B.Sc., A.M.I.C.E., A.M.I.E.E., A.C.G.I.
CHIEF ENGINEER TO MESSRS. TROI.LOPE AND COLLS, LTD., LONDON
P. G. BOWIE,
ASSISTANT ENGINEER TO MESSRS. TKOLLOPE AND COLLS, LTD., LONDON
LONGMANS, GREEN & CO,
LONDON : EDWARD ARNOLD
[All rights reset ved}
THAT the technique of the art of designing reinforced concrete
structures cannot be mastered solely by the study of books
hardly needs to be emphasized, and no one realizes this fact
more fully than the Authors of the present book. Coupled
with study, practice under supervision is also essential. And
this supervision may be of two kinds. There may be the
constant vigilance of a master ready to indicate weak places,
places where material has been wasted, and to suggest other
designs which would be more generally suitable. The alter-
native is the supervision under which the pioneers conducted
their practice, that is, directly under Dame Nature, who still
has to be consulted from time to time. Weak places were
found by collapses of test pieces or structures, places where
material was wasted were indicated by a falling-off in clientele,
and a lack of success in competitive work.
Coupled, then, with practical work and experiment, upon
which more is said in Chap. XIII., it is hoped that this work
may prove helpful. A good deal of the matter is new, and
several important considerations are taken into account which
have hitherto been ignored, as far as the Authors are aware, in
published literature on the subject. Tor example, it has long
been realized that the bending moment for which beams should
be designed cannot adequately be written down by any rigid
formula, such as as suggested by certain reports on rein-
forced concrete, but depends on such considerations as the ratio
of live to dead load, the relative stiffness of beams and columns,
etc. ; yet the present treatise is perhaps the first to subject
these considerations to mathematical treatment and arrive at
simple formulse taking them into account. In the same way,
it has been realized by some that columns are subjected to
some bending action in addition to their direct load, owing
to unequal loading of the floors. Some allowance is made
for this in certain reports by specifying a lower stress in
columns than in beams. It is shown in this book that this
provision is in many cases utterly inadequate, while it is in
a few cases excessive, and the mathematical investigations
lead to comparatively simple formuloe, by which the stress
due to bending may be calculated for any particular case.
The question of resistance of beams to shear is also, among
others, dealt with in a way which has far greater theoretical
justification than commonly accepted methods.
But it is not claimed for the book that it obviates the
necessity of the specialist. Because of the very great number
of variables and the extraordinary choice of alternatives, the
design of reinforced concrete is a hundred times more difficult
than the design of steelwork, which commercial considerations
have standardized to such an extent that the selection, for
example, of a joist to do certain work may be made by
reference to a table. With a concrete beam, you may use
almost any depth and breadth you please, you may use a few
large or many small bars, and no two designers will provide
for shear, adhesion, etc., exactly alike. It is only, therefore,
the fundamental considerations governing design that can be
dealt with in a book, and we hope that our treatment will
bring into prominence the principles underlying the practical
design, which must remain more or less a compromise.
It is obvious that in practice many considerations must
be considered which cannot be dealt with in a book of this
kind, such as standardization of calculations and quantities,
arrangements of reinforcement, and the many similar questions
which are essential to efficient, rapid, and reliable work, and
are matters of importance to the engineering departments of
large firms. Apart, however, from such questions of organiza-
tion, an engineer will always require to be able to make
accurate calculations, and it is hoped that this book may
present the means to the solution of problems hitherto
The authors desire to record their indebtedness to Messrs.
Taylor and Thompson for permission to reproduce Table IV.,
p. 104; to the Council of the EI.B.A. for sanctioning the
inclusion of the second report on Eeinforced Concrete, as an
appendix to this treatise ; and to Prof. W. C. Unwin and W.
Dunn, Esq., for allowing Appendices VII. and VIII. of that
report to be given also.
