Analysis and Design of Slabs

1

Department of Civil Engineering, University of Engineering and Technology Peshawar, Pakistan

Prof. Dr. Qaisar Ali CE 5115 Advance Design of Reinforced Concrete Structures Fall 2011

Lecture-10

Analysis and Design of

Two-way Slab Systems

(Two Way Joist Slabs & Two-way Slab

with Beams)

By: Prof Dr. Qaisar Ali

Civil Engineering Department

UET Peshawar

[email protected]

1



Topics Addressed

Two-Way Joist Slab

Introduction

Behavior

Characteristics

Basic Steps for Structural Design

Example

2

2



Topics Addressed

Moment Coefficient Method for Two Way Slab with

Beams

Introduction

Cases

Moment Coefficient Tables

Reinforcement Requirements

Steps

Example

3


Prof. Dr. Qaisar Ali CE 5115 Advance Design of Reinforced Concrete Structures Fall 2011 4

Two-Way Joist Slab

3



Introduction

A two-way joist system, or waffle slab, comprises evenly

spaced concrete joists spanning in both directions and a

reinforced concrete slab cast integrally with the joists.

Two-Way Joist

Joist



Introduction

Like one-way joist system, a two way system will be called

as two-way joist system if clear spacing between ribs (dome

width) does not exceed 30 inches.

Two-Way Joist

4



Introduction

Two-Way Joist



Introduction

The joists are commonly formed by using standard square

“dome” forms and the domes are omitted around the columns

to form the solid heads.

8

Two-Way Joist

Solid Head

5



Two-Way Joist

Introduction

Standard Dome Data

Generally the dome for waffle slab can be of any size. However the

commonly used standard domes are discussed as follows:

30-inch × 30-inch square domes with 3-inch flanges; from which 6-

inch wide joist ribs at 36-inch centers are formed: these are

available in standard depths of 8, 10, 12, 14, 16 and 20 inches.

19-inch × 19-inch square domes with 2 ½-inch flanges, from which

5-inch wide joist ribs at 24-inch centers are formed. These are

available in standard depths of 8, 10, 12, 14 and 16 inches.

9



Two-Way Joist

Introduction

Standard Dome Data

10

6



Behavior

The behavior of two-way joist slab is similar to a two way flat

Slab system.

Two-Way Joist



Characteristics

Dome voids reduce dead load.

Attractive ceiling (waffle like appearance).

Electrical fixtures can be placed in the voids.

Particularly advantageous where the use of longer spans

and/or heavier loads are desired without the use of

deepened drop panels or supported beams.

Two-Way Joist

7



Basic Steps for Structural Design

Step No. 01 (Sizes): Sizes of all structural and non

structural elements are decided.

Step No. 02 (Loads): Loads on structure are determined

based on occupational characteristics and functionality.

Step No. 03 (Analysis): Effect of loads are calculated on all

structural elements.

Step No. 04 (Design): Structural elements are designed for

the respective load effects following code provisions.

13

Two-Way Joist



Sizes

Minimum Joist Depth

For Joist depth determination, waffle slabs are considered as flat slab

(ACI 13.1.3, 13.1.4 & 9.5.3).

The thickness of equivalent flat slab is taken from table 9.5 (c).

The thickness of slab and depth of rib of waffle slab can be then

computed by equalizing the moment of inertia of equivalent flat slab to

that of waffle slab.

However since this practice is time consuming, tables have been

developed to determine the size of waffle slab from equivalent flat slab

thickness.

14

Two-Way Joist

8



Sizes

Minimum Joist Depth

Equivalent Flat Slab Thickness

ACI 318-05 – Sect. 9.5.3

Minimum thickness = ln/33

15

Two-Way Joist



Sizes

Minimum Joist Depth

Slab and rib depth from equivalent flat slab thickness

Two-Way Joist

Table 01: Waffle flat slabs (19" × 19" voids at 2'-0")-Equivalent thickness

Rib + Slab Depths (in.) Equivalent Thickness te (in.)

8 + 3 8.89

8 + 4 ½ 10.11

10 + 3 10.51

10 + 4 ½ 11.75

12 + 3 12.12

12 + 4 ½ 13.38

14 + 3 13.72

14 + 4 ½ 15.02

16 + 3 15.31

16 + 4 ½ 16.64

Reference: Table 11-2 of CRSI Design Handbook 2002.

Note: Only first two columns of the table are reproduced here.

