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11 — Provisions for slabs and footings

11 — Provisions for slabs and footings

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CHAPTER 11



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For edge columns at points where the slab cantilevers

beyond the column, the critical perimeter will either be

three-sided or four-sided.



11.11.1.3 — For square or rectangular columns,

concentrated loads, or reaction areas, the critical

sections with four straight sides shall be permitted.

11.11.2 — The design of a slab or footing for two-way

action is based on Eq. (11-1) and (11-2). Vc shall be

computed in accordance with 11.11.2.1, 11.11.2.2, or

11.11.3.1. Vs shall be computed in accordance with

11.11.3. For slabs with shearheads, Vn shall be in

accordance with 11.11.4. When moment is transferred

between a slab and a column, 11.11.6 shall apply.

11.11.2.1 — For nonprestressed slabs and footings,

Vc shall be the smallest of (a), (b), and (c):

(a)



2

V c = 0.17 ⎛ 1 + ---⎞ λ f c′ b o d



β⎠



(11-31)



where β is the ratio of long side to short side of the

column, concentrated load or reaction area;

(b)



αs d

V c = 0.083 ⎛ --------- + 2⎞ λ f c′ b o d

⎝ b





(11-32)



o



where αs is 40 for interior columns, 30 for edge

columns, 20 for corner columns; and

(c)



V c = 0.33λ f c′ b o d



(11-33)



11.11.2.2 — At columns of two-way prestressed

slabs and footings that meet the requirements of

18.9.3

V c = ( β p λ f c′ + 0.3f pc )b o d + V p



(11-34)



where βp is the smaller of 3.5 and 0.083(αsd/bo +

1.5), αs is 40 for interior columns, 30 for edge

columns, and 20 for corner columns, bo is perimeter of

critical section defined in 11.11.1.2, fpc is taken as the

average value of fpc for the two directions, and Vp is

the vertical component of all effective prestress forces

crossing the critical section. Vc shall be permitted to

be computed by Eq. (11-34) if the following are satisfied;

otherwise, 11.11.2.1 shall apply:



R11.11.2.1 — For square columns, the shear stress due to

ultimate loads in slabs subjected to bending in two directions is

limited to 0.33λ f c′ . However, tests11.61 have indicated

that the value of 0.33λ f c′ is unconservative when the ratio

β of the lengths of the long and short sides of a rectangular

column or loaded area is larger than 2.0. In such cases, the

actual shear stress on the critical section at punching shear

failure varies from a maximum of about 0.33λ f c′ around

the corners of the column or loaded area, down to

0.17λ f c′ or less along the long sides between the two end

sections. Other tests11.62 indicate that vc decreases as the

ratio bo /d increases. Equations (11-31) and (11-32) were developed to account for these two effects. The words “interior,”

“edge,” and “corner columns” in 11.11.2.1(b) refer to critical

sections with four, three, and two sides, respectively.

For shapes other than rectangular, β is taken to be the ratio

of the longest overall dimension of the effective loaded area

to the largest overall perpendicular dimension of the

effective loaded area, as illustrated for an L-shaped reaction

area in Fig. R11.11.2. The effective loaded area is that area

totally enclosing the actual loaded area, for which the

perimeter is a minimum.

R11.11.2.2 — For prestressed slabs and footings, a

modified form of Code Eq. (11-31) and (11-34) is specified

for two-way action shear strength. Research11.63,11.64 indicates

that the shear strength of two-way prestressed slabs around

interior columns is conservatively predicted by Eq. (11-34).

Vc from Eq. (11-34) corresponds to a diagonal tension

failure of the concrete initiating at the critical section

defined in 11.11.1.2. The mode of failure differs from a

punching shear failure of the concrete compression zone

around the perimeter of the loaded area predicted by Eq.

(11-31). Consequently, the term β does not enter into Eq.

(11-34). Values for f c′ and fpc are restricted in design due

to limited test data available for higher values. When

computing fpc, loss of prestress due to restraint of the slab

by shear walls and other structural elements should be taken

into account.



