Two-Way Flat Slab (Concrete Floor with Drop Panels) System Analysis and Design

 


Two-Way Flat Slab (Concrete Floor with Drop Panels) System Analysis and Design

 

Design the concrete floor slab system shown below for an intermediate floor considering partition weight = 20 psf, and unfactored live load = 60 psf. The lateral loads are independently resisted by shear walls. The use of flat plate system will be checked. If the use of flat plate is not adequate, the use of flat slab system with drop panels will be investigated. Flat slab concrete floor system is similar to the flat plate system. The only exception is that the flat slab uses drop panels (thickened portions around the columns) to increase the nominal shear strength of the concrete at the critical section around the columns. The Equivalent Frame Method (EFM) shown in ACI 318 is used in this example. The hand solution from EFM is also used for a detailed comparison with the model results of spSlab engineering software program.

Figure 1 - Two-Way Flat Concrete Floor System

Contents

1.   Preliminary member sizing. 1

2.   Flexural Analysis and Design. 10

2.1. Equivalent Frame Method (EFM) 10

2.1.1.  Limitations for use of equivalent frame method. 11

2.1.2.  Frame members of equivalent frame. 11

2.1.3.  Equivalent frame analysis. 14

2.1.4.  Factored moments used for Design. 16

2.1.5.  Factored moments in slab-beam strip. 18

2.1.6.  Flexural reinforcement requirements. 19

2.1.7.  Factored moments in columns. 22

3.   Design of Columns by spColumn. 24

3.1. Determination of factored loads. 24

3.2. Moment Interaction Diagram... 26

4.   Shear Strength. 29

4.1. One-Way (Beam action) Shear Strength. 29

4.1.1.  At distance d from the supporting column. 29

4.1.2.  At the face of the drop panel 30

4.2. Two-Way (Punching) Shear Strength. 31

4.2.1.  Around the columns faces. 31

4.2.2.  Around drop panels   34

5.   Serviceability Requirements (Deflection Check) 37

5.1. Immediate (Instantaneous) Deflections. 37

5.2. Time-Dependent (Long-Term) Deflections (Δlt) 49

6.   spSlab Software Program Model Solution. 50

7.   Summary and Comparison of Design Results. 71

8.   Conclusions & Observations. 74

8.1. One-Way Shear Distribution to Slab Strips. 74

8.2. Two-Way Concrete Slab Analysis Methods. 77

 

 

 

                                                                                        

                                                                                        


Code

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

Reference

Concrete Floor Systems (Guide to Estimating and Economizing), Second Edition, 2002 David A. Fanella

Notes on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association.

Simplified Design of Reinforced Concrete Buildings, Fourth Edition, 2011 Mahmoud E. Kamara and Lawrence C. Novak

Control of Deflection in Concrete Structures (ACI 435R-95)

Design Data

Story Height = 13 ft (provided by architectural drawings)

Superimposed Dead Load, SDL =20 psf for framed partitions, wood studs, 2 x 2, plastered 2 sides

                                                                                                                                                     ASCE/SEI 7-10 (Table C3-1)

Live Load, LL = 60 psf                                                                                                              ASCE/SEI 7-10 (Table 4-1)

50 psf is considered by inspection of Table 4-1 for Office Buildings – Offices (2/3 of the floor area)

80 psf is considered by inspection of Table 4-1 for Office Buildings – Corridors (1/3 of the floor area)

LL = 2/3 x 50 + 1/3 x 80 = 60 psf

fc = 5000 psi (for slab)

fc = 6000 psi (for columns)

fy = 60,000 psi

 

Solution

1.       Preliminary member sizing

For Flat Plate (without Drop Panels)

a.        Slab minimum thickness – Deflection                                                                                       ACI 318-14 (8.3.1.1)

In lieu of detailed calculation for deflections, ACI 318 Code gives minimum slab thickness for two-way construction without interior beams in Table 8.3.1.1.

For this flat plate slab systems the minimum slab thicknesses per ACI 318-14 are:

                                                                    ACI 318-14 (Table 8.3.1.1)

But not less than 5 in.                                                                                                               ACI 318-14 (8.3.1.1(a))

                                                                       ACI 318-14 (Table 8.3.1.1)

But not less than 5 in.                                                                                                               ACI 318-14 (8.3.1.1(a))

Where ln = length of clear span in the long direction = 30 x 12 – 20 = 340 in.

Try 11 in. slab for all panels (self-weight = 150 pcf x 11 in. /12 = 137.5 psf)

b.       Slab shear strength – one way shear

At a preliminary check level, the use of average effective depth would be sufficient. However, after determining the final depth of the slab, the exact effective depth will be used in flexural, shear and deflection calculations. Evaluate the average effective depth (Figure 2):

Where:

cclear = 3/4 in. for # 6 steel bar                                                                                    ACI 318-14 (Table 20.6.1.3.1)

db = 0.75 in. for # 6 steel bar                                                                          

Figure 2 - Two-Way Flat Concrete Floor System

 

                                                                                   ACI 318-14 (5.3.1)

 

Check the adequacy of slab thickness for beam action (one-way shear)                                 ACI 318-14 (22.5)

 

at an interior column:  

Consider a 12-in. wide strip. The critical section for one-way shear is located at a distance d, from the face of support (see Figure 3):

                                                                                                                   ACI 318-14 (Eq. 22.5.5.1)                                     

Where λ = 1 for normal weight concrete, more information can be found in “Concrete Type Classification Based on Unit Density” technical article.

