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Full text of "Retaining walls; their design and construction"

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BOUGHT WITH THE INCOME OF THE 

SAGE ENDOWMENT FUND 



' THE GIFT OF 



HENRY W. SAGE 



1891 



Cornell University Library 
TA 770.P12 



Retaining walls; their design and constru 




3 1924 015 698 016 




PI Cornell University 
P Library 



The original of this bool< is in 
the Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



http://www.archive.org/details/cu31924015698016 



RETAINING WALLS 

THEIR DESIGN AND CONSTRUCTION 



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"ykis Qraw-OJillBook (h, Jm 

PUDIISHERS OF BOOKS FO B^ 

Coal Age ' Electric Railway Journal 
Electrical World "'' Engineering News-Record 
American Machinist v Ingenieria Intemacional 
Engineering S Mining Journal ^ Power 
Chemical 6 Metallurgical Engineering 
Electncal Merchandising 




EETAINING WALLS 

THEIR DESIGN 
AND CONSTRUCTION 



BY 
GEORGE PAASWELL, C. E. 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 

NEW YORK: 239 WEST 39TH STREET 

LONDON: 6 & 8 BOUVERIE ST., E. C. 4 

1920 

\ 

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coptkight, 1920, bt the 
McGraw-Hill 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 well-founded 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 

News-Record, 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 — Sub-division of field operations — Mixers — Distribution 
systems — Examples of plant layouts. 

CHAPTER VII 

FOBM WOBK 181 

Sub-division 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 nigger-heads. ' 

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 one-half 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 so-called 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 time-con- 
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 Gauthier-VUlars, Pans, 1917. 




t>j(l+c)hJ 
Fig. 5. — Typical loading Rankine method. 



10 



RETAINING WALLS 



The ratio c is small, generally less than one-third, 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= tan-i (G/P) -b = tan-i (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 Coulomb-Cain 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 so-called 
"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 non-coherent 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 non-uniformly 
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 
well-settled 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^ _ y-a±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 News-Record, 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 News-Record, 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 rip-rap, 



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 ^ 0-42 = 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 one-sixth. The thrust for 
this material is then one-half 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 satura-ted 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 News-Record, 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 
cross-hatched. 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 well-nigh 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 cut-stone 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 News-Record, Vol. 81, p. 890 for a description of the 
iise of dry rubble walls to retain the Hetch-Hetchy 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 non-uniformly 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 - S-2)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) 



^.=^(3fc-l) 



(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 one-third, 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 



M-1 
M-2 
2-4 
2-4 
4-6 
6-8 

8-10 

4r-6 

6-8 
5-200 



1 
2 

4 

6 

10 

4 
75* 



1 
2 
3 

6 

10 

4 
8-40 



1 

t2 



4 
12-20 



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 non-uniform 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.*=^(3fc-l)+3^?^(l-2/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= h-bo = bo [sp^ - l) Since Po =^', 60 = 

and if, finally, Si and S2 are replaced by their values in terms of 
R and k 

. = .p(^iH-i);P = 3-f^;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 one-half 
the distance between the toe and the point Z„_2. Then 

Wo — ln-2 = 2a 
Replacing L-2 by its value from (45), simplifying the resulting 
equation and ehminating the radical and putting 2a/wo = X 

^ -^l-^)+2(l-X), N=^-^ 



3fc - 1 F 

and solving for k 

_ iV-(l-X)(l-3X) 
" 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(l-e)],B=6\'Ml-M); 

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 weU-appearing 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 160-163) 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 pressvu-e. 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 S-i, 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 
one-half. 

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 11-15), 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 self-sustaining 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 

self-sustaining, 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 above-mentioned 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 +44-1) = 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^t-y;--^ 

Fine Sand — t^y^. . 

Piles - A4-^.'"''' 

men A IS over t-ff,Sfep Base t 

as shown. 

ffei'nforce Base when A is 

overZft. 




h 


d 


b 


At 


A? 


A3 


A4 


is 


E-5" 


9'-3' 








1-6' 





20 


Z-S" 


n-i(f 








1-6" 


■1-0' 


25 


3-0' 


14-5- 





0-6" 


2-fi" 


Z-Q~ 


30 


3^4" 


ir-0' 





1-4- 


4'-0' 


3-0 



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 


H-i 


% 



"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 quarter-point to I> is 



and G from (69) is 



(1 + p)hN 
4 

mhHl + 2p)Ar 
2 



The lever arm of G about the quarter-point 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 one-inch 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 one-haK 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. . 

