Reinforced Concrete Structures

What to remember

• Concrete is strong in compression and weak in tension. Steel bars placed in the tension zone make reinforced concrete (RCC). The two materials work together because they bond well and have nearly equal thermal expansion.
• The Indian code is IS 456:2000 (Plain and Reinforced Concrete). It uses the Limit State Method as the main design method. The Working Stress Method is kept in an annex.
• Grade of concrete M20 means a characteristic compressive strength fck = 20 N/mm² at 28 days (150 mm cube). Steel grades are Fe415, Fe500 and Fe550, where the number is the yield strength fy in N/mm².

Materials and basic terms

Concrete is made of cement, fine aggregate (sand), coarse aggregate and water, with admixtures when needed. The water-cement ratio controls strength: lower ratio gives higher strength, as long as the concrete is fully compacted. Workability is measured by the slump test and the compaction factor test. Curing keeps concrete moist so that cement hydration continues, usually for at least 7 days for ordinary Portland cement.

Standard grades start from M20 for reinforced work. Plain concrete may use M15. Higher grades like M25, M30 and M35 are used for heavy members.

Key properties from IS 456:

• Modulus of elasticity of concrete: Ec = 5000 √fck N/mm². For M20 this is about 22,360 N/mm².
• Flexural tensile strength (modulus of rupture): fcr = 0.7 √fck N/mm².
• Modular ratio: m = 280 ÷ (3 σcbc). For M20, σcbc = 7 N/mm², so m = 280 ÷ 21 = 13.33.
• Unit weight of reinforced concrete is taken as 25 kN/m³. Plain concrete is 24 kN/m³.
• Partial safety factors for material strength: 1.5 for concrete and 1.15 for steel.
• Partial safety factor for load: 1.5 for dead and live loads. Factored load = 1.5 × (DL + LL).
• Design strength of concrete in flexure = 0.67 fck ÷ 1.5 = 0.446 fck. Design strength of steel = 0.87 fy.
• Strain in concrete at the extreme fibre in bending at failure = 0.0035.

Types of loads: dead load (self weight and fixed finishes), live (imposed) load as given in IS 875, wind load, seismic load and snow load. The limit states are the limit state of collapse (strength) and the limit state of serviceability (deflection and cracking).

Design of beams

A simply supported beam resists bending and shear. The maximum bending moment for a udl w over the span l is Mu = w l² ÷ 8. For a cantilever it is w l² ÷ 2.

Effective depth (d) is the distance from the extreme compression fibre to the centre of tension steel. Overall depth D = d + clear cover + half the bar diameter.

Limiting depth of the neutral axis (xu,max) as a fraction of d:

A section is under-reinforced if xu < xu,max. This gives a ductile failure, because steel yields first, and is the preferred design. An over-reinforced section fails suddenly by crushing of concrete and is not allowed. A balanced section has xu = xu,max.

Moment of resistance of a singly reinforced section when xu < xu,max: Mu = 0.87 fy Ast d (1 − (Ast fy) ÷ (b d fck)). Depth of neutral axis: xu = 0.87 fy Ast ÷ (0.36 fck b).

Worked example: A beam has b = 230 mm, d = 450 mm, M20 and Fe415. Mu,lim = 0.138 × 20 × 230 × 450² = 128.5 kN·m (about). If Mu exceeds this, the section must be made doubly reinforced by adding steel in the compression zone, or the size must be increased.

T-beams occur where a slab and a beam are cast together. The slab acts as a flange to resist compression. Effective flange width depends on the span, web width and slab thickness. Doubly reinforced beams use compression steel to increase moment capacity, to reduce long-term deflection, and where the depth is restricted.

Shear in beams: nominal shear stress τv = Vu ÷ (b d). The permissible shear stress in concrete τc depends on the percentage of tension steel and grade. If τv is more than τc, shear reinforcement (stirrups) is given. The maximum shear stress τc,max is limited (2.8 N/mm² for M20, 3.1 for M25, 3.5 for M30). Vertical stirrup strength: Vus = 0.87 fy Asv d ÷ sv. Spacing of stirrups should not exceed 0.75 d for vertical stirrups or 300 mm, whichever is less.

Development length is the length of bar needed to develop its full stress through bond: Ld = φ σs ÷ (4 τbd), where φ is the bar diameter, σs is the stress in bar (0.87 fy at design) and τbd is the design bond stress. For deformed bars, τbd is increased by 60% over plain bars. Typical bond stress for plain bars in M20 is 1.2 N/mm².