9



Sizes

Minimum Joist Depth

Slab and rib depth from equivalent flat slab thickness

Two-Way Joist

Table 02: Waffle flat slabs (30" × 30" voids at 3'-0")-Equivalent thickness


8 + 3 8.61

8 + 4 ½ 9.79

10 + 3 10.18

10 + 4 ½ 11.37

12 + 3 11.74

12 + 4 ½ 12.95

14 + 3 13.3

14 + 4 ½ 14.54

16 + 3 14.85

16 + 4 ½ 16.12

20 + 3 17.92

20 + 4 ½ 19.26

Reference: Table 11-2 of CRSI Design Handbook 2002.

Note: Only first two columns of the table are reproduced here.



Sizes

Minimum Width of Rib

ACI 8.11.2 states that ribs shall be not less than 4 inches in width.

Maximum Depth of Rib

ACI 8.11.2 also states that ribs shall have a depth of not more than 3

½ times the minimum width of rib.

Minimum Slab Thickness

ACI 8.11.6.1 states that slab thickness shall be not less than one-

twelfth the clear distance between ribs, nor less than 2 inch.

Two-Way Joist

10



Sizes

Solid Head

Dimension of solid head on either side of column centerline is equal to

l/6.

The depth of the solid head is equal to the depth of the combined

depth of ribs and top slab.

Two-Way Joist



Loads

Floor dead load for two-way joist with certain dome size, dome depth can

be calculated from the table shown for two options of slab thicknesses (3

inches and 4 ½ inches).

Two-Way Joist

Table 03: Standard Dome Dimensions and other Data

Dome Size Dome Depth

(inches)

Volume of Void

(ft3)

Floor Dead Load (psf) per slab

thickness

3 inches 4 ½ inches

30 inches

8 3.98 71 90

10 4.92 80 99

12 5.84 90 109

14 6.74 100 119

16 7.61 111 129

20 9.3 132 151

19 inches

8 1.56 79 98

10 1.91 91 110

12 2.25 103 122

14 2.58 116 134

16 2.9 129 148 Reference: Table 11-1, CRSI Design Handbook 2002

11



Loads

Floor dead load (wdj) for two-way joist can also be

calculated as follows:

Two-Way Joist

36″

8″

3″

30″

Volume of solid:

Vsolid = (36 36 11)/1728 = 8.24 ft3

Volume of void:

Vvoid = (30 30 8)/1728 = 4.166 ft3

Total Load of joists per dome:

wdj = (Vsolid – Vvoid) γconc

= ( 8.24 – 4.166) 0.15 = 0.61 kips/ dome

Total Load of joists per sq. ft:

wdj/ (dome area) = 0.61/ (3 3) = 0.0679 ksf

= 68 psf ≈ 71 psf (from table 03)

The difference is because sloped ribs are not considered. Plan



Loads

Taking a panel out of the system:

Two-Way Joist

l2

l1

wdj

wdj + wsh

l1

a a

b l2

wdj = Dead load at joist (load/ area)

wsh = Dead load of solid head (load/ area)

wdj + wsh

12



Loads

If the complete area l1 × l2 is assumed to occupy joists

alone, then the dead load in the area l1 × l2 will be wdj.

However since there are solid head regions present,

therefore additional dead load due to solid head region shall

be:

wdsh = hsolid γconc - wdj

Two-Way Joist



Loads

Factored loads can be calculated as:

If wL = live load (load/area)

wdj = dead load at joists, then

Factored load due to joists (wj)

wuj = 1.2 wdj + 1.6wL

Factored load due to solid head (wsh)

wush = 1.2(wdsh – wdj)

Two-Way Joist

wj wsh

l1

a a

b

wsh

l2

13



Analysis

ACI code allows use of DDM for analysis of waffle slabs (ACI

R13.1). In such a case, waffle slabs are considered as flat

slabs, with the solid head acting as drop panels (ACI 13.1.3).

25

Two-Way Joist



Analysis

Static moment calculation for DDM analysis:

26

Two-Way Joist

wuj

ln

Moj = wujl2ln2/8

wush

ln

a a

wush

Mosh = wushba2/2

Moj Mosh

Mo = Moj + Mosh

b l2

ln

14



Design

Design of slab for punching shear

The solid head shall be checked against punching shear.

The critical section for punching shear is taken at a section d/2 from face

of the column, where d is the effective depth at solid head.

27

Two-Way Joist



Design

Design of slab for

punching shear

Load on tributary area will

cause punch out shear.

