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Mr. these sections. For, as already explained, the distribution of stresses
will depend on the distribution of resistances. It should be noticed
that, in case a distribution of stresses was found for a uniform plate
by calculation, or by experiment, as was done by Trelease,* and then
the slab was reinforced by belts of steel the resistances of which are
made proportional to the stresses so found, the slab thus reinforced would
no longer be one the rigidities of which were those of the original
uniform plate, and the stresses in the slab would then be distributed
in a manner very different from those in a uniform flat plate. So
that any such basis for design is wholly illusory. This is also true
regarding the subdivision of the total applied bending moment between
the parts of the panel-wide section at mid-span.

The saddle-shaped deformations which occur at the sides of the
panel directly between the column heads produce, in the top of the
slab, tensions across the sides at mid-span and compressions parallel
to the sides. These are stresses on vertical plane sections of the slab
at right angles to each other, and express a state of stress in the
material which may be expressed otherwise by saying that there are
other vertical planes intermediate between these (inclined to them
possibly at nearly 54°) on which the entire resultant stress is a shear
horizontally along each plane, which increases in intensity according
to its distance from mid-depth of slab.

The material of the slab effectively resists these shears, especially
when it is well reinforced near mid-span of the sides, and such resist-
ance, called into play by the load, helps to carry the load, as do all
other resistances; but this is a kind of resistance additional to any
occurring in any simple beam structure, and it enhances the resistance
of this part of the slab above that of simple moment resistance just
as truly as the circumferential and radial action of the steel does
in the convex and concave areas. It is by the help of these saddle-
shaped deformations that the distribution of rigidities across the sides
effect a corresponding distribution of stresses, as may be seen from
the following discussion.

Suppose that little or no tensile resistance existed in the top of
the slab across the edge of the panel near mid-span; then the panel
on each side of that edge would deflect more readily for that reason,
and the deflections at the panel centers on either side of this edge
would be greater than otherwise. This sag on each side and yielding
to bending across mid-span of the edge would transfer the applied
bending moments each way from this point and bring them to bear

• "The Design of Concrete Flat Slabs", by F. J. Trelease, Proceedings, National
Assoc, of Cement Users, Vol. VIII, 1912, p. 218.



on the slab at and near the column heads. If, however, very rigid rein- Mr
forcement is introduced across the edge at and near mid-span, it will
resist the greater portion of the total bending moment across the inner
section of the edge and relieve the tensile stresses across the column
heads by just so much, and so prevent any necessity for piling much
steel directly over the columns. At the same time it will reduce the
central panel deflections as well. Until it is proved that steel across
the column heads resists bending moments more economically than
across the inner sections, any recommendations such as those in the
report, following the Chicago ruling, and intended to control the dis-
tribution of the steel across the margin, etc., are contrary to good policy
and an intolerable restraint on legitimate design.

The truth of this last statement is evident from the following
considerations. The intensities of compressive stresses across panel-
wide sections must be distributed along those sections in a manner
corresponding in the main with the distribution of the tensile stresses
in the steel in these sections. In ordinary design, with steel belts
massed across the supports, it has been frequently asserted that critical
compressive stresses occur in the concrete around the column capitals.
If this is true, it is due to the concentration of steel in the side belts,
and can be obviated by such a disposition of the steel as to transfer
a suitable fraction of the bending moments across the panel edges
to the inner sections, and thereby relieve any excessive compressions
in the outer sections. Any uncertainty which now exists respecting
the existence of too large compressive stresses at right angles around
the capitals can be thus removed, and the question of controlling the
critical stresses in the concrete around the columns be thereby confined
to the question of the vertical shearing stresses.

8. — Bending Moments Reduced by Size of Capitals. — In the next
place, consider the distance by which the length of the span, L, between
the column centers must be reduced in order to obtain a correct
estimate of the effective span. How great such a reduction should
be depends on several circumstances. In case the capitals are integral
with the slab, and the columns perfectly rigid or nearly so, the effec-
tive span between capitals (each of diameter = C) would be L — C,
and if all the reinforcing steel was to pass over these capitals, the
length of the span would practically he L — C, while the panel load,
W, would remain practically unchanged.

