfies the governing conditions. The
wall should be recalculated, using the dimensions and loadings as actually
found.
Fig. 87 shows the sections of the gravity and reinforced concrete walls.
2. Find the stresses, moments, etc., in a box section as shown in Fig. 88.
It is necessary to make a preliminary assumption in order to proceed with
144 RETAINING WALLS
the analysis of this section uhder the theory of least work. For this reason,
it will be assumed, tentatively, that the moments of inertia of the sidewalls
and roof are equal. Adding two feet to b and one foot to h, gives the dimen
sions along the gravity axes of the section. The value of e is now ^ J^g =
1.69. From Table 25, Zi = 0.23 and Zj = 2.38. The value of c = 1^6
=0.875. J is then taken at its usual value }i.
For roof loading alone
ff=:i^^^^X 0.23 =3.7 kips
M = 1 X .875 X 27^ X 16 ^ ^^^ ^ +19.7 ft. kips.
For sidewall loading alone
H = '^^20 ^^ + ^^^ ^ ^^^^ = 7.8 kips
M =  ,■ an (1  438 + 6.38 X 2.38; 27.0 kips.
o X oU
For simultaneous loading
H = 7.8 — 3.7 =4.1 kips, directed outwards,
il^, = 27 + 20 = 7 kip feet.
At any point x, above the base, where x = kh, the moment is
M^ = 7 + Hkh  ^ [3(1 + c  k)k^ + 2k']
=  7 + 66A;  22.7/cH5.6  k)
For the various values of k, M^ has been tabulated as shown in accompanying
table. The roof moment at any point y, where y = pb, is, taking the last
found value of Afi as given in the table, — 46,
k M,
07 Af =  46 + 510p(l  p)
.1 — 2 A table has been similarly prepared for a set of values
of p, up to the center of the span.
.2
+ 1
.3
+2
.4
.6
3
.6
8
.7
16
.8
24
.9
34
1.0
46
P
M
46
.1
.2
36
.3
61
.4
76
.5
82
The assumption that the roof and sidewalls are simul
taneously loaded does not, necessarily give the maximum
moments. During construction it is quite possible that
the side walls will be loaded up to the roof line, before
any load is placed upon the roof. The only roof load
is then its dead weight, which, with the assumption
that the roof is two feet thick, gives a load of 0.3 kips
per foot. There is a triangular distribution of pressure
along the side wall, with a value of g = 1600/3 = 0.53 kips.
VARIOUS TYPES OF WALLS 145
For roof loaded alone, from (160)
H  3 X 27' X .23 ^ ^ , .
r>06 =0.8 kips
jj. _ .3 X 27^ X .23 .„,. , ^
For side wall loaded alone, from (162)
^r— 2^^^3+2.38) K2.3,
M.==:. ^— ^ (1 +4.76) 13.1lkip_feet
Under the simultaneous loading
H = 1.5 directed outwards.
Ml =  9kip feet.
As before, x = kh, and c =
M^ = 9 + 24A;  22.7i;H3.6  k)
A table of values of M for the side wall is given here.
The roof moment is, with p the same as above,
Af =  44 + lllpd  p)
A table of these moments up to the center is given here.
A further condition of loading may be
anticipated. With time the effect of
cohesion may materially reduce the side
wall pressure, or due to a variety of con
ditions, the side wall pressure may be
considerably less than that assumed.
Let this state of loading be analyzed
upon the assumption of a full roof load
ing and a sidewall pressure as given in
the work immediately preceding.
For roof loading alone, from before
H = 3.7; M = 19.7 ft. kips
For the side wall loading as assumed
H = 2.3 and M = 13.1 ft. kips
The net thrust due to both loadings is 1.4 directed outwards, and the mo
ment is +6.6 ft. kips.
M:, = 6.6  22A;  22.7A;2(3.6  k)
The tabular values for the moments in the sidewall are again shown in the
accompanying table.
k M The roof moment is
+7 74 + 510p(l  v)
.1 +4 The values, for this moment up to the center of the
.2 +1 span are given in the table.
.37 p M
.4 14 74
.5 22 .1 28
.6 31 .2 +8
.7 40 .3 +33
.8 52 .4 +48
.9 63 .5 +53
1.0 74
10
k
M
The roof
 9
.1
 7
A table of tl
.2
 7
V
M
.3
 8
44
.4
11
.1
34
.5
15
.2
26
.6
19
.3
21
.7
24
.4
17
.8
32
.5
16
.9
37
1.0
44
146
RETAINING WALLS
The structure is designed to satisfy the maximum moments shown in the
diagrams. The maximum roof moment is 82 with practically an equal but
opposite moment at the fixed corner. The thickness for balanced reinforce
ment is found to be 2.25 feet. The steel ratio 0.0075, requires 2.4 square
inches per linear foot; too heavy a reinforcement. A thickness of 33", or
3 feet overall is finally adopted, which requires a steel reinforcement of 1
inch square bars spaced 6". The maximum side wall moment will occur
about at A; = 0.9 (since the roof is 3' thick), whence Af = — 63 ft. kips.
Again, although balanced reinforcement needs a 2' slab, to keep the rod
weight within reasonable limits a 27" slab will be used, with an overall
dimension of 2' 6". For this condition 1" bars 6" apart are required.
•^ 60
■ t
/"fflsofe J8 "OoC. between these Poinfs
l"'f!ods,6'afvC.
3' >
E6
Fig. 89.
The moments of inertia of these sections, it is noticed, do not fulfill the
assumed condition. To take the ratio as found for the sections above, will
again prove slightly incorrect in the final analysis, and for this reason an
intermediate value of the moment of inertia ratio, between that first assumed
and that now found will be used. The moments of inertia of rectangular
sections, of the same width are to each other as the cubes of their depths.
The ratio 72//i = 15.6/27 = 0.58. The average of this value and the value
1, first taken is 0.79. The value of e is now 1.3, making Zi and Z2 0.28 and
2.35 respectively.
In tabular form the moments at the three important points, for the three
conditions discussed above are
Condition op Loading
C
Full roof and sidewall — 2
Dead weight roof and light wall — 5
Full roof and fight wall fll
A Center of roof
56 +71
50 25
83 +44
VARIOUS TYPES OF WALLS 147
It is seen that quite a large variation in the assumed values of the moment
of inertia ratio has but sluggish effect upon the moments and it is probably
safe to take both the roof and sidewalls of the same thickness, subject to a
bending moment of 70 foot kips at the center of the roof and at the upper
fixed corners, and to a negative moment of —25 foot kips at the center of
the roof.
The final section must take care of the moments throughout the frame
detailed in accordance with the adhesion requirements and bent in accord
ance with the bearing formulas given in the preceding chapter. Fig. 89
gives a layout of the section, with the rod layouts as indicated by the
previous work.
It must again be emphasized that the stresses existing in a structure of
this character are quite different from those which are found upon analyzing
the structure into its separate members and when a subsurface structure is
built as shown above, provision must be made for the distribution of stresses
as given by the analysis just made.
The Selection of an Economical Type.' — While, clearly, for
some given height, a counterforted wall becomes cheaper than a
cantilever wall, a search of pertinent hterature fails to yield any
method of obtaining such a height, save by actual comparison of
two completed designs. It may be well worth while to establish
some method of obtaining this "critical" height.
It is true, extraneous factors may control the selection of types
of walls and the dimensions of the component members, but
generally, a wall is so designed as to satisfy, most economically,
its stresses.
Again, the bending moment, shear, or bond stress, may each
in turn control the necessary thickness of the several parts of the
wall, as the height is varied. It is to be noted that, with few ex
ceptions, such several stresses usually require about the same
thickness of section, though probably, a greater variation in the
amount of reinforcement required. In assuming that the wall
dimensions follow the theoretical requirements a large percentage
of actual cases are covered and, if, further, these dimensions are
taken in accordance with the stress of simplest expression, no
serious error results. With this in mind, the various thicknesses
of both the cantilever and the counterforted walls are those
selected in accordance with the bendingmoment requirements.
In the work that follows, since it is a comparative estimate of
the cost of the two types that is sought, it is justifiable to select
as a type for the present analysis, that involving the least mathe
matical analysis. It is quite clear that variations in the toe
' Reprinted from Engineering and Contracting, Feb. 26, 1919.
148 RETAINING WALLS
length or in the assumed position of the resultant, will not affect,
to any material extent, the comparative estimate. For this
reason, the condition for economy as given on page 82 is adopted
here, with a further provision, that e = ^^, the usual soil pres
sure distribution. With these conditions (91) then becomes
k
"2Vt
+ 3c
The dimensions for the "T" cantilever are taken as follows:
the thickness of the base of the vertical arm, from (112) is
d, = 0.0185 h^Vl + 3c = C^h^
and the thickness of the top of the arm is taken at its usual mini
mum value one foot. For the footing, from (119) /is about 0.7
and the required thickness of the footing slab is then y/Tf or 0.84
times the arm base thickness. For the counterfort wall, from
(126) with the usual value of the constants the thickness of the
vertical slab is
d\ = 0.0132m \A(1 + c) = C',mylh
and that of the footing, from (138) is
d'i = ylM\,
The counterfort itself is usually one foot thick and will be so taken
here.
The cost of the steel rods is a small part of the total cost of the
wall and the relative difference of the cost of the steel rods in the
two types of walls would thus be negligible.
The amount of face and rear forms for the vertical arm of both
types is substantially the same and will not enter into the com
parative estimate. The variable factors in the comparative
estimate are then: the amount of concrete in either type and the
forms required for the counterfort itself.
Let L be the total length of wall under consideration, r be
the cost of placing concrete into the forms (the cost is practically
the same for both types) and let t be the cost of the form work and
necessary bracing, per square foot of concrete face supported.
For the counterforted wall the amount of concrete is
Lid\h + khd'i) + ^ ^
m 2
VARIOUS TYPES OF WALLS 149
and its total cost
Lrtjd'„(l + kVS) + ~\
The cost of the face forms for the counterfort i
IS
m 2
making the total variable cost of the counterfort wall
Lrt{d'„(l + fcV3)+^(l+2^)) (167)
The volume of the "T" cantilever is
L (^^/i + khd, ) = m[^ + d„ g + OMk) ]
and its total cost
M'"{~ + rf.(^ + 0.84A;)} (168)
Equating (167) and (168)
d'„(l + WW) + ^(^ + 2~) = 0.5 + ci!„(0.5 + 0.84A;)
Replacing the thicknesses of the sections by their values given
above
C\mVh(l + hy/2,) + ~ (1 + 2") = 0.5 + C.h
(0.5 + 0.84/b) (169)
Later it will be shown that the economic spacing of the counter
forts is given by
m = 3.1 Rhy*
where
With this value (.169) becomes
Cih^  RCihy* + M =
a quadratic in h^
with C2 = .0132 vTTc 3.1 (1 + k^/S) + ~
and Ci = .OlSeVi + 3c (0.5 + 0.84fc)
The value of h^* is — ^
150
RETAINING WALLS
Table 26 gives a series of values of this critical height h for several
values of the cost ratio t/r and the surcharge ratio c.
Table 26
V
K
>i
K
1
15
22
28
33
M
11
17
22
27
K
10
15
19
23
Economic Spacing of Counterfort. — To determine the spacing
of the counterforts to give the most economic wall sections, it is
seen that (167) is the required expression for the variable cost of
the counterforted wall as the spacing of the counterforts change.
If, by the theory of Maxima and Minima, the derivative of this
expression with respect to m, is put equal to zero, there results,
after replacing the several thicknesses by their values as previously
found
/ feVfeCl + 2t/r) _
\2C'.(1+A;V3)
^/K
1 + 2
.0132\/l + c(l + fc\/3)
With R as given above, and noting that the expression
v;
k
'(HfcV3)\/l + c
after using the value of A; as given in (91) is practically constant
and equal to Mj this expression becomes
__ m = 3.1Rh^*
Table 27 gives a series"of values of m for'the several values of
t/r and the height.
Table 27
H
15
20
25
30
35
40
50
V4
7.5
8.1
8.6
8.9
9.3
9.6
10.2
Vi
8.6
9.3
9.8
10.2
10.7
11.0
11.6
%
9.6
10.4
11.0
11.5
12.0
12.4
13.1
1
10.6
11.4
12.0
12.6
13.1
13.5
14.3
it is reasonable to expect that the laws governing the theory of
probabilities hold here^ and that, therefore, the small errors
introduced in the above approximations are fairly compensatory.
Plate II
» . *■»
• "' *'''«''fF '
'^ ,' 7!
#
(F'irhifi page ir,0)
Plate III
^ ■■;' ';i»i»<
l'i.i. C. (;rack :i(. sharp corner of wall due to tension component of thrust.
CHAPTER V
TEMPERATURE AND SHRINKAGE STRESSES, EXPANSION JOINTS,
WALL FAILURES
In the setting and curing of concrete and in the seasonal varia
tions in temperature, stresses are induced in retaining walls
which, because of the longitudinal continuity of the wall, must
be resisted by the material itself. Plain concrete monoliths, un
reinforced, will crack at well defined intervals because of failure
of the material through tension. It is quite difficult, despite
the insertion of rods to prevent cracks. It is possible, however,
by properly introducing rods, to concentrate the tendency to
cracking at assigned intervals and then, to avoid unsightly
breaks, to place an actual joint at such places. Reinforced
walls are at times built without any joints and seem to have
such proper reinforcement that no cracks are apparent.
A theoretical discussion of the temperature changes that may
be expected within masonry masses may be interesting as indicat
ing the expected amount of stresses to be anticipated by rod
reinforcements.
It is patent, that the further from the exposed surface a point
is within the mass, the smaller will be the variation of tempera
ture at that point for any given surface range of temperature.
Experiments have been made to determine this range at various
points, covering quite long periods of time' and in recent masonry
dam construction, automatic temperature recording devices
have been incorporated in the work so that an exhaustive record
of the variation of temperature is available.
It seems desirable to attempt to express, mathematically, this
distribution of temperature and, in view of the fact that the
theoretical results so obtained are reasonably in accord with the
experimental results, they should prove of service in making
provision for temperature stresses in masonry structures.
' Trans. A.S.C.E., Vol. Ixxix, p. 1226.
151
152 RETAINING WAI.LS
The variation of seasonal temperatures at the surface may be
given by an expression of the form,
u = A + Bcos^t (170)
in which u is the temperature, A and B are constants, T is the
period of change and t is the time.
In the distribution of heat through large masses, where the
temperature at the surface is a function of the time, it can be
shown! that the temperature u at any distance x from the surface
at the time t is
u = A + Be''^cos {2t/T  kx) _ (171)
in which e is the base of natural logarithms and fc =v/^ o^ is
known as the coefficient of thermal diffusivity, which, for concrete
(Smithsonian Physical Tables) is 0.0058 in the C.G.S. system.
The maximum range of temperature occurs between t equal
any integer say n and t = n + ^i At the surface this range
becomes, from (170) 2B; at any point x from the surface the
range is from (171) 2Be~'"' cos kx. The ratio of the range at any
point X to that at the surface is
e''" cos kx = Ix (172)
and if U is the surface range, that at any plane x away from the
surface is UIx
In discussing seasonal changes, the period T is one year, which
must be expressed in seconds in accordance with the diffusivity
constant a^. For this period, and for concrete fc = 0.00413.
Table 28 shows a comparison with the results from the formula
and those experimentally found in the records quoted above. ^
The daily range may in itself be taken as periodic and expressed
by (170) and (171). For this period, one day expressed in sec
onds fc = 0.079. Table 29 gives a parallel comparison between
the theoretical and the experimentally determined range.
It is seen, from a study of the daily variation of temperatures
that the surface range is rapidly decreased a few inches from
the surface. In designing masonry structures it is sufficient, in
making provision for the temperature range to take a seasonal
range based on about weekly averages. For climates in the
'W. E. Btbrly, "Fouriers Series and Spherical Harmonics," p. 89.
2 Tables for r" are to be found in Pierce, "A Short Table of Integrals."
TEMPERATURE AND SHRINKAGE STRESSES 153
Middle Atlantic States, this range is about 40° either way from
the mean.
Table 28
Table 29
Theoretical
Actual
range
range
0.0
1.00
75
75
1.0
.87
65
2.0
.76
57
3.5
.57
43
32
5.0
.42
31
10.0
.09
7
12
20.0
.04
3
X
Ix
Theoretical
range
Actual
range
0.
1.00
50
50
.25
.45
22
.50
.11
5
1.0
.07
3
2
1.5
.02
1
2.0
.01
1
1
2.6
.002
3.0
.000
3.5
If the unit stress developed by a change of one degree in the
temperature is s and if the surface range is U, then the stress at
any x is sUIx and the total stress across a section of thickness
w and unit width is
sUX"'Ux = sUf,"" e^'^cos kxdx
= sUcw, (173)
where
cw
gT le^^Csin kw  cos fci«) + 1  (174)
and the average unit stress over the section is csU. Table 30
gives the value of c for various values of w.
Table 30
Seasonal change
w
c
i
1
.95
.48
2
.87
.47
3
.82
.46
4
.75
.43
5
.70
.42
6
.65
.41
7
.60
.39
8
.55
.37
9
.51
.35
10
.47
.33
154 RETAINING WALLS
If E denotes the modulus of elasticity for masonry and n the
coefficient of expansion,
s = nE (175)
For concrete this value of s is about ten pounds per square inch,
for every degree change in temperature (Fahrenheit).
Replacing w in (173) by the area of the concrete section Ac,
the total stress across a section is
csUAc. (176)
Let the range of temperature where the steel rod is to be placed
be U' and let the area of steel be As, with the ratio of steel to
concrete area, as before p. The stress developed in the steel by a
change of one degree is s' and will be ns, with n the ratio of the
two moduli (see page 86). The total stress across a section
because of a surface range of U is then
csUAc + Ass'U'. (177)
The concrete can take fc pounds per square inch before failure
and the steel can take /s pounds per square inch up to its elastic
limit. The resisting section to the above temperature stress
is thus
fsAs + fcA, = fsvAo + fcAa (178)
Equating (177) and (178) and solving for p
For example, take a range from the mean, as above of 40°, and
average slab thickness of two feet, /<, = 200 pounds, and /, =
45,000 pounds. From the Table 30 c = 0.87, and since for a
cantilever wall, where the vertical rods are at the rear face it is
customary to Ukewise place the check rods (for convenience of
construction) at the rear face from Table 28 7^ = 0.76, whence
V = 0.76 X 40° = 30°. The required ratio of steel is then, from
(179) with s' = 15 X 11 = 165
^ 0.87 X 10 X 4 ^_j00 _
^ 45,000  165 X 30° ~ '""'^^
Specifications usually require about }i of one per cent, of steel
for temperature reinforcement, which agrees fairly well with the
above value just found. It is seen that a steel of high elastic
TEMPERATURE AND SHRINKAGE STRESSES 155
limit should be specified. The expansion coefficients of both
steel and concrete are fairly alike so that there is no stress in
duced between steel and concrete because of this temperature
change.
Shrinkage.^Unlike temperature stresses, the stress due to
shrinkage is induced in the steel by the action of the concrete
in curing and drying out. While there is little definite regarding
the theory of shrinkage experimental data has shown* that the
shrinkage of concrete is about 0.0004 of the length. In the same
paper the stress due to the shrinkage is given by the expression
fc = CE, ^ (180)
* 1 + np
C is the coefficient of shrinkage (given above) E the concrete
modulus, n and p the usual concrete functions. The stress
induced in the steel is then
/. = fc/p (181)
With the amount of reinforcement as specified for tempera
ture stresses, the concrete stress is seen to be, from (181) 40
pounds per square inch and the corresponding steel stress about
12,000 pounds per square inch.
To provide for temperature and shrinkage stresses the rods
should be placed at right angles to those put into take care of the
earth pressure stresses. Since the maximimi temperature ranges
occur at the surface, it is desirable but not necessary that the
rods be placed at the surface. It has been seen that for the canti
lever walls it is not feasible to place the rods at the face. Gener
ally these rods are woven in with the vertical stress rods.
Settlement. — The settlement of a wall is intimately connected
with the character of its foundation. From the discussion on
foundations in Chapter 2, it was seen that certain types of soil
require a distinct distribution of loading; the more yielding the
soil was, the more urgent it became that the distribution of soil
pressure be a uniform one. It is generally agreed, that, within
reasonable limits (these limits determined by the structures
adjacent to or supported by the wall) a uniform settlement of
the wall is harmless, since, with a proper spacing of expansion
joints, or with carefully distributed reinforcement, no cracking
will occur in the wall body. Unequal settlement produces
' See Bulletin No. 30, Iowa State Agricultural College.
156
RETAINING WALLS
cracks, which not only prove unsightly, but may indicate incipi
ent failure.
Unequal settlement may be expected on yielding soils where
the distribution of pressure is not a uniform one; where the char
acter of the soil changes, one type yielding more than the other
type; at junctions of new and old work, the old work having
settled with the soil, the new, in gradually taking up its settle
Deformed
Bars "■
'Hailroad Kails
Fig. 90. — Bottoms reinforced because of threatened settlement.
ment, necessarily destroying the bond between the new and old
work. The remedies for these are quite obvious. For the first
case it has been sufficiently emphasized that there must be a uni
form distribution of pressure. A joint should be placed in the
wall wherever the character of the soil changes and especially
between a yielding and nonyielding soil. Joints should also be
placed between new and old work. It is a good detail, where
Rods in Vertical
Arm
Fig. 91.
settlement is expected, to reinforce the bottom of the footing
with longitudinal rails or rods as shown in Fig. 90. Such rein
forcement will tend to distribute any impending movement and
thus prevent a crack.
While of common occurrence it is poor practice to make a
wing wall monoUthic with the abutment, save on unyielding soils.
The character of loading for each type is radically different mak
TEMPERATURE AND SHRINKAGE STRESSES 157
ing unequal settlement inevitable. Reinforcement across the
junction of the two walls is uncertain and cracking may occur
despite such rods. A photograph (Plate No. 2a) and Fig. 91 are
given illustrative of this.
"While settlement is an uncertain problem, careful attention to
the foregoing points will reduce to a minimum the chances of
cracks on these accounts. Where the face of the wall is to re
ceive special treatment or is to be panelled, it is vital that every
precaution be taken against unsightly cracks. As in the case
of foundations, the provisions to be made against expected set
tlement demand most mature engineering judgment. A large
crack in a wall is usually an indication of lack of engineering
foresight and where such work is adjacent to public highways,
becomes unpardonable.
E^ansion Joints. — Where movement is expected in a wall, due
to any of the interior or exterior changes discussed in the fore
going pages, it is customary to attempt to localize such movement
to small sections of the wall. For this purpose, vertical joints
are placed in the wall at regular intervals and are constructed so
that no movement can be carried vertically or longitudinally
across them. Since it is desirable that a wall be kept in
good line, the joints are usually so built to prevent transverse
movement.
In a monolithic gravity wall, joints are essential and are cus
tomarily spaced at from 30 to 50 feet intervals. This makes
ample provision for temperature and shrinkage stresses and makes
it possible to have complete concrete pours from joint to joint.
An excellent detail of such a joint is shown in Fig. 92, giving
Fig. 92. — Expansion joints.
freedom of movement in every direction except a transverse one.
One section of wall is poured completely between the joints.
After the joints are given a coat of some tar or asphalt prepara
tion the adjoining sections are then poured. To prevent seepage
of water into the joint, several layers of fabric and tar are placed
over the back of the joint and extend about l}i feet on either side
of it and from the row of weep holes at the bottom of the wall up
to the top of the wall.
158 RETAINING WALLS
While, theoretically, steelconcrete walls can so be reinforced
that expansion joints are unnecessary, such implicit confidence
in the theoretical action of such rods is not wholly warranted
and expansion joints are usually placed with about the same
frequency as in plain concrete walls. The check rod system then
distributes all movement to these> joints and the wall is surely
safe against cracking. Mr. Gustav LindenthaP has stated that
expansion joints are a source of danger because of the possible
accumulation of water in them with a threatened wedge action
due to ice formation. Accordingly, in the walls of the New
York Connecting Railroad, described on page 127, no joints were
used, full dependence having been placed in J^ per cent, of rein
forcement to take up whatever secondary stresses were induced
by temperature changes, shrinkage and settlement. General
engineering practice is, however, not in accord with this view
and expansion joints are almost universally used in reinforced
concrete walls.
The details of an expansion joint for the cantilever wall are
simple and may be made the same as the detail for the gravity
wall shown in Fig. 92. For the counterforted and other slab
types of wall, a break cannot be made in the face without provid
ing a special detail. It is, of course, possible, in the case of
counterforted walls, to build two adjoining counterforts with the
:ExpansionJomi XantikverArms ^
^'Rods to fake
Canfi lever Momenh'
Fig. 93. Fig. 94.
joint immediately between them as shown in Fig. 93, but such a
detail is necessarily a costly one and to be avoided. Generally
the joint is made midway between the two buttresses and the
slab in between is made up of two cantilevers as shown in Fig. 94.
The bottom slab, buried in the ground can usually be made con
tinuous and the expansion joint need only extend to the bottom
of the vertical slab. This applies equally well to the cantilever
type of wall.
In stone masonry walls it is inexpedient to place any joints in
the wall, but where the stones have carefully been bedded any
1 Engineering News, Vol. 73, p. 886.
TEMPERATURE AND SHRINKAGE STRESSES 159
movement is usually taken up and distributed by the mortar
joints. It is essential, of course, that there be the proper
ratio of headers to stretchers to effectively distribute all such
movements.
Construction Joints. — Any break in the continuity of pouring
a wall, other than at an expansion joint, leaves a joint in a wall,
which is usually termed a construction joint. It is not generally
possible to pour a section of a wall between expansion joints
completely in one continuous operation. It is impractical,
usually, to, indicate such construction joints in advance, due to
the exigencies of field conditions. The steps in pouring are
generally: the bottom slab is poured; the vertical is later poured
in as few operations as possible. While such a sequence does not
give the. ideal location for such joints, by the proper keying and
cleaning of the construction joints, the strength of a wall may be
satisfactorily maintained. It may be interesting to note a series
of tests on the efficiency of various modes of treating a construc
tion joint to insure a proper bond between the old and new work.
H. St. G. Robinson, Minutes of the Proceedings, Inst, of C. E.,
Vol. clxxxix, 19111912, Part III, p. 313, has performed the fol
lowing series of tensile tests taking the efficiency of a solid prism
as 100 per cent. A series of five tests upon this solid prism gave
an average ultimate strength, in tension, of 329 pounds per square
inch.
For the abutting faces (newand old) merely wetted, the effi
ciency of such a joint was 38.3 per cent, of the solid. A series
of five tests gave an average ultimate strength of the joint of 126
pounds per square inch.
For the abutting faces roughened and wetted the efficiency was
56.2 per cent, of the sohd. A series of six tests gave an average
ultimate strength of the joint of 185 pounds per square inch.
For the abutting faces treated with acid the efficiency of the
joint was 82 per cent, of the solid. An average of six tests gave
an ultimate strength of 270 pounds per square inch.
For the abutting faces roughened and grouted the efficiency
of the joint was 85.5 per cent, of the solid. An average of four
tests gave an ultimate strength of the joint of 281 pounds per
square inch.
From the above it is evident, that by cleaning and grouting the
surface on which the new concrete is to rest almost the full effi
ciency of the joint will be attained.
160
RETAINING WALLS
It must be noted that construction joints in the face of a wall
leave a permanent, and often unsightly mark. This matter is
discussed somewhat in detail in a later chapter.
It is now possible to complete the reinforced concrete design of
Chapter 3. The secondary rod system for temperature, shrink
age and settlement may now be added to the sections shown
in that chapter. For simplicity of construction the rods are
usually attached to the primary system of the wall. In the
"L" and "T" walls the rods are horizontal as shown in Fig. 95.
If the distance between expansion joints is too large, or if there
are no expansion joints, it becomes necessary to splice these rods.
The rods are carried beyond the point of splice each a distance
sufficient to develop the rod in adhesion.
yCheckFfods
Chuck Rods.
Fig. 95.
Fig. 96.
While strictly, such rods are unnecessary in the footing, they
will act as a distributing system in case of threatened
settlement.
For the counterfort and other slab sectioned walls, the check
rods are vertical and placed at the outer face, see Fig. 96.
Small size rods are desirable for this secondary system, both
on account of the adhesion area and because of the ease in hand
ling the long lengths. A high elastic limit steel should be spe
cified (see specifications at end of book) .
Wall Failures. — It was a famous maxim of Sir Benjamin Baker,
that no engineer could claim to be experienced in the design and
construction of retaining walls until he had several failures to his
credit. Such, however, is not the viewpoint of the modern engi
neer. It is today clearly apparent that walls, when they do
fail, fail for definite reasons that can generally be anticipated and
for which provision can be made. It is necessary, not only to
find a proper foundation for a wall, but also to take extreme pre
caution that such a foundation will be maintained permanently
in its proper condition. It is essential to guard against possible
TEMPERATURE AND SHRINKAGE STRESSES 161
saturation of the bottom and against erosion of the soil beneath
the toe by streams of water which, if long continued, reduce the
bearing capacity of the soil and lead to subsequent failure. A
majority of partial and complete wall failures are clearly at
tributable to foundation weakness developed subsequently to
the construction of the wall.
Cases of failure due to excess of overturning moment over
stability moment are rare. It is possible that in placing the fill
behind the wall, material may be dropped from some height,
either striking the wall or setting up vibrations in the retained
mass that may exert an excessive action upon the wall. A failure
of a barge canal wall in New York State^ is alleged to be due to
this cause. The fill behind the wall was saturated and in a quak
ing condition. The material was dropped behind the wall by a
clam shell, from considerable height, setting up heavy vibrations
in the mushy mass, which eventually destroyed the wall.
Care should be observed in dropping big stone from trestles or
from the partially built embankment against the back of the wall.
While complete failure is unlikely, small cracks, due to the im
pact may be developed. At first not serious, later, due to frost
and other weathering action, they
become unsightly, marring the
face and eventually develop erosive
gullies.
The improper and insufficient
attention to drainage (discussed in
a later chapter) may permit the
accumulation of water behind a
wall increasing the pressure to
such a degree as to push the wall
out of line.
Among minor instances of possible causes of failure, complete
or partial, may be mentioned the following.
Lack of expansion joints, or joints spaced too far apart.
The junction of radically different types of walls without a
proper joint. Thus a wing wall tq an abutment; a very light
section wall to a heavy section wall. Walls on different founda
tions. Walls carrying a building load. A sharp angle in a
gravity wall, so that there is a component of the earth pressure
acting in tension (see Fig. 97, and Photograph Plate No. 3a).
1 Engineenng News, Vol. 67, p. 384.
11
Fig. 97.
162
RETAINING WALLS
In the Trans. Engineer's Society of Western Pennsylvania,
Vol. 26, it was noted in gravity walls, where the base varied from
M to 3^ the height, that : /
"Such failures as have occurred have been due, to the most part
to poor construction and lack of drainage."
In discussing the action of clay, both as a fill and as a foun
dation material. Bell, Minutes of the Proceedings, Inst. C. E.,
Vol. cxcix, 19145, Part 1, p. 233, notes that:
"it was disquieting to note the high percentage of failures in works
constructed in clay. Taking all the available records of works subject
to earth pressure, which had failed, it appears that 70 to 80 per cent,
referred to works constructed in clay. While every one recognizes
that clay is a treacherous material and that it will always claim a
substantial percentage of total failures, still this preponderance is
remarkable and would perhaps of itself indicate that there is something
wrong with existing methods."
Some Wall Failures.— Chas. Baillarge^ has pointed out that
the life of the retaining walls in Quebec has been but a brief one.
They were designed upon the assumption of a dry granular fill
and the base, accordingly was made from onefifth to onethird
the height. Subsequently the filling became waterlogged and
since no weep holes or other drainage had been provided to dis
pose of such accumulations of water, the excessive pressures
developed caused the failure of the walls.
■ ^ ■  .„
Fig. 100.
