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the paraoola. Such an equation of relation between the effective thick-
ness and the live load has been derived in what follows.













■ Table 2

Fig. 10.

The equation for d through P^ and P^

may then be written in the


in which I is the span, from center to center of supports, in feet, and
Wq is the total design working load, in pounds per square foot. The

total thickness is , 7 , -. r^ /ia\

r = a H- 1.5 CIO;

in which, if d is the effective depth of the centers of gravity of the steel
below the compression surface of the concrete, 1.5 in. is the remaining
part of the total thickness, including one-half of the layer of steel and
its fire-proof covering. In the report this is taken as 1.5 in., of which
the fire-proofing is required to be 1 in. in the clear.


Mr. In Fig. 10 the two points, P^ and P.,, at which it was assumed that

^' the two corresponding values of d were known, were both for the same
span, I, but the values of the constants, & and c in Equation (9), can
be determined equally well in case the known values of d correspond to
any different known values of I and w^ whatever. For example, assume
that it is known from experience that, for a 17-ft. panel and a design
live load of 200 lb. per sq. ft., the total thickness, t = 7.6 in., nearly,
and for a 23-ft. panel and a design load of 400 lb. per sq. ft., t = 11.6
in., nearly. Substituting these values in Equation (9), gives the fol-
lowing equations from which to determine the values of 6 and c :

I .b — l.;j = 1/ I

. C J

Hence, fe = 700 and c = 2 500, nearly, and Equation (9) becomes

a = I "•» + ""' 1

2 500

a ^ I

4 ^Oo + 2 800

Y (12)

10* J

the latter form being more convenient for numerical computation.

In case the examples previously assumed as a basis for the nu-
merical values of h and c used in Equation (12) seem to be unaccept-
able, then any two other trustworthy and satisfactory numerical
examples may be used instead, to arrive at an equation with values of h
and c slightly different from those used in Equation (12). Table
3, giving the numerical values of d, has been computed from Equa-
tion (12),

The differences in the successive numbers in any row or column of
Table 3 are constant, and are noted in the margins at the right and
bottom of the table, so as to admit of ready interpolation and exterpola-
tion by proportional parts, which, in this case, is exact and not merely
approximate. If, in Equation (12), w^ and I are regarded as rect-
angular co-ordinates in a horizontal plane, and d is erected perpen-
dicular to it, the locus of the extremity of d will be a hyperbolic
paraboloid which by its height will represent graphically the thickness
of the slab. The numbers at the top and' left of Table 3 measure such
co-ordinates in the plane of the paper, and the numbers in the body of
the table express the heights to be laid off' at the positions where the
numbers are written. The graphical surface thus constructed is a
doubly-ruled surface such that a vertical section of it, for any given
constant value of either w^ or I, will be a uniformly sloping straight


The total thickness of the slab will be obtained, as in Equation (10), Mr.
l)y adding to d one-half the total thickness of the layers of reinforcing ' ' ^'
steel plus the thickness of the fire-proofing, which all together may be
either more or less than the average value of 1.5 in. recommended in
the report.

The values of d in Table 3 are such as to make the thickness, t,
throughout the entire table fall vsdthin the limit of one thirty-
second of the span, as recommended in the report, excepting
only the last number at the bottom of the first column, that is,
t = 10.24 + 1.5 = 11.74, which is i in. below the assigned limit.

TABLE 3. — Effective Thickness of Flat Slabs.
^^ 4yi-„ + 2S00.



I =

= span, in

feet; iv^

= live load, in pounds per square foot.

in feet.

Wo = 100





























































13.. ^2




10. OS




14. .56


















A =







It will be noticed that Tables 2 and 3 give practically identical
values oi d = 4.5 in., or t = 4.5 -\- 1.5 = Q in., at the upper left
corner, where I = 14 ft., w^ = 100 lb. ; and at the upper right and left
corners the values of d in Table 2 exceed those in Table S by about
25%, and at the lower right corner by nearly 50 per cent. These

enormous differences are due to the increase in the value of p with —

involved in Equation (5), when a is constant.