As a mean value, under ordinary conditions of flexibility of
columns, etc., the mean effective span may be conservatively taken as

L — - C, although this maj' be subject to change in special cases. This

may be stated explicitly as:


Mr. Proposition IV. — When a correction for the size of capitals is

Eddy. jY L

introduced, the constant applied moment in the span, — ; — ,

becomes approximately


If the panel, however, is one of a tier of panels supported on a
series of successive transverse walls, each of width = C, instead of
on separated supports, the load, W, would also need to be reduced
in the same ratio as the span, and the constant applied moment per
span thus corrected would become

1 / 2G'


due to an effective span, L (l = ) , and an effective panel load,

1 — ^ — - I . It is evident, however, that this last result, which has

been adopted in the report, is inapplicable to the case of the panels
of a slab on separate supports.

Common practice makes C not less than 0.2L, while C = 0.225
is a usual value, and we have, consequently

0.87 > (l — tty) > 0.85 and 0.76 > (l — tt^ ) > 0.72.

Hence, it appears that, by reason of the size of the capitals alone,
the constant applied bending moment may possibly be reduced to

0.85 —

independently of any other reductions that may occur, such, for
example, as that treated in the previous sections. However, the
amount of reduction proposed in the report, which would make the

WL WL ., ,

total constant applied about 0.75 — :; — ,or about , is evidently

unwarranted by . theory, even in case the circumstances were such
that the span were reduced by the entire diameter of the capital,
for the minimum span would then be

L — C = L(1 — 0.225) = 0.775 L.
9. — Recapitulation. — First. — The formula proposed in the report
for the thickness of slabs does not conform to accepted usage in
reinforced concrete design, which increases the steel ratio with
increase in the ratio of thickness to span. Consequently, the formula
for thickness makes the proposed minimum thickness too large for
large loads and spans.


Second. — A continuous flat slab on separated supports is an inde- Mr.
terminate structure in which the distribution and relative intensities
of the stresses depend on the relative rigidities of the parts acting,
and, in particular, on the massing and location of the steel, so that
the statical limitations applicable to ordinary statically determinate
structures are not valid in such a slab.

Third. — On account of the necessary sub-division of the total
shear arising from the panel load which is carried in the two directions
parallel to the sides independently, the total applied bending moment
acting parallel to each side is only 50% of that stated in the report
in case of inside panels, and 60% of it in case of outside panels,
provided the latter be taken to be 20% greater than the former, as
recommended in the report.

Fourth. — The recommendation that the dip of the reinforcing
rods from top to bottom of slab be distributed at various distances
. from the edges, is not well founded. A dip common to all neighboring
rods should be used to fix the position of the lines of inflection and,
at the same time, to fix the sub-division of the constant total applied
moment between mid-span and margins.

Fifth. — The recommendation that the cross-section of steel across
panel-wide sections at mid-span and margins be apportioned in certain
proportions between the inner sections and the sections at the column
heads, is without basis in sound theory, and should be cancelled.

Sixth. — The coefficient of reduction of the total applied moment

/ 2 C\

due to the usual diameter of capitals is approximately (1 — ^—- 1 =0.85

/ 2 C\ ^

and not (1 — ^— ^ ) = 0.7.5, as recommended in the report.

C. A. P. Turner,* M. Am. Soc. C. E. (by letter).— Mr. Scott Mr.
analyzes the report of the Joint Committee in respect to the modulus '''^"°^'"-
of elasticity of steel embedded in concrete beams on the assumption
that the steel furnishes the entire tensile resistance due to the moment
of the applied load. This assumption is not correct. Furthermore,
the assumption that the tensile strength of the concrete furnishes that
portion of the tensile resistance necessary to produce a static balance
of the moments not provided for by the steel is likewise incorrect.

