TwoWay Flat Slab (Concrete Floor with Drop Panels) System Analysis and Design
TwoWay Flat Slab (Concrete Floor with Drop Panels) System Analysis and Design
Design the concrete floor slab system shown below for an intermediate floor considering partition weight = 20 psf, and unfactored live load = 60 psf. The lateral loads are independently resisted by shear walls. The use of flat plate system will be checked. If the use of flat plate is not adequate, the use of flat slab system with drop panels will be investigated. Flat slab concrete floor system is similar to the flat plate system. The only exception is that the flat slab uses drop panels (thickened portions around the columns) to increase the nominal shear strength of the concrete at the critical section around the columns. 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.
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
8.1. OneWay Shear Distribution to Slab Strips
8.2. TwoWay Concrete Slab Analysis Methods
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
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)
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 710 (Table C31)
Live Load, LL = 60 psf ASCE/SEI 710 (Table 41)
50 psf is considered by inspection of Table 41 for Office Buildings – Offices (2/3 of the floor area)
80 psf is considered by inspection of Table 41 for Office Buildings – Corridors (1/3 of the floor area)
LL = 2/3 x 50 + 1/3 x 80 = 60 psf
f_{c}’ = 5000 psi (for slab)
f_{c}’ = 6000 psi (for columns)
f_{y} = 60,000 psi
Solution
For Flat Plate (without Drop Panels)
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 flat plate slab systems the minimum slab thicknesses 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 = 30 x 12 – 20 = 340 in.
Try 11 in. slab for all panels (selfweight = 150 pcf x 11 in. /12 = 137.5 psf)
b. Slab shear strength – one way shear
At a preliminary check level, the use of average effective depth would be sufficient. However, after determining the final depth of the slab, the exact effective depth will be used in flexural, shear and deflection calculations. 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)
Where λ = 1 for normal weight concrete, more information can be found in “Concrete Type Classification Based on Unit Density” technical article.
Slab thickness of 11 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 11 in. is not adequate for twoway shear. It is good to mention that the factored shear (V_{u}) used in the preliminary check does not include the effect of the unbalanced moment at supports. Including this effect will lead to an increase of V_{u} value as shown later in section 4.2.
Figure 3 – Critical Section for OneWay Shear Figure 4 – Critical Section for TwoWay Shear
In this case, four options could be used: 1) to increase the slab thickness, 2) to increase columns cross sectional dimensions or cut the spacing between columns (reducing span lengths), however, this option is assumed to be not permissible in this example due to architectural limitations, 3) to use headed shear reinforcement, or 4) to use drop panels. In this example, the latter option will be used to achieve better understanding for the design of twoway 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 onefourth of the adjacent slab thickness.
ACI 31814 (8.2.4(a))
Since the slab thickness (h_{s}) 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), h_{dp} = 4.25 in. > h_{dp, min}_{ }= 2.5 in.
The total thickness including the slab and the drop panel (h) = h_{s }+ h_{dp} = 10 + 4.25 = 14.25 in.
Nominal Lumber Size, in. 
Actual Lumber Size, in. 
Plyform Thickness, in. 
h_{dp}, 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 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 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 31914. 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 flat plate slab systems 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 = 30 x 12 – 20 = 340 in.
Try 10 in. slab for all panels
Selfweight for slab section without drop panel = 150 pcf x 10 in. /12 = 125 psf
Selfweight 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:
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 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 7)
ACI 31814 (Eq. 22.5.5.1)
Slab thickness of 14.25 in. 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 support (see Figure 7)
ACI 31814 (Eq. 22.5.5.1)
Slab thickness of 10 in. is adequate for oneway shear for the second critical section (from the edge of the drop panel).
Figure 7 – 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 8):
ACI 31814 (Table 22.6.5.2(a))
Slab thickness of 14.25 in. 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 8):
ACI 31814 (Table 22.6.5.2(a))
Slab thickness of 10 in. is adequate for twoway shear for the second critical section (from the edge of the drop panel).
Figure 8 – 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 five story building
Assume 20 in. square column with 4 – No. 14 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 = 0.578 PCA Notes on ACI 31811 (Table A1)
PCA Notes on ACI 31811 (Table A1)
Uniform load fixed end moment coefficient, m_{NF1} = 0.0915
Fixed end moment coefficient for (ba) = 0.2 when a = 0, m_{NF2} = 0.0163
Fixed end moment coefficient for (ba) = 0.2 when a = 0.8, m_{NF3} = 0.0163
b. Flexural stiffness of column members at both ends, K_{c}.
Referring to Table A7, Appendix 20A,
For the Bottom Column (Below):
PCA Notes on ACI 31811 (Table A7)
ACI 31814 (19.2.2.1.a)
l_{c} = 13 ft = 156 in.
For the Top Column (Above):
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)
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 9 – Torsional Member Figure 10 – Column and Edge of Slab
d. Slabbeam joint distribution factors, DF.
At exterior joint,
At interior joint,
COF for slabbeam =0.578
Figure 11 – 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 
0.551 
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, kft 
331.7 
807.6 
721.9 
721.9 
807.6 
331.7 
Midspan M, ftkips 
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 l_{1} from the centers of supports. ACI 31814 (8.11.6.1)
Figure 12  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 30/30 = 1.0 < 2.0. ACI 31814 (8.10.2.3)
4. Column are not offset. ACI 31814 (8.10.2.4)
5. Loads are gravity and uniformly distributed with service livetodead ratio of 0.41 < 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.
All limitation of ACI 31814 (8.10.2) are satisfied and the provisions of ACI 31814 (8.11.6.5) may be applied:
ACI 31814 (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 ftkips
Adjusted positive design moment = 197.3 × 0.884 = 174.4 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 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 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 
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 d_{avg} = 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, 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.
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).
The weighted slab thickness:
ACI 31814 (24.4.3.2)
ACI 31814 (8.7.2.2)
Provide 10  #6 bars with A_{s} = 4.40 in.^{2} and s = 180/10 = 18 in. ≤ s_{max}
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 
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 
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 (γ_{f} x 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} x 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 13).
b_{2} = Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 13).
b_{b} = Effective slab width = ACI 31814 (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.
c_{1} + d/2 = 20 + 13.13/2 = 26.56 in.
c_{2} + 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 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 
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 
 

