RCC Slabs, Columns and Footings

What to remember

• One-way slab: ly/lx > 2, main steel in the short direction; two-way slab: ly/lx ≤ 2, steel in both directions with moments Mx = αx wu lx² and My = αy wu lx².
• Short column: le/D < 12. Axial strength Pu = 0.4 fck Ac + 0.67 fy Asc. Longitudinal steel must be between 0.8% and 6% of the gross area.
• Footing design: size the base from the unfactored load and safe bearing capacity; check one-way shear at distance d and two-way (punching) shear at d/2 from the column face; bending moment is taken at the column face.

1. Slabs

A slab is a flat plate supported on beams or walls. Its depth is small compared with the span, so it is designed for a 1 m wide strip.

Load calculation (per m²): self weight = thickness (m) × 25 kN/m³; add floor finish, plaster and live load. Factored load wu = 1.5 × (DL + LL). Example: D = 150 mm gives 3.75 kN/m²; with finish 1.0 and live load 3.0 the factored load is 1.5 × 7.75 = 11.6 kN/m².

Effective span: the lesser of centre-to-centre distance of supports and clear span + effective depth.

One-way slab: ly/lx > 2. For a simply supported slab, Mu = wu l² / 8 and Vu = wu l / 2. Main bars are along the short span at the bottom; distribution (secondary) bars are placed perpendicular to them. Example: wu = 12 kN/m², l = 4 m gives Mu = 12 × 16/8 = 24 kN·m per metre width.

Two-way slab: ly/lx ≤ 2. IS 456 gives moment coefficients αx, αy for different edge conditions. For a simply supported square slab with corners free to lift, αx = αy = 0.062. Example: wu = 10 kN/m², lx = 4 m: Mx = 0.062 × 10 × 16 = 9.92 kN·m per metre. Where a slab edge is restrained and corners are prevented from lifting, torsion reinforcement is provided at corners over a length of lx/5, in four layers, each with area equal to three-quarters of the steel required for the maximum mid-span moment.

Cantilever slab: main steel at the top; basic l/d is 7.

Detailing rules for slabs:

Slabs are generally safe in shear and no shear reinforcement is normally provided if τv is below τc (with factor k for thin slabs).

Flat slab: a slab supported directly on columns without beams. Drop panels (thickened slab) and column heads (capitals) reduce punching shear and negative moments. The critical section for punching shear is at d/2 from the column (or capital) face. Advantages: flat soffit, easy formwork, more headroom. Disadvantage: low resistance to lateral load.

2. Columns

A column is a vertical compression member whose effective length exceeds three times its least lateral dimension (a shorter one is a pedestal).

Effective length le (theoretical values for a column of length L):

Short and long columns: short if le/D < 12 (fails by crushing); long (slender) if le/D ≥ 12 (fails by buckling). For slender columns an additional moment Ma = Pu D/2000 × (le/D)² is added. Example: Pu = 1000 kN, D = 400 mm, le = 6 m, le/D = 15 so Ma = 1000 × 0.4/2000 × 225 = 45 kN·m.

Minimum eccentricity: emin = L/500 + D/30, not less than 20 mm. Example: L = 3 m, D = 300 mm gives 6 + 10 = 16 mm, so use 20 mm.

Axial strength of a short column (emin ≤ 0.05 D): Pu = 0.4 fck Ac + 0.67 fy Asc, where Ac is the concrete area (gross minus steel). Example: 400 × 400 mm, M20, Fe 415, Asc = 1% = 1600 mm². Ac = 158,400 mm². Pu = 0.4 × 20 × 158,400 + 0.67 × 415 × 1600 = 1267 + 445 kN ≈ 1712 kN.

Working stress form: P = σcc Ac + σsc Asc, with steel stress limited using the modular ratio.

Detailing of columns:

• Longitudinal steel: minimum 0.8%, maximum 6% of gross area (about 4% recommended where bars lap). Minimum 4 bars in a rectangular column and 6 in a circular column. Minimum bar diameter 12 mm.
• Lateral ties: diameter not less than one-quarter of the largest longitudinal bar and not less than 6 mm. Pitch is the least of (a) the least lateral dimension, (b) 16 times the smallest longitudinal bar diameter and (c) 300 mm.
• Cover: 40 mm or bar diameter, whichever is greater (nominal).
• Helical (spiral) columns: pitch not more than 75 mm and not more than one-sixth of core diameter; not less than 25 mm and not less than 3 times the wire diameter. Strength of a helically reinforced column is taken as 1.05 times that of a tied column. Helical columns are more ductile.

