TwoWay Concrete Floor Slab with Beams Design and Detailing
TwoWay Concrete Floor Slab with Beams Design and Detailing
Design the slab system shown in Figure 1 for an intermediate floor where the story height = 12 ft, column crosssectional dimensions = 18 in. x 18 in., edge beam dimensions = 14 in. x 27 in., interior beam dimensions = 14 in. x 20 in., and unfactored live load = 100 psf. The lateral loads are resisted by shear walls. Normal weight concrete with ultimate strength (fc’= 4000 psi) is used for all members, respectively. And reinforcement with Fy = 60,000 psi is used. Use the Equivalent Frame Method (EFM) and compare the results with spSlab model results.
Figure 1 – TwoWay Slab with Beams Spanning between all Supports
Contents
1. Preliminary Slab Thickness Sizing
2. TwoWay Slab Analysis and Design – Using Equivalent Frame Method (EFM)
2.1. Equivalent frame method limitations
2.2. Frame members of equivalent frame
2.3. Equivalent frame analysis
2.5. Distribution of design moments
2.6. Flexural reinforcement requirements
3. Design of Interior, Edge, and Corner Columns
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})
6. spSlab Software Program Model Solution
7. Summary and Comparison of Design Results
Code
Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14)
Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 710)
International Code Council, 2012 International Building Code, Washington, D.C., 2012
References
Notes on ACI 31811 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association.
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 = 12 ft (provided by architectural drawings)
Columns = 18 x 18 in.
Interior beams = 14 x 20 in.
Edge beams = 14 x 27 in.
w_{c} = 150 pcf
f_{c}’ = 4,000 psi
f_{y} = 60,000 psi
Live load, L_{o} = 100 psf (Office building) ASCE/SEI 710 (Table 41)
Solution
Control of deflections. ACI 31814 (8.3.1.2)
In lieu of detailed calculation for deflections, ACI 318 Code gives minimum thickness for twoway slab with beams spanning between supports on all sides in Table 8.3.1.2.
Beamtoslab flexural stiffness (relative stiffness) ratio (α_{f}) is computed as follows:
ACI 31814 (8.10.2.7b)
The moment of inertia for the effective beam and slab sections can be calculated as follows:
Then,
For Edge Beams:
The effective beam and slab sections for the computation of stiffness ratio for edge beam is shown in Figure 2.
For NorthSouth Edge Beam:
For EastWest Edge Beam:
For interior Beams:
The effective beam and slab sections for the computation of stiffness ratio for interior beam is shown in Figure 4.
For NorthSouth Interior Beam:
For EastWest Interior Beam:
Since α_{f} > 2.0 for all beams, the minimum slab thickness is given by:
ACI 31814 (8.3.1.2)
Where:
Use 6 in. slab thickness.
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. The solution per DDM can be found in the “TwoWay Plate Concrete Floor System Design” 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, K_{sb}.
PCA Notes on ACI 31811 (Table A1)
PCA Notes on ACI 31811 (Table A1)
Where I_{sb} is the moment of inertia of slabbeam section shown in Figure 6 and can be computed with the aid of Figure 7 as follows:
Carryover factor COF = 0.507 PCA Notes on ACI 31811 (Table A1)
PCA Notes on ACI 31811 (Table A1)
Figure 7 – Coefficient C_{t} for Gross Moment of Inertia of Flanged Sections
b. Flexural stiffness of column members at both ends, K_{c}.
Referring to Table A7, Appendix 20A:
For Interior Columns:
PCA Notes on ACI 31811 (Table A7)
For Exterior Columns:
PCA Notes on ACI 31811 (Table A7)
c. Torsional stiffness of torsional members, K_{t}.
ACI 31814 (R.8.11.5)
For Interior Columns:
Where:
ACI 31814 (Eq. 8.10.5.2b)
x_{1} = 14 in 
x_{2} = 6 in 
x_{1} = 14 in 
x_{2} = 6 in 
y_{1} = 14 in 
y_{2} = 42 in 
y_{1} = 20 in 
y_{2} = 14 in 
C_{1} = 4738 
C_{2} = 2,752 
C_{1} = 10,226 
C_{2} = 736 
∑C = 4738 + 2,752 = 7,490 in^{4} 
∑C = 10,226 + 736 x 2 = 11,698 in^{4} 

