Code
Building
Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI
318R-14)
Reference
Concrete
Floor Systems (Guide to Estimating and Economizing), Second Edition, 2002 David
A. Fanella
Notes
on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth
Edition, 2013 Portland Cement Association.
Simplified
Design of Reinforced Concrete Buildings, Fourth Edition, 2011 Mahmoud E. Kamara
and Lawrence C. Novak
Control
of Deflection in Concrete Structures (ACI 435R-95)
Design Data
Story Height = 13 ft
(provided by architectural drawings)
Superimposed Dead Load,
SDL =20 psf for framed partitions, wood studs, 2 x 2, plastered 2 sides
ASCE/SEI
7-10 (Table C3-1)
Live Load, LL = 60 psf
ASCE/SEI
7-10 (Table 4-1)
50 psf is considered
by inspection of Table 4-1 for Office Buildings – Offices (2/3 of the floor
area)
80 psf is considered
by inspection of Table 4-1 for Office Buildings – Corridors (1/3 of the floor
area)
LL = 2/3 x 50 +
1/3 x 80 = 60 psf
fc’ = 5000 psi (for slab)
fc’ = 6000 psi (for columns)
fy = 60,000 psi
Solution
For Flat Plate (without Drop Panels)
a.
Slab minimum
thickness – Deflection ACI
318-14 (8.3.1.1)
In lieu of
detailed calculation for deflections, ACI 318 Code gives minimum slab thickness
for two-way construction without interior beams in Table 8.3.1.1.
For this flat plate slab
systems the minimum slab thicknesses per ACI 318-14 are:
ACI 318-14 (Table 8.3.1.1)
But not less than 5 in. ACI
318-14 (8.3.1.1(a))
ACI 318-14 (Table 8.3.1.1)
But not less than 5 in. ACI
318-14 (8.3.1.1(a))
Where ln
= length of clear span in the long direction = 30 x 12 – 20 = 340 in.
Try 11 in. slab for all
panels (self-weight = 150 pcf x 11 in. /12 = 137.5 psf)
b. Slab shear strength – one way shear
Evaluate the average
effective depth (Figure 2):
Where:
cclear = 3/4 in. for
# 6 steel bar ACI
318-14 (Table 20.6.1.3.1)
db = 0.75 in. for # 6 steel bar
Figure 2 - Two-Way Flat Concrete Floor System
ACI
318-14 (5.3.1)
Check the adequacy of
slab thickness for beam action (one-way shear) ACI
318-14 (22.5)
at an interior column:
Consider a 12-in. wide
strip. The critical section for one-way shear is located at a distance d,
from the face of support (see Figure 3):
ACI
318-14 (Eq. 22.5.5.1)
Slab thickness of 11 in.
is adequate for one-way shear.
c.
Slab shear
strength – two-way shear
Check the adequacy of
slab thickness for punching shear (two-way shear) at an interior column (Figure
4):
ACI
318-14 (Table 22.6.5.2(a))
Slab thickness of 11 in. is not adequate
for two-way shear.
Figure 3 –
Critical Section for One-Way Shear
Figure 4 – Critical Section for Two-Way Shear
In
this case, three options could be used: 1) to increase the slab thickness, 2)
to use headed shear reinforcement, or 3) to use drop panels. In this example,
the latter option will be used to achieve better understanding for the design
of two-way slab with drop panels often called flat slab.
Check the drop panel
dimensional limitations as follows:
1)
The drop panel
shall project below the slab at least one-fourth of the adjacent slab
thickness.
ACI
318-14 (8.2.4(a))
Since the slab thickness (hs) is 10 in. (see page 6), the
thickness of the drop panel should be at least:
Drop panel dimensions are also
controlled by formwork considerations. The following Figure shows the standard
lumber dimensions that are used when forming drop panels. Using other depths
will unnecessarily increase formwork costs.
For nominal lumber size (2x), hdp
= 4.25 in. > hdp, min = 2. 5 in.
The total thickness including the
slab and the drop panel (h) = hs + hdp = 10
+ 4.25 = 14.25 in.
|
Nominal Lumber Size, in.
|
Actual Lumber Size, in.
|
Plyform Thickness, in.
|
hdp, in.
|
|
2x
|
1 1/2
|
3/4
|
2 1/4
|
|
4x
|
3 1/2
|
3/4
|
4 1/4
|
|
6x
|
5 1/2
|
3/4
|
6 1/4
|
|
8x
|
7 1/4
|
3/4
|
8
|
Figure 5 – Drop Panel Formwork Details
2)
The drop panel
shall extend in each direction from the centerline of support a distance not
less than one-sixth the span length measured from center-to-center of supports
in that direction.
ACI
318-14 (8.2.4(b))
Based on the previous discussion, Figure
6 shows the dimensions of the selected drop panels around interior, edge
(exterior), and corner columns.
