DESIGN OF TWO WAY SLAB (with beams) BY DDM Fig 1: Two way slab with beams. Problem: A two way slab floor with a total area of 7500 sq ft. is divided into 25 panels with a panel size of 20 ft. x 15 ft. f c ′ = 3000 psi f y = 60000 psi Service Live Load = 120 psf All Column = 14” x 14” Slab Thickness = 6.5” Storey Height = 12’ Long Beam = 14” x 28” Short Beam = 12” x 24” Solution: 1. Calculation of Factored Load
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DESIGN OF TWO WAY SLAB (with beams) BY DDM
Fig 1: Two way slab with beams.
Problem:
A two way slab floor with a total area of 7500 sq ft. is divided into 25 panels with a panel size of 20 ft. x 15 ft.
fc′ = 3000 psify = 60000 psiService Live Load = 120 psfAll Column = 14” x 14”Slab Thickness = 6.5”Storey Height = 12’Long Beam = 14” x 28”Short Beam = 12” x 24”
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 (20’) to shorter (15’) span c/c of supports is (20/15)=1.33<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 20) =6.67’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 (20x12) x 10% = 24” and column width in this direction is 14”, which is less than 24”.
In transverse direction, 10% of the shorter span is (15x12) x 10% = 18” and column width in this direction is 14”, which is less than 18”.
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 = 81.25 psfLL/DL = (120/81.25) = 1.477<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.
Taking l1 and l2 in the long and short directions respectively,
Panel 1, l12
α1= 202
0.5 (18.75+11.02)=26.87
l22
α2= 152
0.5 (7.47+4.44 )=37.78
l12
α1l22
α2
=26.8737.78
=0.738
Panel 2, l12
α1= 202
0.5 (18.75+11.02)=26.87
l22
α2= 152
0.5 (4.44+4.44 )=50.68
l12
α1l22
α2
=26.8750.68
=0.530
Panel 3, l12
α1= 202
0.5 (11.02+11.02 )=36.30
l22
α2= 152
0.5 (7.47+4.44 )=37.78
l12
α1l22
α2
=36.3037.78
=0.961
Panel 4, l12
α1= 202
0.5 (11.02+11.02 )=36.30
l22
α2= 152
0.5 (4.44+4.44 )=50.68
l12
α1l22
α2
=36.3050.68
=0.716
All ratio of (l12/α1) to (l22/α2) lie between 0.2 and 5.0
This problem satisfies all the limitations imposed by ACI 13.6.1 for using DDM.
6. Longitudinal Distribution of Moment
Fig 5: Longitudinal Moment diagram for exterior span
Fig 6: Longitudinal moment diagram for interior span.
Column Strip Moment, Percent of Total Moment at Critical Section l2/l1
0.5 1.0 2.0
Interior Negative Moment (l2/l1) = 0 (l2/l1) ≥ 1.0
7590
7575
7545
Exterior Negative Moment (l2/l1) = 0 t = 0
t ≥ 2.510075
10075
10075
(l2/l1) ≥ 1.0t = 0t ≥ 2.5
10090
10075
10045
Positive Moment (l2/l1) = 0 (l2/l1) ≥ 1.0
6090
6075
6045
i. Percentage of Exterior Negative Moment
Frame A
(l2/l1) t (l2/l1)
8.27
0.5 0.75 1
0 100 100 100
1.30 90.9
2.5 90 82.5 75
Frame B
(l2/l1) t (l2/l1)
14.06
0.5 0.75 1
0 100 100 100
1.30 90.9
2.5 90 82.5 75
Frame C
(l2/l1) t (l2/l1)
5.91
1 1.33 2
0 100 100 100
1.7475.7
12.5 75 65.1 45
Frame D
(l2/l1) t (l2/l1)
9.941 1.33 2
0 100 100 100
1.7475.7
12.5 75 65.1 45
ii. Percentage of Positive Moment
Frame A
(l2/l1) 0.5 0.75 1 (l2/l1) =
8.2790 82.5 75
Frame B
(l2/l1) 0.5 0.75 1 (l2/l1) =
14.0690 82.5 75
Frame C
(l2/l1) 1 1.33 2 (l2/l1) =
5.9175 65.1 45
Frame D
(l2/l1) 1 1.33 2 (l2/l1) =
9.9475 65.1 45
iii. Percentage of Interior Negative Moment
Frame A
(l2/l1) 0.5 0.75 1 (l2/l1) =
8.2790 82.5 75
Frame B
(l2/l1) 0.5 0.75 1 (l2/l1) =
14.0690 82.5 75
Frame C
(l2/l1) 1 1.33 2 (l2/l1) =
5.9175 65.1 45
Frame D
(l2/l1) 1 1.33 2 (l2/l1) =
9.9475 65.1 45
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(20 x 12) = 60”0.25 l2 = 0.25(15 x 12) = 45”
y = 45”
For frame A:Column strip = 2 x 45” = 90”Half middle strip = 2@ [(15 x 12)-90]/2 = 2@45”
For frame B:
Column strip = 45”Half middle strip = 45”
Fig 10: Middle strip and column strip diagram for frame C & DFor frame C & D:
0.25 l1 = 0.25(15 x 12) = 45”0.25 l2 = 0.25(20 x 12) = 60”
y = 45”
For frame C:Column strip = 2 x 45” = 90”Half middle strip = 2@ [(20 x 12)-90]/2 = 2@75”
For frame D:Column strip = 45”Half middle strip = 75”
e. Summary of Calculation
Equivalent Rigid Frame
A B C D
Total Transverse Width (in)
180 90 240 120
Column Strip Width (in) 90 45 90 45
Half Middle Strip (in) 2@45 45 2@75 75
Torsional Constant C (in) 10700 10700 19100 19100
Is (in4) in t 4120 4120 5493 5493
t = C/2Is 1.30 1.30 1.74 1.74
11.02 18.75 4.44 7.47
(l2/l1) 0.75 0.75 1.33 1.33
(l2/l1) 8.27 14.06 5.91 9.94
External (-ve) Moment, % to Column Strip
90.90 90.90 75.71 75.71
Positive Moment, % to Column Strip
82.5 82.5 65.1 65.1
Internal (-ve) Moment, % to Column Strip
82.5 82.5 65.1 65.1
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.