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(a) No portion of the column cross section shall be

closer to a discontinuous edge than four times the

slab thickness;

(b) The value of f c′ used in Eq. (11-34) shall not

be taken greater than 5.8 MPa; and

(c) In each direction, fpc shall not be less than 0.9 MPa,

nor be taken greater than 3.5 MPa.



11

Fig. R11.11.2—Value of β for a nonrectangular loaded area.



In a prestressed slab with distributed tendons, the Vp term in

Eq. (11-34) contributes only a small amount to the shear

strength; therefore, it may be conservatively taken as zero. If

Vp is to be included, the tendon profile assumed in the

calculations should be noted.

For an exterior column support where the distance from the

outside of the column to the edge of the slab is less than four

times the slab thickness, the prestress is not fully effective

around bo, the total perimeter of the critical section. Shear

strength in this case is therefore conservatively taken the

same as for a nonprestressed slab.

11.11.3 — Shear reinforcement consisting of bars or

wires and single- or multiple-leg stirrups shall be

permitted in slabs and footings with d greater than or

equal to 150 mm, but not less than 16 times the shear

reinforcement bar diameter. Shear reinforcement shall

be in accordance with 11.11.3.1 through 11.11.3.4.

11.11.3.1 — Vn shall be computed by Eq. (11-2),

where Vc shall not be taken greater than 0.17λ f c′ bod,

and Vs shall be calculated in accordance with 11.4. In

Eq. (11-15), Av shall be taken as the cross-sectional

area of all legs of reinforcement on one peripheral line

that is geometrically similar to the perimeter of the

column section.

11.11.3.2 — Vn shall not be taken greater than

0.5 f c′ bod.



R11.11.3 — Research11.65-11.69 has shown that shear

reinforcement consisting of properly anchored bars or wires

and single- or multiple-leg stirrups, or closed stirrups, can

increase the punching shear resistance of slabs. The spacing

limits given in 11.11.3.3 correspond to slab shear reinforcement details that have been shown to be effective. Sections

12.13.2 and 12.13.3 give anchorage requirements for stirruptype shear reinforcement that should also be applied for bars

or wires used as slab shear reinforcement. It is essential that

this shear reinforcement engage longitudinal reinforcement at

both the top and bottom of the slab, as shown for typical

details in Fig. R11.11.3(a) to (c). Anchorage of shear

reinforcement according to the requirements of 12.13 is

difficult in slabs thinner than 250 mm. Shear reinforcement

consisting of vertical bars mechanically anchored at each end

by a plate or head capable of developing the yield strength of

the bars has been used successfully.11.69



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(a) single-leg stirrup or bar



(b) multiple-leg stirrup or bar



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(c) closed stirrups

Fig. R11.11.3(a)-(c): Single- or multiple-leg stirrup-type

slab shear reinforcement.



Fig. R11.11.3(d)—Arrangement of stirrup shear reinforcement, interior column.

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11.11.3.3 — The distance between the column face

and the first line of stirrup legs that surround the

column shall not exceed d/2. The spacing between

adjacent stirrup legs in the first line of shear reinforcement shall not exceed 2d measured in a direction

parallel to the column face. The spacing between

successive lines of shear reinforcement that surround

the column shall not exceed d/2 measured in a direction perpendicular to the column face.

11.11.3.4 — Slab shear reinforcement shall satisfy

the anchorage requirements of 12.13 and shall

engage the longitudinal flexural reinforcement in the

direction being considered.



11



Fig. R11.11.3(e)—Arrangement of stirrup shear reinforcement,

edge column.

In a slab-column connection for which the moment transfer

is negligible, the shear reinforcement should be symmetrical

about the centroid of the critical section (Fig. R11.11.3(d)).

Spacing limits defined in 11.11.3.3 are also shown in Fig.

R11.11.3(d) and (e). At edge columns or for interior

connections where moment transfer is significant, closed

stirrups are recommended in a pattern as symmetrical as

possible. Although the average shear stresses on faces AD

and BC of the exterior column in Fig. R11.11.3(e) are lower

than on face AB, the closed stirrups extending from faces

AD and BC provide some torsional strength along the edge

of the slab.

