TwoWay Joist Concrete Slab Floor (Waffle Slab) System Analysis and Design
TwoWay Joist Concrete Slab Floor (Waffle Slab) System Analysis and Design
Design the concrete floor slab system shown below for an intermediate floor with partition weight of 50 psf, and unfactored live load of 100 psf. The lateral loads are independently resisted by shear walls. A flat plate system will be considered first to illustrate the impact longer spans and heavier applied loads. A waffle slab system will be investigated since it is economical for longer spans with heavy loads. The dome voids reduce the dead load and electrical fixtures can be fixed in the voids. Waffle system provides an attractive ceiling that can be left exposed when possible producing savings in architectural finishes. The Equivalent Frame Method (EFM) shown in ACI 318 is used in this example. The hand solution from EFM is also used for a detailed comparison with the model results of spSlab engineering software program from StructurePoint.
Figure 1  TwoWay Flat Concrete Floor System
Contents
2. Flexural Analysis and Design
2.1. Equivalent Frame Method (EFM)
2.1.1. Limitations for use of equivalent frame method
2.1.2. Frame members of equivalent frame
2.1.3. Equivalent frame analysis
2.1.4. Factored moments used for Design
2.1.5. Factored moments in slabbeam strip
2.1.6. Flexural reinforcement requirements
2.1.7. Factored moments in columns
3. Design of Columns by spColumn
3.1. Determination of factored loads
3.2. Moment Interaction Diagram
4.1. OneWay (Beam action) Shear Strength
4.1.1. At distance d from the supporting column
4.1.2. At the face of the drop panel
4.2. TwoWay (Punching) Shear Strength
4.2.1. Around the columns faces
5. Serviceability Requirements (Deflection Check)
5.1. Immediate (Instantaneous) Deflections
5.2. TimeDependent (LongTerm) Deflections (Δ_{lt})
6. spSlab Software Program Model Solution
7. Summary and Comparison of Design Results
Code
Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14)
Reference
Concrete Floor Systems (Guide to Estimating and Economizing), Second Edition, 2002 David A. Fanella, Portland Cement Association.
PCA Notes on ACI 31811 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 435R95), American Concrete Institute
Reinforced Concrete Design .. .Hassoun, McGraw Hill
Design Data
Story Height = 13 ft (provided by architectural drawings)
Superimposed Dead Load, SDL = 50 psf for Frame walls, hollow concrete masonry unit wythe, 12 in. thick, 125 pcf unit density, with no grout
ASCE/SEI 710 (Table C31)
Live Load, LL = 100 psf for Recreational uses – Gymnasiums ASCE/SEI 710 (Table 41)
f_{c}’ = 5000 psi (for slab)
f_{c}’ = 6000 psi (for columns)
f_{y} = 60,000 psi
Solution
Preliminary Flat Plate (without Joists)
a. Slab minimum thickness – Deflection ACI 31814 (8.3.1.1)
In lieu of detailed calculation for deflections, ACI 318 minimum slab thickness for twoway construction without interior beams is given in Table 8.3.1.1.
For flat plate slab system, the minimum slab thickness per ACI 31814 are:
ACI 31814 (Table 8.3.1.1)
But not less than 5 in. ACI 31814 (8.3.1.1(a))
ACI 31814 (Table 8.3.1.1)
But not less than 5 in. ACI 31814 (8.3.1.1(a))
Where l_{n} = length of clear span in the long direction = 33 x 12 – 20 = 376 in.
Use 13 in. slab for all panels (selfweight = 150 pcf x 13 in. /12 = 162.5 psf)
b. Slab shear strength – one way shear
Evaluate the average effective depth (Figure 2):
Where:
c_{clear} = 3/4 in. for # 6 steel bar ACI 31814 (Table 20.6.1.3.1)
d_{b} = 0.75 in. for # 6 steel bar
Figure 2  TwoWay Flat Concrete Floor System
ACI 31814 (5.3.1)
Check the adequacy of slab thickness for beam action (oneway shear) ACI 31814 (22.5)
at an interior column:
Consider a 12in. wide strip. The critical section for oneway shear is located at a distance d, from the face of support (see Figure 3):
ACI 31814 (Eq. 22.5.5.1)
Slab thickness of 13 in. is adequate for oneway shear.
c. Slab shear strength – twoway shear
Check the adequacy of slab thickness for punching shear (twoway shear) at an interior column (Figure 4):
ACI 31814 (Table 22.6.5.2(a))
Slab thickness of 13 in. is not adequate for twoway shear. This is expected as the selfweight an applied loads are very challenging for a flat plate system.
Figure 3 – Critical Section for OneWay Shear Figure 4 – Critical Section for TwoWay Shear
In this case, four options can be considered: 1) increase the slab thickness further, 2) use headed shear reinforcement in the slab, 3) apply drop panels at columns, or 4) use twoway joist slab system. In this example, the latter option will be used to achieve better understanding for the design of twoway joist slab often called twoway ribbed slab or waffle slab.
Check the applicable joist dimensional limitations as follows:
1) Width of ribs shall be at least 4 in. at any location along the depth. ACI 31814 (9.8.1.2)
Use ribs with 6 in. width.
2) Overall depth of ribs shall not exceed 3.5 times the minimum width. ACI 31814 (9.8.1.3)
3.5 x 6 in. = 21 in. à Use ribs with 14 in. depth.
3) Clear spacing between ribs shall not exceed 30 in. ACI 31814 (9.8.1.4)
Use 30 in. clear spacing.
4) Slab thickness (with removable forms) shall be at least the greater of: ACI 31814 (8.8.3.1)
a) 1/12 clear distance between ribs = 1/12 x 30 = 2.5 in.
b) 2 in.
Use a slab thickness of 3 in. > 2.5 in.
Figure 5 – Joists Dimensions
In waffle slabs a drop panel is automatically invoked to guarantee adequate twoway (punching) shear resistance at column supports. This is evident from the flat plate check conducted using 13 in. indicating insufficient punching shear capacity above. Check the drop panel dimensional limitations as follows:
1) The drop panel shall project below the slab at least onefourth of the adjacent slab thickness.
ACI 31814 (8.2.4(a))
Since the slab thickness (h_{MI} – calculated in page 7 of this document) is 12 in., the thickness of the drop panel should be at least:
Drop panel depth are also controlled by the rib depth (both at the same level).For nominal lumber size (2x), h_{dp} = h_{rib} = 14 in. > h_{dp, min}_{ }= 3 in.
The total thickness including the actual slab and the drop panel thickness (h) = h_{s }+ h_{dp} = 3 + 14 = 17 in.
2) The drop panel shall extend in each direction from the centerline of support a distance not less than onesixth the span length measured from centertocenter of supports in that direction.
ACI 31814 (8.2.4(b))
Based on the previous discussion, Figure 6 shows the dimensions of the selected twoway joist system.
Figure 6 – TwoWay Joist (Waffle) Slab
Preliminary TwoWay Joist Slab (Waffle Slab)
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 31814. The critical sections shall be located with respect to:
1) Edges or corners of columns. ACI 31814 (22.6.4.1(a))
2) Changes in slab thickness, such as edges of drop panels. ACI 31814 (22.6.4.1(b))
a. Slab minimum thickness – Deflection ACI 31814 (8.3.1.1)
In lieu of detailed calculation for deflections, ACI 318 Code gives minimum slab thickness for twoway construction without interior beams in Table 8.3.1.1.
