TwoWay Flat Plate Concrete Floor System Analysis and Design
TwoWay Flat Plate Concrete Floor System Analysis and Design
The concrete floor slab system shown below is for an intermediate floor to be designed considering partition weight = 20 psf, and unfactored live load = 40 psf. Flat plate concrete floor system does not use beams between columns or drop panels and it is usually suited for lightly loaded floors with short spans typically for residential and hotel buildings. The lateral loads are independently resisted by shear walls. The two design procedures shown in ACI 31814: Direct Design Method (DDM) and the Equivalent Frame Method (EFM) are illustrated in detail in this example. The hand solution from EFM is also used for a detailed comparison with the analysis and design results of the engineering software program spSlab.
Figure 1  TwoWay Flat Concrete Floor System
Contents
2. TwoWay Slab Analysis and Design
2.1. Direct Design Method (DDM)
2.1.1. Direct design method limitations
2.1.3. Flexural reinforcement requirements
2.1.4. Factored moments in columns
2.2. Equivalent Frame Method (EFM)
2.2.1. Equivalent frame method limitations
2.2.2. Frame members of equivalent frame
2.2.3. Equivalent frame analysis
2.2.5. Distribution of design moments
2.2.6. Flexural reinforcement requirements
3. Design of Interior, Edge, and Corner Columns
3.1. Determination of factored loads
3.2. Column Capacity Diagram (AxialMoment Interaction Diagram)
4. TwoWay Slab Shear Strength
4.1. OneWay (Beam action) Shear Strength
4.2. TwoWay (Punching) Shear Strength
5. TwoWay Slab Deflection Control (Serviceability Requirements)
5.1. Immediate (Instantaneous) Deflections
5.2. TimeDependent (LongTerm) Deflections (Δ_{lt})
7. Summary and Comparison of TwoWay Slab Design Results
8. Comparison of TwoWay Slab Analysis and Design Methods
Code
Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14)
Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 710)
International Code Council, 2012 International Building Code, Washington, D.C., 2012
Reference
Notes on ACI 31811 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association, Example 20.1
Concrete Floor Systems (Guide to Estimating and Economizing), Second Edition, 2002 David A. Fanella
Simplified Design of Reinforced Concrete Buildings, Fourth Edition, 2011 Mahmoud E. Kamara and Lawrence C. Novak
Design Data
FloortoFloor Height = 9 ft (provided by architectural drawings)
Superimposed Dead Load, SDL =20 psf for framed partitions, wood studs plaster 2 sides
ASCE/SEI 710 (Table C31)
Live Load, LL = 40 psf for Residential floors ASCE/SEI 710 (Table 41)
f_{c}’ = 4000 psi (for slabs)
f_{c}’ = 6000 psi (for columns)
f_{y} = 60,000 psi
Required fire resistance rating = 2 hours
Solution
a. Slab minimum thickness  Deflection ACI 31814 (8.3.1.1)
In this example deflection will be calculated and checked to satisfy project deflection limits. Minimum member thickness and depths from ACI 31814 will be used for preliminary sizing.
Using ACI 31814 minimum slab thickness for twoway construction without interior beams in Table 8.3.1.1.
Exterior Panels: in. ACI 31814 (Table 8.3.1.1)
But not less than 5 in. ACI 31814 (8.3.1.1(a))
Interior Panels: in. 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 = 216 – 16 = 200 in.
Try 7 in. slab for all panels (selfweight = 87.5 psf)
b. Slab shear strength – one way shear
Evaluate the average effective depth (Figure 2):
Where:
c_{clear} = 3/4 in. for # 4 steel bar ACI 31814 (Table 20.6.1.3.1)
d_{b} = 0.5 in. for # 4 steel bar
Figure 2  TwoWay Flat Concrete Floor System
Factored dead load, psf
Factored live load, psf ACI 31814 (5.3.1)
Total factored load psf
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)
Tributary area for oneway shear is ft^{2}
kips
ACI 31814 (Eq. 22.5.5.1)
where for normal weight concrete
kips
Slab thickness of 7 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):
Tributary area for twoway shear is ft^{2}
kips
(For square interior column) ACI 31814 (Table 22.6.5.2(a))
kips
Slab thickness of 7 in. is adequate for twoway shear.
d. Column dimensions  axial load
Check the adequacy of column dimensions for axial load:
Tributary area for interior column is
kips
(For square interior column) ACI 31814 (22.4.2)
Column dimensions of 16 in. x 16 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 DDM, EFM, and spSlab software respectively.
Twoway slabs satisfying the limits in ACI 31814 (8.10.2) are permitted to be designed in accordance with the DDM.
There is a minimum of three continuous spans in each direction ACI 31814 (8.10.2.1)
Successive span lengths are equal ACI 31814 (8.10.2.2)
Longtoshort span ratio is 1.29 < 2 ACI 31814 (8.10.2.3)
Columns are not offset ACI 31814 (8.10.2.4)
Loads are uniformly distributed over the entire panel ACI 31814 (8.10.2.5)
Service livetodead load ratio of 0.37 < 2.0 ACI 31814 (8.10.2.6)
Slab system is without beams and this requirement is not applicable ACI 31814 (8.10.2.7)
Since all the criteria are met, Direct Design Method can be utilized.
a. Calculate the total factored static moment:
ftkips ACI 31814 (8.10.3.2)
b. Distribute the total factored moment, , in an interior and end span: ACI 31814 (8.10.4)
Table 1  Distribution of M_{o} along the span 