This latter assumption, although incorrect, is found in almost
every treatise on reinforced concrete published either in America or
abroad. There is one treatise, however, that by Charles F. Marsh,
M. Am. Soc. C. E., which shows that the distribution of stress
in a reinforced beam is not analogous, or even approximately
analogous, to that of a homogeneous beam in respect to any uniform
distribution of intensity of the stress transversely of the beam between

* Minneapolis, Minn.


Mr. reinforcing bars. All agree that there is a co-action between the metal
■ and the concrete in a beam, brought about by the resistance of bond
stress, which prevents the steel from slipping in the concrete matrix.
Under uniform load in a simple beam, all agree that bond stress is
zero at mid-span and increases toward the end of the beam. Now,
this bond force or bond stress is a horizontal shearing stress along
the surface of the steel. Consequently, it sets up tensions and com-
pressions at 45° to the surface of the metal; the intensity of these
indirect stresses is greatest next to the surface of the bar, and
diminishes, according to the law of distribution of force through
mass, with the distance away from the surface. The deformation of
the concrete due to the bond is greatest near the bar, thus forming a
cone of deformation about the bar, and smaller deformations or strains
occur between consecutive bars in the plane of the reinforcing steel
than take place at the surfaces of the bars themselves. This condition
of deformation causes some of the energy of internal work to be per-
formed at an angle to the axis of the bar, or laterally, thus differentiat-
ing the reinforced concrete beam from the beam of homogeneous
material in relation to the storage of internal energy.

It cannot be maintained consistently that burying the steel in the
concrete changes its molecular properties or its modulus of elasticity,
because, when we dig the steel out after it has been embedded in the
concrete, we find that it is just the same kind of steel as it was before
it was embedded, and possesses the same properties; and any conclusion
to the contrary is unwarranted. It cannot be maintained that the
joint action of the combination is equivalent to the resistance fur-
nished by a plain concrete beam without any steel whatever plus the
resistance of the tensile stress in the steel multiplied by its lever arm,
because experiments in which the concrete is divided or separated
at mid-span of the beam by pieces of tin so that no tensile stress can
be transmitted in the tensile zone from one-half of the beam to the
other by the concrete, show to a large extent the same phenomena of
apparent but not true increase in the modulus of elasticity of the
steel when measuring its deformation at mid-span and right across the
line where it is dissociated from the concrete.

The subject of horizontal shears in vertical planes is one that is
little understood by the Engineering Profession, or by theoretical
writers on the subject of reinforced concrete. Lack of familiarity with
this kind of action has led to an erroneous report by the Joint Com-
mittee as to the strength of reinforced flat slab and column construc-
tion, and to the unwarranted assumption in the report that the
tensile strength of plain concrete in the slab can account for the
enormous increase m strength which it exhibits over beam action in
the column-supported flat slab. Less assumption and more experi-


mental work and a careful discussion thereof would have resulted on Mr.
the part of the Joint Committee in a far different report, and one
which would not be questioned, as this one will be by all who have
had any practical experience in the building of structures of this type.

This phenomenon of reinforced concrete is not so simple and
ai^parent as it may nov>f seem, even after the explanation has been
discovered. It required nearly 3 years of hard study and investigation
before this satisfactory explanation was obtained. This question was
discussed in a paper presented to the Boston Society of Civil Engi-
neers, in September, 1914, in. which the phenomenon was compared
in action to a thrust applied at the end of the beam. The law of
conservation of energy applied to the problem would prevent a con-
sideration of the tensile stress in concrete multiplied by its lever arm
as balancing that portion of the applied moment not carried by the
steel, because of the lateral action introduced in the cone-shaped de-
formation through the operation of bond stress about the bar toward
the end. These internal shears seem to be somewhat equivalent in
their action to the effect of a thrust applied at the end of a beam,
and not greatly interfered with by cutting the concrete through the
tensile zone at mid-span.