Mr. Lindsay Duncan^ has described the tilting and settling of
an abutment prior to the setting of the span upon it. The sec
tion of the abutment is shown in Fig. 98. The wall rested upon
an adobe foundation and surface waters gradually softened the
1 Engineering News, Vol. 45, p. 96.
2 Engineering News, Vol. 55, p. 386.
TEMPERATURE AND SHRINKAGE STRESSES 163
adobe, causing the wall to tip forward. An ingenious method
of reinforcing the wall and bringing it back to line is described
in the above article.
Due to the failure of a dam^ the foundation of a wall shown in
Fig. 99 was washed out, and a section of the wall between two
expansion joints was moved out.
A wall of section shown in Fig. 100 was placed in an old creek
bed . ^ The freshet from a spring thaw undermined the foundation
washing away the soil adjacent to the piles. Excessive loads
developed on the piles, and these failed causing the wall to settle
about two feet.
A wall failure due to excessive overturning moment is de
scribed in the Engineering Record, Vol. 41, p. 586 (see Fig. 102).
A wall of rectangular shape, of small stone rubble, supported a
Fig. 101. Fig. 102.
fill slightly surcharged. It had already given evidence of incipi
ent failure by bulging in several places. In grading an adjacent
lot, an additional fill supported by the wall "A" was placed upon
the old embankment, followed by the complete failure of the
wall.
A wall shown in Fig. 101, supported a reservoir embankment
adjacent to a roadway.' The brick pavement lining the road
was taken up, and the wall sUd forward from one to two feet, and
in several places tilted out of fine about 6 in. This seems to
be an instance of insufficient frictional resistance between the
footing and the wall— the brick pavement supplying the neces
sary resistance to prevent the forward movement of the wall.
1 Engineering News, Vol. 63, p. 285.
' Engineering News, "Vol. 61, p. 503.
' Engineering Record, 'Vol. 44, p. 7.
PART II
CONSTRUCTION
CHAPTER VI
PLANT
Plant Expenditure. — ^With the exception of very small con
struction jobs amounting to biit a few hundred dollars in value,
it is necessary to employ tools, machinery and other implements
to supplement and replace manual labor. Such auxiliary ap
pliances are termed plant.
There are no fixed relations between the amounts to be ex
pended on plant and the total value of the work contemplated.
The principal factors of a general nature determining the amount
of plant required are, the yardage of concrete wall, the time given
in which to build the wall and the manner of the distribution of
the wall over the work. Few jobs are exactly alike or sufficiently
similar that the plant requirements become identical and it is a
matter of economy to so buy plant that its cost less its salvage
value, if any, at the completion of the job, is carried by this job
alone. This permits a careful study of the field conditions and
insures a selection of plant most fitted for this work. It is a
slogan of most contractors, that if a job is not worth the plant,
the job is not worth having.
"Inasmuch^ as plant is in reality but a substitute for labor, it would
seem obvious that no more should be invested in plant than will yield
a good return. This relation between plant and labor is apparently
ignored in many instances, and plant charges are incurred out of all
proportion to the volume of work to be done. The ultimate comparison,
whether made directly or indirectly, between hand labor and the pro
posed plant, or between this and that plant, must be made if the selec
tion is to stand the test of experience.
"The selection of plant, the purchase of this or that machinery, has
to a large extent been more or less haphazard. Contractors and engi
1 From "Concrete Plant" issued by Ransome Concrete Machinery.
165
166 RETAINING WALLS
neers, experienced and successful men, have been slow to awake to
the possibilities for loss or gain afforded by plant selection; but it is
nevertheless deserving of careful study.
" There seems to be a strong tendency toward excess in plant expendi
ture and a fact worthy of note is the tendency toward simplicity in plant
upon the part of engineers and contractors whose experience and success
in the field entitles them to be considered as leaders.
"In estimating plant cost, various elements other than first cost of
plant must be carefully considered. Cost of installation, including
freight, cartage, labor, etc., cost of maintenance, cost of removal, interest
upon the investment, must be considered on the one hand, as against
the resultant saving in labor and salvage value of the plant on the other.
"In general the plant best suited to the work is cheapest, regardless
of whether or not it costs a few dollars more than something less suited
to the conditions. First cost is perhaps less important in influence on
final results than cost of operation and maintenance. In many cases a
higher salvage return will offset to a large degree higher first cost. First
cost, too, is a definite constant. It can be positively assessed and proper
allowance made for it in estimating, in this respect differing from main
tenance, which is an unknown quantity subject to great variations."
Standard Layouts. — There are certain types of work, again,
generally speaking, for which the plant layouts are obvious.
Thus a concrete wall in a compact area, all within strategetic
reach of a center not exceeding some maximum distance away,
calls for a central mixing plant and a tower system of distribu
tion. In track elevation work, to eliminate grade crossings, the
availability of a track adjacent to the proposed wall, permits the
use of a compact concreting train. Usually conditions are not
so typical and local topographical conditions, as well as the
character of the work play an important role in determining the
character of the plant best suited for the job.
Arrangement of Plant. — It may be stated as almost axiomatic,
that, that wall is most economically built which, other things
being equal, is most expeditiously built. This necessitates a
certain degree of flexibility in the plant that little time may be
lost in bringing concrete to the forms awaiting it.
"The character^ and arrangement of plant depend to a large extent
upon local conditions, such as contour of ground. The general layout
of the work, while the manner in which the materials are to be delivered
to the site, whether in cars or in wagons, regularly or irregularly, has
an important bearing upon the type of plant. Similarly, the matter of
' lUd.
PLANT
167
total yardage to be placed, of time limit set for the work, of bonus or
penalty, will have a bearing upon plant selection.
"Other considerations which may affect materially the selection is
the amoxint of ground available for material storage, and the time of the
year during which the operation must be carried on, winter work re
quiring very different plant arrangement from summer work.
"Contour of ground is principally effective in determining the loca
tion of the plant with respect to the work and the storage of materials.
For example, a steep slope will often make advisable a system of over
head bins with gravity feed, which under other conditions would not be
advisable.
"The general layout of the work will usually be the determining factor
in the adoption of means for handlingmixedconcrete, subject, of course,
to modifications imposed by total yardage, etc. It may make for the
adoption of two or more separate installations rather than one central
plant or it may cause the adoption of a portable plant rather than a
stationary one.
"Delivery of materials is principally effective in determining the
arrangement for the storage of raw materials.
"Total yardage, time limit, etc. are generally the controlling factors
in determining the amount available for plantage."
Subdivision of Work. — 'It seems natural to divide the plant
necessary for concrete retaining walls into three subdivisions:
(1) the plant to bring the ma
terials to the mixer; (2) the
mixer, (.3) the plant to bring the
materials from the mixer and
place it in the forms.
1. When the layout of the
work is such that one or a few
central plants may be used, this
problem' is comparatively sim
ple. The material is dumped
alongside a storage bin and is
fed to this bin as required, the bin having a hopper to drop
material into the mixer. See Fig. 103. It may be possible,
due to the advantageous location of this bin below the delivery
point, that the material cars or wagons may unload directly
into the bin. This requires a regular and reliable delivery
system to keep the bin constantly supplied, since, with sporadic
delivery of material the concrete work would frequently be
delayed. Usually the material is allowed to accumulate in a
5forage
Pile
Pig. 103. — Loading bin by derrick
from storage pile of aggregate.
168 RETAINING WALLS
storage pile near the bin and is fed from this pile to the hopper
bin by a derrick, with preferably a clam shell, to save the labor
of loading the skips.
When a central plant is not used, the material is distributed
along the site of the work in small piles. It must be remembered
that when the material is distributed in this fashion, there is
considerable loss due to rehandling, to the gathering of foreign
matter such as dirt, etc., and to the inevitable loss of the bottom
portion of the pile on the ground. If the material is to be on the
ground for some time then a large portion of it may be lost on
account of the weather. Such losses may amount to quite a
large percentage of the material ordered and proper allowance
must be made to determine the final net cost of the material
in the concrete.
For this latter mode of the distribution of material the mixer
is usually fed by wheelbarrow from the nearest pile. Other
modes of getting the material to the mixer are easily determinable
from the local environment.
Mixers. — The selection of a proper mixer is comparatively
simple. The requirements of good concreting (as described in a
later chapter) should be noted and a type of mixer chosen that
will make it possible to carry out these requirements. The
necessary capacity of the mixer is readily determined from the
expected daily output required to prosecute the work within the
assigned time limit. Naturally a mixer attached to a central
mixing plant if run continuously will have a greater output
than one of like capacity carried about the work. The catalogues
of the manufacturers of the various types of mixers can be con
sulted to good advantage and, with the advice of their experienced
salesmen, a type most suitable for the work can readily be selected.
"It^ is true that one mixer may have an excess of power with resultant
acceleration of the various operations going to complete the mixing
cycle, one machine may be quicker in mixing or discharging than another;
but these differences will influence the final result less than a defective
organization. For example, it is common practice to employ extra men
to fiU wheelbarrows, a practice which increases the cost of this work
twentyfive to thirtyfive per cent, according to whether or not the
wheeler helps fill his own barrow. Similarly it is common practice to
handle mixed concrete in small wooden or iron barrows holding an average
of two cubic feet. By furnishing substantial runways and the adoption
1 Ibid.
PLANT 169
of carts an average load of 4.5 cubic feet can easily be handled. It is
to such elements of organization that attention should be directed, if
you would cut down the cost of operation. Properly handled, concrete
plant becomes an important factor in setting the pace for the work.
"Cost of installation includes freight, cartage and erection, elements
varying with the character of the plant, location of the work, with
respect to the source of supply, etc. * * *.
* * * "No other class of machinery is subjected to the severe usage
imposed on concrete machinery. The nature of the materials handled
make for excessive wear, to which should be added the fact that the
machinery is ordinarily handled by a class of labor not calculated to
give it the intelligent care and attention to which it is properly entitled.
It is to long experience upon the part of the manufacturer in this special
field that the purchaser must look for protection against failure, under
the severe conditions which actually prevail in the field. The history of
success in this line of work is a history of constant changes in design,
a story of heavier, stronger parts, of adapting the machine to the character
of the work by reducing parts to a minimum.
" The fewer parts your machine has, the less likely it is to get out of
order, and the more readily the operator of ordinary capacity can keep
it in working order.
"Considered broadly, mixers may be divided into Drum Mixers,
Trough Mixers, Gravity Mixers, Pneumatic Mixers.
" Drum Mixers may again be divided into Tilting Mixers (Smith Type)
and NonTilting (Ransome Type). In the former class the mixing
drum is mounted on a swinging frame, and the discharge of the mixed
materials is accomplished by a tipping of the frame and drum. In the
latter class mixed materials are drawn out through a chute inserted in
the drum.
"Trough mixers, as a whole, may be designated as Paddle Mixers,
though the paddles may vary in form from a broken worm, through the
various stages, to the continuous worm and the conveyor flight may be
single or double, of varying or uniform pitch.
"Gravity Mixers are of the same general characteristics, depending
for success upon a series of deflectors, chains, pegs, or conical hoppers, for
the mixing action. They are not adapted to building work in any case
and do not deserve serious consideration here.
"Pneumatic Mixers include the various types of pneumatic mixers
developed during the past two or three years by Wm. L. Canniff, A. W.
Ransome, McMichael, Eichelberger. In the Ransome and Canniff
mixers, the materials are first mixed by air in a container, and the mixed
concrete then forced out through pipes to its ultimate destination.
In the McMichael and Eichelberger machines the materials are assem
bled in a container and forced through pipes without premixing. These
latter machines depend for successful results upon such mixing action
170
RETAINING WALLS
as may take place in transit through the pipe. Pneumatic mixers are
all expensive to operate and cannot be used to advantage except in
special cases."
Distributing Systems. — There is greater latitude in the selec
tion of plant for a distributing system than in the selection of
plant for the two prior operations and since this portion of the
work is the most costly of the three, greater care should be spent
upon the proper selection of the necessary plant.
A retaining wall covers, generally, a long narrow strip, making
a compact, single distributing system from a single central plant
usually out of the question. Nevertheless, heavy walls, with
large concrete yardages within fairly restricted areas may permit,
economically the use of one or more central distributing plants.
"Taiver
TTjnt^
■Mixer
„ Flat Car
\wiiiimivmiiiiminn
FiQ. 104. — Pouring concrete
by tower and mixer mounted
on flat car.
Fig. 105. — Pouring concrete from
platform erected on trestle.
The greater mass of the wall lying above the ground surface,
the concrete must be raised to permit its placement within the
forms. This is accompUshed by several methods. The mixer, a
travelhng one, may be raised and its contents spouted directly, by
gravity, into the form. The mixer may remain on the ground
and its contents raised and delivered into the form. Following
are some possible methods of this latter mode of distribution.
(a) The mixer is on a flat car, with a tower and hoist (see
Fig. 104).
(6) The mixer is on the ground and the concrete taken from it
by cars, or barrows and run over platforms along the top of the
form into the wall (see Fig. 105) .
(c) A derrick takes the bucket from the mixer and dumps its
contents either directly into the form or into a spouting device
leading to the form.
id) Tower distribution.
(e) Cableway distribution.
(/) Pneumatic distribution.
PLANT 111
^ "The handling! of concrete through spouts or chutes is of compara
tively recent development, and as in many other similar developments,
there has been a tendency to overdo. Spouting systems have been
installed on many buildings where the distribution might have been
better done by barrow or cart.
"The installation of a spouting system is expensive, and should not
be undertaken blindly, nor with expectations of abnormal savings.
"Spouting plants may be grouped under Boom plants, Guy line
plants. Tripod plants. In the former, the spouting is mounted on a
swivelled bracket at the tower end, and the outer end supported by a
boom moves freely about the work. A second length of spout ordinarily
completes the unit. This type of plant has a greater freedom of move
ment than either guy line or tower plants, but is not as free moving as
might be desired.
"Many means have been tried to facilitate ready moving of the free
end, none of them, however, proving entirely satisfactory. A sugges
tion has been made to counterbalance the free end, but this has not,
as yet, been tried out thoroughly.
"In guy line plants, the spouting is suspended by ordinary blocks
and falls from guy lines or from special cables set up for the purpose.
In some cases the outer end of the cable is mounted on a portable tower
or "A" frame and the blocks and falls are preferably arranged so that
necessary adjustments in the line may be made from the ground.
"In Tripod plants movable towers are used to support the ends of
various sections of spouting.
"It has been found by practical experience that concrete, thoroughly
mixed and of proper consistency will flow on a slope of eighteen degrees,
with the best results obtained at twentythree degrees. These slopes,
however, are based upon a rigidly supported chute. Where the spouts
are supported from guy lines, the slope must be a little steeper, prefer
ably from twentyseven to thirty degrees. By proper consistency is
meant a mixture with approximately one and a quarter to one and a
half gallons of water to the cubic foot of material. There should be
just as much water, as the material can carry without separation,
so that the stone particles will be carried in suspension in the mass.
There should be a sliding of materials down the spout rather than a
rolling.
"Various types of spouting have been tried, ranging from round pipe
to rectangular troughs. Best results have been secured from the use
of 5inch pipes, or 10inch open troughs, the latter having the preference
for flat slopes and the former where there is necessity for varying pitch,
some of steeper pitch than named above.
"With open spouting, the use of line hoppers in connection with
' "Concrete Plants,'' Ransome Concrete Machinery, p. 23.
172 RETAINING WALLS
flexible spouting accomplishes satisfactorily the necessary changes in
pitch. The greatest items of expense in spouting plants are first cost,
installation and maintenance.
"Maintenance charges are particularly heavy. The ordinary stock
spouting which is made of No. 14 gage metal will seldom handle more
than two thousand yards without renewal . This is due to the abrasive
action of the material, especially as affecting the rivets which join the
various sections.
"In general we would say that whether or not you can use spouting
to advantage must be carefully considered for each job. Where the
work is light and scattered any attempt to spout concrete into place
is foredoomed to failure."
"The economy' of distributing concrete through properly designed
chuting plants has been demonstrated time after time, on all kinds
of construction and it has been conclusively shown that properly
proportioned, thoroughly mixed concrete may be conveyed to any
mechanically practical distance without disintegrating the mass.
"Concrete should flow at a uniform speed of from seventyfive to
onehundred feet per minute. The best results are attained with the
chute line pitched with a fall of one foot in four up to 150 feet radius.
For longer distances the fall should be about one in three, starting
with one foot in four and increasing the grade towards the discharg
ing point."
When it is remembered that a cableway mode of distribution
moves in but two dimensions i.e. in a vertical plane only and that
its cost rapidly increases, and the amount of load to be carried,
decreases with an increase in span, its use as a distributing system
is usually discarded for the methods of distribution previously
mentioned.
Below are given a series of descriptions of various plants
used. While it is impractical to attempt to make a standard
classification of construction problems, the illustrations selected
are thought to be more or less typical and the character of the
plant used probably the most fitted for the environment and
character of the work at hand.
(A) TOWER DISTRIBUTION
Railroad station at Memphis * * * 111. Cent. R. R. (see Fig.
106) for the skeleton layout of the work), Engineering News, Vol.
72, p. 629.
"The construction of the retaining walls and subway bridges was
hampered by the necessity of providing for traffic. There were about
' Bulletin No. 23, The Lakewood Engineering Co.
PLANT 173
60 trains daily, the heaviest traffic being from 7 a.m. to noon and 3 to
a p.M _ The only freight movements over this part of the line were in
switchmg service. The great difficulties encountered were the limited
space available, the handling of concrete whUe keeping clear of the trains
andthe inability of the contractor to get certain parts of the site delivered
to him for work at the time desired. For all work * * * the storage
space for materials was limited and it was necessary to regulate ship
ments of all kinds so as to be able to use the material upon arrival.
h  zfoo'  .
__l ^
Z50'
i
 Jt
JU
Fig. 106. — ^Layout of retaining walls and abutments.
"The concrete was delivered in place by spouting from elevator
towers, using selfsupporting trussed chutes. Two stationary plants
with 100' towers and one portable plant with a 50' tower were used,
each of the former being set up twice (in different locations) and the
latter being shifted as required. Each had its mixer, and, in order to
work to full capacity, a twocompartment material bin or hopper was
erected over the mixer, holding about 30 cubic yards of stone and 15
cubic yards of sand. The materials were brought in railway cars
and unloaded direct to the mixer bin or to small storage piles, there
being little room for storage. A derrick with clam shell bucket took
the materia] from the car or storage pile and dumped it into IJ^ cubic
yard cars, which were hauled up a cable incline and dumped into the
material hopper. The incline had a four rail track in the lower portion
and a three rail track at the top. ***** The maximum output per
day was 550 cubic yards. The entire concrete yardage was 30,000
cubic yards."
(B) CONCRETE TRAINS
As has been previously noted, railway improvement work, such
as track elevation or depression, permits the use of a compact
concrete train. A typical piece of work is the track elevation
work of the Rock Island lines, described in Engineering News
Vol. 73, p. 670; Vol. 74 p. 1275 and Vol. 74 p. 890. The con
crete plant which placed the necessary 30,000 yards of concrete
for this improvement is described as follows :
"Concrete train consists of a mixer car, four to seven stone cars and
two to four cars of sand. ****** The mixer car is a thirtyfive
foot flat car, equipped with a % yard Smith nontilting mixer 10 h.p.
174 RETAINING WALLS
vertical engine, 20 h.p. vertical boiler, 700 gallon storage tank and 60
gallon feed tank for the mixer. The machinery is housed the roof of the
car being higher at the discharging hopper than at the ends of the car,
thus forming an easy incline from the runways on the tops of the gondola
cars to the charging hopper above the mixer. The mixer is located
about 8' from one end of the car and faces the end. It discharges
the concrete into a swivelling chute which may be swung to discharge
the end or either side of the car. This arrangement of pouring from
different angles or from either end of the train eliminates the necessity
of turning the mixer car (as required with the other types) and
makes a considerable saving in working train space.
•'The chute has intermediate openings, so that concrete can be dis
charged at different points. A man on top of the car regulates the
charging of the mixer, the supply of water and the dumping of the
concrete (see Fig. 107). Usually the mixer train stands on trestles
Sfeam Dsn/iej/
Loading
A
Chuhs: dOpirahr's Plaifarm
Fig. 107. — Connecting train.
and the concrete is spouted to the form beneath. For the upper part
of the piers, it has been necessary to elevate the concrete, a crane and
bucket being used to place the concrete in the forms.
" The mixer is designed to carry a tower and hoisting engine if re
quired. ******* A. valuable feature of the car is a powerful
winchhead for a cable, which is anchored ahead. This enables the
mixer car to move the train along as the work progresses, thus
dispensing with the constant attendance of locomotive and crew.
"Each train is placing at the rate of 20 to 30 cubic yards per hour,
with a monthly total for both trains of 11,000 yards of concrete."
Other instances of the use of similar work trains are mentioned
below.
Engineering News, Vol. 76, p. 634. In filling in an old trestle, and build
ing the necessary retaining walls, a concrete train of three cars one mixing,
one stone and one sand, were used.
Engineering News, Vol. 75, p. 1192. The interesting feature of the work
train here was the fact that the hoist was operated by steam from the loco
motive. ,
Engineering News, Vol. 75, p. 494. Fort Wagner Track Elevation. The
concrete train worked on a temporary operating trestle, the track being
out of commission while the concrete train was on it.
Engineering Record, Vol. 70, p. 240. Chicago, Milwavikee and St. PauK
PLANT 175
The concrete train operated upon a trestle. A cableway on the concrete
train took materials from the intermediate cars to the bins. This proved
cheaper than tower cars and hoist cranes.
Cableway. — The use of a cableway for pouring the concrete walls
of a viaduct is described as follows in the Engineering News,
Vol. 72, p. 930 (see Fig. 108).
''Concrete material was delivered in cars on a siding and unloaded
unto stock piles by a stiffleg derrick mounted (with its engine and
hoist) on a tower or platform some 15' high. The same derrick and
clam shell bucket handled the material from the stock piles to the 200
yard bins over the oneyard concrete mixing plant which was located
just east of the structure and on the north side of the tracks.
yffai/roaJ Tracks
\Hmnq M
V
 1190  >
Via. 108. — ^Layout for cableway.
"The cableway was 800 feet long with an 80 foot tower at the mixer
end and a single bent 60 feet high at the further end. It was placed over
each wall in turn and was shifted laterally 80 feet, from one wall to
the other without being dismantled; this was done by placing timber
dollies under the tower. Handling the 12,500 yards of concrete by
cableway was economical as the amount of concrete at the ends of the
walls is small and wheeling it in buckets would have been slow and
expensive."
An interesting comparative analysis of the use of several
different plant layouts for a series of similar pieces of work is
described by Mr. Armstrong in the Journal of the Western Soc. of
Engineers, Vol. 16. New Passenger Terminal: C. & N. W. R. R.
The retaining walls enclosed a rectangular layout, bounded
by two street crossings and the parallel easement lines.
The plant layouts to pour the walls were as follows:
(a) A cableway, placed on movable trucks was used, permit
ting the shifting of the towers to pour each of the walls. This
plant did not prove economical and was of low capacity. The
best run was 24 yards per hour.
Q>) A runway with rails ran around the top of the wall forms.
A derrick hoisted the buckets of concrete to a hopper which
176
RETAINING WALLS
dumped into cars running along the form runway. This was
cheaper than the cableway and had a capacity of about 33 yards
per hour.
(c) In place of the derrick as above a short tower was used
with a hoisting engine. The best average was 37 yards per hour.
The dump cars ran as much as 500 feet away from the tower.
(d) A mixer, elevator and a hoist were mounted on a car and
ran around the forms. This proved very unwieldy and could
not get close to the forms. Less labor was needed here, however,
since the dump cars were eliminated. The best results with this
plant were about 25 yard of concrete per hour.
The following is a trite recommendation by the author of the
above paper:
"It might be stated as a general principle in the design of plant
that the capacity of the mixer should be made the determining factor
in the output. The charging hoisting and conveying appliances should
be designed with such a degree of flexibility as to preclude the possibility
of retarding the mixing process by delay in charging the mixer or delay
in removing the discharged concrete. The most economical mixer,
other things being equal, is the one which discharges its mixed batch
and receives its new batch in the shortest time."
Tower and Trestle. ^ — ^In concreting a high wall, 50 feet in
height, the following description is given of the plant used.
Si'orageffm
and liiur
Railroad
TnsHe along
■■■' tiall
Fig. 109. — Central mixing plant. Combined tower and trestle distribution.
"For concreting the wall a very efficient plant was installed. A Hains
gravity mixer was located about the center of the length of the wall,
where it was easily loaded by derrick, from the adjacent high level
railway. Concrete from the bottom or delivery end of this mixer
was run into an elevator whence it was lifted to be dumped into a
hopper and chute leading to another hopper with a bottom dump located
on a frame just outside of the wall forms. All of the preceding equip
' Engineering News, Vol. 73, p. 776.
PLANT
177
ment was stationary, but alongside of the wall was a trestle which took
concrete from the last noted hopper and dumped it through another
chute to its proper place in the forms (see Fig. 109). The number
of chutings given each batch should be especially noted."
In pouring a retaining wall for the Baltimore and Ohio Im
provementsi the inaccessibility of the site made it necessary to
use a gantry crane device with a platform and stiff leg derrick,
as shown in Fig. 110. A narrow gage railroad ran alongside
the roadway and brought the concrete from a central mixing
plant about onehalf a mile from the work. The gantry served
also to support the wall forms. (This work is also described on
page 211 under winter concreting.)
SHfflej
Derrick:. ^
Consiruciion
Din hst/ Line
Fig. 110.
The following is an interesting description of several methods
of handling the material on a bridge abutment job.^
"Hopper cars, derrick skips, elevator buckets and inclined chutes
were combined in placing 3360 cu. yds. of concrete in abutments and
approach retaining walls for a steel highway bridge across the Chicago
& Northwestern Ry. at Wheaton, 111. To give increased headway the
bridge is at a higher elevation than the old span parallel to it, so that
long inclined approaches were required, practically at right angles
to the bridge, as shown by the accompanying plan (see Fig. 111).
Each approach has a retaining wall on one side, and the wall on the
south side of the railway is about 600 feet along.
"A concretemixing plant was located beyond the end of the cut.
Sand and gravel were unloaded from cars into stock piles on the side
of the adjacent fill, and the stone was loaded into an elevated bin
by a derrick with a grab bucket. The sand was wheeled to the loading
chute. The mixer discharges the concrete into a sidegate hopper car.
Engineering News, Vol. 76, p. 269.
Engineering NewsRecord, March 13, 1919, p. 553.
12
178
RETAINING WALLS
PLANT 179
"Between this plant and the bridge site an elevator tower with a
chute was erected, whUe beyond this and close to the abutment was a
guyed derrick, both tower and derrick being on the narrow strip between
the old road and the top of the cut. A narrowgage track with one
automatic siding extended from the mixer plant to the tower and derrick.
This was operated by an endless cable with a hoisting engine placed near
the derrick and on it the concrete was handled in the hopper cars men
tioned above.
"At first the concrete was delivered to the elevator buckets and
spouted to the forms. The tower chute or spout extended across the
road and delivered the concrete into lateral chute supported directly
above the forms by falsework. This sufficed for about onehalf the
length of the wall.
"For the remainder of the work the cars ran up to the derrick and
discharged the concrete into a homemade wooden skip which was placed
in a pit at the side of the cable track and was handled by the derrick.
A movable gate was fitted to one end of the skip, with inclined boards
on the inside to guide the concrete to the opening and to prevent it
from being pocketed in the corners. The skip was dumped into a feed
hopper at the summit of the inclined chutes carried along and above the
forms for falsework.
"Concrete for the abutment on this side of the railway was placed
directly by the derrick and skip. For the abutment and short wall on
the opposite side and inclined chute was extended across the tracks,
having a feed hopper at its upper end within reach of the derrick. At
its lower end was a vertical drop line leading to the head of the chutes
over the abutment form, these being shifted to deliver the concrete
in the desired portions of the form.
"Baffles were used at the discharge ends of the long chutes to prevent
segregation of the concrete as it was deposited in place. In some cases
these were short troughs secured to the trench bracing or form struts,
being placed opposite the end of the chute and sloping in the opposite
direction, so that the direction of the concrete was reversed just before
its final discharge."
Conclusion. — To summarize, plant is employed solely to effect
an economy in the construction of a wall. To use plant that
does not, in the final analysis, show a saving because of its em
ployment, is unjustifiable. It is understood, of course, that all
economies accompUshed are legitimate ones; not such as are
made at the expense of good construction.
Bearing in mind that most jobs are unique in character, plant
should be bought for the sole requirements of the work at hand
and in proportion to the total cost of the work. Such illustra
180 RETAINING WALLS
tions of actual construction work as have been cited may furnish
an idea of general plant layouts — but each piece of work contem
plated must be studied out individually that advantage may be
taken of all local situations, such as topography, railroad and
highway location and the like.
Naturally some pieces of plant are standard. A good mixer,
hoists, derricks and small plant such as barrows, carts, shovels,
etc., may survive a job and be easily fitted to other work. This
is a matter of judgment. Little mistake is made, however, if
plant is procured for one job and charged off to that one job.
The cost accounting and the preparation of bids for new work
are thus vastly simplified and each job carries itself, the ideal
contracting condition.
In the following chapters some stress is laid upon the require
ments of good form work and of good concrete work. To secure
the proper results as indicated in those chapters requires a co
ordination between the plant and the methods used and plant
that will make it difficult to secure the desired results should not
be employed. It is only just to add that plant manufacturers
are keenly aUve to the demands of modern construction and strive
to cooperate with the engineer and contractor to supply ma
chinery that will aid in turning out flawless work.
Plant Literature
Ransome Concrete Machinery Co., "Concrete Plant."
HooL, "Reinforced Concrete," Vol. II.
Taylor and Thompson, "Concrete Costs," pp. 376380.
"Handbook of Construction Plant," R. T. Dana.
"Concrete Engineers Handbook," Hooi. and Johnson, "Concreting Plant."
CHAPTER VII
FORMS
Panels. — Form work for concrete walls may be divided into
two parts, (a) the form panel proper, consisting of the lagging
with the supporting joists and (b) the necessary bracing to hold
the form panel in place. With the exception of very small jobs
or of intricate and varying shaped walls, forms are usually de
signed to be used several times. To insure maximum economy,
then, it is necessary that the panels be stoutly built, yet of such
dimensions that they be easily set up, stripped and carried about.
The details should be such that the panels can be assembled, put
in place and made grout tight with a minimum of carpentry work.
Concrete Pressure. — That the form panel be properly de
signed, it is necessary that some attempt be made to determine
the amount of the concrete pressure. Both theoretically and
experimentally, it has been found exceedingly difficult to formu
late the action of wet concrete upon the form. At the instant it
is placed in the form, its pressure approximates closely a fluid
pressure, the fluid weighing 150 pounds per cubic foot. Soon
afterwards, both on account of the setting action and of the solids
contained in the concrete, the pressure drops away from the
linear fluid pressure law. For a thin wall with the concrete
level rising with a fair degree of rapidity, this hnear law ( p = wh)
is a good approximation. For a wall of heavy section, such as a
gravity wall and the like, this linear law would give excessive
pressures.
Concrete pressures are quite often underestimated with the
result that the forms yield, or give way entirely, spoiling much
work and entailing an expense far in excess of that required by
the increased amount of material to hold the concrete properly.
Probably the most extensive series of experiments upon con
crete pressures and the one most frequently quoted, were those
performed by Major Shunk.^ His conclusions are as follows:
> A rfeum^ of these experiments is given in Engineering News, Vol. 62, p.
288.
181
Temp.
c
80
20
70
25
60
35
55
42
50
50
40
70
182 RETAINING WALLS
The pressure of concrete follows the linear law
p = wh (182)
with w equal to 150 lb. per cubic foot, until a time T has elapsed,
in minutes,
r = c + 150/E (183)
where c is a constant depending upon the temperature of the
mix (see Table 31) and R is the rate of
Table 31.Conckete j^ ,• g_ ^he rate at which the con
Pkbsstjre Constants '^ ... . ,, » ...
Crete is rismg in the form, in leet per
hour. A series of charts giving the
pressure after the time T has elapsed is
given in the r^sum^ of the report quoted
above.