Figure 8 – Attached Torsional Member at Interior Column
For Exterior Columns:
Where:
ACI 31814 (Eq. 8.10.5.2b)
x_{1} = 14 in 
x_{2} = 6 in 
x_{1} = 14 in 
x_{2} = 6 in 
y_{1} = 21 in 
y_{2} = 35 in 
y_{1} = 27 in 
y_{2} = 21 in 
C_{1} = 11,141 
C_{2} = 2,248 
C_{1} = 16,628 
C_{2} = 1,240 
∑C = 11,141 + 2,248 = 13,389 in^{4} 
∑C = 16,628 + 1,240 = 17,868 in^{4} 

Figure 9 – Attached Torsional Member at Exterior Column
d. Increased torsional stiffness due to parallel beams, K_{ta}.
For Interior Columns:
Where:
For Exterior Columns:
e. Equivalent column stiffness K_{ec}.
Where ∑ K_{ta} 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.
For Interior Columns:
For Exterior Columns:
f. Slabbeam joint distribution factors, DF.
At exterior joint,
At interior joint,
COF for slabbeam =0.507
Determine negative and positive moments for the slabbeams using the moment distribution method.
With an unfactored livetodead load ratio:
The frame will be analyzed for five loading conditions with pattern loading and partial live load as allowed by ACI 31814 (6.4.3.3).
a. Factored load and FixedEnd Moments (FEM’s).
Where (9.3 psf = (14 x 14) / 144 x 150 / 22 is the weight of beam stem per foot divided by l_{2})
PCA Notes on ACI 31811 (Table A1)
b. Moment distribution.
Moment distribution for the five loading conditions is shown in Table 1. Counterclockwise rotational moments acting on 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 for loading (1):
Positive moment span 23 for loading (1):
Table 1 – Moment Distribution for Partial Frame (Transverse Direction) 

Joint 
1 
2 
3 
4 


Member 
12 
21 
23 
32 
34 
43 

DF 
0.394 
0.306 
0.306 
0.306 
0.306 
0.394 

COF 
0.507 
0.507 
0.507 
0.507 
0.507 
0.507 
Loading (1) All spans loaded with full factored live load 

FEM 
148.1 
148.1 
148.1 
148.1 
148.1 
148.1 

Dist 
58.4 
0 
0 
0 
0 
58.4 

CO 
0 
29.6 
0 
0 
29.6 
0 

Dist 
0 
9.1 
9.1 
9.1 
9.1 
0 

CO 
4.6 
0 
4.6 
4.6 
0 
4.6 

Dist 
1.8 
1.4 
1.4 
1.4 
1.4 
1.8 

CO 
0.7 
0.9 
0.7 
0.7 
0.9 
0.7 

Dist 
0.3 
0.5 
0.5 
0.5 
0.5 
0.3 

CO 
0.3 
0.1 
0.3 
0.3 
0.1 
0.3 

Dist 
0.1 
0.1 
0.1 
0.1 
0.1 
0.1 

M 
93.1 
167.6 
153.6 
153.6 
167.6 
93.1 

Midspan M 
89.5 
66.2 
89.5 
Loading (2) First and third spans loaded with 3/4 factored live load 

FEM 
125.4 
125.4 
57.3 
57.3 
125.4 
125.4 

Dist 
49.4 
20.8 
20.8 
20.8 
20.8 
49.4 

CO 
10.6 
25.1 
10.6 
10.6 
25.1 
10.6 

Dist 
4.2 
10.9 
10.9 
10.9 
10.9 
4.2 

CO 
5.5 
2.1 
5.5 
5.5 
2.1 
5.5 

Dist 
2.2 
2.3 
2.3 
2.3 
2.3 
2.2 

CO 
1.2 
1.1 
1.2 
1.2 
1.1 
1.2 

Dist 
0.5 
0.7 
0.7 
0.7 
0.7 
0.5 

CO 
0.4 
0.2 
0.4 
0.4 
0.2 
0.4 

Dist 
0.1 
0.2 
0.2 
0.2 
0.2 
0.1 

M 
86.7 
119 
74.5 
74.5 
119 
86.7 

Midspan M 
83.3 
10.6 
83.3 
Loading (3) Center span loaded with 3/4 factored live load 