Location 
Total Design Strip Moment, 

Exterior Span 
Exterior Negative 
0.26 x M_{o} = 24.3 
Positive 
0.52 x M_{o} = 48.7 

Interior Negative 
0.70 x M_{o} = 65.5 

Interior Span 
Positive 
0.35 x M_{o} = 32.8 
c. Calculate the column strip moments. ACI 31814 (8.10.5)
That portion of negative and positive total design strip moments not resisted by column strips shall be proportionally assigned to corresponding two halfmiddle strips.
ACI 31814 (8.10.6.1)
Table 2  Lateral Distribution of the Total Design Strip Moment, M_{DS} 

Location 
Total Design Strip Moment, M_{DS} (ftkips) 
Column Strip Moment, (ftkips) 
Moment in Two Half Middle Strips, (ftkips) 

Exterior Span 
Exterior Negative^{*} 
24.3 
1.00 x M_{DS} = 24.3 
0.00 x M_{DS} = 0.0 
Positive 
48.7 
0.60 x M_{DS} = 29.2 
0.40 x M_{DS} = 19.5 

Interior Negative^{*} 
65.5 
0.75 x M_{DS} = 49.1 
0.25 x M_{DS} = 16.4 

Interior Span 
Positive 
32.8 
0.60 x M_{DS} = 19.7 
0.40 x M_{DS} = 13.1 
^{*} All negative moments are at face of support. 
a. Determine flexural reinforcement required for column and middle strips at all critical sections
The following calculation is for the exterior span exterior negative location of the column strip.
ftkips
Use average d_{avg} = 5.75 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.
Assumein.
Column strip width, in.
Middle strip width, in.
in^{2}
Recalculate ‘a’ for the actual A_{s} = 0.99 in.^{2}:
in
in
Therefore, the assumption that section is tensioncontrolled is valid.
in^{2}
Minin^{2} in^{2} ACI 31814 (24.4.3.2)
Maximum spacing in in ACI 31814 (8.7.2.2)
Provide 6  #4 bars with in^{2} and in
Based on the procedure outlined above, values for all span locations are given in Table 3.
Table 3  Required Slab Reinforcement for Flexure (DDM) 

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 
24.3 
84 
5.75 
0.96 
1.06 
6#4 
1.2 
Positive 
29 
84 
5.75 
1.15 
1.06 
6#4 
1.2 

