Two-Way Joist Concrete Slab Floor (Waffle Slab) System Analysis and Design

 

 

 

Two-Way Joist Concrete Slab Floor (Waffle Slab) System Analysis and Design

Design the concrete floor slab system shown below for an intermediate floor with partition weight of 50 psf, and unfactored live load of 100 psf. The lateral loads are independently resisted by shear walls. A flat plate system will be considered first to illustrate the impact longer spans and heavier applied loads. A waffle slab system will be investigated since it is economical for longer spans with heavy loads. The dome voids reduce the dead load and electrical fixtures can be fixed in the voids. Waffle system provides an attractive ceiling that can be left exposed when possible producing savings in architectural finishes. 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 from StructurePoint.

Figure 1 - Two-Way Flat Concrete Floor System

Contents

1. Preliminary Member Sizing. 1

2. Flexural Analysis and Design. 13

2.1. Equivalent Frame Method (EFM) 13

2.1.1. Limitations for use of equivalent frame method. 14

2.1.2. Frame members of equivalent frame. 14

2.1.3. Equivalent frame analysis. 17

2.1.4. Factored moments used for Design. 19

2.1.5. Factored moments in slab-beam strip. 21

2.1.6. Flexural reinforcement requirements. 22

2.1.7. Factored moments in columns. 26

3. Design of Columns by spColumn. 28

3.1. Determination of factored loads. 28

3.2. Moment Interaction Diagram... 30

4. Shear Strength. 34

4.1. One-Way (Beam action) Shear Strength. 34

4.1.1. At distance d from the supporting column. 34

4.1.2. At the face of the drop panel 35

4.2. Two-Way (Punching) Shear Strength. 37

4.2.1. Around the columns faces. 37

4.2.2. Around drop panels 40

5. Serviceability Requirements (Deflection Check) 45

5.1. Immediate (Instantaneous) Deflections. 45

5.2. Time-Dependent (Long-Term) Deflections (Δlt) 57

6. spSlab Software Program Model Solution. 58

7. Summary and Comparison of Design Results. 80

8. Conclusions & Observations. 83

 

 

 

 


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, Portland Cement Association.

PCA 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), American Concrete Institute

Reinforced Concrete Design .. .Hassoun, McGraw Hill

Design Data

Story Height = 13 ft (provided by architectural drawings)

Superimposed Dead Load, SDL = 50 psf for Frame walls, hollow concrete masonry unit wythe, 12 in. thick, 125 pcf unit density, with no grout

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

Live Load, LL = 100 psf for Recreational uses Gymnasiums ASCE/SEI 7-10 (Table 4-1)

fc = 5000 psi (for slab)

fc = 6000 psi (for columns)

fy = 60,000 psi

 

Solution

1.       Preliminary Member Sizing

Preliminary Flat Plate (without Joists)

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

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

For flat plate slab system, the minimum slab thickness 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 = 33 x 12 20 = 376 in.

Use 13 in. slab for all panels (self-weight = 150 pcf x 13 in. /12 = 162.5 psf)

 

b.       Slab shear strength one way shear

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)

Slab thickness of 13 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 13 in. is not adequate for two-way shear. This is expected as the self-weight an applied loads are very challenging for a flat plate system.

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

 

In this case, four options can be considered: 1) increase the slab thickness further, 2) use headed shear reinforcement in the slab, 3) apply drop panels at columns, or 4) use two-way joist slab system. In this example, the latter option will be used to achieve better understanding for the design of two-way joist slab often called two-way ribbed slab or waffle slab.

Check the applicable joist dimensional limitations as follows:

1)       Width of ribs shall be at least 4 in. at any location along the depth. ACI 318-14 (9.8.1.2)

Use ribs with 6 in. width.

2)       Overall depth of ribs shall not exceed 3.5 times the minimum width. ACI 318-14 (9.8.1.3)

3.5 x 6 in. = 21 in. Use ribs with 14 in. depth.

3)       Clear spacing between ribs shall not exceed 30 in. ACI 318-14 (9.8.1.4)

Use 30 in. clear spacing.

