Two-Way Concrete Floor Slab with Beams Design and Detailing

 

Two-Way 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 cross-sectional 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 1Two-Way Slab with Beams Spanning  between all Supports

 

 

 

 

 

 


Contents

1.   Preliminary Slab Thickness Sizing. 1

2.   Two-Way Slab Analysis and Design – Using Equivalent Frame Method (EFM) 4

2.1. Equivalent frame method limitations. 5

2.2. Frame members of equivalent frame. 5

2.3. Equivalent frame analysis. 11

2.4. Design moments. 14

2.5. Distribution of design moments. 15

2.6. Flexural reinforcement requirements. 17

2.7. Column design moments. 24

3.   Design of Interior, Edge, and Corner Columns. 25

4.   Two-Way Slab Shear Strength. 25

4.1. One-Way (Beam action) Shear Strength. 25

4.2. Two-Way (Punching) Shear Strength. 26

5.   Two-Way Slab Deflection Control (Serviceability Requirements) 29

5.1. Immediate (Instantaneous) Deflections. 29

5.2. Time-Dependent (Long-Term) Deflections (Δlt) 40

6.   spSlab Software Program Model Solution. 41

7.   Summary and Comparison of Design Results. 63

8.   Conclusions & Observations. 66

 

 


Code

Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)

Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 7-10)

International Code Council, 2012 International Building Code, Washington, D.C., 2012

References

Notes on ACI 318-11 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

Floor-to-Floor Height = 12 ft (provided by architectural drawings)

Columns = 18 x 18 in.

Interior beams = 14 x 20 in.

Edge beams = 14 x 27 in.

wc = 150 pcf

 fc = 4,000 psi

fy = 60,000 psi

Live load, Lo = 100 psf (Office building)                                                                               ASCE/SEI 7-10 (Table 4-1)

Solution

1.       Preliminary Slab Thickness Sizing

Control of deflections.                                                                                                                        ACI 318-14 (8.3.1.2)

In lieu of detailed calculation for deflections, ACI 318 Code gives minimum thickness for two-way slab with beams spanning between supports on all sides in Table 8.3.1.2.

Beam-to-slab flexural stiffness (relative stiffness) ratio (αf) is computed as follows:

 

                                                                                                                               ACI 318-14 (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 North-South Edge Beam:

 

Figure 2 – Effective Beam and Slab Sections (Edge Beam)

 

For East-West Edge Beam:

 

 

 

Figure 3 – Beam Stiffness (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 North-South Interior Beam:

 

Figure 4 – Effective Beam and Slab Sections (Interior Beam)

For East-West Interior Beam:

 

 

 

Figure 5 – Beam Stiffness (Interior Beam)

 

 

Since αf > 2.0 for all beams, the minimum slab thickness is given by:

 

                                                                                          ACI 318-14 (8.3.1.2)

Where:

 

Use 6 in. slab thickness.

 

2.       Two-Way Slab Analysis and Design – Using Equivalent Frame Method (EFM)

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 two-systems from one-way systems is given by ACI 318-14 (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 “Two-Way 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 two-way 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 slab-beam strip, including any beams spanning in the direction of the frame. Different values of moment of inertia along the axis of slab-beams 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 slab-beam at the face of the column, bracket or capital divide by the quantity (1-c2/l2)2 shall be assumed for the calculation of the moment of inertia of slab-beams from the center of the column to the face of the column, bracket or capital.  ACI 318-14 (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 slab-beam 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 318-14 (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 318-14 (8.11.5)

2.1.     Equivalent frame method limitations

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 318-14 (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 318-14 (8.11.2.1)

Panels shall be rectangular, with a ratio of longer to shorter panel dimensions, measured center-to-center of supports, not to exceed 2.                                                                                                                                        ACI 318-14 (8.10.2.3)

2.2.     Frame members of equivalent frame

Determine moment distribution factors and fixed-end 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 fixed-end moment factors FEM for the slab-beams and column members are determined using the design aids tables at Appendix 20A of PCA Notes on ACI 318-11. These calculations are shown below.

a.     Flexural stiffness of slab-beams at both ends, Ksb.