P. G. BOWIE.
5, COLEMAN STREET, LONDON, E.G.,
PREFACE ......... v
NUMBERS OP PRINCIPAL FIGURES .... . xvii
LIST OF SYMBOLS xviii
Properties of Concrete ... 1
Primary Object of Reinforcement 1
Examples of Concrete Structures ........ 2
Advantages of Reinforced Concrete ....... 2
Patent Bars . 10
Concrete ............ 13
Proportions ........... 16
Increase of Strength with Age ........ 18
Consistency ........... 18
Effect of alternate Wetting and Drying Cement Mortar ... 19
Effect of Variation of Stress on the Strength of Concrete and Steel . 22
CALCULATION OF STRESSES UNDER KNOWN
FORCES AND MOMENTS
SIMPLE BENDING AND SIMPLE COMPRESSION
Assumptions ........ 25
Simple Bending 26
T-Bcams . .... 38
Double Reinforcement ... ..... 40
Simple Compression .......... 44
BENDING COMBINED WITH DIRECT FORCES
Bending and Tension .......... 4.7
Bending and Compression ..... . 61
ADHESION AND SHEAR
Adhesion ............ 71
Hooks and Bends ... .... .77
Shear in Beams ........... 79
Shear in Slabs of T-Beams . 90
THE DESIGN OF COLUMNS
STRENGTH OF COLUMNS
Difficulties in the Design ......... 94
Short Columns ........... 97
Value of Hooping and Binding . 97
Eccentric Loading ... 109
Long Columns .... 110
Splices in Columns . . . 113
THE DETERMINATION OP THE DIRECT LOADS
Two Spans 116
Three Spans . ... .... 117
Four or more Spans .......... 120
THE DETERMINATION OF THE ECCENTRIC STRESSES
Two Spans .... 123
Three Spans .... 136
Four or more Spans ......... 141
One Span ........... 143
Two Spans 151
Three or more Spans ......... 152
Value of the Eccentricity ........ 153
THE DESIGN OF BEAMS AND SLABS
Shearing Forces ........... 156
Bending Moments under Different Conditions of Loading . . 157
Variation of Size of Beams ........ 172
General Considerations ... . ... 179
Effect of Settlement of Supports ........ 181
Bent Beams . . . . . 183
Bending Moments ....... 188
Effect of the Deflection of the Supporting Beams ..... 193
Shearing Forces and Adhesion .... 198
APPLICATIONS AND GENEEAL NOTES
Reservoirs below Ground ......... 203
Circular Reservoirs .......... 205
Rectangular Reservoirs ; Bending Moments ...... 210
Water Towers 212
Wind Moments 213
Earth Pressures . .219
Types and Proportions ......... 223
Factor of Safety 224
General . 233
Variations ............ 233
Foundations ... . . .... 233
Column Loads ......... . 234
Floor Loads 234
Test Loads 235
Centering ......... . . 238
Inspection ............ 240
Wiring Bars 240
Leaving Holes ........... 240
Maximum Depth of Beams . . . . . . . . .241
Maximum Size of Columns .... .... 241
Plain Concrete under Footings ...... . 241
Test Blocks .241
Fire-resisting Constructions 242
QUANTITIES AND NOTES ON PRACTICAL APPLICATIONS
Quantities ............ 245
Additional Notes on Applications of Reinforced Concrete
Use of Brick and Concrete Supports ...... 247
Foundation Rafts. ......... 247
Concrete Chimneys . 249
Notes for Students and the Need for Experimental Study . . 253
Electrolytic Corrosion of Reinforcements ..... 254
THE SPECIALIST ENGINEER, AND THE
The Specialist Engineer ... ... .256
The Contractor . 258
BEING MATHEMATICAL ANALYSES OF BEAMS UNDER
VARIOUS CONDITIONS OF LOADING AND FIXING
NO. SUBJECT PAGE
LIST OF SYMBOLS . . . 262
1 Uniform load, slope at the ends given ...... 2G3
2 Load uniformly varying from zero at the ends to a maximum at the
centre, slope at the ends given. ...... 2G6
3 Concentrated load at midspan, slope at the ends given . . . 2G7
4 Two concentrated loads at the third points, slope at the ends given . 270
5 Uniform load, beam monolithic with columns .... 273
G Uniform load, heam not monolithic with columns .... 273
7 Uniform load, beam monolithic with columns .... 274
8 Uniform load, beam monolithic with columns .... 277
9 Any number of spans, uniform loads, spans and loads on different
spans not necessarily equal ....... 280
