Design Of Flat Slab By Ddm

DESIGN OF FLAT SLAB BY DDM

Fig 1: Flat Slab Design.

Problem: A flat slab floor with a total area of 12500 sq. ft. is divided into 25 panels with a panel size of 25’ x 20’. fc′ = 4000 psi fy = 60000 psi Service Live Load = 120 psf Slab Thickness = 7.5” Storey Height = 10’ Solution: 1. Calculation of Factored Load Slab Thickness = 7.5” Neglecting the weight of the drop panel, the service dead load is, DL = (7.5/12) x 150 = 94 psf LL = 120 psf Total factored load, W = 1.4 DL + 1.7 LL

= (1.4 x 94) + (1.7 x 120) = 336 psf = 0.336 ksf 2. Total Factored Static Moment in Equivalent Rigid Frame Total factored moment for flat slab, Mo=18Wl2l12(1- 2c3l1) 2 Where, c = diameter of the column capita. a. In long direction: c/c distance in short direction, l2 = 20’ c/c distance in short direction, l1 = 25’ Mo.long=18×0.336×20×252[1-2×53×25]2=395 k.ft. b. In short direction: c/c distance in short direction, l1= 20’ c/c distance in short direction, l2= 25’ Mo.short=18×0.336×25×202[1-2×53×20]2=292 k.ft. c. Check:

Fig 2: Equivalent Square Column.

Area of column capita= π c24= π ×524=19.63 sq.ft. Equivalent square = √ (19.63) = 4.43’ = 53” Mo.long=18×0.336×20×[25-4.43]2=356 k.ft.<395 k.ft. Mo.short=18×0.336×25×[20-4.43]2=255 k.ft.<292 k.ft. Therefore,

Mo.long = 395 k.ft

and

Mo.short = 292 k.ft

3. Relative Stiffness, α Since there is no edge beam, α = 0 for all. 4. Check for Slab Thickness ACI – Table 9.5(c)

fy (ksi)

40 60 75

Without Drop Panels Exterior Panels Interior α=0 α ≥ 0.8 Panels ln/33 ln/36 ln/36 ln/30 ln/33 ln/33 ln/28 ln/31 ln/31

With Drop Panels Exterior Panels Interior α=0 α ≥ 0.8 Panels ln/36 ln/40 ln/40 ln/33 ln/36 ln/36 ln/31 ln/34 ln/34

a. For interior panel: For α = 0 and fy = 60 ksi and clear span, ln = (25-4.43) = 20.57’ tmin = ln/36 = (20.57*12)/36 = 6.86” < 7.5” b. For exterior panel: For α = 0 and fy = 60 ksi and clear span, ln = (25-4.43) = 20.57’ tmin = ln/33 = (20.57*12)/33 = 7.48” < 7.5” So, given thickness of slab =7.5” (OK) c. Thickness of drop panel: Thickness of drop panel = thickness of slab + ¼ the projection of the drop panel beyond the column capital = 7.5 + ¼(18) = 12”

5. Check for Limitation of DDM 1. There shall be a minimum of three continuous spans in each direction. In this problem, there are five continuous spans in each direction.

2. Panels shall be rectangular, with a ratio of longer to shorter span center-to-center of supports within a panel not greater than 2. Here panels are rectangular and the ratio of longer (25’) to shorter (20’) span c/c of supports is (25/20)=1.25<2.0 3. Successive span lengths center-to-center of supports in each direction shall not differ by more than one-third the longer span. One-third of longer span is (1/3 x 25) =8.33’ In both directions, span lengths are equal. 4. Offset of columns by a maximum of 10 percent of the span (in direction of offset) from either axis between centerlines of successive columns shall be permitted. In longitudinal direction, 10% of the longer span is (25x12) x 10% = 30” and column width in this direction is 18”, which is less than 30”. In transverse direction, 10% of the shorter span is (20x12) x 10% = 24” and column width in this direction is 18”, which is less than 24”. 5. All loads shall be due to gravity only and uniformly distributed over an entire panel. Live load shall not exceed two times dead load. Service LL = 120 psf and service DL = 94 psf LL/DL = (120/94) = 1.277<2.0 6. The relative stiffness ratio of (l12/α1) to (l22/α2) must lie between 0.2 and 5.0 where α is the ratio of the flexural stiffness of the included beam to that of the slab. Since, there is no beam used, this limitation is not required for this problem. This problem satisfies all the limitations imposed by ACI 13.6.1 for using DDM.