One-half 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 four-foot 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 



Ih-ch 




' 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)(l-i)(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)(l-i) _ 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 one-fifth. 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 






— 






























n-lO 


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 concrete-stee 
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 one-half 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 one-third (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-^^[e-iil-2e)]] (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 one-half. 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 one-half. 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 = 
one-third 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 = m-y/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 step-by-step 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 tie-rods 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 = %2-5 = 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 + 1-44) = 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 twenty-two 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 + 1S-C2 _ 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 one-inch 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 5-inch 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 




T-e.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 upon-the 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 

Z-I4 Bars 




14-0 
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 



7-g" 



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 



x|x 



[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 - iy-wHan 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 one-third 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 
one-inch 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 foot-pound 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 one-inch rods on twelve-inch centers, is equal to 
the external bending moment at a point approximately 4.5 feet above the 
footing. The six-inch 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. One-inch 
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. 1-inch 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 foot-walk 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 = 11-4 


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<- 4-0'->| ^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 a-a 



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 a-a 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. 



r-n 



rm 



L 



Fig. 66. — Briek wall with oast iron relieving arches. 

'^ — n 






4' ^ 



4' 

r 1 






i 



^ 



\^ 12-0 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. 



■S7-S- 



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 fifty-feet, 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 non-use 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 a-a 

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 a-a 

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 reinforced-concrete, as economy 

or other factors dictate. The flexibihty of 

reinforced-concrete in permitting slender walls 

with projecting heel and toe indicates that for 

practically every condition a reinforced-concrete 

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 Wing-walls. — 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- 
inforced-concrete 
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 U-band of 
constant cross-section 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 News-Record, 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 box-like 
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. — Sub-surface 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 sub-surface 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(^f5J-j (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 sub-surface 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 Reinforced-Concrete 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 net-work 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 






a-4" 



i_r 



Fig. 87. 



eroundSurface ^--Surcharge of S-O' 

T'iiiiii)iii)>iimi)imimmmi wuui'Kiimmii i iminiiiimmmmr 



-^ 






--2B-0- 



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 side-walls 
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 side-wall loading alone 

H = '^^20 ^^ + ^-^^ ^ ^-^^^ = 7.8 kips 

M = - ,■ an (1 - 4-38 + 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, 

0-7 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. 
.3-7 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. 



•^ 6-0- 






■ t- 



/"fflsofe J8 "OoC. between these Poinfs 



l"'f!ods,6'afvC. 




3' > 






-E-6 






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 -f-ll 



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 bending-moment 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 



'(H-fcV3)\/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 non-yielding 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, steel-concrete 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, 1911-1912, 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 (new-and 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 to-day 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, 1914-5, 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 one-fifth to one-third 
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 




5f-orage 
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 
twenty-five to thirty-five 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 Non-Tilting (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 twenty-three 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 twenty-seven 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 5-inch pipes, or 10-inch 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 seventy-five to 
one-hundred 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-- - z-foo' - . 

__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 self-supporting 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 two-compartment 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 thirty-five 
foot flat car, equipped with a % yard Smith non-tilting 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 



Chu-hs: 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 
winch-head 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 stiff-leg 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 one-yard 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 one-half 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:. ^ 




Consiruci-ion 
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 concrete-mixing 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 News-Record, 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 narrow-gage 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 one-half the 
length of the wall. 

"For the remainder of the work the cars ran up to the derrick and 
discharged the concrete into a home-made 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. 376-380. 

"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, 1911-1912, 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 slow-setting 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 semi-liquid 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 one-eighth 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 one-eighth 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. 

Tie-rods. — The diameter of the tie-rod 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 Re-use. — 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, well-fitting 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. Form-work 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 tie-rods (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 
pock-marked 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'Cen-fvrs 
\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., one-eighth 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^jng|V -^'^^ /' ' '^ 



• ^''^'/;^' 




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 twenty-four hours, the uprights 
and liners remaining as much longer as is necessary before the 
wall is self-supporting. 

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 

r-i-jll 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 tie-rods 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 4-inch by 6-inch studding and 2-inch by 8-inch 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 
center-line 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 T-square. 