INTERIOR BAY OF BEAM OF MANY SPANS
UNIFORMLY DISTRIBUTED LOAD
10 Maximum moment at the centre ....... 283
11 Maximum moment at the supports ...... 284
LOAD UNIFORMLY VARYING FROM ZERO AT THE ENDS TO A MAXIMUM
AT THE CENTRE
12 Maximum moment at the centre 28G
13 Maximum moment at the supports ...... 287
MO. SUBJECT PAGE
CONCENTRATED LOAD AT MIDSPAN
14 Maximum moment at the centre 288
15 Maximum moment at the supports 289
TWO CONCENTRATED LOADS AT THIRD POINTS
10 Maximum moment at the centre ....... 291
17 Maximum moment at the supports ...... 292
18 BEAMS UNDER A LOAD UNIFORMLY VARYING FROM
ZERO AT ONE END TO A MAXIMUM AT THE
EFFECT OF SETTLEMENT OF SUPPORTS ON THE
CENTRE MOMENTS OF CONTINUOUS BEAMS
19 Many Spans 296
20 Two Spans 298
SECOND REPORT OF THE R.I.B.A. ON REINFORCED
Prefatory note . , 304
Fire resistance 305
Methods of calculation ......... 310
Shear reinforcement ......... 315
Pillars and pieces under direct thrust ...... 317
Pillars eccentrically loaded ........ 321
Long pillars ..... . 323
Appendix VII., Bach's theory of slabs .... . 324
Appendix VIII., strength of slabs ... . . 326
PRINCIPAL FIGURE NUMBERS
FIG. SUBJECT PAGE
2 Typical Reinforcement for a Continuous Beam .... 6
6 Variations of strength of concrete with age ..... 17
10 Relation between p and n 28
11 ^ % 30
12 p tj_ 32
13 p a,. 33
14 $1 i 35
16 p R/frZ 2 37
23 t lt p,,, ^~(m = 15) 50
24 t lt p,,, -~(m = 10) 51
32 * p, ^p (m =15) 65
70 Values of constant K in the equation M = KCEa for columns . . 125
82 Values of the bending moment on exterior columns . . .154
gg) Values of the bending moments due to dead and live loads on
beams of two, three, and four spans
99 Typical reinforcement to the junction of a beam to an outside
column . 171
LIST OF SYMBOLS
LENGTHS, DISTANCES, INTENSITY OP LOADS, STRESSES PER
UNIT OF AREA, AND CONSTANTS
a Arm of the couple formed by the compressive and tensile forces in
a t Ratio a/d.
b Breadth of a rectangular beam.
b r Breadth of the rib in a T-beam.
b s Effective breadth of the slab in a T-beam.
c Compressive stress intensity.
c. Compressive stress intensity on concrete.
c s . Compressive stress intensity on steel.
d Effective depth of a beam from top of beam to axis of tensile
d t Total depth.
d s Total depth of a slab in a T-beam.
d c Depth or distance of the centre of compressive reinforcement from the
8 Deflection of a beam.
e Eccentricity of any load.
/ Friction or adhesion between surfaces in units of force per unit of area.
I Effective length or span of a beam or arch.
in Modular ratio, i.e. the ratio between the elastic moduli of steel and
concrete = E S /E C .
n In beams : distance of the neutral axis from the compressed edge of a
n t The ratio n/d, i.e. the distance between the neutral axis and the
compressed edge divided by the effective depth of a beam.
p Percentage of steel, i.e. p = lOOr.
p Intensity of pressure per unit of length or area in any direction.
r Ratio of area of steel to area of concrete in single reinforced beams
t Tensile stress intensity.
LIST OF SYMBOLS xix
t t Ratio of stresses = tjc.
w Weight per unit of length.
Weight per unit of volume.
x Horizontal co-ordinate of a point.
y Vertical co-ordinate of a point.
AREAS, VOLUMES, MOMENTS, TOTAL LOADS, TOTAL FORGES,
A Total cross-sectional area of a pillar.
A L Cross-sectional area of longitudinal steel rods in a pillar.
AE Equivalent area.
Ac Area of compressive reinforcement in beams.
AT Area of tensile reinforcement in beams.
B Ratio I/I for beams, i.e. a measure of stiffness.
C Ratio I/I for columns, i.e. a measure of stiffness.
E Elastic modulus of any material.
EC Elastic modulus of concrete in compression.
Es Elastic modulus of steel.
F Total friction between any two surfaces.
I Inertia moment.
K A constant in the equation M KECa (see Appendix I. 1).
M Bending moment.
P Total pressure on a given area.
