REINFORCED CONCRETE BUILDINGS
Published by the
McGraw-Hill JBooIk. Company
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FIGURE 20.
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Proposes to dispense with the use of ordinary joists and to
make use of wrought-iron tie-rods extending from wall to wall.
"The floor becomes one solid beam, having the tie-rods and hoop-
iron in combination with the concrete to take the tensile strain,
and the concrete to take the compressive action resulting from
the weight of the floor."
Lythgoe & Thornton, 1868, No. 640 (Figure 21).
FIGURE 21.
" The method of constructing floors with bars of J_-iron and
concrete as shown."
Johnson (Coignet) 1869, No. 884 (Figure 22).
An invention relating to the facing of concrete blocks with
cast-iron or steel protecting plates, to be used as street curbs,
etc.
Gedge (Monier) 1870, No. 1999 (Figure 23).
BASIC PATENTS FOR INVENTIONS
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" In short the iron is the skeleton and the cement its
covering."
Tall, 1871, No. 1001 (Figure 24).
Iron hooping, wirework, or netting are interlocked between
FIGURE 22.
FIGURE 23.
the lateral cross bars, and form a close lattice or basketwork.
Portland Cement stucco is applied.
Brannon, 1871, No. 2703 (Figure 25).
FIGURE 24.
" Wirework embedded in concrete, to give cohesive strength
against transverse and tensile strains."
Hyatt, 1871, No. 3124 (Figure 26).
24
REINFORCED CONCRETE BUILDINGS
" The peculiar construction of floor which I designate an
'all-beam' floor, composed of a number of separate tubes laid
side by side."
Turner, 1872, No. 1396.
On the iron beams "I strain my wire from the plates in the
walls; these wires are intended to supersede the use of floor
joists of wood, and will form beds for my concrete floors, and also
answer on the underside instead of laths for the plastered ceil-
ings, which work of plastering may be carried on at the same
time as the laying on of the floors in concrete."
Emmens, 1872, No. 2451 (Figure 27).
FIGURE 25.
FIGURE 27.
" The employment of sheets of corrugated iron as founda-
tion for roadways, paths, steps, and flooring."
Lish, 1873, No. 1621 (Figures 28, 29).
The drawing shows a sectional view of a floor and girder
of concrete with tension .rods embedded therein, as indicated by
the dotted lines.
Hyatt, 1873, No. 3684.
Asbestos combined with perforated, corrugated sheet metal
or with crimped sheet metal or upon a hollow grate bar system.
Coddington, 1873, No. 1004 (Figure 30).
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The figure shows a water pipe or tube, C being the cemented
material, E the interwoven metal.
FIGURE 28.
FIGURE 29.
Hyatt, 1873, No. 3381.
" The system or mode of forming cellular or honeycomb
structures by connecting together single cell blocks by means
FIGURE 30.
of tie-rods or crimped blades of metal, with or without addi-
tional straight tie-rods."
FIGURE 31.
Hyatt, 1874, No. 2550 (Figure 31).
"I form the tie in a way which gives it power to grip and
hold the foreign material in a manner and by a method which
26
REINFORCED CONCRETE BUILDINGS
brings the load and consequent strain upon the tie at the same
instant it is felt by the concrete or foreign material, by which
means the tensile and compressive forces act in harmony with
each other."
Hyatt, 1874, No. 1715 (Figure 32).
FIGURE 32.
" Making hollow metal beams of interlaced lattice or open-
work, as the holder of a tie-rod, to connect the same with con-
crete or equivalent material."
Edwards, 1891, No. 2941 (Figure 33).
1892, No. 1415 (Figure 34).
1894, No. 15,466 (Figure 35).
L
FIGURE 33.
i.--.
FIGURE 34.
Edwards' patents show a remarkable insight into the nature
of reinforced concrete construction. It is proposed to cast
BASIC PATENTS FOR INVENTIONS
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the slabs separately and set them when hard, owing to the great
cost of the centering; the bending up of the principal tension
rods is described at great length, and stress is laid upon the
benefit of many small rather than fewer but larger rods. The
FIGURE 35.
importance of preventing sliding of the reinforcement is shown,
and it is described how the beams may be pierced by openings
in much the same manner as done under the Visintini System.
The benefits as well as troubles arising from the fixing of the
ends of the beams into the walls are perfectly understood, and the
entire argument advanced is illustrated by tests (by Kirkaldy).
While in England the new construction made but scant
headway, a considerable activity took place in Germany, where
the Monier Patents were bought and exploited by G. A. Wayss,
and where M. Koenen advanced the first rational method of
calculation in 1886. The " straight line formula" was fully
discussed by Koenen in " Centralblatt der Bauvervaltung,"
May 14, 1902, and is to this day the commonly accepted
standard.
In Holland, the first ribbed floors were erected in 1886 in
connection with the Public Library in Amsterdam.
In France, it seems that Monier's first patent was taken out
in 1867, but it has been intimated that he had knowledge of
the earlier patent granted to Lambot, who had made a reinforced
concrete boat of small dimensions in 1855. This boat is said
to be in existence today. Monier 's efforts toward the intro-
duction of his inventions were not very successful, partly per-
haps because he failed to realize the necessity of placing the
reinforcement near the bottom; it is told that when Wayss
showed him slabs so reinforced, Monier severely criticized this
arrangement, and abruptly ended the argument by exclaiming,
"Who is the inventor, you or I?" 1 As a matter of fact, little
1 Suenson: Jaernbeton, P. 5.
28
REINFORCED CONCRETE BUILDINGS
was done in building construction until 1892, when Henne-
bique and Coignet took the reinforced concrete construction
up with great success, each introducing his own system.
In the United States, the first indication of anything ap-
proaching reinforced concrete may be found in a patent granted
to P. Summer, 1844, No. 3566 (Figure 36), for a metal lathing,
which was still further improved by J. B. Cornell in 1859, No.
22,939 (Figure 37). At this early date, a number of patents
FIGURE 36.
for cement pipes were granted, as to R. B. Stevenson, 1854,
No. 11,814, for a combination of a pipe of sheet-metal and an
exterior coating of hydraulic-cement mortar of " requisite thick-
ness for strength." In the Wyckoff patent, No. 32,100, of 1861,
the interior pipe is of wood wound with wire of iron or other
metal; in the Knight Patent, No. 32,298, of the same year, a metal
tube is disposed " intermediate between the inner and outer
surfaces" of a cement pipe. In 1868, A. P. Stephens took out a
patent, No. 78,336, on a similar pipe, in which the strengthening
tube was made of corrugated iron; in 1872, Patent No. 127,438,
the tube was changed to a spirally formed sheet metal tube,
and in the same year J. A. Middleton, Patent No. 133,875, pro-
posed to strengthen his cement pipes by a layer of wirecloth
embedded in the cement, thus combining what we now consider
the essential elements of a reinforced concrete pipe (Figure 38).
The first reinforced concrete wall-patent appears to be one
BASIC PATENTS FOR INVENTIONS
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granted to S. T. Fowler, in 1860, No. 28,069, where the concrete
wall is to be strengthened with vertical and horizontal timbers,
to be buried in the concrete; a more rational construction is
FIGURE 38.
proposed in 1862, No. 37,134, by G. H. Johnson, for grain-bins:
" a new construction formed of brick- work tied together by
plates and rods of iron." In 1869, No. 87,569, G. H. Johnson
FIGURE
improved this construction, using " horizontal annular tension-
bars . . . the ends of each bar being so united as that it shall
form an endless, unbroken band ... in the combination . . .
with . . . vertical connecting-rods so as to form a metallic
30
REINFORCED CONCRETE BUILDINGS
frame within the walls of the structure." This invention (Fig-
ure 39) was not the only important improvement of that day;
in 1868, C. Williams, No. 75,098, invented the metal lattice-
reinforcement for concrete walls. The lattice-work was built
up by riveting the slats together (Figure 40).
The first use of concrete in columns must be conceded to
W. H. Wood who, in 1862, No. 36,747, patented an improvement
in piers and bridges. The invention consists in the use of hol-
low cast-iron columns filled with concrete or cement, and sup-
ported on wooden spiles below the surface of the bed of the
river. The first ceiling was proposed by J. Gilbert, 1867, No.
64,659. This patent shows corrugated iron plates filled with
FIGURE 40.
concrete, the concrete to extend an inch or so above the top of
the corrugation (Figure 41). His solution of the problem
" self-centering reinforcement" is not very inferior to those
proposed by more recent inventors. Thus we see that around
the year 1870 the combination of masonry of various kinds
with a strengthening metal work was quite well known. The
patent, No. 88,547, granted to F. Coignet, a Frenchman, in
1869, states the general principles very clearly: " In the body of
artificial stones": "skeletons or metallic framework, linked or
arranged so as to strengthen the same." This is the whole
science of reinforced concrete construction in few words. As
an example, he proposes to use a cylindrical web of small rod-
iron or wire in combination with a cement envelope, for the pur-
pose of resisting the interior pressure in pipes, as well as T- or
BASIC PATENTS FOR INVENTIONS
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L-irons for other purposes. The series of patents granted to
Coignet in 1869 deserve more than usual attention, as they
contain much good advice of value to engineers; they are
No. 88,545, 88,546, 88,547, 88,548, and 88,549.
FIGURE 41.
The brick arch with abutment-shoe and tension bar between
abutments was invented by C. Henderson, in 1871, No. 113,881
(Figure 42); the brick arch reinforced on the cantilever prin-
ciple was invented by F. Alsip, No. 120,608, in the same year.
It is not clear from the description whether Alsip really con-
FIGURE 42.
sidered his invention as a cantilever construction, but the fact
remains that all the essential elements of a cantilever are pres-
ent in this patent. A very interesting patent No. 122,498 is the
one granted to W. H. Smith, in 1872, for a concrete pavement.
On soft ground, the arched pavement is intended to be self-
supporting. Tie-rods are then carried under the pavement from
curb to curb, or "chords may be embedded in the composition
to operate in lieu of abutments to the arch." In the drawing,
the tension rod is shown provided with a large button on the
end, evidently for the purpose of preventing slipping of the
bar (Figure 43).
32
REINFORCED CONCRETE BUILDINGS
The patent issued to Sisson and Wetmore, in 1872, No.
124,453 (Figure 44), shows "a combination of trussed and un-
trussed frames of light bar-iron to form skeleton wall-posts,
girders, etc., in combination with a filling of beton or other
suitable concrete, to be poured in a state more or less liquid.
Our object is to have the beton and iron frames furnish mutual
support and protection to each other." Considered as a beam,
FIGURE 44.
the wall-post of this patent exhibits many of the essential fea-
tures of present day practice: the top bar extending from one
span to another, the trussed bar bent up over the support, the
horizontal lacing of the verticals and the vertical lacing of the
horizontals, etc.
But generally speaking, the reinforced brick-arch continues
to hold the interest of the inventors. In 1872, P. H. Jackson
received a patent, No. 126,396 (Figure 45), for a peculiar con-
struction of abutment-casting to be used in connection with
reinforced arches, and in 1873, No. 137,345, N. Cheney proposes
BASIC PATENTS FOR INVENTIONS
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to make the tension reinforcement of light wires placed close
together and interwoven with cross-wires, to serve the addi-
tional purpose of a metallic lathing. The earthquake-proof
house invented by D. L. Emerson, in 1873, No. 137,833, calls
FIGURE 45.
for vertical rods or plates in the walls, and anchors passing
through them, the plates and anchors being connected with
strap iron. In the same, year J. W. Basset, No. 138,118 (Fig-
ure 46) shows a construction of individual plaster slabs with a
metallic trellis work within, the ends of which extend beyond
the block, for the purpose of locking the various blocks together.
While not strictly within the scope of this paper, attention
is called to the patent, No. 172,641, granted to O. C. Matthews,
n
FIGURE 46.
in 1876, for a foundation, in which piles are driven and again
withdrawn and the holes filled with concrete (Figure 47).
In 1878, T. Hyatt, No. 206,112, ended the " period of dis-
covery " and put the theory of reinforced concrete construction
on a rational basis, and at the same time received a patent of
remarkably broad scope, covering practically the entire field
of reinforced concrete and masonry construction. The general
purport of this invention is set forth in a volume entitled "An
account of some experiments with Portland Cement concrete,
combined with iron," of which a copy was deposited in the
library of the Patent Office, but which was otherwise designed
for private circulation. Hyatt appears to be the first to state
specifically that the steel must be able to resist sufficient tensile
34
REINFORCED CONCRETE BUILDINGS
stress to balance the compressive stresses on the concrete, that
all metal may be dispensed with save the tension rod only, that
both baked bricks and concrete possess in themselves cohesive
power and strength sufficient to perform the functions ordina-
rily performed by the metallic web. He realizes the value of
deformed bars and says: " I prefer to use metal specially rolled
for the purposes, with bosses or raised portions formed upon
FIGURE 47.
FIGURE 48.
the flat faces of the metal. When I make use of common bar
or hoop iron, I stud the slips with pins; or I make use of several
blades threaded upon wires, as represented by Figure 1." In
the book mentioned above, he laid down the results of his ex-
periments which led him to bend some of the bars up, and also to
use a rigidly attached separate " shear member." The analysis
is very complete, both in his book and in his patent speci-
fication. He reinforces his columns with longitudinals or hor-
izontal hoops, as the case may require, or both. He says: "In
BASIC PATENTS FOR INVENTIONS
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constructing the columns or piers wholly of concrete, I make the
structure solid, the concrete then bearing the load, and, giving
way under compression, would naturally incline to yield in the
first place, not from absolute crush of the materials, but from
want of sufficient tensile resistance at the circumference of the
column. But this tendency being resisted by the circular
ties, such a concrete could give way only by the crush of its
particles." In short, the whole theory of hooped columns.
The only difference is that Considere prefers the use of spirally
wound reinforcement, while Hyatt uses the individual bands
(Figure 48).
To what extent Hyatt was familiar with Pasley's tests, if
at all, we do not know; in his book of 1877 he gives a brief ac-
count of the history of fireproof construction, but gives no ref-
erence whatever to the tests just mentioned. It appears that
he had a test made in September, 1855, in New York, under
the general supervision of Mr. R. G. Hatfield; the beam was
about 9" square, and had a tie-rod passing through holes made
for the purpose in the bottoms of the bricks. More important
tests were made by Kirkaldy in London, from 1874 to 1877, on
beams made by Hyatt.
The Period of Improvement. Broadly speaking, the Hyatt
Patent, No. 206,112, shows and describes everything necessary
FIGURE 49.
for the practical use of reinforced concrete, and the patents
of the following period are therefore mainly for improvements,
many of which are due to Hyatt. Most interesting is the one
granted in 1883, No. 290,886, for a concrete floor, showing not
only transverse arches between the ribs, but also the use of web
reinforcement in a continuous sheet along the center of the
beam. In 1881 a patent, No. 237,471, was granted to S. Bis-
sell (Figure 49) for an arch-bridge, showing diagonal straight
reinforcement within the masonry, the object being to construct
36 REINFORCED CONCRETE BUILDINGS
"an arch of limited span without causing any horizontal thrust
upon' the abutments." The Cubbins patent of 1883, No.
285,801, shows a circular cistern cover "of artificial stone,
having a metallic band or tire" (Figure 50) or "consisting of
FIGURE 50.
a concavo-convex or arched disk . . . inclosed by a metallic
band or tire." This appears to be the first slab with " cir-
cular reinforcement." "Expanded metal" was patented,
No. 297,382, in 1884, by J. F. Golding: "metallic screening
formed of slashed and stretched metal." The particular use
to which the invention was to be put is not specified, and at
first it was used exclusively as a metal lath. Its use as
reinforcement for structural concrete is of much later date
(Figure 51).
FIGURE 51.
A number of interesting patents are granted at various
dates to P. H. Jackson. The first, No. 302,338, in 1884, is not
of interest in this connection; it shows principally the usual
tie-rod construction in a brick arch. But the following year,
1885, he took out a patent, No. 314,677, showing, for the first
time, the bent-up or "trussed" arrangement of reinforcement
(Figure 52); the bars are carried to the support where they are
anchored by means of nuts. The concrete and its reinforcement
rest upon corrugated iron plates, and the bars may be secured
BASIC PATENTS FOR INVENTIONS
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or not at intervals to the bottom of the corrugated plates.
Another patent, No. 320,066, of the same year, shows the rein-
forcement continued into the adjacent bay and there hooked
FIGURE 52.
over the tops of the I-beams (Figure 53), which here have the
function of the main girders. The patent, No. 339,296, of 1886,
specifies an expansion joint in the construction of a reinforced
concrete arch; evidently the troubles caused by expansion and
shrinkage were well known at this early date. Two patents,
Nos. 366,839 and 366,840, were taken out in 1887 for " series
FIGURE 53.
of arches composed of concrete, and a longitudinal tie on which
the footings of the said arches are supported and to which they
are fastened," and a construction of arches with longitudinal
reinforcement near the bottom; these arches rest on one side
on the front girder of the building, on the other side upon the
area wall. Also from that year date the following patents:
two, Nos. 367,343 and 370,625, showing the application of dove-
tailed corrugated plates filled with concrete (Figure 54), and
FIGURE 54.
three, Nos. 371,843, 371,844, and ?71,845, showing the use of
I-beam reinforcement in the bottom of the beam, as well as
38
REINFORCED CONCRETE BUILDINGS
compression-reinforcement in the top (Figure 55). The reis-
sued, RIO, 921, and the original patent, No. 375,999, issued in
1888, may be noted in passing.
When we consider the state of the art as it appears from
the patents mentioned above, the Monier patent of 1884, No.
302,664 (Figure 56), cannot be called much of an improvement.
FIGURE 55.
FIGURE 56.
Nevertheless, the name Monier was for many years synonymous
with " reinforced concrete," at least in Europe, where the Mon-
ier patents were bought and greatly developed by German
engineers. "My invention," he says, "relates to the use and
sale of integral elements of construction of metal and concrete
or mortar combined, the mortar forming the covering for a
metal skeleton. This skeleton is composed of longitudinal
bars or rods and transverse ribs, secured together by metal
ligatures." The Monier patent, No. 486,535, of 1892, is practi-
FlGURE 57.
cally nothing but a series of special designs based upon this same
principle, and contains little new material. Yet a great industry
was based both here and in Europe upon the Monier patents.
The Ransome patents have been described in an earlier
chapter and are not referred to here.
The "trussed" arrangement of the steel was, as stated
above, invented by Jackson in 1885. The Gustavino patent,
No. 336,048, of 1886 (Figure 57) shows the same feature, as
BASIC PATENTS FOR INVENTIONS
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well as the rod with a continuous curve between supports. In
addition to this tie-rod which extends from wall to wall, " I
may in practice use a straight tie-rod extending between wall
and wall above the arch." The same year, 1886, saw the origin
of another new type of construction which stands on the border
between reinforced concrete and plaster work. The Rabitz
construction, No. 339,211 (Figure 58), calls for a metallic skele-
FIGURE 58.
ton frame of vertical rods and a reticulated metallic netting,
in combination with a suitable coating of cement mortar or sim-
ilar material. In the patent issued to P. M. Bruner, No. 356,703,
in 1887, something approaching the U-bar (Figure 59) is shown;
FIGURE 59.
although the construction would not be classed as reinforced
concrete at this present time, the rods being disposed princi-
pally on the compression side, from which rods transverse ties
hang down in the beam. A telegraph pole was invented by
D. Wilson, No. 374,103, in 1887; it was to be composed of a skel-
eton frame having rods and horizontal hoops, and a coating or
body of cement inclosing the frame. The same idea was pat-
ented, No. 411,360, in 1889, by O. A. Stempel, who claims a
post, rail-tie, or beam, composed of "a metal frame, the filling
and inclosure of imperishable material that protects said frame
40 REINFORCED CONCRETE BUILDINGS
from the inroads of moisture and rust, and said frame arranged
to protect said structure from breakage." The drawing looks
somewhat like what an engineer would prepare for a column at
this time (Figure 60).
The patents granted to M. F. McCarthy show
again "the combination (with an I-beam supporting
the slab) of the wire strands extending over and
drooped between the same, and the concrete filling
wherein said beams and strands are embedded." This
quotation is from the patent issued in 1891, No.
455,687 (Figure 61); the four patents, Nos. 520,489,
520,490, 520,491, and 520,492, issued in 1894, show
various combinations and variations of the same prin-
ciple. The patent issued to P. Cottancin, No. 459,944,
in 1891, is for a strengthening web " characterized
by the union in a reticulated fabric of a warp and a
weft, each composed of a wire, band, or bar bent on
itself into a sinuous or like shape." This patent
forms the base for a large industry especially in
France. The J. Melan patent, No. 505,054, of 1893
(Figure 62), claims "a vault or arch consisting of
abutments, beams, or girders, arched ribs rigidly con-
nected with said abutments, beams, or girders, and
a filling of concrete or the like between said ribs."
