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of a continuous slab of several tiers of panels on separated supports,
such as was considered previously. As just stated, this shear of ^W
is due to the fact that there are no supporting resistances or rigidities
at the two unsupported sides of this panel.

• This fundamental difference in the distribution of the total vertical
shear at or across panel-wide sections in those two cases is of the
xitmost importance in the theory of flat slabs, because, owing to the
fact already stated that a given panel load, W, can produce a total
shear at the perimeter no greater than W, not only are the total shears
in sections parallel to the sides of the panels of flat slabs on separated
supports 50% less than in the wide beam structure just described,
but, by reason of the necessary relation existing between the shears and
the bending moments, the total applied bending moment produced at
any panel-wide section of an inside panel is likewise reduced to 50%
of that which would be produced if the total load were carried, as in
the beam, by moments across that section and sections parallel to it,
and none of it was carried by moments across sections at right angles
thereto. Now, since in a continuous slab on separated supports equal
rigidities occur in two sets of sections at right angles to each other,
both must act at the same time, and each does 50% only of that wliich
it would do in the absence of the other. In other words, since the
shears in the sections parallel to either one of the sides of the panels
in a continuous slab on separated supports are those that arise from
a uniform panel load of ^W, and equal shears also occur in sections
at right angles thereto, it follows that the applied bending moments
across each of the set of panel-wide sections parallel to the sides of
such a panel are those arising from a uniformly distributed panel load
of only ^W, instead of a load, W, as was the ease in a beam.

However, a wall panel in a flat slab floor that rests on a side-wall
of the building at one side of the panel is prevented by the wall from
having any deflection at that edge, so that the curvature of the panel
under load is more nearly cylindrical near the wall than it would be
with separated column supports at the edge. This makes the action
of the panel more nearly like that of a wide beam with one end rest-
ing on the wall. In other words, the total shear in a section close to
the wall will be greater than iW, but less than iW, while the total
shear in a section at right angles to the wall and at the edge of the
panel will be less than iW. The applied bending moments in these
two directions in the panel will be such as to correspond to those shears.
An outside panel, however, the outer edge of which is supported
on separated columns only, with no additional girder or stiffening


Mr. along its outer edge, undergoes a greater deflection at the outer edge
^' between supports than at other edges, and the total shear at a section
near the outer edge is less than at corresponding sections near the
other edges of the panel, and needs additional stiffening in the slab,
either at the outside edge or parallel thereto. The effect of this kind
of column support is to make the total shear in a section near the
outer edge of the panel less than iW, while it makes that in a section
of the panel at right angles thereto and near the side of the panel
greater than iW. Corresponding changes occur in the total applied
bending moments in the two directions in these panels. This last kind
of outside panel with others contiguous thereto takes on more or less
of the properties of a beam along the outside of the slab in which it
differs materially from an outside panel resting on a wall, which last, it
was seen, had some of the properties of a beam with an end at the wall.

A comparison of the two preceding paragraphs shows that outside
panels supported at the edge of the slab on an outside row of columns
may be designed so that their action will resemble closely that of the
inside panels. In order that this may occur, the reinforcement at
the outside edge must be such that at the heaviest loads to which it is
subjected the deflections will be the same at mid-span between the
columns at the edge as occur under these loads at mid-span between
interior columns. These identical deformations will ensure nearly
equal shears and bending moments in outside and inside panels, because
the saucer-shaped deformations will be nearly the same in each.

4. — Applied Bending Moments in Flat Slabs. — It is a well-known

proposition of beam theory that the sum total of half the numerical

values of the applied negative bending moments at the ends of any

span of length, L, plus the numerical value of the positive bending

moment applied at mid-span, all arising from a uniformly distributed

load, W, amounts to — - — , whether the span is a simple one with end


supports merely, or is continuous, or partly so, and this without regard
to the loading or absence of loading on other spans of the beam. This
proposition holds true regardless of the relative magnitudes of the
moments of inertia of the beam at its successive cross-sections by which
the relative rigidities of the beam in resisting the applied bending
moments is expressed. It likewise is independent of the rigidity or
lack of rigidity of the supports or their connections with the beam in
resisting applied bending moments. The proposition, therefore, considers
such a beam as an indeterminate structure to which the principle of rigid-


ities applies in determining how the total moment, —^, is distributed


between mid-span and ends. It will be convenient to designate this


constant quantity, — ^, as the total applied bending moment in the


span, L, due to the uniform load, W . A proof of this well-known Mr.
proposition may be found, among other places, in a paper by the ^'

In accordance with the conclusions already reached in this dis-
cussion, the total applied bending moment in each direction parallel
to the sides in a panel of a continuous slab on separated supports
depends on the part of the load that is transmitted by the shears in
that direction, so that, in an inside panel of such a slab, only one-half
of the bending moment would be applied in each direction, and there-
fore the sum total of the numerical value of the positive applied bend-
ing moment across a mid-section parallel to one side added to the
numerical value of the negative applied bending moment across that

side is only — — -. In any case, the sum, as above specified, of the ap-
plied bending moments in the two spans lying at right angles in any

panel would together have the constant value, .