4)       Slab thickness (with removable forms) shall be at least the greater of: ACI 318-14 (8.8.3.1)

a)       1/12 clear distance between ribs = 1/12 x 30 = 2.5 in.

b)       2 in.

Use a slab thickness of 3 in. > 2.5 in.

 

Figure 5 Joists Dimensions

In waffle slabs a drop panel is automatically invoked to guarantee adequate two-way (punching) shear resistance at column supports. This is evident from the flat plate check conducted using 13 in. indicating insufficient punching shear capacity above. 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 (hMI calculated in page 7 of this document) is 12 in., the thickness of the drop panel should be at least:

Drop panel depth are also controlled by the rib depth (both at the same level).For nominal lumber size (2x), hdp = hrib = 14 in. > hdp, min = 3 in.

The total thickness including the actual slab and the drop panel thickness (h) = hs + hdp = 3 + 14 = 17 in.

 

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 two-way joist system.

 

Figure 6 Two-Way Joist (Waffle) Slab

Preliminary Two-Way Joist Slab (Waffle Slab)

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 318-14. 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 slab system, 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 = 33 x 12 20 = 376 in.

 

For the purposes of analysis and design, the ribbed slab will be replaced with a solid slab of equivalent moment of inertia, weight, punching shear capacity, and one-way shear capacity.

 

The equivalent thickness based on moment of inertia is used to find slab stiffness considering the ribs in the direction of the analysis only. The ribs spanning in the transverse direction are not considered in the stiffness computations. This thickness, hMI, is given by:

spSlab Software Manual (Eq. 2-11)

Where:

Irib = Moment of inertia of one joist section between centerlines of ribs (see Figure 7a).

brib = The center-to-center distance of two ribs (clear rib spacing plus rib width) (see Figure 7a).

 

Since hMI = 12 in. > hmin = 11.4 in., the deflection calculation can be neglected. However, the deflection calculation will be included in this example for comparison with the spSlab software results.

The drop panel depth for two-way joist (waffle) slab is set equal to the rib depth. The equivalent drop depth based on moment of inertia, dMI, is given by:

spSlab Software Manual (Eq. 2-12)

Where hrib = 3 + 14 12 = 5 in.

Figure 7a Equivalent Thickness Based on Moment of Inertia

 

Find system self-weight using the equivalent thickness based on the weight of individual components (see the following Figure). This thickness, hw, is given by:

spSlab Software Manual (Eq. 2-10)

Where:

Vmod = The Volume of one joist module (the transverse joists are included 11 joists in the frame strip).

Amod = The plan area of one joist module = 33 x 36/12 = 99 ft2

 

Self-weight for slab section without drop panel = 150 pcf x 8 in. /12 = 100.057 psf

Self-weight for drop panel = 150 pcf x (14 + 3 8) in. /12 = 112.44 psf

Figure 7b Equivalent Thickness Based on the Weight of Individual Components

 

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

hs = 17 in. = The drop depth (dMI)

 

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 8)

ACI 318-14 (Eq. 22.5.5.1)

Slab thickness 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 the solid head (see Figure 8)

ACI 318-14 (Eq. 22.5.5.1)

Slab thickness of 12 in. is adequate for one-way shear for the second critical section (at the edge of the drop panel).

Figure 8 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 9):

Tributary area of two-way shear for the slab without the drop panel is:

Tributary area of two-way shear for the slab with the drop panel is:

ACI 318-14 (Table 22.6.5.2(a))

Slab thickness 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 9):

ACI 318-14 (Table 22.6.5.2(a))

Slab thickness of 12 in. is adequate for two-way shear for the second critical section (from the edge of the drop panel).

Figure 9 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 four story building

Assume 20 in. square column with 12 No. 11 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 = 0.54 PCA Notes on ACI 318-11 (Table A1)

PCA Notes on ACI 318-11 (Table A1)

Uniform load fixed end moment coefficient, mNF1 = 0.0911

Fixed end moment coefficient for (b-a) = 0.2 when a = 0, mNF2 = 0.0171

Fixed end moment coefficient for (b-a) = 0.2 when a = 0.8, mNF3 = 0.0016

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

Referring to Table A7, Appendix 20A,

For the Bottom Column:

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:

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)

d.    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 10 Torsional Member Figure 11 Column and Edge of Slab

 

e.     Slab-beam joint distribution factors, DF.