A series of experiments upon the
pressure of liquid concrete has been
given by Hector St. George Robinson.
See Minutes of the Proceeding of the
Institute of Civil Engineers, Vol. clxxxvii, 19111912, Part 1,
"The Lateral Pressure of Liquid Concrete" excerpts of which
are quoted here:
"Numerous experiments were made on different types of concrete
structures. In heavy walls, large piers and other members of fair size
the lateral pressure exerted was found to be fairly uniform and practically
constant for equal heads; but in reinforced concrete columns of small
dimensions, thin walls and other light concrete work, the effect of fric
tion between the more or less rough timber forms and the concrete, to
gether with the arching action, was found to reduce the pressure
considerably.
"The first series of tests were made during the building of a long wall
about three feet thick, constructed of concrete weighing 140 pounds
per cubic foot and composed of slowsetting cement, sand and crushed
granite in the proportions of 1 : 3 : 6 by volume. In mixing sufficient
water was used to bring it to a thoroughly plastic condition, requiring
little or no tamping to consolidate. The concrete was laid more rapidly
than is usual in this class of work, being carried up as rapidly as the
mixing and placing would permit to a height of 8 feet above the center
of the pressure face, during which time a light iron bar with a turned
up end was used for churning the semiliquid mass.
" The second series was carried out on large piers, four feet square, the
concrete in this case being a 1 : 2 : 4 mixture of cement, sand and Thames
ballast, weighing about 145 lbs. per cubic foot. The conditions as
FORMS 183
to mixing and laying were similar to those of the first tests and the con
crete was carried up to a height of 10 feet above the center of the pressure
face.
"In the first series the temperature was fairly uniform throughout,
while in the second considerable variation was experienced; but the
effects of the differences in temperature on the lateral pressure cannot
be traced and would appear to be very small.
"The general conclusions to be drawn from these and other experi
ments is that the lateral pressure of concrete for average conditions
is equivalent to that of a fluid weighing 85 pounds per cubic foot. * * *
For concrete in which little water is used in mixing, the pressure is
rather less, having an equivalent fluid value as low as 70 lbs. per cubic
foot in very dry mixtures."
There is apparently a large divergence of pressures as experi
mentally obtained and until more extensive experimentation has
been performed it is hardly justifiable to use other than an empiric
table of pressures; guided, however, by the results of the above
quoted work. A simple code may be used as indicated below
wherein the pressure is obtained from the equation
p = wh
with p the lateral pressure in pounds per square foot, h is the
concrete head in feet, and w is to be used as follows :
For heights of concrete less than 5', w = 150
For concrete 5 to 10 feet, w = 100
For concrete 10 to 20 feet, w = 75
For concrete over 20 feet, w = 50
These are all safe values and insure, when used, a form that will
not yield.
A comparison of the pressures obtained by using the results as
tabulated by Major Shunk and by using the suggested series of
values just given show quite a divergence in numerical values.
The pressures using the values given by Major Shunk (the curves
giving the maximum pressure for a given C and T are to be found
on p. 448, "American Civil Engineers Pocket Book") are far
lower than those found by the latter method. In view of the
fact, however, that concrete pressures are not readily formulated
and that form failures have demonstrated that such pressures do
reach a high value, it seems better to follow the scheme of pres
sure intensities given above. The forms should be designed
then, using these values in preference to using the experimental
m aximum pressure.
184
RETAINING WALLS
Joist
The extra cost of the stronger forms thus obtained is far less
than the expense entailed in remedying the result of a form
failure.
At the end of the chapter a problem is given illustrating the
application of the preceding formulas to a specific example.
Since a form panel may be placed
at any point of the face of the wall,
it should be designed for the maxi
mum pressure that can come upon
it. The concrete pressure is
carried by the lagging to the joists,
which in turn carry it to the
longitudinal rangers. These carry
the load to tie rods, or where such
rods are not used, to shores placed against the rangers (see
Fig. 112).
Lagging. — Generally tongue and grooved lumber is specified
for the sheeting. The boards are continuous over the joists and
with the support of the tongue and grooving, it is possible to
treat the panel as a plate. Ordinarily, no reUance should be
placed on such plate action and the boards should be designed
as either smple or fixed beams. Another most important fea
Lagging.
Range]
Fig. 112. — Typical form assembly.
Table 32.
—Safe Load
PER Square Foot on Lagging
\a
U%)
\^)
\?«)
l?i
2(.VA)
2}i(2K)
2K(2?^)
2yi(.2H)
3(2J^)
12
1,000
1,700
2,500
3,500
4,700
5,950
7,500
9,200
11,000
14
750
1,250
1,850
2,600
3,450
4,450
5,550
6,750
8,100
16
600
950
1,400
2,000
2,650
3,400
4,250
6,200
6,200
18
450
750
1,100
1,550
2,100
2,650
3,350
4,100
4,900
20
350
600
900
1,250
1,700
2,150
2,700
3,300
3,950
22
300
500
750
1,050
1,400
1,800
2,250
2,750
3,300
24
250
400
650
900
1,200
1,500
1,900
2,300
2,750
26
200
350
550
750
1,000
1,300
1,600
2,000
2,350
28
175
300
450
650
850
1,100
1,400
1,700
2,050
30
160
275
400
550
750
950
1,200
1,500
1,800
33
135
225
350
450
600
800
1,000
1,200
1,500
36
110
200
300
400
500
650
850
1,000
1,200
39
100
150
250
350
450
550
700
850
1,050
42
85
135
200
300
400
500
600
750
900
45
75
125
. 175
250
350
450
550
650
800
48 ■
65
100
160
225
300
400
450
600
700
FORMS 185
Table 33. — Safe Timbeb Stresses for Form Lumber
(Taken from A. R. E. A., railroad timber stresses, the stresses increased
50 per cent, because of the nature of the loading and the temporary
character of the work.)
Douglas fii 1800
Longleaf pine 2000
, Shortleaf pine 1600
White pine 1350
Spruce 1500
Norway pine 1200
Tamarack 1350
Western hemlock 1600
Redwood 1350
Bald cypress 1350
Red cedar 1200
White oak 1600
Table
34. — Safe Loads
0^
Rangers and Joists in Kips
2'0''
3'0''
4'0"
^
2
4
6
8
10
12
2
4
6
8
10
12
2
4
6
8
10 12
2
4
6
8
10
12
0.4
1.8
4.0
7.1
11.1
16.0
0.9
3.5
8.0
14.2
22.1
32.0
1.3
5.3
12.0
21.2
33.2
48.0
1.8
7.1
16.0
28.3
44.4
64.0
2.2
8.8
20.0
35.4
55.3
80.0
2.7
10.6
24.0
42.5
66.2
96.0
0.3
1.2
2.7
4.7
7.4
10.7
0.6
2.4
5.3
9.5
14.8
21.4
0.9
3.6
8.0
14.2
22.2
32.0
1.2
4.7
10.7
19.0
29.6
42.7
1.5
5.9
13.3
23.6
37.1
53.4
1.8
7.1
16.0
28.5
44.5
64.0
0.2
0.9
2.0
3.6
5.6
8.0
0.4
1.8
4.0
7.1
11.1
16.0
0.7
2.7
6.0
10.7
16.7
24.0
0.9
3.6
8.0
14.2
22.2
32.0
1.1
4.4
10.0
17.8
27.8
40.0
1.3
5.3
12.0
21.3
33.3
48.0
5'0"
6'0"
7'0"
2
i
6
8
10
12
0.2
0.7
1.6
2.8
4.4
6.4
0.4
1.4
3.2
5.7
8.9
12.8
0.5
2.1
4.8
8.5
13.4
19.2
0.7
2.8
6.4
11.4
17.8
25.6
0.9
3.6
8.0
14.2
22.2
32.0
1.1
4.3
9.6
17.0
26.7
38.4
0.1
0.6
1.3
2.4
3.7
5.3
0.3
1.2
2.7
4.7
7.4
10.7
0.4
1.8
4.0
8.1
11.1
16.0
0.6
2.4
5.3
9.5
14.8
21.3
0.7
3.0
6.7
11.9
18.7
26.7
0.9
3.6
8.0
14.2
22.2
32.0
0.1
0.5
1.1
2.0
3.2
4.5
0.3
1.0
2.3
4.1
6.3
9.1
0.4
1.5
3.4
6.1
9.6
13.7
0.5
2.0
4.5
8.1
12.7
18.3
0.6
2.5
5.7
10.2
15.8
22.8
0.8
3.0
6.8
12.2
19.0
27.4
8'0"
lO'O"
2
4
6
8
10
12
0.1
0.4
1.0
1.8
2.8
4.0
0.2
0.9
2.0
3.6
5.6
8.0
0.3
1.3
3.0
5.3
8.3
12.0
0.4
1.8
4.0
7.1
11.1
16.0
0.6
2.2
5.0
8.9
13.9
20.0
0.7
2.7
6.0
10.7
16.7
24.0
0.1
0.4
0.8
1.4
2.2
3.2
0.2
0.7
1.6
2.8
4.4
6.4
0.3
1.1
2.4
4.3
6.7
9.6
0.4
1.4
3.2
6.7
8.9
12.8
0.5
1.8
4.0
7.1
11.1
16.0
0.6
2.1
4.8
8.5
13.3
19.2
186
RETAINING WALLS
ture is the amount of defection permissible. It is well to keep the
deflection of the panel within oneeighth of an inch.
Table 32 gives the load per square foot to be carried by a board
12 inches wide, L feet long (L the distance between joists) and h
inches thick. The unit timber stress taken is 1,000 pounds per
square inch. The boards are designed as simple beams. Should
the permissible stress be greater than that used here the load
may be increased in direct proportion to the new stress. Again,
if the board is to be treated as a fixed beam the load to be carried
may be increased 50 per cent. That the deflection may not
exceed oneeighth of one inch, for simple span.
L must be less than 25 y/h
and for a fixed span
L must be less than 45 s/h
Table 33 gives a range of unit timber stresses for several woods.
Table 34 gives the maximum loads to be carried by the joists
for various spacing. The thickness of the joist is b inches
and its depth h inches. The loads may again be increased in the
same proportion for a permissible unit stress greater than one
thousand pounds per square inch and again when the beam is
assumed as fixed in place of simply supported. This same table
may also be used to design the rangers supporting the panels.
Tierods. — The diameter of the tierod depends upon the size
of the panel supported and its position in the form. The con
crete pressures may be taken from the empiric scheme given on
Table 35.
— Loads in Lbs
. ON Tie Rods
Permissible unit stresses
Rod diam
eter
12,000
16,000
20,000
25,000
Va
150
200
250
300
Vi
600
800
1,000
1,200
%
1,320
1,750
2,200
2,700
V2
2,350
3,150
4,000
4,900
H
3,700
4,900
6,100
7,700 .
«
5,300
7,100
8,800
11,000
%
7,200
9,650
12,000
15,000
1
9,400
12,700
15,700
19,600
IM
14,700
19,700
24,500
30,700
FORMS 187
page 183. The unit stress in the steel is usually taken at 16,000
lb. per square inch. Small diameter rods may be pulled out and
this should be borne in mind in selecting the rod spacing. Table
35 gives the load on tie rods for a range of unit steel stresses.
A simple detail carrying the tie rod load is shown in Fig. 112.
This obviates the necessity of boring a large timber to allow the
rod to pass through. The tie rods may be threaded on the
end and fastened to the rangers by nuts and washers, or a patented
support, such as the universal clamp {Universal Clamp Co.)
may be used on a plain round bar.
Rangers. — The rangers themselves may be designed as simple
or fixed beams, with spans between tie rods and carrying the
joists. If the ranger is to be b inches wide and h inches deep,
with span between tie rods L, then
WL/I = pbhye and bh^ = ^ (184)
I may be taken as 8 or 12, depending upon the assumption that
the beam is a fixed or simple one; and p may be taken as the safe
permissible unit stress in the timber.
Form Reuse. — If the panels are built in stout units, carefully
put together, they may be used several times. When the lagging
becomes splintered marring the face of the concrete and making it
very difficult to strip the form, the form should be abandoned.
With care in placing and stripping the forms, a panel maybe used
from 3 to 10 times. Two inch tongue and grooved sheeting makes
a good strong form but its weight Hmits it to small panels. If
plant is available to handle these units, this objection is removed
and as large sections maybe used as is found convenient to assem
ble. Usually a Hmiting section would be about 8' by 10'.
Form Work. — It is essential that a careful study be made of the
form work, taking into consideration the expected daily output
of concrete and the time the forms must remain in place. It must
be remembered that forms of simple shape, quickly assembled,
put in place and stripped, make for large economy on the work.
Skilled carpenters will prepare excellent, wellfitting forms of long
duration. It is a poor economy to substitute for such labor the
ordinary wood butcher, a most competent man in his sphere.
In this connection the use of a portable machine saw, propelled
electrically or by gasoline is a marvellous labor and time saver
and few jobs, however small, can afford to be without one.
188 RETAINING WALLS
The rear and face forms of the wall are kept the proper dis
tance from each other by means of wooden separators called
spreaders. When the tie rods are placed or wire used in place of
tie rods and tension put on them the spreaders are held in place
without any further details. As the concrete is poured and
reaches the lever of a spreader, the spreader is knocked out. The
tie rods and wires must remain until the concrete has set (see
later chapter) .
Bracing. — Bracing, or shoring is necessary to take care of
unbalanced pressures and the possible overturning of the form
due to the vibrations and shocks set up during the pouring of the
concrete. Such stresses are obviously not to be computed and
experience alone dictates the proper amount of bracing to be used.
They are made usually of 4 inch by 4 inch, or 4 in. by 6 in. stock,
nailed to the rangers and held against foot blocks or stakes in the
ground (see Fig. 113). Where concrete is to be poured against
a permanent mass, requiring forms on one side only, no tie rods
or wires can be used through the concrete and the bracing on
the one side must take the full concrete pressure and are to be
designed accordingly. When walls of some height are to be
poured in several hfts, an overlapping of the joists may render
bracing unnecessary above the lower lift.
Fig. 113.— Form Fig. 114.— Holding forms
brace. by bolt in concrete.
Occasionally the environment is such that bracing cannot be
used on either side. It is possible here to concrete eye bolts
into the bottom lift and into each succeeding lift and to anchor
the forms to these (see Fig. 1 14) .
Generally an excessive amount of bracing is used, with a result
ing forest of timber and making it impossible to run plant close
to the forms. Formwork is a fertile field of study for the engineer
and the designing and detailing of such work is worthy of as
serious attention as the design and construction of the wall itself.
Stripping Forms. — It is essential that foTms be stripped as
soon as it is possible to do so. To keep a form in place longer
than is required makes it impossible to get the full economical
FORMS 189
reuse of the form and makes it very difficult to finish and repair
the concrete surface. In the warm summer months the forms
may be stripped after 24 hours. In the spring and fall months
they should be left in place from 48 to 72 hours. When in doubt
as to the hardness of the concrete a small portion of the form
may be taken off and a thumbnail impression made. If there
is no indentation, it is safe to take off the balance of the forms.
If it is possible to remove the tierods (rods K inch or less in
diameter may be economically recovered; rods of larger diameter
are usually left in the wall) these should be taken out before the
forms are stripped. Patent rod pullers^ may be used to take
out the rods. Where the rods are left in the wall, they should
be cut back an inch to an inch and a half and ibhe face of the wall
plastered at these points. Wires are rarely recovered and are
cut off in the same fashion as the rods. The sooner after strip
ping these rods and wires are cut, the easier it is to repair and
finish the face of the wall (see later chapter of wall finish) .
From ten days to two weeks of favorable, warm weather
should elapse before the fill is placed behind the wall. If the
fill is to be placed at a rapid rate, e.g., by dump cars from a tem
. porary trestle and the like, a greater period of time should elapse.
This is especially important for the reinforced concrete walls,
where the concrete will receive the full load immediately after
the completion of the embankment.
Oiling and Wetting Forms. — A dry form will absorb the water
from the concrete, in the process of curing, leaving a pecuUar
pockmarked appearance of the concrete face due to the honey
combing of the surface. The forms should be wetted by pail
or hose immediately before the concrete pour is started. To aid
in the stripping of the form, the inside face of the form is usually
oiled, with a heavy oil, termed a form oil, which is a heavy
sludge. Although this stains the concrete face, the rubbing
and washing of the concrete surface easily removes the oil marks.
Patent Forms. — For a wall of large yardage and of fairly
constant outUne, permitting many reuses of the form panel, the
use of some of the patent forms may show quite an economy,
both in the construction of the form and in the labor of setting
up and stripping the forms.
The two best known types of such forms are the HydrauUc
Pressed Steel Form and the Blaw Form.
1 An excellent rod puller is sold by the Universal Clamp Co, of Chicago.
190
RETAINING WALLS
The Hydraulic Pressed Steel Form consists of two parts: the
bracing and the form panel. The bracing is formed of upright
Us spliced as necessary and held together by tie rods and spacers
or Uners. Fig. 115 shows a sketch of the bracing and its details.
^E^
^ Metal Apron S^ri'p
.■Liners Punched f'Cenfvrs
\forAolJusfmeniof f^
Uprights
I :
Fig. 115. — Hydraulic Pressed Steel Co. form assembly of liners and plates.
The form panel consists of a sheet metal (all metal used in these
forms both panel and uprights are no. 11 gage, i.e., oneeighth inch
metal) backed by 2" boards. Around the periphery of the panel
a U steel edge is put, to which the boards are screwed (see Fig.
116). The panels are held in place against the uprights by
means of stout Us spaced about one foot apart (see Fig. 115).
standard Upright
Standard
nail Plate
"Standard
Wall Plate
■Standard Yfall
Plate Clamp.
FiQ. 116. — Section of Hydraulic Pressed Steel Co. form.
It is claimed by the company that the panels may be reused
about 300 times before wearing out. Where a job will permit a
reuse of the form panel exceeding twenty or thirty times, they
maintain that their form will prove cheaper than the wood form
ordinarily built.
Platk IV
f^.
;%
;*&i^jngV ^'^^ /' ' '^
• ^''^'/;^'
3^
.Z^^^»i~^
=^4
>:
t
., ^i\i%:^
(FariN^j pvin I'.tn
FORMS
191
The advantages of the form are quite obvious. The uprights
may be built up to the top of the wall. After the lower lift of
the wall is poured no further bracing becomes necessary, since
the form is now anchored against the lower half of the wall.
The panels may be removed after twentyfour hours, the uprights
and liners remaining as much longer as is necessary before the
wall is selfsupporting.
The panels need only be put in as the concrete comes near
their level, thus permitting a thorough spading and tamping of
the mass: quite a vital point where the wall is thin or has a
special shape.
Blawform. — The Blawform consists, essentially of a steel
panel, reinforced with angle on the back and held in place with
a steel assembly of joists and rangers. By an ingenious travel
Ung gantry device, the form panels are braced against this travel
ler, which runs on rails alongside the work. A large number of
instances of their use for both heavy and Hght retaining walls
are given in their Catalogue 16.
Supporting the Rod Reinforcement. — Since most of the rod
system in a reinforced concrete wall must be in place before
the concrete pouring is started, some means of support must be
provided. In the "L" or "T"
shaped cantilevers, the heavy
rod system of the vertical arm
extends into the footing and
must, therefore, be set up and
in place before the wall forms
are up. Many simple devices
may be used for this purpose.
Fig. 117, (See Fig. B, Plate
IV) shows a typical and efh
cient method of taking care of these rods. When the footing
has been poured, thereby anchoring these rods, the wall forms
are set in place and the rods are wired and held the required
distance away from the concrete face. The horizontal rod
system is wired to the vertical rods and helps to maintain the
proper spacing of these vertical rods. Patent wire chps may
be used to wire the horizontal and vertical rods together.
The horizontal rods in the footing, itself, are laid in the con
crete when the proper level has been reached. It is preferable to
wire a net of these rods, together before placing in the wet con
SfafKS as
ZounUrmiqkl'S .,
Lon^ihd/nal
Rods
Fia,
117. — Supporting rod reinforce
ment of cantilever wall.
192 RETAINING WALLS
Crete to make sure that the proper spacing as called for on the
plans, will be kept.
The rod systems of the other types of the reinforced concrete
walls are supported and placed in similar fashion. The problem
of supporting the rods extending into the footing for the slab
types of wall is comparatively a simple one, since these rods are
the light system and therefore need little framework to carry
them. The main system (particular stress is placed upon the
counterfort and box types of wall) is suspended to the forms in
the usual manner and kept at the proper distance away from the
face by means of small wooden spreaders which are removed in
pouring as quickly as the concrete reaches their level. The tie
rods form a good sujpport for the horizontal rods and are generally
so used.
It is important that, whatever method of support is employed,
the rods should be held firmly in place. Spading and spouting
of concrete are liable to shift the rods unless they are stoutly
supported. It is understood that in the design of walls involving
intricate rod systems (see Chapter 4) proper consideration has
been given to the practicability of the rod layout and to the
feasibility of supporting the rods and of pouring the concrete.
Simplicity of rod design insures an easy concrete pour and leaves
the engineer with a reasonable assurance that the rods are finally
placed where they were originally designed to go.
The rod system has, presumably, been carefully and economi
ally designed and no variations in spacing should be permitted
in the field, except in isolated instances, where a
proper attempt should then be made to reinforce the
rijll I weak spots resulting. Leaving openings in the walls
"tpr p for construction reasons, as, for example, to permit
placing timbers through the wall, or to place large
Fig. 118. P^'P^ ^**'' "^i^^ result, when the wall is finally patched
in portions being without the proper reinforcement.
The rods should be bent around these openings as shown in
Fig. 118.
Undoubtedly walls are at times designed with excessive rein
forcement due to indifference or carelessness and the knowledge
of such excessive strength has encouraged the engineers in the
field and the contractors constructing such walls to alter the
rod spacing to accommodate minor construction exigencies.
Such acts are, in the main, unfortunate and designs which can
FORMS 193
safely permit many such liberties are to be deplored. Walls
should be designed as economically as possible with due considera
tion for all contingencies and when a design has left the hands
of a competent, conscientious engineer, no changes should be
permitted in the field save with the concurrence of the man
responsible for the design.
Travelling Foims.—Engineenng News, Vol. 73, p. 67. Track
Elevation Rock Island Lines Chicago.
"The walls are built in travelling forms which straddle the site of the
wall and are carried by wheels on either side. Both wood and steel
forms of this type are used, each being long enough for a 35 foot section
and having grooved wheels riding upon two lines of rails. * * *
The abutments are built in fixed forms of the usual type. Plank
sheeting is used in both cases and the two lines of sheeting are held
together by tierods instead of wires. The rods are plain bars, not
threaded, and are fitted with clamps instead of nuts. When a clamp
is in place, a set screw jams the rod against a V slot in the clamp, securing
it rigidly in position. {Engineering News, Sept. 10, 1914). Each
rod is imbedded in a tin tube, so that it can be withdrawn readily,
the holes being then packed with stiff cement grout at each end.
"The retaining walls are built in alternate sections of 35 feet with
the travelling forms. It takes about six hours to fill the form, which
is then left in place about 15 hours. It then takes about 20 hours to
release the travelling form move seventy feet forward and adjust them
and the sheeting ready for the concrete. The use of the travelling forms
has enabled the work to be done in about 25 per cent, of the time re
quired with the ordinary forms (from the building to the removal of
the form) and at about 50 per cent, of the cost (including erecting, pour
ing and dismantling)."
New Passenger Terminal, C. & N. W. R. R. Armstrong,
Journal of the Western Society of Engineers, Vol. 16.
" The forms were buUt iu sections 30 feet long. The footings were
first built and allowed to set. The forms for the super walls were then
built. It was required that an entire section of superwall should be
poured at one continuous run of the mixing plant, in order that no hori
zontal joiats might occur in the walls. The forms were constructed
of 4inch by 6inch studding and 2inch by 8inch dressed and matched
sheeting. The two sides were tied together with J^inch rods which
were passed through iron pipes consisting of old boiler fines. The rods
were drawn out when the forms were removed, but the pipes were left
in place, the opening in the face of the wall being filled with mortar."
194 RETAINING WALLS
Forms Built in Central Yard. — Engineering and Contracting,
June 11, 1913, p. 649. Track Elevation, Chicago, Milwaukee
and St. Paul R. R. For this work the forms were built in a
central yard and were shipped out to the work as required on
flat cars. They were taken from the cars and set in place by
means of locomotive cranes.
Erecting Forms on Curves. — R. H. Brown, Engineering
Record, Vol. 61, p. 714.
"There is nothing more unsightly in concrete work than to see the
impression of the forms running out of level. A great deal of pains
is taken to produce smooth surfaces by spading, but very little attention
is given to the mold itself. This is very noticeable in massive work. On
a straight wall there is no excuse for this, but in building forms on curves
of short radius there is great difficulty in making a symmetrical sur
face and eliminating the segmental effect. If the following method is
carried out a piece of concrete will be produced which is a true curve
in every foot of its length.
" Take a wire about the size of that used in telephone lines and upon
a smooth level surface strike on the board an arc of the radius of the
centerline of the wall or dam. Arc of radius of 150 feet can easily be
handled. Care must be used in doing this that the wire is always
straight. This template is now sawed out on a band saw in about ten
foot lengths. The rear and face templates can be struck from this one
by means of a Tsquare.
"Run out the center line of the wall in chords of 10 feet and put in
permanent plugs at these points. Erect a wellbraced series of batters
around the curve and set the top ledger board at the exact crest of the
wall. Place the centerline templates on these boards and plumb them
over the plugs, cleating them together as fast as they are put in correct
position. With this center to work from, the outside and inside curves
can be set.
"Make two boards four feet long, one edge straight and the other
bevelled to the batter of the front and rear faces respectively. The
studding can now be set, using a carpenter level. The upper end will
rest against the template, the lower end following the inequalities of the
ground.
"Start the bottom board as low as possible and run it along the
curve on both sides making it absolutely level. The rest of the board
ing can now be nailed to the studding, springing each one carefully
into place. The purlins (Rangers) are put in and rods run through
and tightened. After everything is well braced, remove the batter
boards used in lining up. When the forms are removed a true curve
is presented to the eye."
FORMS
195
Problems
It is required to design and construct a set of forms for a wall 30 feet high
above the footing with expansion joints 40 feet apart, of section shown in
Fig. 119. It is figured that the mixer can pour 100 yards of concrete in an
8hour shift, this to govern the lift of concrete poured.
The portion of the wall requiring forms contains a volume between ex
pansion joints of 93 cubic yards. It is thus possible to complete the pouring
of the section in one continuous pour within the time specified — the ideal
arrangement. The forms will be designed upon this basis.
^
>
i
v/
t
am
axk
■l.33k
I.Sk
A'xi'MskSO'ckC /jpMili
2'5heiHng—
S5'\^^^'i?^r^ro '
Reduced LJading'^*/* bj , ' ^
Diagram ^i^^i^^^
Spaomq of Rangers
Fig. 119.
Concrete Pressures. — On the basis of Major Shunk's experiments, the
concrete pressure at the base is determined as follows: (It is assumed that
the concrete enters the form at the temperature of 70°.) Since the con
crete form is 30 feet high and is filled in 8 hours, the rate of filling per hour
is 3.75 feet, the value of R to be used in the work following. From Table 31
with the temperature of 70°, c = 26 and from (183)
r = 25 + 150/3.75 = 65 minutes = 1.1 hrs.
The maximum pressure that can occur is found by employing the curves of
Major Shunk, which can be found in the American Civil Engineers' Pocket
Book page 448. This maximum pressure, with the value of c and T as
above found is 850 pounds per square foot. Using the empiric rule given on
page 183, the pressure function to use is 50 lb. per square foot, which would
give at the base of the wall 30 X 50 or 1500 lb. per square foot. The average
pressures found by Robinson, page 182, of 85 lb. per square foot intensity
would give a base pressure of 85 X 30 = 2550 lbs. far in excess of both of the
pressures just found. The experimental value of 85 lbs. is based upon heads
not exceeding 10 feet— and is therefore of httle appUcation to the case at
hand Again, the experiments of Major Shunk, while most admirably and
extensively performed cannot be made the final basis for concrete pressure
determination. It is therefore logical to employ, awaiting more experi
mental data, the empiric table suggested in the previouspages and the form
work of the given problem will be designed upon the table quoted.
196 RETAINING WALLS
In line with the recommendations of the text, 2ineh tongue and grooved
sheeting will be used. North Carolina spruce dressed all sides will permit
a working stress for the form work of 1500 lbs. per square inch. The sheeting
will be treated as continuous, so that the product hp of Table 32 is 1500 X
12 = 18000. Since the loads on the sheeting of Table 32 employ the con
stant 8000, to use the table directly, the above load of 1500 pounds per
square inch will be reduced in the ratio of ^^''''^ooo, or will become 670
lb. per square inch. For 2" material, the dressed thickness is V/s" and
the table shows that a load of 670 pounds will permit the joists to be spaced
30 inches apart. In view of the fact that the forms are to be used several
times, the panels may be set at any position in the form, and will therefore
all be constructed alike, and of the heaviest dimensions required.
The rangers are set after the panels are in place and may therefore be
spaced to accommodate the concrete pressures. A good working size for a
joist is a 4inch by 6inch stick. Fig. 119 gives the load layout for the 30
inch spacing of the joists. The loads have been divided by the constani
2.25 i.e., the ratio of ^^^^MooOi to permit a direct use of the Tables.
Table 34 is to be used in the design of the joists. Let the lower ranger carry
a threefoot panel of sheeting. From the figure, the lower three feet bring
a tabular equivalent load of 4.8 kips. Table 34 permits a threefoot spacing
of this size joist and accordingly the first ranger having been placed as close
to the bottom as is feasible, the next will be spaced three feet above it. A
similar study of the loading above the lower panel shows that, to maintain
the same size of joist, the next four rangers must be spaced on three feet
centers. The remainder of the spacing is shown on Fig. 119.
The rangers will be made up of two 3inch by 6inch sticks, a handy mer
chantable size. The safe load span upon these two pieces will determine
the tierod spacing. From equation (184) page 187, with 7 = 12; p =
1500 as before, 6=6 and ^ = 6,
WL = 648,000
or if w is the load per linear foot upon the ranger and L is the length expressed
in feet
wL'' = 54,000
The lower ranger will carry 4500 lbs. per linear foot (the actual loads are
used here), whence
L = 3' 6"
The tie rods will accordingly be spaced 3' 6" apart at the lower lift of rangers.
The panel load that a tierod will be called upon to carry is
3.5 X 3 X 1500 = 15,700 lb.
To avoid using large size tierods which cannot be recovered two tie rods
will be used together at the lower lift. From Table 35 with a unit stress of
16,000 pounds per square inch for steel, two J^inch rods will be used.
The other tie rod spacing, and the necessary rod section are both found
by identical means.
CHAPTER Vlll
CONCRETE CONSTRUCTION
Water Content. — Recent years have noted a marked increase
in the knowledge of the proper mode of selecting and mixing
the aggregates necessary to produce good, strong concrete
masonry. Not only must the various aggregates be put in the
correct proportions, but the amount of water used is vitally
important. The excess or deficiency of water seriously affects
the strength of the concrete.
Each element entering into a concrete mix performs a definite
and separate function and each is, accordingly, capable of affect
ing favorably or unfavorably the strength of the concrete.
Concrete is usually so proportioned that each finer material fills,
more or less completely, the voids in the coarser aggregate (see
following pages on Prof. Abrams demonstration that the strength
of the concrete does not require, prima facie, this condition).
The action of water is in part a solvent and in part a chemical one.
The results of Mr. Nathan C. Johnson^ and other laboratory
investigators have strikingly demonstrated the vital importance
of the correct amount of water and it has been shown that con
crete failures, both partial and complete are attributable to excess
of water. The evaporation of this excess amount of water leaves
pockets and crevices in the concrete, materially reducing the
effective area capable of resisting stress. The widely varying
results of concrete tests and the necessary high factors of safety
are thus quite obviously explained.
Prof. Talbot^ has made a series of timely pointers on concrete,
some of which may, with profit, be quoted here.