Interior Negative 
49.6 
84 
5.75 
1.99 
1.06 
10#4 
2 

Middle Strip 
Exterior Negative 
0 
84 
5.75 
0 
1.06 
6#4 
1.2 
Positive 
19.7 
84 
5.75 
0.77 
1.06 
6#4 
1.2 

Interior Negative 
15.9 
84 
5.75 
0.62 
1.06 
6#4 
1.2 

Interior Span 

Column Strip 
Positive 
19.7 
84 
5.75 
0.77 
1.06 
6#4 
1.2 
Middle Strip 
Positive 
13.1 
84 
5.75 
0.51 
1.06 
6#4 
1.2 
b. Calculate additional slab reinforcement at columns for moment transfer between slab and column
The factored slab moment resisted by the column () 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 ACI 31814 (8.4.2.3.1)
Where
ACI 31814 (8.4.2.3.2)
Dimension of the critical section measured in the direction of the span for which moments are determined in ACI 318, Chapter 8 (see Figure 5).
Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 5).
= Effective slab width = ACI 31814 (8.4.2.3.3)
Figure 5 – Critical Shear Perimeters for Columns
Table 4  Additional Slab Reinforcement required for moment transfer between slab and column (DDM) 

Span Location 
M_{u}^{*} (ftkips) 
γ_{f} 
γ_{f} M_{u} (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 
24.3 
0.62 
15.1 
37 
5.75 
0.6 
0.53 
1#4 
Interior Negative 
0.0 
0.60 
0.0 
37 
5.75 
0.0 
0.97 
 

*M_{u} is taken at the centerline of the support in Equivalent Frame Method solution. 
a. Interior columns:
ACI 31814 (8.10.7.2)
ftkips
With the same column size and length above and below the slab,
ftkips
b. Exterior Columns:
Total exterior negative moment from slab must be transferred directly to the column: ftkips. With the same column size and length above and below the slab,
ftkips
The moments determined above are combined with the factored axial loads (for each story) for design of column sections as shown later in this 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:
1) Horizontal slabbeam strip, including any beams spanning in the direction of the frame. Different values of moment of inertia along the axis of slabbeams should be taken into account where the gross moment of inertia at any cross section outside of joints or column capitals shall be taken, and the moment of inertia of the slabbeam at the face of the column, bracket or capital divide by the quantity (1c_{2}/l_{2})^{2} shall be assumed for the calculation of the moment of inertia of slabbeams from the center of the column to the face of the column, bracket or capital. ACI 31814 (8.11.3)
2) Columns or other vertical supporting members, extending above and below the slab. Different values of moment of inertia along the axis of columns should be taken into account where the moment of inertia of columns from top and bottom of the slabbeam at a joint shall be assumed to be infinite, and the gross cross section of the concrete is permitted to be used to determine the moment of inertia of columns at any cross section outside of joints or column capitals. ACI 31814 (8.11.4)
3) Elements of the structure (Torsional members) that provide moment transfer between the horizontal and vertical members. These elements shall be assumed to have a constant cross section throughout their length consisting of the greatest of the following: (1) portion of slab having a width equal to that of the column, bracket, or capital in the direction of the span for which moments are being determined, (2) portion of slab specified in (1) plus that part of the transverse beam above and below the slab for monolithic or fully composite construction, (3) the transverse beam includes that portion of slab on each side of the beam extending a distance equal to the projection of the beam above or below the slab, whichever is greater, but not greater than four times the slab thickness. ACI 31814 (8.11.5)
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 , 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, .
,
For, stiffness factors, PCA Notes on ACI 31811 (Table A1)
Thus, PCA Notes on ACI 31811 (Table A1)
in.lb
where, in^{4}
psi ACI 31814 (19.2.2.1.a)
Carryover factor COF PCA Notes on ACI 31811 (Table A1)
Fixedend moment FEM PCA Notes on ACI 31811 (Table A1)
b. Flexural stiffness of column members at both ends, .
Referring to Table A7, Appendix 20A, in.,in.,
Thus, by interpolation.
PCA Notes on ACI 31811 (Table A7)
in.lb
Where in.
psi ACI 31814 (19.2.2.1.a)
ftin.
c. Torsional stiffness of torsional members, .
ACI 31814 (R.8.11.5)
in.lb
Where ACI 31814 (Eq. 8.10.5.2b)
in^{4}.
in., and ftin.
d. Equivalent column stiffness.
in.lb
Whereis for two torsional members one on each side of the column, andis for the upper and lower columns at the slabbeam joint of an intermediate floor.
e. Slabbeam joint distribution factors, DF.
At exterior joint,
At interior joint,
COF for slabbeam
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).
Factored dead load psf
Factored live load psf
Factored load psf
FEM’s for slabbeams PCA Notes on ACI 31811 (Table A1)
ftkips
b. Moment distribution. Computations are shown in Table 5. Counterclockwise rotational moments acting on the member ends are taken as positive. Positive span moments are determined from the following equation:
(midspan)
Where 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:
ftkips
Positive moment span 23: ftkips 
Table 5 – Moment Distribution for Equivalent Frame 