Figure 6 – Drop Panels Dimensions
For Flat Slab (with Drop Panels)
For slabs with changes in thickness and subjected to
bending in two directions, it is necessary to check shear at multiple sections
as defined in the ACI 319-14. The critical sections shall
be located with respect to:
1) Edges or corners of
columns. ACI 318-14 (22.6.4.1(a))
2) Changes in slab
thickness, such as edges of drop panels. ACI 318-14 (22.6.4.1(b))
a.
Slab minimum
thickness – Deflection ACI
318-14 (8.3.1.1)
In lieu of
detailed calculation for deflections, ACI 318 Code gives minimum slab thickness
for two-way construction without interior beams in Table 8.3.1.1.
For this flat plate slab
systems the minimum slab thicknesses per ACI 318-14 are:
ACI 318-14 (Table 8.3.1.1)
But not less than 4 in. ACI
318-14 (8.3.1.1(b))
ACI 318-14 (Table 8.3.1.1)
But not less than 4 in. ACI
318-14 (8.3.1.1(b))
Where ln
= length of clear span in the long direction = 30 x 12 – 20 = 340 in.
Try 10 in. slab for all
panels
Self-weight for slab
section without drop panel = 150 pcf x 10 in. /12 = 125 psf
Self-weight for slab
section with drop panel = 150 pcf x 14.25 in. /12 = 178 psf
b. Slab shear strength – one way shear
For critical section
at distance d from the edge of the column (slab section with drop
panel):
Evaluate the average
effective depth:
Where:
cclear = 3/4 in. for
# 6 steel bar ACI
318-14 (Table 20.6.1.3.1)
db = 0.75 in. for # 6 steel bar
ACI
318-14 (5.3.1)
Check the adequacy of
slab thickness for beam action (one-way shear) from the edge of the interior
column
ACI
318-14 (22.5)
Consider a 12-in. wide
strip. The critical section for one-way shear is located at a distance d,
from the edge of the column (see Figure 7)
ACI
318-14 (Eq. 22.5.5.1)
Slab thickness of 14.25 in.
is adequate for one-way shear for the first critical section (from the edge of
the column).
For critical section
at the edge of the drop panel (slab section without drop panel):
Evaluate the average
effective depth:
Where:
cclear = 3/4 in. for
# 6 steel bar ACI
318-14 (Table 20.6.1.3.1)
db = 0.75 in. for # 6 steel bar
ACI
318-14 (5.3.1)
Check the
adequacy of slab thickness for beam action (one-way shear) from the edge of the
interior drop panel ACI 318-14 (22.5)
Consider a 12-in. wide
strip. The critical section for one-way shear is located at the face of support
(see Figure 7)
ACI
318-14 (Eq. 22.5.5.1)
Slab thickness of 10 in.
is adequate for one-way shear for the second critical section (from the edge of
the drop panel).
Figure 7 – Critical Sections for One-Way Shear
c.
Slab shear
strength – two-way shear
For critical section at distance d/2 from the edge of the column
(slab section with drop panel):
Check the adequacy of slab thickness
for punching shear (two-way shear) at an interior column (Figure 8):
ACI
318-14 (Table 22.6.5.2(a))
Slab thickness of 14.25 in. is adequate
for two-way shear for the first critical section (from the edge of the column).
For critical section at the edge of the drop panel
(slab section without drop panel):
Check the adequacy of slab thickness
for punching shear (two-way shear) at an interior drop panel (Figure 8):
ACI
318-14 (Table 22.6.5.2(a))
Slab thickness of 10 in. is adequate
for two-way shear for the second critical section (from the edge of the drop
panel).
Figure 8 – Critical Sections for Two-Way Shear
d. Column dimensions - axial load
Check the adequacy of
column dimensions for axial load:
Tributary area for
interior column for live load, superimposed dead load, and self-weight of the
slab is
Tributary area for
interior column for self-weight of additional slab thickness due to the
presence of the drop panel is
Assuming five story
building
Assume 20 in. square
column with 4 – No. 14 vertical bars with design axial strength, φPn,max
of
ACI
318-14 (22.4.2)
Column dimensions of 20 in. x 20
in. are adequate for axial load.
ACI 318 states
that a slab system shall be designed by any procedure satisfying equilibrium
and geometric compatibility, provided that strength and serviceability criteria
are satisfied. Distinction of two-systems from one-way systems is given by ACI
318-14 (R8.10.2.3 & R8.3.1.2).
ACI 318 permits the use of Direct
Design Method (DDM) and Equivalent Frame Method (EFM) for the gravity load
analysis of orthogonal frames and is applicable to flat plates, flat slabs, and
slabs with beams. The following sections outline the solution per EFM and
spSlab software. For the solution per DDM, check the flat plate example.
EFM is the most comprehensive and
detailed procedure provided by the ACI 318 for the analysis and design of
two-way slab systems where the structure is modeled by a series of equivalent
frames (interior and exterior) on column lines taken longitudinally and
transversely through the building.
The equivalent frame consists of three
parts (for a detailed discussion of this method, refer to the
flat plate design example):
1) Horizontal slab-beam strip.