At exterior joint,

At interior joint,

COF for slab-beam =0.576

Figure 12 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 (FEMs).

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

0.640

0.390

0.390

0.390

0.390

0.640

COF

0.576

0.576

0.576

0.576

0.576

0.576

FEM

1146.51

-1146.5

1146.51

-1146.5

1146.51

-1146.5

Dist

-733.6

0.0

0.0

0.0

0.0

733.6

CO

0.0

-422.5

0.0

0.0

422.5

0.0

Dist

0.0

164.8

164.8

-164.8

-164.8

0.0

CO

94.9

0.0

-94.9

94.9

0.0

-94.9

Dist

-60.7

37.0

37.0

-37.0

-37.0

60.7

CO

21.3

-35.0

-21.3

21.3

35.0

-21.3

Dist

-13.7

22.0

22.0

-22.0

-22.0

13.7

CO

12.7

-7.9

-12.7

12.7

7.9

-12.7

Dist

-8.1

8.0

8.0

-8.0

-8.0

8.1

CO

4.6

-4.7

-4.6

4.6

4.7

-4.6

Dist

-3.0

3.6

3.6

-3.6

-3.6

3.0

CO

2.1

-1.7

-2.1

2.1

1.7

-2.1

Dist

-1.3

1.5

1.5

-1.5

-1.5

1.3

CO

0.9

-0.8

-0.9

0.9

0.8

-0.9

Dist

-0.6

0.6

0.6

-0.6

-0.6

0.6

CO

0.4

-0.3

-0.4

0.4

0.3

-0.4

Dist

-0.2

0.3

0.3

-0.3

-0.3

0.2

CO

0.2

-0.1

-0.2

0.2

0.1

-0.2

Dist

-0.1

0.1

0.1

-0.1

-0.1

0.1

CO

0.1

-0.1

-0.1

0.1

0.1

-0.1

Dist

0.0

0.1

0.1

-0.1

-0.1

0.0

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

462.3

-1381.5

1247.5

-1247.5

1381.5

-462.3

Midspan M,

ft-kips

630.0

304.4

630.0

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 13. 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 13 - 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 33/33 = 1.0 < 2.0. ACI 318-14 (8.10.2.3)

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

5. Loads are gravity and uniformly distributed with service live-to-dead ratio of 0.67 < 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.

ACI 318-14 (Eq. 8.10.3.2)

To illustrate proper procedure, the interior span factored moments may be reduced as follows:

Permissible reduction = 1376.9/1552 = 0.887

Adjusted negative design moment = 1247.5 0.887 = 1106.5 ft-kips

Adjusted positive design moment = 304 0.887 = 269.6 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 13), 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

335.1

100

335.1

0

0.0

Positive

630.0

60

378.0

40

252.0

Interior Negative

1207.9

75

905.9

25

302.0

Interior Span

Negative

1097.1

75

822.8

25

274.3

Positive

304.4

60

182.6

40

121.8

 

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 interior negative location:

Use d = 15.88 in. (slab with drop panel where h = 17 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.

Two values of thickness must be considered. The slab thickness in the column strip is 17 in. with the drop panel and 8 in. for the equivalent slab without the drop panel based on the system weight.

ACI 318-14 (24.4.3.2)

ACI 318-14 (24.4.3.3)

Provide 30 - #6 bars with As = 13.20 in.2 and s = 198/30 = 6.6 in. ≤ smax

 

The flexural reinforcement calculation for the column strip of interior span positive location:

Use d = 15.88 in. (slab with rib where h = 17 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.

ACI 318-14 (24.4.3.2)

Since column strip has 5 ribs provide 10 - #6 bars (2 bars/ rib):

Based on the procedures 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.)