For this slab system, the minimum slab thicknesses per ACI 31814 are:
ACI 31814 (Table 8.3.1.1)
But not less than 4 in. ACI 31814 (8.3.1.1(b))
ACI 31814 (Table 8.3.1.1)
But not less than 4 in. ACI 31814 (8.3.1.1(b))
Where l_{n} = length of clear span in the long direction = 33 x 12 – 20 = 376 in.
For the purposes of analysis and design, the ribbed slab will be replaced with a solid slab of equivalent moment of inertia, weight, punching shear capacity, and oneway shear capacity.
The equivalent thickness based on moment of inertia is used to find slab stiffness considering the ribs in the direction of the analysis only. The ribs spanning in the transverse direction are not considered in the stiffness computations. This thickness, h_{MI}, is given by:
spSlab Software Manual (Eq. 211)
Where:
I_{rib} = Moment of inertia of one joist section between centerlines of ribs (see Figure 7a).
b_{rib} = The centertocenter distance of two ribs (clear rib spacing plus rib width) (see Figure 7a).
Since h_{MI} = 12 in. > h_{min} = 11.4 in., the deflection calculation can be neglected. However, the deflection calculation will be included in this example for comparison with the spSlab software results.
The drop panel depth for twoway joist (waffle) slab is set equal to the rib depth. The equivalent drop depth based on moment of inertia, d_{MI}, is given by:
spSlab Software Manual (Eq. 212)
Where h_{rib} = 3 + 14 – 12 = 5 in.
Figure 7a – Equivalent Thickness Based on Moment of Inertia
Find system selfweight using the equivalent thickness based on the weight of individual components (see the following Figure). This thickness, h_{w}, is given by:
spSlab Software Manual (Eq. 210)
Where:
V_{mod} = The Volume of one joist module (the transverse joists are included – 11 joists in the frame strip).
A_{mod} = The plan area of one joist module = 33 x 36/12 = 99 ft^{2}
Selfweight for slab section without drop panel = 150 pcf x 8 in. /12 = 100.057 psf
Selfweight for drop panel = 150 pcf x (14 + 3 – 8) in. /12 = 112.44 psf
Figure 7b – Equivalent Thickness Based on the Weight of Individual Components
b. Slab shear strength – oneway shear
For critical section at distance d from the edge of the column (slab section with drop panel):
Evaluate the average effective depth:
Where:
c_{clear} = 3/4 in. for # 6 steel bar ACI 31814 (Table 20.6.1.3.1)
d_{b} = 0.75 in. for # 6 steel bar
h_{s} = 17 in. = The drop depth (d_{MI})
ACI 31814 (5.3.1)
Check the adequacy of slab thickness for beam action (oneway shear) from the edge of the interior column
ACI 31814 (22.5)
Consider a 12in. wide strip. The critical section for oneway shear is located at a distance d, from the edge of the column (see Figure 8)
ACI 31814 (Eq. 22.5.5.1)
Slab thickness is adequate for oneway 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:
c_{clear} = 3/4 in. for # 6 steel bar ACI 31814 (Table 20.6.1.3.1)
d_{b} = 0.75 in. for # 6 steel bar
ACI 31814 (5.3.1)
Check the adequacy of slab thickness for beam action (oneway shear) from the edge of the interior drop panel ACI 31814 (22.5)
Consider a 12in. wide strip. The critical section for oneway shear is located at the face of the solid head (see Figure 8)
ACI 31814 (Eq. 22.5.5.1)
Slab thickness of 12 in. is adequate for oneway shear for the second critical section (at the edge of the drop panel).
Figure 8 – Critical Sections for OneWay Shear
c. Slab shear strength – twoway 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 (twoway shear) at an interior column (Figure 9):
Tributary area of twoway shear for the slab without the drop panel is:
Tributary area of twoway shear for the slab with the drop panel is:
ACI 31814 (Table 22.6.5.2(a))
Slab thickness is adequate for twoway 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 (twoway shear) at an interior drop panel (Figure 9):
ACI 31814 (Table 22.6.5.2(a))
Slab thickness of 12 in. is adequate for twoway shear for the second critical section (from the edge of the drop panel).
Figure 9 – Critical Sections for TwoWay 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 selfweight of the slab is
Tributary area for interior column for selfweight of additional slab thickness due to the presence of the drop panel is
Assuming four story building
Assume 20 in. square column with 12 – No. 11 vertical bars with design axial strength, φP_{n,max} of
ACI 31814 (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 twosystems from oneway systems is given by ACI 31814 (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 twoway 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 slabbeam 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 31814 (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 31814 (8.11.2.1)
Panels shall be rectangular, with a ratio of longer to shorter panel dimensions, measured centertocenter of supports, not to exceed 2. ACI 31814 (8.10.2.3)
Determine moment distribution factors and fixedend 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 fixedend moment factors FEM for the slabbeams and column members are determined using the design aids tables at Appendix 20A of PCA Notes on ACI 31811. These calculations are shown below.
a. Flexural stiffness of slabbeams at both ends, K_{sb}.
PCA Notes on ACI 31811 (Table A1)
PCA Notes on ACI 31811 (Table A1)
ACI 31814 (19.2.2.1.a)
Carryover factor COF = 0.54 PCA Notes on ACI 31811 (Table A1)
PCA Notes on ACI 31811 (Table A1)
Uniform load fixed end moment coefficient, m_{NF1} = 0.0911
Fixed end moment coefficient for (ba) = 0.2 when a = 0, m_{NF2} = 0.0171
Fixed end moment coefficient for (ba) = 0.2 when a = 0.8, m_{NF3} = 0.0016
b. Flexural stiffness of column members at both ends, K_{c}.
Referring to Table A7, Appendix 20A,
For the Bottom Column:
PCA Notes on ACI 31811 (Table A7)
ACI 31814 (19.2.2.1.a)
l_{c} = 13 ft = 156 in.
For the Top Column:
PCA Notes on ACI 31811 (Table A7)
c. Torsional stiffness of torsional members, .
ACI 31814 (R.8.11.5)
ACI 31814 (Eq. 8.10.5.2b)
d. Equivalent column stiffness K_{ec}.
Where∑ K_{t} is for two torsional members one on each side of the column, and ∑ K_{c} is for the upper and lower columns at the slabbeam joint of an intermediate floor.
Figure 10 – Torsional Member Figure 11 – Column and Edge of Slab
e. Slabbeam joint distribution factors, DF.
At exterior joint,
At interior joint,
COF for slabbeam =0.576
Figure 12 – Slab and Column Stiffness
Determine negative and positive moments for the slabbeams using the moment distribution method. Since the unfactored live load does not exceed threequarters of the unfactored dead load, design moments are assumed to occur at all critical sections with full factored live on all spans. ACI 31814 (6.4.3.2)
a. Factored load and FixedEnd Moments (FEM’s).
For slab:
For drop panels:
PCA Notes on ACI 31811 (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 M_{o} 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 12:
Table 1  Moment Distribution for Equivalent Frame 