Within tributary area, two

types of loads are acting:

Solid head load

Joist load

Both types shall be

considered while calculating

punching shear demand

28

Two-Way Joist

l2 d/2

l1

15



Design

Design of slab for punching

shear

Total area (At) = l1 l2

Solid head area = Asolid

Critical perimeter area = Acp

Vu =At wuj + Asolid (wish − wuj)

Where,

wuj = Factored load considering

joist alone

wsh = Factored solid head dead

load

29

Two-Way Joist

l2 d/2

l1



Design

Shear Strength of Slab in Punching Shear:

ΦVn = ΦVc + ΦVs

ΦVc is least of:

Φ4√ (fc′)bod

(2 + 4/βc) √ (fc′)bod

{(αsd/bo +2} √ (fc′)bod

βc = longer side of column/shorter side of column

αs = 40 for interior column, 30 for edge column, 20 for corner columns

Two-Way Joist

16



Design

Design of Joist for Beam Shear:

Beam Shear Demand

Beam shear is not usually a problem in slabs including waffle slabs.

However for completion of design beam shear may also be

checked. Beam shear can cause problem in case where larger

spans and heavier loads with relatively shallow waffle slabs are

used.

The critical section for beam shear is taken at a section d from face

of the column, where d is the effective depth at solid head.

Two-Way Joist



Design

Design of Joist for Beam Shear:

Beam shear capacity of concrete joist:

ΦVn = ΦVc + ΦVs

ΦVc is least of:

Φ2√ (fc′)bribd

ΦVs = ΦAvfy/bribs

For joist construction, contribution of concrete to shear strength Vc shall be

permitted to be 10 percent more than that specified in Chapter 11.

If required, one or two single legged stirrups are provided in the rib to

increase the shear capacity of waffle slab.

Two-Way Joist

Stirrup

17



Design

Design for Flexure

The design of waffle slab for flexure is done by usual procedures.

However, certain reinforcement requirements apply discussed next.

33

Two-Way Joist



ACI Recommendations on Reinforcement

Requirement of Waffle Slab:

Recommendations for Ribs:

ACI 10.6.7 states that if the effective depth d of a beam or joist

exceeds 36 inches, longitudinal skin reinforcement shall be provided as

per ACI section 10.6.7.

34

Two-Way Joist

18



ACI Recommendations on Reinforcement

Requirement of Waffle Slab:

Recommendations for Slab:

According to ACI 13.3.2, for cellular or ribbed construction reinforcement

shall not be less than the requirements of ACI 7.12.

As per ACI 7.12, Spacing of top bars cannot exceed 5h or 18 inches.

35

Two-Way Joist



Other Important Points:

The amount of reinforcement and, if necessary, the top slab

thickness can be changed to vary the load capacities for

different spans, areas, or floors of a structure.

Each joist rib contains two bottom bars. Straight bars are

supplied over the column centerlines for negative factored

moment.

36

Two-Way Joist

Bottom bar

19




For layouts that do not meet the standard 2-feet and 3-feet

modules, it is preferable that the required additional width be

obtained by increasing the width of the ribs framing into the

solid column head.

37

Two-Way Joist




The designer should sketch out the spacing for a typical panel

and correlate with the column spacing as a part of the early

planning.

38

Two-Way Joist

20



Example: Design the slab system of hall shown in figure as waffle

slab, according to ACI 318. Use Direct Design Method for slab

analysis.

fc′ = 4 ksi

fy = 60 ksi

Live load = 100 psf

39

Two-Way Joist



Solution:

A 108′ × 144′ building, divided into twelve (12) panels, supported at

their ends on columns. Each panel is 36′ × 36′.

The given slab system satisfies all the necessary limitations for Direct

Design Method to be applicable.

40

Two-Way Joist

21



Step No 01: Sizes

Columns

Let all columns be 18″ × 18″.

Slab

Adopt 30″ × 30″ standard dome.

Minimum equivalent flat slab thickness (hf) can be found using ACI Table

9.5 (c):

Exterior panel governs. Therefore,

hf = ln/33

ln = 36 – (2 × 18/2)/12 = 34.5′

hf = (34.5/33) × 12 = 12.45″

41

Two-Way Joist



Step No 01: Sizes

Slab

The closest depth of doom that will fulfill the requirement of equivalent

thickness of flat slab equal to 12.45″ is 12 in. with a slab thickness of 4 ½

in. for a dome size of 30-in.