11.11.4 — Shear reinforcement consisting of structural

steel I- or channel-shaped sections (shearheads) shall

be permitted in slabs. The provisions of 11.11.4.1

through 11.11.4.9 shall apply where shear due to gravity

load is transferred at interior column supports. Where

moment is transferred to columns, 11.11.7.3 shall apply.

11.11.4.1 — Each shearhead shall consist of steel

shapes fabricated by welding with a full penetration

weld into identical arms at right angles. Shearhead

arms shall not be interrupted within the column section.



R11.11.4 — Based on reported test data,11.70 design procedures are presented for shearhead reinforcement consisting

of structural steel shapes. For a column connection transferring

moment, the design of shearheads is given in 11.11.7.3.

Three basic criteria should be considered in the design of

shearhead reinforcement for connections transferring shear

due to gravity load. First, a minimum flexural strength

should be provided to ensure that the required shear strength

of the slab is reached before the flexural strength of the

shearhead is exceeded. Second, the shear stress in the slab at



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11.11.4.2 — A shearhead shall not be deeper than

70 times the web thickness of the steel shape.

11.11.4.3 — The ends of each shearhead arm shall

be permitted to be cut at angles not less than 30 degrees

with the horizontal, provided the plastic moment

strength of the remaining tapered section is adequate

to resist the shear force attributed to that arm of the

shearhead.

11.11.4.4 — All compression flanges of steel shapes

shall be located within 0.3d of compression surface of

slab.



Fig. R11.11.4.5—Idealized shear acting on shearhead.



11

the end of the shearhead reinforcement should be limited.

Third, after these two requirements are satisfied, the

negative moment slab reinforcement can be reduced in

proportion to the moment contribution of the shearhead at

the design section.

11.11.4.5 — The ratio αv between the flexural stiffness of each shearhead arm and that of the

surrounding composite cracked slab section of width

(c2 + d) shall not be less than 0.15.

11.11.4.6 — Plastic moment strength, Mp , required

for each arm of the shearhead shall be computed by

Vu

c

h v + α v ⎛ l v – -----1-⎞

M p = ---------⎝

2φn

2⎠



(11-35)



where φ is for tension-controlled members, n is

number of shearhead arms, and lv is minimum length

of each shearhead arm required to comply with

requirements of 11.11.4.7 and 11.11.4.8.



11.11.4.7 — The critical slab section for shear shall

be perpendicular to the plane of the slab and shall

cross each shearhead arm at three-quarters the

distance [lv – (c1/2)] from the column face to the end

of the shearhead arm. The critical section shall be

located so that its perimeter bo is a minimum, but need

not be closer than the perimeter defined in 11.11.1.2(a).



R11.11.4.5 and R11.11.4.6 — The assumed idealized

shear distribution along an arm of a shearhead at an interior

column is shown in Fig. R11.11.4.5. The shear along each of

the arms is taken as αvφVc /n, where Vc is defined in

11.11.2.1(c). However, the peak shear at the face of the

column is taken as the total shear considered per arm Vu /n

minus the shear considered carried to the column by the

concrete compression zone of the slab. The latter term is

expressed as φ(Vc /n)(1 – αv), so that it approaches zero for a

heavy shearhead and approaches Vu /n when a light shearhead is used. Equation (11-35) then follows from the

assumption that φVc is about one-half the factored shear

force Vu. In this equation, Mp is the required plastic moment

strength of each shearhead arm necessary to ensure that Vu

is attained as the moment strength of the shearhead is

reached. The quantity lv is the length from the center of the

column to the point at which the shearhead is no longer

required, and the distance c1 /2 is one-half the dimension of

the column in the direction considered.