Joint 
1 
2 
3 
4 

Member 
12 
21 
23 
32 
34 
43 
DF 
0.640 
0.390 
0.390 
0.390 
0.390 
0.640 
COF 
0.576 
0.576 
0.576 
0.576 
0.576 
0.576 
FEM 
1146.51 
1146.5 
1146.51 
1146.5 
1146.51 
1146.5 
Dist 
733.6 
0.0 
0.0 
0.0 
0.0 
733.6 
CO 
0.0 
422.5 
0.0 
0.0 
422.5 
0.0 
Dist 
0.0 
164.8 
164.8 
164.8 
164.8 
0.0 
CO 
94.9 
0.0 
94.9 
94.9 
0.0 
94.9 
Dist 
60.7 
37.0 
37.0 
37.0 
37.0 
60.7 
CO 
21.3 
35.0 
21.3 
21.3 
35.0 
21.3 
Dist 
13.7 
22.0 
22.0 
22.0 
22.0 
13.7 
CO 
12.7 
7.9 
12.7 
12.7 
7.9 
12.7 
Dist 
8.1 
8.0 
8.0 
8.0 
8.0 
8.1 
CO 
4.6 
4.7 
4.6 
4.6 
4.7 
4.6 
Dist 
3.0 
3.6 
3.6 
3.6 
3.6 
3.0 
CO 
2.1 
1.7 
2.1 
2.1 
1.7 
2.1 
Dist 
1.3 
1.5 
1.5 
1.5 
1.5 
1.3 
CO 
0.9 
0.8 
0.9 
0.9 
0.8 
0.9 
Dist 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
CO 
0.4 
0.3 
0.4 
0.4 
0.3 
0.4 
Dist 
0.2 
0.3 
0.3 
0.3 
0.3 
0.2 
CO 
0.2 
0.1 
0.2 
0.2 
0.1 
0.2 
Dist 
0.1 
0.1 
0.1 
0.1 
0.1 
0.1 
CO 
0.1 
0.1 
0.1 
0.1 
0.1 
0.1 
Dist 
0.0 
0.1 
0.1 
0.1 
0.1 
0.0 
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, kft 
462.3 
1381.5 
1247.5 
1247.5 
1381.5 
462.3 
Midspan M, ftkips 
630.0 
304.4 
630.0 
Positive and negative factored moments for the slab system in the direction of analysis are plotted in Figure 13. 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 l_{1} from the centers of supports. ACI 31814 (8.11.6.1)
Figure 13  Positive and Negative Design Moments for SlabBeam (All Spans Loaded with Full Factored Live Load)
a. Check whether the moments calculated above can take advantage of the reduction permitted by ACI 31814 (8.11.6.5):
If the slab system analyzed using EFM within the limitations of ACI 31814 (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 M_{o} given by Equation 8.10.3.2 in the ACI 31814.
Check Applicability of Direct Design Method:
1. There is a minimum of three continuous spans in each direction. ACI 31814 (8.10.2.1)
2. Successive span lengths are equal. ACI 31814 (8.10.2.2)
3. LongtoShort ratio is 33/33 = 1.0 < 2.0. ACI 31814 (8.10.2.3)
4. Columns are not offset. ACI 31814 (8.10.2.4)
5. Loads are gravity and uniformly distributed with service livetodead ratio of 0.67 < 2.0
(Note: The selfweight of the drop panels is not uniformly distributed entirely along the span. However, the variation in load magnitude is small).
ACI 31814 (8.10.2.5 and 6)
6. Check relative stiffness for slab panel. ACI 31814 (8.10.2.7)
Slab system is without beams and this requirement is not applicable.
ACI 31814 (Eq. 8.10.3.2)
To illustrate proper procedure, the interior span factored moments may be reduced as follows:
Permissible reduction = 1376.9/1552 = 0.887
Adjusted negative design moment = 1247.5 × 0.887 = 1106.5 ftkips
Adjusted positive design moment = 304 × 0.887 = 269.6 ftkips
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 13), 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 slabbeam 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 31814 (8.11.6.6)
Distribution of factored moments at critical sections is summarized in Table 2.

Slabbeam Strip 
Column Strip 
Middle Strip 

Moment 
Percent 
Moment 
Percent 
Moment 

End Span 
Exterior Negative 
335.1 
100 
335.1 
0 
0.0 
Positive 
630.0 
60 
378.0 
40 
252.0 

Interior Negative 
1207.9 
75 
905.9 
25 
302.0 

Interior Span 
Negative 
1097.1 
75 
822.8 
25 
274.3 
Positive 
304.4 
60 
182.6 
40 
121.8 
a. Determine flexural reinforcement required for strip moments
The flexural reinforcement calculation for the column strip of end span – interior negative location:
Use d = 15.88 in. (slab with drop panel where h = 17 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, tensioncontrolled section will be assumed so the reduction factoris 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 tensioncontrolled is valid.
Two values of thickness must be considered. The slab thickness in the column strip is 17 in. with the drop panel and 8 in. for the equivalent slab without the drop panel based on the system weight.
ACI 31814 (24.4.3.2)
ACI 31814 (24.4.3.3)
Provide 30  #6 bars with A_{s} = 13.20 in.^{2} and s = 198/30 = 6.6 in. ≤ s_{max}
The flexural reinforcement calculation for the column strip of interior span – positive location:
Use d = 15.88 in. (slab with rib where h = 17 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, tensioncontrolled section will be assumed so the reduction factoris 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 tensioncontrolled is valid.
ACI 31814 (24.4.3.2)
Since column strip has 5 ribs à provide 10  #6 bars (2 bars/ rib):
Based on the procedures 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 
M_{u} (ftkips) 
b (in.) 
d (in.) 
A_{s }Req’d for flexure (in.^{2}) 
Min A_{s }(in.^{2}) 
Reinforcement Provided 
A_{s }Prov. for flexure (in.^{2}) 