When the value of a = 25 is used, that necessarily implies that the
slab has steel enough in one direction to carry the entire load without
any stress in the steel at right angles thereto, whereas, in fact, in an
inside panel, the steel in both directions acts simultaneously, which
requires a = 50, and not a = 25. This fact will reduce the value of p
to one-half that previously found.

The steel ratio derived from Equation (5) is :

tv r" (»'o + 12.5 d) f


a t\ i c?^

16 000 X 0.9 a d-



Mr. which gives the range of values of p, shown in Table 4. This
' table gives the numerical value of the ratio of the cross-section of the
steel to the concrete at mid-span, in order that, when a = 25, the
steel will resist the entire applied moment as in a beam, or, when
a = 50, the steel will resist one-half of that amount, as in a two-way
slab. Any intermediate values of I and w^ may be found correctly from
this table by simple interpolation of proportional parts.

TABLE 4. — Values of Steel Ratio, p, by Equation (13).

If a = 25

If a = 50

tt'o = 100

Wq = 600

M'u = 100

Wq = 600

1 = 14
1 = S2





These values of p show that Table 3 requires no excessive steel
ratios, either at mid-span or at panel margins, where, since the values
of a will be about one-half those at mid-span, the values of p will be
about twice as great.

In obtaining the values of d in Table 3, no allowance has been
made for any reduction of span by reason of size of capitals. In order
to make such allowance, it will be sufficient to assume that I is the
effective span, instead of the actual span, and use it as the tabular span
for finding d in Table 3.

It will now be in order to find out, in case slab thicknesses in accord-
ance with Equation (9) and Table 3 are adopted, whether either the
compressions in the concrete will be excessive, or the deflections of the
slab will be greater than is permissible. For, if neither of these things
occurs, then we have established the admissibility of these thicknesses,
which are less than those found by Equation (1) and Table 2, and,
hence, these latter will not be actual minimum values, but should be
replaced by other smaller values found from Equation (9). The com-
pression in the concrete is to be found from the common formula,

f k

f — — -^ —, ill which A; must be less than 0.4, and probably does

"" ?i (1 — k)

not exceed 0.3; and we may assume fg = 16 000, and n = 15, so that

the unit stress, fg, does not exceed 700 lb., and probably is not more

than 500 lb., at any point, unless at the edge of the capital, at which

point a special investigation may be necessary. This shows that, in

general, the compression in the concrete should not be regarded as



An expression for the deflection, z, at the panel center of a flat slab Mr.
has been found to be:

. _ ^^' .13)*

^ "4.75X10^^72^, ^ ^

in which L = 12 Hs the span, in inches (not feet), and ^1^ is the cross-
section of a side belt.


which last was obtained by equating applied and resisting moments as
in Equations (3) and (4). Substitute this value of A^ in Equation
(13), and assume /« = 16 000, a = 25, ; = 0.9, and let the slab be as

thin as permissible, namely,— = 32. We then have the largest pos-
sible relative deflection for a unit steel stress of 16 000 lb. Then, the
relative deflection,

z 2A./aL 1

L 4. 75 X 10^" d 2 060


This relative deflection is so small that it would permit a test load of
more than 2.5 times the sum of the design load plus the dead load,

z 1

without causing -— to exceed — -, provided the steel were not thereby

stressed beyond its elastic limit, which, in this ease, would have to be

at about 40 000 lb. per sq. in. Smaller values of — than 32, such as

occur throughout almost the entire Table 3, will give smaller values

of — than the limiting value computed in Equation (15). On the

other hand, although a value of a larger than 25 would have an effect

L z

to offset the smaller values of — - and so to increase -— , vet the

d L ^

observed values of fg are invariably much smaller than those com-

puted by Equation (14), which serves actually to reduce the value of —

at least to that found by Equation (15). It follows, consequently,
that values of d as small as those in Table 3 may be used with safety,
and that values as large as those in Table 2, as prescribed by Equations
(1) and (2), are not a minimum, and it has been further shown that
they are unnecessarily large.