*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 12, the unbalanced moment at the exterior and interior joints are:
Exterior Joint = +331.7 ftkips
Joint 2= 807.6 + 721.9 = 85.7 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 14  Column Moments (Unbalanced Moments from SlabBeam)
In summary:
For Top column (Above): For Bottom column (Below):
M_{col,Exterior}= 150.61 ftkips M_{col,Exterior}= 157.21 ftkips
M_{col,Interior} = 38.91 ftkips M_{col,Interior} = 40.62 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} 
40.62 
157.21 
157.21 
M_{uy} 
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 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 5 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} = 40.62 ftkips (see the previous Table)
M_{u,y} = 40.62 ftkips (see the previous Table)
Edge (Exterior) Column:
Tributary area for edge column for live load, superimposed dead load, and selfweight of the slab is
Tributary area for edge column for selfweight of additional slab thickness due to the presence of the drop panel is
M_{u,x} = 157.21 ftkips (see the previous Table)
M_{u,y} = 40.62 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} = 157.21 ftkips (see the previous Table)
M_{u,y} = 157.21 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 15 and 16 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)
Note: The calculations below follow one of two possible approaches for checking oneway shear. Refer to the conclusions section for a comparison with the other approach.
Figure 15 – Oneway shear at critical sections (at distance d from the face of the supporting column)
Figure 16 – 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 13.
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 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 zz is:
in.
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.
Note: The twoway shear stress calculations around drop panels do not have the term for unbalanced moment since drop panels are a thickened portion of the slab and are not considered as a support.
a. Exterior drop panel:
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)
ACI 31814 (Table 22.6.5.2)
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)
ACI 31814 (Table 22.6.5.2)
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)
ACI 31814 (Table 22.6.5.2)
Since the slab thickness was selected below the minimum slab thickness tables in ACI 31814, the deflection calculations of immediate and timedependent deflections are required and shown below including a comparison with spSlab model 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 17 – Maximum Moments for the Three Service Load Levels
(No live load is sustained in this example)
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.
PCA Notes on ACI 31811 (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 conservatively excluded from the calculation of I_{cr} since they might not be adequately developed or tied (21 bars are used). 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 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 32 #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)
Figure 19 – I_{g} calculations for slab section near support
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)
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, 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 should be used 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:
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 
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 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 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 31811 (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 Δ_{cx}. Same procedure can be used to find the other terms.
Figure 22 –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 = 37190 in.^{4}
PCA Notes on ACI 31811 (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.
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:
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 (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 = 15000 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 12)
Where:
K_{ec} = effective column stiffness = 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 in this example the panel is squared, Δ_{cx }=_{ }Δ_{cy}= 0.219 in. and Δ_{mx }=_{ }Δ_{my}= 0.125 in.
The average Δ for the corner panel is calculated as follows:
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 
0.0995 
0.1468 
0.00159 
0.00041 
0.05786 
0.01479 
0.219 
0.262 
0.0995 
0.0521 
0.00159 
0.00041 
0.05786 
0.01479 
0.125 