As Reqd for flexure (in.2)

Min As (in.2)

Reinforcement Provided

As Prov. for flexure (in.2)

End Span

Column Strip

Exterior Negative

335.1

198

15.88

4.74

5.18

14-#6 * **

6.16

Positive (5 ribs)

378.0

198

15.81

5.38

2.85

10-#7

(2 bars / rib)

6.00

Interior Negative

905.9

198

15.88

13.05

5.18

30-#6

13.20

Middle Strip

Exterior Negative

0.0

198

15.88

0.0

5.18

14-#6 * **

6.16

Positive (6 ribs)

252.0

198

15.88

3.56

2.85

12-#6

(2 bars / rib)

5.28

Interior Negative

302.0

198

15.88

4.27

5.18

14-#6 * **

6.16

Interior Span

Column Strip

Positive (5 ribs)

182.6

198

15.88

2.57

2.85

10-#6 *

(2 bars / rib)

4.40

Middle Strip

Positive (6 ribs)

121.8

198

15.88

1.71

2.85

12-#6 *

(2 bars / rib)

5.28

* 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 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 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 14).

b2 = Dimension of the critical section measured in the direction perpendicular to in ACI 318, Chapter 8 (see Figure 14).

ACI 318-14 (8.4.2.3.3)

Figure 14 Critical Shear Perimeters for Columns

 

For exterior support:

 

Using the same procedure in 2.1.6.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 reqd

within bb

(in.2)

As prov. For

flexure within bb

(in.2)

Addl

Reinf.

End Span

Column Strip

Exterior Negative

462.3

0.630

291

71

15.88

4.184

2.209

5-#6

Interior Negative

133.4

0.600

80.4

71

15.88

2.029

4.733

-

*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 13, the unbalanced moment at the exterior and interior joints are:

Exterior Joint = +462.3 ft-kips

Joint 2= -1381.5 + 1247.5 = -134 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 15 - Column Moments (Unbalanced Moments from Slab-Beam)

 

In summary:

For Top column: For Bottom column:

Mcol,Exterior= 194.75 ft-kips Mcol,Exterior= 224.97 ft-kips

Mcol,Interior = 56.45 ft-kips Mcol,Interior = 65.21 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

65.21

224.97

224.97

Muy

65.21

65.21

224.97

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 4 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 = 65.21 ft-kips (see the previous Table)

Mu,y = 65.21 ft-kips (see the previous Table)

 

Edge (Exterior) Column:

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

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

Mu,x = 224.97 ft-kips (see the previous Table)

Mu,y = 65.21 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 = 224.97 ft-kips (see the previous Table)

Mu,y = 224.97 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 17 and 19 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)

One-way shear capacity is calculated assuming the shear cross-section area consisting of the drop panel (if any), the ribs, and the slab portion above them, decreased by concrete cover. For such section the equivalent shear width for single rib is calculated from the formula:

spSlab Software Manual (Eq. 2-13)

Where:

b = rib width, in.

d = distance from extreme compression fiber to tension reinforcement centroid.

4.1.1.    At distance d from the supporting column

for middle span with #6 reinforcement.

Figure 16 Frame strip cross section (at distance d from the face of the supporting column)

 

The one-way shear capacity for the ribbed slab portions shown in Figure 16 is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

Figure 17 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

for middle span with #6 reinforcement.

Figure 18 Frame strip cross section (at distance d from the face of the supporting column)

 

The one-way shear capacity for the ribbed slab portions shown in Figure 15 is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

 

Figure 19 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 14.

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 illustration purposes. The analysis procedure must be repeated for the exterior equivalent frame strip 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 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 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.

 

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).

 

Note: For simplicity, it is conservative to deduct only the self-weight of the slab and joists in the critical section from the shear reaction in punching shear calculations. This approach is also adopted in the spSlab program for the punching shear check around the drop panels.

a. Exterior drop panel:

d that is used in the calculation of vu is given by (see Figure 20):

spSlab Software Manual (Eq. 2-14)

Figure 20 Equivalent thickness based on shear area calculation

 

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)

The two-way shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

ACI 318-14 (Table 22.6.5.2)

 

In waffle slab design where the drop panels create a large critical shear perimeter, the factor (bo/d) has limited contribution and is traditionally neglected for simplicity and conservatism. This approach is adopted in this calculation and in the spSlab program (spSlab software manual, Eq. 2-46).