End Span 

Column Strip 
Exterior Negative 
335.1 
198 
15.88 
4.74 
5.18 
14#6 ^{*} ^{**} 
6.16 
Positive (5 ribs) 
378.0 
198 
15.81 
5.38 
2.85 
10#7 (2 bars / rib) 
6.00 

Interior Negative 
905.9 
198 
15.88 
13.05 
5.18 
30#6 
13.20 

Middle Strip 
Exterior Negative 
0.0 
198 
15.88 
0.0 
5.18 
14#6 ^{*} ^{**} 
6.16 
Positive (6 ribs) 
252.0 
198 
15.88 
3.56 
2.85 
12#6 (2 bars / rib) 
5.28 

Interior Negative 
302.0 
198 
15.88 
4.27 
5.18 
14#6 ^{*} ^{**} 
6.16 

Interior Span 

Column Strip 
Positive (5 ribs) 
182.6 
198 
15.88 
2.57 
2.85 
10#6 ^{*} (2 bars / rib) 
4.40 
Middle Strip 
Positive (6 ribs) 
121.8 
198 
15.88 
1.71 
2.85 
12#6 ^{*} (2 bars / rib) 
5.28 
^{*} 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 (γ_{f }M_{sc}) 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 31814 (8.4.2.3)
Portion of the unbalanced moment transferred by flexure is γ_{f} M_{sc} ACI 31814 (8.4.2.3.1)
Where
ACI 31814 (8.4.2.3.2)
b_{1} = Dimension of the critical section b_{o} measured in the direction of the span for which moments are determined in ACI 318, Chapter 8 (see Figure 14).
b_{2} = Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 14).
ACI 31814 (8.4.2.3.3)
Figure 14 – Critical Shear Perimeters for Columns
For exterior support:
Using the same procedure in 2.1.6.a, the required area of steel:
However, the area of steel provided to resist the flexural moment within the effective slab width b_{b}:
Then, the required additional reinforcement at exterior column for moment transfer between slab and column:
Provide 5  #6 additional bars with A_{s} = 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 
M_{sc}^{*} (ftkips) 
γ_{f} 
γ_{f} M_{sc} (ftkips) 
Effective slab width, b_{b} (in.) 
d (in.) 
A_{s} req’d within b_{b} (in.^{2}) 
A_{s} prov. For flexure within b_{b} (in.^{2}) 
Add’l Reinf. 

End Span 

Column Strip 
Exterior Negative 
462.3 
0.630 
291 
71 
15.88 
4.184 
2.209 
5#6 
Interior Negative 
133.4 
0.600 
80.4 
71 
15.88 
2.029 
4.733 
 

*M_{sc} is taken at the centerline of the support in Equivalent Frame Method solution. 
The unbalanced moment from the slabbeams at the supports of the equivalent frame are distributed to the support columns above and below the slabbeam in proportion to the relative stiffness of the support columns. Referring to Figure 13, the unbalanced moment at the exterior and interior joints are:
Exterior Joint = +462.3 ftkips
Joint 2= 1381.5 + 1247.5 = 134 ftkips
The stiffness and carryover factors of the actual columns and the distribution of the unbalanced slab moments (M_{sc}) to the exterior and interior columns are shown in Figure 14.
Figure 15  Column Moments (Unbalanced Moments from SlabBeam)
In summary:
For Top column: For Bottom column:
M_{col,Exterior}= 194.75 ftkips M_{col,Exterior}= 224.97 ftkips
M_{col,Interior} = 56.45 ftkips M_{col,Interior} = 65.21 ftkips
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 