■'The cement and the mixing water may be considered together to
form a paste; this paste becomes the glue which holds the particles of
the aggregate together.
^Engineering News Record, June2&, 1919, p. 1266. Also "Better Concrete—
The Problem and Its Solution," N. C. Johnson, Journal Engineer's Club,
Philadelphia, Pa.
' Engineering NewsRecord, May 1, 1919 for a resum6 of his remarks at
the annual convention of the American Railway Engineering Association.
197
198 RETAINING WALLS
"The volume of the paste is approximately equal to the sum of the
volume of the particles of the cement and the volume of the mixing
water.
" The strength given this paste is dependent upon its concentration — •
the more dilute the paste the lower its strength; the less dilute the greater
its strength.
"The paste coats or covers the particles of aggregate partially or
wholly and also goes to fill the voids of the aggregate partially or wholly.
Full coating of the surface and complete filling of the voids are not
usually obtained.
" The coating or layer of paste over the particles forms the lubricating
materia] which makes the mass workable; that is, makes it mobile and
easily placed to fill a space compactly.
"The requisite mobility and plasticity is obtained only when there
is sufiicient paste to give a thickness of film or layer of paste over the
surface of the particles of aggregate and between the particles suflacient
to lubricate these particles.
"Increase in mobility may be obtained by increasing the thickness
of the layer of paste; this may be accomplished either by adding water
(resulting in a weaker paste) or by adding cement up to a certain point
(resulting in a stronger paste) .
"Factors contributing to the strength of concrete are then, the amount
of cement, the amount of mixing water, the amount of voids in the
combination of fine and coarse aggregate and the area of surface of the
aggregate.
"For a given kind of aggregate the strength of the concrete is largely
dependent upon the strength of the concrete paste used in the mix,
which forms the gluing material between the particles of the aggregate.
"For the same amount of cement and same voids in the aggregate,
that aggregate (or combination of fine and coarse aggregates) will give
the higher strength which has the smaller total area of surface of par
ticles, since it will require the less amount of paste to produce the re
quisite mobility and this amount of paste will be secured with a smaller
quantity of water; this paste being less dilute will therefore be stronger.
The relative surface area of different aggregates or combination of
aggregates may readily be obtained by means of a surface modulus
calculated from the screen analysis of the aggregate.
"For the same amount of cement and the same surface of aggregate,
that aggregate will give the higher strength which has the less voids,
since additional pore space will require a larger quantity of paste and
therefore more dilute paste.
"Any element which carries with it a dilution of the cement paste
may in general be expected to weaken the concrete. Smaller amounts
of cement, the use of additional mixing water to secure increased mo
CONCRETE CONSTRUCTION 199
bUity in the mass, increased surface of aggregate, and increased voids
in the aggregate all operate to lower the strength of the product.
'In varying the gradation of aggregate a point will be reached, how
ever, when the advantage in the reduction of surface of particles is offset
by increased difficulty in securing a mobile mass, the voids are greatly
increased, the mix is not workable and less strength is developed in
the concrete. For a given aggregate and a given amount of cement,
a decrease m the amount of mixing water below that necessary to pro
duce sufficient paste to occupy most of the voids and provide the lubri
cating layer wUl give a mix deficient in mobility and lower in strength.
"A certain degree of mobility is necessary in order to place concrete
m the forms in a compact and solid mass, the degree varying considerably
with the nature of the work and generally it will be found necessary
to sacrifice strength to secure the requisite mobility. It is readily seen,
however, that the effort should be made to produce as strong a cementing
layer of paste as practicable by selecting the proper mixture of ag
gregate and by regulating the amount of mixing water.
"More thorough mixing not only mixes the paste and better coats •
the particles, but it makes the mass mobile with a smaller percentage
of mixing water and this less dilute paste results in higher strength.
Any improvement in methods of mixing which increases the mobility
of the mass will permit the use of less dilute paste and thereby secure
increased strength."
In connection with the above remarks by the Dean of Concrete
Investigators, there may be quoted the conclusions of a classic re
port prepared by the Bureau of Standards. '
"1. No standard of compressive strength can be assumed or guaran
teed for concrete of any particular proportions made with any aggregate
unless all the factors entering into its fabrication are controlled.
"2. A concrete having a desired compressive strength is not neces
sarily guaranteed by a specification requiring only the use of certain
types of materials in stated proportions. Only a fractional part of
the desired strength may be obtained unless other factors are controlled.
"3. The compressive strength of concrete is just as much dependent
upon other factors, such as careful workmanship and the use of the
proper amount of water in mixing the concrete as it is upon the use
of the proper quantity of cement.
"4. The compressive strength of concrete may be reduced by the
use of an excess of water in mixing to a fractional part of what it should
attain with the same materials. Too much emphasis cannot be placed
upon the injurious effect oj the use oj excessive quantities of water in mixing
concrete. [The itahcs are mine.]
1 Technology Papers of the Bureau of Standards, No. 58.
200 RETAINING WALLS
"5. The compressive strength of concrete may be greatly reduced
if, after fabrication, it is exposed to the sun and wind or in any relatively
dry atmosphere in which it loses its moisture rapidly, even though
suitable materials were used and proper methods of fabrication employed.
"6. The relative compressive strengths of concretes to be obtained
from any given materials can be determined only by an actual test
of those materials combined in a concrete.
"7. Contrary to general practice and opinion the relative value of
several fine aggregates to be used in concrete can not be determined by
testing them in mortar mixtures. They must be tested in the combined
state with the coarse aggregate.
"8. Contrary to general practice and opinion the relative value
of several coarse aggregates to be used in concrete cannot be determined
by testing them with a given sand in one arbitrarily selected proportion.
They should be tested in such combination with the fine aggregate as
will give maximum density, assuming the same ratio of cement to
total combined aggregate in all cases.
"9. No type of aggregate such as granite, gravel or limestone can
be said to be generally superior to all other types. There are good
and poor aggregates of each type.
"10. By proper attention to methods of fabricating and curing,
aggregates which appear inferior and may be available at the site of
the work may give as high compressive strength in concrete as the
best selected materials brought from a distance, when the latter are
carelessly or improperly used.
"11. Density is a good measure of the relative compressive strength
of several different mixtures of the same aggregates with the same
proportion of cement to the total aggregate. The mixture having the
highest density need not necessarily have the maximum strength but
it will have a relatively high strength.
"12. Two concretes having the same density but composed of dif
ferent aggregates may have widely different compressive strength.
"13. There is no definite relation between the gradation of the ag
gregates and the compressive strength of the concrete which is applic
able to any considerable number of different aggregates.
"14. The gradation curve for maximum compressive strength,
which is usually the same as for maximum density, differs for each
aggregate.
"15. With the relative volumes of fine and coarse aggregate fixed,
the compressive strength of a concrete increases directly, but not in a
proportionate ratio as the cement content. An increase in the ratio,
of cement to total fine and coarse aggregates when the relative proper"
tions of tie latter are not fixed does not necessarily result in an increase
in strength, but may give even, a lower strength.
CONCRETE CONSTRUCTION 201
" 16. The compressive strength of concrete composed of given
materials, combined in definite proportions and fabricated and exposed
under given conditions can be determined only by testing the concrete
actually prepared and treated in the prescribed manner.
"17. The results included in this paper would indicate that the com
pressive strength of most concretes, as commercially made can be increased
25 to 100 per cent, or more by employing rigid inspection which will insure
proper methods of fabrication of the materials."
In a striking report on how to properly design a concrete
mixture to obtain the utmost strength from the aggregate at hand
by Prof. Duff A. Abrams^ it is shown how little the present day
standard methods of proportioning concrete make for concrete
strength. The importance of the report and its vital conclusions
justify the rather lengthy excerpts below.
The general problem of concrete mixtures has been defined
in the report as follows and some of the principles following a
series of 50,000 tests are noted therein.
■'The design of concrete mixtures is a subject of vital interest to all
engineers and constructors who have to do with concrete work. The
problem involved may be one of the following:
"1. What mix is necessary to produce concrete of proper strength
for a given work?
"2. With given materials what proportion will give the best con
crete at minimum cost?
"3. With different lots of materials of different characteristics which
is best suited for the purpose?
"4. What is the effect on strength of concrete from changes in mix,
consistency or size and grading of aggregate?
"Proportioning concrete frequently involves selection of materials
as well as their combination. In general, the question of relative costs
is also present."
Of the different methods of proportioning concrete, Prof.
Abrams has noted the following as among the most important:
"1. Arbitrary selection, such as 1 :2 :4 mix, without reference to the
size or grading of the fine and coarse aggregate;
"2. Density of aggregates in which the endeavor is made to secure
an aggregate of maximum density;
"3. Density of concrete in which the attempt is made to secure
concrete of maximum density;
1 Design of Concrete Mixtures, Bulletiii 1, Structural Materials Research
Laboratory, Lewis Institute, Chicago.
202 RETAINING WALLS
"4. Sieve analysis, in which the grading of the aggregates is made
to approximate some predetermined sieve analysis curve which is
considered to give the best results;"
"5. Surface area' of aggregates.
"It is a matter of common experience that the method of arbitrary selec
tion in which fixed quantities of fine and coarse aggregates are mixed
without regard to the size or grading of the individual materials, is far
from satisfactory. Our experiments have shown that the other methods
mentioned above are also subject to serious limitations. We have
found that the maximum strength of concrete does not depend on either
an aggregate of maximum density or a concrete of maximum density,
and that the methods that have been suggested for proportioning con
crete by sieve analysis of aggregates are based on an erroneous theory.
All of the methods of proportioning concrete which have been proposed
in the past have failed to give proper attention to the water content
of the mix. Our experimental work has emphasized the importance of
the water in concrete mixtures, and shown that the water is, in fact, the
most important ingredient, since very small variations in water content
produce more important variations in the strength and other properties
of concrete than similar changes in the other ingredients.
After performing a series of over 50,000 tests, covering a
period of three years, Prof. Abrams has established the following
important principles in regard to the correct design of a concrete
mix.
"1. With given concrete materials and conditions of test the quantity
of mixing water determines the strength of the concrete, so long as the
mix is of workable plasticity.
"2. The sieve analysis furnishes the only correct basis for proportion
ing aggregates in concrete mixtures.
"3. A simple method of measuring the effective size and grading of an
aggregate has been developed. This gives rise to a function known as
the "fineness modulus "^ of the aggregate.
"4. The fineness modulus of an aggregate furnishes a rational method
for combining materials of different size for concrete mixtures.
"5. The sieve analysis curve of the aggregate may be widely dif
ferent in form without exerting any influence on concrete strength.
"6. Aggregate of equivalent concretemaking qualities may be
produced by an infinite number of different gradings of a given material.
"7. Aggregates of equivalent concretemaking qualities may be
produced from materials of widely different size and grading.
1 See end of chapter for a definition of Surface Area.
' See end of chapter for a complete definition of the fineness modulus.
CONCRETE CONSTRUCTION ' 203
"8. In general, fine and coarse aggregates of widely different size
or grading can be combined in such a manner as to produce similar
results m concrete.
"9. The aggregate grading which produces the strongest concrete
IS not that givmg the maximum density (lowest voids). A grading
coarser than that giving maximum density is necessary for highest
concrete strength.
" 10. The richer the mix, the coarser the grading should be for an
aggregate of given maximum size; hence, the greater the discrepancy
between maximum density and best grading.
"11. A complete analysis has been made of the water requirements
of concrete mixes. The quantity of water required is governed by the
following factors :
"(a) The condition of "workability" of concrete which must be
used — the relative plasticity or consistency;
" (6) The normal consistency of the cement;
" (c) The size and grading of the aggregate— measured by the fineness
modulus;
"(d) The relative volumes of cement and aggregate— the mix;
" (e) The absorption of the concrete;
"(/) The contained water in aggregate.
"12. There is an intimate relation between the grading of the ag
gregate and the quantity of water required to produce a workable
concrete.
" 13. The water content of a concrete mix is best considered in terms
of the cement — waterratio.
"14. The shape of the particle and the quality of the aggregate
have less influence on the concrete strength than has been reported by
other experimenters."
Prof. Abrams has experimentally determined the relation be
tween the water content and the strength of the concrete and
reports the following most important conclusions together with
an empiric relation between the two.
"It is seen at once that the size and grading of the aggregate and the
quantity of cement are no longer of any importance except in so far
as these factors influence the quantity of water required to produce a
workable mix. This gives us an entirely new conception of the function
of the constituent materials entering into a concrete mix and is the
most basic principle which has been brought out in our studies of
concrete.
"The equation of the curve is of the form
204
RETAINING WALLS
where S is the compressive strength of the concrete and x is the ratio
of the volume of water to the volume of cement in the batch. A and
B are constants whose values depend on the quality of the cement used,
the age of the concrete, curing conditions, etc.
"This equation expresses the law of the strength of concrete so
far as the proportions of materials are concerned. It is'seen that for
given concrete materials the strength depends upon only one factor —
the ratio of water to cement. Equations which have been proposed
in the past for this purpose contain terms which take into account such
factors as quantity of cement, proportions of fine and coarse aggregate,
voids in aggregate, etc., but they have uniformly omitted the only
term which is of any importance; that is, the water.
"A vital function entering into the analysis is the socalled 'fineness
modulus' which may be defined as follows:
"The sum of the percentages in the sieve analysis of the aggregate
divided by 100.
"The sieve analysis is determined by using the following sieve from
the Tyler standard series: 100, 48, 28, 14, 8, 4%, % and 1}4 in. These
sieves are made of squaremesh wire cloth. Each sieve has a clear
Table 36. — Method or Calculating Fineness Modulus of Aggregates
The sieves used are commonly known as the Tyler standard sieves. Each
sieve has a clear opening just double that of the preceding one.
The sieve analysis may be expressed in terms of volume or weight.
The fineness modulus of an aggregate is the sum of the precentages given
by the sieve analysis, divided by 100.
Size of
per cent
Sieve analysis of aggregates
of sample coarser than a given sieve
Sieve
Sand
Pebbles
Concrete
aggregate
square opening
Fine
Medium
Coarse
(C)
Fine
Medium
Coarse
in.
mm.
lOOmesh..,,
.0058
.147
82
91
97
100
100
100
9S
48mesh. . . .
.0116
.295
52
70
81
100
100
100
92
28mesh....
.0232
.69
20
46
63
100
100
100
86
14raesh. . . .
.046
1.17
24
44
100
100
100
81
8nieah
.093
2.36
10
25
100
100
100
78
4mesh
.185
4.70
86
95
100
71
%in
.37
9.4
51
66
86
49
?iin
.75
18.8
9
25
50
19
IMin
1.5
38.1
Fineness m
1.54
2.41
3.10
6.46
6.86
7.36
6.74
* Concrete aggregate G is made up of 25 per cent, of sand B mixed with 76 per cent, of
pebbles E. Equivalent gradings would be secured by mixing 33 per cent, sand B with 67
per cent, coarse pebbles /''; 28 [per cent. A with 72 per cent. F, etc. The proportion coarser
than a given sieve is made up by the addition of these percentages of the corresponding size
of the constituent materials.
CONCRETE CONSTRUCTION 205
opening just double the width of the preceding one. The exact di
mensions of the sieves and the method of determining the fineness mod
ulus will be found in Table 36. It will be noted that the sieve analysis
is expressed in terms of the percentages of material by volume or weight
coarser than each sieve."
Prof. Abrams notes that there is a direct relation' between
the fineness modulus as above defined and the compressive
strength of the concrete, after noting that the "fineness modulus
simply reflects the changes in waterratio necessary to produce a
given plastic condition. " This is, of course, consistent with his
main thesis that the waterratio is the all important function in
determining the concrete strength. It is stated that the relation
between the compressive strength of the concrete, as brought
out by tests and the fineness modulus is to all intents a linear one,
i.e. an increase in the fineness modulus has a proportionate
increase in the compressive strength.
With an assigned compressive strength of concrete, it is now
possible to proceed directly to assemble an aggregate to meet
this strength. The waterratio forming the fundamental basis
of the process, the empiric relation above mentioned is employed
to determine the proper value of x, when S is given and A and
B are known. The details following, showing the method of
obtaining the values of the constants, of the fineness modulus
and of the several combinations possible to satisfy most economic
ally the strength requirements of the concrete are given with
elegance and clearness in the Bulletin just quoted. The noveltj'
of the method and its apparent intricacy (and such intricacy is
only apparent) and the fact that concrete mixes usually just
"grow" and are not scientifically developed may make Prof.
Abrams' procedure seem very cumbersome. A little study of
his methods will show that the contrary is true and that the
correct design of a concrete mix predicated upon his assump
tions (and these assumptions are assuredly based on most valid
premises) is a matter of very simple analysis.
The further comments on the design of a concrete mix, given
at the conclusion of the Bulletin are worthy of quotation here:
"The importance of the waterratio on the strength of concrete will
be shown in the following considerations:
"One pint more water than necessary to produce a plastic concrete
reduces the strength to the same extent as if we should omit 2 to 3
lb. of cement from a onebag batch.
206 RETAINING WALLS
"'Our studies give us an entirely new conception of the function
performed by the various constituent materials. The use of a coarse
wellgraded aggregate results in no gain in strength unless we take
advantage of the fact that the amount of water necessary to produce a
plastic mix can thus be reduced. In a similar way we may say that
the use of more cement in a batch does not produce any beneficial effect
except from the fact that a plastic workable mix can be produced with
a lower waterratio.
"The reason a rich mixture gives a higher strength than a lean one
is not that more cement is used, but because the concrete can be mixed
(and usually is mixed) with a waterratio which is relatively lower
for the richer mixtures than for the lean ones. If advantage is not taken
of the fact that in a rich mix relatively less water can be used, no benefit
will be gained as compared with a leaned mix. In all this discussion
the quantity of water is compared with the quantity of cement in the
batch (cubic feet of water to one sack of cement) and not to the weight
of dry materials or of the concrete as is generally done.
"The mere use of richer mixes has encouraged a feeling of security,
whereas in many instances nothing more has been accomplished than
wasting a large quantity of cement, due to the use of an excess of mixing
water. The universal acceptance of this false theory has exerted a most
pernicious influence on the proper use of concrete materials and has
proven to be an almost insurmountable barrier in the way of progress
in the development of sound principles of concrete proportioning and
construction.
"Rich mixes and wellgraded aggregates are just as essential as ever,
but we now have a proper appreciation of the true function of the
constituent materials in concrete and a more thorough understanding
of the injurious effect of too much water. Rich mixes and wellgraded
aggregates are, after all, only a means to an end; that is, to produce a
plastic, workable concrete with a minimum quantity of water as com
pared with the cement used. Workability of concrete mixes is of
fundamental significance. This factor is the only limitation which
prevents the reduction of cement and water to much lower limits than
are now practicable.
"The above considerations show that the water content is the most
important element of a concrete mix, in that small variation in the
water cause a much wider change in the strength than similar variations
in the cement content or the size or grading of the aggregate. This
shows the absurdity of our present practice in specifying definite grad
ings for aggregates and carefully proportioning the cement, then guessing
at the water. (The italics are mine.) It would be more correct to
carefully measure the water and guess at the cement in the batch.
"The grading of the aggregate may vary over a wide range without
CONCRETE CONSTRUCTION 207
producing any effect on concrete strength so long as the cement and
water remain unchanged. The consistency of the concrete will be
changed, but this will not affect the concrete strength if all mixes are
plastic. The possibility of improving the strength of concrete by
better grading of aggregates is small as compared with the advantages
which may be reaped from using as dry a mix as can be properly placed.
********** f f J f
"Without regard to actual quantity of mixing water the following
rule is^ a safe one to follow: Use the smallest quantity of mixing water
that will produce a plastic or workable concrere. The important of any
method of mixing, handling, placing and finishing concrete which will
enable the builder to reduce the water content of the concrete to a
minimum is at once apparent."
Practical Application.— Some of the details of these copious
excerpts may eventually prove without adequate experimental
basis; yet the fundamental truth conveyed in all the foregoing
must be recognized — namely, the role of the water content of a
concrete mix. The question of paramount importance is the
manner and means of applying these truths to actual concrete
work in the field. Stone, gravel, sand and cement companies
have been educated to furnish products meeting with the require
ments of long continued experimental and field research. These
products are naturally much costlier than are aggregates unre
stricted as to nature, impurities, grading and size. It is essential
then that this added cost be not squandered without any benefit
through oversight of some simple principles.
The proper mixing of the ingredients is conditioned upon the
plant used, both for mixing and for distributing. The character
of such plant has been described both generally and in some detail
in a previous chapter on plant. The average mixer, while a more
or ^ess efficient machine has some difficulty in producing a well
mixed batch of low water content in a shorttimed mix. A little
patience in educating the mixer operator to keep the water con
tents low and an insistence that the concrete be not dumped
until a specified time of mixing has elapsed, will go a long way
towards meeting the experimental requirements of good concrete.
Clearly, it is of no avail to go to the bother, expense and the pos
sible delay of securing specified concrete materials, if little atten
tion is paid to the final steps in concrete mixing.
A batch of concrete must be in the mixer a certain minimum
time before the aggregate has been properly transformed into
208 RETAINING WALLS
concrete. What this time is depends upon the character of the
machine and the number of revolutions it makes per minute.
This time can not be specified in advance nor can good concrete
be expected merely from long time mixing. In this connection
see the Engineering NewsRecord, Nov. 28, 1918, p. 966, and
Jan. 23, 1919, p. 200. The average time of mixing a batch is
about one minute. A little care and study of the particular
machine at hand will determine the correct time for .a batch mix.
Careful inspection will then insure that each batch of concrete
will receive this length of time for its proper mix.
In the use of small mixers, the socalled one or two bag batch
mixers, it is exceedingly hard to get a uniform water ratio for all
the batches. Variations in the piling of the stone and sand, in
the barrows; in the dryness of the aggregate all make it impossible
to apply a constant amount of water and turn out the same con
sistency of mix. However, by a careful attention to the piling
of the carts and by an insistence that water be used in measured
quantity only — preferably from an overhead tank attached to
the machine and certainly not by an indiscriminate use of the
hose or pail — a concrete can be obtained meeting with a fair
degree of success the water requirements of workable plastic
concrete.
It should be definitely predicated that the principles of good
concrete should determine the plant and not, conversely, the
plant determine the mode of concreting (see chapter on Plant).
Concrete Methods. — ^The question of competent labor proves
a most irritating one. It may be set down as axiomatic that
common labor, however willing, and in spite of competent leader
ship cannot mix and place good concrete. A trained concrete
force is necessary for this work. The use of incompetent labor
on concrete work is a most shortsighted policy and here, as in
every other industrial enterprise, the best is decidedly the cheap
est in the end.
The use of poor materials and the employment of lax and in
different methods together with incompetent labor are dependent
upon the laxity of inspection and, unfortunately, the minimum
requirements of the engineer form the maximum goal of the aver
age contractor and, to use the colloquialism of the field, the con
struction superintendent will "get away with" as much as he
can. True, there are many exceptions, but the engineer does
well to prepare for the worst.
CONCRETE CONSTRUCTION 209
To specify a good concrete, especially in light of the above
researches, is, comparatively an easy matter. To assign proper
inspection, tempered by practical judgment and equipped with
a thorough knowledge of good concrete, so that in matters of
field decision the concrete is given the benefit of the doubt, is a
far more difficult matter.
As the details of the requirements of good concrete become more
generally known undoubtedly the common welfare of the con
crete interests, contractors, engineers, plant manufacturers and
the like, will promote a cooperation that will make it a much simpler
matter to secure the maximum strength of concrete from a given
assembly of materials. At present it is necessary to specify in
detail the desired concrete aggregates and the methods by which
these are to be mixed and, in addition, to make ample provision
for carrying out the intent and letter of the specifications.
Distributing Concrete. — Concrete, properly mixed, must like
wise be properly distributed. Poor distribution will nullify the
beneficial results of good mixing. The concrete mix is an aggre
gate of solids in a fluid vehicle and, when transported in any but
a vertical direction, will tend to separate in accordance with
natural laws. The distributing system must aid in overcoming
this separation tendency. For this reason concrete should be
dropped vertically into the forms and spread by shovels and hoes
into thin layers. Spouting a concrete into a form in any direc
tion but the vertical is a serious offence. The mix will separate
and any subsequent hoeing, shovelling or spading will prove inef
fectual. Upon stripping the forms the inevitable pouring streaks
will appear; evidence of poor workmanship and presenting a
most unpleasing appearance.
With a concrete of workable plasticity, properly delivered
into a form, but httle additional work should be necessary to
bring it to its final place in the form. The concrete should be
spaded at the form to permit the grout to collect at the face, in
suring a smooth face and should also be spaded at the rods to aid
in getting a firm grout bond between the steel and the concrete.
The distributing systems have been discussed in detail in the
preceding chapter on plant, which chapter should be read again
in the light of the present observations upon the requirements of
good concrete.
Keying Lifts— If the day's pour is finished before reachmg
the top of the wall, the concrete surface should be brought to a
14
210 RETAINING WALLS
rough level and a long timber to form a longitudinal key should
be imbedded in the top. Dowels may be inserted instead, made
up either of steel rods, or of stones and carried about one foot
into each of the layers. At the pouring of the next layer, the
timber key, if used, is to be removed, the surface to be thoroughly
cleaned and the fresh concrete then placed upon it. For the
eflSciency of various treatments of this joint see "Construction
Joints,'' page 159.
Use of Cyclopean Concrete. — In large concrete walls, it is per
missible to place stones over 12 inches in diameter wherever the
thickness of the concrete mass exceeds 30 inches. The stones
are kept about 12 inches apart and about 6 inches from the face
of the wall. They should be sound, hard rock, wellcleaned
and should be placed by hand into the concrete and not dumped
indiscriminately from a bucket or thrown in at random. A little
care in placing the stone will permit a larger number to be used
and thus cut down the cost of the wall by economizing on the
amount of concrete aggregate required.
In reinforced concrete walls it is questionable whether the use
of such "plums" should be permitted. The rod system makes
it difficult to place the stones, even though the wall exceeds 30
inches in thickness. Since the concrete in this wall is highly
stressed in compression, sound rock must be used. With a care
fully specified aggregate for the concrete, it seems a httle incon
sistent then to permit the use of an indeterminate material.
Local conditions will generally indicate whether good stones are
available. As a general rule, however, for the usual type of
cantilever and counterforted walls, the use of plums is inadvisable.
Winter Concreting. — Quite often the urgent need of a concrete
retaining" wall makes it imperative that its construction proceed
despite winter weather. As the temperature drops, the setting
time of concrete increases. The setting action stops when the
concrete is frozen and does not continue until the concrete has
thawed. It is doubtful whether frost injures a concrete perma
nently. This much, however, is certain — a frozen concrete
must thaw out completely and then be given ample time to set,
before the forms are stripped or any load placed upon the wall.
It is highly desirable and it is generally so specified that concrete
be mixed in such a manner that it reaches the form at a favorable
setting temperature and is then to be suitably protected against
frost until it is thoroughly set.
CONCRETE CONSTRVCTIOX 211
Concrete should not reach the forms at a temperature less than
45° (Fahrenheit) . The aggregate and the water should be heated
when the temperature drops below this mark. While, ordinarily,
concreting is permitted without heating the materials until the
temperature drops below the freezing point, the above tempera
ture should preferably be the controlling one.
A simple method of heating the aggregate is to pile it around
a large metal pipe (a large diameter metal flue, or a water pipe is
just the thing) and have a fire going within the pipe. Old form
lumber is an excellent and cheap fuel for this fire. Another,
similar method is to pile the material on large metal sheets rest
ing on little stone piers, and beneath which sheets fires are kept
burning. In both the methods care must be taken not to burn
the material next to the metal, and not to use such material if it
does become burned. The water may be heated in large con
tainers over fires, or by passing hve steam through the water,
either directly in it or through coils.
An interesting description of a winter concreting job is given
here:^
"The sand and crushed stone used in making the wall concrete were
heated by diffusion of steam from perforations in a coil of a 2" pipe
placed at the bottom of the storage pile. The bottoms of the charging
bin above the mixer were also fitted with perforated piping so that the
heat might be retained in the materials.
" The water used in mixing was maintained at about 100° F. by a
live steam jet discharging at the bottom of a 3000 gallon tank, or
reservoir kept constantly full. The overflow from the tank discharged
into a 50 gallon measuring barrel, being heated to scalding temperature
by another jet of superheated steam.
"The walls forms were insulated with straw and plank on the back
and covered with tongue and grooved flooring on the face, retaining a 2"
space between the steel (metal forms were used) and the wood, through
which low pressure steam from one of the boilers on the deck was diffused
by a perforated 1" pipe. This pipe was at the bottom of the form and
ran longitudinally the entire length connecting with the boiler by a T
connection and vertical pipe at about the middle of the section.
"A stationary mixing plant was installed adjacent to the main line
of the railway about half a mile west of the wall site. The concrete
was conveyed to the wall in buckets on cars drawn by a dinkey on narrow
gage."
1 Retaining Walls, Baltimore & Ohio Railroad, Engineenng News, Vol. 76,
p. 269.
212 RETAINING WALLS
A general note on winter concreting on Miami Conservancy
Work is given here as of interest in connection with the present
topic'
"Concreting has been carried on through the winter in the dam
construction work of the Miami Conservancy District, Ohio, with only
occasional interruption. As the nature of the enterprise demands that
progress be rapid and according to schedule, and as it is important to
keep the working organization intact to avoid losses and delays, it
became necessary to plan reducing the interruptions of concreting to a
minimum.
"Study of the extra costs involved in heating materials and protect
ing deposited concrete led to the conclusion that the greater part of the
extra cost is incurred only at temperatures below 20°, and a general rule
was therefore made that work through the cold season is to be continued
until the thermometer drops below 20°.
"Provision for heating aggregates by steam coils built in the bins has
been made at all three of the dams where concreting has been going on
* * * . Means have also been provided for protecting the surfaces from
freezing by tarpaulins and salamanders, or, in some instances by steam
coils (where steam was available because it was used for other
purposes) .
" Care is taken that no fresh concrete is placed on frozen foundations.
With a view to reducing the liability of freezing also, the amount of
water used in the mixing is closely regulated."
Concrete work in winter, observing the necessary precautions
to prevent freezing, is, of course, more costly, than work at the
seasonable temperatures. Whether this extra cost is less than
the loss involved in the break in the continuity of the work and
the delay in receiving the finished structure, is a matter to be
disposed of uniquely for each piece of work. If the work is to
proceed regardless of the weather, the specifications must so
be drawn, that the precautions to be used when the temperature
falls below a given point (which must be clearly noted) are em
phatically set forth. General specifications as to heating are
unsatisfactory — the details should be given.
Acceleration of Concrete Hardening.— The quicker a concrete
sets, other things being equal, the quicker the forms can be strip
ped and the sooner can the fill be deposited behind the wall.
Under natural conditions, the warmer the concrete is the quicker
it sets. Therefore work in the summer can proceed at a faster
1 Engineering NewsRecord, Vol. 82, p. 618.
CONCRETE CONSTRUCTION 213
rate than work at the other seasons. Some cements are more
quickly setting than others. It is possible, by adding certain
chemicals to accelerate the hardening of the concrete. The
effect of the addition of calcium chloride has been noted as
follows:^
"As the result of some experiments made by the Bureau of Standards
to develop a method to accelerate the rate at which concrete increases
in strength with age, it was found that the addition of small quantities
of calcium chloride to the mixing water gave the most effective results.
A comprehensive series of tests was inaugurated to determine further
the amount of acceleration in the strength of concrete obtained in this
manner and to study the effect of such additions on the durability of
concrete and the effect of the addition of this salt on the liability to corro
sion of iron or steel imbedded in mortar or concrete.
"The results to date indicate that in concrete at the age of two or
three days, the addition of calcium chloride up to 10 per cent, by weight
of water to the mixing water results in an increase in strength, over simi
lar concrete gaged with plain water, of from 30 to 100 per cent., the
best results being obtained when the gaging water contains from 4 to 6
per cent, of calcium chloride.