FEM 
57.3 
57.3 
125.4 
125.4 
57.3 
57.3 

Dist 
22.6 
20.8 
20.8 
20.8 
20.8 
22.6 

CO 
10.6 
11.4 
10.6 
10.6 
11.4 
10.6 

Dist 
4.2 
0.3 
0.3 
0.3 
0.3 
4.2 

CO 
0.1 
2.1 
0.1 
0.1 
2.1 
0.1 

Dist 
0.1 
0.6 
0.6 
0.6 
0.6 
0.1 

CO 
0.3 
0 
0.3 
0.3 
0 
0.3 

Dist 
0.1 
0.1 
0.1 
0.1 
0.1 
0.1 

CO 
0 
0.1 
0 
0 
0.1 
0 

Dist 
0 
0 
0 
0 
0 
0 

M 
28.1 
87.7 
115 
115 
87.7 
28.1 

Midspan M 
27.2 
71.3 
27.2 
Loading (4) First span loaded with 3/4 factored live load and beamslab assumed fixed at support two spans away 

FEM 
125.4 
125.4 
57.3 
57.3 


Dist 
49.4 
20.8 
20.8 
0 

CO 
10.6 
25 
0 
10.6 

Dist 
4.2 
7.7 
7.7 
0 

CO 
3.9 
2.1 
0 
3.9 

Dist 
1.5 
0.6 
0.6 
0 

CO 
0.3 
0.8 
0 
0.3 

Dist 
0.1 
0.2 
0.2 
0 

CO 
0.1 
0.1 
0 
0.1 

Dist 
0 
0 
0 
0 

M 
85.1 
124.1 
86.6 
42.4 

Midspan M 
81.5 
20.6 
Loading (5) First and second spans loaded with 3/4 factored live load 

FEM 
125.4 
125.4 
125.4 
125.4 
57.3 
57.3 

Dist 
49.4 
0.0 
0.0 
20.8 
20.8 
22.6 

CO 
0.0 
25.1 
10.6 
0.0 
11.4 
10.6 

Dist 
0.0 
4.4 
4.4 
3.5 
3.5 
4.2 

CO 
2.2 
0.0 
1.8 
2.2 
2.1 
1.8 

Dist 
0.9 
0.5 
0.5 
0.0 
0.0 
0.7 

CO 
0.3 
0.4 
0.0 
0.3 
0.4 
0.0 

Dist 
0.1 
0.1 
0.1 
0.2 
0.2 
0.0 

CO 
0.1 
0.1 
0.1 
0.1 
0.0 
0.1 

Dist 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

M 
77.6 
146.0 
139.1 
105.7 
84.1 
29.5 

Midspan M 
74.3 
63.7 
28.3 



Max M^{} 
93.1 
167.7 
153.6 
153.6 
167.7 
93.1 
Max M^{+} 
89.4 
71.3 
89.4 
Positive and negative factored moments for the slab system in the direction of analysis are plotted in Figure 13. The negative design moments are taken at the faces of rectilinear supports but not at distances greater than from the centers of supports. ACI 31814 (8.11.6.1)
Figure 13 – Positive and Negative Design Moments for SlabBeam (All Spans Loaded with Full Factored Live Load Except as Noted)
a. Check whether the moments calculated above can take advantage of the reduction permitted by ACI 31814 (8.11.6.5):
Slab systems within the limitations of ACI 31814 (8.10.2) may have the resulting reduced in such proportion that the numerical sum of the positive and average negative moments not be greater than the total static moment M_{o} given by Equation 8.10.3.2 in the ACI 31814:
ACI 31814 (8.11.6.5)
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 22/17.5 = 1.26 < 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 1.33 < 2.0
ACI 31814 (8.10.2.5 and 6)
6. Check relative stiffness for slab panel: ACI 31814 (8.10.2.7)
Interior Panel:
O.K. ACI 31814 (Eq. 8.10.2.7a)
Interior Panel:
O.K. ACI 31814 (Eq. 8.10.2.7a)
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 = 183.7/188.8 = 0.973
Adjusted negative design moment = 117.6 × 0.973 = 114.3 ftkip
Adjusted positive design moment = 71.2 × 0.973 = 69.3 ftkip
M_{o} = 183.7 ftkip
b. Distribute factored moments to column and middle strips:
The negative and positive factored moments at critical sections may be distributed to the column strip and the two halfmiddle strips of the slabbeam according to the Direct Design Method (DDM) in 8.10, provided that Eq. 8.10.2.7(a) is satisfied. ACI 31814 (8.11.6.6)
Since the relative stiffness of beams are between 0.2 and 5.0 (see step 2.4.1.6), the moments can be distributed across slabbeams as specified in ACI 31814 (8.10.5 and 6) where:
Factored moments at critical sections are summarized in Table 2.
Table 2  Lateral distribution of factored moments 