Slabs

A slab is a flat plate that carries load to the beams or walls.

• One-way slab: supported on two opposite sides, or ly ÷ lx > 2. Main steel is in the short span, and distribution (secondary) steel is at right angles to it.
• Two-way slab: supported on all four sides and ly ÷ lx ≤ 2. Bending takes place in both directions, so main steel is provided both ways. The corners tend to lift, so torsion steel is provided at corners when the edges are restrained.
• Flat slab: rests directly on columns without beams. It has column heads or drop panels for shear.

Rules for slabs:

• Minimum reinforcement is 0.15% of gross cross-sectional area for mild steel (Fe250) and 0.12% for high yield strength deformed (HYSD) bars.
• Maximum spacing of main bars is the lesser of 3d or 300 mm. For distribution steel the maximum spacing is the lesser of 5d or 450 mm.
• Diameter of bars should not exceed one-eighth of the slab thickness.
• Deflection control for spans up to 10 m uses a span ÷ effective depth ratio: 7 for a cantilever, 20 for simply supported and 26 for a continuous member. These basic values are modified by a factor for tension steel.

Worked example: A 150 mm thick RCC slab has a self weight of 0.15 × 25 = 3.75 kN/m². A simply supported strip with factored load 12 kN/m² on a 4 m span has Mu = 12 × 4² ÷ 8 = 24 kN·m per metre width.

Columns, footings and detailing

A column is a compression member. If the slenderness ratio le ÷ D (effective length ÷ least lateral dimension) is less than 12, it is a short column. Otherwise it is a long column.

Axial load capacity of a short column with minimum eccentricity: Pu = 0.4 fck Ac + 0.67 fy Asc, where Ac is the net concrete area and Asc is the area of longitudinal steel.

Rules for columns:

• Longitudinal steel: minimum 0.8% and maximum 6% of the gross area. In practice about 4% is used so that laps can be provided.
• A rectangular column needs at least 4 bars, and a circular column needs at least 6 bars. The minimum bar diameter is 12 mm.
• Lateral ties (stirrups) hold the main bars, stop them from buckling and resist shear. The diameter of the tie is not less than one-fourth the main bar diameter and not less than 6 mm. The tie pitch is not more than the least of the least lateral dimension, 16 times the smallest main bar diameter and 300 mm.
• Minimum eccentricity is the larger of l ÷ 500 + D ÷ 30 and 20 mm.
• Helical reinforcement gives higher strength than ordinary ties for the same steel.

Worked example: A 300 × 300 mm short column, M20 and Fe415, with 1% steel. Asc = 900 mm², Ac = 90,000 − 900 = 89,100 mm². Pu = 0.4 × 20 × 89,100 + 0.67 × 415 × 900 = 712,800 + 250,245 = 963,045 N, which is about 963 kN.

Footings transfer column loads to the soil. An isolated footing carries a single column. A combined footing carries two or more. A raft covers the whole area. The base area of a footing = (load + self weight) ÷ safe bearing capacity of soil.

Cover is the distance from the surface of steel to the nearest concrete face. It protects steel from corrosion and fire. The nominal cover for mild exposure is 20 mm, and for columns it is not less than 40 mm or the bar diameter. Cover must be larger for severe exposures. Spacing between bars must allow concrete to flow, normally at least the bar diameter or 5 mm more than the maximum aggregate size, whichever is greater.

Anchorage and laps: hooks and bends are standard (a standard hook adds a fixed number of bar diameters). Laps in bars are staggered. Bars with a diameter over 36 mm are not lapped.

Exam traps

• Fck is characteristic strength; fy is yield strength. A grade M25 has fck = 25 N/mm².
• Partial safety factor is 1.5 for concrete and 1.15 for steel. Do not interchange them.
• Under-reinforced sections are ductile and preferred. Over-reinforced sections are brittle.
• One-way slab: ly ÷ lx > 2. Two-way slab: ly ÷ lx ≤ 2.
• Column steel: minimum 0.8%, maximum 6%. Slab steel: 0.12% or 0.15%.
• Short column has le ÷ D < 12.
• In a one-way slab, main steel is in the short span.
• Stirrups resist shear, while main bars resist bending.