"Run out the center line of the wall in chords of 10 feet and put in 
permanent plugs at these points. Erect a well-braced series of batters 
around the curve and set the top ledger board at the exact crest of the 
wall. Place the center-line 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 
8-hour 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 previous-pages 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, 2-ineh 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 4-inch by 6-inch 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 three-foot panel of sheeting. From the figure, the lower three feet bring 
a tabular equivalent load of 4.8 kips. Table 34 permits a three-foot 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 3-inch by 6-inch sticks, a handy mer- 
chantable size. The safe load span upon these two pieces will determine 
the tie-rod 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 tie-rod will be called upon to carry is 

3.5 X 3 X 1500 = 15,700 lb. 

To avoid using large size tie-rods 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 News-Record, 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 concrete-making qualities may be 
produced by an infinite number of different gradings of a given material. 

"7. Aggregates of equivalent concrete-making 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 — water-ratio. 

"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 so-called '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 square-mesh 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. 


lOO-mesh..,, 


.0058 


.147 


82 


91 


97 


100 


100 


100 


9S 


48-mesh. . . . 


.0116 


.295 


52 


70 


81 


100 


100 


100 


92 


28-mesh.... 


.0232 


.69 


20 


46 


63 


100 


100 


100 


86 


14-raesh. . . . 


.046 


1.17 





24 


44 


100 


100 


100 


81 


8-nieah 


.093 


2.36 





10 


25 


100 


100 


100 


78 


4-mesh 


.185 


4.70 











86 


95 


100 


71 


%-in 


.37 


9.4 











51 


66 


86 


49 


?i-in 


.75 


18.8 











9 


25 


50 


19 


IM-in 


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 water-ratio necessary to produce a 
given plastic condition. " This is, of course, consistent with his 
main thesis that the water-ratio 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 water-ratio 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 water-ratio 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 one-bag 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 
well-graded 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 water-ratio. 

"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 water-ratio 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 well-graded 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 well-graded 
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 short-timed 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 News-Record, 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 so-called 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 short-sighted 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, well-cleaned 
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 News-Record, 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 
smooth-grained 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 
M-in. and l^-in. These sieves are made of square-mesh 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 well-graded torpedo sand up to No. 4 sieve will give a fineness 
modulus of about 3.00 ; a coarse aggregate graded 4-1 ].^-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 drift-sand may have a fineness 
modulus as low as 1.50. 



100 


igi-w^ 


















90 


^; 


I r^ 


^ 






'^^^ 


^ 






»; 80 




N^ 


h^ 




f^^e 




\ "^ 
\ ^ 






> 

Q> 

:? 70 




l\ 


' 




t^ 


^>^ 


\ "^ 

\ 1 "" 




60 






\ 


" ; 






\; > 








^ so 






\ 








!\ 






V 

^ 4-0 






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 single-size 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 News-Record, 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 cement-surface 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 cement-area 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 4-500 



^ 3500 
35 



o- 2500 



1500 



c 


L 








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N 


^ 








^ 




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% 








^ 




c 


[^ 


1 


\N 








H 


s 


1 








\ 


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10 20 30 

Surface Area per5ram of 
Cement inSq.In. 



300 



« 600 



■fc'WO 



200 













> 


\^ 










^ 




<i,. 




t 


L^ 


*^ 






"<%N 


\i^ 


N 






■Xs- 


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1. 























10 20 30 

Surface Area per Oram 
of Cemen+inSqJn. 



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. 124-127 may be used: 



224 



RETAINING WALLS 



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in 


n 




d 




f 1 








1. 

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CO 


o s 




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fl 





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o 

o 



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o 
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■Ulbs ui D3JV s3Dj.jn9 



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 News-Record, 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 4-R 
P 8-R 
P 10-R 



10. 

20. 



Grading, 
per cent. 

15.0 

5.0 

25.0 



P20-R 30 15.0 



P30-R 
P40-R 
P50-R 



40. 
50. 
80. 



P 80-R 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 
Pit-Run 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 sand-aggre- 
gate ratio. In that ideal mix the cement-void 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 fineness-modulus method of 
Abrams and the surface-area 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 News-Record, July 10, 1919. 



Plate V 













■c: c.ji"'^' ^ ■■■' '< 





-^ :7 






F,G t -Mothnd Of layi.ig slone wall hy series of derru-KS. 