Factored Moments 
Column Strip 
Moments in Two 

Percent* 
Moment 
Beam Strip Moment 
Column Strip Moment 

End 
Exterior Negative 
60.2 
75 
45.2 
38.4 
6.8 
15 
Positive 
89.4 
67 
59.9 
50.9 
9.0 
29.5 

Interior Negative 
128.4 
67 
86 
73.1 
12.9 
42.4 

Interior 
Negative 
117.6 
67 
78.8 
67.0 
11.8 
38.8 
Positive 
71.3 
67 
47.8 
40.6 
7.2 
23.5 

*Since α_{1}l_{2}/l_{1} > 1.0 beams must be proportioned to resist 85 percent of column strip per ACI 31814 (8.10.5.7) 

**That portion of the factored moment not resisted by the column strip is assigned to the two halfmiddle strips 
a. Determine flexural reinforcement required for strip moments
The flexural reinforcement calculation for the column strip of end span – interior negative location is provided below:
Assume tensioncontrolled section (φ = 0.9)
Column strip width, b = (17.5 x 12) / 2 = 91 in.
Use average d = 6 – 0.75 – 0.5/2 = 5 in.
in^{2}
Maximum spacing ACI 31814 (8.7.2.2)
Provide 8  #4 bars with A_{s} = 1.60 in.^{2} and s = 91/8 = 11.37 in. ≤ s_{max}
The flexural reinforcement calculation for the beam strip of end span – interior negative location is provided below:
Assume tensioncontrolled section (φ = 0.9)
Beam strip width, b = 14 in.
Use average d = 20 – 0.75 – 0.5/2 = 19 in.
Provide 5  #4 bars with A_{s} = 1.00 in.^{2}
All the values on Table 3 are calculated based on the procedure outlined above.
Table 3  Required Slab Reinforcement for Flexure [Equivalent Frame Method (EFM)] 

Span Location 
M_{u }(ftkip) 
b ^{*}

d ^{**}

A_{s }Req’d

Min A_{s}^{† †† }_{ }(in.^{2}) 
Reinforcement

A_{s }Prov.


End Span 

Beam Strip 
Exterior Negative 
38.4 
14 
19.00 
0.456 
0.608 
4  #4 
0.8 

Positive 
50.9 
14 
18.25 
0.634 
0.852 
5  #4 
1.0 

Interior Negative 
73.1 
14 
19.00 
0.881 
0.887 
5  #4 
1.0 

Column Strip 
Exterior Negative 
6.8 
91 
5.00 
0.304 
0.983 
8  #4 
1.6 

Positive 
9.0 
91 
5.00 
0.403 
0.983 
8  #4 
1.6 

Interior Negative 
12.9 
91 
5.00 
0.580 
0.983 
8  #4 
1.6 

Middle Strip 
Exterior Negative 
15.0 
159 
5.00 
0.672 
1.717 
14  #4 
2.8 

Positive 
29.5 
159 
5.00 
1.331 
1.717 
14  #4 
2.8 

Interior Negative 
42.4 
159 
5.00 
1.926 
1.717 
14  #4 
2.8 

Interior Span 

Beam Strip 
Positive 
40.6 
14 
18.25 
0.503 
0.671 
4  #4 
0.8 

Column Strip 
Positive 
7.2 
91 
5.00 
0.322 
0.983 
8  #4 
1.6 

Middle Strip 
Positive 
23.5 
159 
5.00 
1.057 
1.717 
14  #4 
2.8 

^{*} Column strip width, b = (17.5 × 12)/2  14 = 91 in. 