6. Longitudinal Distribution of Moment

Fig 3: Longitudinal Moment diagram for exterior span.

Fig 4: Longitudinal moment diagram for interior span.

Fig 5: Equivalent rigid frames.

Total factored static moment for equivalent rigid frame A, B, C and D are, Mo.A (frame A) = 395 kip.ft kip.ft

Mo.B (frame B) = ½ Mo.A = ½ x 395 = 198

Mo.C (frame C) = 292 kip.ft kip.ft

Mo.D (frame D) = ½ Mo.C = ½ x 292 = 146

From fig 3 (case 3) and fig 4,

Fig 6: Longitudinal moment distribution.

7. Torsional Constant Torsional constant, C=∑1-0.63xyx3y3 Here, x = Smaller dimension y = Larger dimension Use larger value of C.

Fig 7: Determination of Torsional Constant.

Distance from the outer edge of exterior column to the inner edge of equivalent square = 53” / 2 + ½ x 18” = 35.5”

Fig 8: Imaginary Beam.

For both long and short direction, C=1-0.637.535.57.53×35.53=4330 in4

8. Transverse Distribution of Longitudinal Moment a. Aspect Ratio (l2/l1): For frame A and B: l2/l1 = 20/25 = 0.80 For frame C and D: l2/l1 = 25/20 = 1.25 b. Calculation of βt: βt=C2Is For frame A and B: Is=(20×12)7.5312=8438 in4 Torsional Constant, C = 4330 in4 βt=C2Is=43302×8438=0.26 For frame C and D: Is=(25×12)7.5312=10550in4 Torsional Constant, C = 4330 in4 βt=C2Is=43302×10550=0.21 c. Calculation of Percentage of Moment in Column Strip: Frame C Is β t=C/2Is α l2/l1 α (l2/l1)

A 4330 8438 0.26 0 0.80 0

B 4330 8438 0.26 0 0.80 0

C 4330 10550 0.21 0 1.25 0

D 4330 10550 0.21 0 1.25 0

Column Strip Moment, Percent of Total Moment at Critical Section l2/l1 0.5 1.0 2.0 Interior Negative Moment

α (l2/l1) = 0 α (l2/l1) ≥ 1.0 Exterior Negative Moment α (l2/l1) = 0 α (l2/l1) ≥ 1.0 Positive Moment α (l2/l1) = 0 α (l2/l1) ≥ 1.0

75 90

75 75

75 45

β t= 0 β t ≥ 2.5

100 75

100 75

100 75

β t= 0 β t ≥ 2.5

100 90

100 75

100 45

60 90

60 75

60 45

i. Percentage of Exterior Negative Moment Frame A α (l2/l1)

β

(l2/l1)

t

0.5 0.80 0

1

0

100 100 100

0.26

97.4

2.5

75

75

75

Frame B α (l2/l1)

β

(l2/l1)

t

0.5 0.80 0

1

0

100 100 100

0.26

97.4

2.5

75

75

75

Frame C α (l2/l1)

β

(l2/l1)

t

1 0

1.25

2

0

100 100 100

0.21

97.9

2.5

75

75

75

Frame D α (l2/l1) 0

β

(l2/l1)

t

1 0 0.21

1.25

2

100 100 100 97.9

2.5

75

75

75

ii. Percentage of Positive Moment Frame A (l2/l1) α (l2/l1) = 0

0.5 0.80 1 60 60 60

Frame B (l2/l1) α (l2/l1) = 0

0.5 0.80 1 60 60 60

Frame C (l2/l1) α (l2/l1) = 0

1 1.25 2 60 60 60

Frame D (l2/l1) α (l2/l1) = 0

1 1.25 2 60 60 60

iii. Percentage of Interior Negative Moment Frame A (l2/l1) α (l2/l1) = 0

0.5 0.80 1 75 75 75

Frame B (l2/l1) α (l2/l1) = 0

0.5 0.80 1 75 75 75

Frame C (l2/l1) α (l2/l1) = 0

1 1.25 2 75 75 75

Frame D (l2/l1) α (l2/l1) = 0

1 1.25 2 75 75 75

d. Transverse Distribution of Longitudinal Moment

Fig 9: Middle strip and column strip diagram for frame A & B

For frame A &B: 0.25 l1 = 0.25(25 x 12) = 75” 0.25 l2 = 0.25(20 x 12) = 60” y = 60” For frame A: Column strip = 2 x 60” = 120” Half middle strip = 2@ [(20 x 12)-120]/2 = 2@60” For frame B: Column strip = 60” Half middle strip = 60”