A number of arch-bridges have been constructed
under this patent. A. L. Johnson patented, No.
550,177 (1895), a construction of floors much used
at one time in the West, comprising mainly I-beams
with suspension straps fastened at the tops of the
beams and drooping between the beams; the straps
are flat and support the concrete rib of the beam
(Figure 63), upon which in turn rests the concrete
slab. Another important arch-patent, No. 583,464,
was granted to F. von Emperger, in 1897, for an im-
provement in the Melan patent described above; it
FIGURE 60. consists mainly in using two ribs instead of one, each
rib being placed near one surface of the concrete.
Secondary members connect the top and bottom ribs (Fig-
ure 64).
Recent Patents. The idea of molding reinforced concrete
BASIC PATENTS FOR INVENTIONS
41
members separately and afterwards erecting them in place
appears to be almost as old as the art itself, and a number of
FIGURE 61.
the patents mentioned above refer to this possibility without
going much into the details. In 1898, a patent, No. 606,696,
FIGURE 62.
was issued to G. B. Waite for a beam construction (Figure 65),
the sole object of which is to provide members adapted to be
FIGURE 63.
molded in advance and erected in place after hardening. The
individual sections are made of I-shape and reinforced in top
FIGURE 64.
and bottom, or in the bottom only; " shear" members of vari-
ous forms are used in the beam-webs. The De Man twisted
FIGURE 65.
bar was patented, No. 606,988, in the same year; it consists
42
REINFORCED CONCRETE BUILDINGS
of "a thin flat bar having twists formed therein at intervals"
(Figure 66).
The patents granted to F. Hennebique, in 1898, are three in
number. The first, No. 611,907, shows the now almost univer-
FIGURE 66.
sally used combination of open, U-formed shear members with
horizontal and trussed main reinforcement, with the main bars
extending into the adjacent span (Figure 67). While the
FIGURE 67.
authorities seem to disagree in regard to the value of the pro-
tection afforded by this patent, there is not the slightest reason
to doubt that this construction has been of the greatest benefit
to the art. The second Hennebique patent, No. 611,908, is
for a system of separately molded members, claiming in sub-
stance a combination of joists and " a plurality of slabs having
projecting cores embedded in said joists " (Figure 68); the word
FIGURE 68.
core means here the reinforcing bar, and the slabs are placed
with their ends resting upon the side-forms for the joists, so
BASIC PATENTS FOR INVENTIONS
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that, when concrete is poured in the joist-molds, the project-
ing ends are embedded in the concrete. The third patent,
No. 611,909, is for a pile of reinforced concrete having grooves in
two faces, so that a tight cofferdam may be made by using the
piles for sheet piling, and filling in the grooves with grout. The
structures erected under the Hennebique patents are numbered
by the hundreds in any one of the several civilized countries.
The patent, No. 617,615, issued to E. Thacher, in 1899 (Fig-
ure 69), for an arch construction, claims the combination of the
FIGURE 69.
concrete arch with its abutments, and reinforcing bars in pairs,
one bar near the intrados, and one near the extrados, the two
bars of each pair to be above one another, either both or only
one of these bars to extend well into the abutment, and, in par-
ticular, "each bar of a pair to be independent of the other."
A comparison with the Melan and v. Emperger patents is of
interest, as the bars in the v. Emperger patent extend into the
abutments and are placed one above the other. In the same
year, 1899, a patent (Figure 70), No. 634,986, was granted to
FIGURE 70.
A. Matrai for a system of wire reinforcement embodying many
interesting features. One object of the construction is to unload
as far as possible the middle of the supporting beam or girder,
and these again are reinforced with a number of suspension
cables or wires. This construction is in considerable favor in
Europe. In 1900, a patent, No. 654,683, was issued to I. A.
Shaler, for a construction embodying the use of longitudinal and
transverse rods, the latter welded to the main bars at intervals,
44
REINFORCED CONCRETE BUILDINGS
and in the same year, L. G. Hallberg had a patent, No. 659,967,
issued for a foundation built on the principle of " circular rein-
forcement " (Figure 71) in combination with radial bars. The
Wayss patent, No. 673,310 - 72) of 1901, is of interest, on
account of the rigidly attach ear members and other fea-
tures, the purpose being to ^otain similar advantages as out-
lined for the Hennebique patent without infringing the same;
FIGURE 72.
the construction is dissimilar to Hennebique in the particular
arrangement of the parts. The well-known Thacher bar was
patented in 1902, No. 691,416 (Figure 73), and in the same year
BASIC PATENTS FOR INVENTIONS
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a patent, No. 709,794 (Figure 74), was granted to W. C. Farm-
ley for a concrete arch construction, in which the steel is so
arranged as to make the same bar pass from the tension region
FIGURE 73.
near the intrados to the tension region near the extrados, etc.
The Visintini patent, No. 735,920, of 1903, shows the peculiar
type of construction known under that name; instead of the
FIGURE 74.
ordinary solid beam, a lattice-girder of reinforced concrete is
used. The top and bottom flanges are reinforced with longi-
tudinal bars, and the cross-bars are embedded in the concrete
FIGURE 75.
work of the lattices (Figure 75). The Visintini beam has
been used but little in this country, but abroad a large number
of structures have been erected under this patent. In 1903,
FIGURE 76.
the first Kahn patent, No. 736,602, was issued, to be followed
by many more (Figure 76). The principal features are well
46
REINFORCED CONCRETE BUILDINGS
known: The rigidly attached secondary members are manu-
factured in one piece with the main tension rod, then sheared
loose from the main body along the greater part of the length
FIGURE 77.
of the rod and bent up as desired. The Weber chimney-con-
struction was patented in 1903, No. 748,242; the lower portion
of the chimney is provided with a circumferential air-space
open at its base to the outer air and leading at its upper end into
the chimney flue at the base of the upper single flue (Figure 77).
A. Considere took out a patent, No. 752,523, for his well-
BASIC PATENTS FOR INVENTIONS 47
known column construction, claiming "a solid concrete core
with independent helicoidal coils of metal surrounding said
core, and arranged very close together," and also the combina-
tion of these elements with separate longitudinal rods, in 1904
(Figure 78). With this patent we may consider the period of
invention as coming to an end. A very large number of patents
have been granted since, mostly for slight improvements, and
an enumeration of all these details would be very tedious and
without serious importance, although several patents of the
greatest interest may be found in this great mass of dead
material.
PART II
RATIONAL DESIGN OF REINFORCED
CONCRETE BUILDINGS
BY ALEXIS SAURBREY
CHAPTER III
INTRODUCTION
1. EXPERIENCE teaches that concrete beams may be greatly
strengthened by introducing a comparatively small amount of
steel within the concrete, according to certain principles of which
the following is a discussion. This combination of concrete and
steel is called Reinforced Concrete; the essential peculiarity of
reinforced concrete structures is that both the concrete and the
steel, if alone, would be grossly inadequate for the load which
they will carry when combined; the load carrying capacity is
not the sum of the individual capacities of the concrete and the
steel. This general rule is not without exception, if structures
like the ordinary reinforced concrete column are included;
strictly speaking, only the hooped column is entitled to be clas-
sified as reinforced concrete, because in that case a small amount
of steel added to the concrete changes the structural properties
of the column entirely.
2. The stresses in a reinforced concrete structure are neces-
sarily complicated. Not only is the steel entirely dissimilar
in nature to the concrete which it reinforces, but the concrete
itself is not homogeneous in the strictest sense of the word.
Yet two cubes of large size, cut from different parts of the beam,
must be assumed to be theoretically alike; we make there-
fore the necessary and justified assumption that the lack of homo-
geneity of the concrete is of second order as compared with that
of the structure as a totality: necessary, because otherwise we
cannot advance any theory; justified, because the differences
between the nature of steel and concrete are sufficiently large
to overshadow completely the small differences which un-
doubtedly exist within the concrete itself.
3. Generally, the properties of reinforced concrete are known
when the properties of the two materials are known; there is
51
52 REINFORCED CONCRETE BUILDINGS
no reason for believing that the properties of either material
are changed in any way by the presence of the other. It is,
however, necessary to expand the limits of our research when
dealing with a combination of two materials, because the prop-
erties of the combination depend primarily upon the ability of
the two materials to co-operate, and only in second line upon
their individual properties such as strength, elasticity, etc.
This co-operative ability is of a somewhat obscure nature;
without making any attempt of explaining it, we must admit
its existence. In the following it is referred to as the "bond"
or the "adhesion." When this bond is broken the structure
fails.
4. The purpose of design is to produce not only a structure
of adequate strength, but one of equal strength in its several
parts. With consistent formulas for the various elements, the
allowable stresses should therefore be the same for all elements
of the structure considered. Experience shows, however, that
the difficulties to be overcome in the erection are different for
different parts; we can readily see that a local deposit of bad
concrete as large as a hand will affect a 10" column and a 6"
floor slab in dissimilar ways. This is the reason for variable
allowable stresses in any case the purpose of fixing certain
maximum stresses is to insure an ample factor of safety. For-
tunately the investigation of stresses in a given beam is very
much simpler under moderate loads than near ultimate failure;
the coefficient of elasticity for steel E a is a constant, and that
for concrete E c varies but slightly. For practical purposes
the ratio E S /E C r is assumed to be a constant up to the limit
of the allowable stresses. Within this same limit we assume
sections plane before the load was put on to remain plane under
load, and we assume proportionality between stress and deform-
ation. The tensile strength of the concrete is entirely disre-
garded. None of these assumptions can be called absolutely
correct; they are, however, no more inaccurate than any other
set of assumptions which we would be able to suggest in our
present state of knowledge; moreover, they are the simplest
possible.
5. As the tensile strength of concrete is much less than its
compressive strength, the principle is to utilize the available
compressive resistance and use steel bars to carry the tension.
INTRODUCTION 53
Sometimes steel is also used in compression, although with less
success, the object being to limit the size of the columns and to
fortify them against excentric loads. We shall see later that
it is possible to construct a column in which the steel is stressed
in tension (Article 12).
6. In any kind of concrete structure the embedded steel
has a tendency to displacement in its own longitudinal direc-
tion under load. The value of the steel as reinforcement de-
pends upon its ability to withstand any forces tending to either
push or pull it out; reinforced concrete is an impossibility
without adhesion between steel and concrete, and destruction
of the bond or adhesion means failure. The law governing
adhesion is therefore the foundation of all theoretical study of
reinforced concrete.
CHAPTER IV
ADHESION
7. THE adhesion is measured in Ibs. per square inch of em-
bedded surface of the rod; its value is different for pulling and
pushing tests. As the latter is somewhat higher it is sufficient to
investigate the laws governing the pulling resistance and apply
these laws to the pushing resistance also. The mathematical
analysis of the bond stresses is impossible with the material on
hand; even the test-data are meager and often contradictory.
We know, however, that the following statements are approxi-
mately true, so that an embedded rod pulls out of the concrete
block :
(a) when the stress in the steel reaches the elastic limit of
the steel.
(6) when the tensile resistance of the concrete, in a lateral
direction, is reached, because the block splits.
(c) when, instead of splitting, the concrete around the rod
expands sufficiently to let the irregularities of the rod
pass through.
(d) when the adhesion is destroyed.
Obviously, then, the designer must keep the steel stress well below
the elastic limit, allowing for this and other reasons an ample
factor of safety, while, at the same time, the concrete must be
strong enough to meet the demands made upon it. Hence the
diameter of the concrete block, or the thickness of the piece, is a
very important factor, but unfortunately nothing is known in
regard to the minimum allowable diameter, except that it is
greater for deformed bars than for plain round or square rods.
We can readily see that both the tensile strength and the coeffi-
cient of elasticity of the concrete has great influence upon the
minimum allowable diameter; with a well-proportioned mortar
and a mixture of say 1 :2:3J, we may perhaps suggest a diameter
of concrete equal to ten diameters of the embedded steel as
' 54
ADHESION 55
reasonably safe. In floor construction the bars usually find
their ultimate anchorage in much larger bodies, the slab bars
passing through the beams, the beam bars through the girders,
and finally the girder bars through the columns. In all these
cases the concrete is reinforced in a direction transverse to the
direction of the pull, and the expansion in a lateral direction is
thus partially or entirely prevented.
8. In the beam theory to be outlined below, great importance
is attached to the length of embedment beyond the supports of the
beam, in fact, this length- represents the ultimate reserve of
strength of the beam. It is usually considered good practice
to imbed the bent bars from twenty-four to thirty-six diameters
i
6 Diameters
Y
J, Diam..
V 1
FIGURE
-f
79.
beyond the support, using the lower figure for deformed bars
stressed to 16,000 Ibs./sq. inch, and the higher figure for deformed
bars stressed to 20,000 Ibs./ sq. inch. For plain bars, an addi-
tional hook is made on the end of the bar, equal in length to six
diameters. In many cases the length of embedment here recom-
mended cannot be obtained for the reason that there is no adjacent
concrete into which the bars may be extended, as, for instance,
in the case of a beam finding its bearing in an outside brick wall.
The bars are then hooked, and the length of embedment calcu-
lated from the center of the seat to the end of the bar, including
the curved end of the hook. Square hooks must be avoided, a
gentle curve of, say, six times the radius of the bar is much more
effective, and the more so the greater the radius of the bent
(Figure 79).
9. The diameter referred to in the preceding paragraph is
not the diameter of each individual rod or bar, unless the rods be
spaced so far apart that each will pull out individually, leaving
the concrete intact between. The diameter is that of a circle or
other curved line in which all the rods may be enclosed if laid
closely together. It follows that it is good practice to spread the
rods out as much as possible; in a beam this is easily obtained by
56 REINFORCED CONCRETE BUILDINGS
bending some of the bars up over the support, as is also done for
other important reasons. It is a common but inexcusable
mistake to use a number of small diameter rods bunched together;
it is almost impossible to concrete such beams properly, and
the fallacy of the argument leading to such construction should
be evident.
CHAPTER V
COMPRESSION AND LATERAL EXPANSION
10. WITH few exceptions, materials submitted to deformation
in one direction undergo deformations in all other directions.
If the principal deformation is a shortening, the lateral deforma-
tion is a swelling, which must be taken as evidence of certain
interior stresses in the body in a direction normal to that of the
principal stress. These transverse stresses are of the greatest
importance for materials like reinforced concrete, because, if not
restrained, they bring about the premature failure of the con-
crete, while, if restrained, they may be used to increase the
strength of the structure. Thus, as pointed out above, the
transverse swelling affects the bond of an embedded rod; if
restrained (by surrounding the bar with a coil of large diameter) ,
the value of the bond may be increased as much as fifty per cent,
or more. Even a loose stirrup circling the tension rod at the
bottom of a beam increases the sliding resistance of the rod, so that
a rod, covered at the most with two inches of concrete, may have
the same sliding resistance as one embedded in a large body of
concrete. Similarly, the Ransome Coil Coupling may be used
with good results when splicing rods, although the rods should
always be made in one continuous piece whenever it is practically
possible. The coupling is made simply of a coiled piece of very
heavy wire or a light bar surrounding the splice for its entire
length, which should be equal to at least fifty diameters of the
rods to be spliced (Figure 14).
11. In figure 80 a short block is shown loaded and compressed
in one direction, thereby shortening the length of the vertical
side from aa to bb. We notice now that the block expands in a
horizontal direction, the diameter increasing from cc to dd.
It requires very careful observation to discover this swelling in a
concrete block, which usually fails along a diagonal such as ae,
but, in any case, experiments with greased surfaces have shown
57
58 REINFORCED CONCRETE BUILDINGS
that when the friction is eliminated, the block fails along vertical
planes such as ff. 1 It is therefore clear that longitudinal rein-
forcement in the direction of the compressive force is not very
efficient, because the longitudinal rods simply add their own
1
,b
strength to that of the concrete. The rods act as slender columns
and have a tendency to buckle, so that if no other provisions are
made, the strength of the rods is practically nil. To prevent
buckling, horizontal ties or " hoops " are introduced, but it is
evident that unless closely spaced the hoops are of little value.
If therefore the column or block is to have vertical reinforce-
ment, it must have closely spaced horizontal hoops, and these in
turn prevent the concrete from breaking apart along the vertical
planes ff described. In this way the hoops become a very
efficient means of reinforcing.
12. In order to understand this fully, let us consider a cylinder
filled with water, one end being equipped with a water-tight but
frictionless piston. This piston will carry an immense weight
on its upper surface ; in fact, the entire system cannot fail before
the water pressure within the cylinder exceeds the capacity of
the cylinder walls, so that the cylinder bursts. The pressure
within the cylinder is the same in all directions per unit of area;
more particularly there is a horizontal (lateral) pressure on each
and every square inch equal to the vertical pressure produced by
the load on the piston. If now the cylinder is filled with sand in-
stead of water, the conditions are only changed to this extent
that the lateral pressure against the walls is less than before, so
that it takes a greater load on the piston to burst the walls.
Finally, if the cylinder is filled with liquid concrete, and the con-
crete is allowed to set hard, the pressure on the walls will be even
1 According to tests by Foeppel and Mesnager. See, for instance, Con-
sidere, "Reinforced Concrete," page 120.
COMPRESSION AND LATERAL EXPANSION 59
less than before, but the concrete will stand much higher pressure
when enclosed in the cylinder than when free. This, then, is the
principle of the " hooped column," that the horizontal metal
jacket prevents the concrete from spreading and thereby in-
creases its carrying capacity. 1
For practical reasons it has been found impossible to use a
continuous sheet of iron around the concrete; the horizontal
reinforcement is always in the shape of hoops encircling the body
of the concrete. Under pressure the concrete is sometimes seen to
ooze out between the hoops, indicating the failure of the column,
but usually the column fails by the bursting of the hoops or the
complete disintegration of the concrete. In practical construc-
tion this need not concern us, as the stresses naturally always are
low; more 'important is the relatively great shortening of the
hooped column under working loads. This objection is overcome
by the rational use of vertical rods, so that the true " hooped
column " contains both hoops and verticals (Figure 81)
FIGURE 81.
13. The computation of a hooped column naturally centers
around the calculation of the lateral pressure against the hoops.
With a given concrete area F and a given load X the unit stress
on the concrete becomes
^ IK / K 5 wnere -X" is m Ibs.
F lbs ' /Sq - mch (and F in square inches.
If we were dealing with water, the horizontal unit pressure would
be the same. For concrete this is not the case; according to
1 Attention is called to some very interesting tests by Prof. Ira H. Wool-
son, Eng. News, 1905, Nov. 2, Steel tubes, 4" in diameter and 12" long,
| " thick walls, were filled with concrete. When seventeen days old, the tubes
were tested in compression under loads as high as 120,000 to 150,000 Ibs.
The tubes bent out of shape, and shortened 3", while the diameter increased
from the original 4" to 5". When the tubes were removed, the concrete was
found unbroken, solid, and perfect.
See also Trautwine, 1909, p. 1160.
60 REINFORCED CONCRETE BUILDINGS
experiment, the ratio between intensity of vertical stress and
transverse stress is as 1 to 1/4.8. In other words, if the load
produces a direct compressive stress of 4801bs./sq. inch, the lateral
pressure would at the same time be 100 Ibs./sq. inch. It is now
a simple matter to write up an expression giving the resistance
due to the hoops, in a granular material having this same coeffi-
cient 4.8. Let us denote by u the ratio between this resistance
and the volume of metal in the hoops, and let us denote by U a
similar ratio obtained between the resistance due to vertical
reinforcement, and the volume of the material in the verticals.
The expressions u and U will then give the effect produced by a
unit of material, used as hoops and as verticals. We find,
assuming the same stress in hoops and verticals:
^-M_24
U 2
which' shows that pound for pound, the steel employed in the
hoops is 2.4 times as effective as steel employed for longitudinal
reinforcement.
14. The question is now to find the effect of the verticals.
Assuming that they are well tied so as to prevent buckling of the
individual rods, the unit stress on the verticals must be r times
the stress on the concrete, if the sections are to remain plane as
assumed in Article 4. It is easy to see that this assumption is
on the safe side, because, if the sections curved, the stress in the
steel might be very much more than r times that on the con-
crete, which latter forms the starting-point for our investigation.