42

Table: Waffle flat slabs (30" × 30" voids at 3'-0")-Equivalent thickness


8 + 3 8.61

8 + 4 ½ 9.79

10 + 3 10.18

10 + 4 ½ 11.37

12 + 3 11.74

12 + 4 ½ 12.95

14 + 3 13.3

14 + 4 ½ 14.54

16 + 3 14.85

16 + 4 ½ 16.12

20 + 3 17.92

20 + 4 ½ 19.26

Two-Way Joist

22



Step No 01: Sizes

Planning of Joist layout

43

Two-Way Joist

l = 36′-0″ = 432″

Standard module = 36″ 36″

No. of modules in 36′-0″:

n = 432/36 = 12

Planning:

First joist is placed on interior column

centerline with progressive placing of

other joists towards exterior ends of

panel. To flush the last joist with

external column, the width of exterior

joist comes out to be 15″ (6″+Column

size /2) as shown in plan view.



Step No 01: Sizes

Solid Head

Solid head dimension from column centerline = l/6 = 36/6 = 6′

Total required length of solid head= 2 6 = 12′

As 3′ 3′ module is selected, therefore 4 voids including joist witdh will

make an interior solid head of 12.5′ 12.5′. (Length of solid head = c/c

distance between rib + rib width )

Depth of the solid head = Depth of standard module = 12 + 4.5 = 16.5′′

44

Two-Way Joist

23



Step No 02: Loads

Floor (joist) dead load (wdj) = 109 psf = 0.109 ksf

45

Table: Standard Dome Dimensions and other Data

Dome Size Dome Depth (in.) Volume of Void

(ft3)

Floor Dead Load (psf) per slab

thickness

3 inches 4 ½ inches

30-in

8 3.98 71 90

10 4.92 80 99

12 5.84 90 109

14 6.74 100 119

16 7.61 111 129

20 9.3 132 151

19-in

8 1.56 79 98

10 1.91 91 110

12 2.25 103 122

14 2.58 116 134

16 2.9 129 148 Reference: Table 11-1, CRSI Design Handbook 2002

Two-Way Joist



Step No 02: Loads

Floor (joist) dead load (wdj) = 109 psf = 0.109 ksf

Solid Head dead load (wsh) = γchsh – wdj

= 0.15 ×{(12 + 4.5)/12} – 0.109 = 0.097 ksf

Two-Way Joist

wdj Wdj+sh

l1

a a

b

Wdj+sh

l2 a = 5.25 ft

b = 12.5 ft

24



Step No 02: Loads

wL = 100 psf = 0.100 ksf

Load due to joists plus LL (wuj)

wuj = 1.2 wdj + 1.6wL

= 1.2 0.109 + 1.60.100

= 0.291 ksf

Load due to solid head dead load (wush)

wush = 1.2wsh

= 1.2 0.097 = 0.1164 ksf

Two-Way Joist

wush

l1

a a

b l2

wuj

wush



Step No 03: Frame Analysis (E-W Interior Frame)

Step 1: Marking E-W Interior Frame:

48

Two-Way Joist

l2 = 36′

25




Step 2: Marking of Column and Middle Strips:

49

Two-Way Joist

l2 = 36′ CS/2 = Least of l1/4 or l2/4

l2/4 = 36/4 = 9′

CS/2 = 9′

MS/2 = 9′

MS/2 = 9′

CS/2 = 9′




Step 03: Static Moment Calculation

Moj (due to joists) = wojl2ln2/8

= 0.291 × 36 × 34.52/8 = 1557.56 ft-kip

Mosh (due to solid head excluding joists) = wush ba2/2

= 0.1164×12.5×5.252/2 = 20 ft-kip

Mo (total static moment) = Moj + Mosh = 1557.56 + 20 = 1577.56 ft-kip

Note: Since normally, Mosh is much smaller than Moj the former can be

conveniently ignored in design calculations.

50

Two-Way Joist

26




Step 03: Static Moment Calculation.

51

Two-Way Joist

l2 = 36′




Step 04: Longitudinal distribution of Total static moment (Mo).

52

Two-Way Joist

l2 = 36′ 0.26

0.52

0.70 0.65

0.35

0.65 0.70 0.26

0.52 Longitudinal

distribution

factors (D.F)L

27





53

Two-Way Joist

l2 = 36′ 410

820

1104 1025

552

1025 1104 410

820

Units: ft-kip

ML = Mo × (D.F)L




Step 05: Lateral Distribution of Longitudinal moment (L.M).