M_{u} 
Column Location 

Interior 
Exterior 
Corner 

M_{ux} 
65.21 
224.97 
224.97 
M_{uy} 
65.21 
65.21 
224.97 
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 710 will be ignored in this example. However, the detailed procedure to calculate the reduced live loads is explained in the “wideModule Joist System” example.
Interior Column:
Assume 4 story building
Tributary area for interior column for live load, superimposed dead load, and selfweight of the slab is
Tributary area for interior column for selfweight of additional slab thickness due to the presence of the drop panel is
Assuming five story building
M_{u,x} = 65.21 ftkips (see the previous Table)
M_{u,y} = 65.21 ftkips (see the previous Table)
Edge (Exterior) Column:
Tributary area for exterior column for live load, superimposed dead load, and selfweight of the slab is
Tributary area for exterior column for selfweight of additional slab thickness due to the presence of the drop panel is
M_{u,x} = 224.97 ftkips (see the previous Table)
M_{u,y} = 65.21 ftkips (see the previous Table)
Corner Column:
Tributary area for corner column for live load, superimposed dead load, and selfweight of the slab is
Tributary area for corner column for selfweight of additional slab thickness due to the presence of the drop panel is
M_{u,x} = 224.97 ftkips (see the previous Table)
M_{u,y} = 224.97 ftkips (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 oneway shear (beam action) and twoway shear (punching) in accordance with ACI 318 Chapter 22.
ACI 31814 (22.5)
Oneway shear is critical at a distance d from the face of the column as shown in Figure 3. Figures 17 and 19 show the factored shear forces (V_{u}) 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 31814 (Eq. 22.5.1.1)
Where:
ACI 31814 (Eq. 22.5.5.1)
Oneway shear capacity is calculated assuming the shear crosssection area consisting of the drop panel (if any), the ribs, and the slab portion above them, decreased by concrete cover. For such section the equivalent shear width for single rib is calculated from the formula:
spSlab Software Manual (Eq. 213)
Where:
b = rib width, in.
d = distance from extreme compression fiber to tension reinforcement centroid.
for middle span with #6 reinforcement.
Figure 16 – Frame strip cross section (at distance d from the face of the supporting column)
The oneway shear capacity for the ribbed slab portions shown in Figure 16 is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
Figure 17 – Oneway shear at critical sections (at distance d from the face of the supporting column)
for middle span with #6 reinforcement.
Figure 18 – Frame strip cross section (at distance d from the face of the supporting column)
The oneway shear capacity for the ribbed slab portions shown in Figure 15 is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
Figure 19 – Oneway shear at critical sections (at the face of the drop panel)
ACI 31814 (22.6)
Twoway shear is critical on a rectangular section located at d/2 away from the face of the column as shown in Figure 14.
a. Exterior column:
The factored shear force (V_{u}) in the critical section is computed as the reaction at the centroid of the critical section minus the selfweight 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, M_{unb}, 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 zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the exterior column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
b. Interior column:
For the interior column in Figure 13, the location of the centroidal axis zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the interior column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (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 twoway shear strength of corner supports usually governs. Thus, the twoway shear strength for the corner column in this example will be checked for illustration purposes. The analysis procedure must be repeated for the exterior equivalent frame strip 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 interior column in Figure 13, the location of the centroidal axis zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the corner column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
Twoway shear is critical on a rectangular section located at d/2 away from the face of the drop panel.
The factored shear force (V_{u}) in the critical section is computed as the reaction at the centroid of the critical section minus the selfweight and any superimposed surface dead and live load acting within the critical section (d/2 away from column face).
Note: For simplicity, it is conservative to deduct only the selfweight of the slab and joists in the critical section from the shear reaction in punching shear calculations. This approach is also adopted in the spSlab program for the punching shear check around the drop panels.
a. Exterior drop panel:
d that is used in the calculation of v_{u} is given by (see Figure 20):
spSlab Software Manual (Eq. 214)
Figure 20 – Equivalent thickness based on shear area calculation
The length of the critical perimeter for the exterior drop panel:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
The twoway shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
ACI 31814 (Table 22.6.5.2)
In waffle slab design where the drop panels create a large critical shear perimeter, the factor (b_{o}/d) has limited contribution and is traditionally neglected for simplicity and conservatism. This approach is adopted in this calculation and in the spSlab program (spSlab software manual, Eq. 246).
The twoway shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
b. Interior drop panel:
The length of the critical perimeter for the interior drop panel:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
The twoway shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
ACI 31814 (Table 22.6.5.2)
spSlab Software Manual (Eq. 246)
c. Corner drop panel:
The length of the critical perimeter for the corner drop panel:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
The twoway shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 31814 (9.8.1.5)
ACI 31814 (Table 22.6.5.2)
spSlab Software Manual (Eq. 246)
To mitigate the deficiency in twoway shear capacity an evaluation of possible options is required:
1. Increase the thickness of the slab system
2. Increasing the dimensions of the drop panels (length and/or width)
3. Increasing the concrete strength
4. Reduction of the applied loads
5. Reduction of the panel spans
6. Using less conservative punching shear allowable (gain of 510%)
7. Refine the deduction of drop panel weight from the shear reaction (gain of 25%)
This example will be continued without the required modification discussed above to continue the illustration of the analysis and design procedure.
Since the slab thickness was selected to meet the minimum slab thickness tables in ACI 31814, the deflection calculations of immediate and timedependent deflections are not required. They are shown below for illustration purposes and comparison with spSlab software results.
The calculation of deflections for twoway slabs is challenging even if linear elastic behavior can be assumed. Elastic analysis for three service load levels (D, D + L_{sustained}, D+L_{Full}) is used to obtain immediate deflections of the twoway 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 31814 (24.2.3)
The effective moment of inertia (I_{e}) is used to account for the cracking effect on the flexural stiffness of the slab. I_{e }for uncracked section (M_{cr} > M_{a}) is equal to I_{g}. When the section is cracked (M_{cr} < M_{a}), then the following equation should be used:
ACI 31814 (Eq. 24.2.3.5a)
Where:
M_{a} = 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 21 – Maximum Moments for the Three Service Load Levels
For positive moment (midspan) section:
ACI 31814 (Eq. 24.2.3.5b)
ACI 31814 (Eq. 19.2.3.1)
y_{t} = Distance from centroidal axis of gross section, neglecting reinforcement, to tension face, in.
Figure 22 – Equivalent gross section for one rib  positive moment section
PCA Notes on ACI 31811 (9.5.2.2)
As calculated previously, the positive reinforcement for the middle span frame strip is 22 #6 bars located at 1.125 in. along the section from the bottom of the slab. Figure 23 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete at midspan.
Figure 23 – Cracked Transformed Section  positive moment section
ACI 31814 (19.2.2.1.a)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
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 45 #6 bars located at 1.125 in. along the section from the top of the slab.
ACI 31814 (Eq. 24.2.3.5b)
ACI 31814 (Eq. 19.2.3.1)
Note: A lower value of I_{g} (60,255 in.^{4}) excluding the drop panel is conservatively adopted in calculating waffle slab deflection by the spSlab software.
Figure 24 – Gross section – negative moment section
ACI 31814 (19.2.2.1.a)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
PCA Notes on ACI 31811 (Table 102)
Note: A lower value of I_{cr} (18,722 in.^{4}) excluding the drop panel is conservatively adopted in calculating waffle slab deflection by the spSlab software.
Figure 25 – 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, I_{e}, was developed to provide a transition between the upper and lower bounds of I_{g} and I_{cr} as a function of the ratio M_{cr}/M_{a}. For conventionally reinforced (nonprestressed) members, the effective moment of inertia, I_{e}, shall be calculated by Eq. (24.2.3.5a) unless obtained by a more comprehensive analysis.
I_{e} 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 31814 (24.2.3.7)
For continuous oneway slabs and beams. I_{e} 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 31814 (24.2.3.6)
For the middle span (span with two ends continuous) with service load level (D+LL_{full}):
ACI 31814 (24.2.3.5a)
Where I_{e}^{} is the effective moment of inertia for the critical negative moment section (near the support).
Where I_{e}^{+} 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 I_{e} and this is considered satisfactory in approximate deflection calculations. Both the midspan stiffness (I_{e}^{+}) and averaged span stiffness (I_{e,avg}) can be used in the calculation of immediate (instantaneous) deflection.
The averaged effective moment of inertia (I_{e,avg}) is given by:
PCA Notes on ACI 31811 (9.5.2.4(2))
PCA Notes on ACI 31811 (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 nonprismatic members and the following expressions are recommended according to ACI 31889:
ACI 435R95 (2.14)
For the middle span (span with two ends continuous) with service load level (D+LL_{full}):
ACI 435R95 (2.14)
For the end span (span with one end continuous) with service load level (D+LL_{full}):
Where:
Note: The prismatic member equations excluding the effect of the drop panel are conservatively adopted in calculating waffle slab deflection by spSlab.
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 
I_{g},

I_{cr}, 
M_{a}, kipsft 
M_{cr}, 
I_{e}, in.^{4} 
I_{e,avg}, in.^{4} 

D 
D + 
D + 
D 
D + 
D + 
D 
D + 
D + 

Ext 
Left 
103622 
15505 
206.5 
206.5 
338.0 
539 
103622 
103622 
103622 
62612 
62612 
29087 
Midspan 
60255 
15603 
298.2 
298.2 
491.8 
276 
50964 
50964 
23482 

Right 
103622 
23029 
626.6 
626.6 
1026.2 
539 
74259 
74259 
34692 

Int 
Left 
103622 
23029 
565.8 
565.8 
926.6 
539 
92620 
92620 
38873 
76437 
76437 
49564 
Mid 
60255 
13647 
132.6 
132.6 
221.0 
276 
60255 
60255 
60255 