The value of r can only be indicated in a general way, as the
properties of concrete vary greatly with the circumstances; let
us assume r = 20. Then, if the unit stress on the concrete is
500 Ibs./sq. inch, the stress on the steel becomes 10,000 Ibs./sq.
inch. Let F be the area of the concrete inside the hoops, and the
allowable stress on this concrete C Ibs./sq. inch. Let p denote
the percentage of the verticals with reference to the volume of
concrete, then the effective concrete area is F- -
1UU
7?
and the area of the longitudinals F T^TT ;
hence the load carried by the concrete is C - F fnf) ^s.
COMPRESSION AND LATERAL EXPANSION 61
Let the stress on the longitudinals be S Ibs./sq. inch, then their
7)
share of the load becomes S F - Ibs.
Disregarding for the moment the influence of the hoops, the total
carrying capacity of the reinforced column is
while if allowance be made for the hoops, the percentage of which
is q with reference to the concrete section, we have an additional
strength due to the hoops equal to 2.4 S F -^
and the total carrying capacity of the column becomes
. (2)
15. The formula (2) above is the true formula for a reinforced
concrete column and should always be used except in localities
where the building code prevents its use, in which case formula
(1) may be used. In any case, hoops must be used, otherwise
the column steel is of no value as reinforcement. For the hooped
column, Considere, the inventor and first experimenter, recom-
mends p = q = 2, which, with C = 600 Ibs./sq. inch, and r = 20,
gives
X 2 = 1400 F.
The hoops are spaced as closely as possible, leaving 1" to 2"
clear space between the hoops to facilitate concreting. The
spacing should under no circumstances exceed 1/6 of the diam-
eter of the core. Finally the core is protected with a suffi-
cient thickness of concrete to prevent rust and fire danger, about
1 " to 2 " of protection being required according to location and
exposure.
The plain column has a vertical reinforcement varying from
one to ten per cent, of the concrete area, although reinforcement
in excess of say five per cent, should be avoided on account of
the uncertainty of the strength of columns reinforced with large
amounts of steel. It is evident that hoops are indispensable also
in these columns; it is quite common to see the hoops spaced
one or even two feet apart; such hoops are of no use. The
steel cannot be depended upon to carry its load unless securely
62 REINFORCED CONCRETE BUILDINGS
tied, say, 1/3 to 1/2 column diameters apart. With p = 4,
8 = 12,000, C = 600, we have, for r = 20:
X l = 1060 F.
16. Owing to difficulties in filling columns of small diameter,
the diameter should not be much less than 10 " in any case,
although there are many 8" columns on record. On account of
the danger of " column failure " the length should not exceed
15 diameters. It is possible to advance a theory for " long "
columns, but experience shows that columns exceeding 15 diam-
eters in length are rare indeed except in roof stories where the calcu-
lations often give very light sections. Moreover, all such theories
depend alone upon theoretical considerations and have never been
conclusively tested in the laboratory, so that in the rare cases
where " long " columns are required it is better to make the col-
umn a little larger and avoid the uncertainties of the theory.
17. In tall buildings, or in warehouses, the column bars become
quite heavy, and it is necessary to join the bars of the column
above with those of the column below in a substantial manner.
The most satisfactory way is to square the ends of the bars
carefully and join them in rather closely fitting sleeves, taking
care that each bar has a full bearing on the bar below. Absolute
certainty is had by cutting threads on both bars and sleeves, and
drawing the bars together tight with the sleeve, but this must
be done with great care and under strict supervision in order to
be at all effective; unless carefully made this joint is worse
than useless. When light bars are used they may be spliced by
lapping by the required number of diameters, say about thirty,
but this method is hardly to be recommended.
Each bar of the story above should find bearing on a bar
below; the number of bars therefore increases downward in the
building. The number of bars in each story should be such that
the bars can be symmetrically arranged in the column, unless
there is some extraordinary reason for arranging them otherwise
(excentric loads). The proper arrangement of the column bars
may sometimes cause the designer to spend a good deal of time in
working out the correct solution, but he may feel assured that
this time is well spent.
The hoops may be made from round or flat stock; the round
stock may be obtained in long lengths and lends itself more readily
to the requirements of the hooped column, especially where the
COMPRESSION AND LATERAL EXPANSION
63
reinforcement is manufactured in the shop, with permanent
devices for coiling and fastening the hoops to the longitudinals.
The hooped reinforcement may also be bought ready-made;
quite frequently the manufacturer overlooks the importance of
having the spiral hoops in one continuous piece from top to
FIGURE 82. COLUMN REINFORCEMENT
Loomis Building, Cleveland, Ohio. Alexis Saurbrey, Consulting
Engineer
bottom, or, where the wire is joined, he makes a flimsy joint.
It must be remembered that the hoops are tension-reinforcement
and subject to all the rules governing the design of such bars.
The best joint is made by simply bending the ends of the wire
to the center of the column, making the loose end long enough
to secure the requisite grip.
The hoops may also be made in individual pieces, slipped
over the previously erected verticals and wired in place. If
the hoops are neatly made an excellent job may be had in this
way (Figure 82).
64 REINFORCED CONCRETE BUILDINGS
18. It follows from what is said above that a hooped column
should preferably be made of a circular cross-section, because
in that case the hoops are subject to direct tension only. In
many cases the expense incidental to the use of circular forms is
prohibitive; the concrete may then be made square or octagon
in section while the circular form is retained for the hoops. In
either case only the concrete within the hoops can be taken into
account in the calculations. Sometimes the hoops are made
square or rectangular, in which case they are less effective, but
we do not know how much.
19. The top and bottom of each column deserves special
attention as the tests made so far seem to indicate that these are
the weakest parts of the column, although there are many ex-
ceptions to this rule. Suitable caps and bases are inexpensive,
improve the appearance and increase the strength. Special
investigation is always necessary at points where the concrete
column finds a bearing on another material; the weight carried
by the reinforcing rods must be distributed over such an area
that the concrete in the column is not over-stressed. This is
particularly true where the column rests on the footing; a steel
base plate must be used to distribute the load on the rods, and
the concrete must be enlarged so as to bring the average pressure
within the allowable. This will be considered in detail under
"footings."
20. Before leaving the subject of hooped columns, attention
is called to the possibility of strengthening existing concrete
columns with hoops wound around the outside of the column.
In many cases it would be impossible to obtain satisfactory results
in this manner, but when the concrete is of good quality, and the
existing reinforcement is such as to give a sufficient amount of
longitudinal reinforcement in the finished column, there should
be no theoretical objections to this procedure. In practice it
would of course be difficult to wrap the core tightly, but this is
not absolutely necessary, as grout rich in cement may be forced
between the hoops and the old concrete. Great care would be
required in this operation, but it is not at all impossible, as has
been shown by actual experiments on a small scale. 1
1 Considere: "Reinforced Concrete," page 175. The prism tested in
this manner was allowed to set for three months, then wrapped with hoops
and covered with cement, and tested after ten more days. The crushing
COMPRESSION AND LATERAL EXPANSION - 65
21. In many cases columns are subject to excentric loads, so
that, in addition to the direct compressive force, a bending
moment exists and must be taken care of. This will be con-
sidered in detail in Article 81.
strength was 10,500 Ibs. per square inch. There were no longitudinals in
this prism.
CL AFTER VI
BENDING
22. THE theory of bending used for reinforced concrete beams
is different from the ordinary " theory of flexure " as used for
homogeneous beams in a few particulars only, and this difference
is more apparent than real. We consider here only the point of
maximum bending moment; this is also the point of maximum
depth, and we may assume both the compressive and tensile
resultant to be normal to a vertical section through this particular
point, under the particular loading described below.
The notations used are as follows (Figures 83, 84) :
d or D = depth from top of concrete to center of steel,
inches.
xd = depth from top of concrete to neutral axis, inches.
xd
x = ~j~ = ratio between the two preceding items.
di = distance center of compression to center of ten-
sion, inches.
E c and E s = coefficients of elasticity for concrete and steel.
Tjl
r = -pr-= ratio between these coefficients.
tic
t or T = thickness of a flange, inches.
5 = width of flange considered, inches.
B = ^iS = width of flange considered, feet.
\2i
n = thickness of stem of beam, inches.
r c and r s = deformations of concrete and steel, at extreme
fiber.
C = unit stress on concrete in outside fiber, compres-
sion, Ibs. per square inch.
S = unit stress in steel, tension, Ibs. per square inch.
a = area of steel, square inches.
St. = total pull in steel in tons.
Clj c 2 , c 3 = coefficients relating to balanced design of the
section.
a, = coefficients relating to T-beams with greater than
minimum depth.
66
BENDING
67
w = dead plus live load on. slab, Ibs. per square foot.
/ = span in feet.
q = factor of continuity.
M = bending moment in tons-inches.
m = 2000 M = bending moment in Ibs.-inches.
23. In regard to the load, we wi^ let all loads act in the same
vertical plane along the center line of the beam as is usually the
case in practical construction. This excludes at once all loads
which would cause the beam to rotate around its longitudinal
axis and all loads which would cause the beam to slide in its own
direction.
24. In regard to the deformations, we will consider these as
very small in comparison with the dimensions of the beam, so
that the stresses are considered as acting upon the original cross-
sections, not upon the deformed cross-sections or upon the
deflected beam.
25. This does not mean that the change of shape of the section
is of no importance. In figure 83 a vertical section is shown
\ Shortening
;= < of Top Fibre,
* Concrete
Elongatio
| of~Steel
FIGURE 83.
with the deformations produced by the bending of the beam;
we assume sections plane before bending to remain plane after
bending. Inspection of the diagram shows that the upper fibers
are shortened, the lower fibers extended under the load; the
neutral axis forms the division line between shortened and ex-
tended fibers. The assumption of plane sections is evidently
equivalent to assuming that the deformation of any fiber is in
68 REINFORCED CONCRETE BUILDINGS
direct proportion to its distance from the neutral axis, and thus
we get the equation:
r -< = ** = _^_ , 3)
r a (l-x)d l-x
26. We further assume that the stress on any small unit is
directly proportional with the deformation; this gives the equa-
tions :
Q
for concrete C = r c E c or r c = ^r
o
for steel S = r 8 E s or r s = 7
___
T S S' EC
(4)
27. We shall later 1 have occasion to use the moment of inertia
of the section. It is therefore necessary to note that the assump-
tions made in the preceding paragraphs are identically the same
as those used in the " common theory of flexure " which leads
to the well-known expression
a _ ^ . e where v = stress per unit (5)
I M = bending moment
/ = moment of inertia
e = distance from neutral axis to fiber
considered.
The new feature in a reinforced concrete beam is now that in
writing up the moment of inertia we have to disregard the con-
crete below the neutral axis entirely, and instead consider the
steel area. To this we shall return later.
28. Combining now equations 3 and 4 we find
S E c
hence x
which expression determines the location of the neutral axis.
29. If now a vertical section is laid across the beam and
stresses added on and in the section to represent the removed
portion of the beam, the beam will remain in equilibrium. Let
us project all forces and stresses on a horizontal line: then the
1 Article 79.
BENDING 69
loads, being vertical, give no projections, and similarly the
stresses acting in the vertical section itself disappear. There
remain only the normal stresses acting against the section; as
equilibrium presupposes that the sum of all the projected forces
and stresses is zero, we have
horizontal component
of stresses
on tension side
horizontal component
of stresses
on compression side.
Referring now to Figure 84, the area stressed in compression is
xd inches high, b inches wide, and the average stress J C Ibs. per
square inch. Hence
total compression = \ C xd b Ibs.
Denoting by s t the total pull in the steel in tons, we have, neglect-
ing the tension in the concrete,
total tension = s t 2000 Ibs.
Hence s t 2000 = i Cxdb
, . , . Cxdb c 2 , ,
which gives st = TTTT or s t = Jo"
where c 2 = - (6)
30. Two more conditions must be fulfilled in order to create
equilibrium: (1) the sum of all stresses and forces must be zero
when projected upon a vertical line (when the loads are vertical,
Article 23); this condition we will consider later under " U-bars."
(2) The sum of all moments around any arbitrary point must be
zero. Select for this point the point of application of the com-
pressive stresses; the moment of the loads is then the " bending
moment " m inch-lbs. The moment of the stresses is 2000 s t d\
inch-lbs. We must then have
= m - 2000 s t . di
but according to the diagram (Figure 84)
d, = (1 - J x) d
hence = m - 2000 s t ' (1 - J x) d.
Eliminating st we find
m = -I Cxb (1 - J x) d*
hence d = - V inches where Ci = V I Cx (1 -| x) (7)
c\ o
Finally the steel area: a = s t square inches.
70
REINFORCED CONCRETE BUILDINGS
31. The formulas apply to all rectangular beams and therefore
also to slabs. As we disregard the tensile resistance of the con-
crete, the concrete below the neutral axis does not in any way
enter into the calculations at this point, and the formulas are
therefore also correct for T-beams where the bottom of the flange*
coincides with the neutral axis. In this case the thickness of
flange simply becomes
t = xd inches.
32. We have now everything required to proceed with the
design:
S t Tons
FIGURE 84.
The depth in inches:
The pull in the steel, tons:
*~^T
a =
The thickness of flange, inches: t = xd
2000
Ihe steel area, square inches: a =
o
(8)
(9)
(10)
(11)
Simple as these formulas are they can only be used when the
values of the coefficients x, ci and c 2 are known, and these values
in turn depend upon the allowable stresses and the factor r.
The Tables I, II, and III give full information in regard to the
values of the coefficients; it will be noticed that the same tables
may be used for any value of r, by simply shifting the position
of the S-column in relation to the values of the coefficients. On
the left the ordinarily used ^-column is indicated, corresponding
to r = 15; while on the right, the ^-columns corresponding to
r = 12 and r = 20 are show r n. Usually existing building codes and
engineers' specifications call forr = 15 in bending-problems, but
BENDING 71
this selection is arbitrary, and other values of r may very well
be used. It is impossible to predict the coefficient of elasticity
of concrete beforehand, and even if determined by careful ex-
periment there is no reason to believe that it would remain the
same on the building to be erected as in the laboratory, while it
is quite certain that it changes materially from day to day as
temperature and moisture affect the mixture used for the
concrete.
In Table IV values of the coefficient c 3 = 1 i x are indicated;
the use of this table will be clear from the analysis above. In
Table V the percentage of steel in a rectangular beam is indicated
corresponding to r = 15; when the allowable stresses are decided
upon, the percentage of steel in the section is a fixed quantity.
33. In the formulas above all dimensions are in inches, the
moment in inch-lbs., the pull s t in tons. In practical design it is
usually convenient to have the bending moment in inch-tons, M,
and the width in feet, B. The formulas then become :
The depth in inches : d = y -^- (8a)
The pull in the steel, tons: s t = c%Bd ; (9a)
The thickness of flange, inches: t = xd (10a)
2000
The steel area, square inches: a = ~ Si (11)
These formulas are different from those given above in this
respect only, that the figures handled are much smaller and there-
fore it becomes easier to avoid mistakes, as figures of two or three
places may be multiplied and divided, etc., approximately, without
the use of paper and pencil, so that all calculations are easily
verified.
34. The formulas given above apply, as stated, to slabs, to
rectangular beams, and to T-beams in which the neutral axis
coincides with the bottom line of the flange. Usually these
two lines do not coincide, so that it becomes necessary to make
further investigation in order to derive a general formula. The
formulas given above have this peculiarity, that, for a given width
of beam, the dimensions derived are minimum dimensions which
cannot be decreased without adding to the stress on the material,
thus exceeding the allowable stresses on which the design was
based. Briefly stated, the problem before us consists in finding
72
REINFORCED CONCRETE BUILDINGS
TABLE I. DEPTH OF NEUTRAL Axis = xd
1
r = 15
TABLE I x
r = 12
r =20
S = 24,000
.158
.200
.238
.272
.304
.333
.360
.385
S = 19,200
22,000
.170
.214
.254
.290
.322
.352
.380
.405
17,600
20,000
.184
.231
.272
.310
.344
.375
.404
.429
16,000
18,000
.200
.250
.294
.333
.369
.400
.429
.454
14,400
16,000
.219
.272
.318
.360
.397
.429
.458
.483
12,800
S = 24,000
14,000
.244
.300
.349
.392
.429
.463
.491
.519
11,200
21,300
12,000
.272
.333
.385
.429
.468
.500
.529
.556
9,600
18,600
10,000
.310
.376
.429
.474
.513
.546
.574
.602
16,000
C =
300
400
500
600
700
800
900
1000
r = 15
r = 12
r= 20
TABLE II. EFFECTIVE DEPTH
12.9 /"M 1 /m
d = 4/ or d = ->. -
ci y B Cl y 6
r = 15
TABLE II ci
r = 12
r = 20
S = 24,000
4.7
6.1
7.4
8.6
9.8
11.0
12.0
13.0
-S = 19,200
_
22,000
4.9
6.3
7.6
8.8
10.0
11.2
12.2
13.2
17,600
20,000
5.1
6.5
7.9
9.1
10.3
11.5
12.5
13.5
16,000
18,000
5.3
6.8
8.1
9.4
10.6
11.8
12.8
13.9
14,400
16,000
5.5
7.0
8.4
9.7
11.0
12.1
13.2
14.2
12,800
S = 24,000
14,000
5.8
7.3
8.8
10.1
11.3
12.5
13.6
14.7
11,200
21,300
12,000
6.1
7.7
9.2
10.5
11.7
12.9
14.0
15.1
9,600
18,600
10,000
6.5
8.2
9.6
11.0
12.2
13.4
14.5
15.5
16,000
C =
300
400
500
600
700
800
900
1000
r = 15
r = 12
r =20
TABLE III. TOTAL PULL IN STEEL
s. = 02 Bd or s f = 02 6d
* 12
r = 15
TABLE III c 2
r = 12
r = 20
S = 24 : 000
.14
.24
.36
.49
.64
.80
.97
1.16
S = 19,200
22,000
.15
.26
.38
.52
.68
.85
1.03
1.22
17,600
20,000
.17
.28
.41
.56
.72
.90
1.10
1.29
16,000
18,000
.IS
.30
.44
.60
.78
.96
1.16
1.36
14,400
16,000
.20
.33
.48
.65
.83
1.03
1.24
1.45
12,800
S = 24,000
14,000
.22
.36
.52
.71
.90
1.12
1.33
1.56
11,200
21,300
12000
.25
.40
.58
.77
.99
1.20
1.43
1.67
9,600
18.600
10,000
.28
.45
.64
.86
1.08
1.31
1.55
1.80
16,000
C =
300
400
500
600
700
800
900
1000
r = 15
r = 12
r = 20
BENDING
73
TABLE IV. ARM OF "COUPLE OF STRESSES."
r = 15
TABLE IV c 3 = 1 - \x
r = 12
r = 20
S = 24,000
.95
.93
.92
.91
.90
.89
.88
.87
5 = 19,200
22,000
.94
.93
.92
.90
.89
.88
.87
.87
17,600
20,000
.94
.92
.91
.90
.89
.88
.87
.86
16,000
18,000
.93
.92
.90
.89
.88
.87
.86
.85
14,400
16,000
.93
.91
.89
.88
.87
.86
.85
.84
12,800
S = 24,000
14,000
.92
.90
.88
.87
.86
.85
.84
.83
11,200
21,300
12,000
.91
.89
.87
.86
.84
.83
.82
.82
9,600
18,600
10,000
.90
.88
.86
.84
.83
.82
.81
.80
16,000
c =
300
400
500
600
700
800
900
1000
r = 15
r = 12
r = 20
Amount of Steel In Section
P i*T * 100
-b =12
TABLE V
r = 15
p
S = 24,000
.098
.167
.247
.339
.442
.553
.672
.801
22,000
.115
.196
.288
.393
.510
.636
.771
.920
20,000
.138
.231
.339
.465
.602
.750
.907
1.07
18,000
.166
.278
.406
.553
.714
.884
1.07
1.26
16,000
.204
.339
.493
.672
.865
1.07
1.27
1.50
14,000
.261
.428
.621
.839
1.07
1.33
1.58
1.85
12,000
.339
.554
.799
1.07
1.36
1.66
1.98
2.31
10,000
.463
.750
1.07
1.42
1.79
2.17
2.57
2.99
C =
300
400
500
600
700
800
900
1000
the effect on the T-beam of an increase in depth, which must, in
order to balance the design, be accompanied by a corresponding
decrease in thickness of flange and amount of steel.
74
REINFORCED CONCRETE BUILDINGS
35. Let, then, Figure 85a represent a section of T-shape of
minimum dimensions, having a depth d, a thickness of flange
t a , and a total pull in the steel of s a tons. Let, further, Figure
856 represent a new section with a new, larger depth D = ad.
b=12B
ta
= xd
/Neutral Axis
\
<: n >
c
9
* r
ad=
/Neutral Axis |
FIGURE 85a.