In view of this, and disregarding for the instant the effect of the
size of the supports, we will state:

Proposition II. — In any square panel, inside or outside, of a con-
tinuous flat slab on rows of separated supports placed at suc-
cessive distances of L from each other and carrying a total
uniformly distributed panel load, ^Y , the sum total of the
numerical values of the positive applied bending moments
acting across the two panel-wide mid-sections drawn at right
angles parallel to the sides, added to half the sum total of the
applied bending moments acting across the four sides of the


panel, is equal to the constant quantity, .

This proposition may be readily demonstrated for an inside panel
in which the rigidities are equal and symmetrical about both of the
mid-sections. For,

Let 8 J, represent the total vertical shear across a panel-wide sec-
tion at a distance, x, from the panel center;
and let M^ represent the total applied bending moment at the same

Then, dMj. = S^dx, for this equation expresses the necessary funda-
mental relation of shear to applied moment along x.

Similarly, for sections at right angles to this, the corresponding equa- .
tion may be written :

dMy = Sydy.

* "A Further Discussion of the Steel Stresses in Flat Slab Floors", Proceedings,
Am. Concrete Institute, 1916, p. 284.

Mr. Now, combining sections at right angles to each other and at


■ equal distances from the panel center at which x ^= y, the sum of the

two vertical shears across these sections is equal to the total load

between either of them and the mid-section at x = 0, or at 1/ = 0;

Wx W

that is, wlien x = y, we have *S' + ^„ = ^ , in which — is tlie total

" L L

panel load per unit of width or length of panel.

In case of equal rigidities in directions at right angles, at equal

distances, x = y, from mid-section, we have S^ = -S'^, but not in case

of unequal rigidities. In any case, at x = y we have S^ -{- Sy = .

Hence, by addition, at x = y,

d (M^ + M,^) = (S, + S^) d X

and integrating between x = and x = L,

W X d X
L '

(^x + ^.)x . ^L-(^4 + ^U


that is, the excess of the sum total of the positive moments across the
two panel-wide' mid-sections drawn at right angles to each other over
the sum total of the negative moments across two sides mutually at


right angles is — — . This excess, however, is the sum of the numerical

values of these moments, which is the result stated in Proposition
II. This holds true in general in case of unsymmetrical rigidities,
although the demonstration is less simple for the more general case.

The distribution of the bending moment along each panel-wide
section depends on the relative resistance to bending of the several
parts of that section, as was pointed out, for example, where the
bending resistance of the crown-sheet of a boiler was considered.

The sub-division of the total applied bending moment, — :;— , between

the two directions at right angles, however, depends not only on their
relative resistances to bending moments at these sections, but also
on their capacity to resist vertical shears as well.

As a result of the preceding investigation, we have consequently
established the following:

Corollary. — In a square interior panel under a uniform load, IF,
the total numerical value of the positive bending moment at
a mid-section, added to the total numerical value of the
negative bending moment at an edge parallel to the mid-

section, is only one-half of — — .


Disregarding for the instant the reduction of span due to the size Mr.
of the capital, the report says : ^'

"Analysis shows that * * * the numerical sum of the positive
moment and the negative moment at the two sections named is given
quite closely"


by — - — , when written m the notation used herein. This shows that,


for an inside panel, the analysis adopted in the report wotdd make
the applied moments twice as great as has just been shown really to
exist. It is stated in the report, however, that the use of moment
coefficients somewhat less than those derived by this analysis is believed
to be warranted, and the reason assigned for this is the tensile resist-
ance of the concrete. How small the allowance to be made on this

account is, may be estimated from the recommended coefficients of —
at mid-span and — at a margin, the sum of which is practically —

in place of -— , as given by analysis. It is clear, therefore, that the

recommendations in the report practically require provision to be made,
so far as resisting applied moments are concerned, sufficient to carry
the entire load by resistance in one direction only and then require,
in addition, that provision shall also be made to carry the entire load
by resistance in a section direction at right angles to the first, when,
in fact, one-half must be carried in each direction. It is perhaps
unnecessary to remark that since moment magnitudes are of the
nature of directed quantities, moments at right angles to each other
are mutually independent of each other.