54

Two-Way Joist

l2 = 36′ 0.60

Lateral

distribution

factors (D.F)Lat

1.00 0.75 0.75 0.75

0.60 0.60

1.00 0.75

28





55

Two-Way Joist

l2 = 36′ 492

410 828 769 769

331 492

410 828 MLat = ML × (D.F)Lat

ML,ext- = 410 kip-ft

ML,ext+ = 820 kip-ft

ML,int- = 1104 kip-ft

ML,- = 1025 kip-ft

ML,+ = 552 kip-ft

328/2 328/2

328/2 328/2

0 0

0 0

276/2

276/2

276/2

276/2

221/2

221/2

256/2

256/2

256/2

256/2





56

Two-Way Joist

l2 = 36′ 27.3

22.8 46 42.7 42.7

18.38 27.3

22.8 46 MLat per foot = Mlat/strip width




ML,- = 1025 kip-ft

ML,+ = 552 kip-ft

18.22 18.22

18.22 18.22

0 0

0 0

15.33

15.33

15.33

15.33

12.3

12.3

14.22

14.22

14.22

14.22

29



Step No 03: Frame Analysis (E-W Exterior Frame)

Step 03: Static Moment Calculation

Moj (due to joists) = wojl2ln2/8

= 0.291 × 18.75 × 34.52/8 = 811.78 ft-kip

Mosh (due to solid head excluding joists) = wush ba2/2

= 0.1164×7×5.252/2 = 12.83 ft-kip

Mo (total static moment) = Moj + Mosh = 811.78 + 12.83 = 825 ft-kip

Note: Since normally, Mosh is much smaller than Moj the former can be

conveniently ignored in design calculations.

57

Two-Way Joist

l2 = 18 + c/2

= 18 + (18/12)/2

= 18.75′

c = column dimension




Step 03: Static Moment Calculation.

58

Two-Way Joist

l2 = 18.75′

30





59

Two-Way Joist

0.26

0.52

0.70 0.65

0.35

0.65 0.70 0.26

0.52

Longitudinal

distribution

factors (D.F)L





60

Two-Way Joist

215

429

578 536

289

536 578 215

429

Units: ft-kip

ML = Mo × (D.F)L

31





61

Two-Way Joist

0.60

Lateral

distribution

factors (D.F)Lat

1.00 0.75 0.75 0.75

0.60 0.60

1.00 0.75





62

Two-Way Joist

215 434 402 769 215 402

MLat = ML × (D.F)Lat




ML,- = 536 kip-ft

ML,+ = 289 kip-ft

172 172 0 0 144 144 116 134 134

257 173 257

32





63

Two-Way Joist

23.8 48.2 44.7 44.7 23.8 48.2




ML,- = 536 kip-ft

ML,+ = 289 kip-ft

17.64 17.64 0 0 14.76 14.76 11.89 13.74 13.74

28.6 19.2 28.6

MLat per foot = Mlat/strip width



Step No 03: Frame Analysis

Analysis of N-S Interior and Exterior Frame will be same as E-W respective

frames due to square panels.

64

N-S

Interior

Frame

l2 = 36′-0″

N-S

Exterior

Frame

l2 = 18′-9″

Two-Way Joist

33



Step No 04: Design

For E-W Interior Slab Strip:

65

Two-Way Joist

l2 = 36′ 27.3

22.8 46 42.7 42.7

18.38 27.3

22.8 46

18.22 18.22

18.22 18.22

0 0

0 0

15.33

15.33

15.33

15.33

12.3

12.3

14.22

14.22

14.22

14.22

Selective values

are used for design.



Step No 04: Design

E-W Exterior Frame.

66

Two-Way Joist

23.8 48.2 23.8 48.2

17.64 17.64 0 0 14.76 14.76 11.89

28.6 19.2 28.6

Selective values

are used for

design

34



Step No 04: Design

Design of N-S Interior and Exterior Frame will be same as E-

W respective frames due to square panels and also for the

reason that davg is used in design.

davg = 16.5 – (0.75 inch (cover) + ¾ inch (Assumed bar

diameter) = 15 inch

This will be used for both directions positive as well as

negative reinforcement.