                                     PCA Notes on ACI 318-11 (Table A1)

                                                           PCA Notes on ACI 318-11 (Table A1)

Figure 6 – Cross-Section of Slab-BeamWhere Isb is the moment of inertia of slab-beam section shown in Figure 6 and can be computed with the aid of Figure 7 as follows:

Carry-over factor COF = 0.507                                                                   PCA Notes on ACI 318-11 (Table A1)

                                                 PCA Notes on ACI 318-11 (Table A1)

Figure 7 – Coefficient Ct for Gross Moment of Inertia of Flanged Sections

 

b.    Flexural stiffness of column members at both ends, Kc.

Referring to Table A7, Appendix 20A:

For Interior Columns:

 

                                                                                                 PCA Notes on ACI 318-11 (Table A7)

For Exterior Columns:

 

 

 

                                                                                                 PCA Notes on ACI 318-11 (Table A7)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c.     Torsional stiffness of torsional members, Kt.

                                                                                                                    ACI 318-14 (R.8.11.5)

For Interior Columns:

 

Where:

                                                                                                     ACI 318-14 (Eq. 8.10.5.2b)

x1 = 14 in

x2 = 6 in

x1 = 14 in

x2 = 6 in

y1 = 14 in

y2 = 42 in

y1 = 20 in

y2 = 14 in

C1 = 4738

C2 = 2,752

C1 = 10,226

C2 = 736

∑C = 4738 + 2,752 = 7,490 in4

∑C = 10,226 + 736 x 2 = 11,698 in4

 

Figure 8 – Attached Torsional Member at Interior Column

 

 

 

For Exterior Columns:

 

Where:

                                                                                                     ACI 318-14 (Eq. 8.10.5.2b)

 

x1 = 14 in

x2 = 6 in

x1 = 14 in

x2 = 6 in

y1 = 21 in

y2 = 35 in

y1 = 27 in

y2 = 21 in

C1 = 11,141

C2 = 2,248

C1 = 16,628

C2 = 1,240

∑C = 11,141 + 2,248 = 13,389 in4

∑C = 16,628 + 1,240 = 17,868 in4

 

Figure 9 – Attached Torsional Member at Exterior Column

 

 

 

 

 

d.    Increased torsional stiffness due to parallel beams, Kta.

For Interior Columns:

Where:

For Exterior Columns:

Figure 10 – Slab-Beam in the Direction of Analysis

 

e.     Equivalent column stiffness Kec.

Where ∑ Kta is for two torsional members one on each side of the column, and ∑ Kc is for the upper and lower columns at the slab-beam joint of an intermediate floor.

For Interior Columns:

Figure 11 – Equivalent Column Stiffness

For Exterior Columns:

f.     Slab-beam joint distribution factors, DF.

At exterior joint,

 

 

 

 

At interior joint,

Figure 12 – Slab and Column Stiffness

COF for slab-beam =0.507

 

 

 

2.3.     Equivalent frame analysis

Determine negative and positive moments for the slab-beams using the moment distribution method.

With an unfactored live-to-dead load ratio:

The frame will be analyzed for five loading conditions with pattern loading and partial live load as allowed by ACI 318-14 (6.4.3.3).

 

a.     Factored load and Fixed-End Moments (FEM’s).

Where (9.3 psf = (14 x 14) / 144 x 150 / 22  is the weight of beam stem per foot divided by l2)

                                                           PCA Notes on ACI 318-11 (Table A1)

b.    Moment distribution.

Moment distribution for the five loading conditions is shown in Table 1. Counter-clockwise rotational moments acting on member ends are taken as positive. Positive span moments are determined from the following equation:

Where Mo 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 1-2 for loading (1):

Positive moment span 2-3 for loading (1):

 

Table 1 – Moment Distribution for Partial Frame (Transverse Direction)

Joint

1

2

3

4

Member

1-2

2-1

2-3

3-2

3-4

4-3

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 beam-slab 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

 

2.4.     Design moments

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 318-14 (8.11.6.1)

 

Figure 13 – Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Factored Live Load Except as Noted)

2.5.     Distribution of design moments                                                   

a. Check whether the moments calculated above can take advantage of the reduction permitted by ACI 318-14 (8.11.6.5):

Slab systems within the limitations of ACI 318-14 (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 Mo given by Equation 8.10.3.2 in the ACI 318-14:

                                ACI 318-14 (8.11.6.5)

 

Check Applicability of Direct Design Method:

 