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 nigger-hends. 



CHAPTER IX 
WALLS OTHER THAN CONCRETE 

Plant. — Rubble and cut-stone 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 one-half 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 

" Third-class 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 text-books {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 lamp-black 

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 Bush-hammers 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. 

{Fi-icing page 23G) 



Plate VIII 




Fic;. A. — (h-iiiimcMilal 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 stucco-finished 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 run-off 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 
weep-hole drain. The pipes should be spaced from twenty -five to 
ten-feet 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 6-inches 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 weep-hole, 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 6-inch porous tile on a 
grade of 0.5 per cent, extending from subgrade level to 6-inch pipes, 
which are imbedded in the rear part of the waUs and discharge to the 
drainage wells. Each well has an 8-inch connection to the catch-basin 
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 6-inch 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 water-proof. 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. 

Sub-surface 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 well-tried specification for a tar coating for the 
back of the wall, may read as follows : 

Coal-tar shall be straight-run pitch containing not less than twenty-five 
percentum (25%) and not more than thirty-two 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 ninety-five per- 
centum (95%) of bitumen soluble in cold carbon disulphide, and at least 
ninety-eight and one-half 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 one-half 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 one-half 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 cross-section 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 field-work. 
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 twenty-five 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 
one-eighth of an inch. For example, by reference to the table, 
a radius of 800 feet makes it necessary to stake out the wall in 
eight-foot chord lengths, while a radius of 8,000 feet permits the 
use of twenty-five-foot 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 net-Kne 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 two-dimension 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 News-Record, 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 Pyramid-Base BMHE 
AH-itude 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 worth-while 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 raih-oad 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 one-quarter 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 one-half 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 three-eighths {%) of an inch in di- 
ameter up to one and three-quarters (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 three-quarters (^) 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 water-tight 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 water-tight and shall be carefully inspected while 
the pouring is in progress to prevent the escape of any grout. 

Concrete shall set at least twenty-four (24) hours before the tie-rods 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. 70-86, 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. 155-158.) Mathematical solution. 
Allen Kenneth. Design of Retaining Walls. 1892. {Engineering Rec- 
ord V 26, p. 341-342, 356-357, 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. 1317-1337.) 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. 288-290.) 

AuDB. Nouvelles Expfirieiices sur la Pouss^e des Terres. 1849. {Comptes 
Rendus Hebdomadaires des Stances de I'Acad^mie des Sciences, v. 28, 
p. 565-566.) 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. 140-186.) Discussion, p. 187- 
241. Aims to present data on actual lateral pressure of earthwork, as 
distinguished from "text-book'' pressures, which latter the author 
holds to be genera,lly incorrect 

1881. {Van Nostrand's Engineering Magazine, v. 25, p. 333-342, 353- 

371, 492-505.) 

Bard WELL, F. W. Note on the "Horizontal Thrust of Embankments." 
1861. {"Mathematical Monthly, v. 3, p. 6-7.) 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. 166-169.) 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. 448-452.) Calculations of earth pressures, 
and diagrams. 

BoussiNESQ, J. Calcul Approch6 de la Pouss^e et de la Surface de Rupture, 
dans un Terre-plein Horizontal Homogfene, Contenu par un Mur 
Vertical. 1884. {Comptes Rendus Hebdomadaires des Stances de 
I'Academie des Sciences, v. 98, p. 790-793.) 

BoussiNESQ, J. Complement k de PriScedentes Notes sur la Pouss^e des 
Terres. 1884. (AnnoZesdesPowiseiC/iauss^es, ser. 6, v. 7, p. 443-481.) 

BoussiNESQ, J. Equilibrium of Pulverulent Bodies. 1 diag. 1877. 
{Minutes of Proceedings, Inst. C. E., v. 51, p. 277-283.) 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. 107-110.) 

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. 751-754.) 

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. C25-643.) Application of the theory of 
earth-pressure, as developed by Rankine and Darwin, to design of 
vertical walls. 



SPECIFICATIONS 259 

BotJSsiNESQ, J. Note sur la Mfithode de M. Macquorn-Rankine 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. 169-187.) 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. 720-723.) 

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. 667-670.) 

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. 901-904, 975-978.) 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. 757-759.) 