Slab thickness of 11 in. is adequate for one-way shear.

 

c.        Slab shear strength – two-way shear

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior column (Figure 4):

                                                       ACI 318-14 (Table 22.6.5.2(a))

                                                                                                           

Slab thickness of 11 in. is not adequate for two-way shear. It is good to mention that the factored shear (Vu) used in the preliminary check does not include the effect of the unbalanced moment at supports. Including this effect will lead to an increase of Vu value as shown later in section 4.2.

 

Figure 3 – Critical Section for One-Way Shear                    Figure 4 – Critical Section for Two-Way Shear

In this case, four options could be used: 1) to increase the slab thickness, 2) to increase columns cross sectional dimensions or cut the spacing between columns (reducing span lengths), however, this option is assumed to be not permissible in this example due to architectural limitations, 3) to use headed shear reinforcement, or 4) to use drop panels. In this example, the latter option will be used to achieve better understanding for the design of two-way slab with drop panels often called flat slab.

Check the drop panel dimensional limitations as follows:

1)       The drop panel shall project below the slab at least one-fourth of the adjacent slab thickness.

                                                                                                                                                ACI 318-14 (8.2.4(a))                                                                                                                           

Since the slab thickness (hs) is 10 in. (see page 6), the thickness of the drop panel should be at least:

Drop panel dimensions are also controlled by formwork considerations. The following Figure shows the standard lumber dimensions that are used when forming drop panels. Using other depths will unnecessarily increase formwork costs.

For nominal lumber size (2x), hdp = 4.25 in. > hdp, min = 2.5 in.

 

The total thickness including the slab and the drop panel (h) = hs + hdp = 10 + 4.25 = 14.25 in.

Nominal Lumber Size, in.

Actual Lumber Size, in.

Plyform Thickness, in.

hdp, in.

2x

1 1/2

 3/4

2 1/4

4x

3 1/2

 3/4

4 1/4

6x

5 1/2

 3/4

6 1/4

8x

7 1/4

 3/4

8   

 

Figure 5 – Drop Panel Formwork Details

 

2)       The drop panel shall extend in each direction from the centerline of support a distance not less than one-sixth the span length measured from center-to-center of supports in that direction.                         

                                                                                                                                                           ACI 318-14 (8.2.4(b))

Based on the previous discussion, Figure 6 shows the dimensions of the selected drop panels around interior, edge (exterior), and corner columns.

Figure 6 – Drop Panels Dimensions

For Flat Slab (with Drop Panels)

For slabs with changes in thickness and subjected to bending in two directions, it is necessary to check shear at multiple sections as defined in the ACI 319-14. The critical sections shall be located with respect to:

1) Edges or corners of columns.                                                                                          ACI 318-14 (22.6.4.1(a))

2) Changes in slab thickness, such as edges of drop panels.                                          ACI 318-14 (22.6.4.1(b))

 

a.        Slab minimum thickness – Deflection                                                                                       ACI 318-14 (8.3.1.1)

In lieu of detailed calculation for deflections, ACI 318 Code gives minimum slab thickness for two-way construction without interior beams in Table 8.3.1.1.

For this flat plate slab systems the minimum slab thicknesses per ACI 318-14 are:

                                                                   ACI 318-14 (Table 8.3.1.1)

But not less than 4 in.                                                                                                               ACI 318-14 (8.3.1.1(b))

                                                                       ACI 318-14 (Table 8.3.1.1)

But not less than 4 in.                                                                                                               ACI 318-14 (8.3.1.1(b))                                

 

Where ln = length of clear span in the long direction = 30 x 12 – 20 = 340 in.

Try 10 in. slab for all panels

Self-weight for slab section without drop panel = 150 pcf x 10 in. /12 = 125 psf

Self-weight for slab section with drop panel = 150 pcf x 14.25 in. /12 = 178 psf

 

b.       Slab shear strength – one way shear

 

For critical section at distance d from the edge of the column (slab section with drop panel):

Evaluate the average effective depth:

Where:

cclear = 3/4 in. for # 6 steel bar                                                                                    ACI 318-14 (Table 20.6.1.3.1)

db = 0.75 in. for # 6 steel bar                                                                          

 

                                                                            ACI 318-14 (5.3.1)                                                            

Check the adequacy of slab thickness for beam action (one-way shear) from the edge of the interior column                            

                                                                                                                                                               ACI 318-14 (22.5)

Consider a 12-in. wide strip. The critical section for one-way shear is located at a distance d, from the edge of the column (see Figure 7)

                                                                                                                  ACI 318-14 (Eq. 22.5.5.1)  

      

Slab thickness of 14.25 in. is adequate for one-way shear for the first critical section (from the edge of the column).

 

For critical section at the edge of the drop panel (slab section without drop panel):

Evaluate the average effective depth:

Where:

cclear = 3/4 in. for # 6 steel bar                                                                                    ACI 318-14 (Table 20.6.1.3.1)

db = 0.75 in. for # 6 steel bar                                                                          

 

                                                                            ACI 318-14 (5.3.1)

 

Check the adequacy of slab thickness for beam action (one-way shear) from the edge of the interior drop panel           ACI 318-14 (22.5)

Consider a 12-in. wide strip. The critical section for one-way shear is located at the face of support (see Figure 7)

                                                                                                                  ACI 318-14 (Eq. 22.5.5.1)                                                 

        

Slab thickness of 10 in. is adequate for one-way shear for the second critical section (from the edge of the drop panel).