Joint 
1 
2 
3 
4 

Member 
12 
21 
23 
32 
34 
43 
DF 
0.389 
0.280 
0.280 
0.280 
0.280 
0.389 
COF 
0.509 
0.509 
0.509 
0.509 
0.509 
0.509 
FEM 
+73.8 
73.8 
+73.8 
73.8 
+73.8 
73.8 
Dist CO Dist CO Dist CO Dist CO Dist 
28.7 0.0 0.0 2.1 0.8 0.3 0.1 0.1 0.0 
0.0 14.6 4.1 0.0 0.6 0.4 0.2 0.1 0.0 
0.0 0.0 4.1 2.1 0.6 0.3 0.2 0.1 0.0 
0.0 0.0 4.1 2.1 0.6 0.3 0.2 0.1 0.0 
0.0 14.6 4.1 0.0 0.6 0.4 .02 0.1 0.0 
28.7 0.0 0.0 2.1 0.8 0.3 0.1 0.1 0.0 
Neg. M 
46.6 
84.0 
76.2 
76.2 
84.0 
46.6 
M at midspan 
44.1 
33.2 
44.1 
Positive and negative factored moments for the slab system in the direction of analysis are plotted in Figure 9. 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 from the centers of supports. ACI 31814 (8.11.6.1)
ftft (use face of support location)
Figure 9  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 value obtained from the following equation:
ftkips ACI 31814 (Eq. 8.10.3.2)
End spans: ftkips
Interior span: ftkips
The total design moments from the Equivalent Frame Method yield a static moment equal to that given by the Direct Design Method and no appreciable reduction can be realized.
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 6.
Table 6  Distribution of factored moments 


Slabbeam Strip 
Column Strip 
Middle Strip 

Moment 
Percent 
Moment 
Percent 
Moment 

End Span 
Exterior Negative 
32.3 
100 
32.3 
0 
0 
Positive 
44.1 
60 
26.5 
40 
17.7 

Interior Negative 
67 
75 
50.3 
25 
16.7 

Interior Span 
Negative 
60.8 
75 
45.6 
25 
15.2 
Positive 
33.2 
60 
19.9 
40 
13.2 
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.
ftkips
Use average d_{avg} = 5.75 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.
Assumein.
Column strip width, in.
Middle strip width, in.
in.^{2}
Recalculate ‘a’ for the actual A_{s} = 1.31 in.^{2}: in.
in.
Therefore, the assumption that section is tensioncontrolled is valid.
in.^{2}
Minin^{2} in.^{2} ACI 31814 (24.4.3.2)
Maximum spacing in. in. ACI 31814 (8.7.2.2)
Provide 7  #4 bars with A_{s} = 1.40 in.^{2} and s = 84/7 = 12 in.
Based on the procedure outlined above, values for all span locations are given in Table 7.
Table 7  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 
32.3 
84 
5.75 
1.28 
1.06 
7#4 
1.4 
Positive 
26.5 
84 
5.75 
1.04 
1.06 
6#4 
1.2 