Right 
103622 
23029 
565.8 
565.8 
926.6 
539 
92620 
92620 
38873 
Deflections in twoway 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 twoway slab systems, the midpanel deflection is computed as the sum of deflection at midspan of the column strip or column line in one direction (Δ_{cx} or Δ_{cy}) and deflection at midspan of the middle strip in the orthogonal direction (Δ_{mx} or Δ_{my}). Figure 26 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 31811 (9.5.3.4 Eq. 8)
Figure 26 – Deflection Computation for a rectangular Panel
To calculate each term of the previous equation, the following procedure should be used. Figure 27 shows the procedure of calculating the term Δ_{cx}. Same procedure can be used to find the other terms.
Figure 27 –Δ_{cx }calculation procedure
For end span  service dead load case:
PCA Notes on ACI 31811 (9.5.3.4 Eq. 10)
Where:
ACI 31814 (19.2.2.1.a)
I_{frame,averaged }= The averaged effective moment of inertia (I_{e,avg}) for the frame strip for service dead load case from Table 6 = 62,612 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 11)
LDF_{c} is the load distribution factor for the column strip. The load distribution factor for the column strip can be found from the following equation:
spSlab Software Manual (Eq. 2114)
And the load distribution factor for the middle strip can be found from the following equation:
spSlab Software Manual (Eq. 2115)
Taking for example the end span where highest deflections are expected, the LDF_{ }for exterior negative region (LDF_{L}¯), interior negative region (LDF_{R}¯), and positive region (LDF_{L}^{＋}) 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:
I_{c,g} = The gross moment of inertia (I_{g}) for the column strip for service dead load = 28,289 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 12)
Where:
K_{ec} = effective column stiffness = 1925 x 10^{6} in.lb (calculated previously).
PCA Notes on ACI 31811 (9.5.3.4 Eq. 14)
Where:
Where
Where:
PCA Notes on ACI 31811 (9.5.3.4 Eq. 9)
Following the same procedure, Δ_{mx} can be calculated for the middle strip. This procedure is repeated for the equivalent frame in the orthogonal direction to obtain Δ_{cy}, and Δ_{my} for the end and middle spans for the other load levels (D+LL_{sus} and D+LL_{full}).
Since this example has square panels, Δ_{cx }=_{ }Δ_{cy}= 0.222 in. and Δ_{mx }=_{ }Δ_{my}= 0.128 in.
The average Δ for the corner panel is calculated as follows:
The calculated deflection can now be compared with the applicable limits from the governing standards or project specified limits and requirements. Optimization for further savings in materials or construction costs can be now made based on permissible deflections in lieu of accepting the minimum values stipulated in the standards to avoid deflection calculations.
Table 7 – Immediate (Instantaneous) Deflections in the xdirection 

Column Strip 
Middle Strip 

Span 
LDF 
D 
LDF 
D 

Δ_{framefixed}, in. 
Δ_{cfixed}, in. 
θ_{c1}, rad 
θ_{c2}, rad 
Δθ_{c1}, in. 
Δθ_{c2}, in. 
Δ_{cx}, in. 
Δ_{framefixed}, in. 
Δ_{mfixed}, in. 
θ_{m1}, rad 
θ_{m2}, rad 
Δθ_{m1}, in. 
Δθ_{m2}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.094 
0.147 
0.00129 
0.0004 
0.058 
0.017 
0.222 
0.262 
0.094 
0.052 
0.00129 
0.0004 
0.058 
0.017 
0.128 

Int 
0.675 
0.077 
0.110 
0.0004 
0.0004 
0.014 
0.014 
0.082 
0.325 
0.077 
0.053 
0.0004 
0.0004 
0.014 
0.014 
0.025 

Span 
LDF 
D+LL_{sus} 
LDF 
D+LL_{sus} 

Δ_{framefixed}, in. 
Δ_{cfixed}, in. 
θ_{c1}, rad 
θ_{c2}, rad 
Δθ_{c1}, in. 
Δθ_{c2}, in. 
Δ_{cx}, in. 
Δ_{framefixed}, in. 
Δ_{mfixed}, in. 
θ_{m1}, rad 
θ_{m2}, rad 
Δθ_{m1}, in. 
Δθ_{m2}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.094 
0.147 
0.00129 
0.0004 
0.058 
0.017 
0.222 
0.262 
0.094 
0.052 
0.00129 
0.0004 
0.058 
0.017 
0.128 

Int 
0.675 
0.077 
0.110 
0.0004 
0.0004 
0.014 
0.014 
0.082 
0.325 
0.077 
0.053 
0.0004 
0.0004 
0.014 
0.014 
0.025 

Span 
LDF 
D+LL_{full} 
LDF 
D+LL_{full} 

Δ_{framefixed}, in. 
Δ_{cfixed}, in. 
θ_{c1}, rad 
θ_{c2}, rad 
Δθ_{c1}, in. 
Δθ_{c2}, in. 
Δ_{cx}, in. 
Δ_{framefixed}, in. 
Δ_{mfixed}, in. 
θ_{m1}, rad 
θ_{m2}, rad 
Δθ_{m1}, in. 
Δθ_{m2}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.316 
0.497 
0.0021 
0.0006 
0.205 
0.060 
0.762 
0.262 
0.316 
0.177 
0.0021 
0.0006 
0.205 
0.060 
0.442 

Int 
0.675 
0.186 
0.267 
0.0006 
0.0006 
0.035 
0.035 
0.196 
0.325 
0.186 
0.128 
0.0006 
0.0006 
0.035 
0.035 
0.057 

Span 
LDF 
LL 
LDF 
LL 

Δ_{cx}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.540 
0.262 
0.314 

Int 
0.675 
0.114 
0.325 
0.032 
The additional timedependent (longterm) deflection resulting from creep and shrinkage (Δ_{cs}) may be estimated as follows:
PCA Notes on ACI 31811 (9.5.2.5 Eq. 4)
The total timedependent (longterm) deflection is calculated as:
CSA A23.304 (N9.8.2.5)
Where:
ACI 31814 (24.2.4.1.1)
For the exterior span
= 2, consider the sustained load duration to be 60 months or more. ACI 31814 (Table 24.2.4.1.3)
= 0, conservatively.
Table 8 shows longterm deflections for the exterior and interior spans for the analysis in the xdirection, for column and middle strips.
Table 8  LongTerm Deflections 