2) Columns or other vertical supporting
members.
3) Elements of the structure (Torsional
members) that provide moment transfer between the horizontal and vertical
members.
In EFM, live load shall be arranged in accordance with 6.4.3 which
requires slab systems to be analyzed and designed for the most demanding set of
forces established by investigating the effects of live load placed in various
critical patterns. ACI 318-14 (8.11.1.2 & 6.4.3)
Complete analysis must include representative interior
and exterior equivalent frames in both the longitudinal and transverse
directions of the floor. ACI 318-14 (8.11.2.1)
Panels
shall be rectangular, with a ratio of longer to shorter panel dimensions, measured
center-to-center of supports, not to exceed 2. ACI 318-14 (8.10.2.3)
Determine moment distribution factors and fixed-end
moments for the equivalent frame members. The moment distribution procedure
will be used to analyze the equivalent frame. Stiffness factors k, carry over factors COF, and fixed-end moment factors
FEM for the slab-beams and column members are determined using the design aids tables
at Appendix 20A of PCA Notes on ACI 318-11. These
calculations are shown below.
a. Flexural stiffness of slab-beams at
both ends, Ksb.
PCA
Notes on ACI 318-11 (Table A1)
PCA
Notes on ACI 318-11 (Table A1)
ACI 318-14 (19.2.2.1.a)
Carry-over factor COF = 0.578 PCA
Notes on ACI 318-11 (Table A1)
PCA
Notes on ACI 318-11 (Table A1)
Uniform load fixed end moment coefficient, mNF1
= 0.0915
Fixed end moment coefficient for (b-a) = 0.2 when a =
0, mNF2 = 0.0163
Fixed end moment coefficient for (b-a) = 0.2 when a = 0.8, mNF3
= 0.0163
b.
Flexural
stiffness of column members at both ends, Kc.
Referring to Table A7, Appendix 20A,
For the Bottom Column:
PCA
Notes on ACI 318-11 (Table A7)
ACI 318-14 (19.2.2.1.a)
lc = 13 ft = 156 in.
For the Top Column:
PCA
Notes on ACI 318-11 (Table A7)
c. Torsional stiffness of torsional
members, 
.
ACI 318-14 (R.8.11.5)
ACI 318-14 (Eq. 8.10.5.2b)
Equivalent column stiffness Kec.
Where∑
Kt is for two torsional
members one on each side of the column, and ∑ Kc is for
the upper and lower columns at the slab-beam joint of an intermediate floor.
Figure
9 – Torsional Member Figure 10 – Column and Edge of Slab
d.
Slab-beam joint distribution
factors, DF.
At exterior joint,
At interior joint,
COF for slab-beam =0.578
Figure 11 – Slab and Column Stiffness
Determine negative and positive moments for the slab-beams using the
moment distribution method. Since the unfactored live load does not exceed
three-quarters of the unfactored dead load, design moments are assumed to occur
at all critical sections with full factored live on all spans. ACI
318-14 (6.4.3.2)
a. Factored load and Fixed-End Moments
(FEM’s).
For slab:
For drop panels:
PCA
Notes on ACI 318-11 (Table A1)
b. Moment distribution. Computations are
shown in Table 1. Counterclockwise rotational moments acting on the member ends
are taken as positive. Positive span moments are determined from the following
equation:
Where Mo is the
moment at the midspan for a simple beam.
When the end moments are not equal, the maximum moment
in the span does not occur at the midspan, but its value is close to that
midspan for this example.