 

The two-way shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

[IMA1] 

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)

The two-way shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

ACI 318-14 (Table 22.6.5.2)

spSlab Software Manual (Eq. 2-46)

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)

The two-way shear capacity for the ribbed slab is permitted to be increased by 10%. ACI 318-14 (9.8.1.5)

ACI 318-14 (Table 22.6.5.2)

spSlab Software Manual (Eq. 2-46)

To mitigate the deficiency in two-way shear capacity an evaluation of possible options is required:

1.       Increase the thickness of the slab system

2.       Increasing the dimensions of the drop panels (length and/or width)

3.       Increasing the concrete strength

4.       Reduction of the applied loads

5.       Reduction of the panel spans

6.       Using less conservative punching shear allowable (gain of 5-10%)

7.       Refine the deduction of drop panel weight from the shear reaction (gain of 2-5%)

This example will be continued without the required modification discussed above to continue the illustration of the analysis and design procedure.


 

5.       Serviceability Requirements (Deflection Check)

Since the slab thickness was selected to meet the minimum slab thickness tables in ACI 318-14, the deflection calculations of immediate and time-dependent deflections are not required. They are shown below for illustration purposes and comparison with spSlab software 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 21 Maximum Moments for the Three Service Load Levels

 

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.

Figure 22 Equivalent gross section for one rib - positive moment section

 

PCA Notes on ACI 318-11 (9.5.2.2)

As calculated previously, the positive reinforcement for the middle span frame strip is 22 #6 bars located at 1.125 in. along the section from the bottom of the slab. Figure 23 shows all the parameters needed to calculate the moment of inertia of the cracked section transformed to concrete at midspan.

Figure 23 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 45 #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)

Note: A lower value of Ig (60,255 in.4) excluding the drop panel is conservatively adopted in calculating waffle slab deflection by the spSlab software.

Figure 24 Gross section negative 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)

Note: A lower value of Icr (18,722 in.4) excluding the drop panel is conservatively adopted in calculating waffle slab deflection by the spSlab software.

Figure 25 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 are recommended 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:

Note: The prismatic member equations excluding the effect of the drop panel are conservatively adopted in calculating waffle slab deflection by spSlab.

 

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

103622

15505

206.5

206.5

338.0

539

103622

103622

103622

62612

62612

29087

Midspan

60255

15603

298.2

298.2

491.8

276

50964

50964

23482

Right

103622

23029

626.6

626.6

1026.2

539

74259

74259

34692

Int

Left

103622

23029

565.8

565.8

926.6

539

92620

92620

38873

76437

76437

49564

Mid

60255

13647

132.6

132.6

221.0

276

60255

60255

60255

Right

103622

23029

565.8

565.8

926.6

539

92620

92620

38873

 

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 26 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 26 Deflection Computation for a rectangular Panel

To calculate each term of the previous equation, the following procedure should be used. Figure 27 shows the procedure of calculating the term Δcx. Same procedure can be used to find the other terms.

 

Figure 27 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 = 62,612 in.4

 

 

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

 

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:

 

spSlab Software Manual (Eq. 2-114)

 

And the load distribution factor for the middle strip can be found from the following equation:

 

spSlab Software Manual (Eq. 2-115)

 

Taking for example the end span where highest deflections are expected, the LDF for exterior negative region (LDFL), interior negative region (LDFR), 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 = 28,289 in.4

 

 

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

 

Where:

 

 

 

Kec = effective column stiffness = 1925 x 106 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 this example has square panels, Δcx = Δcy= 0.222 in. and Δmx = Δmy= 0.128 in.

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

 

The calculated deflection can now be compared with the applicable limits from the governing standards or project specified limits and requirements. Optimization for further savings in materials or construction costs can be now made based on permissible deflections in lieu of accepting the minimum values stipulated in the standards to avoid deflection calculations.