Interior Negative 
50.3 
84 
5.75 
2.02 
1.06 
11#4 
2.2 

Middle Strip 
Exterior Negative 
0 
84 
5.75 
0 
1.06 
6#4 
1.2 
Positive 
17.7 
84 
5.75 
0.69 
1.06 
6#4 
1.2 

Interior Negative 
16.7 
84 
5.75 
0.65 
1.06 
6#4 
1.2 

Interior Span 

Column Strip 
Positive 
19.9 
84 
5.75 
0.78 
1.06 
6#4 
1.2 
Middle Strip 
Positive 
13.2 
84 
5.75 
0.51 
1.06 
6#4 
1.2 
b. Calculate additional slab reinforcement at columns for moment transfer between slab and column by flexure
The factored slab moment resisted by the column () 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 ACI 31814 (8.4.2.3.1)
Where
ACI 31814 (8.4.2.3.2)
Dimension of the critical section measured in the direction of the span for which moments are determined in ACI 318, Chapter 8 (see Figure 5).
Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 5).
= Effective slab width = ACI 31814 (8.4.2.3.3)
Table 8  Additional Slab Reinforcement required for moment transfer between slab and column (EFM) 

Span Location 
M_{u}^{*} (ftkips) 
γ_{f} 
γ_{f} M_{u} (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 
46.6 
0.60 
28.9 
37 
5.75 
1.17 
0.62 
3#4 
Interior Negative 
7.8 
0.60 
4.7 
37 
5.75 
0.18 
0.97 
 

*M_{u} 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 9, the unbalanced moment at joints 1 and 2 are:
Joint 1= +46.6 ftkips
Joint 2= 84.0 + 76.2 = 7.8 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 10a.
Figure 10a  Column Moments (Unbalanced Moments from SlabBeam)
In summary:
M_{col,Exterior}= 22.08 ftkips
M_{col,Interior} = 3.66 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. Figure 10b shows the moment diagrams in the longitudinal and transverse direction for the interior and exterior equivalent frames. Following the previous procedure, the moment values at the face of interior, exterior, and corner columns from the unbalanced moment values can be obtained. These values are shown in the following table.
Figure 10b – Moment Diagrams (kipsft)
M_{u} 
Column number (See Figure 10b) 