R11.11.4.7 — The test results11.70 indicated that slabs

containing under-reinforcing shearheads failed at a shear

stress on a critical section at the end of the shearhead

reinforcement less than 0.33 f c′ . Although the use of overreinforcing shearheads brought the shear strength back to

about the equivalent of 0.33 f c′ , the limited test data suggest

that a conservative design is desirable. Therefore, the shear

strength is calculated as 0.33 f c′ on an assumed critical

section located inside the end of the shearhead reinforcement.



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Fig. R11.11.4.7—Location of critical section defined in

11.11.4.7.

The critical section is taken through the shearhead arms

three-fourths of the distance [lv – (c1 /2)] from the face of

the column to the end of the shearhead. However, this

assumed critical section need not be taken closer than d/2 to

the column. See Fig. R11.11.4.7.

11.11.4.8 — Vn shall not be taken greater than

0.33 f c′ bod on the critical section defined in

11.11.4.7. When shearhead reinforcement is provided,

Vn shall not be taken greater than 0.58 f c′ bod on the

critical section defined in 11.11.1.2(a).

11.11.4.9 — Moment resistance Mv contributed to

each slab column strip by a shearhead shall not be

taken greater than

φα v V u ⎛

c

- l v – -----1-⎞

M v = ---------------⎝

2n

2⎠



(11-36)



R11.11.4.9 — If the peak shear at the face of the column

is neglected, and φVc is again assumed to be about one-half

of Vu , the moment resistance contribution of the shearhead

Mv can be conservatively computed from Eq. (11-36), in

which φ is the factor for flexure.



where φ is for tension-controlled members, n is

number of shearhead arms, and lv is length of each

shearhead arm actually provided. However, Mv shall

not be taken larger than the smallest of:

(a) 30 percent of the total factored moment required

for each slab column strip;

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(b) The change in column strip moment over the

length lv ;

(c) Mp computed by Eq. (11-35).

11.11.4.10 — When unbalanced moments are

considered, the shearhead must have adequate

anchorage to transmit Mp to the column.

11.11.5 — Headed shear stud reinforcement, placed

perpendicular to the plane of a slab or footing, shall be

permitted in slabs and footings in accordance with

11.11.5.1 through 11.11.5.4. The overall height of the

shear stud assembly shall not be less than the thickness of the member less the sum of: (1) the concrete

cover on the top flexural reinforcement; (2) the

concrete cover on the base rail; and (3) one-half the

bar diameter of the tension flexural reinforcement.

Where flexural tension reinforcement is at the bottom

of the section, as in a footing, the overall height of the

shear stud assembly shall not be less than the thickness of the member less the sum of: (1) the concrete

cover on the bottom flexural reinforcement; (2) the

concrete cover on the head of the stud; and (3) one-half

the bar diameter of the bottom flexural reinforcement.



R11.11.4.10 — See R11.11.7.3.



R11.11.5 — Headed shear stud reinforcement was introduced in the 2008 Code. Using headed stud assemblies, as

shear reinforcement in slabs and footings, requires specifying the stud shank diameter, the spacing of the studs, and

the height of the assemblies for the particular applications.

Tests11.69 show that vertical studs mechanically anchored as

close as possible to the top and bottom of slabs are effective in

resisting punching shear. The bounds of the overall specified

height achieve this objective while providing a reasonable

tolerance in specifying that height as shown in Fig. R7.7.5.

Compared with a leg of a stirrup having bends at the ends, a

stud head exhibits smaller slip, and thus results in smaller

shear crack widths. The improved performance results in

larger limits for shear strength and spacing between peripheral

lines of headed shear stud reinforcement. Typical

arrangements of headed shear stud reinforcement are shown

in Fig. R11.11.5. The critical section beyond the shear

reinforcement generally has a polygonal shape. Equations

for calculating shear stresses on such sections are given in

Reference 11.69.



11.11.5.1 — For the critical section defined in

11.11.1.2, Vn shall be computed using Eq. (11-2), with

Vc and Vn not exceeding 0.25λ f c′ bod and

0.66 f c′ bod, respectively. Vs shall be calculated

using Eq. (11-15) with Av equal to the cross-sectional

area of all the shear reinforcement on one peripheral

line that is approximately parallel to the perimeter of

the column section, where s is the spacing of the

peripheral lines of headed shear stud reinforcement.