"Compressive strength tests of concretes gaged with water containing
up to 10 per cent, calcium chloride, at the age of one year gave no indi
cation that the addition of this salt had a deleterious effect on the dura
bility of the concrete.
" Corrosion tests that have been completed indicate that the presence
of calcium chloride, although the amount used is relatively small, in
mortar slabs exposed to the weather, causes appreciable corrosion of the
metal within a year. This appears to indicate that calcium chloride
should not be used in stuccos and warns against the unrestricted use of
this salt in reinforced concrete exposed to weather or water."
Concrete Materials. — Concrete aggregates and cement have
been so well classified and placed under standard specifications
that any typical specification will serve as a model for the charac
ter of the material to enter into the construction of a retain
ing wall. A brief description may be given of the essential
requirements of these concrete constituents. It may be well to
read once more the previous pages upon the bearing of the type
of the aggregate on the concrete strength and the relative im
portance of the character and proportions of the aggregates
(including water) as compared with the methods of preparation
1 Engineering News Record, Vol. 82, p. 507.
214
RETAINING WAILS
and distributing. The amounts of the material required depend
upon the proportions specified. Table 37 is given here based
upon the standard proportion and shows the amount of cement,
sand and stone required for the various mixes. These are the
theoretical requirements. It must be borne in mind that the
method of distributing the material, whether in central bins or
in local piles (see chapter preceding on "Plant") will involve a
certain amount of wastage which must be taken into consider
ation in ordering the aggregate. Properly constructed shacks for
the storage of cement will reduce to a minimum the loss of ce
ment through accidental weathering, etc.
Table 37. — Proportions for Mixing Concrete
Mixtures
Yardages of materials for one cubic
in the form
yard of concrete
Specification stone '
up to 2 in.
Gravel,
^ in. size
Cement
Sand
stone
1 ;
Cement,
bble.
2.6
Sand, ' Stone,
yds. yds.
Cement,
bbls.
Sand,
yds.
Stone,
yds.
1.0
2
.4 .8
2.3
.4
.7
1.0
3
2.1 1 .3
9
1.9
.3
9
1.5
3
1.9 .4
8 ' 1.7
.4
8
1.5
4
1.6 1 .4 1
1.5
.3
9
2.0
3
1.7
.5
8
1.5
.5
7
2.0
4
1.5
.4
9
1.3
.4
8
2.0
5
1.3
.4
1
1.2
.4
9
2.5
5
1.2
.5
9
1.1
.4
8
3.0
4
1.3
.6
8
1.2
.5
7
3.0
6
1.0
.5
9
.9
.4
8
3.5
5
1.1
.6 '
8
1.0
.5
8
3.5
7
0.9 .5
9 : .8
A
9
4.0
6
0.9 .6 ■
8 .8
.5 i
8
4.0
8
0.8 .5 1
9 .7
.4 1
9
Cement. — (Portland cement, alone is discussed here.) It is
usual to specify that cement will meet the requirements of the
Committee of the American Society of Civil Engineers on, "Uni
form Tests of Cement." It is usual to insist that the brand of
cement used is one that has been employed on large engineering
works for at least five years.
Portland cement has been defined as the finely pulverized
product resulting from the calcination to incipient fusion of the
CONCRETE CONSTRUCTION 215
properly proportioned mixture of argillaceous and calcareous
materials to which no addition greater than 3 per cent, has been
made subsequent to calcination.
Its fineness shall be determined and limited as follows: The
cement shall leave by weight a residue of not more than 8 per
cent, on a No. 100 sieve and not more than 25 per cent, on a No.
200 sieve, the wires of the sieve being respectively 0.0045 and
0.0024 of an inch in diameter.
The time of setting shall be as follows: The cement shall
develop initial set in not less than 30 minutes, and shall develop
hard set in not less than 1 hour, nor more than 10 hours.
The minimum requirements for tensile strength for briquettes
one inch square in minimum section shall be as follows :
Heat Cement
Age Strengtn
24 hours in moist air 175 lb.
7 days (1 day in moist air, 6 days in water) 500 lb.
28 days (1 day in moist air, 27 days in water) 600 lb.
One Part Cement, Theee Parts Standard Sand
7 days (1 day in moist air, 6 days in water) 170 lb.
28 days (1 day in moist air, 27 days in water) 225 lb.
Neat briquettes shall show a minimum increase in strength
of 10 per cent, and sand briquettes 20 per cent, from the tests at
the end of 7 days, to those at 28 days.
Tests for constancy of volume will be made by means of pats
of neat cement about 3 inches in diameter, }i inch thick at the
center and tapering to a thin edge. These pats to satisfactorily
answer the requirements shall remain firm and hard and show
no signs of distortion, checking, cracking, or disintegrating.
The cement shall contain not more than 1.75 per cent, of anhy
drous sulphuric acid (SO,), or more than 4 per cent, of magnesis
(MgO).
The cement shall have a specific gravity of not less than 3.10
nor more than 3.25 after being thoroughly dried at a temperature
of 212°F. The color shall be uniform, bluish gray, free from yel
low or brown particles.
Sand.— Sand for concrete shall be clean, contammg not more
than 3 per cent, of foreign matter. It should be reasonable free
from loam and dirt. When rubbed between the palm the hand
should be left clean. It should be well graded from coarse to
fine No grains should be left on a ^inch sieve and not more
216 RETAINING WALLS
than 6 per cent, should pass through a 100 mesh sieve. Fine
sand is undesirable and its presence in a quantity greater than
that just specified will materially weaken the concrete. A coarse
smoothgrained sand is not objectionable and will produce, with
other things being equal, an effective and strong concrete. In
connection with the selection of the aggregate and the proportion
ing of the coarse and fine particles, a note in the appendix is
given on the selection and mixing of aggregates by the surface
area method and by the fineness modulus method and the rela
tion between these two modes of selection and the strength of
the concrete. "^
Crushed Stone and Gravel. — Crushed stone should be made
from trap or limestone. Stone from local quarries, or from
rock cuts encountered in the work should be used only after
tests have been made on concrete containing this stone. For
ordinary gravity walls, the size of the crushed stone or of the
gravel may vary from ^i inch to 1% inch in diameter. For the
thin reinforced concrete walls the stone should not exceed %
inch in size.
Occasionally the sand and the stone are delivered already
mixed in the required proportions. Parallel to this method,
the run of a gravel bank may be taken, including the gravel
with the finer sands. Either method of supplying the aggregate
is far from ideal and does not lend itself well to a conscientious
proportioning of the materials. It is preferable to supply the
coarse and the fine aggregates separately and mix them in the
required proportions in the mixer.
A resume of the above methods of selecting the aggregates
and cement is presented in the appendix in the shape of a standard
specification for retaining walls, including the proper specifying
of the materials entering into its composition.
Fineness Modulus of Aggregate.^— The experimental work car
ried out in the laboratory has given rise to what we term the
fineness modulus of the aggregate. It may be defined as fol
lows: The sum of the percentages in the sieve analysis of the
aggregate divided by 100.
The sieve analysis is determined by using the following sieves
iSee preceding pages on the fineness modulus; also Engineering News
Record, June 12, 1919, pp. 1142 to 1149.
^Bulletin No. 1, Structural Materials Research Laboratory, Lewis
Institute, Chicago, D. A. Abrams.
CONCRETE CONSTRUCTION 217
from the Tyler standard series: 100, 48, 28, 14, 8, 4, ^^in
Min. and l^in. These sieves are made of squaremesh wire
cloth. Each sieve has a clear opening just double the width
ot the preceding one. The exact dimensions of the sieves and
the method of determining the fineness modulus will be found in
Table 36. It will be noted that the sieve analysis is expressed in
terms of the percentages of material by volume or weight coarser
than each sieve.
A wellgraded torpedo sand up to No. 4 sieve will give a fineness
modulus of about 3.00 ; a coarse aggregate graded 41 ].^in. will give
fineness modulus of about 7.00; a mixture of the above materials
m proper proportions for a 1 :4 mix will have a fineness modulus of
about 5.80. A fine sand such as driftsand may have a fineness
modulus as low as 1.50.
100
igiw^
90
^;
I r^
^
'^^^
^
»; 80
N^
h^
f^^e
\ "^
\ ^
>
Q>
:? 70
l\
'
t^
^>^
\ "^
\ 1 ""
60
\
" ;
\; >
^ so
\
!\
V
^ 40
i ^
Ix
^\
[ ' \
O
+
O
L
^ 20
V!
\^
\
^
^
\
;o
4e
2
3 \A
e
4
3/
B
i ''/e
Sieve Size(Lco,Socile^
Fig. 120. — From Bulletin No. 1. D. A. Abrams, Structural Materials Research
Laboratory, Lewis Institute, Chicago.
Sieve Analysis of Aggregates. — There is an intimate relation
between the sieve analysis curve for the aggregate and the fineness
modulus; in fact, the fineness modulus enables us for the first
time to properly interpret the sieve analysis of an aggregate.
218
RETAINING WALLS
If the sieve analysis of an aggregate is platted in the manner
indicated in Fig. 120 that is, using the per cent, coarser than a
given sieve as ordinate, and the sieve size (platted to logarithmic
scale) as abscissa, the fineness modulus of the aggregate is mea
sured by the area below the sieve analysis curve The dotted
rectangles for aggregate "G" show how this result is secured.
Each elemental rectangle is the fineness modulus of the material
of that particular size. The fineness modulus of the graded
aggregate is then the summation of these elemental areas. Any
other sieve analysis curve which will give the same total area
corresponds to the same fineness modulus and will require the
same quantity of water to produce a mix of the same plasticity
and gives concrete of the same strength, so long as it is not too
coarse for the quantity of cement used.
The fineness modulus may be considered as an abstract num
ber; it is in fact a summation of volumes of material. There are
several different methods of computing it, all of which will give
the same result. The method given in Table 38 is probably the
simplest and most direct.
Table 38. — Tables Showing Mixtures op Test Mobtaks
Test Series No. 1. Cement Content — 1 G.: 13 Sq. In.
Sand letter
Surface area per
1000 g., sq. in.
Cement, g
Water, cc.
Ratio of cement
to aggregate by
weight
A.
5,856.6
5,106.1
7,683.7
6,758.4
12,816.4
6,769.1
4,182.0
6,564.6
6,564.6
450.5
392.0
591.0
520.0
986.0
521.0
321.5
505.0
505.0
128.0
111.5
134.5
148.0
280.5
148.0
91.5
143.5
143.5
1:2.22
B
1:2.55
c
1 : 1 . 69
D
1:1 92
E
1:1.12
F
1:1 92
G
1:3 11
H
1:1 98
I
1:1.98
Test Series No. 2. Cement Content— 1 G.: 10, 15, 20 and 26 Sq. In.
6,769
6,769
6,769
6,769
677.0
451.0
338.5
270.5
183.0
132.5
105.5
92.5
1:1.47
1:2.21
1:2.95
1:3.61
CONCRETE CONSTRUCTION 219
Some of the mathematical relations involved are of interest,
ihe tollowmg expression shows the relation between the fineness
modulus and the size of the particle:
m = 7.94 + 3.32 log d
Where m = fineness modulus
d = diameter of particle in inches
This relation is perfectly general so long as we use the standard
set of sieves mentioned above. The constants are fixed by the
particular sizes of sieves used and the units of measure. Loga
rithms are to the base 10.
This relation appUes to a singlesize material or to a given
particle. The fineness modulus is then a logarithmic function
of the diameter of the particle. This formula need not be used
with a graded material, since the value can be secured more
easily and directly by the method used in Table 36. It is appU
cable to graded materials provided the relative quantities of each
size are considered, and the diameter of each group is used. By
applying the formula to a graded material we would be calculating
the values of the separate elemental rectangles shown in Fig.
120.
Proportioning Concrete by Surface Areas of Aggregates.' —
Volumetric proportioning of concrete is notoriously unsatis
factory. Many investigators have been studying other propor
tioning methods which will at the same time be practical and will
insure a maximum strength of concrete with any given material.
The latest of such methods and one which in the tests gives
promise of some success is that devised by Capt. L. N. Edwards,
U.S.E.R., testing engineer of the Department of Works, Toronto,
Ontario, which was explained in some detail in a paper entitled
'Proportioning the Materials of Mortars and Concrete by Sur
face Areas of Aggregates," presented to the American Society for
Testing Materials at its annual meeting in June.
Briefly, Captain Edwards' principle is that the strength of mor
tar is primarily dependent upon the character of the bond exist
ing between the individual particles of the sand aggregate, and
that upon the total surface area of these particles depends the
quantity of cementing material. Reduced to practical terms,
this means that a mixture of mortar for optimum strength is a
^Engineering NewsRecord, Aug. 15, 1918, p. 317 et seg.
220 RETAINING WALLS
function of the ratio of the cement content to the total surface
area of the aggregate regardless of the volumetric or weight
ratios of the two component materials. As a corollary to his
investigations, Captain Edwards also lays down the principle
that the amount of water required to produce a normal uniform
consistency of mortar is a function of the cement and of the sur
face area of the particles of the sand aggregate to be w^etted.
Some of the tests deduce the fact, already demonstrated in a
number of previous tests, that strength of mortars and concrete
is a definite function of the amount of water used in the mix.
In demonstrating the cementsurface area relation, the test
procedure was as follows : First, a number of different sands were
graded through nine sieves, varying from 4 to 100 meshes per
inch, and the material passing one sieve and retained on the next
lower was separated into groups. From each group, then, an
actual count was made of the average number of particles of sand
per gram. For the larger sizes 8 to 10 grams or more, medium
sizes 3 to 5 grams, and for the smallest sizes 34 to 1 gram were
counted. For six sands counted, including a standard Ottawa
which is composed of grams passing a 20 and retained on a 30
mesh sieve, the following averages were obtained for the number
of sand particles per gram:
Passing 4, retained on 8 14
Passing 8, retained on 10 55
Passing 10, retained on 20 350
Passing 20, retained on 30 1,500
Passing 30, retained on 40 4,800
Passing 40, retained on 50 16,000
Passing 50, retained on 80 40,000
Passing 80, retained on 100 99,000
With a specific gravity of sand of 2.689, which had been deter
mined by a number of tests, the average volume per particle of
sand was determined for each group, and assuming that the shape
of the particles of sand was spherical, which is approximately
correct, the surface area per gram of sand was determined for
each group. The results are shown in Fig. 121. This gave a
basis of surface areas for the various groups of sand in hand.
The sands were then regarded to different granulometric
analyses in order to get representative and different kinds of
aggregate for the tests. Using these sands for the aggregate,
numerous briquets and cylinders were made up and tested in
CONCRETE CONSTRUCTION
221
tension and in compression, varying the mix according to the
ratio of the weight of cement to the surface area of the sand
aggregate. The basis of the ratio of grams of cement to square
inches of surface area were 1 :10, 1 :15, 1 : 20 and 1 : 25. The con
sistency throughout was controlled so that the water content
would not affect the relative strengths of the different specimens.
eO 40 60 80 100 120
Diameter of Particle of Sand in 0.001 Inch
Fig 121 — Capt. Edwards' method of surface areas. {From Engineering News
Record, Aug. 15, 1918, p. 317.)
Test mortars were then made, first, by keeping the cement
surface area ratio constant and varying the kinds of sand; second,
by varying the ratio and using the same and. These two series
are shown in the accompanying table. As will be n.oted from
Table 38, in test series No. 1 the cement content is one gram for
thirteen square inches of surface area, but the sand has such a
different grading and therefore total surface area that the ratio
o cementtoaggregatebyweightvarysfrom 1:1. 12to 1:3.11. In
spite of this wide variation in weight and therefore in volumetric
relation of the cement to the aggregate, the strength values, as
shown in Fig. 122, were markedly constant. In series No. 2
the cement constant varied from 1 gram to 10 sq. in. to 1 gram to
25 sq. in. of sand surface, and, as shown in Fig 123, the strength
curves are proportionate te the cementarea ratio.
Further tests were made by Captain Edwards extendmg this
investigation to concrete, and while these showed the same gen
222
RETAINING WALLS
eral results, the tests were not sufficiently elaborate to warrant an
abstract of them here.
It might seem offhand that there is no practical occupation to
the method. Certainly, the very considerable labor involved
in counting 125,000 sand grams for one sieve group alone would
deter anyone from contemplating such a program for practical
5500
C D E F
Sand Lether
Fig. 122. — Capt. Edwaida' method of surface areas. (From Engineering News
Record, Aug. 15, 1918, p. 317.)
work, if such a count had to be made very often. However,
Captain Edwards points out that this elaborate counting is
required only as a prehminary to his method and once done need
not be repeated. He says :
"The adaptation of the surface area method of proportioning mortars
and concretes to both laboratory investigation and field construction
CONCRETE CONSTRUCTION 223
SSw w' """ 'f °^' ^'^'''''^ ^^' outstanding feature of
^TtTn^lfl '!^*' "'^''^"^ application is concerned, is the im
portance of knowing the granulometric composition of the aggregate
il amount f?'^ '" '"/"* ^"^^ ^'^°^'^^*^ ^°l« ^ comparatfvely
eS« vTl / ^^V!^ ^^ ™y °^ equipment the use of only the nec
essary scales, standard sieves and screens. The time element involved
IS comparatively negligible, since the computation work of determining
areas and quantities of cement may be largely reduced to the most
simple mathematical operation by the use of tables and diagrams "
5500
g 4500
^ 3500
35
o 2500
1500
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Fig. 123.
Capt. Edwarda' method of surface areas. {From Engineering News
Record, Aug. 15, 1918, p. 317.)
Diagrams for Laboratory and Field Use. — For use in the labora
tory and in the field, diagrams drawn to a large scale increase
accuracy and reduce labor. Fig. 124 is designed for use in de
termining the surface area of sand aggregate. It is intended for
laboratory use. Fig. 125 is the same sort of diagram intended
for both laboratory and field use. The diagrams are derived
from information obtained in the tests. Fig. 126 is designed
for use in determining the surface of stone aggregate, and is in
tended for both field and laboratory use, and Fig. 127 shows the
conversion diagram for determining the relative quantity of
cement in pounds per 100 lb. of sand, and the corresponding
relation of cement in grams to the surface area of 1,000 grams of
sand, and vice versa. The author then gives the following ex
ample of how the diagrams shown in Figs. 124127 may be used:
224
RETAINING WALLS
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CONCRETE CONSTRUCTION
225
160
140
IZO
10
o
o
o
100
5 6 7 8 9 10 11 12 13 14 15
Surface Area of lOOOg. of Sand; Thousoinol Sq.In.
Fig. 127. — (From Engineering NewsRecord, Aug. 15, 1918. Capt. Edwaras.)
Example No. 1.— Required to find the composition of a batch
of mortar using 1,000 g. of sand A and a cement content propor
tioned: 1 g. cement to 15 sq. in. sand area.
Sand Area
Sieve
P 4R
P 8R
P 10R
10.
20.
Grading,
per cent.
15.0
5.0
25.0
P20R 30 15.0
P30R
P40R
P50R
40.
50.
80.
P 80R 100.
Totals . . . .
15.0
10.0
10.0
5.0
Weight, g.
150
50
250
150
150
100
100
50
Area {Fig. 4),
sq. in.
142
75
694
676
997
992
1,348
932
100.0
1,000
5,856
5856
Cement (g.) = yg'
390.5
Water (c.c.)
= J 390 5 X 22.25 per cent, (normal consistency)  +
5856
210
= 115
226 RETAINING WALLS
The author does not give anywhere what he considers to be
proper ratio of the cement to the sand surface area. That would
presumably have to be determined by investigations of the ag
gregates involved in any case.
Ratio of Fine to Coarse Aggregate Basis for Concrete Mix
ture.' — ^Another method of proportioning concrete mixtures is
proposed by R. W. Crum, in a paper, read before the American
Society for Testing Materials, and entitled "Proportioning of
PitRun Gravel for Concrete." The method was devised for
and is specially applicable to Middle Western gravels which occur
in assorted gradings. By its use a proper concrete can be had
with any pit gravel by the addition of the correct amount of
cement, to be determined by the method. Basically, the author's
scheme rests on the assumption that the ratio of cement to air
and water voids is an indication of strength. In other words,
the nearer the cement content approaches the volume of the
voids the greater is the strength of the concrete. He assumes
that for certain classes of concrete — ^that is, for concrete to be
used under certain conditions — there is an optimum sandaggre
gate ratio. In that ideal mix the cementvoid ratio is computed
and the amount of cement necessary to bring the actual mix up to
that ratio is found. This gives the best mixture — reducing to
loose volume — ^for that particular aggregate. Although the au
thor states that the proper grading depends upon the consistency
or amount of water in the mixture, and although he says specifi
cally that one must get a concrete which will yield a workable
mixture for the conditions under which it is placed, he does not
tell in the paper just what degree of workability is reached by his
method nor the standard of consistency or workabiUty which was
used in making the tests. He claims that the method gives
results about midway between the finenessmodulus method of
Abrams and the surfacearea method of Edwards. Analyses of
prospective aggregates may be readily made in the field for the
method, inasmuch as it requires only to be known the gradings
above and below a No. 4 sieve.
' Engineering NewsRecord, July 10, 1919.
Plate V
■c: c.ji"'^' ^ ■■■' '<
^ :7
F,G t Mothnd Of layi.ig slone wall hy series of derruKS.
lie 1^27)
Plate VI
I'^iG A. — UncoLirsed rubble wall with coursed effect given by false pointing
J' lu. b. — Itu
jDie wail (Los Angelew) wntli face formed by niggerhends.
CHAPTER IX
WALLS OTHER THAN CONCRETE
Plant. — Rubble and cutstone walls up to 5 or 6 feet in height
are built of stone of such size that they are easily raised and set
by hand. No special plant is therefore required and the wall is
built entirely by hand labor. As the walls increase in height,
good construction requires the use of larger stone, to insure a
wall properly bonded together and it becomes necessary then
to employ plant to raise and set the large stone. A derrick,
either a guy or a stiff leg, is probably the most serviceable and
efficient piece of plant to use in setting stone walls. It is op
erated by a hoist run by steam, electricity or air. A guy derrick
is possibly preferable in that it permits a greater swing of the
boom. It is limited however by the fact that it requires ample
room to anchor its guys, room not always available, especially
in city work. A stiffleg derrick is a self contained unit, the
weight of the hoist and power plant providing the necessary
anchorage.
In setting a derrick care should be observed that it is placed
back from the wall a distance sufficient to ensure 'topping' out
the wall. When the yardage of masonry permits, it is most
economical and proves most time saving to set up a chain of
derricks at such intervals that no gaps are left in the wall. This
continuity of the work will obviate the tendency to cracks caused
by joining up new work with old work (see Chapter V, "Settle
ment"). The derricks, when set up in sequence, are easily dis
mantled and set up in their new positions by aid of the adjoining
derricks. Photographic Plate No. V, Fig. A shows the method
of constructing high rubble walls (over 32 feet high) by means
of such plant.
Mortar —The mortar for use in the rubble masonry walls is
mixed alongside the wall and is deHvered to the working gangs
in bucket loads as required. The usual mortar is mixed in pro
portions of one cement to three sand. For work of large size
228 RETAINING WALLS
conveniently located, it may prove economical to mix the mortar
by machine in a central plant and deliver by cart or otherwise
over the work. Usually, however, it has proven most efficient
to mix the mortar by hand for each gang, or for two adjacent
gangs. The cement required for a rubble masonry wall of
fairly large size (varying from twelve to forty feet in height)
will average about one and onehalf bags to the finished yard of
wall. Due care in dressing the stone and chinking up the
interstices with spalls will help to keep the amount of cement
required to a minimum. With mortar mixed in the proportion
of one cement to three sand, the finished wall should contain from
15 to 20 per cent, mortar.
Construction of Wall. — In constructing the wall the largest
stone should be placed at the bottom course. If the soil is a
shghtly yeilding one the stones may be dropped from a height
of two to three feet to insure their thorough imbedment. The
bottom course may consist of a lean concrete in place of the
rubble stone. The wall should have a proper proportion of
headers (stones lying transversely) usually about }i of the total
yardage. The stones should be most carefully bedded, and all
the interstices filled with spalls and if the wall is a mortar one,
finished off with the cement mortar. The construction of a
rubble masonry wall, both dry and cement requires a most
conscientious cooperation between the engineer and the con
tractor and it is only by such mutual aid that a good masonry
wall can be built. When a section of wall is to be finished some
time before the adjoining section is to be built it is well to "rack"
back the sides to insure a good bond between the old and new
work. It must be remembered that a masonry wall has no
expansion joints and that all movement of the wall, must be
taken up by the masonry itself. Cracks will therefore a pear
along the plane of weakness and unless great care has been
exercised in the laying of the wall, these cracks will become very
disfiguring.
The stone should be good, sound stone, thoroughly cleaned
and roughly dressed to take off the soft and cracked edges. It
should be wet before setting, especially in hot weather. Friable
and soft stone should not be used. An excellent example of
rubble masonry specifications, is quoted here.'
1 Track Elevation, Philadelphia, Germantown and Norristown R. R.,
S. T. Wagner, Trans. A.S.C.E., Vol. Ixxvi, p. 1833.
WALLS OTHER THAN CONCRETE 229
" Thirdclass masonry shall be formed of approved quarry stone of good
shape and of good flat beds. No stone shall be used in the face of the
walls less than 6 inches thick or less than 12 inches in their least hori
zontal dimension.
Headers shall generally form about }4 of the faces and backs of the
walls with a similar proportion throughout the mass when they do not
interlock, and the face stones shall be well scabbed or otherwise worked
so that they may be set close and chinking with small stone avoided.
In walls five feet thick or less, the stones used shall average 6 to 8
cubic feet in volume and the length of the headers shall be equal to two
thirds the thickness of the wall. In walls more than five feet in thick
ness the stones used shall average 12.cubic feet in volume and the headers
shall not be less than four feet long. Generally no stones shall be used
having a less volume than four cubic feet except for filling the interstices
between the large stones.
In no case shall stones be used having a greater height or build than
30 inches and these stones must bond the joints above and below at least
18 inches; in all other cases the smaller stone must bond the joints above
and below at least 10 inches.
The stones in the foundation shall generally not be less than 10 inches
in thickness and contain not less than 10 square feet of surface. The
foundation shall consist of 1:3:6 concrete, if so directed by the Chief
Engineer."
Coping. — The wall, either dry or cement, is usually topped
with a coping. Expansion joints in this coping should be
placed at intervals of about five to ten feet. The sections may
be separated by plain paper or may be tarred. The coping
should preferably be placed after the wall has been constructed
for some time. This permits settlement to take place and where
definite cracks appear in the wall, expansion joints may be placed,
to avoid unsightly cracking of the coping itself. When built of
concrete, the coping should be about one foot thick and offset
from the face of the wall about 3 or 4 inches. The form for
the coping should be well built and carefully Uned. Any care
lessness in lining the coping forms shows in a wavy broken coping
fine and proves unsightly. The forms should be built of 2 inch
stock, carefully wired and braced. This will prevent the bulging
of the coping face and the thickness of the form will permit a
frequent reuse of the form.
If a stone coping is desired, a blue stone flaggmg from 4 to
8 inches in thickness makes an effective top finish for the wall
230 RETAINING WALLS
Face Finish. — The face of a rubble or other masonry wall,
receives such treatment as the environment of the wall requires
(see Chapter X on "Architectural Treatment"). With care
in the selection of face stone and with a fair attempt to dress
these stones, the wall needs but httle other work upon it except
some pointing of the joints. As the demand for special face
treatment increases, more attention must be paid to the selection
of face stones and to the pointing of the joints. Face treatment
may, roughly be divided into the following classifications : Rough
pointing; special or false pointing; selection of special face stone;
plaster finishes.
Rough Pointing. — After laying the wall, the stones are cleaned
of whatever mortar has accidentally dropped upon them. The
joints are raked and then brought to the rough face plane with
mortar. For walls as generally built in the outlying districts,
this type of treatment is sufficient.
False Pointing. — To obtain a somewhat more pleasing and
decorative effect, rough, uncoursed masonry is pointed falsely,
to give the appearance of coursed masonry. After cleaning the
face stones the face of the wall is brought to a rough plane and is
then coursed with the trowel into rectangles. Work of this
nature is not of great permanence, the mortar slowly spalling
off with the weather. (To secure the coursed masonry effect,
more surface of the wall must receive a mortar coat than is
necessary otherwise.) It is of questionable taste to attempt
to mask the nature of the wall by such face treatment. This
mode of treatment is usually hmited to small walls forming the
street walls of residential plots. A photograph of this class of
wall is shown here (Photo Plate VI, Fig. A).
Special Stone. — The character of the masonry comprising the
wall body may be completely masked by fbrming the face of the
wall with specially selected stone. The rough masonry may then
be considered a backing for the selected stone masonry. For
walls entering into a costly and decorative scheme of landscape
work, the face may be made an ashlar, or other coursed masonry
effect, using Hmestone, sandstone or granite. When the walls
are of considerable thickness it is usual to build them thus, with
the expensive stone at the surface only. Walls of this type are
the most costly of all walls, yet present the most imposing and
pleasing types of masonry construction. The details of construc
tion of these walls are thoroughly discussed in a number of
WALLS OTHER THAN CONCRETE 231
standard textbooks {e.g. Baker's "Masonry Construction")
and need not be mentioned here.
A very pleasing effect secured by the use of boulders or "nigger
heads " is shown here (see Photo Plate VI, Fig. B) (used extensively
in Los Angeles). Various modifications of work of this kind
are readily adapted to local environments with exceptionally
pleasing results.
Plaster Coats. — This is probably the least desirable of surface
finishes, both in effect and in duration of life. Because of its
limited permanence great care must be exercised in applying
these coats to the face of rough masonry walls. Plaster or
stucco coats, when applied to the face of a wall, are rough cast
or stippled. No trowelhng is done upon the face, the mortar
being placed with the usual wooden mortar board. To insure
permanence some form of wire mesh or other netting should be
fastened to the face of the wall to hold the plaster coat. The
netting may be attached to wooden plugs inserted in the mortar
while the wall is in the course of construction.
Cost Data. — The following is an analysis of the cost of a wall
36 feet high, averaging about 13 cubic yards to the running foot.
It is merely a labor charge and does not include the cost of obtain
ing the stone, etc.
Cement Rubble Wall. 2750 Cubic Yards
Foreman, 114 days at $6.00 per day S684.00
Masons, 167 days at $4.50 per day 751.50
Hoistrunner, 113 days at $6.00 per day 678.00
Signalman, 90 days at $2.50 per day 225 . 00
Laborers, 625 days at $2.50 per day ^562.50
Total cost 53901.00
The average cost per yard, exclusive of all overhead, insurance, plant
charges, materials, etc., is $1.42 per yard.
CHAPTER X
ARCHITECTURAL DETAILS, DRAINAGE, WATERPROOFING
Architectural Treatment. — Concrete retaining walls form a
class of engineering structures for which ornate decorations are
of questioned taste. Occasionally, however, some special face
treatment becomes necessary to permit the wall to enter into
the general landscape improvement involving a particular archi
tectural scheme. Thus, for example, retaining walls forming an
approach to a bridge, especially a concrete arch are usually
made to' follow the general viaduct architecture. Walls for a
railroad station, where the main hne is on the fill, must be in
keeping with the architectural motive of the building itself.
Walls in parks must receive such treatment as will make them
harmonize with the park landscape work. In general, however,
simphcity of treatment is essential, to conform with good taste.
Concrete walls are finished on top with a coping; usually about
one foot thick and projecting 3 to 6 inches beyond the face of
the wall. In addition a hand rail, picket fence, or concrete
parapet wall is placed on top of the wall of plain or ornamental
effect as conditions indicate. The face of the wall receives such
treatment as will remove the unavoidable blemishes of
construction.
Face Treatment. — The concrete face of the retaining wall may
either be rubbed, tooled or receive a special composition surface.