R Resistance moment of the internal stresses in a beam at a given
R Reaction of a beam on its support.
S Total shearing force across a section.
T Total tensile force.
W Weight or load.
ANGLES, CONSTANTS, AND MISCELLANEOUS
a Slope of a beam or column at the end produced by bending.
6 An angle.
j. Coefficient of friction.
BEINEORCED CONCRETE DESIGN
BEFORE going deeply into any part of the subject, a general
reconnaissance of the field to be covered will be made.
Concrete is a mixture of cement, sand, and stone, which is
wetted until it forms a plastic mass that will take the shape of
any mould in which it is placed and tamped. The usual
proportions are approximately 4 parts of stone, 2 parts of sand,
and 1 part of cement, measured by volume, though these
proportions are not to be adhered to in all cases. Such a
concrete will set under favourable conditions, and will gradually
harden with age, producing a mass resembling stone in many
of its properties.
The most important property of concrete which underlies
the desirability of reinforcing it at all, is the fact that its
tensile strength is only a fraction (approximately one-tenth) of
its compressive strength. Its tensile strength, besides being
low, is also very unreliable, since it may be entirely lost by a
sudden jar, by vibration, or by the contraction produced either
in setting and drying or during a fall of temperature. For
this reason concrete unreinforced can only be used under such
conditions that no tensile stresses are produced in it. This is a
very serious limitation, which precludes its use in any form of
beam or girder, and practically limits its application to arches,
piers, and such massive constructions as solid dams and retain-
The primary object of reinforcing concrete is to remove this
limitation, and the great success which has attended the
; J2V RKiNFORCED CONCRETE DESIGN [CHAP.
scientific achievement of this object has widened the field for
which concrete is suitable to such a remarkable extent that
there are now few engineering structures in which it may not,
with advantage, be substituted for steel or timber. The mere
enumeration of a few such examples, which could be largely
multiplied, will bear out this statement :
Large buildings of all kinds complete, including fioors,
beams, girders, stanchions, footings, and walls.
Bridges, whether of the arch or girder type.
Eetaining walls of very thin and economical section.
Water towers, including the tank, columns, bracing, etc.
Wharves and piers, including piles, columns, bracing,
Self-supporting chimneys of very light construction, in
which the necessary stability is not produced by the weight of
In all these and many other types of structures, reinforced
concrete has, in numerous cases, shown itself to possess a
combination of the following advantages over the material
which was formerly more usual :
Eesistance to fire.
Eesistance to rot and to the attack of pests, such as the
Teredo navalis in marine structures, and the white ant and such
vermin above ground. In some cases Homo humanus might justly
be included in this list, as he not infrequently plays havoc
with any removable timber.
Eesistance to air and water without requiring painting or
Increase of strength with age.
Eeduced first cost.
The authors only claim the above properties for the material
when proper percautions are taken in the design and execution,
and the claim of reduced first cost cannot be made in all cases.
It may be taken as granted, however, in this age of com-
mercialism, that where it has been adopted, it has always had
a reduced ultimate cost, when its absence of upkeep and other
properties are taken into account. It is not now the custom to
erect a structure in the best possible material as well as we
know how, simply for the joy of doing a thing supremely well
i.] GENERAL PRINCIPLES 3
this belonged to the old-world civilization of Greece, and is,
unfortunately, foreign to us.
To supply the requisite tensile strength to our material,
steel bars are embedded in the concrete where tensile stresses are
anticipated, for instance, at the underside of a freely supported
beam. If this is done consistently, we have, qualitatively, the
key to the greater portion of reinforced concrete design, and it
merely remains to calculate, from ordinary scientific principles,
the quantity of such reinforcement required. It should be noted,
however, that the mere embedding of steel bars in the concrete
would not produce a reliable composite material, except for two
extremely fortunate circumstances, which were probably not
realized by the pioneers of its use. The first is the fact that
concrete contracts slightly during setting in air, and, in contract-
ing round a steel bar, holds it tightly in such a way as to
prevent the steel from slipping even when no hooks or even
roughnesses exist on the bar. The second is the fact that steel
and concrete have practically identical coefficients of expansion,
and consequently a uniform change of temperature does not
involve temperature stresses in the two component materials.