Column Strip 

Span 
(Δ_{sust})_{Inst}, in. 
λ_{Δ} 
Δ_{cs}, in. 
(Δ_{total})_{Inst}, in. 
(Δ_{total})_{lt}, in. 
Exterior 
0.222 
2.000 
0.445 
0.762 
1.207 
Interior 
0.082 
2.000 
0.164 
0.196 
0.360 
Middle Strip 

Exterior 
0.128 
2.000 
0.255 
0.442 
0.698 
Interior 
0.025 
2.000 
0.050 
0.057 
0.107 
spSlab program utilizes the Equivalent Frame Method described and illustrated in details here for modeling, analysis and design of twoway concrete floor slab systems with drop panels. spSlab uses the exact geometry and boundary conditions provided as input to perform an elastic stiffness (matrix) analysis of the equivalent frame taking into account the torsional stiffness of the slabs framing into the column. It also takes into account the complications introduced by a large number of parameters such as vertical and torsional stiffness of transverse beams, the stiffening effect of drop panels, column capitals, and effective contribution of columns above and below the floor slab using the of equivalent column concept (ACI 31814 (R8.11.4)).
spSlab Program models the equivalent frame as a design strip. The design strip is, then, separated by spSlab into column and middle strips. The program calculates the internal forces (Shear Force & Bending Moment), moment and shear capacity vs. demand diagrams for column and middle strips, instantaneous and longterm deflection results, and required flexural reinforcement for column and middle strips. The graphical and text results are provided below for both input and output of the spSlab model.
Table 9  Comparison of Moments obtained from Hand (EFM) and spSlab Solution (ftkips) 

Hassoun (DDM)^{#} 
Hand (EFM) 
spSlab 

Exterior Span 

Column Strip 
Exterior Negative^{*} 
370.0 
335.1 
323.8 
Positive 
444.0 
378.0 
400.6 

Interior Negative^{*} 
748.0 
905.9 
907.3 

Middle Strip 
Exterior Negative^{*} 
 
0.0 
0.0 
Positive 
 
252.0 
267.1 

Interior Negative^{*} 
 
302.0 
302.4 

Interior Span 

Column Strip 
Interior Negative^{*} 
 
822.8 
823.7 
Positive 
 
182.6 
180.4 

Middle Strip 
Interior Negative^{*} 
249 
274.3 
274.6 
Positive 
296 
121.8 
120.2 

^{* }negative moments are taken at the faces of supports ^{# }Direct design method does not distinguish between interior and exterior spans nor explicitly address the effect of column contribution at joints. 
Table 10  Comparison of Reinforcement Results 

Span Location 
Reinforcement Provided for Flexure 
Additional Reinforcement Provided for Unbalanced Moment Transfer 
Total
Reinforcement 

Hassoun 
Hand 
spSlab 
Hassoun 
Hand 
spSlab 
Hassoun 
Hand 
spSlab 

Exterior Span 

Column Strip 
Exterior Negative 
14#6 
14#6 
14#6 
 
5#6 
5#6 
14#6 
19#6 
19#6 
Positive 
10#8 2 bars / rib 
10#7 2 bars / rib 
10#7 2 bars / rib 
 
n/a 
n/a 
10#8 2 bars / rib 
10#7 2 bars / rib 
10#7 2 bars / rib 

Interior Negative 
28#6 
30#6 
31#6 
 
 
 
28#6 
22#6 
21#6 

Middle Strip 
Exterior Negative 
10#6 
14#6 
14#6 
 
n/a 
n/a 
10#6^{*} 
14#6 
14#6 
Positive 
12#7 2 bars / rib 
12#6 2 bars / rib 
12#6 2 bars / rib 
 
n/a 
n/a 
12#7 2 bars / rib 
12#6 2 bars / rib 
12#6 2 bars / rib 

Interior Negative 
10#6 
14#6 
14#6 
 
n/a 
n/a 
10#6^{*} 
14#6 
14#6 

Interior Span 

Column Strip 
Positive 
10#7 2 bars / rib 
10#6 2 bars / rib 
10#6 2 bars / rib 
 
n/a 
n/a 
10#7 2 bars / rib 
10#6 2 bars / rib 
10#6 2 bars / rib 
Middle Strip 
Positive 
10#6 2 bars / rib 
12#6 2 bars / rib 
12#6 2 bars / rib 
 
n/a 
n/a 
10#6 2 bars / rib 
12#6 2 bars / rib 
12#6 2 bars / rib 
^{*} Max spacing requirement exceeded (not checked) 
Table 11 – Comparison of OneWay (Beam Action) Shear Check Results 

Span 
V_{u} @ d, kips 
V_{u} @ drop panel, kips 
φV_{c }@ d , kips 
φV_{c }@ drop panel, kips 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
188.8 
195.5 
145.7 
146.1 
336.0 
336.0 
148.5 
148.5 
Interior 
160.9 
167.2 
117.8 
117.8 
337.4 
337.4 
149.2 
149.2 
^{*} Oneway shear check is not provided in the reference (Hassoun and AlManaseer) 
Table 12  Comparison of TwoWay (Punching) Shear Check Results (around Columns Faces) 

Support 
b_{1}, in. 
b_{2}, in. 
b_{o}, in. 
V_{u}, kips 
c_{AB}, in. 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
27.94 
27.94 
35.88 
35.88 
91.76 
91.75 
153.97 
174.86 
8.51 
8.51 
Interior 
35.88 
35.88 
35.88 
35.88 
143.52 
143.50 
393.83 
414.86 
17.94 
17.94 
Corner 
27.94 
27.94 
27.94 
27.94 
55.88 
55.88 
92.10 
92.43 
6.99 
6.98 
Support 
J_{c}, in.^{4} 
γ_{v} 
M_{unb}, ftkips 
v_{u}, psi 
φv_{c, }psi 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
144,092 
143,990 
0.370 
0.370 
341.27 
316.33 
195.2 
203.1 
212.1 
212.1 
Interior 
512,956 
512,570 
0.400 
0.400 
134.00 
135.09 
195.3 
204.8 
212.1 
212.1 
Corner 
81,483 
81,428 
0.400 
0.400 
181.47 
181.19 
178.5 
178.8 
212.1 
212.1 
Table 13  Comparison of TwoWay (Punching) Shear Check Results (around Drop Panels) 

Support 
b_{1}, in. 
b_{2}, in. 
b_{o}, in. 
V_{u}, kips 
c_{AB}, in. 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
89.94 
89.94 
159.88 
159.88 
339.76 
339.75 
123.33 
143.28 
23.81 
23.81 
Interior 
159.88 
159.88 
159.88 
159.88 
639.52 
639.50 
337.73 
357.54 
79.94 
79.94 
Corner 
89.94 
89.94 
89.94 
89.94 
179.88 
179.87 
75.22 
75.17 
22.49 
22.48 
Support 
J_{c}, in.^{4} 
v_{u}, psi 
φv_{c, }psi 


Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 


Exterior 
971,437 
970,500 
109.3 
127.1 
116.7 
116.7 


Interior 
9,046,406 
9,037,700 
159.1 
168.5 
116.7 
116.7 


Corner 
503,491 
503,010 
126.0 
126.0 
116.7 
116.7 


General notes: 1. Red values are exceeding permissible shear capacity 2. Hand calculations fail to capture analysis details possible in spSlab like accounting for the exact value of the moments and shears at supports and including the loads for the small slab section extending beyond the supporting column centerline. 