FIGURE 856.
The given M and B remain the same; we wish to determine the
new values s b and T pertaining to Figure 856.
We observe, then, that the proportionate depth x of the neutral
axis is the same in the two beams, because the allowable stresses
are the same, so that the depth of neutral axis is calculated as
t a = xd in the first beam and as t b = xD in the second. The
" effective depth" in the first beam is
di = (1 | x) d
and in the second approximately
Di = (1 - %X) D = (1 - J X) ad
The approximation consists in disregarding the tendency of the
center of compression to rise on account of the removal of the
concrete near the neutral axis; the discrepancy is negligible in
most cases and on the safe side. Since now
M ,
-- and
M
we get the equation s a = t*s b where s a =
and, by reference to Figure 86
T-6 +
4000
C T
4000
Introducing these values in s a
an equation
(at - T) - n.
4000
sb we get after some reduction
BENDING
75
ST\
Solving for ( ) and denoting by
\t a J
p b ~H i t
a 2 1 r ] ' ta r
\ a ta/
ft the value of this ratio we find
n (L ^ }
b'
1 3FC |- K"~
*>*^
s
Neutral Axis
-Jn <&-
1
^k
C =r at a-T j
T C ~^T- |
FIGURE 86.
Showing the stresses in the beam of Figure 856.
The thickness of flange in our new beam is now
T = pt a = Pxd = --x-D
a.
and the new total pull in the steel is
a. a a*
corresponding to the new depth D = ad.
36. We have now the following general formulas for any re-
inforced concrete T-section:
12.9 I'M
The depth, in inches
D = a.
IM
VB
The total pull in the steel, in tons s< = -|
The thickness of flange, in inches
The steel area
a =
2000
(13)
(14)
(15)
s (
where M is the bending moment in tons-inches, a an arbitrary
coefficient larger than unit, while the width B feet may be given
or selected. The coefficient ft is derived from by formula 12
but to facilitate calculations, Table VI has been prepared giving
the values of fi for various combinations of a and n/b. This
latter ratio has little influence on the result within the ordinary
limits, and Table IX may also be used in cases where n/b is
76
REINFORCED CONCRETE BUILDINGS
different from 1/4, if the variation is not too large, although
prepared especially for n/b = 1/4.
37. The theory of T-beams is of great importance as all the
floor systems in common use involve this principle. Lately,
beamless floors have come into use, and to these we shall return
later; the beam and slab floors may be divided into two groups,
the first including solid concrete floors, the second what is known
as " tile-concrete " floors. The first of these two is by far the
oldest, but the " tile-concrete " is gaining in favor with every
day, and justly so, as its cost is less for light buildings owing
primarily to the simplicity of the form work. The long flat
ceilings are well adapted to modern store building and office-
structures, especially where the loads are light and distributed.
These floors have a flat portion supported on main girders A
(Figure 87), the flat portion consisting of ribs B built between
I c I cTc I I I
Section A-A
FIGURE 87.
rows of hollow tiles C and a top covering of two or more inches of
concrete, thus forming series of comparatively light T-beams
side by side. The main girders are also of T-shape, the flanges
being formed by leaving out the requisite number of tiles next
to the stem of the girder. Sometimes lighter tiles D are used
near the stem, in which case the flange becomes thinner than when
the tiles are omitted entirely, Figure 88. The commercial sizes
FIGURE 88.
of tiles are usually 12" x 12" in plan, the depth ranging from 4"
to 12" or even 16". When designing, it becomes necessary to
proportion the depth of floor so as to allow for these commercial
BENDING 77
sizes; the function of the tiles is simply to create a void in the
concrete, and they do not enter into the calculated strength of
the floor. The calculations require considerable time if exact,
and tables VII and VIII have therefore been prepared for C = 700
and 5 = 20,000 and 16,000 respectively. These tables show at
a glance the depth of tile and thickness of concrete required for
any given bending moment, together with the corresponding
pull in the steel. Note, however, that the bending moment must
be calculated for a width b of slab equal to the distance between
centers of ribs. If other allowable stresses are assumed than
those for which the tables have been prepared, we may easily
prepare new tables. We have in all the preceding formulas,
that the bending moment is directly proportional to the square
of the coefficient ci, while the total pull in the steel is directly
proportional to the coefficient c 2 . But we have
- = a coefficient times c 3
A glance at Table IV shows that c 3 itself is practically a constant
within fairly wide limits, so that, for the allowable stresses in
ordinary use, we may make
= constant.
c 2
It follows that the new tables are prepared from the tables here
given by multiplying both the bending moment and the pull in
the steel of the old table with a factor; this factor is the same
for both items and is
the new value of c 2
the old value of c 2
A completed floor of this kind is shown in Figure 89.
38. Flat Slabs. If, in formulas Sa and 9a, we make B = l,
we have the slab formulas
d = ^?i IM and s t
But the load on the slab is usually given and in Ibs./sq. foot;
denoting by w the total dead and live load in Ibs./sq. foot, the
bending moment per foot width becomes
M = - - I 2 - 12 tons-inches
78
REINFORCED CONCRETE BUILDINGS
TABLE VI.
INCREASING THE DEPTH FROM d TO D = ad
See Table IX for special case .- = *
o
a = 1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.1
.64
.62
.59
.56
.50
.44
.38
.25
.08
1.2
.54
.50
.46
.41
.34
.26
.15
1.3
.47
.42
.37
.31
.23
.12
1.4
.42
.37
.30
.23
.14
.01
1.5
.38
.32
.25
.16
1.6
.35
.28
.21
.11
1.7
.32
.25
.16
.06
'
1.8
.30
.22
.13
.01
1.9
.28
.20
.09
2.0
.27
.18
.06
n/b =
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
TABLES OF =
S t TONS
TABLE VII S = 20,000
15
700
T " =7.0
393/20.6
6.5
372/20.0
6.0
350/19.3
5.5
276/17.1
329/18.6
5.0
204/14.9
255/16.5
308/17.8
4.5
189/14.3
236/15.7
284/16.9
4.0
62/8.3
93/10.1
132/12.1
174/13.6
216/14.8
259/15.9
3.5
55/7.7
85/ 9.6
120/11.4
158/12.7
196/13.8
235/14.8
3.0
26/5.3
48/7.2
76/ 9.1
108/10.6
141/11.7
175/12.2
210/13.6
2.5
21/4.8
42/6.8
67/ 8.4
94/ 9.7
123/10.6
153/11.5
184/12.3
2.0
17/4.4
37/6.2
57 / 7.6
80/ 8.6
104/ 9.4
130/10.2
158/10.9
1.5
14/3.9
29/5.4
46/ 6.5
65/ 7.3
85/ 8.0
107/ 8.7
132/ 9.3
1.0
10/3.2
21/4.4
35/ 5.0
50/ 5.7
66/ 6.4
84/ 6.9
105/ 7.5
0.5
6/2.3
13/3.0
23/ 3.5
33/ 4.0
46/ 4.5
60/ 5.0
76/ 5.5
0.0
1/0.6
4/1.1
9/ 1.6
15/ 2.0
23/ 2.5
33/ 3.0
45/ 3.5
H " =
4
6
8
10
12
14
16
BENDING
79
q 2000
TABLE VIII S = 16,000
C = 700
T " = 7.0
_
443/23.9
6.5
420/23.0
6.0
396/22.2
5.5
312/19.7
372/21.4
5.0
231/17.1
288/19.0
348/20.5
4.5
214/16.4
266/18.0
320/19.4
4.0
70/9.5
105/11.6
149/13.9
197/15.6
244/17.0
292/18.3
3.5
62/8.9
98/11.0
136/13.1
178/14.6
221/15.9
266/17.0
3.0
29/6.1
55/8.3
86/10.5
122/12.2
159/13.5
198/14.6
238/15.7
2.5
24/5.5
47/7.8
75/ 9.7
106/11.1
138/12.2
173/13.2
208/14.2
2.0
19/5.1
41/7.1
64/ 8.7
90/ 9.9
118/10.8
147/11.7
179/12.5
1.5
16/4.5
33/6.2
52/ 7.5
73/ 8.4
98/ 9.2
121/10.0
149/10.7
1.0
11/3.7
24/5.1
40/ 5.8
56/ 6.6
74/ 7.4
94/ 7.9
119/ 8.6
0.5
7/2.6
15/3.5
26/ 4.0
38/ 4.6
52/ 5.2
68/ 5.8
86/ 6.3
0.0
1/0.7
5/1.3
10/ 1.8
17/ 2.3
26/ 2.9
38/ 3.5
51/ 4.0
H" =
4
6
8
10
12
14
16
which gives the very convenient formulas (16)
d =-.J inches
Ci V q
and s t = c%d tons. (17)
Here the span I is expressed in feet, and the factor q is equal to
8 for non-continuous construction, and from 10 to 16 for con-
tinuous construction.
39. If reinforced in both directions, and supported on all
four sides, the slab is calculated by the formulas above, dividing
the total load w into two portions w\ and w 2 where wi -f- w 2 = w.
If we denote by LI and L 2 the span in each direction, we have the
arbitrary formulas for the division of the load:
and
= W
w<> = w
Li 4 + I* 4
The heaviest of these is now assigned to the shortest span, and
determines the depth and the reinforcement running the short
way, the cross reinforcement is designed in a similar manner
using the other load. In case of square panels the two loads
become equal, each one-half of the total.
The formula is entirely irrational and the only reason it is
80
REINFORCED CONCRETE BUILDINGS
TABLE IX - =|
o
/S = a - \/M 2 - 1)
a 2
a
jS
/3/a
1.00
.000
1.000
1.00
1.01
.005
0.890
0.89
1.02
.099
0.847
0.84
.03
.015
0.815
0.80
.04
.020
0.789
0.77
.05
.025
0.767
0.75
.06
.030
0.747
0.73
.08
1.039
0.712
0.69
1.10
1.049
0.684
0.65
1.12
1.058
0.658
0.62
1.14
1.068
0.636
0.60
1.16
1.077
0.616
'0.57
1.18
1.086
0.596
0.55
1.20
1.096
0.579
0.53
1.25
1.118
0.541
0.49
.30
1.140
0.508
0.44
.35
1.162
0.479
0.41
.40
1.183
0.452
0.38
.45
1.204
0.430
0.36
.50
1.225
0.409
0.33
.60
1.265
0.371
0.29
.70
1.304
0.338
0.26
.80
1.342
0.308
0.23
.90
1.378
0.283
0.21
2.00
1.414
0.258
0.18
2.25
1.500
0.209
0.14
2.50
1.581
0.167
0.11
2.75
1.658
0.130
0.078
3.00
1.732
0.099
0.058
3.50
1.870
0.045
0.024
4.00
2.000
0.000
0.000
a 2
a
/3/a
given here is that it is on the safe side and better than other
existing formulas.
The supporting girders are designed with reference to the
load brought upon them by the particular direction which they
support, and the bending moment is often increased above the
BENDING 81
calculated because the load seems rather more concentrated
towards the center.
FIGURE 89. TILE CONCRETE CONSTRUCTION, READY FOR
PLASTERER
Wise Building, Cleveland, Ohio. Alexis Saurbrey, Consulting
Engineer
Discussion of Tables I to IX.
1
40. TABLE I. x =
-3
rC
The value of x determines the location of the neutral axis,
xd being the distance from compression face to neutral axis.
82 REINFORCED CONCRETE BUILDINGS
We have seen that the position of the neutral axis within the
section of a T-shaped beam leads to the division of T-beams into
two groups, according to whether the neutral axis falls, above
or below the bottom line of the flange. In this latter case we
introduce the coefficients a and /?, and the problems contain an
arbitrary element which is absent in beams of the first type,
where the dimensions depend mutually upon one another as in
formula (8). The table shows that the neutral axis only in
exceptional cases approaches the middle of the beam where it is
located in all symmetrical beams following Hooke's Law (steel
for example). It is obvious that a greater amount of steel is
required for low steel stresses than for high; we therefore see
that the neutral axis is lowered by increasing the amount of steel.
41. TABLE II. Cl = \J\Cx (\ - x
, 12.9 I'M
We have d = -
The smallest possible value of d is obtained when a large value
of Ci is used, or, in other words, when high concrete stresses are
combined with low steel stresses. The influence of the concrete
stresses is much more pronounced than that of the steel stresses;
it is, therefore, not economy to increase the amount of steel in
order to save on the concrete. It is not impossible to analyze
this problem mathematically, but owing to variation in unit
prices it seems hardly worth while. The possibility of decreas-
ing the depth of construction by using high concrete stresses
and low steel stresses may, however, be of importance in special
cases where the head room is limited.
Cx
42. TABLE III. C2 = 333
The total pull in the steel is st = 02. ^ d.
The total amount of steel is a = s t
12
2000
S
2000 bd fa\fS
so that -- ' C2 ' 12 r C2 = (bd) VT66
The coefficient c 2 , then, is a measure of the amount of steel used
BENDING 83
for a given cross-section, bd being the area of the cross-section
in square inches. We note that
^-, times 100
bd
is the percentage of steel required for the beam; if we denote
the percentage by p we have
16600
This expression has been used in calculating Table V.
43. TABLE IV. c 3 = 1 - I x
In general the total pull in the steel is obtained by dividing
the bending moment by a certain lever arm di, equal in length
to the distance between the centers of compression and tension.
Reference to Figure 84 gives at once
di = | xd + (1 - x) d = (1 - \x) d = c,d
When d is known we get d\ as c$d; Table IV gives the values
of c 3 for various combinations of stresses.
44. TABLE V. p = ^ X 100 = c 2
The percentage of steel has but little interest for the prac-
tical designer as the problems usually present themselves.
The table is added for the convenience of those who are in the
habit of selecting the percentage of steel rather than determine
the allowable stresses. The table is correct only for such beams
where a = 1 and r = 15.
45. TABLE VI will be found useful when designing T-beams
of larger than minimum depth. When we have selected , as
explained in connection with formula 12, etc., the correspond-
ing ft is found by Table VI for any value of n/b. The method
of design will be clearly evident from the example in Article 47.
Table IX is a more extensive table for the special case where
n/b = J, which is a common value in practice. The variation
of n/b does not affect the values very much, so that for small
values of a Table IX may be used for other values of n/b than J.
46. TABLES VII and VIII. Tile-concrete floors.
The use of these tables is best explained by an example.
The span of the flat portion is 20 feet; the total dead and live
84 REINFORCED CONCRETE BUILDINGS
load is assumed to be 250 Ibs. per square foot. With 4" ribs
we get the width of beam
b = 4 + 12 = 16"
and the corresponding bending moment in inch-tons
250
M = * x ^ x 2 2 X 16 = 100 inch-tons.
If the allowable stresses are S = 20,000 and C = 700, we must
use Table VII, and we see at once that we can use either a 10 "
tile with about 2f" concrete, or a 12" tile with 2" concrete.
As we do not wish to have less than 2" of concrete over the tiles,
we cannot use the larger tiles economically. If we select 12"
tiles and 2" concrete, the corresponding pull in the steel is 9.4
tons according to the table, requiring .94 square inches of steel,
for instance, one f " square and one f " square bar.
It should not cause surprise that the moments tabulated in
Table VIII are larger than the corresponding values of Table
VII, although the allowable stress on the steel is smallest in
Table VIII. The explanation is given in the remarks under
Table II, Article 41, and, in accordance with the statements
made there, it will be seen that the larger moments of Table
VIII are obtained only by increasing the steel areas.
TABLE IX. See Table VI, Article 45.
47. EXAMPLE 1. T-Sections. Continuing the example given
above under the discussion of Tables VII and VIII, we proceed
as follows to design the girder:
The load on the floor is 250 Ibs. per square foot, the span of
the flat portion on each side of the girder is 20' 0", and the
girder therefore carries a load of 250 X 20 = 5,000 Ibs. per
lineal foot, to which should be added the weight of the girder
itself. Assuming this item to be included in the 5,000 Ibs., and
assuming a span of 24' 0" for the girder, the bending moment
on the girder becomes
M = J X x 24 2 X 12 = 2160 inch-tons.
We decide to use high tension steel, for which S = 20,000, and
we allow C = 700 Ibs. per square inch on the concrete. We get
then from Table II : Ci = 10.3; from Table III : c 2 = .72; and from
Table I: x = .344, and we may now proceed with the design,
using formulas (8a), (9a), (lOa), and (11). The width of flange,
BENDING 85
B, may be selected arbitrarily. Let us make B = 4' 0".
Then, by
(9a) ......... s = c 2 BD = .72 X 4 X 29.1 = 84 tons.
(11) ......... a = ? - s t = X 84 = 8.4 square inches.
(10a) t = xd = .344 X 29.1 = 10"
We have to make the stem of the beam wide enough to accom-
modate 8.4 square inches of steel, say n = 12", and the girder
is then designed as far as concerns the bending moment. Ques-
tions pertaining to shear, etc., will be considered later.
We can, if we desire, reduce the thickness t of the flange by
increasing the depth d. While this operation is not always neces-
sary, or even desirable, we will nevertheless continue the ex-
ample to show the method of procedure.
If, then, we increase the depth from 29.1" to, say, 35", we get
a = J^_ = a6a>2
the coefficient a indicating the proportionate increase in the
depth. The value of ft is next obtained by Table IX, remem-
bering that
i _ OK
6 " *~
the stem being 12" wide and the flange 48".
For - = .25 and a = 1.2, Table IX gives ft = .43; then,
6
by the theory outlined for this case, Article 36, we have
The new depth D = ad = 1.2 X 29.1 = 35".
s 84
The new pull in steel s& = = y-x = 70 tons.
The new thickness of flange T = ftt = .43 X 10 = 4.3".
It is of course unnecessary to calculate the dimensions of
the " minimum " beam first, as done here, unless we expressly
desire to have these dimensions. Let us, for instance, again
consider a given bending moment of 2,160 inch-tons; let us
86
REINFORCED CONCRETE BUILDINGS
further select arbitrarily the width B = 4' 0", and let us
finally choose the coefficient a = 1.2, then, by Table IX, we
77
get ft = .43 for an estimated value of = .25; and we also
o
have a2 = 1.44. We may now find the dimensions directly, by
12.9 /M 12.9 2l60
'^'V- =
12.9
1, above the bottom of the flange.
If we now wish to determine the stresses in a given beam, we
begin by selecting r, next we determine the value of the cri-
terion, so that, if equal to unit, we use formula (21), while if
larger than unit, we use formula (19), and if smaller, the orig-
inal formula. Then the location of the neutral axis is calcu-
lated from
1
and the coefficient c 3 = 1 f x is determined from the value
of x just found. The effective depth is then
where d is the depth from ultimate compression fiber to the cen-
ter of the steel. We have then
s =-
and, by (11)
2000
90
REINFORCED CONCRETE BUILDINGS
We know now S and S/C, and it is a simple matter to determine
C.
50. EXAMPLE 3. T-SECTION.
Given the beam shown in Figure 92, find the stresses, when
the bending moment is 2,160 inch-tons. We have
T 2
4.5 X 4.5
= 20.25
2 VH ~ 2 X 2.19 X 30.5 " 113.5
V 7 sq. inches
FIGURE 92.
which is -evidently < 1. Using therefore formula (18) we find
325.4
_ 15X
C ~ 5 X 168.4
Now
x =
1
^ = .342
and c 3 = 1 - J a; = 0.886.
Hence di = c 3 d = 0.886 X 35 = 31"
Bending moment 2,160 inch-tons, then
2160
31
= 69.6 tons.
69.6 = 20,100
20100 ,
c = W = 695 '
or about 20,000 and 700 Ibs. /square inch for steel and concrete,
respectively.
51. EXAMPLE 4. T-SECTION. Special case.
Given the beam shown in Figure 93.
We have V = = 15 X ~ = 2.62.
10 X 10
= 100
2 VH ~ 2 X 2.62 X 19.1 == 100
= 1.
BENDING
91
Use formula (21) which gives
S 10 X48
C =
= 28.6
2 X8.4
The balance of the calculations may now be continued ex-
actly as in the preceding example.
j-8.4 sq. inches
FIGURE 93.
52. EXAMPLE 5. RECTANGULAR BEAM. Slabs.
Given the rectangular beam shown in Figure 94; we use
formula (19) which gives
C 2 + 2\
V_0.85 sq. inches
FIGURE 94.