Although the operation of horizontal shears in vertical planes is
not to be depended on in a beam, because the utility of this action is
destroyed by the cracking of the concrete toward the end of the beam,
the conditions are different in the continuous flat slab floor, where
the tendency to rotate is predominantly about a vertical instead of a
horizontal axis, and, for that reason, is dependable.

A further point of interest lies in the fact that although every
pound of pull in the steel is brought on it by bond stress, the
work of internal deformation in tension is largely stored within the
steel, a conclusion which was far from being apparent in the first
consideration of the subject. Static balance merely requires equal
opposing forces, with no restriction as to the amount of work performed
directly by these forces, so that although the steel may store very
largely the internal work of tensile deformation, horizontal shears in
vertical planes may have an equal effect in reducing deflection without
performing the same relative amount of work, since they operate in
a radically different manner and through a much shorter range of

The writer presented some tests at a meeting of the Concrete Insti-
tute relative to this subject several years ago, but has not been able
to supplement them by further experimental work, as he had hoped
to do by this time.

Mr. Scott deserves the thanks of the Profession for clearly bringing
out the incongruities of the Joint Committee's theory which he notes



and which the writer has endeavored to explain. Figs. 11 and 12 show,
in an exaggerated manner, the condition discussed.

The impossibility of assigning any dependable value to the tensile
strength of the concrete in a floor may be made apparent by the
following experiment conducted by the writer. A slab, about 24 ft.
square and 5J in. thick, supported on the edges on beams and piers
and by a center pier, was loaded as shown in Fig. 13. It carried
this load, consisting of concrete barrels filled with water, without





Ceutcr ol Co-wp.
Neutral Plane

Irregularity of deformation in Vert. Planes.

Deformaticns fi-oni bond esay^ers.tcd.



Fig. 11,

Condition after Concrete Cracks.

Steel lever arm, Jd, reduced at eadj

ii,lso lever arm, Jd( reduced at end,.

Fig. 12.

signs of failure when first applied, but a drop of temperature of about
25° caused the failure shown in Fig. 14. If a change or di-op in
temperature of only 25° is suiScient to overcome the tensile strength
of concrete, even when assisted by a little wire netting, as was the
case here, the error of relying on such resistance as furnishing any
dependable load-carrying capacity in the average building, which is
subjected to greater reduction of temperature than this test slab was,
is apparent, and needs no further comment.

Fig. 13. — Slab Carrying Load Without Distress, Very Largely by Virtue
OF THE Tensile Strength of the Concrete.

Fig. 14. — Failure of Slab Shown in Fig. IJ


F. E. TuRNEAURE,* M. Am. Soc. C. E. (by letter). — Professor Mr.

Eddy's discussion of the report of the Joint Committee raises certain
fundamental questions which should be answered. It appears to the
writer that his analysis at certain points is absolutely incorrect, and
should not go unchallenged. In the first place, on page 1178, he
criticizes the Committee for applying the theory of statics to the flat
slab, and for assuming flat slabs "to be statically determinate struc-
tures." The Committee makes no such assumption, and neither does
Mr. Nichols. The Committee is well aware that such structures,
like continuous beams, are not statically determinate, but it does not
follow that the law of statics cannot be applied to them. To be
statically indeterminate simply means that the unknown quantities
in the problem are too numerous to be determined by the equations
of statics alone. Certain total shears and moments, however, can
be determined by statics, although the exact distribution of these
shears and moments cannot be thus determined. This is the position
of the Committee.

