A flat slab sits directly on columns, with no beams. The usual hand method cuts the floor into frames along each column line, one way and then the other. Each frame (a slab strip plus its columns) is analysed for the full load; its moments are then shared between a column strip over the columns (which takes most of them) and a middle strip between. Punching at the columns and deflection finish the design. Every number below comes from the free STATIX flat slab calculator with the inputs in step 1.
1. The problem
Input
Value
Design code
SANS 10100-1
Spans along x m, comma list
6, 6, 6
Spans along y m, comma list
6, 6, 6
Slab depth h
275 mm
Column cx × cy
450 mm
Column size along y mm
450 mm
Storey above / below 0 = none
3 m
Storey height below m
3 m
Extra dead gk + self-weight
1.5 kPa
Imposed qk
2.5 kPa
Concrete fcu
30 MPa
Steel fy
450 MPa
Cover
40 mm
Exposure
Moderate
2. Frames along each column line SANS 10100-1 4.6.5.1
Each frame is the slab strip between the mid-lines of the panels on either side, with the columns above and below. It is analysed with the full design load to get the moment at every column and in every span. Then the same is done the other way.
spans along x: 6.00 + 6.00 + 6.00 m
spans along y: 6.00 + 6.00 + 6.00 m
slab h = 275 mm, cover 40 mm, columns 450 × 450 mm
dead g_k = 8.10 kPa (self weight included), imposed q_k = 2.50 kPa
3. Share the moments: column strip and middle strip
Over a column the slab is much stiffer and the moment crowds in: the column strip takes 75 % of the hogging moment and 55 % of the sagging moment, the middle strip the rest. Each strip is then designed like a wide beam.
X direction: top, column strip, at the columns:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
column 2 (at 6 m): M = 220.6 kNm over 6 m; 75 % in a 3000 mm strip: 55.1 kNm/m; A_s,req 647 mm2/m -> Y16 @ 300 = 670 mm2/m
647 ≤ 670 mm²/m OK
X direction: top, middle strip, at the columns:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
column 2 (at 6 m): M = 220.6 kNm over 6 m; 25 % in a 3000 mm strip: 18.4 kNm/m; A_s,req 216 mm2/m -> Y12 @ 300 = 377 mm2/m
216 ≤ 377 mm²/m OK
X direction: bottom, column strip, in the span:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
span 1 (6 m): M = 166.1 kNm over 6 m; 55 % in a 3000 mm strip: 30.5 kNm/m; A_s,req 358 mm2/m -> Y12 @ 300 = 377 mm2/m
358 ≤ 377 mm²/m OK
X direction: bottom, middle strip, in the span:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
span 1 (6 m): M = 166.1 kNm over 6 m; 45 % in a 3000 mm strip: 24.9 kNm/m; A_s,req 293 mm2/m -> Y12 @ 300 = 377 mm2/m
293 ≤ 377 mm²/m OK
4. The other direction
The same frames, strips and steel the other way. The two layers of bars sit at different depths, so the steel differs a little even for a square grid.
Y direction: top, column strip, at the columns:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
column 2 (at 6 m): M = 220.6 kNm over 6 m; 75 % in a 3000 mm strip: 55.1 kNm/m; A_s,req 683 mm2/m -> Y16 @ 275 = 731 mm2/m
683 ≤ 731 mm²/m OK
Y direction: top, middle strip, at the columns:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
column 2 (at 6 m): M = 220.6 kNm over 6 m; 25 % in a 3000 mm strip: 18.4 kNm/m; A_s,req 228 mm2/m -> Y12 @ 300 = 377 mm2/m
228 ≤ 377 mm²/m OK
Y direction: bottom, column strip, in the span:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
span 1 (6 m): M = 166.1 kNm over 6 m; 55 % in a 3000 mm strip: 30.5 kNm/m; A_s,req 377 mm2/m -> Y10 @ 200 = 393 mm2/m
377 ≤ 393 mm²/m OK
Y direction: bottom, middle strip, in the span:
[SANS 10100-1 4.3.3.4, 4.11.4; SANS 10100-1 4.6.5.1 (equivalent frame; strip division as BS 8110-1 Table 3.18)]
span 1 (6 m): M = 166.1 kNm over 6 m; 45 % in a 3000 mm strip: 24.9 kNm/m; A_s,req 309 mm2/m -> Y12 @ 300 = 377 mm2/m
309 ≤ 377 mm²/m OK
5. Edge frames and the moment into an edge column
A frame along the edge of the floor is narrower. At an edge or corner column the slab can only pass a limited moment into the column; the rest is redistributed into the span.
X direction: moment to the edge column <= M_t,max:
[SANS 10100-1 4.6.5.3.2, 4.6.5.3.3]
end moment 97.8 kNm; M_t,max = 0.15 f_c b_e d^2 = 0.15 x 30 x 900 x 229^2 = 212.4 kNm (b_e = c_par + c_perp, slab edge flush with the column)
97.8 ≤ 212.4 kNm OK
Y direction: moment to the edge column <= M_t,max:
[SANS 10100-1 4.6.5.3.2, 4.6.5.3.3]
end moment 97.8 kNm; M_t,max = 0.15 f_c b_e d^2 = 0.15 x 30 x 900 x 217^2 = 190.7 kNm (b_e = c_par + c_perp, slab edge flush with the column)
97.8 ≤ 190.7 kNm OK
X edge frame: moment to the corner column <= M_t,max:
[SANS 10100-1 4.6.5.3.2, 4.6.5.3.3]
end moment 68.9 kNm; M_t,max = 0.15 f_c b_e d^2 = 0.15 x 30 x 675 x 229^2 = 159.3 kNm (b_e = c_y + c_x/2, slab edge flush with the column)
68.9 ≤ 159.3 kNm OK
Y edge frame: moment to the corner column <= M_t,max:
[SANS 10100-1 4.6.5.3.2, 4.6.5.3.3]
end moment 68.9 kNm; M_t,max = 0.15 f_c b_e d^2 = 0.15 x 30 x 675 x 217^2 = 143 kNm (b_e = c_y + c_x/2, slab edge flush with the column)
68.9 ≤ 143.0 kNm OK
6. Punching at the columns
Every column is checked for punching, as in the punching calculator: the effective shear (raised for the moment at edge and corner columns) on perimeters around the column.
interior column: V = 524 kN, β = 1.15, u1 = 4,476 mm
v = 0.603 vs v_c = 0.448 MPa → links: 3 rings of 17 legs
edge column: V = 277 kN, β = 1.40, u1 = 2,688 mm
v = 0.646 vs v_c = 0.448 MPa → links: 3 rings of 10 legs
corner column: V = 126 kN, β = 1.25, u1 = 1,569 mm
v = 0.451 vs v_c = 0.448 MPa → links: 2 rings of 5 legs
7. Deflection
The longer span over the effective depth is checked against the basic ratio for a flat slab (reduced, because a flat slab is more flexible than a slab on beams), with the steel stress factor.
governing span 1: span/d actual / allowed: 26.2 ≤ 45.9 OK
Drops, column heads, openings and cantilever edges change the frames; the calc sheet lists every row.
Run this example yourselfOpen the free STATIX flat slab calculator and press Load the example (3 × 3 bays of 6 m) under "How it is calculated". Change any value and every step updates, with a calc sheet. Open the flat slab calculator
A worked example for learning, not a design. Check every value against the current code and your own project, and have the design checked and signed by a registered professional engineer.
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