Int 
0.675 
0.0886 
0.1197 
0.00041 
0.00041 
0.01319 
0.01319 
0.093 
0.325 
0.0886 
0.0576 
0.00041 
0.00041 
0.01319 
0.01319 
0.031 

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 
0.0995 
0.1468 
0.00159 
0.00041 
0.05786 
0.01479 
0.219 
0.262 
0.0995 
0.0521 
0.00159 
0.00041 
0.05786 
0.01479 
0.125 

Int 
0.675 
0.0886 
0.1197 
0.00041 
0.00041 
0.01319 
0.01319 
0.093 
0.325 
0.0886 
0.0576 
0.00041 
0.00041 
0.01319 
0.01319 
0.031 

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 
0.2128 
0.2775 
0.00224 
0.00057 
0.12316 
0.0315 
0.432 
0.262 
0.2128 
0.0985 
0.00224 
0.00057 
0.12316 
0.03125 
0.253 

Int 
0.675 
0.1819 
0.2455 
0.00057 
0.00057 
0.02693 
0.02693 
0.192 
0.325 
0.1819 
0.1182 
0.00057 
0.00057 
0.02693 
0.02693 
0.064 

Span 
LDF 
LL 
LDF 
LL 

Δ_{cx}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.213 
0.262 
0.128 

Int 
0.675 
0.098 
0.325 
0.033 
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 
0.219 
2.000 
0.439 
0.432 
0.871 
Interior 
0.093 
2.000 
0.187 
0.192 
0.378 
Middle Strip 

Exterior 
0.125 
2.000 
0.250 
0.253 
0.503 
Interior 
0.031 
2.000 
0.062 
0.064 
0.127 
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) 

Hand (EFM) 
spSlab 

Exterior Span 

Column Strip 
Exterior Negative^{*} 
246.5 
244.8 
Positive 
209.8 
219.7 

Interior Negative^{*} 
521.9 
517.6 

Middle Strip 
Exterior Negative^{*} 
0.0 
0.0 
Positive 
139.8 
146.5 

Interior Negative^{*} 
174.0 
172.5 

Interior Span 

Column Strip 
Interior Negative^{*} 
467.6 
463.6 
Positive 
118.4 
120.1 

Middle Strip 
Interior Negative^{*} 
155.9 
154.5 
Positive 
78.9 
80.1 

^{* }negative moments are taken at the faces of supports 
Table 10  Comparison of Reinforcement Results 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior Span 

Column Strip 
Exterior Negative 
10#6 
10#6 
5#6 
5#6 
15#6 
15#6 
Positive 
13#6 
13#6 
n/a 
n/a 
13#6 
13#6 

Interior Negative 
22#6 
21#6 
 
 
22#6 
21#6 

Middle Strip 
Exterior Negative 
10#6 
10#6 
n/a 
n/a 
10#6 
10#6 
Positive 
10#6 
10#6 
n/a 
n/a 
10#6 
10#6 