Figure 7 – Critical Sections for One-Way Shear

 

c.        Slab shear strength – two-way shear

For critical section at distance d/2 from the edge of the column (slab section with drop panel):

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior column (Figure 8):

                                                      ACI 318-14 (Table 22.6.5.2(a))

 

Slab thickness of 14.25 in. is adequate for two-way shear for the first critical section (from the edge of the column).

 

For critical section at the edge of the drop panel (slab section without drop panel):

Check the adequacy of slab thickness for punching shear (two-way shear) at an interior drop panel (Figure 8):

                                                      ACI 318-14 (Table 22.6.5.2(a))

                                                                                                           

Slab thickness of 10 in. is adequate for two-way shear for the second critical section (from the edge of the drop panel).

Figure 8 – Critical Sections for Two-Way Shear


 

d.       Column dimensions -  axial load

Check the adequacy of column dimensions for axial load:

Tributary area for interior column for live load, superimposed dead load, and self-weight of the slab is

Tributary area for interior column for self-weight of additional slab thickness due to the presence of the drop panel is

Assuming five story building

Assume 20 in. square column with 4 – No. 14 vertical bars with design axial strength, φPn,max of

                                                                           ACI 318-14 (22.4.2)

Column dimensions of 20 in. x 20 in. are adequate for axial load.

2.       Flexural Analysis and Design

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. For the solution per DDM, check the flat plate example.

2.1.     Equivalent Frame Method (EFM)

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 (for a detailed discussion of this method, refer to the flat plate design example):

1)       Horizontal slab-beam strip.

2)       Columns or other vertical supporting members.

3)       Elements of the structure (Torsional members) that provide moment transfer between the horizontal and vertical members.

2.1.1.   Limitations for use of equivalent frame method

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)

 

2.1.2.   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 k, 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)

 

                                                     ACI 318-14 (19.2.2.1.a)

Carry-over factor COF = 0.578                                                                   PCA Notes on ACI 318-11 (Table A1)

                                           PCA Notes on ACI 318-11 (Table A1)      

Uniform load fixed end moment coefficient, mNF1 = 0.0915

Fixed end moment coefficient for (b-a) = 0.2 when a = 0, mNF2 = 0.0163

Fixed end moment coefficient for (b-a) = 0.2 when a = 0.8, mNF3 = 0.0163

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

Referring to Table A7, Appendix 20A,

For the Bottom Column (Below):

                                                                                      PCA Notes on ACI 318-11 (Table A7)

 

                                                     ACI 318-14 (19.2.2.1.a)

lc = 13 ft = 156 in.

 

For the Top Column (Above):

                                                                                              PCA Notes on ACI 318-11 (Table A7)

c.     Torsional stiffness of torsional members, .

                                                                                                                      ACI 318-14 (R.8.11.5)

                                                                                       ACI 318-14 (Eq. 8.10.5.2b)

 Equivalent column stiffness Kec.

 

Where∑ Kt 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.

                     Figure 9 – Torsional Member        Figure 10 – Column and Edge of Slab

 

d.    Slab-beam joint distribution factors, DF.

At exterior joint,

At interior joint,

COF for slab-beam =0.578

Figure 11 – Slab and Column Stiffness

 

2.1.3.   Equivalent frame analysis

Determine negative and positive moments for the slab-beams using the moment distribution method. Since the unfactored live load does not exceed three-quarters of the unfactored dead load, design moments are assumed to occur at all critical sections with full factored live on all spans.                                                                             ACI 318-14 (6.4.3.2)

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

For slab:

For drop panels:

                                       PCA Notes on ACI 318-11 (Table A1)

 

                                  

b.    Moment distribution. Computations are shown in Table 1. Counterclockwise rotational moments acting on the member ends are taken as positive. Positive span moments are determined from the following equation:

Where 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:

Table 1 - Moment Distribution for Equivalent Frame

Joint

1

2

3

4

Member

1-2

2-1

2-3

3-2

3-4

4-3

DF

0.551

0.355

0.355

0.355

0.355

0.551

COF

0.578

0.578

0.578

0.578

0.578

0.578

FEM

677.6

-677.6

677.6

-677.6

677.6

-677.6

Dist

-373.1

0.0

0.0

0.0

0.0

373.1

CO

0.0

-215.7

0.0

0.0

215.7

0.0

Dist

0.0

76.6

76.6

-76.6

-76.6

0.0

CO

44.3

0.0

-44.3

44.3

0.0

-44.3

Dist

-24.4

15.7

15.7

-15.7

-15.7

24.4

CO

9.1

-14.1

-9.1

9.1

14.1

-9.1

Dist

-5.0

8.2

8.2

-8.2

-8.2

5.0

CO

4.8

-2.9

-4.8

4.8

2.9

-4.8

Dist

-2.6

2.7

2.7

-2.7

-2.7

2.6

CO

1.6

-1.5

-1.6

1.6

1.5

-1.6

Dist

-0.9

1.1

1.1

-1.1

-1.1

0.9

CO

0.6

-0.5

-0.6

0.6

0.5

-0.6

Dist

-0.4

0.4

0.4

-0.4

-0.4

0.4

CO

0.2

-0.2

-0.2

0.2

0.2

-0.2

Dist

-0.1

0.2

0.2

-0.2

-0.2

0.1

CO

0.1

-0.1

-0.1

0.1

0.1

-0.1

Dist

-0.1

0.1

0.1

-0.1

-0.1

0.1

CO

0.0

0.0

0.0

0.0

0.0

0.0

Dist

0.0

0.0

0.0

0.0

0.0

0.0

M, k-ft

331.7

-807.6

721.9

-721.9

807.6

-331.7

Midspan M,

ft-kips

349.6

197.4

349.6

 