225 + ^ '
0.85
CHAPTER VII
TRANSVERSE STRESSES. U-BARS
53. IN addition to the longitudinal stresses examined in
the preceding articles, transverse stresses exist in reinforced
concrete beams as well as in beams of other materials. But
the transverse stresses are different in trusses and in solid beams :
in the truss, each individual member is stressed in its longi-
tudinal direction only, and there is no shear. In the solid beam,
longitudinal stresses exist in the top and bottom chords or
fibers, and the web is then subject to shear stresses both longi-
tudinally and transversely. In special cases these shear stresses
may vanish, as for instance in the I-shaped steel beam of vari-
able depth, when the ratio
bending moment at any point
depth at the same point
is a constant. This is the case in a parabolic girder loaded
over its entire length with a uniformly distributed load.
54. In view of this difference between trussed beams and
solid beams, it becomes necessary to decide whether to treat
the reinforced concrete beam as the one or the other. To the
eye a reinforced concrete beam certainly appears solid enough,
and such is indeed the case when the beam is first made and the
load is being put on. But when the load reaches a certain in-
tensity the " solidity" of the beam is destroyed. Slight cracks
soon become evident, at least when arrangement has been made
to observe them, and that under loads corresponding to a steel
stress of from 4,000 to 6,000 Ibs./square inch, or a concrete
stress of 350 Ibs./square inch. It follows that under the ordi-
nary working load our reinforced concrete beam is perforated
with cracks extending from the bottom fiber up toward the
neutral axis, without quite reaching the neutral axis, so that,
under any circumstances, the beam is certainly not a "solid"
beam. These hair cracks have been noted by all who have
92
TRANSVERSE STRESSES. U-BARS 93
taken the trouble to look for them with but one exception
(Considere) ; they are not an occasional occurrence, but a uni-
versally recognized phenomenon of the greatest importance for
our understanding of the stresses within a reinforced concrete
beam. The presence of these cracks is accounted for by the
simple fact that concrete is unable to stretch as much as steel
before cracking, so that, under a certain load, the concrete refuses
to follow the steel in its elongation and goes to pieces. The
cracks of this class appear throughout the length of the beam,
fairly uniformly spaced, and increase in size with increasing
load.
55. The crack of course is an open space existing between
surfaces which at some earlier time were in close contact and
united. We must now understand as a fundamental principle
that stresses cannot be transmitted through open cracks. Com-
pression may be transmitted through a contact only, and fric-
tion may exist on surfaces pressed together, but no kind of
stress will jump across an open space. It follows that shear
in the ordinary sense of the word cannot exist in a reinforced
concrete beam loaded above a certain limit, because the nature
of shear requires equal intensity on a horizontal and a vertical
plane, and this is of course impossible when the beam has ver-
tical cracks. Or, we may simply say that the vertical shear
cannot exist in the crack itself. Where a crack occurs there is
therefore nothing but the compression flange and the tension
steel to carry the shear, a distribution of the shear which is,
to say the least, not easily reconciled with current ideas of shear
in solid beams.
56. Entirely different from these hair cracks are the much
larger, pronounced failure cracks which predict the approach-
.1 t
FIGURE 95.
ing collapse of the test beam. If located at or near the point
of maximum bending moment, they are undoubtedly due to
excessive elongation of the steel disclosing a failure by tension
in the steel; if near the end, the crack usually takes the shape
shown in Figure 95, either with or without the horizontal crack
94 REINFORCED CONCRETE BUILDINGS
D. The vertical crack E is wide open, especially at the bottom,
decreasing in width as it approaches the top of the beam. As
the steel stress at this point certainly cannot exceed the steel
stress at the point of maximum bending moment, this crack is
not due to excessive tensile stresses in the steel. It must be
due to sliding of the reinforcement: the steel is pulling out of
the end of the beam at the same time bursting its concrete
envelope, and causing the horizontal crack. Let it be under-
stood that no amount of shear will cause a gaping crack, but once
sliding sets in and causes the vertical crack, it is clear that the
one end of the beam will be compelled to revolve around the
other end, causing in the first place the double-curved line
of cleavage, and, secondly, great friction on the surfaces of
contact.
57. The above remarks lead to the conclusion that a concrete
beam is a solid beam up to a certain load at which point the
tensile resistance of the concrete is exhausted, and a readjust-
FIGURE 96.
ment of stresses takes place within the beam. This readjustment
is different for different types of beams.' In a rectangular
beam (Figure 96) we may well assume that the com-
pression follows lines as AC and BC when the load is placed
FIGURE 97.
at C; if the load moves to D, the lines change to AD and DB.
Under these circumstances there is no shear, at least not in the
ordinary sense of the word. We may compare a system of this
TRANSVERSE STRESSES. U-BARS
95
kind to a triangular frame with hinged corners (Figure 97).
The chords AC and BC will be in compression, and the chord
A B in tension, hence at A and B the hinges are subject to severe
stresses. The same is the case at A and B in the reinforced
concrete beam, so that the " length of embedment " AE and BF
in Figure 96 must be made long enough to prevent sliding of
the rod. The shear existing in a system of this kind is that
negligible quantity caused by the stiffness of the system as a
whole, a kind of friction caused by the lack of flexibility at the
supposed hinges.
The system A BC is an equilibrium curve for the load C
and the reactions A and B', this same argument would of course
hold true for any number of forces, or even for uniformly dis-
tributed loads, in which latter case the compression curve would
be a continuously arched curve from A to B (when the load
covers the entire span). But if the beam under consideration
is a T-beam instead of a rectangular beam it becomes impos-
sible to make the compression line curve down to the support-
ing points, except for a width equal to that of the stem. A
T-beam (Figure 98a) may be considered as consisting of two
FIGURE 98a.
FIGURE 986.
FIGURE 98c.
beams side by side; a T-beam proper (Figure 986) and a rect-
angular beam (Figure 98c). In this rectangular portion it is
quite possible for the compression lines to dip down at the sup-
ports, but not so for the T-beam portion, there being no con-
crete left to carry the stresses down to the steel. This leads
to the idea of bending the steel up over the support to meet
the compression flange, reversing the conditions shown in
Figure 96.
58. Let us consider a portion of a reinforced concrete beam
between two points a and b (Figure 99). The bending moment
at a is denoted by M a , the distance between center of com-
96
REINFORCED CONCRETE BUILDINGS
pression and center of tension by d a , then the total pull in the
steel at a is
M a
Sa = -T-
and at b
s b =
Mi
d b
FIGURE 99.
The difference between these two is
_M a _ M b
' Sa Sb - T a ~d b
The prefix A simply denotes the difference in the item con-
sidered so that As means the variation of s between the points
in question.
It is now evident that the portion abed is subject to two
pulling forces acting near its lower end cd: one force s a pulling
toward the left, another pulling toward the right, s b . If s a is
larger than s b , the end cd must have a tendency to move toward
the left in precisely the same manner as if pulled that way by
a force equal to the difference of the two pulling forces; we
may therefore consider the force As = s a s b as acting alone.
This condition is represented in Figure 100, and this diagram
5
*f
f
F
f 1
f )
h
As^
1
I
f
B
^j ^ Al
FIGURE 100.
shows at once that cdef is a cantilever fixed at its base ef, and
loaded near its end with a load As. The depth ce we do not
know at the present time; let us indicate this unknown quan-
TRANSVERSE STRESSES. U-BARS
97
tity by h. The bending moment on the cantilever is then h As;
the arm of " the couple of stresses " in the cantilever is c 3 AZ;
hence if a vertical reinforcing rod is disposed near bd the pull on
this rod becomes
But this pull k can exist only when counterbalanced by a cor-
responding compression, so that the beam becomes a trussed
beam as shown in Figure 101. The vertical reinforcement
J /]
xx K |>
h K
XI Fl
X X 1 ^ 1
I N N
* 1 x 1^
\t \| \j
A t
R a
B
C!R C
FIGURE 101.
designed in this manner is usually made of a bar bent to U-
shape and circling the main tension rod (Figure 102a, 6); they
are therefore called U-bars or stirrups. The U-bar is unneces-
FIGURE 102a.
FIGURE 1026.
sary when k = 0, which is always the case when As = 0: i.e.,
when either the tension chord, or the compression chord, or
both together, follow the equilibrium curve. As shown above,
this is always the case in a rectangular beam with well-anchored
reinforcement, and it is also the case for such parts of a T-beam
in which the reinforcement is bent up to follow the equilibrium
curve. In all other cases k has a definite value. For straight
reinforcement and straight top chord, we have (Figure 103) :
d a = db = Cj,d
M a - M b
hence
As =
98
REINFORCED CONCRETE BUILDINGS
where
and
hence
or
M a = Ra -
M b = R (a - AQ - 2P (p - AZ)
M a - M b = AZ (R -
A S = - (R -
and k = ^ (R -
It is now easy to understand that the length h cannot exceed
1r.
r sr
a-Al
FIGURE 103.
the distance from the center of the steel to the neutral axis.
This gives
h (1 - x)d 1 - x
The value of this expression cannot exceed unit; for ordinary
cases its value is about three-fourths. Hence the maximum
possible value of k, for the conditions named, is:
kmax == K Zr. (23)
N
A ,
mi:
FIGURE 104.
59. It is now interesting to note that this same expression
may be obtained directly in the simplest manner. Let, in
Figure 104, the section A B remove the right end of the beam
TRANSVERSE STRESSES. U-BARS 99
leaving the main tension bar and the U-bar projecting. The
stress-resultants acting upon A B are then, when the chords are
parallel: the horizontal compression X, the horizontal pull
Y'j and the vertical force k in the U-bar. The loads are PI, P 2 ,
etc., and the reaction R. If we now project on a vertical line
MN, the horizontal stresses vanish and we have
R- (P! + P 2 + P 3 + ...... ) =k
or k = R - 2P.
60. The beam with straight top- and bottom-chords is an
exception. Usually the stem of a T-beam may be considered
as in equilibrium, and in addition some of the bars are bent
up in the T-beam portion to approximate the equilibrium curve,
so that a material reduction in the value of k takes place in all
practical beams. With the notations of Figure 98 we have,
on account of the stem, a reduction equal to
b n , j b n ro vrn
r hence k = r [R 2P]
If out of the total number x of bars in the T-beam, a certain
number y follow the equilibrium curve, we have a further re-
duction equal to
fc = ?LH . *r_ . [R _ 2P] (24)
x x b
Thus, if b = 48" and n = 12", we have
b - n _ 48 - 12 _ 3
~b~ 48 ~ 4
so that, out of a total number of say eight bars, the six belong to
the T-beam. If out of these six, two are bent as required, we
have x = 6 and y = 2, hence
k = i X i X [R - 2P] = } - [R - SP]
61. Thus, in order to calculate the stress on the U-bars,
it becomes necessary to know the properties of the curve of equi-
librium for the system. When the loads are stationary, the
curve is drawn as a force polygon to the actual loads and reac-
tions. For a uniform load, covering the entire span, this curve
is a parabola; it is not practical to bend the bars to this shape,
but it may be closely approximated by a system of bars with
straight portions between the several bents. A uniformly
100 REINFORCED CONCRETE BUILDINGS
distributed, moving load has no definite curve of equilibrium,
so that in that case the most dangerous position of the load
must be found and the U-bars proportioned according to For-
mula 23 above, while the bent bars are arranged to meet the
requirements of some particular type of loading, for instance,
the total load. Similarly, concentrated loads may be either
stationary or moving. In buildings the concentrated loads
are usually stationary. The given load is a uniform load, so
that the beams are loaded as explained above; these beams
in turn frame into the girders, one, two, or three beams to each
span, and these concentrated beam loads are stationary. It
is a simple matter to bend the main tension bars to conform to
this type of loading; examples are given in Figures 105 and 106.
I I
FIGURE 105. FIGURE 106.
The moving concentrated load is usually found only in structures
like highway bridges, subject to steam-roller traffic, in crane-
track girders, etc. In such cases, the live load is large in pro-
portion to the dead weight of structure and covering, so that
the T-beams are usually not economical structures for this class
of girders. They may be constructed by using the adequate
number of U-bars; or rectangular beams may be used of the
required cross-section.
62. The problem of designing a T-beam under a uniform
load confronts the reinforced concrete designer every day. It
is customary to consider the load as covering the entire span,
except in cases where it is expressly stipulated that the most
dangerous position of the load shall form the basis for the cal-
culation of the U-bars. Arguments may be advanced pro et
con., usually the load specified is a maximum load which
seldom, if ever, covers the entire beam, and the designer will
have to use his best judgment as to what constitutes proper
practice in each individual case. It is hardly necessary to say
that in other lines of engineering the most dangerous condition
is always considered in making the calculations as a matter of
course, and there is no reason why other professional ethics
should prevail when dealing with reinforced concrete.
TRANSVERSE STRESSES. U-BARS
101
In Figure 107 the moment-curve is shown corresponding to
a uniformly distributed load covering the entire span. The
maximum moment is taken as unit, and the several ordinates
of the curve are given under the assumption that there is no
continuity. The reinforcement must be made to conform to
n 12
.75
.89
1X0
.97
FIGURE 107.
this curve as closely as possible, hence we see that at points
3 and 9, only f of the total number of bars is required, at 2 and
10, slightly more than J, and less than J is required at points
1 and 11. The quota of bars not required may and should be
bent up at the points specified, provided that no other kinds of
loading can occur. In Figure 108 the corresponding curve is
FIGURE 108.
shown when the beam is considered as continuous, with q = 10.
63. The entire theory outlined for the calculation of the
U-bars is based upon the assumption that sliding of the steel
cannot take place. In such cases where the anchorage beyond
or at the supports is insufficient to prevent sliding of the main
tension bars the factor of reduction must be decreased, so that a
correspondingly larger amount of vertical reinforcement is used
for the U-bars. In the present state of our knowledge this must
be taken care of by judgment alone, there being no way of cal-
culating a beam with inefficient anchorage. It must here be
sufficient to point to the fact that the U-bars retard the sliding
of the reinforcement, and that, for that reason, light U-bars should
always be used even in cases where the theoretical considerations
show that they may be dispensed with. This applies particularly
to rectangular beams.
102 REINFORCED CONCRETE BUILDINGS
64. Spacing of the U-Bars. It will be noted that the entire
line of argument advanced in the preceding paragraph is based
principally upon the inability of the concrete to resist tensile
stresses, and that the entire problem finally resolves itself into
one of tension carried entirely on the steel, and compression
carried entirely on the concrete. The word " shear " is referred
to incidentally only, and this is a natural consequence of the
fundamental principle of disregarding the tensile stresses in
the concrete. As this development leads to rather important
results, it may be well to consider these matters a little more
in detail.
65. Figure 109 shows the simplest conceivable system of
material units, i.e., three particles, A, B, and C. Whatever
the nature of the force uniting these par-
tides, if the particle C is moved to the
position D through the influence of some
\ external force, the displacement CD rep-
. \ resents in all cases the result of the influ-
__ _\; ence of that force and is called the "shear
A B deformation " if parallel with the line
FIGURE 109. . T , . ,., , ,,
AD. It is readily seen, however, that
the more direct and more readily understood deformations
are (1) the lengthening of AC to AD, and (2) the shortening of
BC to BD. Hence this shear deformation CD is nothing but
the resultant of the deformations along the original lines AC
and BC, and we perceive that even in the most complicated
system of particles any deformation may be reduced to a sys-
tem of lengthenings and shortenings, that is, tension and com-
pression, if we speak of stresses instead of deformations. The
word " shear," therefore, has no real or material meaning,
except as a pure figure of speech to express in one short word
a rather intricate condition of tensile and compressive rela-
tions, in precisely the same manner as the word " bending
moment " is used to indicate a mathematical conception of the
mutual condition of a number of forces acting upon a beam.
Needless to say that nobody has ever seen, or will ever see, a
bending moment in the realm of things as they are, and that
whoever undertakes to explain the so-called " shear stresses "
in a solid body will ultimately have to account for pure tensional
and compressional stresses.
TRANSVERSE STRESSES. U-BARS 103
66. If two material bodies are in contact, the stresses act-
ing in the contact surface are termed frictional stresses which,
as far as the materials themselves are concerned, are compres-
sive stresses with no possibility of accompanying tensile stresses
in the direction perpendicular to the contact surface.
67. Of the nature and extent of frictional stresses we know
next to nothing. A force acting parallel with the contact sur-
face will cause sliding of one body in relation to the other; if
the force is inclined, the sliding becomes increasingly difficult
as the angle of the force increases, and the sliding becomes
impossible when the angle at which the force acts exceeds the
" angle of friction," which has a definite value for each material,
depending in part upon the character of the surface. For con-
crete upon concrete, this angle appears to be near 41.
68. In certain types of reinforced concrete construction the
floor beams are not made in one continuous operation with the
floor slab resting upon the beams, and U-bars or similar mechani-
cal devices are then resorted to in order to tie the slab and stem
together, and to so unite them that they may be considered
as acting as one piece. In this case, the slab would form the
upper flange of a T-beam, and in order to insure this action,
sliding between flange and stem must be prevented. Figure
110 represents a portion of a beam, the lines AC, CD, and CB
c P
'Mf ;|i *" |, \M
FIGURE 110.
indicate the directions of the principal stresses. If now the
line of diagonal compression BC is inclined so that the angle
BCD is less than the angle of friction, the flange would slide
in relation to the stem, on account of the joint along the line
MM; the U-bars AC and BD would resist this tendency by
virtue of their " shear " resistance (and this resistance we know
is very small, and cannot exceed the compressive edge resist-
ance of the concrete; see Figures 111, 112, where the black areas
indicate the crushed concrete). If, on the other hand, the angle
BCD is larger than the angle of friction, then there can be no
104
REINFORCED CONCRETE BUILDINGS
sliding, and therefore no shear stresses on the U-bars, which
will act directly in tension as described above. The rule derived
from this argument may be briefly expressed thus: The spacing
of the U-bars must not exceed the depth of the beam, in which
case the angle of forces would be about 45.
FIGURE 111.
FIGURE 112.
69. If we now turn to the T-beam manufactured in one
continuous operation, where no separation exists between stem
and slab, we note that, theoretically at least, this beam is in
the same condition as the one just considered, owing to the orig-
inal assumption whereby the tensile stresses in the concrete
are considered as non-existing. Each and every horizontal
stratum must be considered as isolated and influenced by its
neighbor through the medium of frictional resistance only,
and the direction of the diagonal compression must be such
that no sliding can take place. The same rule must therefore
be imposed in this case.
But this rule gives the maximum spacing possible: owing to
the usual considerations of a margin of safety, the spacing must be
made smaller, and we would therefore recommend that the spacing
of the U-bars must in no case exceed one-half the effective depth of
the beam.
70. Tensile Stresses in Concrete Disregarded. The ready-
made reinforced concrete beam formulas now in common use
are derived under the apparent assumption that the steel rein-
forcement takes all the tensile stresses, and this is also the case
in this book. In reality, we cannot wholly disregard these ten-
sile stresses in the concrete, or, at least, we cannot deny their
existence, because if we did, we would also rob the concrete
of its cohesion, and we would have a granular mass such as sand
or crushed stone, wholly unsuitable for our purpose. The true
statement is that we disregard the tensile stresses in certain
directions and for certain purposes. In this book, we have
considered the concrete as fractured vertically along the planes
TRANSVERSE STRESSES. U-BARS 105
of the U-bars (1) because the cracks in probability will appear
in the weakest plane, there being less concrete to resist the ten-
sion where the concrete is displaced by the steel of the U-bar;
(2) because the U-bar encircling the main tension rod in a meas-
ure acts as a washer on the rod, causing the somewhat resilient
concrete to crack immediately behind this point of gripping;
and (3) because such tests as throw any light upon the location
of the cracks indicate that they occur very largely at just these
points.
71. NOTE: For the gripping action of a loose U-bar encir-
cling the tension rod, see Morsch, page 47.
For the location of the cracks, see the same book, page 155. *
72. We have also considered the stem of our T-beam ' as
composed of horizontal layers acting upon one another by con-
tact only, and thereby determined the spacing of the U-bars.
But between these vertical and horizontal lines of weakness,
we have assumed the concrete to be solid. Hence, we have
assigned to the concrete a certain amount of tensile resistance
in certain locations and directions.
73. It follows that with increasing loads the compressive
stresses in the beam do not increase as rapidly as the load,
especially not in beams where the slab and the stem are sep-
arately manufactured. In such beams, the compression at
rupture must in many cases be uniformly distributed over the
entire compressive zone, and we find here the explanation of
the fact sometimes observed that the compressive strength of
concrete is much higher in a beam test than in a cube test. An
analysis of these conditions would be interesting and of great
value practically.