1 
2 
3 
4 

M_{ux} 
3.66 
22.08 
2.04 
12.39 
M_{uy} 
2.23 
1.28 
12.49 
6.79 
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 (Column #1):
Assume 4 story building
Tributary area for interior column is
kips
M_{u,x} = 3.66 ftkips (see the previous Table)
M_{u,y} = 2.23 ftkips (see the previous Table)
Edge (Exterior) Column (Column #2):
Tributary area for interior column is
kips
M_{u,x} = 22.08 ftkips (see the previous Table)
M_{u,y} = 1.28 ftkips (see the previous Table)
Edge (Exterior) Column (Column #3):
Tributary area for interior column is
kips
M_{u,x} = 2.04 ftkips (see the previous Table)
M_{u,y} = 12.49 ftkips (see the previous Table)
Corner Column (Column #4):
Tributary area for interior column is
kips
M_{u,x} = 12.39 ftkips (see the previous Table)
M_{u,y} = 6.79 ftkips (see the previous Table)
The factored loads are then input into spColumn to construct the axial load – moment interaction diagram.
Interior Column (Column #1):
Edge Column (Column #2):
Edge Column (Column #3):
Corner Column (Column #4):
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. Figure 11 shows the factored shear forces (V_{u}) at the critical sections around each column. 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)
for normal weight concrete
kips
Because at all the critical sections, the slab has adequate oneway shear strength.
Figure 11 – Oneway shear at critical sections (at distance d from the face of the supporting column)
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 5.
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).
kips
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.
kipsft
For the exterior column in Figure 5, the location of the centroidal axis zz is:
in.
The polar moment J_{c} of the shear perimeter is:
in.^{4}
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the exterior column:
in.
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
psi
ACI 31814 (Table 22.6.5.2)
psi psi
psi
Since at the critical section, the slab has adequate twoway shear strength at this joint.
b. Interior column:
kips
kipsft
For the interior column in Figure 5, the location of the centroidal axis zz is:
in.
The polar moment J_{c} of the shear perimeter is:
in.^{4}
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the interior column:
in.
ACI 31814 (R.8.4.4.2.3)
psi
ACI 31814 (Table 22.6.5.2)
psi psi
psi
Since at the critical section, the slab has adequate twoway shear strength at this joint.
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.
kips
kipsft
For the corner column in Figure 5, the location of the centroidal axis zz is:
in.
The polar moment J_{c} of the shear perimeter is:
in.^{4}
ACI 31814 (Eq. 8.4.4.2.2)
Where:
ACI 31814 (8.4.2.3.2)
The length of the critical perimeter for the exterior column:
in.
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
psi
ACI 31814 (Table 22.6.5.2)
psi = 253 psi
psi
Since at the critical section, the slab has adequate twoway shear strength at this joint.
Since the slab thickness was selected based on the minimum slab thickness tables in ACI 31814, the deflection calculations are not required. However, the calculations of immediate and timedependent deflections are covered in this section for illustration and 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 12.
Figure 12 – Maximum Moments for the Three Service Load Levels
M_{cr} = cracking moment.
ACI 31814 (Eq. 24.2.3.5b)
f_{r} = Modulus of rapture of concrete.
ACI 31814 (Eq. 19.2.3.1)
I_{g} = Moment of inertia of the gross uncracked concrete section
I_{cr} = moment of inertia of the cracked section transformed to concrete. PCA Notes on ACI 31811 (9.5.2.2)
The calculations shown below are for the design strip (frame strip). The values of these parameters for column and middle strips are shown in Table 9.
As calculated previously, the exterior span frame strip near the interior support is reinforced with 17 #4 bars located at 1.25 in. along the section from the top of the slab. Figure 13 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete.
Figure 13 – Cracked Transformed Section
E_{cs} = Modulus of elasticity of slab concrete.
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)
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, Ie, shall be calculated by Eq. (24.2.3.5a) unless obtained by a more comprehensive analysis.
Ie shall be permitted to be taken as the value obtained from Eq. (24.2.3.5a) at midspan for simple and continuous spans, and at the support for cantilevers. ACI 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 exterior span (span with one end 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. The averaged effective moment of inertia (I_{e,avg}) is given by:
PCA Notes on ACI 31811 (9.5.2.4(1))
Where:
For the interior span (span with both ends continuous) with service load level (D+LL_{full}):
ACI 31814 (24.2.3.5a)
The averaged effective moment of inertia (I_{e,avg}) is given by:
PCA Notes on ACI 31811 (9.5.2.4(2))
Where:
Table 9 provides a summary of the required parameters and calculated values needed for deflections for exterior and interior equivalent frame. It also provides a summary of the same values for column strip and middle strip to facilitate calculation of panel deflection.
Table 9 – Averaged Effective Moment of Inertia Calculations 

For Frame Strip 

Span 
zone 
I_{g},

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

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

Ext 
Left 
4802 
499 
26.10 
26.10 
35.78 
54.23 
4802 
4802 
4802 
4802 
4802 
4554 
Midspan 
465 
24.95 
24.95 
34.25 
4802 
4802 
4802 