^{*} Middle strip width, b = 22*12(17.5*12)/2 = 159 in. 

^{*} Beam strip width, b = 14 in. 

^{**} Use average d = 6 – 0.75 – 0.5/2 = 5.00 in. for Column and Middle strips 

^{**} Use average d = 20  1.5  0.5/2 = 18.25 in. for Beam strip Positive moment regions 

^{**} Use average d = 20  0.75  0.5/2 = 19 in. for Beam strip Negative moment regions 

^{†} Min. A_{s} = 0.0018 × b × h = 0.0108 × b for Column and Middle strips ACI 31814 (7.6.1.1) 

^{†} Min. A_{s} = min (3(f_{c}')^0.5/f_{y}*b*d , 200/f_{y}*b*d) for Beam strip ACI 31814 (9.6.1.2) 

^{††} Min. A_{s} = 1.333 × As Req'd if As provided >= 1.333 × As Req'd for Beam strip ACI 31814 (9.6.1.3) 

s_{max} = 2 × h = 12 in. < 18 in. ACI 31814 (8.7.2.2) 
b. Calculate additional slab reinforcement at columns for moment transfer between slab and column by flexure
Portion of the unbalanced moment transferred by flexure is γ_{f} x M_{u}
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.
b_{2} = Dimension of the critical section b_{o} measured in the direction perpendicular to b_{1} in ACI 318, Chapter 8.
b_{o} = Perimeter of critical section for twoway shear in slabs and footings.
ACI 31814 (8.4.2.3.3)
For Exterior Column:
Figure 14 – Critical Shear Perimeters for Columns
Additional slab reinforcement at the exterior column is required.
Table 4  Additional Slab Reinforcement at columns for moment transfer between slab and column [Equivalent Frame Method (EFM)] 

Span Location 
Effective slab width, b_{b} (in.) 
d 
γ_{f} 
M_{u}^{*}

γ_{f}
M_{u} 
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 
36 
5 
0.614 
93.1 
57.14 
2.973 
1.187 
10#4 
Interior Negative 
36 
5 
0.600 
44.5 
26.70 
1.265 
1.387 
 

*M_{u} is taken at the centerline of the support in Equivalent Frame Method solution. 
b. Determine transverse reinforcement required for beam strip shear
The transverse reinforcement calculation for the beam strip of end span – exterior location is provided below.
Figure 15 – Shear at critical sections for the end span (at distance d from the face of the column)
The required shear at a distance d from the face of the supporting column V_{u_d}= 31.64 kips (Figure 15).
ACI 31814 (22.5.5.1)
∴ Stirrups are required.
Distance from the column face beyond which minimum reinforcement is required:
ACI 31814 (22.5.10.1)
O.K.
ACI 31814 (22.5.10.1)
ACI 31814 (22.5.10.5.3)
ACI 31814 (9.6.3.3)
ACI 31814 (9.7.6.2.2)
Select s_{provided} = 8 in. #4 stirrups with first stirrup located at distance 3 in. from the column face.
The distance where the shear is zero is calculated as follows:
The distance from support beyond which minimum reinforcement is required is calculated as follows:
The distance at which no shear reinforcement is required is calculated as follows:
All the values on Table 5 are calculated based on the procedure outlined above.
Table 5  Required Beam Reinforcement for Shear 

Span Location 
A_{v,min}/s 
A_{v,req'd}/s 
s_{req'd} 
s_{max} 
Reinforcement 
End Span 

Exterior 
0.0117 
0.0090 
34.28 
9.13 
8  #4 @ 8 in^{*} 
Interior 
0.0117 
0.0225 
17.76 
9.13 
10  #4 @ 8.6 in 
Interior Span 

