Lecture 25 - Design of Two-Way Floor Slab System
August 6, 2003
CVEN 444
Lecture Goals
Example of DDM
Panel Design
Example 1 Design an interior panel of the two-way
slab for the floor system.The floor
consists of six panels at each direction,
with a panel size 24 ft x 20 ft. All
panels are supported by 20 in square
columns. The slabs are supported by
beams along the column line with cross
sections. The service live load is to be
taken as 80 psf and the service dead
load consists of 24 psf of floor
finishing in addition to the self-weight.
Use fc = 4 ksi and fy = 60 ksi
Example 1 –Previous Example
The cross-sections are:
h = 7 in.
Example 1 –Previous Example
The resulting cross section:
Example 1 –Previous Example
The thickness was calculated in an earlier example.
Generally, thickness of the slab is calculated at the
for the external corner slab. So use h = 7 in.
Example 1- Loading
The weight of the slab is given as.
2 3 2
u 2 2
2 2
lb 1 ft lb lb24 7.0 in. 150 111.5
ft 12 in. ft ft
lb lb1.2 1.6 1.2 111.5 1.6 80
ft ft
lb kips262 0.262
ft ft
DL
w DL LL
Example 1 – calculation d
Compute the average depth, d for the slab. Use an
average depth for the shear calculation with a #4 bar
(d = 0.5 in)
bcover / 2
7.0 in. 0.75 in. 0.5 in./ 2 6.0 in.
d h d
d
Example 1 – One-way shear
The shear stresses in the slab are not
critical. The critical section is at a
distance d from the face of the beam.
Use 1 ft section.
u u
2
beam width12 ft. 1 ft.
2
1 ft16 in.
1 ft12 in.0.262 k/ft 12 ft. 6 in. 1 ft.
2 12 in.
2.84 k
V w d
Example 1 – One-way shear
The one way shear on the face of the beam.
c c2
1 kip0.75 2 3000 12 in. 6 in.
1000 lb
5.92 k 2.84 k OK.
V f bd
Example 1 – Strip size
Determine the strip sizes for the column and middle
strip. Use the smaller of l1 or l2 so l2 = 20 ft
in 168ft 14 ft 52 ft 24
in 120ft 10 ft 52 ft 20
s
l
b
b
ft 5
4
ft 20
4
2 l
l
Therefore the column strip b = 2( 5 ft) = 10 ft (120 in)
The middle strips are
Example 1 – Strip Size
Calculate the strip sizes
Example 1 – Static Moment Computation
Moment Mo for the two directions.
long direction
short direction
n
222
u 2 n
ol
20 in. 1 ft.24 ft. 2 22.333 ft.
2 12 in.
0.262 k/ft 20 ft. 22.333 ft.M
8 8
326.7 k-ft
l
w l l
n
222
u 2 n
os
20 in. 1 ft.20 ft. 2 18.333 ft.
2 12 in.
0.262 k/ft 24 ft. 18.333 ft.M
8 8
264.2 k-ft
l
w l l
Example 1 – Internal Panel Moment distribution
Interior panel
0.35Mo
0.65Mo
Example 1 – Moments (long)
The factored components
of the moment for the
beam (long).
Negative - Moment
Positive + Moment
0.65 326.7 k-ft 212.4 k-ft
0.35 326.7 k-ft 114.4 k-ft
Example 1- - Moment (long)
Coefficients
The moments of inertia about beam, Ib = 22,453 in4 and
Is = 6860 in4 (long direction) are need to determine the
distribution of the moments between the column and
middle strip.
71.28333.0*27.3
27.3
in 6860
in 22453
E
E
8333.0
ft 24
ft 20
1
2l
4
4
ss
bbl
1
2
l
l
I
I
l
l
Example 1- Moment (long) Factors (negative)
8.0
5.08333.0
0.15.0
75.09.09.0
Factor
Need to interpolate to
determine how the
negative moment is
distributed.
Example 1 - Moment (long) Factors (positive)
8.0
5.08333.0
0.15.0
75.09.09.0
Factor
Need to interpolate
to determine how
the positive moment
is distributed.