Prehminary to applying the face treatment, the tie rods, wires,
etc. are cut back, and the face patched where necessary, employ
ing a stiff mortar for this purpose. To insure a successful surface
finish, it is imperative that the wall be well built. A surface
finish cannot conceal poor work and poor work will eventually
destroy the best surface finish. The less a wall is patched or
otherwise repaired, the more certain it is that the surface treat
ment will be of pleasing and permanent character. Board marks
are left after the forms are stripped which may be more or less
masked by careful treatment. It may be set down as almost
axiomatic' that board marks can never be entirely eradicated,
232
ARCHITECTURAL DETAILS 233
no matter what face treatment is applied. For this reason care
must be taken in the continued use of the same set of forms, so
that no panel is used in the face after its edges become sphntered
or frayed.
It has been pointed out in a previous chapter that construction
joints leave a distinct cleavage mark. To make sure, for walls
that will occupy a position of more or less architectural promi
nence, that there shall be no construction joints, it is specified
that the section of wall between the expansion joints shall be
poured completely in one operation. This is a praiseworthy
mandate and is worthy of adoption for all character of work,
regardless of merely the insistence of an architectural finish.
The distance between expansion joints may be made such that it
is practicable to pour a section complete with ordinary plant in
one pour.
Defective concrete work appearing at the surface must be
removed immediately upon stripping the forms and a rich mortar
concrete iaserted. Haphazard patchwork will not do. It is
but a temporary expedient and the patch will soon spall off
leaving a disfigured wall. A photograph of a wall so treated is
shown here (See Fig. A, Plate VII) and is eloquent of the effects
of poor concrete work and poor patch work.
If the forms are not held tight, or are not carefully caulked
above work aheady completed, the yielding of the form, even to a
minute degree, will permit the grout to run down coating and
disfiguring the concrete work.
Briefly stated, conscientious vigUance in the observance of the
edicts of good concrete work is the price of a good surface finish
and using the analogy of pathology, diseases of the concrete
body of a wall are usually exhibited by symptoms of facial
blemishes.
Rubbing. — The face of concrete mirrors most faithfully the
inside face of the form, bringing out the dehneations of the board
marks, the hps of the panels, etc. Immediately upon stripping
the forms, and after cutting the rods and wires where necessary,
and after making such patches as are indicated, the face is rubbed
down with an emery block, and a thin grout wash is appHed at
the same time. The fresher the concrete, the easier it is to
remove the facial blemishes by rubbing and it is therefore im
perative that the forms be stripped as soon as good construc
tion permits. For the average environment, and over 90 per cent.
234 RETAINING WALLS
of retaining walls are built in such environment, rubbing a wall
presents finally a surface that is sufficiently pleasing.
In applying the grout wash, care must be taken to use a con
stant proportion of the cement and water. It is quite possible,
where the rubbing is not done on one day, to use grout mixes of
different strengths leaving the surface finished in two shades.
Tooling. — If the cement skin of a concrete wall is removed by
sharp bits, the abraded surface gives a rough stone appearance
quite pleasing in effect. This skin may be removed by hand with
an ordinary wedge bit, or with special two, four and six edged
bits. If there is a large amount of surface to be so treated, it is
a matter of economy to use an air drill to work the hammer.
The hammer is passed Hghtly over the surface, apphed just long
enough to remove the grout skin, care being taken not to start
ravelling the stone. A gravel concrete seems to give a better
appearance than a broken stone concrete, the sparkhng effect
of the pebbles presenting an excellent appearance, especially
in the direct sunlight. When broken stone is used, the size of the
stone should be hmited to % inch stone, the ordinary commercial
stone. With larger stone it is difficult, in tooling the wall to
prevent ravelHng.
It is understood that toohng is much more expensive than rub
bing (roughly about ten to fifteen times) and, ordinarily is only
specified to effect a special architectural feature.
As in the case of rubbing, the concrete wall must be carefully
patched and construction devices, such as rods, wires, etc., removed
or cut back several inches from the surface.
It is usual to finish the edges of a tooled surface by means of a
rubbed border of one or more inches in width. Care must be
taken not to tool too near an edge as the concrete may be broken
off.
Special Finishes. — To enhance the architectural appearance
of a retaining wall, a special face finish is appHed to the wall,
masking its construction finish. An ordinary plaster coat may
be applied to the wall, or a granolithic or other fine grit finish may
be placed upon its surface. In applying such a coat it is essential
that due appreciation should be had of the proper bond between
the wall and the coat. To apply a coat of mortar or other finish
after the forms have been stripped and the wall set gives Uttle
assurance of a permanent finish. The coefficients of expansion
between the wall concrete and the rich mortar are unlike, produc
ARCHITECTURAL DETAILS 235
ing eventually voids between the wall and coat. The action of
frost and the other destructive elements finally cause the coat to
spall. It is therefore usually specified that the finish coat shall be
applied simultaneously with the pouring of the wall, so that the
coat is a part of the wall itself, and is therefore more or less immune
to the weathering actions. An excellent specification for a grano
lithic coat is quoted here and may be used as a model clause for
all grit finishes.^
"Surface of concrete exposed to the street shall be composed of one
part cement, two parts coarse sand or gravel and two parts granolithic
grit, made into a stiff mortar. Granolithic grit shall be granite or trap
rock crushed to pass a J^ inch sieve and screened of dust. For vertical
surfaces the mixture shall be deposited against the face forms to a least
thickness of one inch by skilled workmen, as the placing of concrete
proceeds and thus form a body of the work. Care shall be taken to
prevent the occurrence of air spaces or voids in the surface. The face
forms shall be removed as soon as the concrete has sufficiently hardened
and any voids that may appear shall be filled up with the mixture.
"The surface shall then be immediately washed with water until the
grit is exposed and rinsed clean and protected from the sun and kept
moist for three days. For horizontal surfaces the granolithic mixture
shall be deposited on the concrete to a least thickness of 1.5 inches
immediately after the concrete has been tamped and before it has set
and shall be trowelled to an even surface and after it has set sufficiently
hard shall be washed until the grit is exposed.
"AU concrete surfaces exposed to the street shall be marked off into
courses in such detailed manner as may be directed by the Chief
Engineer."
Finishes of various colors may be secured by the use of properly
colored grit. A red finish may be secured by the use of brick
grit; a gray by bluestone screenings, etc. Below is a method
of obtaining still another type of surface finish.^
"A surface finish for concrete, whereby a sand coating is applied may
be secured by the following method, outlined by Mr. Albert Moyer
of the Vulcanite Portland Cement Co. Erect forms of rough boards
m courses of three feet or less and plaster the insides with wet clay
worked to a plastic consistency. While the clay is wet apply evenly
loose buff, red or other colored sand and then pour in the concrete.
1 S. T. Wagner, Track Elevation, Philadelphia, Germantown and Norris
town Railroad, Trans. A.S.C.E., Vol. Ixxvi, p. 1836.
' Engineering Record, Vol. 61, p. 454.
3
236
RETAINING WALLS
After removing the forms, wash off the clay with water and if necessary
scrub lightly with a brush. The sand, Mr. Moyer states will adhere to
the concrete and givs a surface of pleasing color and texture.
The following table gives the proportion of coloring matter
to use to secure a desired shade of concrete finish. The table
is taken from "Concrete Construction for Rural Communities,"
by Roy A. Seaton, page 148.
Color of hardened
mortar
Mineral to be used
Pounds of color
to each bag of
cement
Gray
Black
Black
Blue
Green
Red
Bright red
Brown. . . ,
Buff
Germantown lampblack
Manganese dioxide
Excelsior carbon black
Ultramarine blue
Ultramarine green
Iron oxide
Pompeian or English red
Roasted iron oxide or brown ochre
Yellow ochre
>'2
12
3
5
6
6
6
6
6
"Colors wUl usually be considerably darker while the concrete is
wet than after it dries out and the colors are likely to grow somewhat
lighter with age. Hence considerably more pigment should be used
than is necessary to bring wet concrete or mortar to the desired shade."
Artistic Treatment of Concrete Surfaces in General. — The
treatment of concrete surfaces of all types is ably discussed in a
book by Lewis and Chandler, "Popular Hand Book for Cement
and Concrete Users" (see chapter "Artistic Treatment of Con
crete Surfaces"). The various methods of finishing a concrete
surface are classified as follows :
"1. Spading and trowelling the surface.
"2. Facing with Stucco.
" 3. Facing with Mortar.
"4. Grouting.
"5. Scrubbing and washing.
"6. Etching with Acid.
"7. Tooling the Surface with Bushhammers or other tools.
"8. Surfacing with gravel or pebbles.
"9. Tinting the surface.
"10. Panelling, Mosaic, carving etc."
Plate VJl
Fig. .1. — Showing effects ol poor concrete work
Fia. B. — Ornamental parapet wall. Tooled willi ruljl.ed borders.
{Fiicing page 23G)
Plate VIII
Fic;. A. — (hiiiimcMilal hmiJiail — aijproacli
1
^;;;r:;:;:rZ::Z~^^^n.n, ope. cut a„„n,a.h to ,:,ep,e.ed ...eet
(Facino P''ffe ■"■'''
Plate IX
Fig. a. — Ornamental concrete hanJiail approach to connote aitli.
ARCHITECTURAL DETAILS 237
The methods specially applicable to retaining walls have been
analyzed in detail in the present chapter.
In connection with the artistic treatment of retaining wall
surfaces, it may prove of interest to note that an exhaustive
study of a special surface was made by John J. Earley, Proceedings
American Concrete Institute, 1918, in a paper entitled "Some
Problems in Devising a New Finish For Concrete. " The wall
under discussion was built in Meridian Hill Park, Washington,
D.C. The original plans called for a stuccofinished wall. A
sample of wall with such a finish was built. "The result was a
plaster wall, nothing more ********** ^j^g ^g^y ^^^
without scale. It did not give the appearance of strength or
size equal to its task as a retaining wall. " It was finally decided
to strip the forms of the wall as soon as possible after pouring
(from 24 to 48 hours) and scrub the surface with steel brushes
"until the aggregate was exposed as evenly as possible. "
"This method of treating the surfaces at once supplied the sense of
strength and size that was lacking before. The wall was no longer a
plastered one, but was reinforced concrete and nothing else, and it
seemed big and strong enough to suit all demands that would be made
upon it."
The face was panelled and the piers were treated differently,
to afford a contrast to the tooled surfaces.
Hand Rails. — To prevent accidents and trespassing or to lend
a pleasing finish to a retaining wall a raihng of some kind is
built into the coping of the wall, of a character in conformity
with the needs of the environment. When a wall retains an em
bankment rising above the surrounding country, the raihng is
reqtdred as a protection to those walking along the edge of the em
bankment. If the environment demands a raihng more ornate in
character, the raihng may be made of concrete, stone, concrete
blocks, etc. Some photographs of raihngs of this latter character
are shown here (see Plates VII, Fig. B, VIII, Fig. A and IX, Fig. A).
To prevent trespassing, by cKmbing over low walls, or walls which
Hne cuts along a highway, it is usual to build a picket fence.
A photograph of a standard type of such fence is shown on Plate
VIII, Fig. B.
The metal raihngs are anchored to the wall by bolts. Holes
are drilled in the wall coping to fit the raihng bolts and the bolts
are fastened in by means of grout, lead or sulphur. To properly
238 RETAINING WALLS
and securely fasten concrete railings to the wall reinforcing rods
should be incorporated in the coping while it is being poured and
should project a distance above the top of the coping to obtain a
good bond to the hand rail. For all types of raihng provision
should be made for the expansion due to temperature changes.
Drainage. — The presence of water in a retained fill increases
the earth thrust in an uncertain but considerable amount.
Again, to insure a well founded roadbed, water must not be
permitted to accumulate in the fill. For these reasons means
are provided for the removal of any water that may collect in
the fill behind the wall. The simplest method of accomplishing
this is to insert pipes in the walls at frequent intervals, permitting
the water to drain through them and out on the surrounding
ground. To insure ample provision for the runoff of the water
and to prevent the pipe from silting up, a large size pipe, about
4 inches in diameter has proven to be most satisfactory as a
weephole drain. The pipes should be spaced from twenty five to
tenfeet intervals depending upon the anticipated conditions of
water accumulation. That water may be permitted to reach
these openings in the wall, some rough drainage must be placed
at the back of the wall. A well planned wall will provide for a
layer of broken stone, from 6inches to a foot in thickness upon
the back of the wall and extending down to the level of theweep
holes. If this method is considered too expensive, or unneces
sary for the conditions at hand, a layer of broken stone may be
placed immediately around the weephole, preventing the silt
from accumulating at the opening and permitting the water to
drain off. Under no circumstances should the fill be placed
immediately against the wall drains.
It is sometimes objectionable or impossible to dispose of the
water through drains leading out from the face of the wall,
because of private property, or important public thoroughfares
adjoining and a regular sewerage system must be installed to
dispose of the water through the neighboring sewers. For
example in the track elevation work of the Rock Island Lines. ^
"An unusual feature is the provision of drainage wells in the ends of
the retaining walls adjacent to the abutments at the subway bridges.
These are 3 feet by 3 feet and extend to the bottom of the wall (see Fig.
128). There are no weep holes through the retaining walls, but along
' Engineering News, Vol. 73, p. 671.
4'Tile Drain
6' Tile Drain
Fig. 128.— Drainage of retaioed fill,
carried to sewer system.
ARCHITECTURAL DETAILS 239
the backs of the walls are laid inclined drains of 6inch porous tile on a
grade of 0.5 per cent, extending from subgrade level to 6inch pipes,
which are imbedded in the rear part of the waUs and discharge to the
drainage wells. Each well has an 8inch connection to the catchbasin
of a city sewer as shown."
Again in the track elevation work of the Philadelphia, German
town and Norristown R. R. i The walls were on private property
and a layer of loose stone made up in sizes varying from % inch
to two feet were placed along the
back of the wall. A 6inch vitri
fied tile pipe was laid along the
bottom of the wall below this
stone layer, on a 1 per cent,
grade, with open joints and led
to sewers on the cross streets.
Another efficient method of
securing a well drained fill is to
place wells of broken stone at
each weep hole extending from
the subgrade of the fill to' the weep holes. In the construction
of the retaining walls for the Hell Gate Arch Approach (see
page 127) it was vital that no water be allowed to accumulate
in the fill and wells were built at each weep hole to insure the
drainage of the earth work.
Waterproofing. — The presence of water in the wall body,
aside from that left originally from the concrete mix, has a harm
ful effect both on the concrete mass itself and upon the face of
the wall. Generally it is specified that some means shall be
taken to keep the water out of the wall. Retaining walls are
not made of very rich mixes so that the wall cannot be said to be
inherently waterproof. It is an easy matter to coat the back
of the wall with tar or asphalt preparation. While it is exceed
ingly difficult to get an intact skin and to keep it intact, care
exercised in placing the waterproofing and in preserving it from
accidental abrasion after it has been placed will give a membrane
of sufficient integrity to save the face of the wall. It is much
better practice to place two coats of waterproofing upon the
back of the wall, thus insuring that there are no bare spots on
the wall back.
1 Trans. American Society of Civil Engineers, Vol. Ixxvi, S. T. Wagnek.
240 RETAINING WALLS
Before placing the membrane of tar, it is absolutely necessary
that the wall be dry, free from frost and well cleaned. After
the tar has been placed the fill should be deposited with care and
large boulders should not be permitted to roll down and against
the back of the wall. Where a mixed fill, rock and earth is used,
it is good practice to carry up the soft fill against the back of the
wall (unless a stone drainage well has been placed against the
back of the wall) to act as a cushion for the rock fill.
Where expansion joints occur, several layers of fabric coated
with hot tar are placed across the joint to insure its water
tightness, extending about a foot or two on either side of the
joint.
Subsurface walls and walls whose exterior face receive special
architectural treatment to which any moisture is damaging,
must, of course, receive more detailed waterproofing, involving
the extensive use of fabrics, of brick laid in an asphaltic mastic,
or the possible additions of chemicals to the concrete mix itself
(the integral method of waterproofing) all of which fall without
the province of the present text.
A typical and welltried specification for a tar coating for the
back of the wall, may read as follows :
Coaltar shall be straightrun pitch containing not less than twentyfive
percentum (25%) and not more than thirtytwo percentum (32%) of free
carbon, and shall soften at approximately 70° F., and melt at 120° F., deter
mined by the cube (in water) method, being a grade in which distillate oils
distilled therefrom shall have a specific gravity of 1.05.
Asphalt shall consist of fluxed natural asphalt, or asphalt prepared by the
careful distillation of asphaltic petroleum and shall comply with the follow
ing requirements :
The asphalt shall contain in its refined state not less than ninetyfive per
centum (95%) of bitumen soluble in cold carbon disulphide, and at least
ninetyeight and onehalf percentum (98.6%) of the bitumen soluble in the
, cold carbon disulphide shall be soluble in cold carbon tetrachloride. The
remaining ingredients shall be such as not to exert an injurious effect on the
work.
The asphalt shall not flash below 350 degrees Fahr., when tested in the
New York State Closed Oil Tester. When twenty (20) grams of the mate
rial are heated for five (5) hours at a temperature of 325 degrees F., in a tin
box two and onehalf inches in diameter it shall lose not over five percentum
(5%) by weight nor shall the penetration at 77 degrees Fahr. after such
heating be less than onehalf of the original penetration.
The melting point of the material shall be between 115 degrees and 135
degrees Fahr., as determined by the Kraemer and Sarnow method.
ARCHITECTURAL DETAILS 241
The consistency shall be determined by the penetration which be between
75 and 100 at 77° F.
A briquette of solid bitumen of crosssection of one square centimeter
shall have a ductility of not less than twenty centimeters at 77° F., the
material being elongated at the rate of five (5) centimeters per minute.
(Dow moulds.)
The penetrations indicated" herein refer to a depth of penetration in hun
dredth centimeters of a No. 2 cambric needle weighted to one hundred
grams at 77° F., acting for five seconds.
16
CHAPTER XI
LINES AND GRADES. COMPUTATION OF VARIOUS SECTIONS.
ISOMETRIC WORKING SKETCHES. COST DATA
Surve3ring.^ — As an engineering structure, a retaining wall
requires but little more special field work than other masonry
structures. The trenches within which the wall is to rest must
be staked out, the face of the wall must be laid out on the con
crete bottom of the wall in its correct location with respect to
the property, or other governing hne, and finally the forms must
be checked as to correct section and location. As the wall is
essentially a longitudinal strip, a preliminary Hne, parallel to
the face, or other important line of the wall, is staked out. This
forms the base line of the wall location work, and the accuracy
with which this Hne is laid out determines all the accuracy of the
Hnes subsequently staked out from this Hne. The degree of
exactness which must be employed in laying out the wall is
conditioned upon several factors. The presence of adjacent
structures, the nearness of the wall to important easement Hnes,
either public or private, the necessity of tying other structures
to the retaining wall (or abutment), the proposed permanence of
the wall, will each control the permissible error in the fieldwork.
An allowable error of one in 25,000 is sufiiciently exact for any
type of wall, regardless of the degree of exactness required and
larger error factors should be used for less important structures.
The importance of the base Hne with reference to the field
work which foUows and is dependent upon it makes it necessary
that it be laid out at a distance away from the work that will
keep it safely out of the construction way, and yet close enough
that it can readily be employed as a reference line. If th loca
tion of the work permits the Hne should be about 25 feet away
from the waU Hne and referenced at frequent intervals to fixed
land marks. It should be tied in to other important Hnes of
permanent nature, such as city monument Hnes, the main rail
road survey Hnes and such Hnes as control more or less, the loca
tion of the easement Hnes of the wall.
242
LINES AND GRADES 243
In conjunction with the location of the base Une, a run of
benches is made, safely established, so that the progress of con
struction will not disturb them. The accuracy of this run need
not be high, unless steel structures are to be tied into the wall
{e.g. abutments supporting steel bridges; retaining walls carry
ing building walls upon them, etc.).
It is patent, that in the establishment of both the base line and
the bench run, points must be selected that can readily be found
and used for the construction work. This is a matter of judg
ment, tempered by much field experience and vexatious delays
must occur through poor selection of important surveying points.
Construction Lines. — The base line as above described is not
used directly to stake out the construction work. It is cus
tomary to place a Une about five feet from the face of the wall,
and where possible, another line ten feet from the face, and
both parallel to the base line, which lines are directly employed
by the mechanics to lay out the excavation lines and the
concrete lines. As the lines are destroyed in the ordinary course
of construction, they may easily be restored, where necessary,
by recourse to the permanent base Une. On tangent walls, net
line stakes (i.e. the actual wall lines) may be placed at twenty
to twentyfive foot intervals. On curves, they should be placed
close enough, that the chords do not diverge more than the per
missible Umit from the true arc
of the waU. For the excavation ^ ^ ^_ U'""t ^
lines, rough work is, of course ■^''*^^^^.>i /^^
permissible. For the concrete  Rad
lines more refinement is re j.,^^ i29.Length of chord for per
quired. To determine the proper missibie amount of flattening.
chord length, L, to be used in i, ^ ^
staking out the waU, so that its middle ordmate wiU not exceed
the permissible allowance a (see Fig. 129), note that from the
approximate paraboUc relation that the offset y to an arc, at a
distance x from the point of tangency is given by the formula
y = x^/2R
where R is the radius of the arc. Employing this formula in the
present case
o = LysR
It is generally specified that the flatness of the waU shaU not
244
RETAINING WALLS
exceed }4 oi an inch, or 0.01 feet. This last equation when solved
for L, using the value 0.01 for a is then
r _ V2R
To aid in the use of this equation, Table No. 39 is given here
with showing the necessary chord length to be used for any
assigned radius of arc, that the chord offset shall not exceed
oneeighth of an inch. For example, by reference to the table,
a radius of 800 feet makes it necessary to stake out the wall in
eightfoot chord lengths, while a radius of 8,000 feet permits the
use of twentyfivefoot chords.
Table 39
— Maximum Choed Lengths
B
i
R
L
B
L
B
L
B
L
50
2.0
325
5.1
700
7.5
1250
10.0
4500
19.0
75
2.5
350
5.3
750
7,8
1300
10.2
5000
20,0
100
2.8
375
5.5
800
8.0
1350
10.4
5500
21.0
125
3.2
400
5.7
850
8.2
1400
10.6
6000
21.9
150
3.5
425
5.8
900
8.5
1450
10.8
6500
22.8
175
3.7
450
6.0
950
8.7
1500
11.0
7000
23.7
200
4.0
475
6.2
1000
9.0
2000
12.7
7500
24,5
225
4.2
500
6.3
1050
9.2
2500
14.2
8000
25,0
250
4.6
550
6.6
1100
9.4
3000
15.5
275
4.7
600
6.9
1150
9.6
3500
16.7
300
4.9
650
7.2
1200
9.8
4000
17.9
The bottom of the wall, whether of concrete or other masonry
should not necessarily fill the trench unless this has been trimmed
with unusual care. If the netKne stakes have been lost in the
excavation, these should be restored, and the proper bottom lines
given for the masonry footing. For grades, stakes may be driven
into the side of the cut at the required elevation, or at a stated
distance above this hne. For the elevation of the bottom of the
wall the same stakes may be employed.
Forms. — "With the bottom concrete in, it is necessary to give
some line to commence the form work. A very serviceable
method is that of nailing a molding strip to the concrete bottom,
marking the inside of the lagging of the form (see Fig. 130).
With this line in place the face forms may be set and the rear
forms placed at the required distance away as specified on the
LINES AND GRADES 245
plans. After the forms are assembled, wired and braced, they
may be rechecked from the reference line, and then plumbed to
see that the section meets that theoretically required. The
proper grades at which to make the breaks in the wall section, if
there be any, and the grade for the top of the wall, are most
commonly given by nails driven in the side of the form at these
elevations.
CpncreU Farms
■• Sfripaf Molding
Hailed to Foofing
Fig. 130. — ^Method of lining in concrete forms.
Computation of Volumes. — When the section of a retaining
wall remains constant between two given points, its volume is the
product of the area of the section by the distance between the two
points. Generally the section of the wall varies, the top of the
wall following a given grade. Breaks in the width of the wall, or
in other but the vertical dimensions, are made at the expansion
joints, so that between two adjacent expansion joints the width
of wall at the coping and at the base remain constant. The
volume of a wall, whose coping and base widths are respectively
a and h, and whose heights at the beginning and end of the section
are Ai and 'hi, respectively, is
Y = ^{a^ b)ihi + hi)
To get the volume of sections of the wall which are irregular
because of breaks in the wall, or because of intersections with
other walls, it is essential that a careful and detailed drawing be
made. It is difficult to show clearly the volume in question
when the drawing merely gives a twodimension section. For this
reason isometric drawings may serve to bring out clearly and
exactly all the dimension necessary to obtain the volume of the
portion sought. To make the isometric drawing correct to scale
and to be able to interpret mathematically the lengths scaled
from the isometric drawing the following matter gives some
formulas and tables which should serve to make the isometric
layout as easy to handle as the plane detail drawing. ^
1 See Engineering NewsRecord, April 3, 1919, p. 661.
246 RETAINING WALLS
It is assumed that the isometric taxes of the figure have so been
chosen that all the important Hnes of the figure he in planes
parallel to the axes. The following theorems apply solely to
such lines. Lines parallel to the axes are shown correctly to
length by the principles of isometric projection. Lines not
parallel are not shown correctly to length. To obtain the angles
which these lines make in actual space, and the actual lengths
of such lines and conversely, the lengths of such hnes in isometric
projection and the angles which they make with the isometric
axes, refer to Figs. 131 and 132. In Fig. 131, the UneL has pro
c c c
Fig. 131. Fig. 132.
The plane and isometric triangles.
jections b and c and makes an angle <t> with the projection c.
In isometric projection the length L becomes either Li or L,
depending upon whether it subtends an angle of 120° or 60°.
The angle <j) is again 4>i or 4>j in isometric projection. The lengths
b and c remain unchanged, tan ^ = b/c. Referring to Figure
132, by the law of sines
b/c = sin <^i/sin (180°  120°  <^i)
b/c = sin ^,/sin (180°  ' 60°  0,)
From which two equations,
/ I + 2 cot <t>\
,_, /I + 2 cot <^\
= cot'
, / 2 cot <^  1 \
\ Vs )
Table 40 gives the values of 0f and ^, for the several values of <t>.
Referring to Figs. 131, 132.
jr,2 = 52 ^ c2
Li^ = &2 + c2  26c cos 120° = b^ + c^ + be
Lj^ = = 62 + c2  6c
b = L cos tj) and c = L sin <^
Substituting these values in the preceding equations there is
finally
Li = kL; Lj = jL
where A;^ = 1 + sin cos 0; j'' = 1 — sin ^ cos ^.
LINES AND GRADES
247
Table 40 gives a series of values of k and j for the run of
values of 4>.
Table 40. — Isometric Functions
*;
1
0°
0°
0° ,
1.00
1.00
5
4
5
1.04
0.95
10
8
10
1.09
0.90
15
12
15
1.12
0.87
20
15
21
1.15
0.82
25
18
28
1.18
0.79
30
21
35
1.20
0.75
35
24
43
1.21
0.73
40
27
51
1.22
0.71
45
30
60
1.23
0.71
Fig. 133 gives an illustration of some wall details shown iso
metrically and properly scaled and dimensioned (all dimensions
Far Lines Parallel foAF
true Dimensions are equal
ft? Scaled Dimensions
Divided by h' in
EiHmah Volume at Large Sedion toPlaneDHej
EsHmale Wlume of Small Section foPlcneCABD
EsHmafe Volume of IrregalarJunf ion as Follows
Volume of lfightPrism{AE6K:MModeDE
Las Volume of Righf PyramidBase BMHE
AHitude ED
F,o 133 The isometric detail and its application to the computation of
volumes.
248 RETAINING WALLS
shown are the true ones, the isometric lengths as shown having
been corrected by means of the tables above.
Cost Data. — The compilation of worthwhile cost data is
conditioned upon the proper valuation of the relative operations
involved in the piece of work under analysis as well as a correct
understanding as to how much of the work is standard in con
nection with retaining wall construction and how much is peculiar
to the individual piece of work in question. Merely gathering
cost statistics without an intelligent interpretation of the opera
tions affecting or controlling costs is a valueless and time wasting
procedure.
Cost analysis in general may be said to serve two purposes.
It furnishes an accounting of work already done, in order that
proper disbursements may finally be made and a correct financial
history compiled of the job in question. In this sense it is
properly an accounting job, based upon payroll and material
forms prepared by the timekeeper. It may also be an antici
patory analysis of work to be done and then comes within
the province of an engineer preparing such an estimate. Proper
attention to the former purpose of cost data is of course essential
that the latter purpose may be efficiently carried out and the
more voluminous the files of cost accounts (intelligently kept)
the better able is the engineer to make a scientific prediction of
the cost of future work.
That a true comparison may be made of the relative value
of the various types of retaining walls, it is apparent that the
elements entering into the cost data must be properly weighted, so
that items of cost unique to a peculiar environment be disregarded.
For this purpose, it is best that cost data be reduced, as far as is
practicable, to fundamental and elemental operations, independ
ent, more or less, of the peculiar character of any piece of
construction.
Cost may be divided into several general subdivisions : Labor
cost; material cost; plant cost and general administrative
expenses. The first item, the labor cost, is the uncertain item,
and one requiring experience and judgment in its proper deter
mination. Material costs are simple, are easily compiled; can
easily be anticipated and with a proper allowance for the wastage
involved in the several operation are estimated with a high
degree of accuracy. Plant cost, while possibly not so easily
compiled or anticipated as material cost, should not, at least
LINES AND GRADES 249
to the engineer with a moderate amount of experience, prove
difficult of computation. In a previous chapter the character
and the distribution of plant employed for a number of pieces of
typical retaining wall construction may furnish a good working
clue to the type most suited to the work under analysis. General
administrative expenses will cover office expenses, salaries of the
executives, insurance upon the labor, miscellaneous casualty and
public liability insurance, minor expenses in connection with the
prosecution of the work, such as telephone, fares, taxes, etc.
This item is usually termed the overhead of the work and is
spread over all the items entering into the construction of a wall.
While of an indefinite character, it must be properly ascertained
or anticipated in order to be included in the estimated cost.
It must be remembered that it is a constant charge carried
continuously, regardless of the weather or other delays and in
work of long duration, may effect materially the cost of the opera
tions. Blanket percentages added to cover items of this nature,
while excusable in small work, are apt to work hardships upon
large work unless the percentage factor so applied is the result
of data compiled from several jobs of similar nature. Naturally
the number of items of uncertain amount appearing in an esti
mate of future work will be in inverse proportion to the amount
of experience of the engineer preparing such estimates.
Labor Costs. — Without entering into a detailed analysis of
the various labor elements involved in wall construction, ^ some
general labor costs may be presented to guide an estimator in
preparing a bid for contemplated work. Before employing
such data it is well to read again (chapter on "Plant") the impor
tant bearing of plant selection and arrangement upon the cost
of labor. A good bid is not one that contains merely a carefully
and detailed analysis of the cost of the labor. It must plan a
scheme of the work together with the amount of plant to be had
and the character of the labor to operate it. Haphazard bidding
or snap judgment estimates are unpardonable in all but the most
experienced of contractors and engineers, and must eventually
lead to financial disaster. Such figures and quoted estimates of
the cost of work as are given below must be used in light of the
above remarks.
The material for the wall is taken from the point of delivery
1 See Dana, "Cost Data," Gillette "Handbook of Cost Data;" Taylor
and Thomson, "Concrete Costs."
250 • RETAINING WALLS
and brought to the site of the work either at a contracted price
per yard (which price may be ascertained at the time of preparing
the bid) or if dehvered F.O.B. nearest raihoad station or
hghterage dock may be hauled by hired team or auto truck.
With the latter method, the length of haul will determine the
average number of trips that the trucks can make, and knowing
the load that can be carried, the price per yard for delivering
the material can be computed with no great difficulty. An
analysis of the cost of several pieces of work, follows. The
files of the Engineering Press may be used to examine the cost
of numerous pieces of work.
From Taylor and Thompson "Concrete Costs," p. 16: •
Cantilever wall, 16 feet high, 250 feet long; common labor $2.00 per day,
carpenters $3.82 per day. Concrete yardage 277 cubic yards.