Interior 
0.0117 
0.0158 
25.37 
9.13 
9  #4 @ 8.6 in 
^{*} Minimum transverse reinforcement governs 
The unbalanced moment from the slabbeams at the supports of the equivalent frame are distributed to the actual columns above and below the slabbeam in proportion to the relative stiffness of the actual columns. Referring to Fig. 9, the unbalanced moment at joints 1 and 2 are:
Joint 1 = +93.1 ftkip
Joint 2 = 119 + 74.5 = 44.5 ftkip
The stiffness and carryover factors of the actual columns and the distribution of the unbalanced moments to the exterior and interior columns are shown in Fig 9.
Figure 16  Column Moments (Unbalanced Moments from SlabBeam)
In summary:
Design moment in exterior column = 55.81 ftkip
Design moment in interior column = 24.91 ftkip
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. A detailed analysis to obtain the moment values at the face of interior, exterior, and corner columns from the unbalanced moment values can be found in the “TwoWay Flat Plate Concrete Floor Slab Design” example.
The design of interior, edge, and corner columns is explained in the “TwoWay Flat Plate Concrete Floor Slab Design” example.
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.
Oneway shear is critical at a distance d from the face of the column. Figure 17 shows the V_{u} at the critical sections around each column. Since there is no 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)
λ = 1 for normal weight concrete
Because φV_{c} > V_{u} at all the critical sections, the slab is o.k. in oneway shear.
Figure 17 – Oneway shear at critical sections (at distance d from the face of the supporting column)
Twoway shear is critical on a rectangular section located at d_{slab}/2 away from the face of the column. The factored shear force V_{u} in the critical section is calculated 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.
The factored unbalanced moment used for shear transfer, M_{unb}, is calculated 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:
For the exterior column in Figure 18, 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:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
O.K.
For the interior column:
For the interior column in Figure 19, 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:
ACI 31814 (Table 22.6.5.2)
O.K.
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 20.
Figure 20 – Maximum Moments for the Three Service Load Levels
For positive moment (midspan) section of the exterior span:
ACI 31814 (Eq. 24.2.3.5b)
ACI 31814 (Eq. 19.2.3.1)
y_{t} = Distance from centroidal axis of gross section, neglecting reinforcement, to tension face, in.
Figure 21 – I_{g} calculations for slab section near support
PCA Notes on ACI 31811 (9.5.2.2)
As calculated previously, the positive reinforcement for the end span frame strip is 22 #4 bars located at 1.0 in. along the slab section from the bottom of the slab and 4 #4 bars located at 1.75 in. along the beam section from the bottom of the beam. Five of the slab section bars are not continuous and will be excluded from the calculation of I_{cr}. Figure 22 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete at midspan.
Figure 22 – Cracked Transformed Section (positive moment section)
ACI 31814 (19.2.2.1.a)
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 27 #4 bars located at 1.0 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 23 – 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 24 – Cracked Transformed Section (interior negative moment section for end span)
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 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 6 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 6 – Averaged Effective Moment of Inertia Calculations 

For Frame Strip 

Span 
zone 
I_{g}, in.^{4} 
I_{cr}, in.^{4} 
M_{a}, ftkip 
M_{cr}, kft 
I_{e}, in.^{4} 
I_{e,avg}, in.^{4} 

D 
D + LL_{Sus} 
D + L_{full} 
D 
D + LL_{Sus} 
D + L_{full} 
D 
D + LL_{Sus} 
D + L_{full} 

Ext 
Left 
9333 
7147 
30.61 
30.61 
66.92 
36.89 
9333 
9333 
7513 
22761 
22761 
22693 
Midspan 
25395 
2282 
27.19 
27.19 
59.43 
63.14 
25395 
25395 
25395 

Right 
9333 
7331 
58.35 
58.35 
127.56 
36.89 
7837 
7837 
7380 

Int 
Left 
9333 
7331 
52.93 
52.93 
115.73 
36.89 
8009 
8009 
7396 
20179 
20179 
19995 
Mid 
25395 
1553 
18.06 
18.06 
44.57 
63.14 
25395 
25395 
25395 

