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’. f c′ = 4000 psi f y = 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

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= (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 long direc directio tion: n:

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 shor shortt direct direction ion::

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.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)

(ksi)

Without Drop Panels Exterior Panels Interior α=0 α ≥ 0.8 Panels

40 60 75

ln ln ln

f y

/33 /30 /28

n n n

l /36 l /33 l /31

n n n

l /36 l /33 l /31

With Drop Panels Exterior Panels Interior α=0 α ≥ 0.8 Panels n n n

l /36 l /33 l /31

n n n

l /40 l /36 l /34

/40 l /36 l n /34 l n n

a. For interior panel: For α = 0 and f y = 60 ksi and clear span, ln = (25-4.43) = 20.57’ tmin = ln /36 /36 = (20.57*12)/36 = 6.86” < 7.5” b. For ext exteri erior or panel: panel:

For α = 0 and f y = 60 ksi and clear span, ln = (25-4.43) = 20.57’ tmin = ln /33 /33 = (20.57*12)/33 = 7.48” < 7.5” So, given thickness of slab =7.5” (OK) c. Thickn Thickness ess of drop drop panel: 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. Ther There e shal shalll be a mini minimu mum m of thre three e cont contin inuo uous us span spans s in each each







2. Panels Panels shall be rectangular rectangular,, with a ratio of longer to shorter 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 s horter (20’) span c/c of supports is (25/20)=1.25<2.0 3. Succ Succes essiv sive e span span leng length ths s cent center er-t -too-ce cent nter er of supp support orts s in each 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. Offs Offset et of colu column mns s by a maxim maximum um of 10 perce percent nt of the the span span (in dire direct ctio ion n of offs offset et)) from from eith either er axis axis betw betwee een n cent center erli line nes s 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 shall be due to gravity gravity only and uniform uniformly ly distribu distributed ted 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 ( l 12/α1) to (l 22/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. Sinc Since, e, ther there e is no beam beam used, used, this this limi limita tati tion on is not not requ require ired d for for this this problem. This problem satisfies all the limitations imposed by ACI 13.6.1 for using DDM.







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 M o.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”







8. Transverse Distribution of Longitudinal Moment a. Aspect Ratio (l 2 /l 1):

For frame A and B: l2 /l /l1 = 20/25 = 0.80 For frame C and D: l2 /l /l1 = 25/20 = 1.25 b. Calculation of β t:

βt=C2Is For frame A and B: Is=(20×12)7.5312=8438 in4 Torsional Constant, C = 4330 in 4 βt=C2Is=43302×8438=0.26 For frame C and D: Is=(25×12)7.5312=10550in4 Torsional Constant, C = 4330 in 4 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

A 4330 8438 0.26 0 0 80

B 4330 8438 0.26 0 0 80

C 4330 10550 0.21 0 1 25

D 4330 10550 0.21 0 1 25







α (l2 /l1) = 0 α (l2 /l1) ≥ 1.0

Exterior Negative 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

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

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

1 25

2







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









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”







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”







Equivalent Rigid Frame Total Transverse Width (in)

A

B

C

D

240

120

300

150

Column Strip Width (in)

120

60

120

60

Half Middle Strip (in)

2@60

60

2@90

90

Torsional Constant C (in) Is (in4) in β

4330

4330

4330

4330

8438

8438

10550

10550

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

β

t

t

= C/2Is







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





Exterior Span -ve +ve -ve Momen Momen Momen t t t 6 2 20

Interior Span -ve +ve -ve Momen Momen Momen t t t 90 03 90

Total Moment in







Equivalent Rigid Frame D





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

Design of Flat Slab by DDM

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