Interior Negative 
11#6 
11#6 
n/a 
n/a 
11#6 
11#6 

Interior Span 

Column Strip 
Positive 
10#6 
10#6 
n/a 
n/a 
10#6 
10#6 
Middle Strip 
Positive 
10#6 
10#6 
n/a 
n/a 
10#6 
10#6 
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 
123.7 
126.7 
96.9 
96.7 
392.91 
392.97 
339.10 
338.88 
Interior 
107.8 
110.9 
81.0 
81.0 
392.91 
392.97 
339.10 
338.88 
^{*} x_{u }calculated from the centerline of the left column for each span 
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 
26.56 
26.56 
33.13 
33.13 
86.26 
86.25 
103.6 
114.6 
8.18 
8.18 
Interior 
33.13 
33.13 
33.13 
33.13 
132.52 
132.50 
256.4 
263.0 
16.57 
16.56 
Corner 
26.56 
26.56 
26.56 
26.56 
53.13 
53.12 
60.3 
60.6 
6.64 
6.64 
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 
98,315 
98,239 
0.374 
0.374 
259 
249.5 
188.3 
194.4 
212.1 
212.1 
Interior 
330,800 
330,520 
0.400 
0.400 
85.70 
85.07 
167.9 
171.7 
212.1 
212.1 
Corner 
56,292 
56,249 
0.400 
0.400 
137.65 
137.40 
164.4 
164.8 
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 
74.44 
74.44 
128.88 
128.88 
277.76 
277.75 
87.65 
98.24 
19.95 
19.95 
Interior 
128.88 
128.88 
128.88 
128.88 
515.52 
515.5 
225.5 
233.91 
64.44 
64.44 
Corner 
74.44 
74.44 
74.44 
74.44 
148.88 
148.87 
51.54 
51.53 
18.61 
18.61 


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 
1,468,983 
1,468,000 
N.A. 
N.A. 
N.A. 
N.A. 
35.5 
39.9 
156.9 
156.9 
Interior 
12,688,007 
12,679,000 
N.A. 
N.A. 
N.A. 
N.A. 
49.7 
51.1 
142.6 
142.6 
Corner 
767,460 
766,930 
N.A. 
N.A. 
N.A. 
N.A. 
38.98 
39.00 
169.3 
169.3 
Note: Shear stresses from spSlab are higher than hand calculations since it considers the load effects beyond the column centerline known in the model as right/left cantilevers. This small increase is often neglected in simplified hand calculations like the one used here. 
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 
0.219 
0.207 
0.219 
0.207 
0.432 
0.395 
0.213 
0.188 
Interior 
0.093 
0.089 
0.093 
0.089 
0.192 
0.185 
0.098 
0.096 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.125 
0.120 
0.125 
0.120 
0.253 
0.218 
0.128 
0.098 
Interior 
0.031 
0.030 
0.031 
0.030 
0.064 
0.071 
0.033 
0.040 
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 
2.0 
2.0 
0.439 
0.414 
0.871 
0.808 
Interior 
2.0 
2.0 
0.187 
0.178 
0.378 
0.363 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.0 
2.0 
0.250 
0.241 
0.503 
0.459 
Interior 
2.0 
2.0 
0.062 
0.060 
0.127 
0.131 
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. Excerpts of spSlab graphical and text output are given below for illustration.
In oneway shear checks above, shear is distributed uniformly along the width of the design strip (30 ft.). StructurePoint finds it necessary sometimes to allocate the oneway shears with the same proportion moments are distributed to column and middle strips.
spSlab allows the oneway shear check using two approaches: 1) calculating the oneway shear capacity using the average slab thickness and comparing it with the total factored oneshear load as shown in the hand calculations above; 2) distributing the factored oneway shear forces to the column and middle strips and comparing it with the shear capacity of each strip as illustrated in the following figures. An engineering judgment is needed to decide which approach to be used.
Figure 23a – Distributing Shear to Column and Middle Strips (spSlab Input)
Figure 23b – Distributed Column and Middle Strip Shear Force Diagram (spSlab Output)
Figure 23c – Tabulated Shear Force & Capacity at Critical Sections (spSlab Output)
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}). 