Positive moment in span 1-2:
|
Table 1 - Moment Distribution for Equivalent Frame
|
|
|
|
Joint
|
1
|
2
|
3
|
4
|
|
Member
|
1-2
|
2-1
|
2-3
|
3-2
|
3-4
|
4-3
|
|
DF
|
0.551
|
0.355
|
0.355
|
0.355
|
0.355
|
0.551
|
|
COF
|
0.578
|
0.578
|
0.578
|
0.578
|
0.578
|
0.578
|
|
FEM
|
677.6
|
-677.6
|
677.6
|
-677.6
|
677.6
|
-677.6
|
|
Dist
|
-373.1
|
0.0
|
0.0
|
0.0
|
0.0
|
373.1
|
|
CO
|
0.0
|
-215.7
|
0.0
|
0.0
|
215.7
|
0.0
|
|
Dist
|
0.0
|
76.6
|
76.6
|
-76.6
|
-76.6
|
0.0
|
|
CO
|
44.3
|
0.0
|
-44.3
|
44.3
|
0.0
|
-44.3
|
|
Dist
|
-24.4
|
15.7
|
15.7
|
-15.7
|
-15.7
|
24.4
|
|
CO
|
9.1
|
-14.1
|
-9.1
|
9.1
|
14.1
|
-9.1
|
|
Dist
|
-5.0
|
8.2
|
8.2
|
-8.2
|
-8.2
|
5.0
|
|
CO
|
4.8
|
-2.9
|
-4.8
|
4.8
|
2.9
|
-4.8
|
|
Dist
|
-2.6
|
2.7
|
2.7
|
-2.7
|
-2.7
|
2.6
|
|
CO
|
1.6
|
-1.5
|
-1.6
|
1.6
|
1.5
|
-1.6
|
|
Dist
|
-0.9
|
1.1
|
1.1
|
-1.1
|
-1.1
|
0.9
|
|
CO
|
0.6
|
-0.5
|
-0.6
|
0.6
|
0.5
|
-0.6
|
|
Dist
|
-0.4
|
0.4
|
0.4
|
-0.4
|
-0.4
|
0.4
|
|
CO
|
0.2
|
-0.2
|
-0.2
|
0.2
|
0.2
|
-0.2
|
|
Dist
|
-0.1
|
0.2
|
0.2
|
-0.2
|
-0.2
|
0.1
|
|
CO
|
0.1
|
-0.1
|
-0.1
|
0.1
|
0.1
|
-0.1
|
|
Dist
|
-0.1
|
0.1
|
0.1
|
-0.1
|
-0.1
|
0.1
|
|
CO
|
0.0
|
0.0
|
0.0
|
0.0
|
0.0
|
0.0
|
|
Dist
|
0.0
|
0.0
|
0.0
|
0.0
|
0.0
|
0.0
|
|
M, k-ft
|
331.7
|
-807.6
|
721.9
|
-721.9
|
807.6
|
-331.7
|
|
Midspan M,
ft-kips
|
349.6
|
197.4
|
349.6
|
Positive and
negative factored moments for the slab system in the direction of analysis are
plotted in Figure 12. The negative moments used for design are taken at the
faces of supports (rectangle section or equivalent rectangle for circular or
polygon sections) but not at distances greater than 0.175 l1
from the centers of supports. ACI
318-14 (8.11.6.1)
Figure 12 - Positive
and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Factored
Live Load)
a. Check whether the moments calculated above can take
advantage of the reduction permitted by ACI 318-14 (8.11.6.5):
If the slab
system analyzed using EFM within the limitations of ACI 318-14
(8.10.2), it is permitted by the ACI code to reduce the calculated
moments obtained from EFM in such proportion that the absolute sum of the
positive and average negative design moments need not exceed the total static
moment Mo given by Equation 8.10.3.2 in
the ACI 318-14.
Check Applicability
of Direct Design Method:
1.
There is a
minimum of three continuous spans in each direction. ACI
318-14 (8.10.2.1)
2.
Successive span
lengths are equal. ACI
318-14 (8.10.2.2)
3.
Long-to-Short
ratio is 30/30 = 1.0 < 2.0. ACI
318-14 (8.10.2.3)
4.
Column are not
offset. ACI
318-14 (8.10.2.4)
5.
Loads are gravity
and uniformly distributed with service live-to-dead ratio of 0.41 < 2.0
(Note: The self-weight of the drop
panels is not uniformly distributed entirely along the span. However, the
variation in load magnitude is small).
ACI
318-14 (8.10.2.5 and 6)
6.
Check relative
stiffness for slab panel. ACI
318-14 (8.10.2.7)
Slab
system is without beams and this requirement is not applicable.
All limitation of ACI 318-14
(8.10.2) are satisfied and the provisions of ACI 318-14 (8.11.6.5)
may be applied:
ACI 318-14 (Eq. 8.10.3.2)
To illustrate proper procedure, the interior span
factored moments may be reduced as follows:
Permissible reduction = 812.8/919.3 = 0.884
Adjusted negative
design moment = 721.9 × 0.884 = 638.2 ft-kips
Adjusted positive
design moment = 197.3 × 0.884 = 174.4 ft-kips
ACI 318 allows the reduction of the moment values
based on the previous procedure. Since the drop panels may cause gravity loads
not to be uniform (Check limitation #5 and Figure 12), the moment values
obtained from EFM will be used for comparison reasons.
b. Distribute factored moments to column and middle
strips:
After the negative and positive moments have been
determined for the slab-beam strip, the ACI code permits the distribution of
the moments at critical sections to the column strips, beams (if any), and
middle strips in accordance with the DDM. ACI
318-14 (8.11.6.6)
Distribution of factored moments at critical sections
is summarized in Table 2.
|
Table 2 - Distribution of
factored moments
|
|
|
Slab-beam Strip
|
Column Strip
|
Middle Strip
|
|
Moment
(ft-kips)
|
Percent
|
Moment
(ft-kips)
|
Percent
|
Moment
(ft-kips)
|
|
End Span
|
Exterior Negative
|
246.5
|
100
|
246.5
|
0
|
0.0
|
|
Positive
|
349.6
|
60
|
209.8
|
40
|
139.8
|
|
Interior Negative
|
695.9
|
75
|
521.9
|
25
|
174.0
|
|
Interior Span
|
Negative
|
623.4
|
75
|
467.6
|
25
|
155.9
|
|
Positive
|
197.3
|
60
|
118.4
|
40
|
78.9
|
a. Determine flexural reinforcement required for strip
moments
The flexural reinforcement calculation for the column
strip of end span – exterior negative location is provided below.