It is interesting to note that, after criticizing the Committee for
applying the laws of statics to this case. Professor Eddy, on page 1179,
proceeds to do the same thing, where, in Paragraph 3, he applies the
laws of "equilibrium" to determine total shears — a perfectly correct
procedure, and exactly the same sort of thing as that done by the
Committee. The same process can be used to determine total bending
moments along either axis, and it is quite unnecessary to make use
of any involved mathematical process to do this. Although the total
bending moments can be determined in this way, their distribution
along the slab cannot, and, in this degree, the problem is statically

Professor Eddy finds that the total shear around the edges of a
square slab is equal to the total load — a result with which the writer
fully agrees — but the results obtained by his analysis for total bending
moments are quite erroneous. For a square panel, he finds that the
numerical sum of the positive and negative bending moments along

the center line, and along one edge, is — ;- W I, whereas the correct


value is - >r Z (neglecting the eft"ect of size of column in reducing

slightly the bending moment). As this question is a very fundamental
one, and as the conditions of the problem may easily result in con-
fusion in applying the ordinary laws of statics, it will not be out
of place to discuss this matter on a rather elementary basis.


* Madison, Wis.



Consider a square panel, Fig. 15, loaded uniformly with a total
load, W. Assume that the columns are equivalent to square columns,
in accordance with Professor Eddy's conclusions on pages 1179-80. For
simplicity, consider that the columns are small as compared to the
size of the panel, so that the size of the columns need not enter into
the expression for bending moments. The total shear at the columns
will be W, and the shear on each of the surfaces, AB, BC, etc., will

be — W. There will be no shear on the lines, A G, CD, etc. These


statements agree with Professor Eddy's conclusions.

Now proceed to the question of moments: Consider the half -panel

to the left of the center line, JK, Fig. 16. There will be bending

moments at all points of the peri-
phery, these moments acting in a di-
rection at right angles to JK along
the surface, JK, and also along the
surfaces, AB, GD, and EF. Mo-
ments along the surfaces, AJ, GB,
DE, and FJ, will act in a direction
parallel to the axis, JK. The load

on the half-slab is — TF" and the

Shear= A

j\ No Sliear G

K\ No Shear /f

shears are -, W on each of the sur-

FiG. 15.


A -M \j




faces, AB, BG, etc. Now, taking

moments about the axis, JK, the

external moment is that of the couple formed

by the load, - W, and the four shears of - 1^

•' 2 8



-.V \K

Fig. 16.

This moment is practically equal to
Wl. For equilibrium, the sum of the in-

ternal momeuts must also equal ^ W I, and the


internal moments in question are the positive

moments along JK and the negative moments

along AB, GD, and EF. That is to say, the

numerical sum of the negative moments along a line through the

columns and the positive moment along the center is equal to ^ W I, as

given by the Committee.

This method of analysis is a simple, direct application of the laws
of equilibrium, quite as definite as the application of these laws in
determining shears. No amount of mathematics will enable the con-
clusion to be reached that the internal moments acting perpendicular


to the surfaces, AJ , CB, etc., can help balance the bending moments Mr.
about an axis, JK, at right angles thereto. Tumeaure.

^ W I, Professor Eddy arrives at

In contrast with this result of

No Moment


, Total Load

3^-' },.r I


—- W I hy a. process of analysis in which he omits the effect of the

shears on the surfaces, CB and DE, considering that these shears
affect only the bending moments parallel to JK. As a matter
of fact, the shears on CB and DE are a little more effective in
producing bending moments about the axis, JK, than those on
AB and CF, on account of the increased lever arm. The fallacy of
Professor Eddy's position may be brought out clearly by noting what
happens if we consider the portion of a slab limited by the lines,
AF and GH, Fig. 15. Neglecting the small load
applied directly to the strip, BCDE, and the sim-
ilar strip on the right, the total shears on the lines,
AF and GH, must be equal to the load, W , by the
same process of reasoning as that heretofore used.
Then, if we consider one-half of this slab and take ^{"IB
moments at right angles to the axis, JK, as before,

the external moment will be plainly - W I, which


must be equal to the total internal moments along
the lines, AF and JK.

The inevitable conclusion, therefore, is that the
numerical sum of the negative and positive mo-
ments along the edge and center line of a square

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