Example 1 - Moment (long) column/middle strips
Components on the beam (long).
Negative – Moment
Positive + Moment
Column Strip
0.80 212.4 k-ft 169.9 k-ft
0.80 114.4 k-ft 91.5 k-ft
Negative – Moment
Positive + Moment
Middle Strip
0.20 212.4 k-ft 42.5 k-ft
0.20 114.4 k-ft 22.9 k-ft
Example 1 - Moment (long)-beam/slab distribution (negative)
When 1 (l2/l1) > 1.0, ACI Code Section 13.6.5 indicates
that 85 % of the moment in the column strip is assigned
to the beam and balance of 15 % is assigned to the slab
in the column strip.
Beam Moment
Slab Moment
Column Strip - Negative Moment (169.9 k-ft)
0.85 169.9 k-ft 144.4 k-ft
0.15 169.9 k-ft 25.5 k-ft
Example 1 - Moment (long)-beam/slab distribution (positive) When 1 (l2/l1) > 1.0, ACI Code Section 13.6.5
indicates that 85 % of the moment in the column strip
is assigned to the beam and balance of 15 % is
assigned to the slab in the column strip.
Beam Moment
Slab Moment
Column Strip - Positive Moment (91.5 k-ft)
0.85 91.5 k-ft 77.8 k-ft
0.15 91.5 k-ft 13.7 k-ft
Example 1- Moment (short)
The factored components
of the moment for the
beam (short).
Negative – Moment
Positive + Moment
0.65 264.2 k-ft 171.7 k-ft
0.35 264.2 k-ft 92.5 k-ft
Example 1 - Moment (short) coefficients
The moments of inertia about beam, Ib = 22,453 in4 and
Is = 8232 in4 (short direction) are need to determine the
distribution of the moments between the column and
middle strip.
1
2
4
b b1 4
s s
11
2
24 ft1.22222
20 ft
22453 in2.73
8232 in
2.73* 1.2222 3.333
l
l
E I
E I
l
l
Example 1 - Moment (short) Factors (negative)
6833.0
0.12222.1
0.20.1
45.075.075.0
Factor
Need to interpolate
to determine how
the negative
moment is
distributed.
Example 1 - Moment (short) Factors (positive)
Need to interpolate
to determine how
the positive
moment is
distributed.
6833.0
0.12222.1
0.20.1
45.075.075.0
Factor
Example 1- Moment (short) column/middle strip
Components on the beam (short).
Negative – Moment
Positive + Moment
Column Strip
0.683 171.7 k-ft 117.3 k-ft
0.683 92.5 k-ft 63.2 k-ft
Negative – Moment
Positive + Moment
Middle Strip
0.317 171.7 k-ft 54.4 k-ft
0.317 92.5 k-ft 29.3 k-ft
Example 1 - Moment (short) beam/slab distribution (negative)
When 1 (l2/l1) > 1.0, ACI Code Section 13.6.5 indicates
that 85 % of the moment in the column strip is assigned
to the beam and balance of 15 % is assigned to the slab
in the column strip.
Beam Moment
Slab Moment
Column Strip - Negative Moment (117.3 k-ft)
0.85 117.3 k-ft 99.7 k-ft
0.15 117.3 k-ft 17.6 k-ft
Example 1 - Moment (short) beam/slab distribution (positive)
When 1 (l2/l1) > 1.0, ACI Code Section 13.6.5
indicates that 85 % of the moment in the column strip
is assigned to the beam and balance of 15 % is
assigned to the slab in the column strip.
Beam Moment
Slab Moment
Column Strip - Positive Moment (63.2 k-ft)
0.85 63.2 k-ft 53.7 k-ft
0.15 63.2 k-ft 9.5 k-ft
Example 1 - Summary
Example 1- Reinforcement calculation
Use same procedure to do the reinforcement on the
concrete. Calculate the bars from the earlier
version of the problem.