Right 
9333 
7331 
52.93 
52.93 
115.73 
36.89 
8009 
8009 
7396 
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 25 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 25 – Deflection Computation for a rectangular Panel
To calculate each term of the previous equation, the following procedure should be used. Figure 26 shows the procedure of calculating the term Δ_{cx}. same procedure can be used to find the other terms.
Figure 26 –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 6 = 22761 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}Æ), interior negative region (LDF_{R}Æ), and positive region (LDF_{L}^{＋}) are 0.75, 0.67, and 0.67, 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 T section) = 20040 in.^{4}
I_{frame,g} = The gross moment of inertia (I_{g}) for the frame strip (for T section) = 25395 in.^{4}
PCA Notes on ACI 31811 (9.5.3.4 Eq. 12)
Where:
K_{ec} = effective column stiffness for exterior column.
= 764 x E_{c} = 2929 x 10^{6} in.lb (calculated previously).
PCA Notes on ACI 31811 (9.5.3.4 Eq. 14)
Where:
Where
K_{ec} = effective column stiffness for interior column.
= 631 x E_{c} = 2419 x 10^{6} in.lb (calculated previously).
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}= 0.009 in. and Δ_{mx }=_{ }Δ_{my}= 0.021 in.
The average Δ for the corner panel is calculated as follows:
Table 7  Instantaneous Deflections 

Column Strip 
Middle Strip 

Span 
LDF 
D 
LDF 
D 

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

Ext 
0.69 
0.0063 
0.0055 
0.00012 
0.00003 
0.0033 
0.0007 
0.009 
0.31 
0.0063 
0.0172 
0.00012 
0.00003 
0.0033 
0.0007 
0.021 

Int 
0.67 
0.0071 
0.0060 
0.00003 
0.00003 
0.0008 
0.0008 
0.004 
0.33 
0.0071 
0.0207 
0.00003 
0.00003 
0.0008 
0.0008 
0.019 

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

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

Ext 
0.69 
0.0063 
0.0055 
0.00012 
0.00003 
0.0033 
0.0007 
0.009 
0.31 
0.00627 
0.01724 
0.00012 
0.00003 
0.00330 
0.00072 
0.021 

Int 
0.67 
0.0071 
0.0060 
0.00003 
0.00003 
0.0008 
0.0008 
0.004 
0.33 
0.00707 
0.02069 
0.00003 
0.00003 
0.00081 
0.00081 
0.019 

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

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

Ext 
0.69 
0.0137 
0.0120 
0.00027 
0.00006 
0.0072 
0.0016 
0.021 
0.31 
0.01374 
0.03780 
0.00027 
0.00006 
0.00724 
0.00158 
0.047 

Int 
0.67 
0.0156 
0.0132 
0.00006 
0.00006 
0.0018 
0.0018 
0.010 
0.33 
0.01559 
0.04566 
0.00006 
0.00006 
0.00179 
0.00179 
0.042 

Span 
LDF 
LL 
LDF 
LL 

Δ_{cx}, 
Δ_{mx}, 

Ext 
0.69 
0.011 
0.31 
0.025 

Int 
0.67 
0.005 
0.33 
0.023 
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 
0.009 
2.000 
0.019 
0.021 
0.040 
Interior 
0.004 
2.000 
0.009 
0.010 
0.018 
Middle Strip 

Exterior 
0.021 
2.000 
0.043 
0.047 
0.089 
Interior 
0.019 
2.000 
0.038 
0.042 
0.080 
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 will be provided from the spSlab model in a future revision to this document. For a sample output refer to “TwoWay Flat Plate Concrete Floor Slab Design” example.
Table 9  Comparison of Moments obtained from Hand (EFM) and spSlab Solution (ftkip) 