 

 

 

 

 


 

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

0.094

0.147

0.00129

0.0004

0.058

0.017

0.222

0.262

0.094

0.052

0.00129

0.0004

0.058

0.017

0.128

Int

0.675

0.077

0.110

-0.0004

-0.0004

-0.014

-0.014

0.082

0.325

0.077

0.053

-0.0004

-0.0004

-0.014

-0.014

0.025

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

0.094

0.147

0.00129

0.0004

0.058

0.017

0.222

0.262

0.094

0.052

0.00129

0.0004

0.058

0.017

0.128

Int

0.675

0.077

0.110

-0.0004

-0.0004

-0.014

-0.014

0.082

0.325

0.077

0.053

-0.0004

-0.0004

-0.014

-0.014

0.025

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

0.316

0.497

0.0021

0.0006

0.205

0.060

0.762

0.262

0.316

0.177

0.0021

0.0006

0.205

0.060

0.442

Int

0.675

0.186

0.267

-0.0006

-0.0006

-0.035

-0.035

0.196

0.325

0.186

0.128

-0.0006

-0.0006

-0.035

-0.035

0.057

Span

LDF

LL

LDF

LL

Δcx,

in.

Δmx,

in.

Ext

0.738

0.540

0.262

0.314

Int

0.675

0.114

0.325

0.032

 

 


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

0.222

2.000

0.445

0.762

1.207

Interior

0.082

2.000

0.164

0.196

0.360

Middle Strip

Exterior

0.128

2.000

0.255

0.442

0.698

Interior

0.025

2.000

0.050

0.057

0.107

 

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)

Hassoun (DDM)#

Hand (EFM)

spSlab

Exterior Span

Column Strip

Exterior Negative*

370.0

335.1

323.8

Positive

444.0

378.0

400.6

Interior Negative*

748.0

905.9

907.3

Middle Strip

Exterior Negative*

---

0.0

0.0

Positive

---

252.0

267.1

Interior Negative*

---

302.0

302.4

Interior Span

Column Strip

Interior Negative*

---

822.8

823.7

Positive

---

182.6

180.4

Middle Strip

Interior Negative*

249

274.3

274.6

Positive

296

121.8

120.2

* negative moments are taken at the faces of supports

# Direct design method does not distinguish between interior and exterior spans nor explicitly address the effect of column contribution at joints.

 

Table 10 - Comparison of Reinforcement Results

Span Location

Reinforcement Provided for Flexure

Additional Reinforcement

Provided for Unbalanced Moment Transfer

Total Reinforcement
Provided

Hassoun

Hand

spSlab

Hassoun

Hand

spSlab

Hassoun

Hand

spSlab

Exterior Span

Column Strip

Exterior

Negative

14-#6

14-#6

14-#6

---

5-#6

5-#6

14-#6

19-#6

19-#6

Positive

10-#8

2 bars / rib

10-#7

2 bars / rib

10-#7

2 bars / rib

---

n/a

n/a

10-#8

2 bars / rib

10-#7

2 bars / rib

10-#7

2 bars / rib

Interior

Negative

28-#6

30-#6

31-#6

---

---

---

28-#6

22-#6

21-#6

Middle Strip

Exterior

Negative

10-#6

14-#6

14-#6

---

n/a

n/a

10-#6*

14-#6

14-#6

Positive

12-#7

2 bars / rib

12-#6

2 bars / rib

12-#6

2 bars / rib

---

n/a

n/a

12-#7

2 bars / rib

12-#6

2 bars / rib

12-#6

2 bars / rib

Interior Negative

10-#6

14-#6

14-#6

---

n/a

n/a

10-#6*

14-#6

14-#6

Interior Span

Column Strip

Positive

10-#7

2 bars / rib

10-#6

2 bars / rib

10-#6

2 bars / rib

---

n/a

n/a

10-#7

2 bars / rib

10-#6

2 bars / rib

10-#6

2 bars / rib

Middle Strip

Positive

10-#6

2 bars / rib

12-#6

2 bars / rib

12-#6

2 bars / rib

---

n/a

n/a

10-#6

2 bars / rib

12-#6

2 bars / rib

12-#6

2 bars / rib

* Max spacing requirement exceeded (not checked)

