4 — Openings in slab systems
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solutions based on discrete elements, or yield-line analyses,
including, in all cases, evaluation of the stress conditions
around the supports in relation to shear and torsion as well
as flexure. The design of a slab system involves more than
its analysis, and any deviations in physical dimensions of
the slab from common practice should be justified on the
basis of knowledge of the expected loads and the reliability
of the calculated stresses and deformations of the structure.
13.5.1.1 — Design of a slab system for gravity loads,
including the slab and beams (if any) between
supports and supporting columns or walls forming
orthogonal frames, by either the Direct Design Method
of 13.6 or the Equivalent Frame Method of 13.7, shall
be permitted.
R13.5.1.1 — For gravity load analysis of two-way slab
systems, two analysis methods are given in 13.6 and 13.7.
The specific provisions of both design methods are limited
in application to orthogonal frames subject to gravity loads
only. Both methods apply to two-way slabs with beams as
well as to flat slabs and flat plates. In both methods, the
distribution of moments to the critical sections of the slab
reflects the effects of reduced stiffness of elements due to
cracking and support geometry.
13.5.1.2 — For lateral loads, analysis of frames shall
take into account effects of cracking and reinforcement
on stiffness of frame members.
R13.5.1.2 — During the life of a structure, construction
loads, ordinary occupancy loads, anticipated overloads, and
volume changes will cause cracking of slabs. Cracking reduces
stiffness of slab members, and increases lateral flexibility when
lateral loads act on the structure. Cracking of slabs should be
considered in stiffness assumptions so that drift caused by
wind or earthquake is not grossly underestimated.
The structure may be modeled for lateral load analysis using
any approach that is shown to satisfy equilibrium and
geometric compatibility and to be in reasonable agreement
with test data.13.10,13.11 The selected approach should
recognize effects of cracking as well as parameters such as
l2/l1, c1/l1, and c2/c1. Some of the available approaches are
summarized in Reference 13.12, which includes a discussion
on the effects of cracking. Acceptable approaches include
plate-bending finite-element models, the effective beam
width model, and the equivalent frame model. In all cases,
framing member stiffnesses should be reduced to account
for cracking.
For nonprestressed slabs, it is normally appropriate to
reduce slab bending stiffness to between one-half and onequarter of the uncracked stiffness. For prestressed construction,
stiffnesses greater than those of cracked, nonprestressed
slabs may be appropriate. When the analysis is used to
determine design drifts or moment magnification, lowerbound slab stiffnesses should be assumed. When the analysis is
used to study interactions of the slab with other framing
elements, such as structural walls, it may be appropriate to
consider a range of slab stiffnesses so that the relative
importance of the slab on those interactions can be assessed.
13.5.1.3 — Combining the results of the gravity load
analysis with the results of the lateral load analysis
shall be permitted.
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13.5.2 — The slab and beams (if any) between
supports shall be proportioned for factored moments
prevailing at every section.
13.5.3 — When gravity load, wind, earthquake, or
other lateral forces cause transfer of moment between
slab and column, a fraction of the unbalanced moment
shall be transferred by flexure in accordance with
13.5.3.2 through 13.5.3.4.
13.5.3.1 — The fraction of unbalanced moment not
transferred by flexure shall be transferred by eccentricity of shear in accordance with 11.11.7.
13.5.3.2 — A fraction of the unbalanced moment
given by γf Mu shall be considered to be transferred by
flexure within an effective slab width between lines that
are one and one-half slab or drop panel thickness
(1.5h) outside opposite faces of the column or capital,
where Mu is the factored moment to be transferred and
13
1
γ f = ----------------------------------------------1 + ( 2 ⁄ 3 ) b1 ⁄ b2
R13.5.3 — This section is concerned primarily with slab
systems without beams. Tests and experience have shown
that, unless measures are taken to resist the torsional and
shear stresses, all reinforcement resisting that part of the
moment to be transferred to the column by flexure should be
placed between lines that are one and one-half the slab or
drop panel thickness, 1.5h, on each side of the column. The
calculated shear stresses in the slab around the column are
required to conform to the requirements of 11.11.2. See
R11.11.1.2 and R11.11.2.1 for more details on application
of this section.
(13-1)
13.5.3.3 — For nonprestressed slabs with unbalanced
moments transferred between the slab and columns, it
shall be permitted to increase the value of γf given by
Eq. (13-1) in accordance with the following:
(a) For edge columns with unbalanced moments
about an axis parallel to the edge, γf = 1.0 provided
that Vu at an edge support does not exceed
0.75φVc , or at a corner support does not exceed
0.5φVc.
(b) For unbalanced moments at interior supports,
and for edge columns with unbalanced moments
about an axis perpendicular to the edge, increase γf
to as much as 1.25 times the value from Eq. (13-1),
but not more than γf = 1.0, provided that Vu at the
support does not exceed 0.4φVc. The net tensile
strain εt calculated for the effective slab width
defined in 13.5.3.2 shall not be less than 0.010.
The value of Vc in items (a) and (b) shall be calculated
in accordance with 11.11.2.1.
R13.5.3.3 — The 1989 Code procedures remain unchanged,
except that under certain conditions it is permitted to adjust the
level of moment transferred by shear without revising member
sizes. Tests indicate that some flexibility in distribution of
unbalanced moments transferred by shear and flexure at both
exterior and interior supports is possible. Interior, exterior, and
corner supports refer to slab-column connections for which the
critical perimeter for rectangular columns has four, three, or
two sides, respectively. Changes in the 1995 Code recognized,
to some extent, design practices before the 1971 Code.13.13
At exterior supports, for unbalanced moments about an axis
parallel to the edge, the portion of moment transferred by
eccentricity of shear γv Mu may be reduced provided that the
factored shear at the support (excluding the shear produced
by moment transfer) does not exceed 75 percent of the shear
strength φVc as defined in 11.12.2.1 for edge columns or
50 percent for corner columns. Tests13.14,13.15 indicate that
there is no significant interaction between shear and
unbalanced moment at the exterior support in such cases.
Note that as γv Mu is decreased, γf Mu is increased.
Evaluation of tests of interior supports indicate that some
flexibility in distributing unbalanced moments transferred
by shear and flexure is possible, but with more severe
limitations than for exterior supports. For interior supports,
the unbalanced moment transferred by flexure is permitted
to be increased up to 25 percent provided that the factored
shear (excluding the shear caused by the moment transfer)
at the interior supports does not exceed 40 percent of the
shear strength φVc as defined in 11.11.2.1.
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When the factored shear for a slab-column connection is
large, the slab-column joint cannot always develop all of the
reinforcement provided in the effective width. The modifications for interior slab-column connections in 13.5.3.3 are
permitted only when the reinforcement (within the effective
width) required to develop the unbalanced moment γf Mu
has a net tensile strain εt not less than 0.010. The use of
Eq. (13-1) without the modification permitted in 13.5.3.3
will generally indicate overstress conditions on the joint.
The provisions of 13.5.3.3 are intended to improve ductile
behavior of the slab-column joint. When a reversal of
moments occurs at opposite faces of an interior support,
both top and bottom reinforcement should be concentrated within the effective width. A ratio of top to bottom
reinforcement of approximately 2 has been observed to
be appropriate.
For the 2008 Code, two changes were introduced to
13.5.3.3: (1) the limitation for the amount of reinforcement
in the effective slab width to 37.5 percent of the balanced
steel ratio was updated to refer to a minimum net tensile
strain of 0.010 to be consistent with the unified design
approach adopted in the 2002 Code, and (2) the requirement
for the minimum net tensile strain was eliminated for
moment transfer about the slab edge for edge and corner
connections based on the original recommendations from
Joint ACI-ASCE Committee 352.13.15
13.5.3.4 — Concentration of reinforcement over the
column by closer spacing or additional reinforcement
shall be used to resist moment on the effective slab
width defined in 13.5.3.2.
13.5.4 — Design for transfer of load from slabs to
supporting columns or walls through shear and torsion
shall be in accordance with Chapter 11.
13.6 — Direct design method
R13.6 — Direct design method
The direct design method consists of a set of rules for
distributing moments to slab and beam sections to satisfy
safety requirements and most serviceability requirements
simultaneously. Three fundamental steps are involved as
follows:
(1) Determination of the total factored static moment (see
13.6.2);
(2) Distribution of the total factored static moment to
negative and positive sections (see 13.6.3);
(3) Distribution of the negative and positive factored
moments to the column and middle strips and to the
beams, if any (see 13.6.4 through 13.6.6). The distribution
of moments to column and middle strips is also used in
the equivalent frame method (see 13.7).
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13.6.1 — Limitations
R13.6.1 — Limitations
Design of slab systems within the limitations of
13.6.1.1 through 13.6.1.8 by the direct design method
shall be permitted.
The direct design method was developed from considerations
of theoretical procedures for the determination of moments
in slabs with and without beams, requirements for simple
design and construction procedures, and precedents
supplied by performance of slab systems. Consequently, the
slab systems to be designed using the direct design method
should conform to the limitations in this section.
13.6.1.1 — There shall be a minimum of three
continuous spans in each direction.
R13.6.1.1 — The primary reason for the limitation in this
section is the magnitude of the negative moments at the
interior support in a structure with only two continuous
spans. The rules given for the direct design method assume
that the slab system at the first interior negative moment
section is neither fixed against rotation nor discontinuous.
13.6.1.2 — Panels shall be rectangular, with a ratio
of longer to shorter span center-to-center of supports
within a panel not greater than 2.
R13.6.1.2 — If the ratio of the two spans (long span/short
span) of a panel exceeds 2, the slab resists the moment in
the shorter span essentially as a one-way slab.
13.6.1.3 — Successive span lengths center-tocenter of supports in each direction shall not differ by
more than one-third the longer span.
R13.6.1.3 — The limitation in this section is related to the
possibility of developing negative moments beyond the
point where negative moment reinforcement is terminated,
as prescribed in Fig. 13.3.8.
13.6.1.4 — Offset of columns by a maximum of 10
percent of the span (in direction of offset) from either
axis between centerlines of successive columns shall
be permitted.
R13.6.1.4 — Columns can be offset within specified
limits from a regular rectangular array. A cumulative total
offset of 20 percent of the span is established as the upper
limit.
13.6.1.5 — All loads shall be due to gravity only and
uniformly distributed over an entire panel. The
unfactored live load shall not exceed two times the
unfactored dead load.
R13.6.1.5 — The direct design method is based on
tests13.16 for uniform gravity loads and resulting column
reactions determined by statics. Lateral loads such as wind
or seismic require a frame analysis. Inverted foundation
mats designed as two-way slabs (see 15.10) involve application
of known column loads. Therefore, even where the soil
reaction is assumed to be uniform, a frame analysis should
be performed.
In the 1995 Code, the limit of applicability of the direct
design method for ratios of live load to dead load was
reduced from 3 to 2. In most slab systems, the live to dead
load ratio will be less than 2 and it will not be necessary to
check the effects of pattern loading.
13.6.1.6 — For a panel with beams between
supports on all sides, Eq. (13-2) shall be satisfied for
beams in the two perpendicular directions
R13.6.1.6 — The elastic distribution of moments will
deviate significantly from those assumed in the direct design
method unless the requirements for stiffness are satisfied.
2
α f 1l2
0.2 ≤ -------------≤ 5.0
2
α f 2l1
(13-2)
where αf1 and αf 2 are calculated in accordance with
Eq. (13-3).
ACI 318 Building Code and Commentary