74. Details of Reinforcement. The various arguments
advanced above will lead to rational design of the steel if con-
sistently applied, and there is but little new to add. The great
principle in all beam construction is that there is a compres-
1 Morsc^i: Concrete Steel Construction, 1909. While the cracks do not
all occur at the U-bars, the tendency is fairly pronounced, especially in the
beams with U-bars in one half only, see Figure 149, Beam V; Figure 153,
Beam VIII; Fig. 154, Beam IX; Fig. 157, Beam X; and compare the cracks
in the U-bar end with those of the other end of the same beam. The draw-
ings of all these beams show them just before final collapse, while our calcu-
lations have reference to a much earlier stage, viz., under the working load,
or at the most a load not more than twice the working load.
106 REINFORCED CONCRETE BUILDINGS
sion and a tension, separate from one another, but with hori-
zontal projections of equal intensity or magnitude, provided
the loads are vertical. Whatever the arrangement, the com-
pression and tension must ultimately meet one another and
annihilate one another, whether this takes place gradually by
increments, as in the plate girder of constant depth; or in one
operation, as in the King truss, where the tension chord meets
the compression chord at the ends of the beam; or in a number
of places, all well defined, as in the Howe truss. We have seen
that the rectangular beam is somewhat similar to the King
truss, and that the T-beam is very similar to the Howe truss;
we have also pointed out that the theory of stress-transmission
by gradual increments is not tenable under high loads owing
to the slight tensile resistance of the concrete. We must
assume that the sooner the compression and the tension are
brought to annihilate one another the better will our beams
withstand the loads, hence the necessity of bending the rods
up as soon as possible, and the desirability of closely spaced
U-bars. A simple and effective way of bending the bars is
shown in Figure 113. The point of bending should be deter-
FIGURE 113.
mined by the bending moment, so that there is steel enough to
meet the requirements at all points. In this beam and the fol-
lowing we must suppose that there are some straight bars, but
these are not shown in the figures. Hence the principal stresses
in Figure 113, disregarding the straight bars, are: a constant
compression along the slab, a constant tension in the rod, and
certain vertical resultants. The rod has a curve under the load
A, against which the concrete is pressing. The resultant of
all these pressures should go through the point of application
of A, hence the rod should be bent to a circle with center in the
point of application. The same applies to the reaction, B, and
in addition the rod should be extended beyond the support to
develop the full adhesive resistance.
A somewhat more complicated method is shown in Figure 114
TRANSVERSE STRESSES. U-BARS 107
where there are two systems of bent rods (aside from some
straight ones). The " first " rod, AC, is curved under the load
P for the reasons explained above; in addition, the resultants
C and D must be made to meet one another in the same point
and with the same direction and same force. Hence the num-
ber of rods in each chord should be the same. The length of
rod in compression flange ab should be sufficient to develop the
full strength of the bond, in the same way as for the " second "
\ \
FIGURE 114.
chord over the point of support. The slope or angle of the bent
bars would seem to be of no importance; but many authorities
are of another opinion and recommend an angle of about 45
degrees. (In practice the bars are seldom bent to such large
radii as shown in Figure 114, this diagram being purposely
exaggerated.)
75. The shape of the U-bars should be as shown in Figure
102a, 6, with curved top and bottom, and hooked over. The
downward projection of the end makes it easy to support the
U-bar on the form work, and the entire U-bar is firmly anchored
against sliding, both top and bottom : the top on account of the
curves, the bottom because it passes around the reinforcement.
The direction of the U-bar should be vertical. The sloping or
slanting U-bar is said to strip the concrete away from the ten-
sion rod, as we might expect if our theory is correct, and it does
not give as efficient reinforcement in the small cantilevers as
the vertical U-bar. Round U-bars appear to be better than
flat bars; but there is a great amount of information along this
and similar lines which will have to be furnished before rein-
forced concrete design can be perfected. But our lack of in-
formation in this and similar cases is not different from that
existing in other lines of engineering.
76. When we now finally combine all these elements to one
beam, Figure 115, we have a structure of a very complicated
nature, and we must ask ourselves if all these stresses can travel
through and between one another as here assumed without
108
REINFORCED CONCRETE BUILDINGS
upsetting our calculations and assumptions entirely. To this
we must answer that we do not know, but if we compare our
problem with those met in other lines of engineering we must
admit that there is no fundamental difference between the diffi-
culties. Thus a combination of two simple Pratt trusses is
treated as if the two trusses were really present individually
instead of combined into one structure, and many other instances
FIGURE 115.
could be cited to show that we often have to dissolve a struc-
ture into its apparent elements in order to solve its problems.
Assuming the reinforced concrete beam to be similar to a Howe
truss, as here proposed, seems to be no more of a mistake than to
assume the connections in a riveted truss to be frictionless,
movable joints. But the approximations made in steel con-
struction are so old that they seem almost part and parcel of
the art, while the comparatively new assumptions made for
reinforced concrete have hardly had time to solidify, and they
are therefore supposed to be of a more questionable nature
than the older ones, which have indeed had the profit of the
test of time. Yet there is a number of reinforced concrete
buildings about thirty years old which stand up as well as any-
body could wish, and the modern steel sky-scraper is of no older
date.
CHAPTER VIII
APPLICATIONS OF THE BENDING THEORY
77. Continuity of Reinforced Concrete Beams. The dif-
ference between the beam with simple supports and the
continuous beam is that the continuous beam is subject to a
" reverse" bending moment over the support, while in the simple
beam there is no such reverse moment. The cantilever beam
is an example of the beam in which only reverse moments exist,
and as we have found it feasible to construct reinforced con-
crete cantilevers we cannot deny that continuity may exist in
reinforced concrete beams. In fact, unless special precautions
are taken to eliminate reverse moments over the supports, we
know that continuity must exist and should be taken into ac-
count. The question is then: to what extent are the ends of
a reinforced concrete beam restrained? When this question
is answered we must make the beam strong enough to resist
the bending moment at the column, and then it is a matter for
further investigation to decide in how far the beam is actually
benefited by the restraint to such extent, that the moment at
the middle of the beam may be reduced.
In Figure 116 a beam is shown in which the ends are per-
p Ibs. lin. foot
FIGURE 116.
fectly restrained, and where the uniform load covers the entire
span. The bending moments are over the supports:
MA = MB = & pi 2
at the center: Me = aV pi 2 -
Hence M A + M c = M B + M c = ( T V + A) P? = i P?;
109
110 REINFORCED CONCRETE BUILDINGS
or: the total amount of bending moment to be taken care of in
the beam with " built in " ends is the same as in a simply sup-
ported beam. The bending moment carried by a reinforced
concrete beam is
m a constant X bd 2 (Formula 8);
hence for constant depth the allowable bending moment is
directly proportional to the width 6. At C, Figure 116, the
width is = 6, but at the support where the reverse moment
must be taken care of, the width of beam is only that of the
stem = n. Hence if we assign a moment M c to the middle of
the beam, the end will only carry a moment
M A =
so that M A + M c = \M C + M c = \ pi 2 ',
u o
b 4
we have MC = % i pi 2 = iV pi 2
and M A = MB = I- T V pi 2 =
T, n 1
For 6 = 6
we have M c = f 1 pi 2 = - pi 2
and M A = MB = J - -& pi 2 = A pi 2 -
The moment at the center of the span, in the case of a T-beam,
will therefore be about
and at the end ^ pi 2 .
i 4U
If greater depth is provided near the support the reverse moment
may be increased and the moment at the center of the span may
be decreased a corresponding amount. 1 In Europe it is quite
Attention is called to the obvious fact that no degree of "restraint"
can be allowed at wall ends; this is especially true for beams resting in brick
work.
APPLICATIONS OF THE BENDING THEORY
111
common to make the beams and girders deeper at the columns;
in America the beams and girders are usually of the same depth
throughout. The American practice is to be preferred, because
the continuous effect depends entirely upon the stiffness of the
supports: the slightest yielding of the footings, or even the com-
pressibility of the columns may destroy the continuity entirely,
and too much dependence upon the continuous effect may lead
to serious trouble.
In a slab, the depth and the " width of beam "is the same
at the middle of the span and at the supports. If the supports
are unyielding there may be some excuse for allowing a higher
degree of continuity for slabs than for beams; the more so
because tests on reinforced concrete buildings point distinctly
to such effects. Let us assume a degree of continuity leading
to the following bending moment:
M e = - p I 2 .
In Figure 117 the equivalent system of construction is shown
FlGURE 117.
in which the center portion is considered as a simple beam
resting upon cantilevers of span R. We have then
(L - 2 R)* = - L\
hence R = % L 1 -
(25)
It is hardly necessary to say that we have no absolute certainty
that the slab will adjust itself to conform to this arbitrary divi-
sion of the bending moment. Yet if the cantilever is made
strong enough to carry its load, and the central portion strong
enough to carry its share, it is difficult to see why such a system
should not be perfectly safe. Other assumptions may be made
and carried through in the same manner; this analysis will be
used later for the calculation of the " mushroom " system as
invented by Mr. Turner.
112 REINFORCED CONCRETE BUILDINGS
The formulas usually given for continuous beams depend
upon the factor EL The value of / for a reinforced concrete
beam is not a constant; in Article 79 we shall consider this
in detail. We will find that the moment of inertia depends
upon the maximum unit stresses in the point considered, and
we cannot expect these stresses to be uniform throughout the
length of the beam. The usual application of the formulas
for continuous beams presupposes that the moment of inertia
is constant throughout the length of beam, and we cannot there-
fore apply the formulas used for homogeneous beams to the
reinforced concrete beams with any degree of certainty.
78. While, then, the exact degree of continuity cannot be
determined, continuity does nevertheless exist in many cases
if not in all, and the stresses thus created must be taken care
of. These are, primarily, tensile stresses over the supports,
requiring reinforcement in the top of the girders over the col-
umns, in the beams over the girders, in the slabs over the beams.
The top bars may be loose bars, but it is rather difficult to main-
tain such bars in their proper position; the bent-up bars may
be utilized as top reinforcement with good results, especially
as they extend a distance into the next bay in any case. It is
evident from the remarks made above that the top reinforce-
ment over the support should not be less than 25 per cent, of
the bottom reinforcement; usually more bars are bent up, but
they need not all extend as far beyond the support as the bars
designed to resist the reverse moment. For a uniform load
covering the entire span, the point of inflexion is evidently deter-
mined by the Formula 25:
R = JLfl
v/S
so that 25 per cent, of steel mentioned above should be carried
at least that distance out from the center of the support. The
bars must be embedded in a sufficient amount of concrete to
develop the bond, not less than four diameters from the face of
the concrete, or, if closer to the face of the concrete, they should
be provided with inverted U-bars. The stress on these U-bars
cannot be calculated, it is their presence rather than their
strength which benefits the beam.
79. Moment of Inertia. The moment of inertia in a rein-
APPLICATIONS OF THE BENDING THEORY
113
forced concrete beam is of interest only because certain prob-
lems connected with continuity of the beam, deflection, etc.,
cannot be solved except through a knowledge of its value. The
expression given below is of indirect value only, showing that
the ordinary formulas for continuity do not apply to reinforced
concrete beams, because the moment of inertia is not a con-
stant for the length of the beam, as is usually assumed in the
solution of such problems.
The moment of inertia with reference to the neutral axis
may be found as the sum of two moments: /i, referring to the
concrete above the neutral axis, and J 2 , referring to the steel
below the neutral axis, the concrete below this line being dis-
regarded as usual. We have then (Figure 118)
"a sq. inches
FIGURE 118.
and
Jl=l
7 2 = rad 2 (1 - x) 2
the steel being considered as equal to ra square inches of con-
crete. But according to the formulas given in Articles 25 ff. we
have
i Cxdb = aS
a _xdbC_ Mb .
2 S 2(l-x) r'
hence (after some reduction) :
7 = /i + 7 2 = J fed 8 (1 - J a) x 2 (26)
By means of Formula 5 in Article 27 the expressions derived
above for d and s t , etc., may now be verified. The real import-
ance of Expression 26 is, however, that it shows that the moment
114
REINFORCED CONCRETE BUILDINGS
of inertia depends upon the location of the neutral axis which
again changes with the stresses in the various points of the
beam.
80. Beams with Reinforcement in the Compression Side.
Sometimes it is found impossible to make the compression
flange of the beam wide enough to bring the concrete stress down
to the allowable maximum. In that case some engineers use
compression reinforcement, but as a matter of fact, our knowl-
edge of the properties of such beams is very slight, and there
is grave doubt as to the advisability of using this method of
construction in important cases. The calculations are simple:
to the bending moment sustained by the beam with its ordi-
nary amount of reinforcement is added another bending moment
due to extra reinforcement in top and bottom, this latter cal-
culated as for an ordinary steel beam, but with quite low stresses
(not to exceed 10,000 Ibs./square inch). The compression bars
must be laced carefully to r the tension bars, but under any cir-
cumstances it seems hardly possible to provide properly for the
excessive shear stresses set up in this kind of beams. A steel
I-beam is cheaper and better in places where this kind of con-
struction is actually necessary.
81. Combined Bending and Compression. The section
is best designed by trial. In the case of an arch ring, the sec-
tion is rectangular, and the symmetrical reinforcement is of
small area compared with the concrete sec-
tion. The bars on the compression side
must therefore be disregarded, as it would
require too many hoops to make this rein-
forcement effective in compression. We must
select the depth and the reinforcement by
judgment; the stresses due to the bending
moment alone are then easily found by For-
mula 19 in Article 49. Let Figure 119 rep-
resent the section; let C m and S m denote the
stresses just found due to the moment alone.
If now in Figure 119 gh is made equal to
S m /r and ef is made equal to C m , then the line gf will represent
the distribution of stresses on the section due to the bending
moment alone. The stress due to the pressure P is now P/bd
Ibs./square inch; this is represented by the line ik parallel with
FIGURE 119.
APPLICATIONS OF THE BENDING THEORY 115
fg. The total pull in the steel is then equal to the area of
the triangle khl times the width 6 of the section. For slabs or
arches the width is usually taken as 12 ". The final concrete
stress ei must not exceed the allowable stress; we can therefore
arrive at a preliminary estimate of the dimensions required by
Formula 16, assuming a materially lower " allowable stress "
for the concrete, and a higher stress for the steel, when making
the first trial.
If the section is one in a column the calculations are essen-
tially different. The eccentrically loaded column is of fre-
quent occurrence; in fact, few columns are always loaded
centrally. In practical cases it is almost always impossible to
calculate the eccentricity of the load, and elaborate formulas are
therefore of little or no use. Tension should never occur in the
column; if there is tension with the selected arrangement it is
better to change the lay-out. The percentage of steel will
always be much greater than in the case considered above,
and, as there is no tension, we may perhaps calculate our col-
umn as a homogeneous section, using, however, for the moment
of inertia the expression
/ = I e + r I s (27)
where ] T c ~
i. l s =
I c = mom. of inertia of concrete alone,
mom. of inertia of steel alone.
The cases where the condition of loading can be ascertained
with any degree of certainty are very few indeed, and when
they do occur the bending moment is likely to be very small.
If such is the case it is simpler and probably as correct to cal-
culate the column as a pure column, using a correspondingly
higher factor of safety, and then, if necessary, finally investi-
gate the problem assuming the neutral axis to be disposed at
the center of the section, and take the moment of inertia with
reference to the center line.
82. Chimneys. As an example of approximate methods
of calculating a piece subject to bending and compression, let
us consider a single shell chimney of uniform thickness. The
diameter d (in feet) of the flue is given, and so also the height H
(in feet). Let the outside diameter be D (in feet); the area
presented to the wind pressure (w Ibs. per sq. ft.) is then DH
116
REINFORCED CONCRETE BUILDINGS
square feet, and the total pressure DHw Ibs. Hence the bend-
ing moment at the base (the overturning moment) becomes
DHw times J H = \ wDH 2 Ibs. X feet.
If now the total allowable compressive stress on the concrete
is C Ibs./sq. in. and the compressive stress due to the weight
of a column of concrete 1" square and H feet high is (approxi-
mately) H Ibs./sq. in., then the compressive stress due to
the overturning moment must not exceed C -- H Ibs./sq. in.
Assuming the neutral axis to go through the center of the sec-
tion, which indeed is not true, and disregarding further the bene-
fit derived from the steel in the compressive side (which is on
the safe side), the moment of inertia of the ring is
hence
(C - H) 144 =
64
(D 4 - d 4 )
D
2
which, when solved, gives the outside diameter
D
J
V -
|
187T (C - H) ' V V187T C - H,
and the tension per inch of circumference becomes
\ (C - 2 H) (D - d) Ibs.
83. Footings. In Figure 120, 2R (inches) denotes the side
of the footing, 2r (inches) the side of the column. The bending
FIGURE 120.
moment on side ab (considering the footing as a cantilever-
slab) corresponds to the loaded area dabc. We have, for a
load p Ibs./sq. in. :
Load dcef = (R r)2Rp; arm of bending moment around
ef - i (R - r).
APPLICATIONS OF THE BENDING THEORY
117
Hence bending moment
A = pR (R - r) 2 ;
load aed plus bcf = (R r) 2 p ; arm of bending moment around
ef = 402-0.
Hence bending moment
B = I p (R - r) 3
The total bending moment due to the area abed is then the
difference between A and B;
and the depth of footing becomes, according to Formula 8, for
a width of beam 2r = b
' *--'-v*/ +1I
-R-rJp
ci VG
to which corresponds a pull s, in the steel, for the distance ab
s t = 77: 2 r
1-
It is, however, quite necessary to provide reinforcement for
the portions ae and bf ; for this reason the amount found above
may be multiplied by a factor estimated at about 2, which
gives :
s = C ^rd (29)
for each layer of steel (Figure 121). The radius of the column
i
314
FIGURE 121.
should be made as large as possible, because a material saving
in depth of footing is obtained thereby; usually the column
must have an enlarged base for other reasons as well. In Ar-
ticles 14 and 15 we found the cross-sectional area of column:
X = 1400 F for a hooped column
X = 1060 F for a plain reinforced column,
118
REINFORCED CONCRETE BUILDINGS
so that the average pressure, under the conditions assumed, is
1400 and 1060 Ibs./sq. in., respectively. With higher per-
centages of reinforcement these pressures may become mate-
rially higher; the column base is. therefore enlarged so that the
pressure on top of the footing does not exceed the allowable
unit pressure, and a steel plate is put under the bars in order
to distribute the pressure over the requisite area. According
to tests by Bach this allowable pressure may be somewhat
increased, owing to the reinforcing effect of the surrounding
concrete of the footing, but it does not seem wise to exceed say
1000 Ibs. per square inch. The thickness of the plate may be
approximately determined by means of a formula by Grashof:
t = ^Vp (30)
where
t = thickness of plate, in inches,
r = radius of reinforcement (= | of diameter of column, less 2"),
in inches,
p = pressure on plate, in Ibs. per square inch.
The dimension Q in Figure 129 may be found by Formula
FIGURE 122.
28 above, using for p the allowable pressure on the concrete.
84. Circular Reinforcement in Plates. The circular plate
in Figure 122 is supported on a central column. The load is
APPLICATIONS OF THE BENDING THEORY 119
uniformly distributed over its surface, or symmetrically and
continuously disposed along the circular circumference. A
segment, Oab, will then be subject to a certain bending moment,
which moment determines the depth D at the circumference of
the column. It is now easy to show that when the load is uni-
formly distributed over the entire surface, the same formula
applies in regard to depth as was derived above for a square
footing; the calculations are practically the same and need not
be repeated here. When the load is distributed along the
edge, the Expression 32 in the following article may be used.
In any case, we will assume that the depth is known in the
thickest part of the plate (at the edge of the column). If now
the distance dc is one inch long, we have by Formula 9
s t = T V c 2 D,
which expression leads to the amount of steel required along
the radii, the bars being I" apart on the circumference of the
column. Imagine now that all these radial bars be cut asunder
over the top of the column disregarding the tensile strength
of the concrete, each bar will then have a tendency to move
outward, so that if a steel ring surrounded the entire plate,
each bar would exert a pressure s t against the inner face of the
ring. If now dc equals one inch, then db equals one inch times
R/r , hence the pressure on the ring, measured in pounds per
lineal inch of circumference, equals s t X r /R. The tension
on the ring is then
SR = s t ' r ^.R = ^czr D. (31)
It is now evident that the ring with radius R and designed to
resist the tension S R is, mathematically, sufficient reinforce-
ment, so that the radial bars may be dispensed with. In actual
practice this is somewhat modified owing to the fact that con-
crete shrinks when setting, so that it would pull away from the
ring; the ring would therefore exert no pressure against the
concrete until a substantial, and perhaps dangerous, deforma-
tion had taken place. But when the ring is used in combina-
tion with a radial reinforcement and when at the same time the
depth D is not too small compared with the radius, say D larger
than ^ R, then the ring would seem to be a very efficient rein-
forcement. Direct proof of this statement is indeed missing,
120 REINFORCED CONCRETE BUILDINGS
but the " Mushroom " floors furnish at least some indirect
information in this respect, as they probably owe their strength
in a great measure to the intelligent use of circular reinforce-
ment. That this type of reinforcement is successful in other
types of structures may be seen from the remarks made under
" columns," where hoops are extensively used to take care of
stresses somewhat similar to those existing in a plate, although
the plate at the same time acts as a beam. Exact analysis is
of course difficult in these structures which border upon the
class where reinforcement may sometimes be omitted entirely : it
is well known that tapering footings are often constructed with-
out steel, and the same may be true of columns in special cases.