Avfyt /(bos) shall not be less than 0.17 f c′ .



R11.11.5.1 — When there is unbalanced moment

transfer, the design will be based on stresses. The maximum

shear stress due to a combination of Vu and the fraction of

unbalanced moment γvMu should not exceed φvn, where vn

is taken as the sum of 0.25λ f c′ and Av fyt /(bos).



11.11.5.2 — The spacing between the column face

and the first peripheral line of shear reinforcement

shall not exceed d/2. The spacing between peripheral

lines of shear reinforcement, measured in a direction

perpendicular to any face of the column, shall be

constant. For prestressed slabs or footings satisfying

11.11.2.2, this spacing shall not exceed 0.75d; for all

other slabs and footings, the spacing shall be based

on the value of the shear stress due to factored shear

force and unbalanced moment at the critical section

defined in 11.11.1.2, and shall not exceed:



R11.11.5.2 — The specified spacings between peripheral

lines of shear reinforcement are justified by experiments.11.69

The clear spacing between the heads of the studs should be

adequate to permit placing of the flexural reinforcement.



(a) 0.75d where maximum shear stresses due to

factored loads are less than or equal to 0.5φ f c′ ; and

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Fig. R11.11.5—Typical arrangements of headed shear stud reinforcement and critical sections.

(b) 0.5d where maximum shear stresses due to

factored loads are greater than 0.5φ f c′ .

11.11.5.3 — The spacing between adjacent shear

reinforcement elements, measured on the perimeter of

the first peripheral line of shear reinforcement, shall

not exceed 2d.

11.11.5.4 — Shear stress due to factored shear

force and moment shall not exceed 0.17φλ f c′ at the

critical section located d/2 outside the outermost

peripheral line of shear reinforcement.



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11.11.6 — Openings in slabs



R11.11.6 — Openings in slabs



When openings in slabs are located at a distance less

than 10 times the slab thickness from a concentrated

load or reaction area, or when openings in flat slabs

are located within column strips as defined in

Chapter 13, the critical slab sections for shear defined

in 11.11.1.2 and 11.11.4.7 shall be modified as follows:



Provisions for design of openings in slabs (and footings) were

developed in Reference 11.3. The locations of the effective

portions of the critical section near typical openings and free

edges are shown by the dashed lines in Fig. R11.11.6.

Additional research11.61 has confirmed that these provisions

are conservative.



11.11.6.1 — For slabs without shearheads, that part

of the perimeter of the critical section that is enclosed

by straight lines projecting from the centroid of the

column, concentrated load, or reaction area and

tangent to the boundaries of the openings shall be

considered ineffective.

11.11.6.2 — For slabs with shearheads, the ineffective

portion of the perimeter shall be one-half of that

defined in 11.11.6.1.



11



11.11.7 — Transfer of moment in slab-column

connections



R11.11.7 — Transfer of moment in slab-column

connections



11.11.7.1 — Where gravity load, wind, earthquake,

or other lateral forces cause transfer of unbalanced

moment Mu between a slab and column, γf Mu shall be

transferred by flexure in accordance with 13.5.3. The

remainder of the unbalanced moment, γv Mu , shall be

considered to be transferred by eccentricity of shear

about the centroid of the critical section defined in

11.11.1.2 where



R11.11.7.1 — In Reference 11.71 it was found that where

moment is transferred between a column and a slab, 60 percent

of the moment should be considered transferred by flexure

across the perimeter of the critical section defined in

11.11.1.2, and 40 percent by eccentricity of the shear about

the centroid of the critical section. For rectangular columns,

the portion of the moment transferred by flexure increases as

the width of the face of the critical section resisting the

moment increases, as given by Eq. (13-1).



γv = (1 – γf )



(11-37)



Fig. R11.11.6—Effect of openings and free edges (effective

perimeter shown with dashed lines).

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Fig. R11.11.7.2—Assumed distribution of shear stress.