Pedestal: a compression member whose height does not exceed three times its least lateral dimension.

3. Footings

A footing spreads the column load over a wide soil area so that the bearing pressure is less than the safe bearing capacity (SBC).

Types: isolated (square, rectangular, circular; sloped or stepped), wall (strip) footing, combined footing (two or more columns), strap (cantilever) footing, raft or mat, and pile foundation.

Sizing an isolated footing (use unfactored loads): A = (P + about 10% for self weight) / SBC. Example: P = 900 kN, SBC = 150 kN/m²: A = 1.1 × 900/150 = 6.6 m². For structural design use factored load/area (net upward pressure); self weight of the footing is ignored in bending and shear because it is balanced by the soil reaction.

Pressure under eccentric load: p = (P/A)(1 ± 6e/L) for eccentricity e along length L. There is no tension if e ≤ L/6 (middle-third rule). Example: P = 600 kN, 2 m × 2 m, e = 0.2 m: p = 150 × (1 ± 0.6) = 240 and 60 kN/m².

Design checks:

Permissible punching shear stress τc' = ks τc, where ks = 0.5 + βc (βc = short side / long side of column), ks ≤ 1, and τc = 0.25 √fck. For M20 and a square column, τc' = 0.25 × √20 = 1.12 N/mm².

Footing rules: minimum cover 50 mm (earth-face concrete); minimum edge thickness 150 mm on soil (300 mm on piles); dowels from column into footing have minimum area of 0.5% of the column cross-section.

Combined footing: used when columns are close or one column is near a property line. The centroid of the footing area should coincide with the resultant of column loads. Example: loads of 600 kN and 400 kN, 4 m apart: resultant is 4 × 400/1000 = 1.6 m from the heavier column.

Strap footing: two footings joined by a strap beam so that an eccentric edge column does not tilt. Raft: used for weak soil or when isolated footings would cover more than about half the plan area.

Exam traps

• One-way slab limit is ly/lx > 2; at exactly 2 the slab is two-way.
• Slab minimum steel is 0.15% for Fe 250 but 0.12% for Fe 415 and Fe 500.
• Distribution bar limit is 5d or 450 mm, but main bar limit is 3d or 300 mm.
• Column slenderness limit is le/D = 12; do not confuse with L/r values of steel columns.
• Short-column formula uses 0.4 fck and 0.67 fy (not 0.446 and 0.87).
• Footing bending is at the column face; punching shear is at d/2 and one-way shear at d.
• Footing size uses working loads; footing thickness design uses factored loads.
• Fixed-free column has le = 2L; both ends fixed has 0.65 L.

One-liners

• 1. A slab is one-way if ly/lx > 2.
• 2. Maximum bar spacing: 3d or 300 mm (main); 5d or 450 mm (distribution).
• 3. Minimum slab steel is 0.12% of gross area for Fe 415.
• 4. Maximum bar diameter in a slab is D/8.
• 5. Short column means le/D < 12.
• 6. Minimum eccentricity is L/500 + D/30 or 20 mm, whichever is greater.
• 7. Pu = 0.4 fck Ac + 0.67 fy Asc.
• 8. Column steel: minimum 0.8%, maximum 6%.
• 9. Tie spacing is the least of least lateral dimension, 16 φ and 300 mm.
• 10. Punching shear section is at d/2; one-way shear section is at d.
• 11. ks = 0.5 + βc, not more than 1.
• 12. Middle-third rule: e ≤ L/6 avoids tension in soil pressure.

Practice questions

1. A rectangular slab supported on all four edges behaves as a one-way slab when ly/lx is:

   1. Less than 1.5
   2. Greater than 2
   3. Equal to 1
   4. Between 1 and 2
   Answer

   B. Greater than 2

   For ly/lx > 2 bending in the long direction is negligible, so main steel runs in the short span.

2. A simply supported slab of 3 m span needs a minimum effective depth (basic l/d = 20) of:

   1. 150 mm
   2. 100 mm
   3. 300 mm
   4. 200 mm
   Answer

   A. 150 mm

   d = 3000/20 = 150 mm.