Fig 10: Middle strip and column strip diagram for frame C & D

For frame C & D: 0.25 l1 = 0.25(20 x 12) = 60” 0.25 l2 = 0.25(25 x 12) = 75” y = 60” For frame C: Column strip = 2 x 60” = 120” Half middle strip = 2@ [(25 x 12)-120]/2 = 2@90” For frame D: Column strip = 60” Half middle strip = 90”

e. Summary of Calculation

Equivalent Rigid Frame Total Transverse Width (in) Column Strip Width (in)

A

B

C

D

240

120

300

150

120

60

120

60

Half Middle Strip (in)

2@60

60

2@90

90

Torsional Constant C (in)

4330

4330

4330

4330

8438

8438

10550

10550

β t = C/2Is

0.26

0.26

0.21

0.21

α

0

0

0

0

(l2/l1)

0.80

0.80

1.25

1.25

α (l2/l1)

0

0

0

0

External (-ve) Moment, % to Column Strip

97.4

97.4

97.9

97.9

Positive Moment, % to Column Strip

60

60

60

60

Internal (-ve) Moment, % to Column Strip

75

75

75

75

Is (in4) in β

t

9. Distribution of Factored Moment in Column Strip and Middle Strip All the moments are divided into three parts, percentage to column strip (of which 85% goes to the beam and 15% to the slab) and rest to the middle strip slab. Equivalent Rigid Frame A Total Width = 240” Moments at Vertical Section (kip.ft) Total Moment in Frame A % to Column Strip Moment in Beam Moment in Column Strip Slab Moment in Mid

Column Strip = 120”

Middle Strip = 120”

Exterior Span Interior Span -ve +ve -ve -ve +ve -ve Momen Momen Moment Moment Momen Momen t t t t -103 206 -277 -257 139 -257 97.4 -86 -15

60 105 19

75 -177 -32

75 -164 -29

60 71 13

75 -164 -29

-3

83

-70

-65

56

-65

Strip Slab Equivalent Rigid Frame B Total Width = 120”

Column Strip = 60”

Middle Strip = 60”

Moments at Vertical Section (kip.ft)

Exterior Span -ve +ve -ve Momen Momen Momen t t t -52 103 -139

Interior Span -ve +ve -ve Momen Momen Momen t t t -129 70 -129

Total Moment in Frame B % to Column Strip Moment in Beam Moment in Column Strip Slab Moment in Mid Strip Slab

97.4 -43 -8

60 53 10

75 -89 -16

75 -83 -15

60 36 7

75 -83 -15

-2

42

-35

-33

28

-33

Equivalent Rigid Frame C Total Width = 300”

Column Strip = 120”

Middle Strip = 180”

Moments at Vertical Section (kip.ft)

Exterior Span -ve +ve -ve Momen Momen Momen t t t -76 152 -205

Interior Span -ve +ve -ve Momen Momen Momen t t t -190 103 -190

Total Moment in Frame C % to Column Strip Moment in Beam Moment in Column Strip Slab Moment in Mid Strip Slab

97.9 -64 -12

60 78 14

75 -131 -23

75 -122 -22

60 53 10

75 -122 -22

-2

61

-52

-48

42

-48

Equivalent Rigid Frame D Total Width = 150”

Column Strip = 60”

Middle Strip = 90”

Moments at Vertical Section (kip.ft)

Exterior Span -ve +ve -ve Momen Momen Momen t t t -38 76 -103

Interior Span -ve +ve -ve Momen Momen Momen t t t -95 52 -95

Total Moment in Frame D % to Column Strip Moment in Beam Moment in Column Strip Slab Moment in Mid Strip Slab

97.9 -32 -6

60 39 7

75 -66 -12

75 -61 -11

60 27 5

75 -61 -11

-1

31

-26

-24

21

-24

Fig: Moment Distribution.