Right 
629 
46.76 
46.76 
64.17 
4802 
4802 
3148 

Int 
Left 
629 
42.47 
42.47 
58.27 
4802 
4802 
3993 
4802 
4802 
4559 

Mid 
465 
18.47 
18.47 
25.34 
4802 
4802 
4802 

Right 
629 
42.47 
42.47 
58.27 
4802 
4802 
3993 
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 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 14 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 14 – Deflection Computation for a rectangular Panel
To calculate each term of the previous equation, the following procedure should be used. Figure 15 shows the procedure of calculating the term Δ_{cx}. same procedure can be used to find the other terms.
Figure 15 –Δ_{cx }calculation procedure
For exterior 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 9 = 4802 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 11)
Where 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 6 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 = 2401 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 12)
Where:
K_{ec} = effective column stiffness = 553.7 x 10^{6} in.lb (calculated previously).
PCA Notes on ACI 31811 (9.5.3.4 Eq. 14)
Where:
Where
= rotation of the span right support.
= Net frame strip negative moment of the right support.
in.
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}).
Assuming square panel, Δ_{cx }=_{ }Δ_{cy}= 0.076 in. and Δ_{mx }=_{ }Δ_{my}= 0.039 in.
The average Δ for the corner panel is calculated as follows:
Table 10 – 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 
0.0386 
0.0570 
0.00059 
0.00010 
0.0161 
0.0027 
0.076 
0.262 
0.0386 
0.0202 
0.00059 
0.00010 
0.0161 
0.0027 
0.039 

Int 
0.675 
0.0386 
0.0521 
0.00010 
0.00010 
0.0027 
0.0027 
0.047 
0.325 
0.0386 
0.0251 
0.00010 
0.00010 
0.0027 
0.0027 
0.020 

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 
0.0386 
0.0570 
0.00059 
0.00010 
0.0161 
0.0027 
0.076 
0.262 
0.0386 
0.0202 
0.00059 
0.00010 
0.0161 
0.0027 
0.039 

Int 
0.675 
0.0386 
0.0521 
0.00010 
0.00010 
0.0027 
0.0027 
0.047 
0.325 
0.0386 
0.0251 
0.00010 
0.00010 
0.0027 
0.0027 
0.020 

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 
0.0559 
0.0825 
0.00082 
0.00014 
0.02333 
0.00388 
0.110 
0.262 
0.0559 
0.0293 
0.00082 
0.00014 
0.02333 
0.00388 
0.017 

Int 
0.675 
0.0558 
0.0753 
0.00014 
0.00014 
0.00387 
0.00387 
0.068 
0.325 
0.0558 
0.0363 
0.00014 
0.00014 
0.00387 
0.00387 
0.009 

Span 
LDF 
LL 
LDF 
LL 

Δ_{cx}, in. 
Δ_{mx}, in. 

Ext 
0.738 
0.034 
0.262 
0.017 

Int 
0.675 
0.021 
0.325 
0.009 
From the analysis in the transverse direction the deflection values below are obtained:
For DL loading case:
For DL+LL_{sust} loading case:
For DL+LL_{full} loading case:
These values for the xdirection are shown in Table 10. Then, the total midpanel deflection is calculated by combining the contributions of the column and middle strip deflections from the X and Y directions:
PCA Notes on ACI 31811 (9.5.3.4 Eq. 8)
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 11 shows longterm deflections for the exterior and interior spans for the analysis in the xdirection, for column and middle strips.
Table 11  LongTerm Deflections 

Column Strip 

Span 
(Δ_{sust})_{Inst}, in 
λ_{Δ} 
Δ_{cs}, in 
(Δ_{total})_{Inst}, in 
(Δ_{total})_{lt}, in 
Exterior 
0.076 
2.000 
0.152 
0.110 
0.261 
Interior 
0.047 
2.000 
0.095 
0.068 
0.162 
Middle Strip 

Exterior 
0.039 
2.000 
0.078 
0.056 
0.134 
Interior 
0.020 
2.000 
0.041 
0.029 
0.069 
spSlab program utilizes the Equivalent Frame Method described and illustrated in details here for modeling, analysis and design of twoway concrete floor slab systems. 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 12  Comparison of Moments obtained from Hand (EFM) and spSlab Solution 