3. For a slab of overall depth 150 mm using Fe 415 steel, the minimum steel per metre width is:

   1. 120 mm²
   2. 225 mm²
   3. 180 mm²
   4. 300 mm²
   Answer

   C. 180 mm²

   0.12% × 1000 × 150 = 180 mm².

4. The maximum spacing of main bars in a slab with effective depth 120 mm is:

   1. 360 mm
   2. 450 mm
   3. 240 mm
   4. 300 mm
   Answer

   D. 300 mm

   Lesser of 3d = 360 mm and 300 mm is 300 mm.

5. The maximum spacing of distribution bars in a slab with effective depth 80 mm is:

   1. 300 mm
   2. 400 mm
   3. 450 mm
   4. 240 mm
   Answer

   B. 400 mm

   Lesser of 5d = 400 mm and 450 mm is 400 mm.

6. The maximum diameter of reinforcing bar allowed in a slab of overall thickness 160 mm is:

   1. 20 mm
   2. 25 mm
   3. 16 mm
   4. 32 mm
   Answer

   A. 20 mm

   Bar diameter should not exceed D/8 = 20 mm.

7. A simply supported one-way slab of 4 m span carries a factored load of 12 kN/m². The design moment per metre width is:

   1. 48 kN·m
   2. 36 kN·m
   3. 24 kN·m
   4. 12 kN·m
   Answer

   C. 24 kN·m

   Mu = wu l²/8 = 12 × 16/8 = 24 kN·m.

8. A 150 mm slab (25 kN/m³) carries finish 1.0 kN/m² and live load 3.0 kN/m². The factored load is nearly:

   1. 10.1 kN/m²
   2. 7.75 kN/m²
   3. 12.4 kN/m²
   4. 11.6 kN/m²
   Answer

   D. 11.6 kN/m²

   DL = 3.75 + 1.0; with LL 3.0 total 7.75; 1.5 × 7.75 = 11.6.

9. For a simply supported square two-way slab (αx = 0.062) with wu = 10 kN/m² and lx = 4 m, Mx per metre is nearly:

   1. 9.92 kN·m
   2. 20 kN·m
   3. 5 kN·m
   4. 12.5 kN·m
   Answer

   A. 9.92 kN·m

   Mx = 0.062 × 10 × 4² = 9.92 kN·m.

10. Torsion reinforcement at corners of a two-way slab is provided over a length of:

    1. One-tenth of the longer span
    2. One-fifth of the shorter span
    3. One-third of the shorter span
    4. Half of the longer span
    Answer

    B. One-fifth of the shorter span

    Corner torsion steel extends lx/5 in each direction.

11. The main purpose of distribution bars in a one-way slab is to:

    1. Increase the neutral axis depth
    2. Carry the entire bending moment
    3. Distribute loads and resist shrinkage and temperature stresses
    4. Resist shear
    Answer

    C. Distribute loads and resist shrinkage and temperature stresses

    Distribution steel runs at right angles to main bars and controls cracks.

12. In a cantilever slab the main reinforcement is placed:

    1. Near the bottom face
    2. At mid depth
    3. Equally top and bottom
    4. Near the top face
    Answer

    D. Near the top face

    Hogging moment causes tension at the top.

13. In a flat slab, the critical section for punching shear is at a distance from the column face of:

    1. d/2
    2. Zero
    3. 2d
    4. d
    Answer

    A. d/2

    Punching shear perimeter is taken at d/2 from the column or capital face.

14. A column is classed as short when the ratio of effective length to least lateral dimension is:

    1. Less than 3
    2. Less than 12
    3. More than 12
    4. Equal to 20
    Answer

    B. Less than 12

    Short columns have le/D < 12; slender columns have le/D ≥ 12.

15. The minimum eccentricity for a column of length 3.0 m and lateral dimension 300 mm is:

    1. 6 mm
    2. 10 mm
    3. 20 mm
    4. 16 mm
    Answer

    C. 20 mm

    L/500 + D/30 = 6 + 10 = 16 mm, but not less than 20 mm.

