Part 5 - One way and two-way slab design
To design the slab, we specify the reinforcement detail according to the span length by either one-way slab or two-way slab.
The major difference between one way and two-way slab is the tendency of a slab to bend in a different direction.
One way slab bend in one direction and two-way slab bend in both the direction.
If the ratio of longer span to a shorter span of the slab is less than 2 then the slab is called a two-way slab. and, if this ratio is higher than two then the slab is called a one-way slab.
For better understanding, every designer or civil engineer should visit the site and check the slab reinforcement arrangement, the following is an image from my site for the slab.
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| On-site, one-way slab |
One way slab
One way slab is mostly used in balcony, chajjas, passage, etc. where there is a huge difference between the shorter and longer span and the ratio is more than two.
One way slab is more economical than the two-way slab and but also the two-way slab transfers the load in both the direction.
The simple one-way slab reinforcement details are as follows.
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| One-way slab, bend tendency |
The design of a one-way slab can be done by using the Indian standard code. ( IS: 456-2000 )
There are some basic steps to calculate and design the one-way simply supported slab. which are given below.
One way slab design steps
- Effective depth,
- Effective span,
- Load calculation,
- Bending moment calculation,
- Check for effective depth,
- Reinforcement detail,
- Design checks.
Design calculation
To design the one-way slab, we assume the 1-meter strip. which means in our design calculation the breadth, b will be equal to 1000 mm.
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| S-6 one-way slab plan |
We will calculate the S-6 slab, which is a one-way slab, and the usage will be the passage.
The loads will transfer from S-6 to beams B-7, 8, and 5.
Following is the simplified version of the above plan.
The dimension of the above plan is the c/c distance. the slab is resting on the beam with a size of 230 mm in all the directions.
To calculate the S-6, we have the following value,
fck = 20
fy = 415 N/mm2
Shorter span = lx = 0.953 m (c/c distance)
Longer span = ly = 4.76 m (c/c distance)
Live load = 3 kN/㎡
Floor finish = 1 kN/㎡
Effective depth
(From IS: 456-2000, P.-37 & 38)
`\frac{l}{d} = 20 X M.F`
M.F = 1.15 ( IS: 456-2000, P.- 38, fig. 4 )
fs = 240 N/ mm2
`d = \frac{953}{20 X 1.15} = 41.43 mm`
Provide, d = 76 mm and assume 8 mm bars
Total depth, D = d + Φ/2 + clear cover
= 76+ 4 + 20
D = 100 mm
Effective span
( from IS: 456-2000, p- 34, Cl. 22.2 (a) )
(1) lx + d = 953 + 76 = 1029 mm
(2) c/c of support = 953 + 230 = 1183
Whichever is smaller,
Effective span, l = 1029 mm.
Load calculation
We have considered 1 m strip,
(1) dead load = (0.100 X 1 X 25) = 2.5 kN/m
(2) floor finish = ( 1 X 1) = 1 kN/m
(3) live load = ( 3 X 1) = 3 kN/m
Total load = 2.5 + 1 + 3 = 6.5 kN/m
Factored load w = 6.5 X 1.5 = 9.75 kN/m
Bending moment
`M_{u} = \frac{wl^{2}}{8}`
`M_{u} = \frac{9.75 X 1.02^{2}}{8}`
Mu = 1.26 kN⋅m
Check for effective depth
( SP. 16, P- 10, table - C)
`M_{u} = 0.138 f_{ck} b d^{2}`
`1.26 X 10^{6} = 0.138 X 20 X 1000 X d^{2}`
d = 21.37 mm < 76 mm ....... O.K.
Steel
Main steel
`p_{t} = 50 \frac{f_{ck}}{f_{y}} [ 1 - \sqrt{1 - \frac{4.6 M_{u}}{f_{ck}b d^{2}}}]`
`= 50 X \frac{20}{415} [ 1 - \sqrt{1 - \frac{4.6 X 1.26 X 10^{6}}{20 X 1000 X 76^{2}}}]`
pt = 0.061 %
According to IS standard, the minimum percentage of steel for Fe 415 is 0.12 %.
pt = 0.12 %
`A_{st} = \frac{p_{t}}{100} bd`
`A_{st} = \frac{0.12}{100} X 1000 X 76`
Ast = 91.2 mm2
For 8 mm bars,
`a_{st} = \frac{\pi}{4} X d^{2} = 50.24 mm^{2}`
Spacing,
`\frac{50.24}{91.2} X 1000 = 550.87 mm`
According to IS specification, the minimum spacing between bars shall be,
(1) 3 d = 3 x 76 = 228 mm
(2) 300 mm
Whichever is smaller,
Provide, 228 mm c/c.