Hand (EFM) 
spSlab 

Exterior Span 

Column Strip 
Exterior Negative^{*} 
32.20 
32.66 
Positive 
26.50 
26.87 

Interior Negative^{*} 
50.30 
50.21 

Middle Strip 
Exterior Negative^{*} 
0.00 
0.20 
Positive 
17.70 
17.91 

Interior Negative^{*} 
16.70 
16.74 

Interior Span 

Column Strip 
Interior Negative^{*} 
45.60 
45.47 
Positive 
19.90 
19.90 

Middle Strip 
Interior Negative^{*} 
15.20 
15.16 
Positive 
13.20 
13.27 

^{* }negative moments are taken at the faces of supports 
Table 13  Comparison of Reinforcement Results with Hand and spSlab Solution 

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 
7#4 
7#4 
3#4 
3#4 
10#4 
10#4 
Positive 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 

Interior Negative 
11#4 
11#4 
 
 
11#4 
11#4 

Middle Strip 
Exterior Negative 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 
Positive 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 

Interior Negative 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 

Interior Span 

Column Strip 
Positive 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 
Middle Strip 
Positive 
6#4 
6#4 
n/a 
n/a 
6#4 
6#4 
*In the EFM, the unbalanced moment (M_{sc}, M_{unb}) at the support centerline is used to determine the value of the additional reinforcement as compared with DDM using the moments at the face of support. 
Table 14  Comparison of OneWay (Beam Action) Shear Check Results Using Hand and spSlab Solution 

Span 
V_{u} , kips 
x_{u}^{*} , in. 
φV_{c}, kips 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
23.3 
23.3 
1.15 
1.15 
91.64 
91.64 
Interior 
21.2 
21.2 
16.85 
16.85 
91.64 
91.64 
^{*} x_{u }calculated from the centerline of the left column for each span 
Table 15  Comparison of TwoWay (Punching) Shear Check Results Using Hand and spSlab Solution 

Support 
b_{1}, in. 
b_{2}, in. 
b_{o}, in. 
A_{c}, in.^{2} 
V_{u}, kips 
v_{u}, psi 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
18.88 
18.88 
21.75 
21.75 
59.5 
59.5 
342.00 
342.13 
21.65 
22.79 
63.3 
66.6 
Interior 
21.75 
21.75 
21.75 
21.75 
87 
87 
500.00 
500.25 
50.07 
50.07 
100.1 
100.1 


Support 
c_{AB}, in. 
J_{c}, in.^{4} 
γ_{v} 
M_{unb}, kipsft 
v_{u}, psi 
φv_{c, }psi 

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
5.99 
5.99 
14109 
14110 
0.383 
0.383 
37.8 
37.2 
137.0 
139.2 
189.7 
189.7 
Interior 
10.88 
10.88 
40131 
40131 
0.400 
0.400 
7.8 
7.7 
110.2 
110.1 
189.7 
189.7 
Table 16  Comparison of Immediate Deflection_{ }Results Using Hand and spSlab Solution (in.) 

Column Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.076 
0.072 
0.076 
0.072 
0.110 
0.103 
0.034 
0.031 
Interior 
0.047 
0.045 
0.047 
0.045 
0.068 
0.064 
0.021 
0.019 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.039 
0.038 
0.039 
0.038 
0.056 
0.054 
0.017 
0.016 
Interior 
0.020 
0.019 
0.020 
0.019 
0.029 
0.027 
0.009 
0.008 
Table 17  Comparison of TimeDependent Deflection_{ }Results Using Hand and spSlab Solution 

Column Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.0 
2.0 
0.152 
0.145 
0.261 
0.248 
Interior 
2.0 
2.0 
0.095 
0.089 
0.162 
0.153 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.0 
2.0 
0.078 
0.076 
0.134 
0.129 
Interior 
2.0 
2.0 
0.041 
0.038 
0.069 
0.065 
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.
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 flat plate 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}). 