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.
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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.
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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.
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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.
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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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