1. There is a minimum of three continuous spans in each direction                                 ACI 318-14 (8.10.2.1)

2. Successive span lengths are equal                                                                                      ACI 318-14 (8.10.2.2)

3. Long-to-Short ratio is 22/17.5 = 1.26 < 2.0                                                                      ACI 318-14 (8.10.2.3)

4. Column are not offset                                                                                                           ACI 318-14 (8.10.2.4)

5. Loads are gravity and uniformly distributed with service live-to-dead ratio of 1.33 < 2.0

                                                                                                                                       ACI 318-14 (8.10.2.5 and 6)                                                                                                                        

6. Check relative stiffness for slab panel:                                                                              ACI 318-14 (8.10.2.7)

Interior Panel:

                     O.K.                    ACI 318-14 (Eq. 8.10.2.7a)                   

Interior Panel:

                  O.K.                    ACI 318-14 (Eq. 8.10.2.7a)                   

 

All limitation of ACI 318-14 (8.10.2) are satisfied and the provisions of ACI 318-14 (8.11.6.5) may be applied:

                                                          ACI 318-14 (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 ft-kip

Adjusted positive design moment = 71.2 × 0.973 = 69.3 ft-kip

Mo = 183.7 ft-kip

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 half-middle strips of the slab-beam according to the Direct Design Method (DDM) in 8.10, provided that Eq. 8.10.2.7(a) is satisfied.                                                                                                                                        ACI 318-14 (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 slab-beams as specified in ACI 318-14 (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
(ft-kips)

Column Strip

Moments in Two
Half-Middle Strips**
 (ft-kips)

Percent*

Moment
(ft-kips)

Beam Strip

Moment
(ft-kips)

Column Strip

Moment
(ft-kips)

End
Span

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
Span

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 α1l2/l1 > 1.0 beams must be proportioned to resist 85 percent of column strip per ACI 318-14 (8.10.5.7)

**That portion of the factored moment not resisted by the column strip is assigned to the two half-middle strips

 

2.6.     Flexural reinforcement requirements

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 tension-controlled 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.

in2

Maximum spacing                                                                ACI 318-14 (8.7.2.2) 

Provide 8 - #4 bars with As = 1.60 in.2 and s = 91/8 = 11.37 in.  ≤  smax

The flexural reinforcement calculation for the beam strip of end span – interior negative location is provided below:

Assume tension-controlled 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 As = 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

Mu
(ft-kip)

b *
(in.)

d **
(in.)

AReq’d
for flexure
(in.2)

Min As† ††
 (in.2)

Reinforcement
Provided

AProv.
for flexure
(in.2)

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. As = 0.0018 × b × h = 0.0108 × b for Column and Middle strips                                                       ACI 318-14 (7.6.1.1)

Min. As = min (3(fc')^0.5/fy*b*d , 200/fy*b*d) for Beam strip                                                                   ACI 318-14 (9.6.1.2)

†† Min. As = 1.333 × As Req'd if As provided  >= 1.333 × As Req'd for Beam strip                                   ACI 318-14 (9.6.1.3)

smax = 2 × h = 12 in. < 18 in.                                                                                                                          ACI 318-14 (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 γf x Mu

Where:

                                                                                                           ACI 318-14 (8.4.2.3.2)

b1 = Dimension of the critical section bo measured in the direction of the span for which moments are determined in ACI 318, Chapter 8.

b2 = Dimension of the critical section bo measured in the direction perpendicular to b1 in ACI 318, Chapter 8.

bo = Perimeter of critical section for two-way shear in slabs and footings.

                                                                                    ACI 318-14 (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, bb  (in.)

d
(in.)

γf

Mu*
(ft-kip)

γf Mu
(ft-kip)

As req’d within bb (in.2)

As prov. for

flexure within bb (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

-

*Mu 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 Vu_d= 31.64 kips (Figure 15).

                                                                                                        ACI 318-14 (22.5.5.1)

                             Stirrups are required.

Distance from the column face beyond which minimum reinforcement is required:

                                                                                                                      ACI 318-14 (22.5.10.1)

                                                                                              O.K.