Cost of labor of forms per cubic yard of concrete .... $2.75
Total cost of forms per cubic yard of concrete $3 . 91
Cost of material per cubic yard of concrete $3 . 57
Cost of mixing and placing concrete per cubic yard . . $1 . 35
Total cost of concrete in place (including superin
tendence) $12 , 03
Cantilever wall 16 feet high. Labor 20 cents per hour; carpenters 50 cents
per hour.
Total cost of forms per cubic yard of concrete $3 . 60
Cost of concrete material per cubic yard 4 . 75
Cost of mixing and placing the concrete 1 . 25
Cantilever wall 8 feet high. Labor 20 cents per hour; carpenters 50 cents
per hour.
Total cost of forms per cubic yard of concrete $6 . 23
Total cost of material per cubic yard of concrete .... 4 . 75
Cost of mixing and placing per cubic yard 1 . 25
A resume of the total labor cost of pouring retaining walls of
both gravity and reinforced "L" type, averaging about 35 feet
in height is as follows:
Gravity Type 1935, cubic yards of concrete. Plant used was
two small batch mixers, the concrete wheeled to the forms and
poured in.^
1 See "Enlarging an Old Retaining Wall," for a detailed description of the
methods and plant used, Engineering News, Sept. 8, 1915.
LINES AND GRADES 251
The forms were used on the average about four times.
Foreman, 175 days at $5.00 per day $875.00
Carpenters, 190 days at $3.50 per day $665 '. 00
Engineer, 46 days at $5.00 per day 230,00
Laborers, 926 days at $1.75 per day 1620 . 50
Teams, 21 days at $3.50 per day 73.50
Timbermen, 20 days at $3.00 per day 60.00
Masons, 37 days at $4.00 per day 148 . 00
Riggers, 14 days at $3.00 per day 42.00
Watchmen, 33 days at $1.00 per day 33.00
Total labor cost $3747 00
This makes the labor cost per yard, exclusive of all overhead
insurance, plants charges etc., $1.94 per cubic yard of concrete.
A similar detailed labor cost to pour a "L" shaped cantilever wall, in
volving a yardage of 1697 cubic yards is:
Foremen, 197 days at $5.00 per day $985 . 00
Carpenters, 503 days at $3.50 per day 1760.50
Engineer, 37 days at $5.00 per day 185 . 00
Riggers, 24 days at $3.00 per day 72 . 00
Laborers, 1197 days at $1.75 per day 2094.75
Masons, 55 days at $4.00 per day 220 . 00
Teams, 51 days at $3.50 per day 178.50
Watchmen, 124 days at $1.00 per day 124.00
Total labor cost $5619 . 75
The unit labor cost per cubic yard for pouring this type of
wall, exclusive of all overhead charges as above enumerated is
$3.31 per cubic yard.
While endless data might be furnished of the cost of existing
work, conditions are usually too unique to make such data of
general usefulness. Unit costs as quoted above may fill in
uncertain data in a bid, when properly altered to take care of
changed labor rates. The labor cost on a retaining wall, roughly,
averages about onequarter the total cost of the wall. Barring
unforseen contingencies an estimator with a fair knowledge of
construction work should be able to anticipate the labor cost
within 20 per cent, of its correct final value. Should the dis
crepancy amount to the hmiting value of 20 per cent., in the
final data it will amount to merely 5 per cent, of the total cost
of the work. Estimates of work can hardly be expected to
reach a higher degree of accuracy than this.
As an example of the analysis of a proposed piece of work, let
252 RETAINING WALLS
it be required to determine the cost of constructing a retaining
wall about 1,000 feet long, 40 feet high, with a yardage of about
10,000 cubic yards. One year is the allotted time in which to
construct the wall. The wall is a cantilever type.
Plant. — A mixer of about 100 yards per day capacity (a 3^^ to
}2 yard batch mixer will easily satisfy this requirement) should
pour the required yardage of concrete with an ample time margin.
This mixer should be obtained in the neighborhood of about
$1,000. The other plant requirements, such as wheelbarrows,
shovels, etc.; shanties for storing cement and tools, for temporary
offices; lumber for runways for pouring the concrete etc., should
not cost more than an additional $1,000 making the total plant
charge $2,000.
Materials. — Assuming that the wall is a 1:2.5:5 mixture of
concrete, there will be required about 1.2 barrels of cement for
each yard of concrete placed. Theoretically about 10,000 3rards
of stone and 5,000 yards of sand will be required. To allow for
wastage of all kinds these quantities will be increased 10 per
cent. It will be assumed that the materials will be delivered
on the job, where required for the following unit prices; cement
$3.50 per barrel (net, no allowance for bags); stone for $2.50
per yard and sand for $2.00 per yard. The material totals are
then
13,200 bbls. cement at $3.50 $46,200
11,000 yards stone at $2.50 27,500
5,500 yards sand at $2.00 11,000
The total material will cost $84,700
Form Lumber. — Assume that 2 inch tongue and grooved
sheeting will be used to make the form panels. Allow about
20 per cent, wastage of forms each time the forms are stripped
(this is equivalent to a form use of five times). The area of
wall surface that must be coVered with new form lumber is then
(allowing a footing thickness of four feet)
36 X 2 X 1000 ^ . ^^^ . ,
z = 14,400 square feet.
To allow for the joists, rangers, bracing etc., and to allow for was
tage in material due to cutting it to required lengths, it is cus
tomary to double the board feet required for the sheeting.
(Exactly, the forms may be designed as outUned in the chapter
LINES AND GRADES 253
on FORMS, and detailed as shown in the problem accompanying
the chapter, and the required amount of timber taken from these
estimates. An estimate of the cost of the work, does not, how
ever, justify such refinement, and it is better to use the rule of
thumb method just stated.) Since the sheeting is to be 2
inches thick, the total lumber requirements are 4 board feet for
every square foot of new lumber surface. With a price of 175
per M for timber delivered on the job, tongue and grooved, the
timber cost is
14.4 X 4 at $75 = $4,320
Labor Costs. — To get the total labor costs on the wall, the
analysis of the cost of the reinforced concrete wall at last outlined
may be used with the following revised rates of labor: Foreman
$8.00 per day, Engineer and Carpenters, $7.00 per day; laborers
$4.00 per day and the other items in keeping. This will prac
tically double the unit cost of labor as given. The unit cost is
then about $6.75 per yard, or the total cost is $67,500. To this
must be added the item of insurance, amounting to about 10 per
cent, of the labor total, or $6,750.
Overhead. — ^The work will require the employment of a super
intendent for one year ($4,000) and a timekeeper ($1,500)
Miscellaneous expenses around the work should not exceed
$1,000, making the field overhead about $6,500.
The ofifice overhead is indeterminate, depending upon the
number of jobs going on at one time. This factor will be omitted
here.
The rods are usually quoted at a separate unit price and are
not mentioned here.
To summarize:
Plant $2,000
Materials 84,700
Lumber 4,320
Labor (and Ins.) 74,250
Overhead 6,500
Total $171,770
With an allowance for profit the wall will be estimated in the
neighborhood of $200,000, or at a unit cost of $20.00 per cubic
yard.
SPECIFICATIONS
General Layout of Work. — The retaining walls to be constructed under
this contract are shown on Plans Nos. to inclusive. These specifica
tions and the plans are intended to be consistent and where any apparent
inconsistency appears the interpretation shall convey the intent of the best
work and construction.
Classes of Work. — The retaining walls shall be classified for payment as
follows :
Class A. — Walls without reinforcement, marked A on the plans, of what
ever height indicated.
Class B. — Reinforced concrete walls up to but not including twenty (20)
feet in height from subgrade to top of coping.
Class C. — Reinforced concrete walls from twenty (20) feet up to but not
including thirty (30) feet from subgrade to top of coping.
Class D. — ^Reinforced concrete walls over thirty (30 feet) in height from
subgrade to top of coping.
Class E. — Walls of cement rubble masonry of whatever height indicated.
Payment. — Payment for the walls as indicated shall include the furnish
ing of all labor and materials necessary, including the cost of all scaffolding,
forms and the cost ot removing the same; also the cost of finishing the face
of the wall where a rubbed finish is indicated.
Concrete Proportions. — Concrete for class A walls shall be mixed in the
proportions of one part cement, two and onehalf parts of sand and five
parts of stone or gravel, by volume.
Concrete for reinforced concrete walls (classes B, C and D) shall be mixed
in the proportions of one part cement, two parts sand and four parts ot
stone or gravel, by volume.
Cement. — The cement shall be Portland Cement of a brand that has been
on the market for the last five years.
(Insert here the details of che propertie,s of cement as has been given
on pages 214 to 215.)
Sand. — Sand for use in making the concrete shall be clean and well
graded, not exceeding Ji inch in size. Not more than six per centum (6%)
by weight shall pass a 100 mesh screen. It shall contain not more than three
per centum (3%) by weight of foreign matter.
Broken Stone. — Stone for concrete shall be a clean sound, hard broken
limestone or trap rock and graded from threeeighths {%) of an inch in di
ameter up to one and threequarters (l^i) inches in diameter. Where the
thickness of the concrete wall is twelve inches or less in thickness the size
of the stone shall not exceed threequarters (^) of an inch in diameter.
It shall be screened and washed to remove all impurities and shall be care
fully stored along the site of the work to prevent the gathering of any foreign
matter in it.
254
SPECIFICATIONS 255
Gravel.— Gravel shall be screened, cleaned and graded in the same man
ner as the broken stone.
Use of Large Stone.— In Class A walls (and in these walls only) where
the thickness of the wall exceeds thirty (30) inches the contracter will be
permitted to imbed stones of at least 12 inches in thickness not closer than
four (4) inches to the face of the form and not closer than six (6) inches to
each other. The stones shall be sound, clean stones and shall be carefully
placed in the concrete.
Concrete.— Concrete shall be mixed by machine. In case of emergency
it shall be within the discretion of the Engineer to state whether the mixing
shall proceed by hand.
It is the very essence of these specifications that the water content of
the concrete mix by kept low. No machine mixer shall be used that is not
equipped with a tank or other device for supplying a measured amount of
water to each batch of concrete and a competent operator shall be in atten
dance upon the machine.
The Engineer, or his duly authorized representative shall decide upon
the length of time each batch shall be mixed and upon the amount of water
that shall go into each batch.
The contractor shall permit the Engineer to take samples of the concrete
mix to be tested and no charges shall be made for material taken for such
purposes .
The use of a continuous mixer is forbidden and a mixer that is found
incapable of delivering a concrete in conformity with the specifications shall
be removed from the work and a mixer substituted for it that is capable of
mixing concrete in accordance with these specifications.
Concrete shall be conveyed to the forms in watertight conveyances and
shall be dropped vertically into the forms. It shall then be shovelled into
place and thoroughly compacted and rammed to insure a concrete of uniform
density.
Spades or other special tools shall be used on the concrete to insure a free
circulation of the grout around the reinforcing bars and against the face of
the forms.
Forms. — The forms for concrete shall be made of stout tongue and grooved
sheeting, properly supported and braced and of strength sufficient to meet
the concrete pressures. If so required the contractor shall submit to the
engineer plans of the form work and bracing.
Before pouring the forms shall be oiled, or thoroughly wetted and before
reusing shall be cleaned of all adhering cement, dirt, etc., to insure a smooth
face on all exposed concrete work.
The joints shall be watertight and shall be carefully inspected while
the pouring is in progress to prevent the escape of any grout.
Concrete shall set at least twentyfour (24) hours before the tierods are
loosened or any of the sheeting removed. This time shall be increased when
the temperature of the air drops below sixty (60) degrees Fahr. Forms shall
be stripped in the presence of the Engineer, it the contractor is so directed.
Placing Fill.— No fill shall be deposited behind the walls until ten days
have elapsed since the walls were poured and not until the assent of the
Engineer or his duly authorized representative has been obtained.
256 RETAINING WALLS
Defective Work. — If upon stripping the forms there is evidence of any
defective work, such defective work shall immediately be repaired and the
surface of the wall finished in a manner that will present as little evidence
of such defective work as possible.
Evidence of extensive defective work shall be sufficient cause to order the
contractor to remove portions of the work showing such defective work and
all such repairs and reconstruction work shall be made at the contractor's
own expense.
Concrete Work in Winter Weather. — When the temperature of the air
drops below 45 degrees Pahr. it shall be within the discretion of the Engineer
to order the contractor to heat the concrete materials before pouring them
into the forms.
No concrete shall be deposited in the forms in freezing temperature that
has not been mixed with materials heated by means of suitable appliances
so that the temperature of the concrete upon being placed in the forms shall
not be less than 60 degrees F. Concrete deposited in freezing weather shall
be protected while setting by means of salt hay, tarpaulin, canvas, or by
other devices which will maintain the temperature of the concrete above
freezing until it has set.
No concrete shall be deposited in the forms when the temperature drops
below 20 degrees Fahr., unless such forms have been constructed in a manner
approved by the Engineer, to prevent freezing of the concrete mix.
Joints. — Where a break occurs in the day's pour, no additional concrete
shall be deposited on such a joint when work is subsequently started until
the joint has been thoroughly scrubbed to remove all laitanoe and other
foreign matter. If so directed a layer of cement grout shall be deposited
upon the joint immediately before placing fresh concrete.
It is the intent of these specifications to secure a section of wall between
expansion joints free of all joints as above and the contractor shall use
plant of such capacity that a section can be poured complete in a regular
day's operation. When, due to an emergency, such a construction joint is
unavoidable, the Engineer, or his duly authorized representative shall in
struct the Contractor as to what details of construction must be adopted
to obtain the full efficiency of such a joint and to prevent, as far as possible,
any unsightly appearance of the face of the wall after the forms have been
stripped.
Drains. — There shall be incorporated in the wall, tile drains of spacing
and diameter shown on the plans. Immediately back of the drains shall
be placed one cubic yard of broken stone.
Waterproofing. — The back of the retaining walls shall be given two coats
of hot asphalt or pitch. The back of the wall, before the tar is applied shall
be thoroughly dried and free of all frost.
(Insert specifications for tar as given on page 240.)
Extreme care shall be exercised in placing the fill back of the wall so that
the coats of tar shall not be abraded.
If, after the fill has been in place the face of the wall shows evidence of
water leaking through it, the contractor, if so directed by the Engineer,
shall excavate back of the wall to the indicated position of the defective
SPECIFICATIONS 257
waterproofing and shall make such repairs as are necessary, no additional
payment to be made for this work
Concrete Finish. — Where no special face finish is indicated, the Contractor
shall, immediately upon removing the forms, remove all wires, rods, etc.,
or cut them back to about two inches from the face of the wall. He shall
then point up these places with a rich mortar or concrete. The face of the
wall will then be rubbed down with suitable appliances as approved by the
Engineer and the entire surface given a coat of thin grout wash.
Reinforcing Bars. — Reinforcing bars shall be placed in the concrete walls
of dimensions and spacing as shown on the plans. Payment for these rods
includes all labor and material required for their installation as indicated.
Rods shall be deformed as approved by the Engineer. Plain bars may
not be used.
Rods shall be bent to radii as indicated and shall generally be delivered
in the full length as required on the plans.
Rods shall be made by the open hearth process with the following maxi
mum impurities :
Phosphorus, not more than 0.04 per cent.
Sulphur, not more than 0.05 per cent.
The elastic Hmit or yield point shall not be less than 40,000 pounds per
square inch.
Test specimens for bending shall be bent under the following conditions
without fracture on the outside of the bent portion:
Around twice their own diameter.
1 in. in diam., 80 degrees.
^i in. in diam., 90 degrees.
}i in. in diam., 110 degrees.
Around their own diameter.
J.^ in. in diam., 130 degrees.
%6 in. in diam., 140 degrees.
}4 in. or less in diam., 180 degrees
Retaining Walls, Including Lateral Earth Pressure'
Alexander, T., and Thomson, A. W. Elementary Apphed Mechanics.
675 p. 1902. Contains chapters, " Apphcation of the Ellipse of Stress
to the Stability of Earthwork," p. 7086, and "The Scientific Design
of Masonry Retaining Walls."
Allen J RoMiLLr. Investigation of the Question of the Thrust of Earth
Behind a Retaining Wall. 3 diag. 1877. {Van Nostrand's Engineer
ing Magazine, v. 17, p. 155158.) Mathematical solution.
Allen Kenneth. Design of Retaining Walls. 1892. {Engineering Rec
ord V 26, p. 341342, 356357, 374, 393.) On practical design of
retaining walls, sea walls, and dock walls. Illustrated with actual
designs.
American Railway Engineering and Maintenance op Way Associa
tion. [Report of Committee on] Retaining Walls and Abutments.
1909. (Proceedings, Tenth Annual Convention, Am. Ry. Eng. and
iprom Report Spec. Comm. on Soils A.S.C.E.
17
258 RETAINING WALLS
Maintenance of Way Assoc, p. 13171337.) Gives information show
ing practice of various railroads in the designing of retaining walls.
Committee submits method of determining earth pressures based on
Rankine's formula.
Condensed. 1909. {Engineering Record, v. 60, p. 288290.)
AuDB. Nouvelles Expfirieiices sur la Pouss^e des Terres. 1849. {Comptes
Rendus Hebdomadaires des Stances de I'Acad^mie des Sciences, v. 28,
p. 565566.) Short review of Audi's work presented by Poncelet.
Baker, Benjamin. Actual Lateral Pressure of Earthwork. 1881. (Min
utes of Proceedings, Inst. C. E., v. 65, p. 140186.) Discussion, p. 187
241. Aims to present data on actual lateral pressure of earthwork, as
distinguished from "textbook'' pressures, which latter the author
holds to be genera,lly incorrect
1881. {Van Nostrand's Engineering Magazine, v. 25, p. 333342, 353
371, 492505.)
Bard WELL, F. W. Note on the "Horizontal Thrust of Embankments."
1861. {"Mathematical Monthly, v. 3, p. 67.) Finds the formula de
■ rived by D. P. Woodbury to be correct.
BoARDMAN, H. P. Concerning Retaining Walls and Earth Pressures.
1905. {Engineering News, v. 54, p. 166169.) Concludes that in
formation regarding earth pressures is quite inexact. Suggests con
ducting series of tests on large scale.
Bone, Evan P. Reinforced Concrete Retaining Wall Design. 1907.
{Engineering News, v. 57, p. 448452.) Calculations of earth pressures,
and diagrams.
BoussiNESQ, J. Calcul Approch6 de la Pouss^e et de la Surface de Rupture,
dans un Terreplein Horizontal Homogfene, Contenu par un Mur
Vertical. 1884. {Comptes Rendus Hebdomadaires des Stances de
I'Academie des Sciences, v. 98, p. 790793.)
BoussiNESQ, J. Complement k de PriScedentes Notes sur la Pouss^e des
Terres. 1884. (AnnoZesdesPowiseiC/iauss^es, ser. 6, v. 7, p. 443481.)
BoussiNESQ, J. Equilibrium of Pulverulent Bodies. 1 diag. 1877.
{Minutes of Proceedings, Inst. C. E., v. 51, p. 277283.) Abstract
translation of "Essai Th6orique sur I'Equilibre des Massifs Pulvfiru
lents. Compare k celui de Massifs Solides et sur la Poussfie des Terres
sans Cohesion." Brussels. 1876.
1881. {Van Nostrand's Engineering Magazine, v. 25, p. 107110.)
BoussiNESQ, J. Integration de I'Equation Diff(5rentielle qui pent Donner
une DeuxiSme Approximation, dans le Calcul Rationnel de la Pouss6e
Exerc^e contre un Mur par des Terres D^pourvues de Cohesion. 1
diag. 1870. {Comptes Rendus Hebdomadaires des Stances de I'Acad
6mie des Sciences, v. 70, p. 751754.)
BoussiNESQ, J. Note sur la Determination de I'Epaisseur Minimum que
doit avoir un Mur Vertical, d'une Hauteur et d'une Density Donnfies,
pour Contenir un Massif Terreux, sans Coh&ion, dont la Surface
Sup(3rieure est Horizontale. 1 diag. 1882. {Annates des Ponts et
Chausshes, ser. 6, v. 3, p. C25643.) Application of the theory of
earthpressure, as developed by Rankine and Darwin, to design of
vertical walls.
SPECIFICATIONS 259
BotJSsiNESQ, J. Note sur la Mfithode de M. MacquornRankine pour le
Calcul des Pressions Exerc.ees aux Divers Points d'un Massif Pesant
que Limite, du C6t6 Sup6rieur, une Surface Cylindrique k Generatrices
Horizontales, et qui est Inddfini de Tous les Autres C6t6s. 1874.
(Annales des Fonts et Chaussees, ser. 5, v. 8, p. 169187.) Criticism of
Kankine's theory of earth pressure.
BoussiNESQ, J. Sur la Pouss6e d'une Masse de Sable, a Surface Superieure
Horizontale, Contre une Parol Verticale dans le Voisinage de Laquelle
son Angle de Frottement Interieur est Suppose Croitre Legerement
d'apres une Certaine Loi. 1884. {Comptes Rendus Hebdomadaires de?
Seances de rAcademie des Sciences, v. 98, p. 720723.)
BoussiNESQ, J. Sur la Poussee d'une Masse de Sable, &, Surface Superifiure
Horizontale, Contre une Parol Verticale ou Inclinde. 1884. {Comptes
Rendu? Hehdomadairet^ des Stances de rAcademie des Sciences, v.
98, p. 667670.)
BoussiNESQ, J. Sur le Principe du Prisme de plus grande Poussee Posd par
Coulomb dans la Thdorie de I'Equilibre Limite des Terres. 1884.
{Comptes Rendus Hebdomadaires des Seances de I'Acaddmie des Sciences,
V. 98, p. 901904, 975978.) Critical review.
BoussiNESQ, J. Sur les Lois de la Distribution Plane des Pressions a I'ln
terieur des Corps Isotropes dans I'Etat d'Equilibre Limite. 1874.
{Comptes Rendus Hebdomadaires des Seances de TAcaddmie des Sciences,
V. 78, p. 757759.)
BovBY, Henry T. Theory of Structures and Strength of Materials, ed. 3.
835 p. 1900. Includes section on earthwork and retaining walls.
Bursting Pressure of an Earth Fill. 1912. {Engineering News, v. 68,
p. 593594.) Editorial discussing the causes of failure of a retaining
wall in St. Louis.
Cain, WiLLiAii. Cohesion and the Plane of Rupture in Retaining Wall
Theory. 1 diag. 1912. {Engineering News, v. 67, p. 992.) Letter
to editor discussing Hirschthal's article " Some Contradictory Retaining
Wall Results/' Engineering News, v. 67, p. 799.
Cain, William. Earth Pressure, Retaining Walls and Bins. 287 p. 1916.
Wiley. Contains chapters on the theory of earth friction and cohesion,
of earth thrust, and of bins. Gives special attention to coherent and
noncoherent earths. Emphasizes throughout the presence in earth of
cohesion as well as of friction.
Cain, William. Retaining Walls. 1880. {Van Nostrand's Engineering
Magazine, v. 22, p. 265277.) Considers "the earth as a homogeneous
and incompressible mass, made up of Uttle grains, possessing the resis
tance to sliding over each other called friction, but without cohesion."
Calculations FOR Retaining Walls. 1911. {Architect and Contract Re
porter, V. 86, p. 4344, 5961, 7576, 8587, 9697, 109110.) Takes
all factors into consideration, wind pressure, slides, earth pressure, etc.
"Angles of repose of various earths," p. 109.
Carter, Frank H. Bracing and Sheeting Trenches. 1910. {Engineenng
Contracting, v. 34, p. 7678.) Computes pressures on bracing and shor
ing for well underdrained excavations in virgin soil.
Carter, Frank H. Comparative Sections of Thirty RetainingWalls, and
260 RETAINING WALLS
Some Notes on Retaining Wall Design. 1910. [Engineering News, v.
64, p. 106108.) Discusses theoretical earth pressures, giving formulas.
Clavenad. M6moire sur la Stability, les Mouvements, la Rupture des
Massifs en G6n&al, Coh^rents ou sans Cohesion. Quelques Consider
ations sur la Pouss(5e des Terres, Etude Sp6ciale des Murs de Soutene
ment et de Barrages. 64diag. 1887. (Annales des Fonts et Chaussees,
ser. 6, V. 13, p. 593683.)
Coleman, T. E. Retaining Walls in Theory and Practice; A Textbook for
Students. 160 p. 1909. Design and construction. Avoids advanced
mathematics where possible.
CoNSiDERE. Note sur la Pouss^e des Terres. 1870. (Annales des Fonts
et Chaussees, ser. 4, v. 19, p. 547594.) Extension of Levy's theory of
earthpressure. See Comptes Rendus Hebdomadaires des Seances de
l'Aoad6mie des Sciences, v. 68, p. 1456.
Constable, Casimik. Retaining Walls : An Attempt to Reconcile Theory
with Practice. 3 diag. 1874. {Transactions, Am. Soc. C. E., v. 3,
p. 6775.) Gives results of a number of experiments with models,
using walls made of wood blocks and filling composed of oats and peas.
Abstract. 1873. {Van Nostrand's Engineering Magazine, v. 8, p.
375377.)
Condensed. 1873. {Journal, Frankhn Inst., v. 95, p. 317322.1
CoENiSH, L. D. Earth Pressures : A Practical Comparison of Theories and
Experiments. 1916. {Transactions, Am. Soc. C. E., v. 81, p. 191201.)
Discussion, p. 202221. Endeavors to show graphically the results
obtained in actual wall design by the use of the different formulas
(principally those of Rankine and Cain) and by values obtained in
certain experiments, so that the points of interest may be discussed
without resorting to mathematics.
Cornish, L. D. Fallacies in Retaining Wall Design and the Lateral Pres
sure of Saturated Earth. 1916. (United States Corps of Engineers,
Professional Memoirs, v. 8, p. 161172.) Discussion, p. 173195.
Treats of lateral pressure of saturated soils in connection with the de
sign of retaining walls. Presents considerable mathematical data on
the treatment of saturated soil in such design work.
Couplet. De la Pouss^e des Terres Centre leurs Revestemens et la Force
des Revestemens qu'on Leur Doit Opposer. 8 pi. 17261728. {His
ioire de I'Acad^mie Royale des Sciences, v. 28, p. 106164; v. 29, p.
132141; V. 30, p. 113138.
CotJSiNBRT. Determination Graphique de I'Epaisseur des Murs de Soutene
ment. 1 pi. 1841. {Annates des Fonts et Chauss6es, ser. 2, v. 2, p.
167184.) Develops a method of graphical determination of thickness
of retaining walls. Shows how to apply the theory of earth pressure
in connection with this graphical construction.
Cramer, E. Die Gleitflache des Erddruckprismas und der Erddruck gegen
geneigte Stutzwande. 4 diag. 1879. {Zeitschriftfur Bauwesen,v.29
p. 521526.)
Crelle. Zur Statik unfester Korper. An dem Beispiele des Drucks der
Srde auf Futtermauern. 1 pi. 1850. {Ahhandlungen der Konig
SPECIFICATIONS 261
lichen Akademie der Wissenschaften zu Berlin, v. 34, p. 6197.) To be
found in section " Mathematische Abhandlungen."
CuNO. Die Steinpackungen und Futtermauern der RheinNaheEisenbahn.
1861. {Z&itschrift fiir Bauwesen, v. 11, p. 613626.)
Curie, J. Note sur la Brochure de M. Benjamin Baker Intitulee: "The
Actual Lateral Pressure of Earthwork." 9 diag. 1882. {Annales des
Fonts ei Chaussees, ser. 6, v. 3, p. 558592.) Criticism of Baker's paper
in Mmutes of Proceedings, Inst. C. E., v. 65, p. 140.
Curie, J. Nouvelles Exp(§riences Relatives k la Th6orie de la Poussde des
Terras. 4 diag. 1873. {Comptes Rendus Hebdomadaires des Stances
de I'Acadlmie des Sciences, v. 77, p. 142146.)
Curie, J. Sur la PoussSe des Terres et la Stability des Murs de Revetments.
1868. {Comptes Rendus Hebdomadaires des Stances de rAoad(5mie des
Sciences, v. 67, p. 12161218.) Theoretical paper.
Curie, J. Sur la Th6orie de la Pouss6e des Terres. 1871. {Comptes
Rendus Hebdomadaires des Stances de I'AoadiSmie des Sciences, v. 72,
p. 366369.) Critical review of the theories advanced by Maurice
L6vy.
Curie, J. Sur la Th^orie de la Pouss6e des Terres. 1 diag. 1873. {Comp
tes Rendus Hebdomadaires des Stances de I'AcadSmie des Sciences, v.
77, p. 778781.) Reply to SaintVenant's criticism in same volume.
Curie, J. Sur le D&accord qui Existe entre I'Ancienne Th^orie de la
Pouss6e des Terres et I'Experience. 1 diag. 1873. {Comptes Rendus
Hebdomadaires des Stances de I'Acad^mie des Sciences, v. 76, p. 1579
1582.)
Curie, J. Trois Notes sur la Th^orie de la Pouss^e des Terres. Disaccord
entre I'Ancienne Thfiorie et I'Experience; Nouvelles Experiences; R6
ponse aux Objections. 1873. GauthierVillars. Paris. 1875. {An
nates des Fonts et Chaussees, ser. 5, v. 9, p. 490.) Short review of Curie's
pamphlet.
Daly Cesar. Mur de SoutSnment de la Terrasse du Chateau de Meudon,
l' pi. 1859. {Revue Generate de l' Architecture et des Travaux Fublics,
V. 17, p. 243.)
Diagram for Overturning Moments on Retaining Walls for Earth
or Water. 1907. {Engineering News, v. 57, p. 460.) Diagram was
constructed by Charles H. Hoyt.
DoNATH Ad. Untersuchungen uber den Erddruck auf Stvitzwunde ange
stellt mit der fiir die Technische Hochschule in Berlin erbauten Versuchs
vorrichtung. 1 pi. 1891. {Zeitschriftfilr Bauwesen, v. 41, p. 491518.)
Du Bois A J Upon a New Theory of the Retaining Wall. 14 diag.
1879 {Journal, Franklin Inst., v. 108, p. 361387.) Gives a concise
history of the subject, and develops in detail Weyrauch's theory.
Duncan, Lindsay. Plumbing a Leaning Retaining Wall and Bridge Abut
ment. 1906. {Engineering News, v. 55, p. 386.) .
DYRSSEN, L. Analytische Bestimmung der Lage der Stutzlinie m Futter
mauern. 11 diag. 1885. {Zeitschnftfur Bauwesen, v. i5, p. 101106^)
Dyrssen, L. Ermittlung von Futtermauerquerschnitten. 1 diag. 188b.
{Zeitschriftfilr Bauwesen, v 36, p. 389392.)
262 RETAINING WALLS
Dyrssen, L. Ermittlung von Futtermauerquerschnitten mit gebogener
Oder gebroohener vorderer Begrenzungslinie. 3 diag. 1886. (Zeit
schrift fur Bauwesen, v. 36, p. 127130.)
Eddy, Henry T. New Constructions in Graphical Statistics. 1877. {Van
Nostrand's Engineering Magazine, v. 17, p. 110.) Contains section on
"Retaining Walls and Abutments," p. 510.
Enqesser, Fr. Geometrische ErddruckTheorie. 1880. {Zeitschrift fur
Bauwesen, v. 30, p. 189210.)
Everest, J. H. Treatise on Retaining Wall Design. 1911 {Canadian Engi
neer. V. 21, p. 192193, 237, 264265.) Considers earth Pressure, slope,
weights of materials, etc.
Flamant, a. Formules Simples et trfes Approch6es de la Pouss6e des Terres,
pour les Besains de la Pratique. 1884. {Comptes Rendus Hebdoma
daires des Stances de I'Academie des Sciences, v. 99, p. 11511153.)
Flamant, A. Note sur la Poussde des Terres. 1 pi. 1872. {Annales des
Fonts et Chaussees, ser. 5, v. 4, p. 242275.) Expounds Rankine's
theory.
Flamant, A. Note sur la Pouss6e des Terres. 1882. {Annates des Fonts
et Chaussees, ser. 6, v. 3, p. 616624.) Mostly a review of Baker's paper
in Minutes of Froceedings, Inst. C. E., v. 65, p. 140.