5. — Lines of Inflection. — As ordinarily understood, lines of inflec-
tion or lines of contraflexure are those lines drawn on the slab at
which successive panel-wide sections of the slab cut out by vertical
planes parallel to the sides of the panel change curvature from convex
to concave, or vice versa. It is evident that since there are two such
sets of panel-wide sections at right angles to each other, there must
be two sets of lines of inflection crossing each other. There would
also be other lines of inflection for any other sets of sections, as, for
example, sections parallel to the diagonals, etc. Considering, how-
ever, for the present, only those lines of inflection arising from panel-
wide sections parallel to the sides, and in order to arrive at a more
complete understanding of them and their relations to the deforma-
tions of the panels, let us take, for the sake of simplicity, the case
of a homogeneous uniform flat plate, such, for example, as a steel
plate of indefinite extent, resting on rows of equidistant separated
supports in square array, and let it be subjected to equal concentrated


Mr. loads, applied at the panel centers. Then, the upward resistance at
^* each support is equal in magnitude to each of the concentrated loads.
The top and bottom of the plate are then subjected to sets of forces
which are alike in every particular except position and opposite direc-
tion, and the plate is bent in a perfectly symmetrical manner. On
the surface of the plate draw lines parallel to the sides of the panels
located half way between the supports and the panel centers. This
will divide up the entire surface, checker-board fashion, into squares,
the sides of which are one-half the distance between supports. These
lines will be the inflection lines of the plate. The squares over the
supports will be convex upward, those in which the panel centers lie
will be convex downward, the others will be saddle-shaped. The
bending moments across the lines of inflection will be zero.

If, now, the loading is supposed to be changed from concentrated
loads to a uniformly distributed load, the curvatures in the central
part of the panel will not be so sharp as in the case of concentrated
loads, and the lines of inflection will thereby be removed slightly
toward the supports, but will still remain straight. If, however, the
plate is replaced by a slab which is reinforced so as to be stiller over
the supports than elsewhere, that will move those parts of the lines
of inflection which are around the supports farther away from the
supports, so that they no longer will be straight. If, in addition, the
stiffness of the slab across the saddle-shaped areas is made less than
elsewhere, those parts of the lines of inflection which lie around the
panel centers will move nearer to the sides of the panel. In such a
slab, the lines of inflection will divide up the area of the slab irregu-
larly, in checker-board fashion, into areas which are only roughly
rectangular, in which the convex and concave areas have bulging
sides, but the saddle-shaped areas will have hollowing sides. The
sides of these irregular areas are the loci of no bending across them
along sections perpendicular to the sides of the panel, and, conse-
quently, of no bending moment in that direction, but that does not
necessarily signify that the stresses in either the steel or the concrete
is zero across these lines. This is a fact, because the steel rods which
are known to be in tension both in the tension areas around the
supports and in those around the panel centers, are, no doubt, in
tension throughout their entire lengths, not only in the tension areas,
but across the lines of inflection as well, which fact would require
the entire cross-section of the concrete to be in compression across
lines of inflection, because, at any vertical section subjected to bending
moment only, the sum of the tensions and compressions must be
numerically equal.

From what has been already stated, it is evident that the location
of the lines of inflection is fixed by the rigidity of the slab, and is


largely within the control of the designer, so that it will be prac- Mr.
tically where the steel dips down below the neutral surface of the '''
slab as it passes from the tensile zones at the top to those at the
bottom of the slab.

'Now, compare this with a design in which all the steel does not
dip down at the same line. The effect of this would be to cause the
lines of inflection to lie at one place for one concentration of loading
and at another place for a different concentration.

The difference of action between these arrangements of steel is
like that existing between the cantilever bridge and the continuous
bridge of several spans. In the continuous bridge, the inflection
point moves along the bridge to some extent as a train crosses, whereas,
in the cantilever bridge, the point of zero moments is located definitely
by a joint at the junction of the cantilever and suspended span. The
same advantages are secured in flat slabs as in bridges by fixing
the length of the cantilever as well as the relative size of cantilever
head and suspended span by having all the steel dip down at a
given line, for this arrangement reduces the bending resistance at
the line so as to make it the line of inflection at all times.

It is evidently possible to make the cantilever extend out from
the supports to any distance, even as far as mid-span, by sufficiently
reinforcing the top of the cantilever throughout and, at the same time,
reducing the resisting moment of inertia of the section at the edge
of the cantilever. Now, this will be practically accomplished by
making all rods dip together at the same line of inflection prede-
termined in such position as to separate the total applied bending
moment of the panel into such parts as may be desired. This may
be formally stated in the following :

Proposition III. — In a continuous flat slab on separated supports
the relative size of the cantilevers over the supports and the
suspended spans within the panels, as well as the relative
magnitudes of the total applied bending moments across the
panel- wide sections at mid-span and at the sides of the panel,
may be controlled by the location of the dip in the reinforcing
rods across the neutral surface at predetermined lines of

Now, it is recommended in the report to have the steel dip down
one part of it at one distance from the edge of the panel and another
part at another distance, thus making the position of the lines of
inflection absolutely impossible of determination for any loading, and
different for different dispositions of load, and also making the sub-
division of the total moment between mid-span and margin uncertain.
The actual subdivision will then depend on several factors besides
the positions where the steel dips down, such as the relative moments


Mr. of inertia of the cross-sections at mid-span and margin, and the
^' stiffness of columns, etc. The recommendation is bad in principle
and more to be honored in the breach than in the observance, because
making the rods dip down at different points renders bending moments
indeterminate which, otherwise, would be determinate. This is dis-
cussed somewhat more in detail in the next section.

6. — Column Rigidities and Relative Slah Rigidities across Panel-
wide Sections, as Ajfecting the Suh-division between Mid-Span and
Margins of the Total Applied Bending Moment in the Panel. — In
any beam of constant moment of inertia, fixed horizontally at the
ends and uniformly loaded with a total load, W, the applied bending


moment at mid-span is - — - and at each end is — -^ttt- I" case the
24 12

moment of inertia or resistance to bending is made greater at and

near the ends than elsewhere, the numerical value of the applied

bending moment at each end will be increased at the expense of that


■ at mid-span, but their numerical sum will still be — -— . If, however,

the restraint at the ends is reduced so as to be insufficient to fix them
horizontally at the supports, the positive moment at mid-span will
increase numerically, while the numerical value of the negative moment
at each end will be decreased by the same amount. Similar principles
apply to the moments at panel-wide sections of inside panels of a
continuous slab over separated supports, except that the uniformly
distributed load in the case of the slab, which causes the applied
moment in each direction, will be only one-half of the total panel
load, W, as previously shown. The column supports of a continuous
slab are not usually sufficiently rigid to afford perfectly horizontal
restraint at the supports, but there seem to be clear indications that,
under ordinary circumstances, the columns resist about one-half the
unequalized bending moments at the supports. That, however, is partly
dependent on the rigidity of the connections between the slab and
the supports.

In order to obtain an outside estimate of the divergence between
the effect of the complete restraint at the ends previously considered
and free tipping over the supports, let us suppose a uniform beam of
three equal spans resting on four supports, with a uniform load on
the middle span only, but free to tip at all four supports. The theorem
of three moments shows that, in this case, the applied moments at


the support would be — . while that at mid-span would be about

-L If, however, the beam were fixed horizontallv at its two ends,

but could tip freely over the two intermediate supports, the amounts


over these would each be — — — , while that at mid-span would be about Mr

ly Eddy.


. These results, compared with those for the sino^le spaa with


fixed ends, show that, although lack of fixity at the ends of the span, L,
might reduce the numerical value of the applied moment by one-third
or more at the edges of the panel, at the same time it might nearly

double the moment, , at mid-span, even when no account is taken

of any additional effect due to increased stiffness, such as is usual at
the edges of the panel.

As previously shown, the foregoing results for beams should be
divided by two, in order to make them applicable to inside panels of
continuous slabs of equal moments of resistance at mid-span and

How little the results thus obtained can be reconciled with the

recommendation in the report which proposes to use the coefficient — ,

at the margin and — at mid-span, regardless of relative column and

slab rigidities, is left to the meditation of the reader.

7. — The Distribution of Bending Moments Across Panel-wide
Sections. — As already stated, in connection with the bending of the
crown-sheet of a boiler, the distribution of the rigidities in any panel-
wide section completely determines and controls that of the intensities
of the resisting moments in that section. Bending moments are by-
products and secondary phenomena in a beam or slab, and depend
mathematically on the shear, which is, in fact, the primary action
produced by the load. The rigidities which affect the distribution of
shears are mostly independent of those which affect the distribution
of the moments. The rigidities which control the distribution of the
bending moments do this in two distinct ways, one is by the relative
resistance across different parts of any panel-wide section, which deter-
mines the distribution and relative intensities of the stresses at the
several parts of that section, the other is by the relative total resistances
across the parallel sections at mid-span and margins of the panel,
which determines the relative total moments across those sections.
The effect of the latter has been considered previously.

We are concerned here, therefore, with the sub-division of the total
bending moment at one margin of a panel into two parts, one part
across the two middle quarters, or inner section of the edge as it is
called, and the other across the rest of the margin, or the column-head
sections. The only possible basis for any sub-division or partition
of the total marginal moment between these sections of the margin
must be some existing or assumed distribution of the resistances across

Online LibraryAmerican Society of Civil EngineersTransactions of the American Society of Civil Engineers (Volume 81) → online text (page 100 of 167)