85. Theory of plates. The " Mushroom " System.
A reinforced concrete floor without beams or girders is first
indicated and patented by Mr. C. A. P. Turner of Minneapolis.
As far as known there is no perfectly satisfactory way of finding
the stresses in constructions of this kind, although buildings
actually constructed on this principle have given good satis-
faction, according to the published records. The stresses must
necessarily be of a very complicated nature, especially under
concentrated or unsymmetrical loads; the following analysis
does not pretend to solve the problem in anything approaching
a general way, and the formulas apply only in case the entire
building is loaded with a uniformly distributed load. The
formulas are not inconsistent with the assumptions made for
reinforced concrete construction, and they are therefore pre-
sumably a step in the right direction. It is well known that
most of the proposed formulas are based upon the theoretical
strength of the plates with equal tensile and compressive resist-
ance, and reinforced concrete does not possess any such qualities.
Figure 123 shows the general scheme for a floor of this kind:
the floor slab is simply a flat plate resting upon columns, the tops
of which are enlarged. Let the uniformly distributed load be w
Ibs./sq. foot and the span I feet. The slab is divided into six
strips: two diagonal strips AD and BC, and four strips along the
sides AB, BD, DC, and AC. If we now suppose the panel to be
square, the load on each of the crossing diagonals may be taken
as \w, while the span AD = BC = I V2. Then, by Formula 30
for AB: d = lJ and for BC: d= ^ . t/SL 1 - 1/?
Ci V q a V q a V q
APPLICATIONS OF THE BENDING THEORY
121
so that the depth is uniform, and our problem centers around the
design of a side strip like AB. The notations are shown in Figure
124, where
If
FIGURE 123.
FIGURE 124.
L inches is the span between column centers.
p the load in Ibs./sq. inch.
R the radius in inches of a certain circular plate.
r the radius in inches of the support under the plate, here
referred to as the " cap."
p the radius of the column in inches.
d the depth of the slab in inches.
D the depth of the cap in inches.
We will now proceed as follows : We consider the floor slab as
supported on the edge of the circular plate with radius R; this
plate will then have a uniform load on its surface and a concen-
trated load along its circumference. Finally the " cap " with
radius r will be designed for a load concentrated on its circum-
ference, disregarding the uniform load on its surface.
The total area of the floor panel between the four column
centers is L 2 square inches; the total weight corresponding to
this area is pL 2 Ibs. The area of the circular plate with radius
R is irR 2 , the total weight on same ptrR 2 . Hence the weight
of the portion outside the circular plate becomes p (L 2 irR 2 ) Ibs.
122
REINFORCED CONCRETE BUILDINGS
In the following computations, L 2 is always large compared with
TrR 2 so that this quantity may be neglected, which is also on the
safe side. The load is therefore pL 2 , and as the circumference
of the circular plate is 2 TrR, the load per lineal inch of circum-
ference becomes
producing a bending moment equal to
_
27TR
(R r ). If measured per lineal inch of the circumference
of the cap with radius r it becomes, by multiplication with R/r c
(Figure 125),
p
L 2 -7T R 2
27TK
FIGURE 125.
FIGURE 126.
and the corresponding depth, for 6=1"
L
R - r ) (32)
for the circular plate. For the slab portion we have, by
Formula 16:
7 LT, T. ' l~
V < 33 )
q c,\ q
'The value of r must now be such that the two depths become
alike, which gives
q
r n
(34)
at the same time, the value of R is determined by the selected
value of q by formula
APPLICATIONS OF THE BENDING THEORY 123
see Article 67, Formula 25. The depth of the cap. is found by
the formula above, as the load again is pL 2 , substituting only r
for R and p for r ot hence
D--J^-(r.-P). (35)
Ci V 2-n-p
According to Article 84 the reinforcement may be disposed in a
ring with radius r ; the tension in this ring becomes:
S T = ^ c,r D (36)
LA
The arrangement is shown in Figure 126, where the thickness t
should be about 4" so as to cover the ring thoroughly. The cap
should be cast in one piece with the column, but there is no
reason why a joint may not be made between the top of the cap
and the bottom of the flat portion along line a a in Figure 126.
The reinforcement for the flat portion is designed as for any
other slab. We have the depth
- (33)
q
and the corresponding pull in the steel $/ tons, for a band one
foot wide, is therefore, according to (17)
S f = c z d (37)
This reinforcement should be disposed near the bottom of the
slab at the center of the span, and near the top over the columns.
It will be seen that this leaves a considerable space around the
column without reinforcement near the bottom, which should
be avoided. We may therefore follow the prevailing practice
and bend every alternate bar up, leaving the balance of the
steel straight near the bottom. The reinforcement over the
column is then inadequate, and we will have to introduce addi-
tional steel at that point; if we decide to use rings we may
use one ring with radius R, the strength of which is determined
according to Article 84 by the formula
(38)
We have now (Figure 127)
Thickness of slab, in inches d = y - (33)
Pull in slab steel per foot wic
Radius of upper ring, inches
Tension in upper ring, tons
Radius of cap-ring, inches
Tension in cap-ring
Depth of cap, inches
q is the factor of continuity,
L- r - -
ith, tons Sf = c . , ft (
Beam
Type
B
No.
ent
Straight
1\
fl o
gs
U-bars
Type of
Loading
"S-S u 03
111 I
P>3 3
Diam.
in
No.
Diam.
in
m/m
m/m
IS -2
H2
IV
T
3
15 and 1
18
14
none
Uniform
42.0
VI
T
3
15 and 1
18
14
full supply
load
37.8
V
S
-2
15 and 2 16
14
one end only
covering
31.0
entire span
Two
VII
T
3
16 and 1
16
14
full supply
concentrated
34.0
VIII
S
2
16 and 2 16
10
one end only
loads
23.4
IX
S
2
16 and 2 16
14
one end only
at third
25.6
points
X
T
3
16 and 1
16
14
one end only
One
27.0
XI
S
2
16 and 2 16
14
none
concentrated
26.0
XII
T
3
16 and 1
16
14
none
load
26.0
at center
The tests naturally divide themselves into three groups,
according to the manner of loading:
(1) Uniform load, beams IV, VI, and V. It is at once
apparent, by comparing beam IV (without U-bars) with beam
VI (with U-bars), that the influence of the U-bars is very slight,
if any, the difference in ultimate load being accounted for by
the fact that the ends of the straight bar in beam IV were hooked,
while those in beam VI had no hooks. Both of these beams were
of the trajectory type, and if we compare them with beam V of
the suspension type, the superiority of the trajectory type seems
clearly established. But we must not lose sight of the fact that
in the two first beams three of the four rods were bent up, while
in the latter, only two of the four rods were bent up. This
beam failed in the end without U-bars, and while therefore
this group does not prove the author's theories, as outlined in a
preceding chapter, it does not disprove them, and still leaves
the question open whether or not the bending of one additional
rod, or the proper use of U-bars, would not have changed the
results materially. It will be remembered that in a T-beam,
the straight reinforcement is effective only as reinforcement of
the stem, while the bent bars correspond to the flange; if the rods
are not so arranged, stirrups must be introduced to again bal-
212 REINFORCED CONCRETE BUILDINGS
ance the design, the size of the U-bars being in direct ratio to
the violation of the principle outlined.
(2) Two concentrated loads, beams VII, VIII, and IX.
Here again, beam VII of the trajectory type, with a full
supply of U-bars, is compared with two beams of the suspension
type, the two latter being without U-bars. Again the traject-
ory type seems superior to the suspension type, and again we
find the reason to be that in the trajectory beam, the proper
amount of rods have been bent up, while in the suspension type,
only two of the four rods have been bent, and no U-bars have
been introduced to overcome the deficiency.
It is rather interesting to note that the difference in width
of stem between beam VIII (10 cm.) and beam IX (14 cm.) affects
the ultimate strength but slightly, both beams being of the
suspension type.
In his discussion of these tests, Prof. Morsch has taken
occasion to criticize the suspension, or Hennebique, type. It
is to be regretted that the tests were not carried out so as to
have the same number of bent-up bars in both of the types
considered, in which case the suspension type would probably
have stood up as well as the trajectory beams. It is only fair
to note that the U-bars or stirrups have always been considered
as an essential part of the Hennebique system, and that such
tests as these, however valuable otherwise, give no indication
whatever as to the merits of this system.
(3) One concentrated load at center, beams X, XI, XII.
In this group, the two systems give the same carrying
capacity, owing undoubtedly to the fact that in no one of these
beams the reinforcement is arranged according to the equilib-
rium curve, while in no case U-bars have been introduced to
compensate for the deviation.
Bent Bars in T-Sections Author's Tests. The beam tests
just referred to were published by Prof. Morsch in "Deutsche
Bauzeitung," April 13, 1907. It occurred to the author of the
theory of this present volume that the description of the action
of the suspension rods was subject to doubt, for the reasons
outlined above, and that additional information might possibly
be gained by tests on beams with trussed rods only. The author
designed a series of nine test beams which were tested in the
winter 1907-1908 at Case School in Cleveland, in co-operation
THEORY OF BEAMS AS ILLUSTRATED BY TESTS 213
Type A
ams 1,4,7
n
TypeB-
ams 2,8.
\
>> c
"I
r U
omag jo JlH 9U I 5"
^ ^H^
|<
214 REINFORCED CONCRETE BUILDINGS
with Prof. F. H. Neff. It will be seen from Figure 151 that
these beams had no straight reinforcement, and that the sloping
stem terminated at the supports, so as to make the system one
of equilibrium under two concentrated loads. The results were
first published in the Engineering Record, August 22, 1908, from
which the following is an extract:
" Three different molds were made, types A, B, and C,
respectively, each one of which was used three times with a
different percentage of steel for reinforcement of the beam.
In this way, three beams of type A were made, one of which was
reinforced with 0.5 per cent., one with 0.75 per cent., and one
with 1.0 per cent. In the same way three beams B and three
beams C were made, reinforced as described, so that of the total
number of nine beams no two were alike in all respects, but any
one beam would have a corresponding one which was different in
one detail only. In this way, it would be possible to compare the
beams and find the exact effect of a certain change, which is a
safer way than to try to obtain absolute results from so few tests.
"All the beam swere provided with U-bars in one end only,
the object being to show that the stirrups were of no conse-
quence at all. The stirrups made no difference in the results
obtained, four of the nine beams failing in the end equipped
with U-bars.
"Two short cross bars were placed in the slab at the points
where the loads were applied, and three similar bars were placed
in the slab near the support. These bars were i-inch square
twisted bars. The main tension bars were 1-inch square twisted
Ransome bars. It was found that the elastic limit of these
bars averaged about 56,000 Ibs. per square inch, and their ulti-
mate breaking strength was 73,600 Ibs. per square inch.-
" The concrete was made quite wet and very carefully placed.
The mixture used was 1:2:3^, Lake Erie sand and Euclid bluestone
being used for the aggregates. The strength of the cubes was low,
as might be .expected with the aggregates used, and the average
of the 6-inch cubes in pounds per square inch was as follows:
Age, days 7 14 28 60
Strength, pounds 660 1,065 1,440 1,787
"The beams were all tested when sixty days old. In the
table here below the results are given, and this table, together
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 215
with the diagrams of the beams, should give all the information
needed. Attention is called to the ways of supporting beams 5
and 6. While all the other beams are supported at the point
where the sloping stem begins, these two beams are supported
further out from the stem, making the overhang shorter for
them than for the similar beams of same type.
"As to the column headings used in the table, the percen-
tage of reinforcement is calculated with reference to the ' enclos-
ing rectangle' proposed by Professor Talbot. Under 'lever' the
distance from point of support to point of application of the
load is given, while 'overhang' means the length of the pro-
jecting end beyond the support.
"The bending moment given in this table is found by mul-
tiplying the 'lever' by one-half of the ultimate load, disre-
garding entirely the weight of the beam itself. The lever arm
of the internal stresses is assumed to be 0.85 times the distance
from the top fiber to the center of the steel, which distance is
approximately 9 in., giving a lever arm of 7.65 in. This, of
course, is not quite correct, as the position of the neutral axis
varies with the percentage of steel and the coefficient of elas-
ticity, which latter again depends upon the stress on the
concrete. It is, however, sufficiently accurate considering the un-
avoidable variations in the position of the steel bars and in the
elastic properties of the concrete, and the 'total stress in the
steel' may therefore be found by dividing the bending moment
by 7.65, giving the values shown in the table as well as the
stress in the steel per square inch of its cross-section.
RESULTS OF TESTS AT CASE SCHOOL.
Beam. Type. Per cent. Lever. Overhang. Ultimate
load.
0.5
0.5
0.5
0.75
0.75
0.75
1.00
1.00
1.00
30 in
26 in
22 in
30 in
30 in
30 in
30 in
26 in
22 in
10 in.
14 in.
18 in.
10 in.
10 in.
10 in.
10 in.
14 in.
18 in.
12,500
16,000
27,800
11,900
16,200
16,000
13,950
22,000
28,900
Bending
- Stress in steel
moment.
Total.
Per sq. in.
187,500
24,500
49,000
208,000
27,200
54,400
305,800
39,900
79,800
178,500
23,300
31,000
243,000
31,800
42,400
240,000
31,400
41,800
209,250
27,300
27,300
286,000
37,400
37,400
317,900
41,600
41,600
"Eef erring now to the several photographs of the beams
after failure, it will be noticed that the failures are of uniform
nature. Comparing the figures given in the table above, it
will be seen that the ultimate load varies greatly as well as the
total stress and the stress per square inch. If the failure has
216
REINFORCED CONCRETE BUILDINGS
Beam 1
Beam 2
Beam 4
FIGURE 152.
FIGURE 153.
FIGURE 154.
Beam 5 FIGURE 155.
THE CASE SCHOOL BEAMS AFTER TESTING.
TtiE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 217
a common cause in all these beams it cannot be due to either
tension or compression in the usual sense of the word. It may
also be assumed that shear had little to do with the failure.
Beam 6
FIGURE 156.
Beam 8
FIGURE 157.
Beam 9 FIGURE 158.
THE CASE SCHOOL BEAMS AFTER TESTING.
On account of the trussed form of the beams, the steel follows
the curve of equilibrium of the external forces acting upon the
beam, and the only stresses possible are tension in the steel and
compression in the concrete.
218 REINFORCED CONCRETE BUILDINGS
"This is also evident from the behavior of the beams under
load. The cracks started on the tension side and opened slowly
with increasing load, at the same time becoming longer, until
finally the compressive area left above the top of the crack became
too small to carry the stress on it and crushed. A shear crack
cannot grow in this manner. It is well known that the maxi-
mum shear stress does not occur at any fiber near the extreme
top or bottom of a beam. Therefore, when the crack extends
up into the stem and reaches the neutral axis, the shear resist-
ance of the beam is practically exhausted.
" The beams also made it evident in other ways that no ver-
tical shear was active. In some cases the beams had received
a vertical crack in handling, the crack being located about 3 in.
inside the support, and extending clear through the concrete.
At first, it was believed that these beams would not give a fair
test, and it was taken under consideration to leave these beams
out. It proved, however, that the crack closed up as soon as
the load was put on, and after the load was increased to a cer-
tain amount, the cracks were hardly visible, while the final
failure took place some distance from the injured section. If
there had been any vertical shear acting on the beam, the ulti-
mate load would have reached a comparatively small value only,
and in all probability the injured section would have sheared
off at once.
"The tension in the steel must be constant from end to end
of the beam between the supports. The steel would have a
tendency to pull out of the overhanging ends with a force equal
to the total pull in the steel, which is the same near the supports
as at the center of the beam. The overhanging ends furnish
the necessary anchorage for the bars on account of the grip
of the concrete around the bars, which increases with the com-
pression in the concrete, and, therefore, also with the load, the
horizontal cross-bars giving the required horizontal restraint of
the concrete to produce the desired effect. The numerical
value of the length of the anchorage may therefore be expressed
in figures by simply dividing the length of the overhang into
the total pull on the steel, the quotient giving the value of the
bond in pounds per lineal inch of embedment, regardless of the
amount of steel. This figure is given in the accompanying
table, the length of the anchorage being the length of the over-
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 219
hang, and disregarding the extra length of the hook at the
ends of the bars. Beams 5 and 6 are not included in the table,
as these beams had an overhang of only 10 in., leaving a hori-
zontal space inside the support, and this, of course, makes it
impossible to compare these two beams directly with the rest.
VALUES OF BOND OBTAINED
Total pull Bond
Beam Type Overhang in steel per lin. in.
1 A 10 in. 24,500 2,450
2 B 14 in. 27,200 1,940
3 C 18 in. 39,900 2,230
4 A 10 in. 23,300 2,330
7 A 10 in. 27,300 2,730
8 B 14 in. 37,400 2,670
9 C 18 in. 41,600 2,310
"This table, it is believed, is remarkable when the uniform-
ity of the results is considered. The beams tested here had
reinforcement varying from ^ of 1 per cent, to 1 per cent.,
spans varying from 74 to 80 in., and tension stresses in steel
varying from 27,300 to 79,800 Ibs. per square inch. It seems safe
to say that these beams all failed by sliding of the steel.
" So far, no attention has been paid to beams 5 and 6. The
overhang for these beams was 10 in. in each case, the slab con-
tinuing for a distance inside the supports. The bond stress
developed in the overhang, if figured as for beams above, be-
comes 3,180 and 3,140 Ibs. per linear inch, or quite high when
compared with the results of the table above. Remembering,
however, that the straight portion of the bar is continued inside
the supports for a distance of 4 and 8 in., respectively, the bond,
if distributed over the total distance of 14 in. for No. 5 and 18
in. for No. 6, becomes:
Total stress Bond
Beam Type Overhang in steel per lin. in.
5 B 10" + 4" = 14" 31,800 2,270
6 C 10" + 8" = 18" 31,400 1,745
" If any importance can be given these two isolated results,
they would show that the bond inside the support is quite as
effective as that outside the support, but for a short distance
only, and that its value decreases rapidly with the distance
inside the support."
220 REINFORCED CONCRETE BUILDINGS
The lessons to be drawn from these tests are:
(1) That, with the arrangement used, the presence or ab-
sence of U-bars does not influence the strength of the beam.
(2) That "shear," properly understood, does not exist in
beams of this kind.
(3) That, with proper arrangement of the end supports
and of the anchorage, such beams will not fail until the com-
pressive strength of the concrete, or the tensile strength of the
steel, is exhausted.
(4) That such beams are rational structures capable of prac-
tical and economical use.
(5) That the sliding resistance of the steel does not depend
upon the number or size of the individual rods, but only upon
the anchorage of the group of rods, the length of embedment
being much more important than the diameter of either each
rod or of the group of rods.
Effect of Joint between Slab and Stem, Tests by Professor
Johnson. In connection with the introduction of the Ransome
Unit System (p. 162 ff.) in Boston, a series of very interesting
tests were made on T-beams of both the monolithic and unit
types, reinforced with straight bars only, and with both straight
and bent bars. All the beams had U-bars. In the "Unit"
beams, the slab was cast from four to nine days later than the
stem. A total of twenty-eight beams were prepared, of which
eleven have so far been tested, the balance being held for a
longer-time test. 1 The beams were all reinforced with Ran-
some steel, that is, cold-twisted squares. The U-bars were
round except in Type C, where square twisted U-bars had been
used.
Type A, Beams 1, 2, 4, 5, 9, 10, and 12. See Figure 159.