Flamant, A. R6sum6 d' Articles Pubhes par la Soci6t6 des Ing^nieurs
Civils de Londres sur la Poussee des Terres. 1883. {Annates des
Fonts et ChaussSs, ser. 6, v. 6, p. 477532.) Review of Darwin's, Gau
dard's and Boussinesq's papers in Minutes of Proceedings, Inst. C. E.,
V. 71 and 72.
Flamant, A. Tables Numeriques pour le Calcul de la Pouss6e des Terres.
2 diag. 1885. {Annates des Fonts et Chaussees, ser. G, v. 9, p. 5155'tO.)
Gives many tables of constants for the relations derived by Boussinesq
and based on the experiments of Darwin in England and Gobin in
France.
Glauber, J. Bestimmung der Starke geneigter Stutz — und Futtermauern
mit Riicksicht auf die Incoharenz ihrer Masse. 1880. {Zeitschrift far
Bauwesen, v. 30, p. 6372.)
Gobin, A. Determination Precise de la Stability des Murs de Soutfenement
et de la Poussee des Terres. 71 diag. 1883. {Annates des Fonts et
Chaussees, ser. 6, v. 6, p. 98231.) Points out some faults in Rankine's
theory, develops his own theory, and gives various applications and
results of experiments.
Godfrey, Edward. Design of Reinforced Concrete Retaining Walls.
1906. {Engineering News, v. 56, p. 402403.) Considers lateral pres
sure of different materials, angles of repose, and necessary calculations.
Goodrich, Ernest P. Lateral Earth Pressures and Related Phenomena.
44 diag., 3 dr., 1 ill. 1904. {Transactions, Am. Soc. C. E., v. 53, p.
272304.) Discussion, p. 305321. Experimentally determines ratio of
lateral to vertical pressure. Gives series of conclusions. See also edi
torial, "Lateral Earth Pressure," Engineering Record, v. 49, p. 633634.
Abstract. 1904. {Minutes of Proceedings, Inst. C. E., v. 158, p. 460
451.)
SPECIFICATIONS 263
Gould, E. Sherman. Retaining Walls. 13 diag. 1877. {VanNoslrand's
Engineering Magazine, v. 16, p. 1117.) Methods of design.
Gould, E. Sherman. Retaining Walls. 2 diag. 1883. {Van Nostrand's
Engineering Magazine, v. 28, p. 204207.) Gives the theory of J.
Dubosque.
Graff, C. F. High Reinforced Concrete Retaining Wall Construction at
Seattle, Wash. 1905. {Engineering News, v. 53, p. 262264.)
HiRSCHTHAL, M. Some Contradictory Retaining Wall Results. 1 diag.
1912. {Engineering News, v. 67, p. 799800.) Letter to editor re
viewing some accepted formulas of earth pressure on retaining walls.
See also Cain, Engineering News, v. 67, p. 992.
HisELY. Constructions Diverses pour Determiner la Poussfie des Terres sur
un Mur de Soutenement. 1899. {Annates des Fonts et Chaussees, ser.
7, V. 17, p. 99120.) Develops a general graphical solution apphcable
to a load of any character.
HosKiNG. On the Introduction of Constructions to Retain the Sides of
Deep Cuttings in Clays, or Other Uncertain Soils. 14 dr. 1844.
{Minutes of Proceedings, Inst. C. E., v. 3, p. 355372.)
Condensed. 1846. {Journal, Franklin Inst., v. 41, p. 7379.)
Howe, Malvbrd A. Retaining Walls for Earth, Including the Theory of
Earthpressure as Developed from the Ellipse of Stress, with a Short
Treatise on Foundations, Illustrated with Examples from Practice,
ed. 4. 167 p. 1907.
Hughes, Thomas. Description of the Method Employed for Draining some
Banks of Cuttings on the London and Croydon, and London and Bir
mingham Railways; and a Part of the Retaining Wall of the Euston
Incline, London and Birmingham Railway. 4 ill. 1845. {Minutes of
Proceedings, Inst. C. E., v. 4, p. 7886.)
International Correspondence Schools. Railroad Location, Railroad
Construction, Track Work, Railroad Structures. [473 p.] (Inter
national Library of Technology, v. 34B.) Includes section on theory
and design of retaining walls, p. 899912.
Jacob, Arthur. On Retaining Walls. 27 diag. 1873. {Van Nostrand's
Engineering Magazine, v. 9, p. 194204.) Reprint, with a few emenda
tions, of author's original essay on "Practical Designing of Retaining
Walls." Takes up design. Considerable attention to earth pressure.
^1873. {Building News, v. 25, p. 421422, 465466, 478479.)
Jacquier. Note sur la Determination Graphique de la Pouss^e des Terres.
5 diag. 1882. (Annates des Ponte e< C/ia«ssees, ser. 6, v. 3, p. 463472.)
Bases his graphical construction on Rankine's theory, as developed by
Levy, Consid^re, and others.
Kirk, P. R. Graphic Methods of Determining the Pressure of Earth on
Retaining Walls. 1899. {Builder, London, v. 77, p. 233235.)
Klein, Albert. Die Form der Winkelstutzmauern aus Eisenbeton mit
Ruoksioht auf Bodendruck und Reibung in der Fundamentfuge. 1909.
{Beton und Eisen, v. 8, p. 384387.)
Kleitz. Determination de la Poussfe des Terres et Etablissement des
Murs de Soutenement. 1884. {Annates des Fonts et Chaussees, ser. 2,
V. 7, p. 233256.) Theoretical discussion.
264 RETAINING WALLS
Klbmperer, p. Graphische Bestimmung des Erddruckes an eine ebene
Wand mit Riicksicht auf die Cohasion des Erdreiches. 1 pi. 1870.
(Zeitschrift, Oesterreichischen Ingenieurund ArchitektenVereines, v.
31, p. 116120.)
Krantz, J. B. Study on Reservoir Walls; Translated from the French by
F. A. Mahan. 64 p. 1883.
Lachbr, Walter S. Retaining Walls on Soft Foundations. 1915. (Jour
nal, Western Soo. of Engrs., v. 20, p. 232265.) Experiments gave the
following conclusions as to types of walls and their advantages: (1)
The block wall is economical, and may be constructed in several stages,
but it does not possess as great a potential factor of safety as a mono
lithic wall; (2) the heavy batter mass waU is economical, but is open to
the same objections as the block wall; (3; the cellular wall offers great
resistance to overturning or sliding, but occupies considerable space
before fUling and may thus interfere with use of tracks; (4) the mass
wall on piles gives maximum security, but is expensive and may givd
trouble because of damage to adjacent buildings on insecure founda
tions.
LaPont, de. Mfimoire sur la Poussfie des Terres et sur les Dimensions k
Donner, Suivant leurs Profils, aux Murs de SoutenSment et de Reser
voirs d'Eau. 1 pi. 1866. (Annales des Fonts et Chaussies, ser. 4, v. 12,
380462.) Gives in tabulated form experiments performed and con
stants arrived at by Aud6, Domergue, and SaintGuilhem, p. 397400.
LaFont, de. Note sur la Repartition des Pressions dans les Murs de
Souten^ment et de R&ervoirs, Nouvelles Formules pour le Calcul de
ces Murs. 1868. (Annales des Fonts et Chaussees, ser. 4, v. 15, p.
199203.)
Lagrene, H, de. Note sur la Poussfie des Terres Avec ou Sans Surcharges.
8 diag., 2 dr. 1881. (Annales des Fonts et Chaussees, ser. 6, v. 2, p.
441471.) Gives calculations for ejirth pressure of level surfaces on
vertical retaining walls.
Abstract. 1882. (Minutes of Froceedings, Inst. C. E., v. 68, p. 336
337.)
Lateral Earth Pressure. 1904. (Engineering Record, v. 49, p. 633634.)
Editorial comment on "Lateral Earth Pressure and Related Phenom
ena," by Ernest P. Goodrich.
Lethieh and Jozan. Note sur la Consolidation des Terrassements du
Chemin de Fer de Gien a Auxerre. 2 pi. 1888. (Annales des Fonts
et Chaussees, ser. 6, v. 16, p. 518.) Consolidation of treacherous slopes
in heavy cuts by means of rubble spurs perpendicular to face of slopes.
Abstract translation. 1889. (Miniaes of Froceedings, Inst. C. E., v.
95, p. 466468.)
L'EvBiLLE. De I'Emploi des Contreforts. 1844. (Annales des Fonts et
Chaussees, ser. 2 , v. 7, p. 208232.) Derives formulas for proper
design.
Levy, Maurice. Essai sur une Th6orie Rationnelle de I'Equilibre des
Terres Fratchement RemuiSes et ses Applications au Calcul de la Stabil
it6 des Murs de Soutenement. 1869. (Comptes Rendus Hebdomadaires
des Stances de I'Acadfimie des Sciences, v. 68, p. 14561458.) Develops
SPECIFICATIONS 265
a theory of earth pressure, and shows its application in design of retain
ing walls.
Leygije. Notice sur les grands Murs de Souttaement de la Ligne de
Mazamet a BMarieux. 2 pi. 1887.. (Annales des Fonts et Chaussees,
ser. 6, V. 13, p. 98114.) Considerable attention is given to design.
Maconchy, G. C. Earthpressures on Retaining Walls. 1898. {Engi
neenng, v. 66, p. 256257, 484^185, 641643.) Gives simple method
for calculating overturning moments.
Main, J. A. Graphic Determination of Pressures on Retaining Walls.
1912. {The Engineer, London, v. 113, p. 220.)
Mebm, J. C. Bracing of Trenches and Tunnels, with Practical Formulas
for Earth Pressures. 2 diag., 5 ill., 13 dr. 1908. {Transactions, Am.
Soc. C. E., V. 60, p. 123.) Discussion, 10 diag., 5 ill. 54 dr., p. 24
100. Develops a theory of earth pressure, and basis of this theory
deduces analytical relations.
Abstract. 1908. {Minutes of Proceedings, Inst. C. E., v. 171, p.
435^36.)
Abstract. 1 ill., 3 dr. 1907. {Engineering Record, v. 56, p. 4:MiQ6.)
See also editorial 'JSheet Piling and Earth Pressure," p. 528, and letter
to editor, p. 608.
Mehriman, Mansfield. Textbook on Retaining Walls and Masonry
Dams. 122 p, 1893.
MoFFBT, J. S. D. Mistaken Ideas with Reference to the Resultant Force
and the Maximum Pressure in Retaining Wall Calculations. 1903.
{Feilden's Magazine, v. 9, p. 197199.)
MoHLER, C. K. Tables for the Determination of Earth Pressures on Re
taining Walls. 1909. {Engineering News, v. 62, p. 588589.)
MullerBeeslatj, Heinrich. Erddruck auf Stutzmauern. 159 p. 1906.
"Literatur," p. 158159. Contains a thorough discussion of the theory
of the lateral pressure of sand and loose earth, and a full description of
the author's extensive experiments.
Pearl, James Warren. Retaining Walls; Failures, Theories and Safety
Factors. 1914. {Journal, Western Soc. of Engrs., v. 19, p. 113172.)
Discusses foundation soil of retaining walls, and calculates design
mathematically.
Petterson, Harold A. Design of Retaining Walls. 1908. {Engineering
Record, v. 57, p. 757769, 777778.) Diagrams are given. See also
letter by C. E. Day, Engineering Record, v. 58, p. 56.
Pichault, S. Calcul des Murs de Soutenement des Terres en Cas de Sur
charges Quelconques. 1899. {Mimoires et Compte Rendu des Travaux
de la Society des Ing^nieurs Civils de France, 1899, pt. 2, p. 210266,
844846.) Bibliography, p. 264266. Mathematical treatment of
earth pressures on retaining walls.
Poncelet. M6moir6 sur la Stabilite des RevStements et de leurs Fondations.
1840. {Comptes Rendus Hebdomadaires des Stances de I'Academie des
Sciences, v. 11, p. 134140.) Review of the author's 270page essay
published in Memorial de I'Officier du Genie, No. 13. Author is an able
supporter of Coulomb's theorv.
266 RETAINING WALLS
Abstract. 1840. {Revue Oenerale de V Architecture et des Travaux
Publics, V. 1, p. 482483.)
Pbelini, Chaelbs. Graphical Determination of Earth Slopes, Retaining
Walls and Dams. 129 p. 1908. Elementary treatment, for students
rather than professional engineers. Graphical methods are given for
solving problems concerning the slopes of earth embankments, the
lateral pressure of earth, and the thickness of retaining walls and
dams.
PuRVER, George M. Design of Retaining Walls, Adapted from Georg
Christoph Mehrtens, " Vorlesungen iiber Static der Baukonstructionen
und Festigkeitslehre." 1910. {EngineeringContracting, v. 34, p. 388
395.) Includes "Tables for Allowable Pressure, Adopted by the Public
Service Convention [Commission?], First District, State of New York."
Ramisch. Neue Versuche zur Bestimmung des Erddrucks. 1910. {Zeit^
schrift, Oesterreichischen Ingenieur und ArchitektenVereines, v. 62,
p. 233240; v. 63, p. 323425.) Mathematical calculations.
Rbbhann, Gbokg. Theorie des Erddruckes und der Futtermauern mit
besonderer Rticksioht auf das Bauwesen. 1871. {Zeitschrift, Oester
reichischen Ingenieurund ArchitektenVereines, v. 23, p. 211.) Review,
by O. Baldermann, of Rebhann's book, published in 1870 in Vienna by
Carl Gerold's Son.
Reissner, H. Theorie des Erddrucks. 1910. (Enzyklopadie der Mathe
matischen Wissenschaften, v. 4, pt. 4, p. 386417.) "Literatur,'' p. 387.
Reppert, Charles M. Recent Retaining Wall Practice, City of Pitts
burgh. 1910. {Proceedings, Engrs. Soc. of Western Pennsylvania,
V. 26, p. 316354.) Discussion, p. 355367. Givesatte ntion to calcu
lation of earth pressures as affecting design.
Rbsal, Jean. Pouss6e des Torres. 2 v. 19031910. (Enzyklopadie des
Travaux Publics.) v. 1. Stability des Murs de Soutenement. v. 2.
Theorie des Terres Coh&entes. — Applications. — Tables Num&iques.
Purely theoretical work on earth pressures as affecting the design of
structures, v.l deals entirely with soils lacking cohesion.
Reutbrdahl, Arvid. From the Soil Up: A New Method of Designing.
1914. {EngineeringContracting, v. 42, p. 581585.) Considers espe
cially retaining wall design. Advocates starting with the bearing capac
ity of the soil, and working from that basis.
Rose, W. H. Formulas for the Design of Gravity Retaining Walls. 1910.
{EngineeringContracting, v. 34, p. 115117.) From Professional Mem
oirs, Corps of Engineers, U. S. Army.
SaintVenant, de. Examen d'un Essai de Th6orie de la Pouss6e des Terres
Contre les Murs Destines k les Soutenir. 1873. {Comptes Rendus
Hebdomadaires des S(5ances de I'Acaddmie des Sciences, v. 73, p. 234
241.) Criticizes Curie's theory, and defends the socalled rational
theory developed by Levy.
SaintVenant, db. PoussiSe des Terres. Comparaison de ses Evaluations
au Moyen de la Consid6ration Rationnelle de I'Equilibrelimite, et au
Moyen de I'Emploi du Principe dit de Moindre Resistance, de Moseley.
1870. {Comptes Rendus Hebdomadaires des Stances de rAcad6mie des
Sciences, v. 70, p. 894897.)
SPECIFICATIONS 267
SaintVenant, de. Rapport sur un Memoire de M. Maurice Levy, Prd
senU le 3 Juin, 1867, Reproduit le 21 Juin, 1869, et Intitule: Essai sur.
une Tli6orie Rationnelle d'Equilibre des Terras Fraiohements Remu^es,
et ses Applications au Calcul de la Stabilitd des Murs de Soutfenement.
1870. {Comptes Rendus Hebdomadaires des Seances de I'Acad^mie des
Sciences, v. 70, p. 217235.) Report of a committee, giving a historical
review of the works on earth pressure, and discussing in detail Maurice
Levy's theory.
SaintVenant, de. Recherche d'une Deuxifeme Approximation dans le
Calcul Rationnel de la Poussee, Exerc^e, Contre un Mur dont la Face
Posterieure a une Inclinaison quelconque, par des Terres non Coh6
rentes dont la Surface Superieure s'Eleve en un Talus Plan quelconque
El Partir du Haut de Cette Face du Mur. 1 diag. 1870. {Comptes
Reiidus Hebdomadaires des Stances de rAcaderaie des Sciences, v. 70,
p. 717724.) Based on Levy's theory.
SaintVenant, de. Sur une Determination Rationnelle, par Approxima
tion, de la Poussee qu' Exercent des Terres Depourvues de Cohesion,
Contre un Mur ayant une Inclinaison quelconque. 3 diag. 1870.
{Comptes Rendus Hebdomadaires des Stances de I'AcadSmie des Sciences
v. 70, p. 229235, 281286.) Development of Levy's theory.
SaintVenant, de. Sur une Evaluation, ou Exacte ou d'une Ties Grande
Approximation, de la Poussee des Terres Sablonneuses Contre un Mur
Destine k les Soutenir. 1884. (Comptes Rendus Hebdomadaires des
Stances de I'Acadc^mie des Sciences, v. 98, p. 850852.) Based on
Boussinesq's works.
Schaffeb. Erddruck und Stiitzwande. 1 diag., 1 pi. 1878. {Zeitschnft
fur Bauwesen, v. 28, p. 527548.)
ScHMiTT, Edu ABD . Empirische Formeln zur Bestimmung der Starke der Fut
termauern. 1871. {Zeitschrift, Oesterreichisohen Ingenieurund Archi
tektenVereines, v. 23, p. 336338.) Mathematical calculations on the
basis of Rebhann's tables.
Sohwedleb, J. W. [Unterschnittene Futtermauern.] 1871. {Zeitschrift
fur Bauwesen, v. 21, p. 280282.) Discussion of the formula derived
by Schwedler at a meeting of the ArchitektenVerein zu Berlin.
Seebek, D. C. Stability of Sea Walls. 15 diag. 1906. {Engineering
News, V. 56, p. 198200.) Gives method of design.
Brief abstract. 1906. {Le Genie Civil, v. 50, p. 32.)
Sheetpiling and Eabth Pebssube. 1907. {Engineering Record, v. 56,
p. 528.) Refers particularly to paper on "The Bracing of Trenches
and Tunnels," by J. C. Meem. „„, ,  ,
SiEGLER. Experiences Nouvelles sur la Poussee du Sable. 1887. {Annates
des Fonts et Chaussees, ser. 6, 13, p. 488505.) Experimental method
for studying reactions between masses of earth and their supporting
walls. Friction dynamometer was used to determine intensity of
Condensed translation. "New Experiments on the Thrust of Sand."
1887 {Scientific American Supplement, v. 34, p. 972497^5.)
SiNOEK, MAX. Fliessende ffinge. 1902. (ff ^''f ' ^f ™';'^te
Ingenieur und ArchitektenVereines, v. 54, pt. 1, p. 190196.) JJe
268 RETAINING WALLS
scribes yielding of sides of railway cutting in valley of the Eger, Austria,
with methods used for retaining embankment.
Sinks, F. F. Analysis and Design of a Reinforced Concrete Retaining Wall.
1905. {Engineering News, v. 53, p. 89.)
Sinks, F. F. Design for Reinforced Concrete Retaining Wall. 1904.
{Railroad Gazette, v. 37, p. 676677.) Letter.
Skibinski, Carl. Ueber Sttitzmauerquerschnitte. 1898. {Zeitschrift,
Oesterreichischen Ingenieur und ArchitektenVereines, v. 45, p. 666
670.)
Skibinski, Karl. Theorie des Erddrucks auf Grund der neueren Versuchen.
■ 1 diag., 1 pi. 1885. {Zeitschrift, Oesterreichischen Ingenieur und
ArchitektenVereines, v. 37, p. 6577.) Develops his own theory of
earth pressure based on the experimental work of Forchheimer, Gobin,
and Darwin. Gives a graphical construction of his theory, and methods
of practical application.
Spillnek, E. Stiitzmauern. 1904. (Handbuch der Architektur, od. 3.
V. 3, pt. 6, p. 182197.) "Literatur," p. 196.
Strukel, M. Beitrag zur Kenntniss des Erddruckes. 2 diag., 4 dr. 1888.
(Zeitschrift, Oesterreichischen Ingenieur und ArchitektenVereines, v.
40, p. 119125.) Critical review of the salient points of the earth
pressure theory as developed by Coulomb, Rebhann, and others. In
support of his own views, gives results of some experiments.
Sylvester, J. J. On the Pressure of Earth on Revetment Walls. 1 diag.
1860. {London, Edinburgh and Dublin Philosophical Magazine and
Journal of Science, ser. 4, v. 20, p. 489499.) Criticism of theories of
Coulomb and Rankine.
Tate, Jambs S. Surcharged and Different Forms of Retaining Walls. 59
p. 1873. VanNostrand. Theoretical calculations for retaining walls.
1873. (Van Nostrand's Engineering Magazine, v. 9, p. 481494.)
Thornton, William M. Retaining Walls. 7 diag. 1879. {Van Nos
trand' sC Engineering Magazine, v. 20, p. 313318.) Concise and simpli
fied account of the theory of earth pressure and its application to the
design of retaining walls.
Van BuREN, John D., Jr., Quay and Other Retaining Walls. 6 diag. 1872.
{Transactions, Am. Soc. C. E., v. 2, p. 193221.) Establishes practical
formulas for the dimensions of walls of various shapes and under various
conditions. Follows Coulomb's theory. An appendix gives a number
of mathematical relations.
Vedel, p. Theory of the Actual Earth Pressure and Its Application to
Four Particular Cases. 1894. {Journal, Franklin Inst., v. 138, p.
139148, 189198.) Mathematical calculation.
Walmisley, a. T. Retaining Walls. 1907. {The Builder, London, v. 93,
p. 647648.) Discusses calculations of earth pressure, foundations, etc.
Wbingartbn. [Die Theorie des Erddrucks.] 1 diag. 1870. {Zeitschrift
fur Bauwesen, v. 20, p. 122124.) Abstract of a paper read before the
ArchitektenVerein zu Berlin.
Weston, W. E. Tables for Use in Determining Earth Pressure on Retain
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Winkler, E. Neue Theorie des Erddruckes. 19 diag. 1871. {Zeit
SPECIFICATIONS 269
schrift, Oesterreicliischen Ingenieur und ArchitektenVereines, v. 23,
p. 7989, 117122.)
Woodbury, D. P. On the Horizontal Thrust of Embankments. 1 diag.
1859. {Mathematical Monthly, v. 1, p. 175177.) Mathematical paper.
WooDBTTRT, D. P. Remarks on Barlow's Investigation of "the Pressure of
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Franklin Inst., v. 40, p. 17.)
INDEX
Numbers refer to pages
Abrams, D. A., concrete strength,
201, 216
Abutments, general theory, 128
highway, 132
problems, 140
settlement cracks, 156
types of, 130
Adhesion, reinforced concrete, 89
Aggregates, effect on concrete
strength, 200
fineness modulus, 202, 216
heating, 211
proportions, 214
ratio, fine to coarse, 226
surface area, 202, 219
Architectural treatment, 232
Arm, vertical, 91 •
Asphalt, waterproofing, 240
B
Baker, Sir Benjamin, 3, 18
Bars, see Rods.
Bearing, concrete stress, 90
Belidor, 2
Bell, cohesion, 23
Bernoulli, theory of flexure, 85
Bibliography, 41, 77, 120.
Special Committee on Soil, Am.
Soc. C. E., 257
Bilger, H. E., standard abutment
sections, 132
Board marks, 232
Bond, see Adhesion.
Boussinesq, J., 2, 8, 31
Box sections, 132
problem, 143
Bracing, 188
Bullet, 1
Bureau of Standards, Report on
Concrete, 199
Cableway, 175
Cain, Wm., cohesion, 20, 22
experimental data, 19
factor of safety, 57
footing of counterfort wall, 99
modification of coulomb the
ory, 5
revetment wall, 65
surcharge, 28
Calcium chloride, hardening con
crete, 213
Cement, effect on concrete strength,
200
Portland, 214
proportions, 214
specifications, 215
Center of gravity, walls, 63
trapezoid, 10
Clay, as a foundation, 50
failures, 162
permissible bearing, 52
CodeS; building, 52
Cofferdam, pressures on, 31
Cohesion, 3, 20
Colors, face treatment, 235
Concrete, acceleration of set, 212
allowable stresses, 90
compressive strength, 200
construction, 197
Cyclopean, 210
distributing, 209
materials, 213
methods, 208
methods of proportioning, 201
271
272
INDEX
Concrete, pressures, 181, 183
proportions, 214
Report Special Committee, Am.
Soc. C. E., 85
Report Tests Bureau of Stand
ards, 199
see also Aggregates, Cement,
Reinforced concrete.
trains, 173
water content, 197, 203
Concreting, winter, 210
Conjugate pressures, 7
Coping, rubble walls, 229
Cost data, 248
labor, 249
rubble walls, 231
Counterfort, design of, 101
economical spacing, 150
Counterfort walls, 96, 107
economic comparison with "T"
walls, 147
Coulomb, 2, 5, 11
Couplet, 2
Crane, erecting, effect upon abut
ment, 129
Cribbing, concrete, 124
timber, 124
Crum, R. W., 226
Curves, permissible flattening, 243
D
Details, wall, 138
Distributing systems, cableway, 170
concrete, 170
pneumatic, 170
spouting, 170
tower, 170, 171, 172
Drainage, 238
E
Earth pressure, history of theory, 1
problems, 36
theories, 5
Eddy, Prof., theory of plates, 108
Edwards, L. N., method of surface
area, 219
Embankment, bounded by two
walls, 126
Embankment, rolled in layers, 5
see also Fill.
Empiric design, 3
Enger, M. L., experiments on trans
mitted pressure, 31
EquiKbrium polygon, use in wall
design, 48
Error, permissible in wall survey,
242
Euler, theory of flexure, 85
Experimental data, 18
F
Face treatment, 232
Factor of safety, 48, 56, 84
Failures, wall, 57, 160
Fill, ideal and actual, 4
sea walls, 35
Fineness modulus, 202, 216
Finish, see Face treatment. Archi
tectural treatment.
Footing, counterfort wall, 98
design of reinforced concrete, 93
Forms, 181, 187
blaw, 191
hydraulic pressed steel, 190
lines and grades, 244
oiling, 189
on curves, 194
patent, 189
problem in, 195
reuse, 187
stripping, 188
traveling, 193
Foundation, character of, 49
problems, 67
see also. Rock, Sand, Clay, Piles.
Frame, stresses in rigid, 132
Friction, between wall and earth, 3,
8, 19
between wall and foundation, 44
G
Grades, 242
Gravel, see also Aggregates.
soil, 50, 52
specifications for concrete, 216
INDEX
273
Gravity wall, center of gravity, 63
direct design, 61
merits, 137
problems, 67
stresses, 48
table of dimensions, 64
types, 65
Godfrey, E., 97
Goodrich, E. P., earth pressure
tests, 18
PI
Hand rail, 237
Hell Gate Arch, see. New York Con
necting Railroad.
Hool, Prof., factor of safet}, 57
Howe, Prof., 8, 23
Husted, A. G., pressure of saturated
soils, 32
Interboro Rapid Transit Co., East
ern Parkway Walls, 127
White Plains Road Extension,
127
Isometric drawing, 216
Lagging, forms, 184
Levy, M., .2
Lines and grades, 242
Loads, see Pressure, Surcharge.
Love, A. E. H., transmitted pressure
through solids, 31
M
Mayniel, 2
Mehrtens, see Purver, G. M.
Middle third, 56
Mixer, concrete, see Plant.
Mixing, proper methods, 207
time of, 208
Mohler, C. K., thrust expression, 17
wingwall, 131
Moments, overturning, 44
resistance of reinforced con
crete, 87
thrust and stability, 47
Mortar, rubble wall, 227
N
Navier, 2
Neutral axis, reinforced concrete, 86
New York Connecting Railroad, re
taining walls of, 21, 127
Johnson, N. C, 197
Joints, construction, 159, 233
details of, 158
efficiency of, 159
expansion, 157
omission of, 158
Joists, forms, 186
K
Kelly, E. F., 131
Keys, concrete, 209
Labor, costs, 249
Lacher, cellular, 123
transmission of liveload, 30
13
Offset, gravity wall, 58
Overturning, criterion against, 44
Passive stress, 23
Piles, 50
problems, 69
proper centering, 52
walls on, 77
Plant, 165, 179
arrangement, 166
central, 168
concrete, see Preface.
rubble walls, 227
standard layout, 166
Plaster coat, 231
274
INDEX
Plates, theory of, 108
Pointing, stone walls, 230
Poncelet, 2, 5
graphic thrust determination,
38
Pressure, base, distribution, 50
cofferdam, 31
permissible soil, 52
toe, criterion against excessive,
44
values of, 51
see also Earth pressure,
transmission of vertical,^ 30
Prior, J. H., abutments, 131
cellular walls, 123
Public Service ' Commission, see
Codes, Building.
Purver, G. M., 26
R
Rangers, 187
Rankine, 2, 3, 5
Reinforced concrete, abutments, 142
constants, 88
theory, 85
walls, 79
base pressure, 83
base ratio, 82
economical width, 82
factor of safety, 84
merits, 137
problems, 104
skeleton outline, 80
tables, 84
Reinforcement, economical, 187
see also Rods,
shrinkage, 155
supports, 192
temperature, 154
Resal, 2
Robinson, concrete pressure experi
ments, 182
Rock, 49, 52
Rods, anchoring, 90
bending, 90
counterfort walls, 101
periphery for adhesion, 89
see also Reinforcement.
Rods, specifications, 257
Rondelet, 2
Rubbing, 223
Rubble, cement, 46
dry, 46
HetchHetchy Railroad, 47
Sallonmeyer, 2
Sand, foundation, 82
see also Aggregates.
specification, 215
Serber, D. C, sea walls, 35
Settlement, 155
Shale, 50
Shear, reinforced concrete, 89
Shrinkage, 155
Shunk, concrete pressure experi
ments, 182
Slabs, face, counterfort walls, 97
see Reinforced concrete.
thin, 139
Sliding, see Friction.
Soils, bearing, 52
plastic, 50
saturated, 32
see also Earth pressure. Fill,
> Foundations, Pressures,
etc.
Specifications, 5'!
Speedway, cellular walls supporting,
124
Stone, broken, 216
see Aggregates.
St. Venant, theory of flexure, 85
Subsurface structures, 136
Surcharge, 25
sea walls, 35
Surface area, 202, 219
Surveying, 242
Sweeny, F. R., see Cofferdam.
Talbot, Prof. A. N., 197
Tar, specifications, 240
Temperature, distribution in large
masses, 152
stresses,151
INDEX
275
Thrust, coulomb expression, 14
fluid expression, 17
Mohler, C. K., 17
Rankine expression, 8
standard form, 9, 10, 15
Tierods, 186
Timber, safe stresses, 185
Toe, offset gravity walls, 59
reinforced concrete walls, 95
Tooling, 234
Tower, concrete, 176
Track elevation, 42, 123
Trapezoid, center of gravity, 8
Trautwine, 3
Vauban, general, 1
Volumes, computation of, 245
W
Walls, Ashlar, 230
backstays, 125
cellular, 122
classes, 43
counterfort, 96
Walls, economic location and height,
42
economy of various types, 137
European practice, 126
hollow cellular, 123
land ties, 125
relieving arches, 125
revetment, 65
rubble coping, 229
construction, 228
cost, 231
face finish, 230
plant, 227
specifications, 229
sea, 34
see also, Gravity walls, Rein
forced concrete walls,
selection of economical type,
147
stone, 45
Washers, rod anchorage, 91
Water content, see Concrete.
Wedge beam, method of Cain, 97
Wedge of maximum sliding, 11
Winter concreting, 210
Work, theory of least, 132
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