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. 593-594.) 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 
non-coherent 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. 265-277.) 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. 43-44, 59-61, 75-76, 85-87, 96-97, 109-110.) 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. 76-78.) Computes pressures on bracing and shor- 
ing for well under-drained 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. 106-108.) 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. 593-683.) 
Coleman, T. E. Retaining Walls in Theory and Practice; A Text-book 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. 547-594.) Extension of Levy's theory of 
earth-pressure. 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. 67-75.) 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. 

375-377.) 

Condensed. 1873. {Journal, Frankhn Inst., v. 95, p. 317-322.1 

CoENiSH, L. D. Earth Pressures : A Practical Comparison of Theories and 
Experiments. 1916. {Transactions, Am. Soc. C. E., v. 81, p. 191-201.) 
Discussion, p. 202-221. 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. 161-172.) Discussion, p. 173-195. 
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. 1726-1728. {His- 
ioire de I'Acad^mie Royale des Sciences, v. 28, p. 106-164; v. 29, p. 
132-141; V. 30, p. 113-138. 
CotJSiNBRT. Determination Graphique de I'Epaisseur des Murs de Soutene- 
ment. 1 pi. 1841. {Annates des Fonts et Chauss6es, ser. 2, v. 2, p. 
167-184.) 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 Erddruck-prismas und der Erddruck gegen 
geneigte Stutzwande. 4 diag. 1879. {Zeitschriftfur Bauwesen,v.29 
p. 521-526.) 
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. 61-97.) To be 
found in section " Mathematische Abhandlungen." 
CuNO. Die Steinpackungen und Futtermauern der Rhein-Nahe-Eisenbahn. 

1861. {Z&itschrift fiir Bauwesen, v. 11, p. 613-626.) 
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. 558-592.) 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. 142-146.) 
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. 1216-1218.) 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. 366-369.) 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. 778-781.) Reply to Saint-Venant'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. Gauthier-Villars. 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. 491-518.) 
Du Bois A J Upon a New Theory of the Retaining Wall. 14 diag. 
1879 {Journal, Franklin Inst., v. 108, p. 361-387.) 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. 101-106^) 
Dyrssen, L. Ermittlung von Futtermauerquerschnitten. 1 diag. 188b. 
{Zeitschriftfilr Bauwesen, v- 36, p. 389-392.) 



262 RETAINING WALLS 

Dyrssen, L. Ermittlung von Futtermauerquerschnitten mit gebogener 
Oder gebroohener vorderer Begrenzungslinie. 3 diag. 1886. (Zeit- 
schrift fur Bauwesen, v. 36, p. 127-130.) 

Eddy, Henry T. New Constructions in Graphical Statistics. 1877. {Van 
Nostrand's Engineering Magazine, v. 17, p. 1-10.) Contains section on 
"Retaining Walls and Abutments," p. 5-10. 

Enqesser, Fr. Geometrische Erddruck-Theorie. 1880. {Zeitschrift fur 
Bauwesen, v. 30, p. 189-210.) 

Everest, J. H. Treatise on Retaining Wall Design. 1911 {Canadian Engi- 
neer. V. 21, p. 192-193, 237, 264-265.) 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. 1151-1153.) 

Flamant, A. Note sur la Poussde des Terres. 1 pi. 1872. {Annales des 
Fonts et Chaussees, ser. 5, v. 4, p. 242-275.) Expounds Rankine's 
theory. 

Flamant, A. Note sur la Pouss6e des Terres. 1882. {Annates des Fonts 
et Chaussees, ser. 6, v. 3, p. 616-624.) 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. 477-532.) 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. 515-5'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. 63-72.) 

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. 98-231.) 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. 402-403.) 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. 
272-304.) Discussion, p. 305-321. Experimentally determines ratio of 
lateral to vertical pressure. Gives series of conclusions. See also edi- 
torial, "Lateral Earth Pressure," Engineering Record, v. 49, p. 633-634. 

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. 11-17.) Methods of design. 

Gould, E. Sherman. Retaining Walls. 2 diag. 1883. {Van Nostrand's 
Engineering Magazine, v. 28, p. 204-207.) Gives the theory of J. 
Dubosque. 

Graff, C. F. High Reinforced Concrete Retaining Wall Construction at 
Seattle, Wash. 1905. {Engineering News, v. 53, p. 262-264.) 