1 The authors are indebted to Prof. L. J. Johnson, M. Am. Soc. C. E., for
the following data, and for permission to publish the same. The beams were
designed by Prof. Johnson, by Mr. J. R. Worcester, M. Am. Soc. C. E., Consult-
ing Engineer, by Mr. J. R. Nichols, Jun., Am. Soc. C. E., by the Concrete
Engineering Co. of Boston, and the Ransome Engineering Co. of New York,
each having designed one series of beams or contributed to the design by
suggestions. Professor Johnson, who made the tests on the testing machine
in the Harvard University laboratory, expects to publish in due season a
complete report of both this series and of the long-time tests. The authors
of this present volume, eye-witnesses of these tests, are solely responsible for
conclusions reached herein.
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 221
In the Unit beams, the top of the stem was either left fairly
smooth, as it would be in usual every-day practice, or corru-
Toggle for lifting beam
8-8
Rods
"13-"^
Bars y \ t 1 4^ ^ 6 * Stirrups 5^
Straight'
FIGURE 159.
gated as shown in Figure 160. The ends of the stem rested in
previously prepared seats (Figure 161), and the joints were
Y
FIGURE 160.
sealed with grout to ensure a similar action as obtained in actual
construction, where the Unit beam rests in a pocket in the gir-
der. Nine days after the casting of the stem, the slab
was put on, while in the monolithic beam, the entire
amount of concrete was, of course, deposited in the
forms in one operation.
Beam No. 5 was a Unit beam, with the top of the
stem corrugated. The age of the stem was forty-
five days, that of the slab thirty-five days. At a
total load of 12,000 Ibs. the first tension crack ap-
peared near the middle of the span. Inclined cracks
became evident near the ends under a load of 23,000
Ibs.; the ultimate load was 41,000 Ibs., when failure
occurred, through compression of the slab between
the loads, and slipping of the straight tension bars
(see beam No. 12 below).
Beam No. 4 was a Unit beam, the top of the
stem being fairly smooth; that is, no attempt had
been made toward getting a particularly .rough sur-
face. The age of the stem was forty-five days, of the slab
t
t
i
1
I
i
1
FIGURE 161.
222
REINFORCED CONCRETE BUILDINGS
thirty-six days; the first crack was observed under a load of
20,000 Ibs.; ultimate failure took place under 48,000 Ibs. in
precisely the same manner as in No. 5.
Beam No. 1 was exactly similar, except that slab and stem
were both one day older than in No. 4; the ultimate load was
49,400 Ibs., and the beam failed in the same manner as the fore-
going.
Beam No. 2 was of the same age and detail as No. 1 ; the first
crack was observed at 10,000 Ibs. loading; the ultimate failure
occurred in the same manner as above under 54,300 Ibs. total.
The higher load on this beam is perhaps due in some measure
to the fact that the rocker-supports for the beam came to a
bearing, making possible some horizontal thrust on the beam.
Beam No. 12 was monolithic, forty-one days old; of same
design as the foregoing Unit beams, except that the fillet between
stem and slab was slightly reduced (see Figure 162). The first
FIGURE 162.
crack occurred at 4,000 Ibs., the ultimate load was 45,800 Ibs.,
and failure occurred through a slip of the straight reinforce-
ment, causing the sudden collapse of the left end.
Beam No. 9 was of the same general design, cast in one
piece, and forty-one days old. The first crack occurred at
3,000 Ibs.; the beam failed suddenly at 50,000 Ibs. by slipping
of the rods at the right end.
Beam No. 10 was also a monolith forty-one days old, show-
ing a tension crack at 6,000 Ibs., with ultimate failure at
52,400 Ibs. from a combination of initial sliding of the tension
rods with compression at the center.
It will be seen from these data that the Unit beams stood up
as well under the load as the monolithic beam, so that the joint
between slab and stem was perfectly adequate, whether cor-
rugated or plain. The general behavior of all these beams up
to the point of failure was so much the same that no one, from
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 223
observation of the beams in the machine, could have pointed out
which beams were unit and which monolithic. In fact, they all
failed in the customary manner, exhibiting the usual inclined
and vertical cracks, and no sliding was noticeable between slab
and stem, although carefully looked for.
Type B, Beams 25 and 27. See Figure 163.
2- 3 /- Bars'T t I Ho Stirrups
2-K" Straight
Ho Stirrups 5K
FIGURE 163.
The beams of Type B were built exactly as the beams of
Type A, except that the tension rods had been reversed, being
2|" bars bent and 2^" bars straight. This reinforcement
would, under the theories- advanced in this book, be more effi-
cient, and the U-bars were therefore reduced from T V' round
stock in Type A to i 3 s " round stock in Type B, thus having about
one-third of the area of the former.
Beam No. 25 of Unit construction had a stem twenty-nine
days old and a slab twenty-five days old; the first crack was
observed under a load of 9,000 Ibs., and ultimate failure took
place under simultaneous compression of the slab and of the
side of the stem, at the point where the tension rod was bent,
under a load of 42,500 Ibs. (See Figure 164.)
2 - Straight
FIGURE 164.
Beam 27
Beam 25
Beam No. 27 was monolithic, of same design, and twenty-
seven days old. The first crack was seen at 10,000 Ibs. ; while
the ultimate load was 47,500 Ibs. Also in this case was com-
pression in both slab and stem evident as shown in Figure 164.
224
REINFORCED CONCRETE BUILDINGS
It is interesting that these two beams carried practically
as much load as the older beams of Type A, in spite of the great
reduction in the weight of the U-bars. The explanation is to
be found in the theory set forth in Chapter VII of this book,
where the relation between the bent bars and the U-bars has
been considered at length; in fact, the design of beams 25
and 27 was made to prove, or disprove, these theories as far as
possible.
Type C, Beams 13 and 20. See Figure 165.
i /i
1 ,,0-10 ,,-J
'oggle f orjif ting beam k-8 >r< 8 >i
9_ S/'
1
-f
*^
/8-X Rods
_^.
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i
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J.-L_L_L_L
_o_L 1 X i J J J J
._ 1
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V 2-1"
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FIGURE 165.
In designing these beams, Professor Johnson had endeavored
to secure a high strength in compression and tension. The
special feature was the absence of bent bars so that the stresses in
the stem and in the joint between slab and stem were especially
severe under the common theory of shear. The beams rested in
concrete supports shown in Figure 166.
Beam No. 20 was of the Unit type with corrugated top
(Figure 167). The stem was forty-two days old, the slab thirty-
4
1 i "^ i 1 1
%"
_/ \ / \ / ' \ / j
t
i Wj
[ ! !
FIGURE 166.
FIGURE 167.
two days old. Tension cracks developed in the usual manner,
beginning under a load of 10,000 Ibs. ; at 22,000 Ibs. small diag-
onal cracks appeared. At 50,000 Ibs. it was noticed that the
visible end of the curved tension rods began to slide, and the
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 225
ultimate failure occurred under a load of 55,600 Ibs., when
the compression area was crushed.
Beam No. 13 was again of the Unit type, with smooth top of
stem, which was forty-three days old; the slab was thirty-four
days old. The first tension craok occurred at 9,000 Ibs., and
the cracks then developed in the usual manner. Failure took
place at 50,000 Ibs., when the adhesion between concrete and
steel was broken; the rods began to pull through, and the slab
was crushed at the center.
The analysis of the stresses follows:
Types A and B
By reference to formula (18), page 38, we have
2 = 1; T = 4"; H = 9"; D = 13"; V = ~ X 1.62 = 2
hence
S 82.0
and
^ = 7^M = ^ = - 436;1 -? = - 8 ^
Now, the bending moment is f L.42 = 21 L, and the arm of
internal stresses approximately
.855 X 13 = 11.1 inches, hence the pull in the steel
s = jj-L = 1.89Llbs.
The beams had 1.62 square inches of tension steel, hence the
unit tension on steel:
S L89 T 1 17 T
= ' L = 1 ' 17 ' L
and the unit compression on the concrete
Type C
| = i ; T = 5"; H = 6"; D = 11"; V = 15 - ^ = 2.5;
226 REINFORCED CONCRETE BUILDINGS
hence
~ = 15.9
and
x =
1 +
15.9
15
- .485; 1 - l -x = .838
Again, the bending moment is \ -L-30 = 15 L, and the arm of
internal stresses approximately
11 X .838 = 9.2", hence the pull in the steel is
The beams had 2.0 square inches of steel, hence the unit tension
on steel
and the unit compression on the concrete
It is evident that these calculations do not give the true
stresses existing at rupture, because r is not equal to 15 at that
time, and the assumption of plane sections probably does not
hold good. For the sake of comparison, however, they may be
useful. The results are indicated in the table. The testing
^
Corresponding
6
Age (days)
calculated stresses
"SI
fc
Ultimate
165 sq. in.
a"
S
How made
Load
>
2
Ibs.
1
pq
Stem
Slab
C
s
A
1
Unit, Smooth Top
46
37
49,400
3060
57,800
A
2
Unit, Smooth Top
46
37
54,300
3370
63,500
A
4
Unit, Smooth Top
45
36
48,000
2980
56,200
A
5
Unit, Corrugated Top
44
35
41,000
2540
48,000
A
9
Monolith
41
41
50,000
3100
58,500
A
10
Monolith
41
41
52,400
3240
61,200
A
12
Monolith
41
41
45,800
2840
53,600
B
B
25
27
Unit, Smooth Top
Monolith
29
27
25
27
42,500
47,500
2640
2950
49,700
55,600
C
13
Unit, Smooth Top
43
34
50,000
2560
40,700
C
20
Unit, Corrugated Top
42
32
55,600
2840
45,300
Harvard Test Beams. Summary of Results obtained
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 227
machine was equipped with means for registering the deflec-
tions automatically; the diagrams are shown in Figures 168,
FIGURE 168.
169, and 170. Generally speaking, there is little difference in
the deflection of the Unit and monolithic beams.
A number of interesting observations were made during
these tests. First, the feasibility of the Unit beam was estab-
lished beyond doubt, contrary to what many engineers would
Deflection
^L
40000
^25
^<^
V
30000
>
\
20000
\
\
1.0*
0.9"
0.8 "
0.7"
0.6"
0.5"
0.4"
0.3"
0.2"
0.1" \
FIGURE 169.
probably have expected. In fact, many building regulations
throughout the country specify positively that the beam and its
superimposed slab must be concreted in one continuous opera-
228
REINFORCED CONCRETE BUILDINGS
tion. Where improperly designed, or otherwise inadequate,
U-bars are used, this rule is undoubtedly highly beneficial, but
where proper U-bars are used, the rule is wholly unnecessary.
The progress report of the special committee of the American
Society of Civil Engineers recommends that the slab be con-
sidered effective in compression when " proper bond" is pro-
vided between slab and stem; it will be appreciated that this
is a much more consistent requirement, although somewhat
indefinite. The beams tested so far have shown that the bond
provided was adequate, whether the more elaborate method of
Deflection 0.9" 0.8"
50000
30000
0.6'
0.5'
13
FIGURE 170.
corrugating the top of the stem was used, or whether the top of
the stem was simply left as it was upon completion. It would
be very interesting to learn what would happen when the bond
was " inadequate," and just where the limit may be found, and
in this particular the present tests furnish no information, as
the bond remained intact in all cases. See Figures 171 and 172,
showing the Unit Beam No. 25.
In the second place, these tests confirm in a remarkable
degree the theories set forth by the author in Chapter VII in
regard to the action of U-bars.
Compression failures of the stem were observed in beams 25
and 27; these are shown in Figure 164 and in Figures 171-174.
It was observed that the compression failure of the stem was
on the same side as the corresponding bent bar, the two bent
bars being each near the opposite face of the beam ; in beam 27
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 229
FIGURE 171. HARVARD BEAM No. 25.
The black lines are ink marks indicating the principal cracks
Photo by Mr. J. R. Nichols, Jr., Am. Soc. C. E.
FIGURE 172. A CLOSER VIEW OF BEAM No. 25, SHOWING CRUSHING
OF THE CONCRETE AT THE ROD.
This beam was a Unit beam, it will be noticed that there was no indication
of slipping between stem and slab.
Photo by Mr. J. R. Nichols, Jr., Am. Soc. C. E.
230
REINFORCED CONCRETE BUILDINGS
FIGURE 173. BEAM No. 27, SHOWING PRINCIPAL CRACKS AT LEFT END,
AND THE CRUSHING OF THE CONCRETE AT THE ROD.
Photo by Mr. J. R. Nichols, Jr., Am. Soc. C. E.
FIGURE 174. CLOSER VIEW OF BEAM No. 27.
Photo by Mr. J. R. Nichols, Jr., Am. Soc. C. E.
THE THEORY OF BEAMS AS ILLUSTRATED BY TESTS 231
crushing took place at both bent bars, one spot on each side,
but in different locations, corresponding to the position of the
curves in the bars. It is self-evident that this upward pres-
sure of the rod must be resisted by an equal downward pres-
sure (from the load) thus dissolving the beam into a number of
well-defined compressive zones in a manner very different from
what takes place in a " solid" homogeneous beam. The same
observation was made by Prof. Morsch in regard to his test
beam No. VI.
Third, a deep, gaping crack was observed in the top of beam
No. 20 (Figure 175), near the support. The explanation of this
FIGURE 175.
crack may be found in the distribution of internal stresses
indicated in the drawing, the horizontal arrow at the steel indi-
cating the pulling of the steel, the inclined arrow indicating the
sum of the compressive forces in the concrete. It will be noted
that if these two do not intersect on the vertical line of the
reaction, a " re verse" bending moment is created at the end
which would cause just such a crack. Here again we have a
fact showing that a reinforced concrete beam cannot be consid-
ered as a " solid" beam, in which such stresses' are impossible.
Considering the beam as a truss, we see at once that the crack
comes outside the "end panel," and so would have no influence
on the load-carrying capacity.
In addition, it must be admitted that "shear," so called,
would have caused the instantaneous collapse of a beam with
such a crack. As an actual matter of fact, this beam, with the
gaping crack in the top, carried a total load of 55,600 Ibs., or
more than any other beam of the entire series. The stem was
perforated with inclined and vertical cracks so that the only
portions of the beam which could actually carry some shear
were the main tension rods. This proposition has been consid-
ered above and cannot be maintained. The truth is that there
232
REINFORCED CONCRETE BUILDINGS
was no active shear in this beam, the system consisting approx-
imately of members as shown in Figure 176. 1
Fourth, it was established that the quarter turn given the
straight tension rods at the ends was not sufficient to develop
the desired amount of sliding resistance. Thus, beams 9 and 12
failed suddenly by the entire separation of the lower rods from
the concrete, while the beams of Type C (13 and 20) showed a
sliding of from i" to f " (in these beams, the ends of the rods
I I
FIGURE 176.
could easily be observed by breaking away a thin shell of con-
crete). The behavior of the balance of the beams, and especially
inspection of the deflection diagrams, makes it, however, appar-
ent that only a very small additional margin of sliding resistance
was required in order to prevent the sudden collapse. Without
doubt, the large turn of the upper bar might profitably have
terminated at its lowest point, as the last fourth of the circle
materially weakened the concrete along the lines of cleavage
1 The authors are aware of the fact that other observations were made
during the testing of the Harvard series which strongly support the theory
advanced in Chapter VII of this volume. We are, however, requested to
withhold this matter from publication at the present time, and we must
refer to the later report to be published by Prof. Johnson.
INDEX
Accidents, 191
Acid, carbonic, for hardening con-
crete, 12
hydrochloric, for joining con-
crete, 10
joint, 10
joint, specifications for, 198
Adhesion, 54
Allowable stresses, 52, 128
Alum, 190
Arches, allowable stresses in, 131
Assumptions, homogeneity, 51
in general, 52
tensile resistance of concrete
disregarded, 104
Basic inventions, 18
Beam formulas, 70, 71, 75, 88
Belt course, Ransome patent, 13
reinforcement of, 153
Bending, 66
combined with compression, 114
Board marks, 176
Brushing, 179
Cement, 137
Chimneys, approximate formula for,
115
Chloride of calcium, 2, 190
of sodium, 2, 190
Clay, effect on concrete, 12
Coil joint, concrete to concrete, 9
for joining rods, 15
Columns, allowable stresses, 130
hooped, 58
least diameter, 62
repairs of defective, 189
strengthening existing, 64
Concrete, dry or wet mixture, 16
mixing and placing, 149
shrinkage and swelling, 126
specifications, 196
Conflagrations, 185
Continuity of beams, 109
Core boxes, permanent, 15
Corners chamfered, 176
Cracks, in cement finish, 181
in slabs and beams, 127
in tile-concrete floors, 182
repairs of, 188
structural significance, 92
Delayed placing of concrete, 7
Design, general remarks, 153
Earthquake, San Francisco, 6
Embedment, required length of, 55
Expansion joint, 2, 128
Facing of concrete, 177
see also Veneer
Factor of safety, 129, 132
Falsework, general design, 156
instructions, 196
standardization, 15
Finish, cement, general, 181
instructions, 198
Fireproofing, 183
effect of salt, 16
Floor coverings, cement finish, 181
wood, 154
Footings, formulas, 116
foundations, 171
Forms. See Falsework
Frost protection, 151
effect of salt, 10
233
234
INDEX
Hooked ends of reinforcement, 55
Hooped columns, 58
Inertia, moment of, 112
Injurious agencies, 17
Illuminating panels, 8
Joining new concrete to old, 9, 198
Laitance, 149, 181, 189
Lateral expansion, 57
Lime, slacked lime in concrete, 12
Mixing and placing of concrete, 149
Molds. See Falsework
Monolithic construction, 156
Notations used in bending theory, 66
Overmixing of concrete, 7, 8
Patentees, Alsip, 31
Aspdin, 19
Basset, 33
Bissell, 35
Brannon, 23
Bruner, 39
Bunnet, 21
Cheney, 32
Coddington, 24
Coignet, 22, 30, 31
Considere, 46
Cornell, 28
Cottancin, 40
Cubbins, 36
De Man, 41
Dennet, 21
Edwards, 26
Emerson, 33
Emmens, 24
Fowler, 29
Gedge, 22
Gilbert, 30
Golding, 36
Gustavino, 38
Hallberg, 44
Henderson, 31
Hennebique, 42
Patentees, Hyatt, 23, 24, 25, 26, 33,
34, 35
Jackson, 32, 36, 37, 38
Johnson, 22, 29, 40
Kahn, 45
Knight, 28
Lambot, 27
Lish, 24
Lythgoe, 22
McCarthy, 40
Matrai, 43
Matthews, 33
Melan, 40
Middleton, 28
Monier, 22, 27, 38
Parker, 19
Parkes, 22
Parmley, 45
Rabitz, 39
Ranger, 21
Ransome, E. L., 2, 3, 5, 8, 9,
10, 13, 15, 16
Ransome, Fk., 22
Shaler, 43
Sisson, 32
Smith, 31
Stempel, 39
Stephens, 28
Stevenson, 28
Summer, 28
Tall, 23
Thacher, 43, 44
Thornton, 22
Turner, 24
Visintini, 45
von Emperger, 40
Waite, 41
Wayss, 44
Weber, 46
Wetmore, 32
Wilkinson, 21
Williams, 30
Wilson, 39
Wood, 30
Wyckoff, 28
Piling, 171
cost of, 174
Plastering, 176
INDEX
235
Plates, concrete, 118, 120
steel base, 118, 171
Reinforced concrete, defined, 51
elements of invention, 18
Reinforcement, details of, 105
double, 114
circular, 118
kinds of, 145, 147
requirements, 147
placing, 159
Repairs to buildings, 188
Rolling of floors, 8
instructions, 196
Rubbing of surfaces, 179
Salt, effect on concrete, 10, 184
instructions for using, 196
Sand, 143
Sidewalk lights, 8
Silicate of lime, 2
of potash, 190
of soda, 2, 190
Slab formulas, 79
Slag, aggregate, 145
for making joints, 9
Sliding, of concrete upon concrete,
103
of reinforcement, 101
see also Adhesion, 54
effect of U-bars, 57
Specifications, superintendents', 195
Stand-pipes, 132
Stone, 144
Steel, 145
specifications, 198
see also Reinforcement
Stresses, longitudinal, 66
transverse, 93
tensile (in concrete), 104
initial, 126
temperature, 126
allowable, 128
Stirrups. See U-bars
Tables, T-beams of minimum depth,
72, 73
T-beams of increased depth, 78,
80
tile concrete floors, 78
discussion of, 81
Tanks, 132
Tensile stresses disregarded, 104
Tile-concrete construction, 76
tables, 78
description of, 148
Tooling, 178
Twisted bars, invention of, 3
effect of twisting, 147
U-bars, 97
spacing of, 102
Unit construction, Ransome's, 16,
162
types of, 161, 162
Veneered buildings, 187
Water, consistency of concrete, 149
consistency of finish, 181
dry versus wet concrete, 16
hot water used, 21
importance of sprinkling, 127
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