When what may be termed the flange stresses due to bend-
ing moments have been guarded against by reinforcing as
suggested, it will be found that the safe load on a member of a
given size has been so largely increased that secondary stresses
due to shear rise to importance, and may produce failure unless
the concrete is reinforced with reference to them also. Consider,
for instance, a beam supported at the ends as in Tig. 1. If
unreinforced, fracture will occur as in 1 (a), by the concrete failing
under the tension flange stress due to the bending moment.
This may be prevented by adequately reinforcing as in 1 (>).
If the safe load is increased in this way up to a certain
point, the tensile stress developed across oblique planes near
the ends will cause failure, as in 1 (c). This may be prevented
by adequately reinforcing across such planes by bending up
some of the bars near the ends as 1 (d), by providing vertical
reinforcement generally called stirrups as at 1 (e), or by a
combination of these methods.
When a beam is continuous over several supports, as it
generally is, it may be designed in one of two ways :
REINFORCED CONCRETE DESIGN
(a) As a continuous beam, in which tension will exist at
the top of the beam near the supports, and will necessitate
suitable reinforcement to take up this tension.
n <*> w
FIG. 1. Stages in development of beam reinforcement.
(&) As a series of non-continuous beams, in which the
negative bending moment at the point of support is neglected
in the calculations.
i.] GENERAL PRINCIPLES 5
In the latter case, owing to the inability of concrete to resist
tensile stresses, a crack will be formed at the top of the beam at
the point of support.
The relative merit of these two methods depends on the
circumstances of any particular case, (b) being almost invariably
adopted in steelwork designs, and being desirable in reinforced
concrete when there is a possibility of the supports subsiding
unequally. The method (a) is generally more economical of
materials, and is therefore adopted whenever permissible, and
even in cases when the authors would consider the other method
Under case (a) it is found a suitable and economical
arrangement to combine in the bent-up bars the provision for
shear and for negative bending over the support, as is done in
Fig. 2, which shows a typical beam reinforced for continuity.
It must be noted at once that the design of a continuous beam
is not a simple matter, since many conditions of loading have to
When the central bay in Fig. 2 is fully loaded, the bars a
and I may be sufficient to provide for all tensile stresses due to
bending moments, but the case has also, in general, to be con-
sidered when the bays to the left and to the right are fully
loaded, and the central bay in Fig. 2 is unloaded. In that case
a little consideration will show that tension may occur at the
top of the beam instead of at the bottom, the bars c being
required to resist such stresses. As the amount of the negative
bending moment causing tension at the top of the beam near
midspan will be counteracted by the dead weight of the floor,
it is obvious that the design of the bars c is not simple. They
are advisable even when tension at the top is not anticipated
near midspan, since they are very convenient for fixing the
main bars and stirrups while concreting, and give a good con-
nection to the upper ends of the stirrups, which is extremely
important, as will be shown later (p. 86).
The design of continuous beams will be fully treated in
If we follow out rigidly the principle that reinforcement is
primarily required to increase the tensile strength of the
material, it is obvious that columns, axially loaded, do not
REINFORCED CONCRETE DESIGN
require reinforcing, and the authors are of opinion that cases
sometimes occur when the reinforcement in a column does not
add appreciably to its strength.
More generally, however, the column is rigidly connected to
beams, which may be unequally loaded in such a way as to load
the column eccentrically, and in such cases longitudinal steel
may be necessary to resist the tensile stresses produced at one
side. Obviously, too, a column with longitudinal reinforce-
ment is much better able to withstand shock or accidental side
It is also found that the columns in the lower tiers of a
building of many floors are frequently called upon to carry
heavy loads, and would require to be of large section if the
concrete were relied upon to carry all the load. In cases where
for satisfactory architectural treatment or from other consider-
ations such large columns would be objectionable, it becomes
necessary to reinforce them to render a smaller section capable
of carrying the load. This may be done by one of two
(a) By using a very high percentage of longitudinal steel.
FIG. 3. Reinforcement for columns.
This must, however, be bound together at short intervals by
adequate binding, or links, since otherwise the column fails by
8 REINFORCED CONCRETE DESIGN [CHAP.
the bars buckling individually and bursting the column. Fig. 3 (a)
shows such a column.
(b) By providing a spiral binding round the column, designed
with a view to preventing the lateral dilatation of the concrete
under vertical compression. It is found that such a spiral
increases very considerably the compressive stress which the