                                            ACI 318-14 (22.5.10.1)

                                                         ACI 318-14 (22.5.10.5.3)

                                                                                                           ACI 318-14 (9.6.3.3)

               ACI 318-14 (9.7.6.2.2)

 

Select sprovided = 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

Av,min/s
in2/in

Av,req'd/s
in2/in

sreq'd
in

smax
in

Reinforcement
Provided

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.7.     Column design moments

The unbalanced moment from the slab-beams at the supports of the equivalent frame are distributed to the actual columns above and below the slab-beam 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 ft-kip

Joint 2 = -119 + 74.5 = -44.5 ft-kip

The stiffness and carry-over 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 Slab-Beam)

 

In summary:

Design moment in exterior column = 55.81 ft-kip

Design moment in interior column = 24.91 ft-kip

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 “Two-Way Flat Plate Concrete Floor Slab Design” example.

3.       Design of Interior, Edge, and Corner Columns

The design of interior, edge, and corner columns is explained in the “Two-Way Flat Plate Concrete Floor Slab Design” example.

4.        Two-Way Slab Shear Strength

Shear strength of the slab in the vicinity of columns/supports includes an evaluation of one-way shear (beam action) and two-way shear (punching) in accordance with ACI 318 Chapter 22.

4.1.     One-Way (Beam action) Shear Strength

One-way shear is critical at a distance d from the face of the column. Figure 17 shows the Vu 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 318-14 (Eq. 22.5.1.1)

Where:

                                                                                                                 ACI 318-14 (Eq. 22.5.5.1)

λ = 1 for normal weight concrete

Because φVc > Vu at all the critical sections, the slab is o.k. in one-way shear.

Figure 17 – One-way shear at critical sections (at distance d from the face of the supporting column)

4.2.     Two-Way (Punching) Shear Strength

Two-way shear is critical on a rectangular section located at dslab/2 away from the face of the column. The factored shear force Vu in the critical section is calculated as the reaction at the centroid of the critical section minus the self-weight and any superimposed surface dead and live load acting within the critical section.

The factored unbalanced moment used for shear transfer, Munb, 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 z-z is:

 

 

Figure 18 – Critical section of exterior support of interior frame
 


The polar moment Jc of the shear perimeter is:

 

 

                                                                                             ACI 318-14 (Eq. 8.4.4.2.2)

The length of the critical perimeter for the exterior column:

                                                                                                                 ACI 318-14 (R.8.4.4.2.3)

 

                                  ACI 318-14 (Table 22.6.5.2)

                                                                                                            O.K.

 

Figure 19 – Critical section of interior support of interior frame

For the interior column:

For the interior column in Figure 19, the location of the centroidal axis z-z is:

 

 

 

 

 

 

The polar moment Jc of the shear perimeter is:

 

 

                                                                                             ACI 318-14 (Eq. 8.4.4.2.2)

The length of the critical perimeter for the exterior column:

 

                                  ACI 318-14 (Table 22.6.5.2)

                                                                                                            O.K.   

5.       Two-Way Slab Deflection Control (Serviceability Requirements)

Since the slab thickness was selected based on the minimum slab thickness tables in ACI 318-14, the deflection calculations are not required. However, the calculations of immediate and time-dependent deflections are covered in this section for illustration and comparison with spSlab model results.

5.1.     Immediate (Instantaneous) Deflections

The calculation of deflections for two-way slabs is challenging even if linear elastic behavior can be assumed. Elastic analysis for three service load levels (D, D + Lsustained, D+LFull) is used to obtain immediate deflections of the two-way 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 318-14 (24.2.3)

The effective moment of inertia (Ie) is used to account for the cracking effect on the flexural stiffness of the slab. Ie for uncracked section (Mcr > Ma) is equal to Ig. When the section is cracked (Mcr < Ma), then the following equation should be used:

                                                                            ACI 318-14 (Eq. 24.2.3.5a)  

Where:

Ma = 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 318-14 (Eq. 24.2.3.5b)

                                                                     ACI 318-14 (Eq. 19.2.3.1)

yt = Distance from centroidal axis of gross section, neglecting reinforcement, to tension face, in.