* See "Concrete Steel Construction", Eddy and Turner, (71), p. 204.


Mr. The report recommends minimum total thicknesses that may be

^'' written as follows:

For a slab without a drop, t — 0.024 I \/ ni + 1.5 = rt + 1.5
For slab with drop, t = 0.02 I \/ w + 1 =: -^ d + 1



For the drop itself, t = 0.3 I \/ v- + 1.5 = ^d + 1.5


and, further, that the width of the drop is 0.4 L, so that its area will be
(0.4 i)" = — , nearly. Hence, the total volume, in cubic inches, of . a

panel without a drop would be:

v = (d-\-1.5) L- (17)

but the volume of a panel together with a drop would be :

' = (I ''' + 1 ^' + (t ^^ + '■') f ' "'^''•^^' "^'

v= (|c? + l.l)i-^ (IH)

From this it is evident that the report discriminates against the flat
slab without a drop and in favor of the slab with a drop by making
the latter so much thinner that the total volume per panel is less.
In the writer's opinion such discrimination is without justification in
theory or practice. In his opinion, in order to make the formulas for
the thicknesses of slabs with a drop comparable with those without a
drop, the volume of the former should be at least as great as the
latter. With this in view, it will be necessary, in order to have the

drop 60% thicker than the rest of the slab and its area = — to write
for the thickness of a slab with a drop,

t = 0.022 I \fw + 1.5 = — r7 + 1.5 1
and for the drop itself, . J> (19)

< =0.0033? \/«'+ 1-5 =— fZ + 1.5 I

or, 1" = {d + 1.5) i^, nearly.

The reason for the foregoing statement is this: In the case of a
given uniform loading, the numerical sum of the bending moments
at mid-span and one margin is constant, but the sub-division of this


constant total moment between mid-span and margin depends on the Mr.
structure of the slab, as follows: Either the line of inflection in the ^'
slab occupies a position fixed by the simultaneous dip of the reinforc-
ing rods as they drop below the neutral surface, or it is undetermined,
due to the irregular dipping of the steel, and, in that case, is de-
termined only by the relative stiffness of the columns and of the panel
at the margin, as compared with that at mid-span. The report does
not contemplate any fixed lines of inflection, for it makes express pro-
vision for irregular dipping of steel rods. However, it recommends
that steel be provided to resist moments of certain given magnitudes
at mid-span and margin separately, regardless of the presence or ab-
sence of a drop. If that recommendation is to be adopted, the sus-
pended portion of the panel must needs be just as thick in case of a
drop as without it. The only excuse for its being made thinner with
a drop would be that the columns and the cantilever portion contain-
ing the drop had been made stiffer and the cantilever made larger
thereby, so that the lines of inflection were thereby made to occupy
positions somewhat more distant from the columns than in the case
where there is no drop. This would require the steel over the drop to
be kept near the top of the slab for a greater distance than in a plain
flat slab, and would also require larger and stiffer columns to resist
the unequalized moments, but would hardly permit the total volume
of the panel to be decreased, as has been done in the report.

It is still an unsolved problem to determine the economical rela-
tive sizes of the suspended portions of the panel and of the cantilever,
and the relative stiffness of the columns, as well as the manner of dis-
tribution of the resisting moments across the margins of the panels.
On the solution of this problem evidently no light has been shed by
the recommendations of the report, since it recommends a difference
in thickness without any difference in the resisting moment — a recom-
mendation hard, if not impossible, to justify.