Hand (EFM) 
spSlab 

Exterior Span 

Beam Strip 
Exterior Negative^{*} 
38.4 
40 
Positive 
50.9 
48.17 

Interior Negative^{*} 
73.1 
80.63 

Column Strip 
Exterior Negative^{*} 
6.8 
7.06 
Positive 
9 
8.5 

Interior Negative^{*} 
12.9 
14.23 

Middle Strip 
Exterior Negative^{*} 
15 
15.36 
Positive 
29.5 
27.55 

Interior Negative^{*} 
42.4 
46.12 

Interior Span 

Beam Strip 
Interior Negative^{*} 
67 
73.15 
Positive 
40.6 
36.65 

Column Strip 
Interior Negative^{*} 
11.8 
12.91 
Positive 
7.2 
6.47 

Middle Strip 
Interior Negative^{*} 
38.8 
41.84 
Positive 
23.5 
20.96 

^{* }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 

Beam Strip 
Exterior Negative 
4  #4 
4  #4 
n/a 
n/a 
4  #4 
4  #4 
Positive 
5  #4 
4  #4 
n/a 
n/a 
5  #4 
4  #4 

Interior Negative 
5  #4 
5  #4 
 
 
5  #4 
5  #4 

Column Strip 
Exterior Negative 
8  #4 
8  #4 
10  #4 
12  #4 
18  #4 
20  #4 
Positive 
8  #4 
8  #4 
n/a 
n/a 
8  #4 
8  #4 

Interior Negative 
8  #4 
8  #4 
 
 
8  #4 
8  #4 

Middle Strip 
Exterior Negative 
14  #4 
14  #4 
n/a 
n/a 
14  #4 
14  #4 
Positive 
14  #4 
14  #4 
n/a 
n/a 
14  #4 
14  #4 

Interior Negative 
14  #4 
14  #4 
n/a 
n/a 
14  #4 
14  #4 

Interior Span 

Beam Strip 
Positive 
4  #4 
4  #4 
n/a 
n/a 
4  #4 
4  #4 
Column Strip 
Positive 
8  #4 
8  #4 
n/a 
n/a 
8  #4 
8  #4 
Middle Strip 
Positive 
14  #4 
14  #4 
n/a 
n/a 
14  #4 
14  #4 
Table 11  Comparison of Beam Shear Reinforcement Results 

Span Location 
Reinforcement Provided 

Hand 
spSlab 

End Span 

Exterior 
8  #4 @ 8 in 
8  #4 @ 8 in 
Interior 
10  #4 @ 8.6 in 
10  #4 @ 8.6 in 
Interior Span 

Interior 
9  #4 @ 8.6 in 
10  #4 @ 8.6 in 
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 
20.5 
20.5 
23.0 
44.0 
64 
64 
43.56 
48.47 
9.09 
9.09 
Interior 
23.0 
23.0 
23.0 
23.0 
92 
92 
104.76 
104.50 
11.50 
11.50 
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 
95338 
95338 
0.386 
0.313 
84.37 
83.49 
76.8 
73.8 
189.7 
189.7 
Interior 
114993 
114990 
0.400 
0.400 
14.10 
16.77 
91.0 
92.1 
189.7 
189.7 
Table 13  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 
0.009 
0.010 
0.009 
0.010 
0.021 
0.023 
0.011 
0.012 
Interior 
0.004 
0.005 
0.004 
0.005 
0.010 
0.011 
0.005 
0.006 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
0.021 
0.022 
0.021 
0.022 
0.047 
0.049 
0.025 
0.026 
Interior 
0.019 
0.020 
0.019 
0.020 
0.042 
0.044 
0.023 
0.024 
Table 14  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 
0.019 
0.021 
0.040 
0.043 
Interior 
2.0 
2.0 
0.009 
0.010 
0.018 
0.020 
Middle Strip 

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

Hand 
spSlab 
Hand 
spSlab 
Hand 
spSlab 

Exterior 
2.0 
2.0 
0.043 
0.044 
0.089 
0.093 
Interior 
2.0 
2.0 
0.038 
0.040 
0.080 
0.084 
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 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 
ž 

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

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

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 
ž 

8.10.2.5 
All loads shall be due to gravity only 
ž 

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


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

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. 
ž 

8.7.4.2 
Structural integrity steel detailing 
ž 
ž 
ž 
8.5.4 
Openings in slab systems 
ž 
ž 
ž 
8.2.2 
Concentrated loads 
Not permitted 
ž 
ž 
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}). 