Use davg = 12.75 in.
To determine the area of steel, assumptions have to be
made whether the section is tension or compression controlled, and regarding
the distance between the resultant compression and tension forces along the
slab section (jd). In this example, tension-controlled section will be
assumed so the reduction factor
is equal to 0.9, and jd will be taken equal to 0.95d.
The assumptions will be verified once the area of steel in finalized.
Therefore, the assumption that section is
tension-controlled is valid.
The slab have two thicknesses in the column strip
(14.25 in. for the slab with the drop panel and 10 in. for the slab without the
drop panel).
ACI
318-14 (24.4.3.2)
ACI 318-14 (8.7.2.2)
Provide 10 - #6 bars with As = 4.40 in.2
and s = 180/10 = 18 in. ≤ smax
Based on the procedure outlined above, values for all
span locations are given in Table 3.
|
Table 3 - Required Slab Reinforcement for
Flexure [Equivalent Frame Method (EFM)]
|
|
Span Location
|
Mu
(ft-kips)
|
b
(in.)
|
d (in.)
|
As Req’d for flexure (in.2)
|
Min As (in.2)
|
Reinforcement Provided
|
As Prov. for flexure (in.2)
|
|
End Span
|
|
Column Strip
|
Exterior Negative
|
246.5
|
180
|
12.75
|
4.357
|
4.157
|
10-#6
|
4.40
|
|
Positive
|
209.8
|
180
|
8.50
|
5.631
|
3.240
|
13-#6
|
5.72
|
|
Interior Negative
|
521.9
|
180
|
12.75
|
9.366
|
4.157
|
22-#6
|
9.68
|
|
Middle Strip
|
Exterior Negative
|
0.0
|
180
|
8.50
|
0.0
|
3.240
|
10-#6 * **
|
4.40
|
|
Positive
|
139.8
|
180
|
8.50
|
3.719
|
3.240
|
10-#6 **
|
4.40
|
|
Interior Negative
|
174.0
|
180
|
8.50
|
4.649
|
3.240
|
11-#6
|
4.84
|
|
Interior Span
|
|
Column Strip
|
Positive
|
118.4
|
180
|
8.50
|
3.141
|
3.240
|
10-#6 * **
|
4.40
|
|
Middle Strip
|
Positive
|
78.9
|
180
|
8.50
|
2.083
|
3.240
|
10-#6 * **
|
4.40
|
|
* Design governed by minimum reinforcement.
** Number of bars governed by maximum allowable spacing.
|
b. Calculate additional slab reinforcement at columns
for moment transfer between slab and column by flexure
The factored slab moment resisted by the column (γf
x Msc) shall be assumed to be transferred by flexure.
Concentration of reinforcement over the column by closer spacing or additional
reinforcement shall be used to resist this moment. The fraction of slab moment
not calculated to be resisted by flexure shall be assumed to be resisted by
eccentricity of shear. ACI 318-14 (8.4.2.3)
Portion of the unbalanced moment transferred by
flexure is γf x Msc ACI
318-14 (8.4.2.3.1)
Where
ACI
318-14 (8.4.2.3.2)
b1
= Dimension of the critical section bo
measured in the direction of the span for which
moments are determined in ACI 318, Chapter 8 (see Figure 13).
b2
= Dimension of the critical section 
measured in the direction perpendicular to 
in ACI 318, Chapter 8 (see Figure 13).
bb
= Effective slab width =
ACI
318-14 (8.4.2.3.3)
Figure 13 – Critical Shear Perimeters for Columns
For exterior support:
d = h – cover – d/2 = 14.25 – 0.75 – 0.75/2 = 13.13 in.
c1 + d/2 = 20 + 13.13/2 = 26.56 in.
c2 + d = 20 + 13.13 = 33.13 in.
in.
Using the same procedure in 2.1.7.a, the required area
of steel:
However, the area of steel provided to resist the
flexural moment within the effective slab width bb:
Then, the required additional reinforcement at
exterior column for moment transfer between slab and column:
Provide 5 - #6 additional bars with As
= 2.20 in.2
Based on the procedure outlined above, values for all
supports are given in Table 4.
|
Table 4 - Additional Slab Reinforcement
required for moment transfer between slab and column (EFM)
|
|
Span Location
|
Msc*
(ft-kips)
|
γf
|
γf Msc
(ft-kips)
|
Effective slab
width, bb
(in.)
|
d
(in.)
|
As req’d
within bb
(in.2)
|
As prov. For
flexure within bb
(in.2)
|
Add’l
Reinf.