HiRSCHTHAL, M. Some Contradictory Retaining Wall Results. 1 diag. 
1912. {Engineering News, v. 67, p. 799-800.) 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. 99-120.) 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. 355-372.) 

Condensed. 1846. {Journal, Franklin Inst., v. 41, p. 73-79.) 

Howe, Malvbrd A. Retaining- Walls for Earth, Including the Theory of 
Earth-pressure 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. 78-86.) 

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. 899-912. 

Jacob, Arthur. On Retaining Walls. 27 diag. 1873. {Van Nostrand's 
Engineering Magazine, v. 9, p. 194-204.) 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. 421-422, 465-466, 478-479.) 

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. 463-472.) 
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. 233-235.) 
Klein, Albert. Die Form der Winkelstutzmauern aus Eisenbeton mit 
Ruoksioht auf Bodendruck und Reibung in der Fundamentfuge. 1909. 
{Beton und Eisen, v. 8, p. 384-387.) 
Kleitz. Determination de la Poussfe des Terres et Etablissement des 
Murs de Soutenement. 1884. {Annates des Fonts et Chaussees, ser. 2, 
V. 7, p. 233-256.) 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 Ingenieur-und Architekten-Vereines, v. 
31, p. 116-120.) 

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. 232-265.) 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, 
380-462.) Gives in tabulated form experiments performed and con- 
stants arrived at by Aud6, Domergue, and Saint-Guilhem, p. 397-400. 

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. 
199-203.) 

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. 
441-471.) 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. 633-634.) 
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. 5-18.) 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. 466-468.) 

L'EvBiLLE. De I'Emploi des Contre-forts. 1844. (Annales des Fonts et 
Chaussees, ser. 2 , v. 7, p. 208-232.) 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. 1456-1458.) 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. 98-114.) Considerable attention is given to design. 

Maconchy, G. C. Earth-pressures on Retaining Walls. 1898. {Engi- 
neenng, v. 66, p. 256-257, 484^185, 641-643.) 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. 1-23.) 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:M-iQ6.) 

See also editorial 'JSheet Piling and Earth Pressure," p. 528, and letter 
to editor, p. 608. 

Mehriman, Mansfield. Text-book 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. 197-199.) 

MoHLER, C. K. Tables for the Determination of Earth Pressures on Re- 
taining Walls. 1909. {Engineering News, v. 62, p. 588-589.) 

Muller-Beeslatj, Heinrich. Erddruck auf Stutzmauern. 159 p. 1906. 
"Literatur," p. 158-159. 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. 113-172.) 
Discusses foundation soil of retaining walls, and calculates design 
mathematically. 

Petterson, Harold A. Design of Retaining Walls. 1908. {Engineering 
Record, v. 57, p. 757-769, 777-778.) 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. 210-266, 
844-846.) Bibliography, p. 264-266. 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. 134-140.) Review of the author's 270-page 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. 482-483.) 

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. {Engineering-Contracting, 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 Architekten-Vereines, v. 62, 
p. 233-240; v. 63, p. 323-425.) Mathematical calculations. 

Rbbhann, Gbokg. Theorie des Erddruckes und der Futtermauern mit 
besonderer Rticksioht auf das Bauwesen. 1871. {Zeitschrift, Oester- 
reichischen Ingenieur-und Architekten-Vereines, 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. 386-417.) "Literatur,'' p. 387. 

Reppert, Charles M. Recent Retaining Wall Practice, City of Pitts- 
burgh. 1910. {Proceedings, Engrs. Soc. of Western Pennsylvania, 
V. 26, p. 316-354.) Discussion, p. 355-367. Givesatte ntion to calcu- 
lation of earth pressures as affecting design. 

Rbsal, Jean. Pouss6e des Torres. 2 v. 1903-1910. (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. {Engineering-Contracting, v. 42, p. 581-585.) 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. 
{Engineering-Contracting, v. 34, p. 115-117.) From Professional Mem- 
oirs, Corps of Engineers, U. S. Army. 

Saint-Venant, 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 so-called rational 
theory developed by Levy. 

Saint-Venant, db. PoussiSe des Terres. Comparaison de ses Evaluations 
au Moyen de la Consid6ration Rationnelle de I'Equilibre-limite, 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. 894-897.) 