67

Two-Way Joist



Step No 05: Detailing (E-W Frames)

68

#6 @ 12″ #6 @ 6″ #6 @ 6″ #6 @ 12″

#6 @ 12″ #6 @ 6″ #6 @ 6″ #6 @ 12″

#6 @ 18″ #6 @ 18″ #6 @ 18″ #6 @ 18″

Two-Way Joist

Negative Reinforcement in

E-W direction

For M = 46 ft-k =46 x12

= 552 in-k

d = 15

fy = 60 ksi

As = 0.7 in2

35



Step No 05: Detailing (N-S Frames)

69

#6 @ 6″

#6 @ 12″

#6 @ 6″

#6 @ 18″

#6 @ 18″

#6 @ 18″

#6 @ 12″

#6 @ 6″

#6 @ 6″

Two-Way Joist

Negative Reinforcement

in N-S direction



Step No 05: Detailing (N-S Frames)

Positive reinforcement

For M = 27 ft-k =27x12 = 324in-k

d = 15

fy = 60 ksi

As = 0.373 in2 This is per foot reinforcement. For 18 feet col strip, this will

be equal to 0.373 x 18 = 6.714 in2

There are 6 joists in 18 feet with. Therefore per rib reinforcement = 1.12

Using # 7 bars, 2 bars per joist rib will be provided.

70

Two-Way Joist

36



Step No 05: Detailing (E-W Interior Frame)

71

Column Strip (Interior Frame); section taken over support

Column Strip (Exterior Frame); section taken over support

#6 @ 12″ c/c

#6 @ 6″ c/c

2 #7 Bars

18′-0″

2 #7 Bars

Two-Way Joist



Step No 05: Detailing (E-W Interior Frame)

72

Middle Strip (Interior Frame); Section taken over column line

Middle Strip (Exterior Frame); Section taken over column line

#6 @ 18″ c/c

#6 @ 18″ c/c

2 #7 Bars

18′-0″

2 #7 Bars

Two-Way Joist

37



Step No 05: Detailing (E-W Exterior Frame)

73

Column Strip (Interior Frame); section over support

#6 @ 6″ c/c

2 #7 Bars

9′-0″

Column Strip (Exterior Frame); section over support

#6 @ 12″ c/c

2 #7 Bars

Two-Way Joist



Step No 05: Detailing (E-W Exterior Frame)

74

Middle Strip (Interior Frame) ; section over support

#6 @ 18″ c/c

2 #7 Bars

9′-0″

Middle Strip (Exterior Frame); section over support

#6 @ 18″ c/c

2 #7 Bars

Two-Way Joist

38



Step No 04: Design

Note: For the completion of design problem, the waffle slab

should also be checked for beam shear and punching shear.

75

Two-Way Joist



Moment Coefficient

Method

39



Two Way Slabs

Moment Coefficient Method (Introduction)

The Moment Coefficient Method included for the first time in

1963 ACI Code is applicable to two-way slabs supported on

four sides of each slab panel by walls, steel beams relatively

deep, stiff, edge beams (h = 3hf).

Although, not included in 1977 and later versions of ACI code,

its continued use is permissible under the ACI 318-08 code

provision (13.5.1). Visit ACI 13.5.1.

77



Two Way Slabs

Moment Coefficient Method

Moments:

Ma, neg = Ca, negwula2

Mb, neg = Cb, negwulb2

Ma, pos, (dl + ll) = M a, pos, dl + M a, pos, ll = Ca, pos, dl × wu, dl × la2 + Ca, pos, ll × wu, ll × la

2

Mb, pos, (dl + ll) = Mb, pos, dl + Mb, pos, ll = Cb, pos, dl × wu, dl × lb2 + Cb, pos, ll × wu, ll × lb

2

Where Ca, Cb = Tabulated moment coefficients

wu = Ultimate uniform load, psf

la, lb = length of clear spans in short and long directions respectively.

78

Ma,neg lb

la

Ma,neg

Ma,pos

Mb,neg Mb,neg Mb,pos

40



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs

Moment Coefficient Method: Cases

Depending on the support conditions, several cases are possible:

79



Two Way Slabs



80

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

41



Two Way Slabs



81

4 spans @ 25′-0″

3 sp

ans @

20

′-0″



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



82

42



Two Way Slabs

Moment Coefficient Tables:

83



Two Way Slabs


84

43



Two Way Slabs


85



Two Way Slabs


86

44



Two Way Slabs


87



Two Way Slabs


88

45



Two Way Slabs

Load Coefficient Table:

89



Two Way Slabs

90

Maximum Spacing and Minimum Reinforcement

Requirement:

Maximum spacing (ACI 13.3.2):

smax = 2 hf in each direction.

Minimum Reinforcement (ACI 7.12.2.1):

Asmin = 0.0018 b hf for grade 60.