|
|
End Span
|
|
Column Strip
|
Exterior Negative
|
331.7
|
0.626
|
207.7
|
62.75
|
13.13
|
3.63
|
1.534
|
5-#6
|
|
Interior Negative
|
85.7
|
0.60
|
51.42
|
62.75
|
13.13
|
0.877
|
3.375
|
-
|
|
*Msc is taken at the
centerline of the support in Equivalent Frame Method solution.
|
The unbalanced moment from the slab-beams at the
supports of the equivalent frame are distributed to the support columns above
and below the slab-beam in proportion to the relative stiffness of the support
columns. Referring to Figure 12, the unbalanced moment at the exterior and
interior joints are:
Exterior Joint = +331.7 ft-kips
Joint 2= -807.6 + 721.9 = -85.7 ft-kips
The stiffness and carry-over factors of the actual
columns and the distribution of the unbalanced slab moments (Msc)
to the exterior and interior columns are shown in Figure 14.
Figure 14 - Column Moments (Unbalanced Moments from
Slab-Beam)
In summary:
For Top column: For
Bottom column:
Mcol,Exterior= 150.61 ft-kips Mcol,Exterior=
157.21 ft-kips
Mcol,Interior = 38.91 ft-kips Mcol,Interior
= 40.62 ft-kips
The moments determined above are combined with the
factored axial loads (for each story) and factored moments in the transverse
direction for design of column sections. The moment values at the face of
interior, exterior, and corner columns from the unbalanced moment values are
shown in the following table.
|
Table 5 – Factored Moments in Columns
|
|
Mu
kips-ft
|
Column Location
|
|
Interior
|
Exterior
|
Corner
|
|
Mux
|
40.62
|
157.21
|
157.21
|
|
Muy
|
40.62
|
40.62
|
157.21
|
This section includes the design of
interior, edge, and corner columns using spColumn
software. The preliminary dimensions for these columns were calculated
previously in section one. The reduction of live load per ASCE 7-10
will be ignored in this example. However, the detailed procedure to calculate
the reduced live loads is explained in the “wide-Module
Joist System” example.
Interior Column:
Assume 5 story building
Tributary
area for interior column for live load, superimposed dead load, and self-weight
of the slab is
Tributary
area for interior column for self-weight of additional slab thickness due to
the presence of the drop panel is
Assuming five story
building
Mu,x = 40.62 ft-kips (see the previous Table)
Mu,y = 40.62 ft-kips (see the
previous Table)
Edge (Exterior) Column:
Tributary area for edge
column for live load, superimposed dead load, and self-weight of the slab is
Tributary area for edge
column for self-weight of additional slab thickness due to the presence of the
drop panel is
Mu,x = 157.21 ft-kips (see the previous Table)
Mu,y = 40.62 ft-kips (see the
previous Table)
Corner Column:
Tributary area for corner
column for live load, superimposed dead load, and self-weight of the slab is
Tributary area for corner
column for self-weight of additional slab thickness due to the presence of the
drop panel is
Mu,x = 157.21 ft-kips (see the previous Table)
Mu,y = 157.21 ft-kips (see the
previous Table)
Interior Column:
Edge Column:
Corner Column:
Shear strength of the slab in the vicinity
of columns/supports includes an evaluation of one-way shear (beam action) and
two-way shear (punching) in accordance with ACI 318 Chapter 22.
ACI
318-14 (22.5)
One-way
shear is critical at a distance d from the face of the column as shown
in Figure 3. Figures 15 and 16 show the factored shear forces (Vu)
at the critical sections around each column and each drop panel, respectively. In
members without shear reinforcement, the design shear capacity of the section
equals to the design shear capacity of the concrete:
ACI 318-14 (Eq. 22.5.1.1)
Where:
ACI 318-14 (Eq. 22.5.5.1)
Figure 15 – One-way shear at critical sections (at
distance d from the face of the supporting column)
in.
Figure 16 – One-way shear at critical sections (at the
face of the drop panel)
ACI
318-14 (22.6)
Two-way shear is critical on
a rectangular section located at d/2 away from the face of the column as
shown in Figure 13.
a. Exterior column:
The
factored shear force (Vu) in the critical section is computed
as the reaction at the centroid of the critical section minus the self-weight
and any superimposed surface dead and live load acting within the critical
section (d/2 away from column face).
The factored
unbalanced moment used for shear transfer, Munb, is computed
as the sum of the joint moments to the left and right. Moment of the vertical
reaction with respect to the centroid of the critical section is also taken
into account.
For the exterior
column in Figure 13, the location of the centroidal axis z-z is:
The
polar moment Jc of the shear perimeter is:
ACI 318-14 (Eq. 8.4.4.2.2)
The
length of the critical perimeter for the exterior column:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
b. Interior column:
For
the interior column in Figure 13, the location of the centroidal axis z-z is:
The
polar moment Jc of the shear perimeter is:
ACI 318-14 (Eq. 8.4.4.2.2)
The
length of the critical perimeter for the interior column:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
c. Corner column:
In
this example, interior equivalent frame strip was selected where it only have
exterior and interior supports (no corner supports are included in this strip).