`A_{st} = \frac{50.24}{228} X 1000`
Ast = 220.35 mm2
Provide, 8 mm Φ - 230 mm c/c
Distribution steel
pt = 0.12 %
Ast = 91.2 mm2
According to IS: 456- 2000 P.- 48, CL. 26.5.2.1,
Minimum spacing for distribution bars is,
(1) 5d = 5 X 76 = 380 mm
(2) 450 mm
Whichever is smaller,
Provide, 380 mm c/c.
Assume 8 mm dia. bars,
`A_{st} = \frac{50.24}{380} X 1000`
Ast = 132.21 mm2
Provide, 8 mm Φ - 380 mm c/c
(6 mm dia. bars also can be used)
Design Checks
Check for cracking
For main steel,
(1) 3 d = 3 x 76 = 228 mm
(2) 300 mm
230 mm provided < 300 mm ..... O.K.
For distribution steel,
(1) 5d = 5 X 76 = 380 mm
(2) 450 mm
380 mm provided < 450 mm ..... O.K.
Check for deflection
Allowable, l/d = 20 X M.F = 20 X 1.15 = 23
Actual, l/d = 1029/76 = 13.53
13.53 < 23 ..... O.K.
Check for development length
`L_{d} \leq 1.3 \frac{M_{1}}{V} + L_{0}`
(1) d = 76 mm
(2) 12 Φ = 12 X 8 = 96 mm
Taking larger of two value,
L0 = 96 mm
Shear force at support,
take, l = clear span
`V_{u} = \frac{wl}{2}`
`V_{u} = \frac{9.75 X 0.953}{2} = 4.64 kN`
50 % are bent up bars near the support,
Ast at support = 50 % of Ast at mid-span
= 220.35 X 50/100
= 110.175 mm2
(From IS: 456-2000, P. - 96 )
`M_{1} = 0.87 f_{y} A_{st} d (1- \frac{f_{y}A_{st}}{f_{ck}bd})`
`M_{1} = 0.87 X 415 X 110.18 X 76 (1- \frac{415 X 110.18}{20 X 1000 X 76})`
M1 = 2.93 kN⋅m
`1.3 \frac{M_{1}}{V} + L_{0} = 1.3 X \frac{2.93 X 10^{6}}{4.64 X 10^{3}} + 96`
= 916.90 mm
(From IS: 456-2000, P.-42)
`L_{d} = \frac{\phi \sigma_{s}}{4 \tau_{bd}}`
`\sigma_{s}` = stress in bar at the section considered at design load.
`\sigma_{s} = 0.87 X f_{y} = 0.87 X 415`
= 361.05 N/mm2
`\tau_{bd} = 1.2 X \frac{160}{100}`
(Increased by 60 % = 100+60 = 160 %)
`\tau_{bd}` = 1.92 N/ mm2
`L_{d} = \frac{\phi \sigma_{s}}{4 \tau_{bd}}`
`L_{d} = \frac{8 X 361.05}{4 X 1.92} = 376.09 mm`
376.09 mm < 916.90 mm ..... O.K.
Check for shear
`\tau_{v} < \tau_{c}'`
`\tau_{v} = \frac{V_{u}}{bd} = \frac{4.64 X 10^{3}}{1000 X 76} = 0.061 mm^{2}`
50 % bent up bars at support,
Ast = 110.175 mm2
`p_{t} = \frac{100 A_{st}}{bd} = \frac{100 X 110.18}{1000 X 76}`
= 0.145 %
( From IS: 456- 2000, P.- 73, table - 1 )
`\tau_{c}` = 0.28 N/ mm2
k = 1.30 ( IS: 456- 2000, P.- 72, CL. 40.2.1.1)
`\tau_{c}' = k \cdot \tau_{c}`
= 1.30 X 0.28
= 0.364 N/ mm2
0.061 < 0.364 ..... O.K.
So, from the above design checks, we can say that our slab is safe.
Reinforcement details
The reinforcement details for the Slab S-6 is given below.
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| One way slab reinforcement detail |
Any doubt regarding this article should be asked.
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