Asmin = 0.002 b hf for grade 40 and 50.

46



Two Way Slabs

Special Reinforcement at Exterior Corner of Slab

The reinforcement at exterior ends of the slab shall be provided as per ACI

13.3.6 in top and bottom layers as shown.

The positive and negative reinforcement in any case, should be of a size and

spacing equivalent to that required for the maximum positive moment (per foot

of width) in the panel.

91



Two Way Slabs

Moment Coefficient Method

Steps

Find hmin = perimeter/ 180 = 2(la + lb)/180

Calculate loads on slab (force / area)

Calculate m = la/ lb

Decide about case of slab,

Use table to pick moment coefficients,

Calculate moments and then design.

Apply reinforcement requirements (smax = 2hf, ACI 13.3.2)

92

47



Two Way Slabs

Moment Coefficient Method: Example 1

A 100′ 60′, 3-storey commercial building is to be designed.

The grids of column plan are fixed by the architect.

93



Two Way Slabs


Complete analysis of the slab is done by analyzing four panels

94

Panel I Panel I

Panel I Panel I

Panel II Panel II

Panel III Panel III

Panel III Panel III

Panel IV Panel IV

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

48



Two Way Slabs


A 100′ 60′, 3-storey commercial building: Sizes and Loads.

Sizes:

Minimum slab thickness = perimeter/180 = 2 (20+25)/180 = 6″

However, for the purpose of comparison, take hf = 7″

Columns = 14″ 14″ (assumed)

Beams = 14″ 20″ (assumed)

Loads:

S.D.L = Nil ; Self Weight = 0.15 x (7/12) = 0.0875 ksf

L.L = 144 psf ; wu = 0.336 ksf

95



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs


Panels are analyzed using Moment Coefficient Method

96

Case = 4

m = la/lb = 0.8

Mb,neg Mb,neg Mb,pos Ma,neg

Ma,pos

Ma,neg

49



Two Way Slabs



97

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016



Two Way Slabs



98

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016

50



Two Way Slabs



99

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016



Two Way Slabs



100

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016

51



Two Way Slabs



101

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016



Two Way Slabs



102

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016

52



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



103

Panel I

Case = 4

m = la/lb = 0.8

Ca,neg = 0.071

Cb,neg = 0.029

Ca,posLL = 0.048

Cb,posLL = 0.020

Ca,posDL = 0.039

Cb,posDL = 0.016

Ma,neg = 9.5 k-ft

Ma,pos = 6.1 k-ft

Mb,neg = 6.1 k-ft

Mb,pos = 3.9 k-ft

Mb,neg Mb,neg Mb,pos Ma,neg

Ma,pos

Ma,neg

For slab supported on Spandrals, Mneg,ext = 1/3Mpos



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



104

Panel II

Case = 9

m = la/lb = 0.8

Ca,neg = 0.075

Cb,neg = 0.017

Ca,posLL = 0.042

Cb,posLL = 0.017

Ca,posDL = 0.029

Cb,posDL = 0.010

Ma,neg = 10.1 k-ft

Ma,pos = 5.1 k-ft

Mb,neg = 3.6 k-ft

Mb,pos = 3.1 k-ft


Ma,neg

Ma,pos

Ma,neg

53



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



105

Panel III

Case = 8

m = la/lb = 0.8

Ca,neg = 0.055

Cb,neg = 0.041

Ca,posLL = 0.044

Cb,posLL = 0.019

Ca,posDL = 0.032

Cb,posDL = 0.015

Ma,neg = 7.4 k-ft

Ma,pos = 5.4 k-ft

Mb,neg = 8.6 k-ft

Mb,pos = 3.7 k-ft


Ma,neg

Ma,pos

Ma,neg



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



106

Panel IV

Case = 2

m = la/lb = 0.8

Ca,neg = 0.065

Cb,neg = 0.027

Ca,posLL = 0.041

Cb,posLL = 0.017

Ca,posDL = 0.026

Cb,posDL = 0.011

Ma,neg = 8.7 k-ft

Ma,pos = 4.9 k-ft

Mb,neg = 5.7 k-ft

Mb,pos = 3.2 k-ft


Ma,neg

Ma,pos

Ma,neg

54



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs


Slab analysis summary

107

3.2

4.9 5.7 5.7

8.7

8.7

7.4

7.4

8.6 8.6 5.4

3.7

3.2

5.1 3.6

10.1

10.1

9.5

9.5

6.1 6.1

3.9

6.1

3.6



Two Way Slabs


Slab Reinforcement Details

108

A

B B B

C

C

C

C C B

A

A

B A

C

C

C

B B

A

A= #4 @ 12″

B = #4 @ 6″

C = #4 @ 4″

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

55



Two Way Slabs


Load On Beams from coefficient tables

109

Table: Load on beam in panel I, using

Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.71 - 2.39

B2 25 10 0.71 - 2.39

B3 20 12.5 - 0.29 1.22

B4 20 12.5 - 0.29 1.22

Panel I

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4



B1

B1

B2

B2

B3 B3 B3 B4 B4

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Panel I

Two Way Slabs



110


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.71 - 2.39

B2 25 10 0.71 - 2.39

B3 20 12.5 - 0.29 1.22

B4 20 12.5 - 0.29 1.22

56



Two Way Slabs



111


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.83 - 2.78

B3 20 12.5 - 0.17 0.714

B4 20 12.5 - 0.17 0.714

Panel II

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4



B1

B1

B2

B2

B3 B3 B3 B4 B4

Panel II

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



112


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.83 - 2.78

B3 20 12.5 - 0.17 0.714

B4 20 12.5 - 0.17 0.714

57



Two Way Slabs



113


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.55 - 1.84

B3 20 12.5 - 0.45 1.89

Panel III

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4



B1

B1

B2

B2

B3 B3 B3 B4 B4

Panel III

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



114


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.55 - 1.84

B3 20 12.5 - 0.45 1.89

58



Two Way Slabs



115


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.71 - 2.39

B3 20 12.5 - 0.29 1.22

Panel IV

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4



B1

B1

B2

B2

B3 B3 B3 B4 B4

Panel IV

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



116


Coefficients

(wu = 0.336 ksf)

Bea

m

Length

(ft)

Width

(bs) of

slab

panel

support

ed by

beam

Wa Wb

Load

due to

slab,

Wwubs

(k/ft)

B1 25 10 0.71 - 2.39

B3 20 12.5 - 0.29 1.22

59



Two Way Slabs


Load On Beams

117

2.39 k/ft

1.22 k/ft

2.39 k/ft

1.22 k/ft

2.77 k/ft

2.77 k/ft

0.71 k/ft 0.71 k/ft

1.84 k/ft

1.89 k/ft

1.84 k/ft

1.89 k/ft

2.39 k/ft

1.22 k/ft

2.39 k/ft

1.22 k/ft

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4



Two Way Slabs


Load On Beams

118

1.45 k/ft

2.62 k/ft

5.39 k/ft

0.94 k/ft

3.34 k/ft

2.07 k/ft

2.16 k/ft

4.46 k/ft

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

B1

B1

B2

B2

B3 B3 B3 B4 B4

Loads

including self

weight of

beams 0.23

kip/ft

60



Two Way Slabs


Similarly, slab analysis results of 25′ × 20′ structure.

119

Panel I Panel I

Panel I Panel I

Panel II Panel II

Panel III Panel III

Panel III Panel III

Panel IV Panel IV

4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Slab thickness = 6″

SDL = 40 psf

LL = 60 psf

fc′ =3 ksi

fy = 40 ksi

wu = 0.234 ksf



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



120

2.0

3.0 3.9 3.9

6.1

6.1

5.1

5.1

6.0 6.0 3.5

2.4

1.9

3.2 2.5

7

7

6.6

6.6

4.2 4.0

2.6

4.2

2.5

61



Two Way Slabs


Similarly, slab analysis results of 20′ × 15′ structure.

121

Panel I Panel I

Panel I Panel I

Panel II Panel II

Panel III Panel III

Panel III Panel III

Panel IV Panel IV

4 spans @ 20′-0″

3 sp

ans @

15

′-0″

Slab thickness = 6″

SDL = 40 psf

LL = 60 psf

fc′ =3 ksi

fy = 40 ksi

wu = 0.234 ksf



4 spans @ 25′-0″

3 sp

ans @

20

′-0″

Two Way Slabs



122

1.0

1.8 2.1 2.1

3.6

3.6

3.2

3.2

3.4 3.4 2.2

1.3

0.9

2.0 1.3

4.1

4.1

4

4

2.2 2.5

1.3

2.2

1.3

62



References

CRSI Design Handbook

ACI 318

Design of Concrete Structures 13th Ed. by Nilson, Darwin and

Dolan.

123



The End

124

Analysis and Design of Slabs

Education

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