Example 1 - Reinforcement calculation
Computing the reinforcement uses:
y
c
c
yc
u2
c
u2
cu
2
uu
2
R*1.747.170.1
0
R*1.770.159.01 R
bd
R
f
wf
f
fw
fw
f
wwwfw
M
Example 1 - Reinforcement calculation for long –middle strip (negative)
Compute the reinforcement need for the negative moment
in long direction. Middle strip width b =120 in. (10 ft),
d =6 in. and Mu = 42.5 k-ft
uu 22
2
12 in.42.5 k-ft
1 ft0.118 ksi
bd 120 in. 6 in.
1.7 0.118 ksi1.70 0
0.9 3 ksi
MR
w w
Compute the reinforcement need for the negative
moment in long direction. Middle strip width b =120 in.
(10 ft) d =6 in. and Mu = 42.5 k-ft
2
c
y
1.70 1.7 4 0.74330.0449
2
0.0449 3 ksi0.00225
60 ksi
w
wf
f
Example 1 - Reinforcement calculation for long –middle strip (negative)
The area of the steel reinforcement for a strip width
b =120 in. (10 ft), d = 6 in., and h = 7 in.
2
s
2
s min
0.00225 120 in. 6 in. 1.62 in
0.0018 0.0018 120 in. 7 in. 1.52 in
A bd
A bh
Example 1 - Reinforcement calculation for long –middle strip (negative)
The area of the steel reinforcement for a strip width
b =120 in. (10 ft), d = 6 in., and As = 1.62 in2. Use a
#4 bar (Ab =0.20 in2 )
Maximum spacing is 2(h) or
18 in.
So 13.33 in < 14 in. OK! Use 10 #4
2
s
2
b
1.62 in# bars 8.08 Use 9 bars
0.2 in
120 in.13.33 in.
9
A
A
s
Example 1 - Reinforcement calculation for long –middle strip (negative)
Example 1 – Long Results
The long
direction using
# 4 bars
Negative Positive Negative Positive
Moment (k-ft) 25.5 13.7 42.5 22.9
b (in) 120 120 120 120
d (in) 6 6 6 6
h (in) 7 7 7 7
fy (ksi) 60 60 60 60
fc (ksi) 3 3 3 3
Ru (ksi) 0.07083 0.03806 0.11806 0.06361
w 0.02665 0.01421 0.04491 0.02390
0.00133 0.00071 0.00225 0.00119
As (in2) 0.96 0.51 1.62 0.86
As(min) (in2) 1.51 1.51 1.51 1.51
# bars req 7.56 7.56 8.08 7.56
spacing (in) 15.00 15.00 13.33 15.00
Use
# bars (#4) 10 10 10 10
spacing (in) 12 12 12 12
Column Strip Middle Strip
Example 1 – Long summary
The long direction
using # 4 bars
Example 1 – Short Results
The short
direction
using # 4
bars
Negative Positive Negative Positive
Moment (k-ft) 17.6 9.5 54.4 29.3
b (in) 120 120 168 168
d (in) 6 6 6 6
h (in) 7 7 7 7
fy (ksi) 60 60 60 60
fc (ksi) 3 3 3 3
Ru (ksi) 0.04889 0.02639 0.10794 0.05813
w 0.01830 0.00983 0.04096 0.02181
0.00092 0.00049 0.00205 0.00109
As (in2) 0.66 0.35 2.06 1.10
As(min) (in2) 1.51 1.51 2.12 2.12
# bars req 7.56 7.56 10.58 10.58
spacing (in) 15.00 15.00 15.27 15.27
Use
# bars (#4) 10 10 14 14
spacing (in) 12 12 12 12
Column Strip Middle Strip
Example 1 – Short Summary
The short direction
using # 4 bars
Example –Two-way Slab (Panels)
Using the direct design method,
design the typical exterior flat-
slab panel with drop down
panels only. All panels are
supported on 20 in. square
columns, 12 ft long. The slab
carries a uniform service live
load of 80 psf and service dead
load that consists of 24 psf of
finished in addition to the slab
self-weight. Use fc = 4 ksi and
fy = 60 ksi
Example –Two-way Slab (Panels)
The thickness of the slab is
found using
in 8.0in 44.7
36
in 20
ft
in 12*24
36
panels Noin 12.8
33
in 20
ft
in 12*24
33
n
n
l
l
Example –Two-way Slab (Panels)
From the ACI Code limitation:
For panels with discontinuous edges, end beams
with a minimum equal to 0.8 must be used;
otherwise the minimum slab thickness calculated
by the equations must be increased by at least 10%.