2.1.4.   Factored moments used for Design

Positive and negative factored moments for the slab system in the direction of analysis are plotted in Figure 12. The negative moments used for design are taken at the faces of supports (rectangle section or equivalent rectangle for circular or polygon sections) but not at distances greater than 0.175 l1 from the centers of supports.           ACI 318-14 (8.11.6.1)                                                                                                                         

Figure 12 - Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Factored Live Load)

2.1.5.   Factored moments in slab-beam strip                                                   

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

If the slab system analyzed using EFM within the limitations of ACI 318-14 (8.10.2), it is permitted by the ACI code to reduce the calculated moments obtained from EFM in such proportion that the absolute sum of the positive and average negative design moments need not exceed the total static moment Mo given by Equation 8.10.3.2 in the ACI 318-14.

 

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 30/30 = 1.0 < 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 0.41 < 2.0

(Note: The self-weight of the drop panels is not uniformly distributed entirely along the span. However, the variation in load magnitude is small).

                                                                                                                                         ACI 318-14 (8.10.2.5 and 6)

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

Slab system is without beams and this requirement is not applicable.

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 = 812.8/919.3 = 0.884

Adjusted negative design moment = 721.9 × 0.884 = 638.2 ft-kips

Adjusted positive design moment = 197.3 × 0.884 = 174.4 ft-kips

 

ACI 318 allows the reduction of the moment values based on the previous procedure. Since the drop panels may cause gravity loads not to be uniform (Check limitation #5 and Figure 12), the moment values obtained from EFM will be used for comparison reasons.

 

b. Distribute factored moments to column and middle strips:

After the negative and positive moments have been determined for the slab-beam strip, the ACI code permits the distribution of the moments at critical sections to the column strips, beams (if any), and middle strips in accordance with the DDM.                                                                                                                                                        ACI 318-14 (8.11.6.6)

Distribution of factored moments at critical sections is summarized in Table 2.

Table 2 - Distribution of factored moments

 

Slab-beam Strip

Column Strip

Middle Strip

 Moment
(ft-kips)

Percent

Moment
(ft-kips)

Percent

Moment
(ft-kips)

End Span

Exterior Negative

246.5

100

246.5

0

0.0

Positive

349.6

60

209.8

40

139.8

Interior Negative

695.9

75

521.9

25

174.0

Interior Span

Negative

623.4

75

467.6

25

155.9

Positive

197.3

60

118.4

40

78.9

 

2.1.6.   Flexural reinforcement requirements

a. Determine flexural reinforcement required for strip moments

The flexural reinforcement calculation for the column strip of end span – exterior negative location is provided below.

 

Use davg = 12.75 in.

To determine the area of steel, assumptions have to be made whether the section is tension or compression controlled, and regarding the distance between the resultant compression and tension forces along the slab section (jd). In this example, tension-controlled section will be assumed so the reduction factoris equal to 0.9, and jd will be taken equal to 0.95d. The assumptions will be verified once the area of steel in finalized.

 

Therefore, the assumption that section is tension-controlled is valid.

The slab have two thicknesses in the column strip (14.25 in. for the slab with the drop panel and 10 in. for the slab without the drop panel).

The weighted slab thickness:

 

 

 

 

 

                                                          ACI 318-14 (24.4.3.2)

                                                                                ACI 318-14 (8.7.2.2)

Provide 10 - #6 bars with As = 4.40 in.2 and s = 180/10 = 18 in. ≤ smax

Based on the procedure outlined above, values for all span locations are given in Table 3.

 

Table 3 - Required Slab Reinforcement for Flexure [Equivalent Frame Method (EFM)]

Span Location

Mu

(ft-kips)

b

(in.)

d (in.)

AReq’d for flexure (in.2)

Min As (in.2)

Reinforcement Provided

AProv. for flexure (in.2)

End Span

Column Strip

Exterior Negative

246.5

180

12.75

4.357

4.157

10-#6

4.40

Positive

209.8

180

8.50

5.631

3.240

13-#6

5.72

Interior Negative

521.9

180

12.75

9.366

4.157

22-#6

9.68

Middle Strip

Exterior Negative

0.0

180

8.50

0.0

3.240

10-#6 * **

4.40

Positive

139.8

180

8.50

3.719

3.240

10-#6 **

4.40

Interior Negative

174.0

180

8.50

4.649

3.240

11-#6

4.84

Interior Span

Column Strip

Positive

118.4

180

8.50

3.141

3.240

10-#6 * **

4.40

Middle Strip

Positive

78.9

180

8.50

2.083

3.240

10-#6 * **

4.40

*   Design governed by minimum reinforcement.