However, the two-way shear strength of corner supports usually governs. Thus,
the two-way shear strength for the corner column in this example will be
checked for educational purposes. Same procedure is used to find the reaction
and factored unbalanced moment used for shear transfer at the centroid of the
critical section for the corner support for the exterior equivalent frame
strip.
For
the interior column in Figure 13, the location of the centroidal axis z-z is:
in.
The
polar moment Jc of the shear perimeter is:
ACI 318-14 (Eq. 8.4.4.2.2)
The
length of the critical perimeter for the corner column:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
Two-way shear is critical on
a rectangular section located at d/2 away from the face of the drop
panel.
a. Exterior drop panel:
The
length of the critical perimeter for the exterior drop panel:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
b. Interior drop panel:
The
length of the critical perimeter for the interior drop panel:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
c. Corner drop panel:
The
length of the critical perimeter for the corner drop panel:
The
two-way shear stress (vu) can then be calculated as:
ACI 318-14 (R.8.4.4.2.3)
ACI 318-14 (Table 22.6.5.2)
Since the slab thickness was
selected below the minimum slab thickness tables in ACI 318-14, the deflection calculations
of immediate and time-dependent deflections are required and shown below
including a comparison with spSlab model results.
The
calculation of deflections for two-way slabs is challenging even if linear
elastic behavior can be assumed. Elastic analysis for three service load levels
(D, D + Lsustained, D+LFull) is used to obtain immediate
deflections of the two-way slab in this example. However, other procedures may
be used if they result in predictions of deflection in reasonable agreement
with the results of comprehensive tests. ACI
318-14 (24.2.3)
The
effective moment of inertia (Ie) is used to account for the cracking
effect on the flexural stiffness of the slab. Ie for
uncracked section (Mcr > Ma) is equal to Ig.
When the section is cracked (Mcr < Ma), then
the following equation should be used:
ACI 318-14 (Eq. 24.2.3.5a)
Where:
Ma = Maximum moment in member due to service loads at stage deflection is
calculated.
The
values of the maximum moments for the three service load levels are calculated
from structural analysis as shown previously in this document. These moments are
shown in Figure 17.
Figure 17 – Maximum Moments for the
Three Service Load Levels
For positive moment (midspan) section:
ACI
318-14 (Eq. 24.2.3.5b)
ACI
318-14 (Eq. 19.2.3.1)
yt = Distance from centroidal axis
of gross section, neglecting reinforcement, to tension face, in.
PCA Notes on ACI 318-11 (9.5.2.2)
As
calculated previously, the positive reinforcement for the end span frame strip
is 23 #6 bars located at 1.125 in. along the section from the bottom of the
slab. Two of these bars are not continuous and will be excluded from the
calculation of Icr. Figure 18 shows all the parameters
needed to calculate the moment of inertia of the cracked section transformed to
concrete at midspan.
Figure 18 –
Cracked Transformed Section (positive moment section)
ACI
318-14 (19.2.2.1.a)
PCA Notes on ACI 318-11 (Table 10-2)
PCA
Notes on ACI 318-11 (Table 10-2)
PCA Notes on
ACI 318-11 (Table 10-2)
PCA
Notes on ACI 318-11 (Table 10-2)
For negative moment section (near the interior
support of the end span):
The
negative reinforcement for the end span frame strip near the interior support
is 32 #6 bars located at 1.125 in. along the section from the top of the slab.
ACI
318-14 (Eq. 24.2.3.5b)
ACI
318-14 (Eq. 19.2.3.1)
Figure 19 – Ig
calculations for slab section near support
ACI
318-14 (19.2.2.1.a)
PCA Notes on ACI 318-11 (Table 10-2)
PCA
Notes on ACI 318-11 (Table 10-2)
PCA Notes on
ACI 318-11 (Table 10-2)
PCA
Notes on ACI 318-11 (Table 10-2)
Figure 20
– Cracked Transformed Section (negative moment section)
The
effective moment of inertia procedure described in the Code is considered
sufficiently accurate to estimate deflections. The effective moment of inertia,
Ie, was developed to provide a transition between the upper
and lower bounds of Ig and Icr as a
function of the ratio Mcr/Ma. For conventionally
reinforced (nonprestressed) members, the effective moment of inertia, Ie,
shall be calculated by Eq. (24.2.3.5a) unless obtained by a more comprehensive
analysis.
Ie
shall be permitted to be taken as the
value obtained from Eq. (24.2.3.5a) at midspan for simple and continuous spans,
and at the support for cantilevers. ACI
318-14 (24.2.3.7)
For
continuous one-way slabs and beams. Ie shall be permitted to
be taken as the average of values obtained from Eq. (24.2.3.5a) for the
critical positive and negative moment sections. ACI
318-14 (24.2.3.6)
For the middle span
(span with two ends continuous) with service load level (D+LLfull):
ACI
318-14 (24.2.3.5a)
Where Ie-
is the effective moment of inertia for the critical negative moment
section (near the support).