When drop panels are used without beams, the
minimum slab thickness may be reduced by 10 %.
The drop panels should extend in each direction
from the centerline of support a distance not less
than one-sixth of the span length in that direction
between center to center of supports and also
project below the slab at least h/4.
1.
2.
Example –Two-way Slab (Panels) From the ACI Code limitation:
Regardless of the values obtained for the equations,
the thickness of two-way slabs shall not be less
than the following:
3.
For slabs without beams or drop panels, 5 in.
for slabs without beams but with drop
panels, 4 in.
for slabs with beams on all four sides with
m > 2.0, 3.5 in. and for m < 2.0, 5 in. (ACI
Code 9.5.3)
1.
2.
3.
Example –Two-way Slab (Panels)
Therefore, the panel thickness is
The panel half width are at least L/6 in length.
in. 10
4
in. 8 in. 8
4
h
h
ft 3.5 ft 33.3
6
ft 20
6
ft 4
6
ft 24
6
L
L
Example –Two-way Slab (Panels)
Therefore, the drop down panel thickness is 10 in.
and has 7 ft x 8 ft.
Example –Two-way Slab (Panels)
The load on the slab is given as:
The load on the panel is
2 3 2
2 2 2
u
1 ftSlab load 24 lb/ft 8 in. 150 lb/ft 124 lb/ft
12 in.
1.2 124 lb/ft 1.6 80 lb/ft 276.8 lb/ftw
2 3 2
2 2 2
u
1 ftPanel load 24 lb/ft 10 in. 150 lb/ft 149 lb/ft
12 in.
1.2 149 lb/ft 1.6 80 lb/ft 306.8 lb/ftw
Example –Two-way Slab (Panels)
The drop panel length is L/3 in each direction, then the
average wu is
2 2
u
2
2 1276.8 lb/ft 306.8 lb/ft
3 3
286.8 lb/ft
w
Example –Two-way Slab (Panels)
The punch out shear at center column is
in. 8.75
in. 5.0in. 75.0 in. 10
d
in. 151
in. 8.75 in. 204 o
b
Example –Two-way Slab (Panels)
The punch out shear at center column is
2
2
u
c c o
u
1 ft0.287 k/ft 24 ft 20 ft 28.75 in.
12 in.
136.1 k
4
0.75 4 4000 115 in. 8.75 in.
190.9 k OK.
V
V f b d
V
Example –Two-way Slab (Panels)
The punch out shear at panel is
in. 6.75
in. 5.0in. 75.0 in. 8
d
in. 387
in. 6.75
ft. 1
in. 12ft. 72
in. 6.75
ft. 1
in. 12ft. 82 o
b
Example –Two-way Slab (Panels)
The punch out shear at panel is
2
u
c c o
u
24 ft 20 ft
0.287 k/ft 1 ft 1 ft102.75 in. 90.75 in.
12 in. 12 in.
119.2 k
4
0.75 4 4000 387 in. 6.75 in.
495.6 k OK.
V
V f b d
V
One way shear is not critical.
Example –Two-way Slab (Panels)
Moment Mo for the two directions are:
Long
direction
22
o
0.287 k/ft 20 ft 22.33 ft357.9 k-ft
8M
Short
direction
22
o
0.287 k/ft 24 ft 18.33 ft289.4 k-ft
8M
Example –Two-way Slab (Panels)
The column strip will be 10 ft. (20 ft /4 = 5ft),
therefore the middle strips for long section is 10 ft and
the middle strip for the short section will be 14 ft.
The average d for
the panel section in. 8.5
in. 5.1 in. 10
d
The average d for
the slab section in. 6.5
in. 5.1 in. 8
d
Example –Two-way Slab (Panels)
The factored
components of the
moment for the beam
(long) is similar to an
interior beam.
Negative – Moment
Positive + Moment
0.65 357.9 k-ft 232.6 k-ft
0.35 357.9 k-ft 125.3 k-ft
Example –Two-way Slab (Panels)
Components on the beam (long) interior.