Equations (19) for thicknesses of slab and drop to replace those
recommended in the report, assume in effect that the cantilever portion
of a slab with a drop will be slightly larger than without a drop, but do
not proceed to such unjustifiable lengths as the corresponding expres-
sions in Equations (16) given in the report, which wholly ignore the
redistribution of the constant total moment between mid-span and
margin and the increase of that part of the unequ.alized moment dis-
tributed to any column due to decreasing the stiffness of the panel at

In adopting Equations (19) for the thickness of the slab and drop,
it is understood that the values of d which are to be inserted, in order
to obtain minimum values of i, are those obtained from Equation (12)
nnd Table 3, although the foregoing criticism of Equations (16) is


Mr. equally valid were we to adopt as correct the values of d derived from
^^^^- Equation (8) and Table 2.

2. — Flat Slabs as Statically Indeterrmnate Structures. — In the
second place, the writer objects to the adoption in the report of any
theory of flat slabs as correct, which assumes them to be statically
determinate structures, because they cannot be so regarded. The
report, however, on page 1138, refers to the paper of John R. Nichols,*
Jun. Am . Soc. C. E., as affording an approximate basis for its formulas.
Now, any assumption of the validity and applicability of statical
analysis to continuous flat slabs is incorrect, and leads to erroneous
results, just as much as in the case of continuous beams or of any other
indeterminate structures.

Any structure in which the magnitude or distribution of the stresses
in any part of it undergoes any alteration by varying the rigidity of
any of its members or elements is statically indeterminate. Hence,
the principles of statics cannot be assumed to be applicable to such a
structure unless there is definite proof that the statical principles
sought to be applied are actually valid for the case in hand, so that it
is beside the mark to adduce any statical limitations or requirements
in flat slabs, such as are adduced in the report, because statical prin-
ciples must here be subordinated to the principle of rigidities, which
is the guiding principle in all indeterminate structures.

That a flat slab continuous over an unlimited array of separated
supports is an indeterminate structure according to the above defini-
tion is evident from comparing the distribution of the stresses across
the sides of the panels, due to the bending moments in a slab having
most of the reinforcing steel concentrated in belts passing over the
supports, with the distribution of the stresses across the sides of the
panels where this is not the case, as it is not in the crown-sheet of a
steam-boiler, for example. For, in the former case, the stresses across
the sides due to the bending moments are largely concentrated at and
near the supports, but it is far otherwise in the crown-sheet, which is
supported on stay-bolts that pass through holes in the sheet, which com-
pletely destroy the radial resistance to bending at the supports. By
reason of this distribution of the rigidity in the crown-sheet, the entire
resistance to bending across any side of a panel is afforded by that part
of the side lying between supports, and none of it is at the supports.

Statical principles, therefore, may be applied to flat slabs only after
due inquiry and after it has been ascertained to what extent they may
be applied to them. Statics and the principle of the lever are so in-
grained in all reasoning about structures that the unconscious assump-
tion of their applicability is likely to vitiate otherwise well-considered
investigations of structures so complex in their interactions as inde-

* "Statical Limitations upon the Steel Requirement In Reinforced Concrete Flat
Slab Floors", Transactions, Am. Soc. C. E., Vol. LXXVII, p. 1670 (1914).


terminate structures must necessarily be, and great care must be Mr.
exercised in order to avoid the conclusions that inevitably follow from ^'^^'^■
yielding to the insidious temptation unconsciously to make this as-
sumption. The following unavoidable conclusion, therefore, may be
formally stated:

Proposition I. — Continuous flat slab floors on separated supports
are indeterminate structures subject to the law of relative
rigidities and its corollary, the law of least work.

In any indeterminate structure the relative rigidities of its parts
control the distribution of the stresses in it strictly in proportion to
the relative rigidities of the paths by which those stresses are carried,
so that the greatest total stresses occur wherever the structure offers
the greatest total resistance to deformation.

The steel reinforcement in a slab affords the principal resistance
to tensile stress, and hence the total tensile resistance is, in general,
where the cross-section of steel is greatest. Consequently, the distri-
bution of the tensile stresses will in general be controlled by the
quantity and position of the steel in the slab.