**  Number of bars governed by maximum allowable spacing.

b. Calculate additional slab reinforcement at columns for moment transfer between slab and column by flexure

The factored slab moment resisted by the column (γf  x Msc) shall be assumed to be transferred by flexure. Concentration of reinforcement over the column by closer spacing or additional reinforcement shall be used to resist this moment. The fraction of slab moment not calculated to be resisted by flexure shall be assumed to be resisted by eccentricity of shear.          ACI 318-14 (8.4.2.3)

Portion of the unbalanced moment transferred by flexure is γf  x Msc                               ACI 318-14 (8.4.2.3.1)

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 (see Figure 13).

b2 = Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 13).

bb = Effective slab width =                                                                                      ACI 318-14 (8.4.2.3.3)

Figure 13 – Critical Shear Perimeters for Columns


 

For exterior support:

          d = h – cover – d/2 = 14.25 – 0.75 – 0.75/2 = 13.13 in.

          c1 + d/2 = 20 + 13.13/2 = 26.56 in.

          c2 + d = 20 + 13.13 = 33.13 in.

          in.

         

         

Using the same procedure in 2.1.7.a, the required area of steel:

           

However, the area of steel provided to resist the flexural moment within the effective slab width bb:

           

Then, the required additional reinforcement at exterior column for moment transfer between slab and column:

         

Provide 5 - #6 additional bars with As = 2.20 in.2

Based on the procedure outlined above, values for all supports are given in Table 4.

 

Table 4 - Additional Slab Reinforcement required for moment transfer between slab and column (EFM)

Span Location

Msc*

(ft-kips)

γf

γf Msc

(ft-kips)

Effective slab

width, bb 

(in.)

d

(in.)

As req’d

within bb

(in.2)

As prov. For

flexure within bb

(in.2)

Add’l

Reinf.

End Span

Column Strip

Exterior Negative

331.7

0.626

207.7

62.75

13.13

3.63

1.534

5-#6

Interior Negative

85.7

0.60

51.42

62.75

13.13

0.877

3.375

-

*Msc is taken at the centerline of the support in Equivalent Frame Method solution.

 

2.1.7.   Factored moments in columns

The unbalanced moment from the slab-beams at the supports of the equivalent frame are distributed to the support columns above and below the slab-beam in proportion to the relative stiffness of the support columns. Referring to Figure 12, the unbalanced moment at the exterior and interior joints are:

Exterior Joint = +331.7 ft-kips

Joint 2= -807.6 + 721.9 = -85.7 ft-kips

The stiffness and carry-over factors of the actual columns and the distribution of the unbalanced slab moments (Msc) to the exterior and interior columns are shown in Figure 14.

Figure 14 - Column Moments (Unbalanced Moments from Slab-Beam)

 

In summary:

For Top column (Above):                                  For Bottom column (Below):

Mcol,Exterior= 150.61 ft-kips                                 Mcol,Exterior= 157.21 ft-kips

Mcol,Interior = 38.91 ft-kips                                    Mcol,Interior = 40.62 ft-kips

The moments determined above are combined with the factored axial loads (for each story) and factored moments in the transverse direction for design of column sections. The moment values at the face of interior, exterior, and corner columns from the unbalanced moment values are shown in the following table.


 

Table 5 – Factored Moments in Columns

Mu
kips-ft

Column Location

Interior

Exterior

Corner

Mux

40.62

157.21

157.21

Muy

40.62

40.62

157.21

3.       Design of Columns by spColumn

This section includes the design of interior, edge, and corner columns using spColumn software. The preliminary dimensions for these columns were calculated previously in section one. The reduction of live load per ASCE 7-10 will be ignored in this example. However, the detailed procedure to calculate the reduced live loads is explained in the “wide-Module Joist System” example.

3.1.  Determination of factored loads

Interior Column:

Assume 5 story building

Tributary area for interior column for live load, superimposed dead load, and self-weight of the slab is

Tributary area for interior column for self-weight of additional slab thickness due to the presence of the drop panel is

Assuming five story building

                Mu,x = 40.62 ft-kips  (see the previous Table)

Mu,y = 40.62 ft-kips  (see the previous Table)

 

Edge (Exterior) Column:

Tributary area for edge column for live load, superimposed dead load, and self-weight of the slab is

Tributary area for edge column for self-weight of additional slab thickness due to the presence of the drop panel is

                Mu,x = 157.21 ft-kips  (see the previous Table)

Mu,y = 40.62 ft-kips  (see the previous Table)

 

Corner Column:

Tributary area for corner column for live load, superimposed dead load, and self-weight of the slab is

Tributary area for corner column for self-weight of additional slab thickness due to the presence of the drop panel is

                Mu,x = 157.21 ft-kips  (see the previous Table)

Mu,y = 157.21 ft-kips  (see the previous Table)


 

3.2.  Moment Interaction Diagram

Interior Column:

Edge Column:

 

Corner Column:

4.        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

                                                                                                                                                               ACI 318-14 (22.5)

One-way shear is critical at a distance d from the face of the column as shown in Figure 3. Figures 15 and 16 show the factored shear forces (Vu) at the critical sections around each column and each drop panel, respectively. In members without shear reinforcement, the design shear capacity of the section equals to the design shear capacity of the concrete:

                                                                                     ACI 318-14 (Eq. 22.5.1.1)

Where:

                                                                                                                   ACI 318-14 (Eq. 22.5.5.1)

Note:    The calculations below follow one of two possible approaches for checking one-way shear. Refer to the conclusions section for a comparison with the other approach.