 


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

188.8

195.5

145.7

146.1

336.0

336.0

148.5

148.5

Interior

160.9

167.2

117.8

117.8

337.4

337.4

149.2

149.2

* One-way shear check is not provided in the reference (Hassoun and Al-Manaseer)

 

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

27.94

27.94

35.88

35.88

91.76

91.75

153.97

174.86

8.51

8.51

Interior

35.88

35.88

35.88

35.88

143.52

143.50

393.83

414.86

17.94

17.94

Corner

27.94

27.94

27.94

27.94

55.88

55.88

92.10

92.43

6.99

6.98

Support

Jc, in.4

γv

Munb, ft-kips

vu, psi

φvc, psi

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

144,092

143,990

0.370

0.370

341.27

316.33

195.2

203.1

212.1

212.1

Interior

512,956

512,570

0.400

0.400

134.00

135.09

195.3

204.8

212.1

212.1

Corner

81,483

81,428

0.400

0.400

181.47

181.19

178.5

178.8

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

89.94

89.94

159.88

159.88

339.76

339.75

123.33

143.28

23.81

23.81

Interior

159.88

159.88

159.88

159.88

639.52

639.50

337.73

357.54

79.94

79.94

Corner

89.94

89.94

89.94

89.94

179.88

179.87

75.22

75.17

22.49

22.48

Support

Jc, in.4

vu, psi

φvc, psi

 

Hand

spSlab

Hand

spSlab

Hand

spSlab

 

Exterior

971,437

970,500

109.3

127.1

116.7

116.7

 

Interior

9,046,406

9,037,700

159.1

168.5

116.7

116.7

 

Corner

503,491

503,010

126.0

126.0

116.7

116.7

 

General notes:

1. Red values are exceeding permissible shear capacity

2. Hand calculations fail to capture analysis details possible in spSlab like accounting for the exact value of the moments and shears at supports and including the loads for the small slab section extending beyond the supporting column centerline.

 

 

 

 

 


 

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

0.222

0.337

0.222

0.337

0.762

0.867

0.540

0.530

Interior

0.082

0.116

0.082

0.116

0.196

0.222

0.114

0.106

Middle Strip

Span

D

D+LLsus

D+LLfull

LL

Hand

spSlab

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

0.128

0.189

0.128

0.189

0.442

0.443

0.315

0.254

Interior

0.025

0.039

0.025

0.039

0.057

0.082

0.033

0.043

 

Table 15 - Comparison of Time-Dependent Deflection Results

Column Strip

Span

λΔ

Δcs, in.

Δtotal, in.

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

2.000

2.000

0.445

0.679

1.207

1.453

Interior

2.000

2.000

0.164

0.245

0.360

0.493

Middle Strip

Span

λΔ

Δcs, in.

Δtotal, in.

Hand

spSlab

Hand

spSlab

Hand

spSlab

Exterior

2.000

2.000

0.255

0.434

0.698

0.902

Interior

2.000

2.000

0.050

0.142

0.107

0.291

 

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. The deflection results from spSlab are, however, more conservative than hand calculations for two main reasons explained previously: 1) Values of Ig and Icr at the negative section exclude the stiffening effect of the drop panel and 2) The Ie,avg used by spSlab considers equations for prismatic members.


 

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.

 

StucturePoints 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

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

8.10.2.3

Panels shall be rectangular, with ratio of longer to shorter panel dimensions, measured center-to-center supports, not exceed 2.

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

8.10.2.5

All loads shall be due to gravity only

8.10.2.5

All loads shall be uniformly distributed over an entire panel (qu)

 

 

8.10.2.6

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

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.

8.7.4.2

Structural integrity steel detailing

8.5.4

Openings in slab systems

8.2.2

Concentrated loads

Not permitted

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).

 


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