3. — Shearing Stresses in Flat Slabs. — Equilibrium requires that
the total vertical shearing stress across the perimeter or margin of
any panel shall be equal to the sum of the total live load resting on
the panel added to the weight of the panel itself. The perimeter of
the panel may be taken approximately at the sides of the rectangle the
corners of which are at the four adjacent column centers, but it may
be found somewhat more exactly by replacing parts of this rectangle
near the columns by parts of the perimeters of the column capitals for
90° around each column center.

Shearing stresses transmit loads horizontally by the help of the bend-
ing moments which they bring into play, and they re-apply those loads
undiminished in total amount at the supports at some distance hori-
zontally from their initial positions.

In order to discuss this matter mathematically, designate the total
uniformly distributed load on one panel by W, then, in case of a square
panel loaded uniformly and supported at its four sides by walls or
equal girders, the total vertical shear across a section close to one side
is iW, as is evident either by symmetry or by the principle of the
equal rigidities of the possible paths, lying in two directions at right
angles to each other, by which the load may be carried to the supports.

Again, take the case of a single inside panel of any equally loaded
panels supported on columns with square capitals, the edges of which
are parallel to the sides of the panels; then, on the same principle, it
is evident that the vertical shear across each side of each capital is
^W. Now, whatever may be the shape of the perimeter of the capitals,
the sections on which vertical shear occurs may be taken to be made up


Mr. of an infinite number of successive elementary vertical faces all of
^' wliich are parallel either to one or other of the sides of the panel.
Then the total shear on all the vertical faces on one side of a capital,
which are parallel to one side of a panel, is IW. This may be regarded
as equivalent to saying that, whatever shape the perimeter of the
capital may actually have, it can be assumed without material error
for the puj'poses of computation to be replaced by a square capital.

It will be noticed that the intensity of the vertical shear across the
perimeter of the panel is distributed in a very different manner, in
the case of supporting walls on four sides, from its distribution across
the edges of square capitals, in the case of supporting columns, for,
in the latter case, the shear is all concentrated at the perimeter of the
capitals and is zero at the other parts of the perimeter of the panel
between capitals, although the total amount in either case must be
equal to the panel load, W.

This concentration of shear is brought about by the bending and
deformation of the slab, in which a shallow saucer-shaped hollow is
produced around each panel center, while around each column center
the slab is bent into the shape of an inverted saucer. The stresses ac-
companying these deformations effect the distribution of the shears
just mentioned, as well as that of the bending moments also, a matter
which will be treated later.

In case of wall supports on the sides, slab deformations are very
different from this, as they produce the distribution of shears and
moments that occur at the walls, which are very different from those
at the perimeter of the capitals, a matter that has been treated by the
writer elsewhere.* In all cases, the distribution of the vertical shears
depends on, and is controlled by, the relative rigidities of the various
parts of the perimeter and supports, as expressed by their capacity
to resist vertical forces without yielding. Any subsidence or yielding
of a support or any part of it will alter the distribution of shears that
otherwise would take place and reduce their intensity, and any in-
equalities in the resistance of the material at the perimeter of a capital
will cause corresponding inequalities in distribution of the shears.

Again, consider a single square panel that is one of a single tier
of equal square panels which constitute a wide beam and are sup-
ported on a succession of equi-distant transverse parallel walls or rigid
girders across the tier. In this case no saucer-shaped hollows are pro-
duced by the load, for simple flexure alone occurs in such a case, and
the deformation is designated as cylindrical, meaning thereby that the
surface produced by flexure is a ruled surface such that any sections
of it by vertical planes parallel to the wall supports are horizontal
straight lines parallel to the walls. In this structure the entire shear-
• "Concrete-Steel Construction", Eddy and Turner, p. 285.


ing rigidity at the sides of a panel is at the walls, and is zero at the two Mr.
unsupported sides. The total shear at each supported side is 4 IF and '^'
the total shear at any parallel vertical section between the walls will
be twice that occurring at the corresponding section of an inside panel

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