4.1.1.    At distance d from the supporting column

Figure 15 – One-way shear at critical sections (at distance d from the face of the supporting column)

4.1.2.    At the face of the drop panel

 

Figure 16 – One-way shear at critical sections (at the face of the drop panel)

 


 

4.2.     Two-Way (Punching) Shear Strength

                                                                                                                                                               ACI 318-14 (22.6)

 

4.2.1.    Around the columns faces

Two-way shear is critical on a rectangular section located at d/2 away from the face of the column as shown in Figure 13.

a. Exterior column:

The factored shear force (Vu) in the critical section is computed 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 (d/2 away from column face).

The factored unbalanced moment used for shear transfer, Munb, is computed as the sum of the joint moments to the left and right. Moment of the vertical reaction with respect to the centroid of the critical section is also taken into account.

 

For the exterior column in Figure 13, the location of the centroidal axis 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:

The two-way shear stress (vu) can then be calculated as:

                                                                                                          ACI 318-14 (R.8.4.4.2.3)

                                                        ACI 318-14 (Table 22.6.5.2)

 

b. Interior column:

 

For the interior column in Figure 13, 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 interior column:

The two-way shear stress (vu) can then be calculated as:

                                                                                                          ACI 318-14 (R.8.4.4.2.3)

                                                      ACI 318-14 (Table 22.6.5.2)

 

c. Corner column:

In this example, interior equivalent frame strip was selected where it only have exterior and interior supports (no corner supports are included in this strip). However, the two-way shear strength of corner supports usually governs. Thus, the two-way shear strength for the corner column in this example will be checked for educational purposes. Same procedure is used to find the reaction and factored unbalanced moment used for shear transfer at the centroid of the critical section for the corner support for the exterior equivalent frame strip.

 

For the interior column in Figure 13, the location of the centroidal axis z-z is:

in.

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 corner column:

The two-way shear stress (vu) can then be calculated as:

                                                                                                          ACI 318-14 (R.8.4.4.2.3)

                                                        ACI 318-14 (Table 22.6.5.2)

 

4.2.2.    Around drop panels

Two-way shear is critical on a rectangular section located at d/2 away from the face of the drop panel.

Note:    The two-way shear stress calculations around drop panels do not have the term for unbalanced moment since drop panels are a thickened portion of the slab and are not considered as a support.

a. Exterior drop panel:

The length of the critical perimeter for the exterior drop panel:

The two-way shear stress (vu) can then be calculated as:

                                                                                                                                  ACI 318-14 (R.8.4.4.2.3)

                                                        ACI 318-14 (Table 22.6.5.2)

b. Interior drop panel:

 

The length of the critical perimeter for the interior drop panel:

The two-way shear stress (vu) can then be calculated as:

                                                                                                                                  ACI 318-14 (R.8.4.4.2.3)

                                                        ACI 318-14 (Table 22.6.5.2)

 

c. Corner drop panel:

 

The length of the critical perimeter for the corner drop panel:

The two-way shear stress (vu) can then be calculated as:

                                                                                                                                  ACI 318-14 (R.8.4.4.2.3)

                                                        ACI 318-14 (Table 22.6.5.2)

 


 

5.       Serviceability Requirements (Deflection Check)

Since the slab thickness was selected below the minimum slab thickness tables in ACI 318-14, the deflection calculations of immediate and time-dependent deflections are required and shown below including a 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 17.


 

Figure 17 – Maximum Moments for the Three Service Load Levels

(No live load is sustained in this example)

 

For positive moment (midspan) section:

                                          ACI 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.

     PCA Notes on ACI 318-11 (9.5.2.2)

As calculated previously, the positive reinforcement for the end span frame strip is 23 #6 bars located at 1.125 in. along the section from the bottom of the slab. Two of these bars are not continuous and will be conservatively excluded from the calculation of Icr since they might not be adequately developed or tied (21 bars are used). Figure 18 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete at midspan.

Figure 18 – Cracked Transformed Section (positive moment section)

 

                                                        ACI 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)

 

For negative moment section (near the interior support of the end span):

The negative reinforcement for the end span frame strip near the interior support is 32 #6 bars located at 1.125 in. along the section from the top of the slab.

                                          ACI 318-14 (Eq. 24.2.3.5b)

                                                                   ACI 318-14 (Eq. 19.2.3.1)

                                                                                                                                                                       

Figure 19 – 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 20 – Cracked Transformed Section (negative moment section)

 

The effective moment of inertia procedure described in the Code is considered sufficiently accurate to estimate deflections. The effective moment of inertia, 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 middle span (span with two ends 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. Both the midspan stiffness (Ie+) and averaged span stiffness (Ie,avg) can be used in the calculation of immediate (instantaneous) deflection.

 

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

 

                                  PCA Notes on ACI 318-11 (9.5.2.4(2))

 

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

 

However, these expressions lead to improved results only for continuous prismatic members. The drop panels in this example result in non-prismatic members and the following expressions should be used according to ACI 318-89:

 

                                                                   ACI 435R-95 (2.14)

 

For the middle span (span with two ends continuous) with service load level (D+LLfull):

 

 

                                                                                          ACI 435R-95 (2.14)

 

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

 

 

Where:

 

Table 6 provides a summary of the required parameters and calculated values needed for deflections for exterior and interior spans.