16. A short 400 mm × 400 mm column has fck = 20, fy = 415 and 1% steel (1600 mm²). The axial strength Pu is nearly:

    1. 1267 kN
    2. 1500 kN
    3. 2200 kN
    4. 1712 kN
    Answer

    D. 1712 kN

    0.4 × 20 × 158,400 + 0.67 × 415 × 1600 = 1267 + 445 = 1712 kN.

17. The percentage of longitudinal steel in a column should lie between:

    1. 0.5% and 4%
    2. 0.15% and 2%
    3. 0.8% and 6%
    4. 1% and 8%
    Answer

    C. 0.8% and 6%

    IS 456 gives minimum 0.8% and maximum 6% of the gross area.

18. The minimum number of longitudinal bars in a circular column is:

    1. 4
    2. 8
    3. 3
    4. 6
    Answer

    D. 6

    Rectangular columns need 4, circular columns 6.

19. The minimum diameter of longitudinal bars in a column is:

    1. 12 mm
    2. 8 mm
    3. 10 mm
    4. 16 mm
    Answer

    A. 12 mm

    IS 456 specifies 12 mm.

20. The minimum diameter of lateral ties for columns with 16 mm longitudinal bars is:

    1. 8 mm
    2. 6 mm
    3. 4 mm
    4. 10 mm
    Answer

    B. 6 mm

    Larger of 16/4 = 4 mm and 6 mm, so 6 mm.

21. A 250 mm × 400 mm column has 16 mm longitudinal bars. The maximum pitch of lateral ties is:

    1. 256 mm
    2. 300 mm
    3. 400 mm
    4. 250 mm
    Answer

    D. 250 mm

    Least of 250 (least dimension), 16 × 16 = 256 and 300 is 250 mm.

22. Compared with a tied column, the strength of a helically reinforced column is taken as:

    1. 1.05 times
    2. Equal
    3. 1.5 times
    4. 1.25 times
    Answer

    A. 1.05 times

    IS 456 allows 5% higher strength for helical reinforcement if conditions are met.

23. The maximum pitch of helical reinforcement in a column is:

    1. 75 mm or one-sixth of core diameter, whichever is less
    2. 25 mm
    3. 150 mm
    4. 300 mm only
    Answer

    A. 75 mm or one-sixth of core diameter, whichever is less

    Pitch should not exceed 75 mm and core diameter/6.

24. For a column of 6 m effective length, D = 400 mm and Pu = 1000 kN, the additional moment Ma = (Pu D/2000)(le/D)² is:

    1. 22.5 kN·m
    2. 90 kN·m
    3. 45 kN·m
    4. 15 kN·m
    Answer

    C. 45 kN·m

    le/D = 15; Ma = 0.2 × 225 = 45 kN·m.

25. The effective length of a column with both ends fixed (theoretical) is:

    1. 1.0 L
    2. 0.65 L
    3. 0.8 L
    4. 0.5 L
    Answer

    B. 0.65 L

    Fixed-fixed ends give 0.65 L.

26. The effective length of a column fixed at one end and free at the other is:

    1. 0.8 L
    2. 1.0 L
    3. 1.5 L
    4. 2.0 L
    Answer

    D. 2.0 L

    A cantilever column buckles with an effective length of twice its height.

27. The effective length of a column hinged at both ends is:

    1. 0.65 L
    2. 1.0 L
    3. 0.8 L
    4. 2.0 L
    Answer

    B. 1.0 L

    Both ends hinged and not sway: le = L.

28. The normal nominal cover to the main bars of a column is:

    1. 25 mm
    2. 75 mm
    3. 40 mm
    4. 20 mm
    Answer

    C. 40 mm

    IS 456 specifies 40 mm or the bar diameter, whichever is greater.

29. The minimum cover to the reinforcement at the earth face of a footing is:

    1. 75 mm
    2. 50 mm
    3. 25 mm
    4. 30 mm
    Answer

    B. 50 mm

    Footing steel in contact with earth gets a minimum cover of 50 mm.

30. A column carries 900 kN. The safe bearing capacity is 150 kN/m² and 10% extra is added for footing weight. The footing area is:

    1. 6.6 m²
    2. 7.2 m²
    3. 6.0 m²
    4. 5.4 m²
    Answer

    A. 6.6 m²

    A = 1.1 × 900/150 = 6.6 m².