One-liners

• 1. IS 456:2000 is the code for plain and reinforced concrete.
• 2. M20 concrete has fck = 20 N/mm².
• 3. Fe415 steel has fy = 415 N/mm².
• 4. Ec = 5000 √fck.
• 5. fcr = 0.7 √fck.
• 6. Unit weight of RCC is 25 kN/m³.
• 7. Design steel stress is 0.87 fy.
• 8. Factored load is 1.5 times the working load.
• 9. Strain in concrete at failure is 0.0035.
• 10. Steel is placed in the tension zone of a beam.
• 11. A bent-up bar and stirrups resist shear.
• 12. Curing keeps concrete moist after placing.

Practice questions

1. Ec for M25 concrete, taken as 5000 √fck, is

   1. 12,500 N/mm²
   2. 22,360 N/mm²
   3. 25,000 N/mm²
   4. 5,000 N/mm²
   Answer

   C. 25,000 N/mm²

   √25 = 5, so Ec = 5000 × 5 = 25,000 N/mm².

2. The flexural tensile strength fcr = 0.7 √fck for M25 concrete is

   1. 1.75 N/mm²
   2. 7.0 N/mm²
   3. 5.0 N/mm²
   4. 3.5 N/mm²
   Answer

   D. 3.5 N/mm²

   0.7 × √25 = 0.7 × 5 = 3.5 N/mm².

3. The modular ratio m = 280 ÷ (3 σcbc) for M20 concrete with σcbc = 7 N/mm² is about

   1. 10.0
   2. 13.33
   3. 18.67
   4. 40.0
   Answer

   B. 13.33

   280 ÷ (3 × 7) = 280 ÷ 21 = 13.33.

4. The limiting moment of resistance of a beam with b = 230 mm, d = 450 mm, M20 and Fe415 steel (Mu,lim = 0.138 fck b d²) is about

   1. 137.9 kN·m
   2. 123.9 kN·m
   3. 128.5 kN·m
   4. 257.1 kN·m
   Answer

   C. 128.5 kN·m

   0.138 × 20 × 230 × 450² = 128.5 × 10⁶ N·mm = 128.5 kN·m.

5. A 300 × 300 mm short column of M20 and Fe415 has 1% longitudinal steel (900 mm²). Using Pu = 0.4 fck Ac + 0.67 fy Asc, the axial capacity is about

   1. 250 kN
   2. 963 kN
   3. 1,213 kN
   4. 713 kN
   Answer

   B. 963 kN

   Ac = 89,100; 0.4 × 20 × 89,100 = 712.8 kN; 0.67 × 415 × 900 = 250.2 kN; total = 963 kN.

6. The self weight of an RCC slab 150 mm thick (unit weight 25 kN/m³) is

   1. 3.75 kN/m²
   2. 1.5 kN/m²
   3. 2.5 kN/m²
   4. 6.0 kN/m²
   Answer

   A. 3.75 kN/m²

   0.15 m × 25 kN/m³ = 3.75 kN/m².

7. A simply supported beam of span 6 m carries a factored udl of 20 kN/m. The maximum factored bending moment is

   1. 120 kN·m
   2. 60 kN·m
   3. 180 kN·m
   4. 90 kN·m
   Answer

   D. 90 kN·m

   Mu = w l² ÷ 8 = 20 × 36 ÷ 8 = 90 kN·m.

8. A cantilever of 3 m span carries a udl of 10 kN/m. The maximum bending moment at the support is

   1. 90 kN·m
   2. 30 kN·m
   3. 15 kN·m
   4. 45 kN·m
   Answer

   D. 45 kN·m

   M = w l² ÷ 2 = 10 × 9 ÷ 2 = 45 kN·m.

9. The minimum steel in a 150 mm thick slab using HYSD bars (0.12%) per metre width is

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

   B. 180 mm²

   0.12% of 1000 × 150 = 0.0012 × 150,000 = 180 mm².

10. For a slab with effective depth 120 mm, the maximum spacing of main bars is

    1. 120 mm
    2. 450 mm
    3. 300 mm
    4. 360 mm
    Answer

    C. 300 mm

    The limit is the lesser of 3d (360 mm) and 300 mm, so it is 300 mm.

11. For a beam with d = 500 mm and Fe415 steel, the maximum depth of the neutral axis xu,max is

    1. 240 mm
    2. 200 mm
    3. 230 mm
    4. 265 mm
    Answer

    A. 240 mm

    xu,max = 0.48 d = 0.48 × 500 = 240 mm.