SPECIFICATIONS 267 

Saint-Venant, 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. 217-235.) Report of a committee, giving a historical 
review of the works on earth pressure, and discussing in detail Maurice 
Levy's theory. 

Saint-Venant, 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. 717-724.) Based on Levy's theory. 

Saint-Venant, 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. 229-235, 281-286.) Development of Levy's theory. 

Saint-Venant, 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. 850-852.) Based on 
Boussinesq's works. 

Schaffeb. Erddruck und Stiitzwande. 1 diag., 1 pi. 1878. {Zeitschnft 
fur Bauwesen, v. 28, p. 527-548.) 

ScHMiTT, Edu ABD . Empirische Formeln zur Bestimmung der Starke der Fut- 
termauern. 1871. {Zeitschrift, Oesterreichisohen Ingenieur-und Archi- 
tekten-Vereines, v. 23, p. 336-338.) Mathematical calculations on the 
basis of Rebhann's tables. 

Sohwedleb, J. W. [Unterschnittene Futtermauern.] 1871. {Zeitschrift 
fur Bauwesen, v. 21, p. 280-282.) Discussion of the formula derived 
by Schwedler at a meeting of the Architekten-Verein zu Berlin. 

Seebek, D. C. Stability of Sea Walls. 15 diag. 1906. {Engineering 
News, V. 56, p. 198-200.) Gives method of design. 

Brief abstract. 1906. {Le Genie Civil, v. 50, p. 32.) 

Sheet-piling 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. 488-505.) 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. 9724-97^5.) 

SiNOEK, MAX. Fliessende ffinge. 1902. (ff -^''f ' ^f ™';'^te- 
Ingenieur- und Architekten-Vereines, v. 54, pt. 1, p. 190-196.) 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. 8-9.) 

Sinks, F. F. Design for Reinforced Concrete Retaining Wall. 1904. 
{Railroad Gazette, v. 37, p. 676-677.) Letter. 

Skibinski, Carl. Ueber Sttitzmauerquerschnitte. 1898. {Zeitschrift, 
Oesterreichischen Ingenieur- und Architekten-Vereines, v. 45, p. 666- 
670.) 

Skibinski, Karl. Theorie des Erddrucks auf Grund der neueren Versuchen. 
■ 1 diag., 1 pi. 1885. {Zeitschrift, Oesterreichischen Ingenieur- und 
Archi-tekten-Vereines, v. 37, p. 65-77.) 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. 182-197.) "Literatur," p. 196. 

Strukel, M. Beitrag zur Kenntniss des Erddruckes. 2 diag., 4 dr. 1888. 
(Zeitschrift, Oesterreichischen Ingenieur- und Architekten-Vereines, v. 
40, p. 119-125.) 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. 489-499.) 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. 481-494.) 

Thornton, William M. Retaining Walls. 7 diag. 1879. {Van Nos- 
trand' sC Engineering Magazine, v. 20, p. 313-318.) 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. 193-221.) 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. 
139-148, 189-198.) Mathematical calculation. 

Walmisley, a. T. Retaining Walls. 1907. {The Builder, London, v. 93, 
p. 647-648.) Discusses calculations of earth pressure, foundations, etc. 

Wbingartbn. [Die Theorie des Erddrucks.] 1 diag. 1870. {Zeitschrift 
fur Bauwesen, v. 20, p. 122-124.) Abstract of a paper read before the 
Architekten-Verein zu Berlin. 

Weston, W. E. Tables for Use in Determining Earth Pressure on Retain- 
ing Walls. 1911. {Engineering News, v. 65, p. 756-757.) 

Winkler, E. Neue Theorie des Erddruckes. 19 diag. 1871. {Zeit- 



SPECIFICATIONS 269 

schrift, Oesterreicliischen Ingenieur- und Architekten-Vereines, v. 23, 

p. 79-89, 117-122.) 
Woodbury, D. P. On the Horizontal Thrust of Embankments. 1 diag. 

1859. {Mathematical Monthly, v. 1, p. 175-177.) Mathematical paper. 
WooDBTTRT, D. P. Remarks on Barlow's Investigation of "the Pressure of 

Banks, and Dimensions of Revetments." 2 diag. 1845. (Journal, 

Franklin Inst., v. 40, p. 1-7.) 



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 

wing-wall, 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 live-load, 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 

Hetch-Hetchy 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 
Sub-surface 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 

Tie-rods, 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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