31. The critical section for one-way shear in an isolated footing is at a distance from the column face of:

    1. d/2
    2. 2d
    3. Zero
    4. d
    Answer

    D. d

    Beam-type shear is checked at d from the column face.

32. The permissible punching shear stress τc for M20 concrete and a square column is nearly (τc' = 0.25√fck):

    1. 1.5 N/mm²
    2. 0.56 N/mm²
    3. 1.12 N/mm²
    4. 2.0 N/mm²
    Answer

    C. 1.12 N/mm²

    0.25 × √20 = 1.118 N/mm² (ks = 1 for square column).

33. The critical section for bending moment in an isolated footing is at:

    1. Distance d/2 from the column face
    2. The face of the column
    3. Distance d from the column face
    4. Centre of the footing
    Answer

    B. The face of the column

    Moment is calculated at the column face.

34. A 2 m × 2 m footing carries 600 kN with eccentricity 0.2 m in one direction. The maximum soil pressure is:

    1. 240 kN/m²
    2. 180 kN/m²
    3. 300 kN/m²
    4. 150 kN/m²
    Answer

    A. 240 kN/m²

    p = (600/4)(1 + 6 × 0.2/2) = 150 × 1.6 = 240 kN/m².

35. For a combined footing under two columns of 600 kN and 400 kN spaced 4 m apart, the resultant load is at what distance from the 600 kN column?

    1. 2.0 m
    2. 1.0 m
    3. 1.6 m
    4. 2.4 m
    Answer

    C. 1.6 m

    Moment about 600 kN column: 400 × 4/1000 = 1.6 m.

36. The minimum thickness at the edge of a footing resting on soil is:

    1. 200 mm
    2. 100 mm
    3. 300 mm
    4. 150 mm
    Answer

    D. 150 mm

    IS 456: 150 mm on soil and 300 mm on piles.

37. In bearing at a column base the limit on √(A1/A2) is:

    1. 1
    2. 2
    3. 3
    4. 4
    Answer

    B. 2

    Allowable bearing stress is 0.45 fck √(A1/A2) with √(A1/A2) ≤ 2.

38. The minimum area of dowel bars across a column-footing junction is:

    1. 0.15% of the footing area
    2. 0.8% of the column cross-section
    3. 1% of the footing area
    4. 0.5% of the column cross-section
    Answer

    D. 0.5% of the column cross-section

    IS 456 requires dowels of not less than 0.5% of the supported column area.

39. Which of the following statements are correct? 1. Slabs normally do not need shear reinforcement. 2. The minimum steel percentage in a slab depends on the steel grade.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are true: τv is usually less than τc and minimum steel is 0.15% for Fe 250 and 0.12% for Fe 415.

40. Which of the following statements are correct? 1. Short columns fail mainly by crushing. 2. Slender columns fail mainly by buckling.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Short columns crush; long columns buckle and need an additional moment.

41. Which of the following statements are correct? 1. A pedestal has height not exceeding three times its least lateral dimension. 2. A pedestal must always carry at least 0.8% longitudinal steel.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Statement 2 is false: it is the column rule; pedestals are generally plain.

42. A footing used when one column lies close to the property line and its footing cannot be centred is a:

    1. Isolated square footing
    2. Circular footing
    3. Pedestal footing
    4. Strap (cantilever) footing
    Answer

    D. Strap (cantilever) footing

    A strap beam joins the eccentric footing to an interior footing to balance the moment.

43. Which statement about footing design is correct?

    1. Both size and thickness use factored loads
    2. Both use working loads
    3. Size is found from working loads; thickness is designed using factored loads
    4. Self weight is added in bending design
    Answer

    C. Size is found from working loads; thickness is designed using factored loads

    Soil pressure is checked at working load while shear and bending use factored net pressure.

44. Which is NOT an advantage of a flat slab compared with a beam-slab floor?

    1. Greater resistance to lateral loads
    2. Simple formwork
    3. Flat soffit
    4. More headroom
    Answer

    A. Greater resistance to lateral loads

    Flat slabs lack beams and resist lateral loads poorly.

45. In two-way slabs with ly/lx ≤ 2, the larger moment occurs per metre width in the:

    1. Longer span direction
    2. Diagonal direction only
    3. Both equal in all cases
    4. Shorter span direction
    Answer

    D. Shorter span direction

    The stiffer short direction carries more load, giving the larger moment coefficient.