 

Table 6 – Averaged Effective Moment of Inertia Calculations

For Frame Strip

Span

zone

Ig,
in.4

Icr,
 in.4

Ma, kips-ft

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

53445

7170

-179.72

-179.72

-252.8

401.42

53445

53445

53445

37190

37190

24576

Midspan

30000

3797

197.23

197.23

278.14

265.17

30000

30000

26503

Right

53445

10471

-434.41

-434.41

-611.14

401.42

44379

44379

22649

Int

Left

53445

10471

-388.46

-388.46

-546.48

401.42

53445

53445

27503

41723

41723

28752

Mid

30000

3317

107.56

107.56

152.03

265.17

30000

30000

30000

Right

53445

10471

-388.46

-388.46

-546.48

401.42

53445

53445

27503

 

Deflections in 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 in 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 21 shows the deflection computation for a rectangular panel. The average Δ for panels that have different properties in the two direction is calculated as follows:

 

                                                                  PCA Notes on ACI 318-11 (9.5.3.4 Eq. 8) 

 

                          

Figure 21 – Deflection Computation for a rectangular Panel

 

To calculate each term of the previous equation, the following procedure should be used. Figure 22 shows the procedure of calculating the term Δcx. Same procedure can be used to find the other terms.

 

Figure 22 –Δcx calculation procedure


 

For end 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 = 37190 in.4

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

For this example and like in the spSlab program, the effective moment of inertia at midspan will be used.

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 (LDFLÆ), interior negative region (LDFRÆ), and positive region (LDFL) are 1.00, 0.75, and 0.60, respectively (From Table 2 of this document). Thus, the load distribution factor for the column strip for the end span is given by:

Ic,g = The gross moment of inertia (Ig) for the column strip for service dead load = 15000 in.4

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

Where:

Kec = effective column stiffness =  in.-lb (calculated previously).

      

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

Where:

 

Where

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).

Since in this example the panel is squared, Δcx = Δcy= 0.219 in. and Δmx = Δmy= 0.125 in.

The average Δ for the corner panel is calculated as follows:


 

Table 7 – Immediate (Instantaneous) Deflections in the x-direction

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

0.738

0.0995

0.1468

0.00159

0.00041

0.05786

0.01479

0.219

0.262

0.0995

0.0521

0.00159

0.00041

0.05786

0.01479

0.125

Int

0.675

0.0886

0.1197

0.00041

0.00041

-0.01319

-0.01319

0.093

0.325

0.0886

0.0576

0.00041

0.00041

-0.01319

-0.01319

0.031

Span

LDF

D+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

0.738

0.0995

0.1468

0.00159

0.00041

0.05786

0.01479

0.219

0.262

0.0995

0.0521

0.00159

0.00041

0.05786

0.01479

0.125

Int

0.675

0.0886

0.1197

0.00041

0.00041

-0.01319

-0.01319

0.093

0.325

0.0886

0.0576

0.00041

0.00041

-0.01319

-0.01319

0.031

Span

LDF

D+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

0.738

0.2128

0.2775

0.00224

0.00057

0.12316

0.0315

0.432

0.262

0.2128

0.0985

0.00224

0.00057

0.12316

0.03125

0.253

Int

0.675

0.1819

0.2455

0.00057

0.00057

-0.02693

-0.02693

0.192

0.325

0.1819

0.1182

0.00057

0.00057

-0.02693

-0.02693

0.064

Span

LDF

LL

LDF

LL

Δcx,

 in.

Δmx,

in.

Ext

0.738

0.213

0.262

0.128

Int

0.675

0.098

0.325

0.033


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

0.219

2.000

0.439

0.432

0.871

Interior

0.093

2.000

0.187

0.192

0.378

Middle Strip

Exterior

0.125

2.000

0.250

0.253

0.503

Interior

0.031

2.000

0.062

0.064

0.127

 

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 with drop panels. spSlab uses the exact geometry and boundary conditions provided as input to perform an elastic stiffness (matrix) analysis of the equivalent frame taking into account the torsional stiffness of the slabs framing into the column. It also takes into account the complications introduced by a large number of parameters such as vertical and torsional stiffness of transverse beams, the stiffening effect of drop panels, column capitals, and effective contribution of columns above and below the floor slab using the of equivalent column concept (ACI 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 are provided below for both input and output of the spSlab model.


7.       Summary and Comparison of Design Results

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

Hand (EFM)

spSlab

Exterior Span

Column Strip

Exterior Negative*

246.5

244.8

Positive

209.8

219.7

Interior Negative*

521.9

517.6

Middle Strip

Exterior Negative*

0.0

0.0

Positive

139.8

146.5

Interior Negative*

174.0

172.5

Interior Span

Column Strip

Interior Negative*

467.6

463.6

Positive

118.4

120.1

Middle Strip

Interior Negative*

155.9

154.5

Positive

78.9

80.1

* negative moments are taken at the faces of supports

 