Negative – Moment
Positive + Moment
Column Strip
0.75 232.6 k-ft 174.5 k-ft
0.60 125.3 k-ft 75.2 k-ft
Example –Two-way Slab (Panels)
Components on the beam (long) interior.
Negative – Moment
Positive + Moment
Middle Strip
0.25 232.6 k-ft 58.2 k-ft
0.40 125.3 k-ft 50.2 k-ft
Example –Two-way Slab (Panels)
Computing the reinforcement uses:
y
c
c
yc
u2
c
u2
cu
2
uu
2
R*1.747.170.1
0
R*1.770.159.01 R
bd
R
f
wf
f
fw
fw
f
wwwfw
M
Example –Two-way Slab (Panels)
Compute the reinforcement need for the internal moment
in long direction. Strip width b =120 in. (10 ft) d =8.5 in.
and Mu = 174.5 k-ft
uu 22
2
12 in.174.5 k-ft
1 ft0.242 ksi
bd 120 in. 8.5 in.
1.7*0.242 ksi1.70 0
0.9 4 ksi
MR
w w
Example –Two-way Slab (Panels)
Compute the reinforcement need for the internal moment
in long direction. Strip width b =120 in. (10 ft) d =8.5 in.
and Mu = 174.5 k-ft
2
c
y
1.70 1.7 4 0.11410.0700
2
0.0700 4 ksi0.00466
60 ksi
w
wf
f
Example –Two-way Slab (Panels)
The area of the steel reinforcement for a strip width
b =120 in. (10 ft), d = 8.5 in., and h = 10 in.
2
s
2
s min
0.00466 120 in. 8.5 in. 4.76 in
0.0018 0.0018 120 in. 10 in. 2.16 in
A bd
A bh
Example –Two-way Slab (Panels)
The area of the steel reinforcement for a strip width
b =120 in. (10 ft), d = 8.5 in., and As = 4.76 in2. Use
a #5 bar (Ab = 0.31 in2 )
Maximum spacing is 2(h)
or 18 in.
So 7.5 in. < 18 in. OK
2
s
2
b
4.76 in# bars 15.3 Use 16 bars
0.31 in
120 in.s 7.5 in.
16
A
A
Example –Two-way Slab (Panels)
The long
direction
Negative Positive Negative Positive
Moment (k-ft) 174.5 75.2 58.2 50.2
b (in) 120 120 120 120
d (in) 8.5 6.5 6.5 6.5
h (in) 10 8 8 8
fy (ksi) 60 60 60 60
fc (ksi) 4 4 4 4
Ru (ksi) 0.24152 0.17799 0.13775 0.11882
w 0.06997 0.05097 0.03917 0.03367
0.00466 0.00340 0.00261 0.00224
As (in2) 4.76 2.65 2.04 1.75
As(min) (in2) 2.16 1.73 1.73 1.73
# bars req 15.35 13.25 10.18 8.75
spacing (in) 7.50 8.57 10.91 13.33
Use
# bars 16 #5 15 #4 10 #4 10 #4
spacing (in) 7.5 8 12 12
Column Strip Middle Strip
Example –Two-way Slab (Panels)
The short
direction
Negative Positive Negative Positive
Moment (k-ft) 141.1 60.8 47.1 40.5
b (in) 120 120 168 168
d (in) 8.5 6.5 6.5 6.5
h (in) 10 8 8 8
fy (ksi) 60 60 60 60
fc (ksi) 3 3 3 3
Ru (ksi) 0.19529 0.14391 0.07963 0.06847
w 0.07570 0.05508 0.03002 0.02575
0.00379 0.00275 0.00150 0.00129
As (in2) 3.86 2.15 1.64 1.41
As(min) (in2) 2.16 1.73 2.42 2.42
# bars req 12.45 10.74 12.10 12.10
spacing (in) 9.23 10.91 12.92 12.92
Use
# bars 16 #5 12 #4 14 #4 14 #4
spacing (in) 7.5 10 12 12
Column Strip Middle Strip
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