Most of the data in Reference 11.71 were obtained from

tests of square columns, and little information is available

for round columns. These can be approximated as square

columns. Figure R13.6.2.5 shows square supports having

the same area as some nonrectangular members.

11.11.7.2 — The shear stress resulting from

moment transfer by eccentricity of shear shall be

assumed to vary linearly about the centroid of the critical

sections defined in 11.11.1.2. The maximum shear

stress due to Vu and Mu shall not exceed φvn:

(a) For members without shear reinforcement,

φvn = φVc /(bod)



R11.11.7.2 — The stress distribution is assumed as illustrated in Fig. R11.11.7.2 for an interior or exterior column.

The perimeter of the critical section, ABCD, is determined

in accordance with 11.11.1.2. The factored shear force Vu

and unbalanced factored moment Mu are determined at the

centroidal axis c-c of the critical section. The maximum

factored shear stress may be calculated from

γ v M u c AB

V

v u ( AB ) = ------u- + -----------------------Ac

Jc



(11-38)



where Vc is as defined in 11.11.2.1 or 11.11.2.2.

(b) For members with shear reinforcement other

than shearheads,

φvn = φ(Vc + Vs)/(bod)



or

V

γ v M u c CD

v u ( CD ) = ------u- – -----------------------Ac

Jc



(11-39)



where Vc and Vs are defined in 11.11.3.1. The

design shall take into account the variation of shear

stress around the column. The shear stress due to

factored shear force and moment shall not exceed

φ(0.17λ f c′ ) at the critical section located d/2

outside the outermost line of stirrup legs that

surround the column.



where γv is given by Eq. (11-37). For an interior column, Ac

and Jc may be calculated by

Ac =

=

Jc =



area of concrete of assumed critical section

2d (c1 + c2 + 2d)

property of assumed critical section analogous to

polar moment of inertia



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3



3



2



d ( c2 + d ) ( c1 + d )

d ( c1 + d )

( c 1 + d )d

- + ------------------------ + -------------------------------------------= -----------------------6

6

2

Similar equations may be developed for Ac and Jc for

columns located at the edge or corner of a slab.

The fraction of the unbalanced moment between slab and

column not transferred by eccentricity of the shear should

be transferred by flexure in accordance with 13.5.3. A

conservative method assigns the fraction transferred by

flexure over an effective slab width defined in 13.5.3.2.

Often column strip reinforcement is concentrated near the

column to accommodate this unbalanced moment. Available

test data11.71 seem to indicate that this practice does not

increase shear strength but may be desirable to increase the

stiffness of the slab-column junction.

Test data11.72 indicate that the moment transfer strength of a

prestressed slab-to-column connection can be calculated

using the procedures of 11.11.7 and 13.5.3.

Where shear reinforcement has been used, the critical section

beyond the shear reinforcement generally has a polygonal

shape (Fig. R11.11.3(d) and (e)). Equations for calculating

shear stresses on such sections are given in Reference 11.69.

11.11.7.3 — When shear reinforcement consisting

of structural steel I- or channel-shaped sections

(shearheads) is provided, the sum of the shear

stresses due to vertical load acting on the critical

section defined by 11.11.4.7 and the shear stresses

resulting from moment transferred by eccentricity of

shear about the centroid of the critical section defined

in 11.11.1.2(a) and 11.11.1.3 shall not exceed

φ0.33λ f c′ .



R11.11.7.3 — Tests11.73 indicate that the critical sections

are defined in 11.11.1.2(a) and 11.11.1.3 and are appropriate

for calculations of shear stresses caused by transfer of

moments even when shearheads are used. Then, even

though the critical sections for direct shear and shear due to

moment transfer differ, they coincide or are in close proximity

at the column corners where the failures initiate. Because a

shearhead attracts most of the shear as it funnels toward the

column, it is conservative to take the maximum shear stress

as the sum of the two components.

Section 11.11.4.10 requires the moment Mp to be transferred to the column in shearhead connections transferring

unbalanced moments. This may be done by bearing within

the column or by mechanical anchorage.



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