Table 14  Comparison of Immediate Deflection_{ }Results (in.) 

Column Strip 

Span 
D 
D+LL_{sus} 
D+LL_{full} 
LL 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.222 
0.337 
0.222 
0.337 
0.762 
0.867 
0.540 
0.530 
Interior 
0.082 
0.116 
0.082 
0.116 
0.196 
0.222 
0.114 
0.106 
Middle Strip 

Span 
D 
D+LL_{sus} 
D+LL_{full} 
LL 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.128 
0.189 
0.128 
0.189 
0.442 
0.443 
0.315 
0.254 
Interior 
0.025 
0.039 
0.025 
0.039 
0.057 
0.082 
0.033 
0.043 
Table 15  Comparison of TimeDependent Deflection_{ }Results 

Column Strip 

Span 
λ_{Δ} 
Δ_{cs}, in. 
Δ_{total}, in. 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.000 
2.000 
0.445 
0.679 
1.207 
1.453 
Interior 
2.000 
2.000 
0.164 
0.245 
0.360 
0.493 
Middle Strip 

Span 
λ_{Δ} 
Δ_{cs}, in. 
Δ_{total}, in. 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.000 
2.000 
0.255 
0.434 
0.698 
0.902 
Interior 
2.000 
2.000 
0.050 
0.142 
0.107 
0.291 
In all of the hand calculations illustrated above, the results are in close or exact agreement with the automated analysis and design results obtained from the spSlab model. The deflection results from spSlab are, however, more conservative than hand calculations for two main reasons explained previously: 1) Values of I_{g} and I_{cr} at the negative section exclude the stiffening effect of the drop panel and 2) The I_{e,avg} used by spSlab considers equations for prismatic members.
A slab system can be analyzed and designed by any procedure satisfying equilibrium and geometric compatibility. Three established methods are widely used. The requirements for two of them are described in detail in ACI 31814 Chapter 8 (8.2.1).
Direct Design Method (DDM) is an approximate method and is applicable to twoway slab concrete floor systems that meet the stringent requirements of ACI 31814 (8.10.2). In many projects, however, these requirements limit the usability of the Direct Design Method significantly.
StucturePoint’s spSlab software program solution utilizes the Equivalent Frame Method to automate the process providing considerable timesavings in the analysis and design of twoway slab systems as compared to hand solutions using DDM or EFM.
Finite Element Method (FEM) is another method for analyzing reinforced concrete slabs, particularly useful for irregular slab systems with variable thicknesses, openings, and other features not permissible in DDM or EFM. Many reputable commercial FEM analysis software packages are available on the market today such as spMats. Using FEM requires critical understanding of the relationship between the actual behavior of the structure and the numerical simulation since this method is an approximate numerical method. The method is based on several assumptions and the operator has a great deal of decisions to make while setting up the model and applying loads and boundary conditions. The results obtained from FEM models should be verified to confirm their suitability for design and detailing of concrete structures.
The following table shows a general comparison between the DDM, EFM and FEM. This table covers general limitations, drawbacks, advantages, and costtime efficiency of each method where it helps the engineer in deciding which method to use based on the project complexity, schedule, and budget.
Applicable ACI 31814 Provision 
Limitations/Applicability 
Concrete Slab Analysis Method 

DDM (Hand) 
EFM (Hand//spSlab) 
FEM (spMats) 

8.10.2.1 
Minimum of three continuous spans in each direction 
þ 

8.10.2.2 
Successive span lengths measured centertocenter of supports in each direction shall not differ by more than onethird the longer span 
þ 

8.10.2.3 
Panels shall be rectangular, with ratio of longer to shorter panel dimensions, measured centertocenter supports, not exceed 2. 
þ 
þ 

8.10.2.4 
Column offset shall not exceed 10% of the span in direction of offset from either axis between centerlines of successive columns 
þ 

8.10.2.5 
All loads shall be due to gravity only 
þ 

8.10.2.5 
All loads shall be uniformly distributed over an entire panel (q_{u}) 
þ 


8.10.2.6 
Unfactored live load shall not exceed two times the unfactored dead load 
þ 

8.10.2.7 
For a panel with beams between supports on all sides, slabtobeam stiffness ratio shall be satisfied for beams in the two perpendicular directions. 
þ 

8.7.4.2 
Structural integrity steel detailing 
þ 
þ 
þ 
8.5.4 
Openings in slab systems 
þ 
þ 
þ 
8.2.2 
Concentrated loads 
Not permitted 
þ 
þ 
8.11.1.2 
Live load arrangement (Load Patterning) 
Not required 
Required 
Engineering judgment required based on modeling technique 
R8.10.4.5^{*} 
Reinforcement for unbalanced slab moment transfer to column (M_{sc}) 
Moments @ support face 
Moments @ support centerline 
Engineering judgment required based on modeling technique 

Irregularities (i.e. variable thickness, nonprismatic, partial bands, mixed systems, support arrangement, etc.) 
Not permitted 
Engineering judgment required 
Engineering judgment required 
Complexity 
Low 
Average 
Complex to very complex 

Design time/costs 
Fast 
Limited 
Unpredictable/Costly 

Design Economy 
Conservative (see detailed comparison with spSlab output) 
Somewhat conservative 
Unknown  highly dependent on modeling assumptions: 1. Linear vs. nonlinear 2. Isotropic vs nonisotropic 3. Plate element choice 4. Mesh size and aspect ratio 5. Design & detailing features 

General (Drawbacks) 
Very limited applications 
Limited geometry 
Limited guidance nonstandard application (user dependent). Required significant engineering judgment 

General (Advantages) 
Very limited analysis is required 
Detailed analysis is required or via software (e.g. spSlab) 
Unlimited applicability to handle complex situations permissible by the features of the software used (e.g. spMats) 

^{*} The unbalanced slab moment transferred to the column M_{sc} (M_{unb}) is the difference in slab moment on either side of a column at a specific joint. In DDM only moments at the face of the support are calculated and are also used to obtain M_{sc }(M_{unb}). In EFM where a frame analysis is used, moments at the column center line are used to obtain M_{sc }(M_{unb}). 
[IMA1]Update note