Table 10 - Comparison of Reinforcement Results

Span Location

Reinforcement Provided for Flexure

Additional Reinforcement

Provided for Unbalanced Moment Transfer*

Total Reinforcement
Provided

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior Span

Column Strip

Exterior

Negative

10-#6

10-#6

5-#6

5-#6

15-#6

15-#6

Positive

13-#6

13-#6

n/a

n/a

13-#6

13-#6

Interior

Negative

22-#6

21-#6

---

---

22-#6

21-#6

Middle Strip

Exterior

Negative

10-#6

10-#6

n/a

n/a

10-#6

10-#6

Positive

10-#6

10-#6

n/a

n/a

10-#6

10-#6

Interior Negative

11-#6

11-#6

n/a

n/a

11-#6

11-#6

Interior Span

Column Strip

Positive

10-#6

10-#6

n/a

n/a

10-#6

10-#6

Middle Strip

Positive

10-#6

10-#6

n/a

n/a

10-#6

10-#6

 


 

Table 11 - Comparison of One-Way (Beam Action) Shear Check Results

Span

Vu @ d, kips

Vu @ drop panel, kips

φVc @ d , kips

φVc @ drop panel, kips

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

123.7

126.7

96.9

96.7

392.91

392.97

339.10

338.88

Interior

107.8

110.9

81.0

81.0

392.91

392.97

339.10

338.88

* xu calculated from the centerline of the left column for each span

 

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

26.56

26.56

33.13

33.13

86.26

86.25

103.6

114.6

8.18

8.18

Interior

33.13

33.13

33.13

33.13

132.52

132.50

256.4

263.0

16.57

16.56

Corner

26.56

26.56

26.56

26.56

53.13

53.12

60.3

60.6

6.64

6.64

Support

Jc, in.4

γv

Munb, ft-kips

vu, psi

φvc, psi

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

98,315

98,239

0.374

0.374

259

249.5

188.3

194.4

212.1

212.1

Interior

330,800

330,520

0.400

0.400

85.70

85.07

167.9

171.7

212.1

212.1

Corner

56,292

56,249

0.400

0.400

137.65

137.40

164.4

164.8

212.1

212.1

 

Table 13 - Comparison of Two-Way (Punching) Shear Check Results (around Drop Panels)

Support

b1, in.

b2, in.

bo, in.

Vu, kips

cAB, in.

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

74.44

74.44

128.88

128.88

277.76

277.75

87.65

98.24

19.95

19.95

Interior

128.88

128.88

128.88

128.88

515.52

515.5

225.5

233.91

64.44

64.44

Corner

74.44

74.44

74.44

74.44

148.88

148.87

51.54

51.53

18.61

18.61

 

Support

Jc, in.4

γv

Munb, ft-kips

vu, psi

φvc, psi

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

1,468,983

1,468,000

N.A.

N.A.

N.A.

N.A.

35.5

39.9

156.9

156.9

Interior

12,688,007

12,679,000

N.A.

N.A.

N.A.

N.A.

49.7

51.1

142.6

142.6

Corner

767,460

766,930

N.A.

N.A.

N.A.

N.A.

38.98

39.00

169.3

169.3

Note: Shear stresses from spSlab are higher than hand calculations since it considers the load effects beyond the column centerline known in the model as right/left cantilevers. This small increase is often neglected in simplified hand calculations like the one used here.

 

 

 

 

 

 


 

 

Table 14 - Comparison of Immediate Deflection Results (in.)

Column Strip

Span

D

D+LLsus

D+LLfull

LL

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

0.219

0.207

0.219

0.207

0.432

0.395

0.213

0.188

Interior

0.093

0.089

0.093

0.089

0.192

0.185

0.098

0.096

Middle Strip

Span

D

D+LLsus

D+LLfull

LL

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

0.125

0.120

0.125

0.120

0.253

0.218

0.128

0.098

Interior

0.031

0.030

0.031

0.030

0.064

0.071

0.033

0.040

 

Table 15 - Comparison of Time-Dependent Deflection Results

Column Strip

Span

λΔ

Δcs, in.

Δtotal, in.

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

2.0

2.0

0.439

0.414

0.871

0.808

Interior

2.0

2.0

0.187

0.178

0.378

0.363

Middle Strip

Span

λΔ

Δcs, in.

Δtotal, in.

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

2.0

2.0

0.250

0.241

0.503

0.459

Interior

2.0

2.0

0.062

0.060

0.127

0.131

 

In all of the hand calculations illustrated above, the results are in close or exact agreement with the automated analysis and design results obtained from the spSlab model. Excerpts of spSlab graphical and text output are given below for illustration.


 

8.       Conclusions & Observations

8.1.     One-Way Shear Distribution to Slab Strips

In one-way shear checks above, shear is distributed uniformly along the width of the design strip (30 ft.). StructurePoint finds it necessary sometimes to allocate the one-way shears with the same proportion moments are distributed to column and middle strips.

 

spSlab allows the one-way shear check using two approaches: 1) calculating the one-way shear capacity using the average slab thickness and comparing it with the total factored one-shear load as shown in the hand calculations above; 2) distributing the factored one-way shear forces to the column and middle strips and comparing it with the shear capacity of each strip as illustrated in the following figures. An engineering judgment is needed to decide which approach to be used.

Figure 23a – Distributing Shear to Column and Middle Strips (spSlab Input)

 

Figure 23b – Distributed Column and Middle Strip Shear Force Diagram (spSlab Output)

 

Figure 23c – Tabulated Shear Force & Capacity at Critical Sections (spSlab Output)


 

8.2.     Two-Way Concrete Slab Analysis Methods

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

ž

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

ž

8.10.2.3

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

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, slab-to-beam 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 (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).