Where Ie+
is the effective moment of inertia for the critical positive moment section
(midspan).
Since
midspan stiffness (including the effect of cracking) has a dominant effect on
deflections, midspan section is heavily represented in calculation of Ie
and this is considered satisfactory in approximate deflection calculations.
Both the midspan stiffness (Ie+) and averaged span
stiffness (Ie,avg) can be used in the calculation of
immediate (instantaneous) deflection.
The averaged effective
moment of inertia (Ie,avg) is given by:
PCA Notes on ACI 318-11 (9.5.2.4(2))
PCA
Notes on ACI 318-11 (9.5.2.4(1))
However,
these expressions lead to improved results only for continuous prismatic
members. The drop panels in this example result in non-prismatic members and
the following expressions should be used according to ACI 318-89:
ACI
435R-95 (2.14)
For the middle span
(span with two ends continuous) with service load level (D+LLfull):
ACI
435R-95 (2.14)
For the end span (span
with one end continuous) with service load level (D+LLfull):
Where:
Table
6 provides a summary of the required parameters and calculated values needed
for deflections for exterior and interior spans.
|
Table 6 – Averaged Effective Moment of
Inertia Calculations
|
|
For Frame Strip
|
|
Span
|
zone
|
Ig,
in.4
|
Icr,
in.4
|
Ma, kips-ft
|
Mcr,
k-ft
|
Ie, in.4
|
Ie,avg, in.4
|
|
D
|
D +
LLSus
|
D +
Lfull
|
D
|
D +
LLSus
|
D +
Lfull
|
D
|
D +
LLSus
|
D +
Lfull
|
|
Ext
|
Left
|
53445
|
7170
|
-179.72
|
-179.72
|
-252.8
|
401.42
|
53445
|
53445
|
53445
|
37190
|
37190
|
24576
|
|
Midspan
|
30000
|
3797
|
197.23
|
197.23
|
278.14
|
265.17
|
30000
|
30000
|
26503
|
|
Right
|
53445
|
10471
|
-434.41
|
-434.41
|
-611.14
|
401.42
|
44379
|
44379
|
22649
|
|
Int
|
Left
|
53445
|
10471
|
-388.46
|
-388.46
|
-546.48
|
401.42
|
53445
|
53445
|
27503
|
41723
|
41723
|
28752
|
|
Mid
|
30000
|
3317
|
107.56
|
107.56
|
152.03
|
265.17
|
30000
|
30000
|
30000
|
|
Right
|
53445
|
10471
|
-388.46
|
-388.46
|
-546.48
|
401.42
|
53445
|
53445
|
27503
|
Deflections
in two-way slab systems shall be calculated taking into account size and shape
of the panel, conditions of support, and nature of restraints at the panel
edges. For immediate deflections in two-way slab systems, the midpanel deflection is computed as the sum of deflection at
midspan of the column strip or column line in one direction (Δcx
or Δcy) and deflection at midspan of the middle strip in
the orthogonal direction (Δmx or Δmy).
Figure 21 shows the deflection computation for a rectangular panel. The average
Δ for panels that have different properties in the two direction is
calculated as follows:
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 8)
Figure 21
– Deflection Computation for a rectangular Panel
To calculate each
term of the previous equation, the following procedure should be used. Figure 22
shows the procedure of calculating the term Δcx. Same
procedure can be used to find the other terms.
Figure 22 –Δcx calculation
procedure
For end span - service
dead load case:
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 10)
Where:
ACI
318-14 (19.2.2.1.a)
Iframe,averaged =
The averaged effective moment of inertia (Ie,avg)
for the frame strip for service dead load case from Table 6 = 37190 in.4
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 11)
For
this example and like in the spSlab program, the effective moment of inertia at
midspan will be used.
LDFc is the load distribution factor for the column strip.
The load distribution factor for the column strip can be found from the
following equation:
And the load
distribution factor for the middle strip can be found from the following
equation:
For
the end span, LDF for exterior negative region (LDFL¯), interior
negative region (LDFR¯), and positive region (LDFL＋) are 1.00, 0.75, and 0.60, respectively (From Table 2 of this document).
Thus, the load distribution factor for the column strip for the end span is
given by:
Ic,g = The gross moment of inertia (Ig) for
the column strip for service dead load = 15000 in.4
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 12)
Where:
Kec = effective column stiffness = 
in.-lb (calculated previously).
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 14)
Where:
Where
Where:
PCA
Notes on ACI 318-11 (9.5.3.4 Eq. 9)
Following
the same procedure, Δmx can be calculated for the middle
strip. This procedure is repeated for the equivalent frame in the orthogonal
direction to obtain Δcy, and Δmy
for the end and middle spans for the other load levels (D+LLsus
and D+LLfull).
Since
in this example the panel is squared, Δcx = Δcy=
0.219 in. and Δmx = Δmy=
0.125 in.
The
average Δ for the corner panel is calculated as follows:
