American Society of Civil Engineers.

Transactions of the American Society of Civil Engineers (Volume 81) online

. (page 95 of 167)
Online LibraryAmerican Society of Civil EngineersTransactions of the American Society of Civil Engineers (Volume 81) → online text (page 95 of 167)
Font size
QR-code for this ebook

When a reinforced concrete beam is subjected to flexural action,
diagonal tensile stresses are set up. A beam without web reinforce-
ment will fail if these stresses exceed the tensile strength of the
concrete. When web reinforcement, made up of stirrups or of diagonal
bars secured to the longitudinal reinforcement, or of longitudinal
reinforcing bars bent up at several points, is used, new conditions
prevail, but, even in this case, at the beginning of loading the diagonal
tension developed is taken principally by the concrete, the deforma-
tions which are developed in the concrete permitting but little stress
to be taken by the web reinforcement. When the resistance of the
concrete to the diagonal tension is overcome at any point in the
depth of the beam, greater stress is at once set up in the web rein-

For homogeneous beams, the analytical treatment of diagonal tension
is not very complex — the diagonal tensile stress is a function of the hori-
zontal and vertical shearing stresses and of the horizontal tensile stress
at the point considered, and as the intensity of these three stresses
varies from the neutral axis to the remotest fiber, the intensity of the
diagonal tension will be different at different points in the section,
and will change with different proportionate dimensions of length
to depth of beam. For the composite structure of reinforced concrete
beams, an analysis of the web stresses, and particularly of the diagonal
tensile stresses, is very complex; and when the variations due to a
change from no horizontal tensile stress in the concrete at remotest
fiber to the presence of horizontal tensile stress at some point below
the neutral axis are considered, the problem becomes more complex
and indefinite. Under these circumstances, in designing, recourse is
had to the use of the calculated vertical shearing stress, as a means
of comparing or measuring the diagonal tensile stresses developed,
it being understood that the vertical shearing stress is not the numerical
equivalent of the diagonal tensile stress, and that there is not even
a constant ratio between them. It is here recommended that the
maximum vertical shearing stress in a section be used as the means of


comparison of the resistance to diagonal tensile stress developed in the
concrete in beams not having web reinforcement.

Even after the concrete has reached its limit of resistance to
diagonal tension, if the beam has web reinforcement, conditions of
beam action will continue to prevail, at least through the com-
pression area, and the web reinforcement will be called on to resist
only a part of the web stresses. From experiments with beams it is
concluded that it is safe practice to use only two-thirds of the
external vertical shear in making calculations of the stresses that
come on stirrups, diagonal web pieces, and bent-up bars, and it is
here recommended for calculations in designing that two-thirds of
the external vertical shear be taken as producing stresses in web

It is well established that vertical members attached to or looped
about horizontal members, inclined members secured to horizontal
members in such a way as to insure against slip, and the bending
of a part of the longitudinal reinforcement at an angle, will increase
the strength of a beam against failure by diagonal tension, and
that a well-designed and well-distributed web reinforcement may,
under the best conditions, increase the total vertical shear carried
to a value as much as three times that obtained when the bars are
all horizontal and no web reinforcement is used.

When web reinforcement comes into action as the principal tension
web resistance, the bond stresses between the longitudinal bars and
the concrete are not distributed as uniformly along the bars as they
otherwise would be, but tend to be concentrated at and near stirrups,
and at and near the points where bars are bent up. When stirrups
are not rigidly attached to the longitudinal bars, and the proportioning
of bars and stirrup spacing is such that local slip of bars occur a1
stirrups, the effectiveness of the stirrups is impaired, though the
presence of stirrups still gives an element of toughness against diagonal
tension failure.

Sufficient bond resistance between the concrete and the stirrups
or diagonals must be provided in the compression area of the beam.

The longitudinal spacing of vertical stirrups should not exceed
one-half the depth of beam, and that of inclined members should
not exceed three-fourths of the depth of beam.

Bending of longitudinal reinforcing bars at an angle across the
web of the beam may be considered as adding to diagonal tension
resistance for a horizontal distance from the point of bending equal
to three-fourths of the depth of beam. Where the bending is made
at two or more points, the distance between points of bending should
not exceed three-fourths of the depth of the beam. In the case of a
restrained beam, the effect of bending up a bar at the bottom of the
beam in resisting diagonal tension may not be taken as extending


beyond a section at the point of inflection, and the effect of bending
down a bar in the region of negative moment may be taken as extend-
ing from the point of bending down of bar nearest the support to a
section not more than three-fourths of the depth of beam beyond the
point of bending down of bar farthest from the support, but not
beyond the point of inflection. In case stirrups are used in the
beam away from the region in which the bent bars are considered
effective, a stirrup should be placed not farther than a distance equal
to one-fourth the dejjth of beam from the limiting sections defined
above. In case the web resistance required through the region of
bent bars is greater than that furnished by the bent bars, sufficient
additional web reinforcement in the form of stirrups or attached
diagonals should bo provided. The higher resistance to diagonal
tension stresses given by unit frames having the stirrups and bent-up
bars securely connected together both longitudinally and laterally
is worthy of recognition. It is necessary that a limit be placed
on the amount of shear which may be allovi'ed in a beam; for when
web reinforcement sufficiently efficient to give very high web resistance
is used, at the higher stresses the concrete in the beam becomes checked
and cracked in such a way as to endanger its durability as well as its

The section to be taken as the critical seciHon in the calculation
of shearing stresses will generally be the one having the maximvim
vertical shear, though experiments show that the section at which
diagonal tension failures occur is not just at a support, even though
the shear at the latter point be much greater.

In the case of restrained beams, the first stirrup or the point of
bending down of bar should be placed not farther than one-half of
the depth of beam away from the face of the support.

It is important that adequate bond strength or anchorage be pro-
vided to develop fully the assumed strength of all web reinforcement.

Low bond stresses in the longitudinal bars are helpful in giving
resistance against diagonal tension failures, and anchorage of longi-
tudinal bars at the ends of the beams or in the supports is advantageous.

It should be noted that it is on the tension side of a beam that
diagonal tension develops in a critical way, and that proper connection
should always be made between stirrups or other web reinforcement
and the longitudinal tension reinforcement, whether the latter is on
the lower side of the beam or on its upper side. Where negative
moment exists, as is the case near the supports in a continuous beam,
web reinforcement, to be effective, must be looped over or wrapped
around, or be connected with, the longitudinal tension reinforcing
bars at the top of the beam in the same way as is necessary at the
bottom of the beam at sections where the bending moment is positive.

Inasmuch as the smaller the longitudinal deformations in the


horizontal reinforcement are, the less the tendency for the formation
of diagonal cracks, a beam will be strengthened against diagonal
tension failure by so arranging and proportioning the horizontal rein-
forcement that the unit stresses at points of large shear shall be
relatively low.

It does not seem feasible to make a complete analysis of the action
of web reinforcement, and more or less empirical methods of calcu-
lation are therefore employed. Limiting values of working stresses
for different types of web reinforcement are given in Chapter VIII,
Section 5. The conditions apply to cases commonly met in design.
It is assumed that adequate bond resistance or anchorage of all web
reinforcement will be provided.

When a flat slab rests on a column, or a column bears on a footing,
the vertical shearing stresses in the slab or footing immediately
adjacent to the column are termed punching shearing stresses. The
element of diagonal tension, being a function of the bending moment
as well as of shear, may be small in such cases, or may be other-
wise provided for. For this reason the permissible limit of stress
for punching shear may be higher than the allowable limit when the
shearing stress is used as a means of comparing diagonal tensile stress.
The working values recommended are given in Chapter VIII, Section 5.


By colmnns are meant compression members of which the ratio
of unsupported length to least width exceeds about four, and which
are provided with reinforcement of one of the forms hereafter described.

It is recommended that the ratio of unsupported length of column
to its least width be limited to 15.

The effective area of hooped columns or columns reinforced with
structural shapes shall be taken as the area within the circle enclosing
the spiral or the polygon enclosing the structural shapes.

Columns may be reinforced by longitudinal bars; by bands, hoops,
or spirals, together with longitudinal bars; or by structural forms
which are sufficiently rigid to have value in themselves as columns.
The general effect of closely spaced hooping is to greatly increase the
toughness of the column and to add to its ultimate strength, but
hooping has little effect on its behavior within the limit of elasticity.
It thvis renders the concrete a safer and more reliable material, and
should permit the use of a somewhat higher working stress. The
beneficial effects of toughening are adequately provided by a moderate
amount of hooping, a larger amount serving mainly to increase the
ultimate strength and the deformation possible before ultimate failure.

Composite columns of structural steel and concrete, in which the
steel forms a column by itself, should be designed with caution.
To classify this type as a concrete colunni reinforced with structural


steel is hardly permissible, as the steel generally will take the greater
part of the load. When this type of column is used, the concrete
should not be relied upon to tie the steel units together nor to
transmit stresses from one unit to another. The units should be
adequately tied together by tie-plates or lattice bars, which, together
with other details, such as splices, etc., should be designed in con-
formity with standard practice for structural steel. The concrete may
exert a beneficial effect in restraining the steel from lateral deflection
and also in increasing the carrying capacity of the col\imn. The
proportion of load to be carried by the concrete will depend on the
form of the column and the method of construction. Generally, for
high percentages of steel, the concrete will develop relatively low
unit stresses, and caution should be used in placing dependence on
the concrete.

The following recommendations are made for the relative working
stresses in the concrete for the several types of columns :

(a) Columns with longitudinal reinforcement to the extent of
not less than 1% and not more than 4%, and with lateral
ties of not less than J in. in diameter, 12 in. apart, nor
more than 16 diameters of the longitudinal bar: the \init
stress recommended for axial compression, on concrete piers
having a length not more than four diameters, in Chapter
VIIT, Section 3.

(6) Columns reinforced with not less than 1% and not more than
4% of longitudinal bars and with circular hoops or spirals
not less than 1% of the volume of the concrete and as here-
inafter specified: a unit stress 55% higher than given for
(a), provided the ratio of unsupported length of column ta
diameter of the hooped core is not more than 10.

The foregoing recommendations are based on the following condi-
tions :

It is recommended that the minimum size of columns to which the
working stresses may be applied be 12 in., out to out.

In all cases longitudinal reinforcement is assumed to carry its
proportion of stress in accordance with Section 3 (c) 6 of this chapter.
The hoops or bands are not to be counted on directly as adding to the
strength of the column.

Longitudinal reinforcement bars should be maintained straight,
and shall have sufficient lateral support to be securely held in place
until the concrete has set.

Where hooping is used, the total amount of such reinforcement
shall be not less than 1% of the volume of the column, enclosed. The
clear spacing of such hooping shall be not greater than one-sixth the
diameter of the enclosed colunan, and preferably not greater than one-


tenth, and in no case more than 2^ in. Hooping is to be circular and
the ends of bands must be united in such a way as to develop their
full strength. Adequate means must be provided to hold bands or
hoops in place so as to form a column, the core of which shall be
straight and well centered. The strength of hooped columns depends
very much upon the ratio of length to diameter of hooped core, and
the strength due to hooping decreases rapidly as this ratio increases
beyond five. The working stresses recommended are for hooped
columns with a length of not more than ten diameters of the hooped
core. The Committee has no recommendation to make for a formula
for working stresses for columns longer than ten diameters.

Bending stresses due to eccentric loads, such as unequal spans of
beams, and to lateral forces, must be provided for by increasing the
section until the maximum stress does not exceed the values above
specified. Where tension is possible in the longitudinal bars of the
column, adequate connection between the ends of the bars must be
provided to take this tension.


When areas of concrete too large to expand and contract freely as
a whole are exposed to atmospheric conditions, the changes of form
due to shrinkage and to action of temperature are such that cracks
may occur in the mass unless precautions are taken to distribute the
stresses so as to prevent the cracks altogether or to render them very
small. The distance apart of the cracks, and consequently their size,
will be directly proportional to the diameter of the reinforcement and
to the tensile strength of the concrete, and inversely proportional to
the percentage of reinforcement and also to its bond resistance per unit
of surface area. To be most efl^ective, therefore, reinforcement (in
amount generally not less than one-third of 1% of the gross area) of a
form which will develop a high bond resistance should be placed near
the exposed surface and be well distributed. Where openings occur
the area of cross-section of the reinforcement should not be reduced.
The allowable size and spacing of cracks depends on various consider-
ations, such as the necessity for water-tightness, the importance of
appearance of the surface, and the atmospheric changes.

The tendency of concrete to shrink makes it necessary, except where
expansion is provided for, to thoroughly con^^ect the component parts
of the frame of articulated structures, such as floor and wall members
in buildings, by the use of suitable reinforcing material. The amount
of reinforcement for such connection should bear some relation to the
size of the members connected, larger and heavier members requiring
stronger connections. The reinforcing bars should be extended beyond
the critical section far enough, or should be sufficiently anchored to
develop their full tensile strength.



The continuous flat slab reinforced in two or more directions and
built monolithically with the supporting columns (without beams or
girders) is a type of construction which is now extensively used and
which has recognized advantages for certain types of structures as,
for example, warehouses in which large, open floor space is desired.
In its construction, there is excellent opportunity for inspecting the
position of the reinforcement. The conditions attending depositing
and placing of concrete are favorable to securing uniformity and
soundness in the concrete. The recommendations in the following
paragraphs relate to flat slabs extending over several rows of panels
in each direction. Necessarily the treatment is more or less empirical.

The coefficients and moments given relate to imiformly distributed

(a) Column Capital. — It is usual in flat slab construction to
enlarge the supporting columns at their top, thus forming column
capitals. The size and shape of the column capital affect the strength
of the structure in several ways. The moment of the external forces
which the slab is called upon to resist is dependent upon the size of
the capital; the section of the slab immediately above the upper
periphery of the capital carries the highest amount of punching shear;
and the bending moment developed in the column by an eccentric or
unbalanced loading of the slab is greatest at the under surface of the
slab. Generally, the horizontal section of the column capital should
be round or square with rounded corners. In oblong panels the sec-
tion may be oval or oblong, with dimensions proportional to the panel
dimensions. For computation purposes, the diameter of the column
capital will be considered to be measured where its vertical thickness
is at least li in., provided the slope of the capital below this point
nowhere makes an angle with the vertical of more than 45 degrees.
In case a cap is placed above the column capital, the part of this cap
within a cone made by extending the lines of the column capital
upward at the slope of 45° to the bottom of the slab or dropped panel
may be considered as part of the column capital in determining the
diameter for design purposes. Without attempting to limit the size
of the column capital for special cases, it is recommended that the
diameter of the column capital (or its dimension parallel to the edge of
the panel) generally be made not less than one-fifth of the dimension
of the panel from center to center of adjacent columns. A diameter
equal to 0.225 of the panel length has been used quite widely and
acceptably. For heavy loads or large panels, especial attention should
be given to designing and reinforcing the column capital with respect
to compressive stresses and bending moments. In the case of heavy
loads or large panels, and where the conditions of the panel loading


or variations in panel length or other conditions cause high bending
stresses in the column, and also for column capitals smaller than the
size herein recommended, especial attention should be given to design-
ing and reinforcing the column capital with respect to compression
and to rigidity of connection to floor-slab.

(h) Dropped Panel. — In one type of construction the slab is thick-
ened throughout an area surrounding the column capital. The sqiiare
or oblong of thickened slab thus formed is called a dropped panel or a
drop. The thickness and the width of the dropped panel may be
governed by the amount of resisting moment to be provided (the
compressive stress in the concrete being dependent upon both thick-
ness and width), or its thickness may be governed by the resistance
to shear required at the edge of the column capital and its width by
the allowable compressive stresses and shearing stresses in the thinner
portion of the slab adjacent to the dropped panel. Generally, however,
it is recommended that the width of the dropped panel be at least
four-tenths of the corresponding side of the panel as measured from
center to center of columns, and that the offset in thickness be not
more than five-tenths of the thickness of the slab outside the dropped

(c) Slah Thickness. — In the design of a slab, the resistance to
bending and to shearing forces will largely govern the thickness, and,
in the case of large panels with light loads, resistance to deflection
may be a controlling factor. The following formulas for minimum
thicknesses are recommended as general rules of design when the
diameter of the column capital is not less than one-fifth of the dimen-
sion of the panel from center to center of adjacent columns, the larger
dimension being used in the case of oblong panels. For notation, let

t = total thickness of slab, in inches ;
L = panel length, in feet ;
w = sum of live load and dead load, in pounds per square foot.

Then, for a slab without dropped panels,

minimum t = 0.024 L \/ w -f 1^ ;

for a slab with dropped panels,

minimum t = 0.02 L \/ w -\- 1 :,

for a dropped panel whose width is four-tenths of the panel length,

minimum t = 0.0.3 L sj w -f 1^.

In no case should the slab thickness be made less than 6 in., nor
should the thickness of a floor-slab be made less than one-thirty-
second of the panel length, nor the thickness of a roof slab less than
one-fortieth of the panel length.



(d) Bending and Resisting Moments in Slabs. — If a vertical sec-
tion of a slab be taken across a panel along a line midway between
columns, and if another section be taken along an edge of the panel
parallel to the first section, but skirting the part of the periphery of
the column capitals at the two corners of the panels, the moment of
the couple formed by the external load on the half panel, exclusive of
that over the column capital (sum of dead and live loads) and the
resultant of the external shear or reaction at the support at the two
column capitals (see Fig. 1), may be found by ordinary static analysis.
It will be noted that the edges of the area here considered are along
lines of zero shear, except around the column capitals. This moment
of the external forces acting on the half panel will be resisted by the
numerical sum of (a) the moment of the internal stresses at the section
of the panel midway between columns (positive resisting moment)

Position of resultant !
of shear on quarter |
peripiieries of two I
column capitals. I

Center of gravity of
load on half panel.

Fig. 1.

and (&) the moment of the internal stresses at the section referred to
at the end of the panel (negative resisting moment). In the curved
portion of the end section (that skirting the column), the stresses
considered are the components which act parallel to the normal stresses
on the straight portion of the section. Analysis shows that, for a
uniformly distributed load, and round columns, and square panels, the
numerical sum of the positive moment and the negative moment at
the two sections named is given quite closely by the equation

3f = — wl (I — ~c) .
^8 V 3 /

In this formula and in those which follow relating to oblong panels,

w = Sum of the live and dead loads per unit of area ;

I = Side of a square panel measured from center to center

of columns;
l^ = One side of the oblong panel measured from center

to center of columns;


Zj = Other side of obloBg panel measured in the same

c = Diameter of the column capital ;
i/a, = Numerical sum of positive moment and negative

moment in one direction;
My = Numerical sum of positive moment and negative
moment in the other direction.

(See paper and closure, "Statical Limitations upon the Steel Requirement in
Reinforced Concrete Flat Slab Floors", by John R. Nichols, Jun. Am. Soc. C. E.,
Transactions, Am. Soc. C. E., Vol. LXXVII.)

For oblong panels, the equation for the numerical sima of the posi-
tive moment and the negative moment at the two sections named

Online LibraryAmerican Society of Civil EngineersTransactions of the American Society of Civil Engineers (Volume 81) → online text (page 95 of 167)