Figure 21 – Ig calculations for slab section near support

    PCA Notes on ACI 318-11 (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 Icr. 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 318-14 (19.2.2.1.a)

                                                                         PCA Notes on ACI 318-11 (Table 10-2)

                                                                        

                                                                                                

 

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 318-14 (Eq. 24.2.3.5b)

                                                                     ACI 318-14 (Eq. 19.2.3.1)

                                                                                                                                                                         

Figure 23 – Ig calculations for slab section near support

 

                                                         ACI 318-14 (19.2.2.1.a)

                                                                         PCA Notes on ACI 318-11 (Table 10-2)

                                            PCA Notes on ACI 318-11 (Table 10-2)

                                  PCA Notes on ACI 318-11 (Table 10-2)

                                                            PCA Notes on ACI 318-11 (Table 10-2)

 

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, Ie, was developed to provide a transition between the upper and lower bounds of Ig and Icr as a function of the ratio Mcr/Ma. 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 318-14 (24.2.3.7)

For continuous one-way slabs and beams. Ie 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 318-14 (24.2.3.6)

For the exterior span (span with one end continuous) with service load level (D+LLfull):

                                                                                                                                                      ACI 318-14 (24.2.3.5a)

Where Ie- is the effective moment of inertia for the critical negative moment section (near the support).

 

Where Ie+ 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 Ie and this is considered satisfactory in approximate deflection calculations. The averaged effective moment of inertia (Ie,avg) is given by:

                                                        PCA Notes on ACI 318-11 (9.5.2.4(1))

 

Where:

 

For the interior span (span with both ends continuous) with service load level (D+LLfull):

                                                                                                                                                      ACI 318-14 (24.2.3.5a)

 

The averaged effective moment of inertia (Ie,avg) is given by:

                                  PCA Notes on ACI 318-11 (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

Ig,

in.4

Icr,

in.4

Ma, ft-kip

Mcr,

k-ft

Ie, in.4

Ie,avg, in.4

D

D +

LLSus

D +

Lfull

D

D +

LLSus

D +

Lfull

D

D +

LLSus

D +

Lfull

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 two-way 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 two-way slab systems the midpanel deflection is computed as the sum of deflection at midspan of the column strip or column line in one direction (Δcx or Δcy) and deflection at midspan of the middle strip in the orthogonal direction (Δmx or Δmy). 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 318-11 (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 Δcx. same procedure can be used to find the other terms.

Figure 26 –Δcx calculation procedure

For exterior span - service dead load case:

                                                          PCA Notes on ACI 318-11 (9.5.3.4 Eq. 10)

Where:

 

 

                                                            ACI 318-14 (19.2.2.1.a)

Iframe,averaged = The averaged effective moment of inertia (Ie,avg) for the frame strip for service dead load case from Table 6 = 22761 in.4

                                                   PCA Notes on ACI 318-11 (9.5.3.4 Eq. 11)

Where LDFc 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 (LDFL¯), interior negative region (LDFR¯), and positive region (LDFL) 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:

Ic,gThe gross moment of inertia (Ig) for the column strip (for T section) = 20040 in.4

Iframe,gThe gross moment of inertia (Ig) for the frame strip (for T section) = 25395 in.4

                                                                                PCA Notes on ACI 318-11 (9.5.3.4 Eq. 12)

Where:

 

 

Kec = effective column stiffness for exterior column.

        = 764 x Ec = 2929 x 106 in.-lb (calculated previously).

 

      

                                                                      PCA Notes on ACI 318-11 (9.5.3.4 Eq. 14)

 

Where:

 

 

 

Where

 

 

 

Kec = effective column stiffness for interior column.

        = 631 x Ec = 2419 x 106 in.-lb (calculated previously).

 

 

 

Where:

 

 

                                                               PCA Notes on ACI 318-11 (9.5.3.4 Eq. 9)

 

 

Following the same procedure, Δmx can be calculated for the middle strip. This procedure is repeated for the equivalent frame in the orthogonal direction to obtain Δcy, and Δmy for the end and middle spans for the other load levels (D+LLsus and D+LLfull).

Assuming square panel, Δcx = Δcy= 0.009 in. and Δmx = Δmy= 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

Δframe-fixed,
 in

Δc-fixed,
 in

θc1,
 rad

θc2,
 rad

Δθc1,
 in

Δθc2,
 in

Δcx,
 in

Δframe-fixed,
 in

Δm-fixed,
 in

θm1,
 rad

θm2,
 rad

Δθm1,
 in

Δθm2,
 in

Δmx,
 in

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+LLsus

LDF

D+LLsus

Δframe-fixed,
 in

Δc-fixed,
 in

θc1,
 rad

θc2,
 rad

Δθc1,
 in

Δθc2,
 in

Δcx,
 in

Δframe-fixed,
 in

Δm-fixed,
 in

θm1,
 rad

θm2,
 rad

Δθm1,
 in

Δθm2,
 in

Δmx,
 in

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+LLfull

LDF

D+LLfull

Δframe-fixed,
 in

Δc-fixed,
 in

θc1,
 rad

θc2,
 rad

Δθc1,
 in

Δθc2,
 in

Δcx,
 in

Δframe-fixed,
 in

Δm-fixed,
 in

θm1,
 rad

θm2,
 rad

Δθm1,
 in

Δθm2,
 in

Δmx,
 in

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

Δcx,
 in

Δmx,
 in

Ext

0.69

0.011

0.31

0.025

Int

0.67

0.005

0.33

0.023


5.2.     Time-Dependent (Long-Term) Deflections (Δlt)

The additional time-dependent (long-term) deflection resulting from creep and shrinkage (Δcs) may be estimated as follows:

 

                                                                                  PCA Notes on ACI 318-11 (9.5.2.5 Eq. 4)

 

The total time-dependent (long-term) deflection is calculated as:

 

                                                       CSA A23.3-04 (N9.8.2.5)                                                                                                                           

 

Where:

 

 

                                                                                                                               ACI 318-14 (24.2.4.1.1)

 

 

 

 

For the exterior span

 

= 2, consider the sustained load duration to be 60 months or more.                    ACI 318-14 (Table 24.2.4.1.3)

 

= 0, conservatively.

                    

Table 8 shows long-term deflections for the exterior and interior spans for the analysis in the x-direction, for column and middle strips.

 

 

Table 8 - Long-Term Deflections

Column Strip

Span

sust)Inst, in

λΔ

Δcs, in

total)Inst, in

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

 

 

6.       spSlab Software Program Model Solution

spSlab program utilizes the Equivalent Frame Method described and illustrated in details here for modeling, analysis and design of two-way 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 318-14 (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 long-term 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 Two-Way Flat Plate Concrete Floor Slab Design” example.


7.       Summary and Comparison of Design Results

Table 9 - Comparison of Moments obtained from Hand (EFM) and spSlab Solution (ft-kip)

 

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
Provided

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 Two-Way (Punching) Shear Check Results (around Columns Faces)

Support

b1, in.

b2, in.

bo, in.

Vu, kips

cAB, 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

Jc, in.4

γv

Munb, ft-kips

vu, psi

φvc, 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+LLsus

D+LLfull

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+LLsus

D+LLfull

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 Time-Dependent Deflection Results

Column Strip

Span

λΔ

Δcs, in.

Δtotal, 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

λΔ

Δcs, in.

Δtotal, 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.

 

 

 

 

8.       Conclusions & Observations

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 318-14 Chapter 8 (8.2.1).

 

Direct Design Method (DDM) is an approximate method and is applicable to two-way slab concrete floor systems that meet the stringent requirements of ACI 318-14 (8.10.2). In many projects, however, these requirements limit the usability of the Direct Design Method significantly.

 

The Equivalent Frame Method (EFM) does not have the limitations of Direct Design Method. It requires more accurate analysis methods that, depending on the size and geometry can prove to be long, tedious, and time-consuming.

 

StucturePoint’s spSlab software program solution utilizes the Equivalent Frame Method to automate the process providing considerable time-savings in the analysis and design of two-way 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 cost-time 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

318-14

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 center-to-center of supports in each direction shall not differ by more than one-third the longer span

þ

8.10.2.3

Panels shall be rectangular, with ratio of longer to shorter panel dimensions, measured center-to-center 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 (qu)

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, slab-to-beam 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 (Msc)

Moments @ support face

Moments @ support centerline

Engineering judgment required based on modeling technique

 

Irregularities (i.e. variable thickness, non-prismatic, 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. non-linear

2. Isotropic vs non-isotropic

3. Plate element choice

4. Mesh size and aspect ratio

5. Design & detailing features

General (Drawbacks)

Very limited applications

Limited geometry

Limited guidance non-standard 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 Msc (Munb) 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 Msc (Munb). In EFM where a frame analysis is used, moments at the column center line are used to obtain Msc (Munb).