12. Vertical stirrups with Asv = 100 mm², Fe415 steel, d = 400 mm and spacing 150 mm give a shear capacity Vus = 0.87 fy Asv d ÷ sv of about

    1. 48.1 kN
    2. 144.4 kN
    3. 96.3 kN
    4. 192.6 kN
    Answer

    C. 96.3 kN

    0.87 × 415 × 100 × 400 ÷ 150 = 96,280 N = 96.3 kN.

13. The development length of a 16 mm deformed bar with σs = 0.87 × 415 = 361 N/mm² and design bond stress 1.92 N/mm² is about

    1. 752 mm
    2. 376 mm
    3. 1,128 mm
    4. 565 mm
    Answer

    A. 752 mm

    Ld = φ σs ÷ (4 τbd) = 16 × 361 ÷ (4 × 1.92) = 5,776 ÷ 7.68 = 752 mm.

14. A column 3 m long (effective), 300 mm in least dimension, has a minimum eccentricity of

    1. 10 mm
    2. 20 mm
    3. 30 mm
    4. 16 mm
    Answer

    B. 20 mm

    l ÷ 500 + D ÷ 30 = 6 + 10 = 16 mm, which is less than 20 mm, so 20 mm governs.

15. Which Indian Standard is the code of practice for plain and reinforced concrete?

    1. IS 800
    2. IS 1893
    3. IS 875
    4. IS 456
    Answer

    D. IS 456

    IS 456:2000 covers plain and reinforced concrete.

16. Concrete grade M20 means a characteristic compressive strength of

    1. 200 N/mm² at 28 days
    2. 20 N/mm² at 7 days
    3. 20 N/mm² at 28 days
    4. 20 kN/m² at 28 days
    Answer

    C. 20 N/mm² at 28 days

    M is the mix and the number is the 28-day cube strength in N/mm².

17. The partial safety factor for the strength of concrete in the limit state method is

    1. 1.5
    2. 1.0
    3. 0.87
    4. 1.15
    Answer

    A. 1.5

    1.5 for concrete and 1.15 for steel.

18. The partial safety factor for the strength of reinforcing steel is

    1. 1.5
    2. 1.15
    3. 1.25
    4. 0.67
    Answer

    B. 1.15

    Design stress in steel is fy ÷ 1.15 = 0.87 fy.

19. The maximum strain in concrete at the extreme compression fibre at failure in bending is

    1. 0.002
    2. 0.01
    3. 0.0055
    4. 0.0035
    Answer

    D. 0.0035

    IS 456 limits the strain to 0.0035.

20. An under-reinforced beam section fails by

    1. yielding of steel first, giving warning
    2. sudden crushing of concrete
    3. shear only
    4. buckling of the web
    Answer

    A. yielding of steel first, giving warning

    Steel yields before concrete crushes, so failure is ductile.

21. Compression steel is added to a beam (doubly reinforced) mainly when

    1. the span is under 2 m
    2. the concrete grade is high
    3. the beam is very long
    4. the moment exceeds Mu,lim and the depth is restricted
    Answer

    D. the moment exceeds Mu,lim and the depth is restricted

    Compression steel supplements the concrete when singly reinforced capacity is not enough.

22. A slab is designed as a one-way slab when ly ÷ lx is

    1. about 1.5
    2. less than 1
    3. greater than 2
    4. equal to 1
    Answer

    C. greater than 2

    For ratios above 2 the slab bends mainly in the short span.

23. In a one-way slab the main reinforcement is placed

    1. along the long span
    2. diagonally
    3. only at the corners
    4. along the short span
    Answer

    D. along the short span

    Distribution steel is along the long span.

24. A column is classed as a short column when le ÷ D is

    1. more than 12
    2. less than 12
    3. equal to 50
    4. more than 30
    Answer

    B. less than 12

    Slenderness ratio below 12 is a short column.

25. The minimum percentage of longitudinal steel in a column is

    1. 0.8%
    2. 0.12%
    3. 1.5%
    4. 2.5%
    Answer

    A. 0.8%

    Columns need at least 0.8% and at most 6% of the gross area.

26. Lateral ties in a column mainly

    1. provide the bond
    2. reduce the cover
    3. prevent buckling of main bars and confine concrete
    4. resist bending moment
    Answer

    C. prevent buckling of main bars and confine concrete

    Ties hold the bars in position and stop them from buckling outwards.

27. Stirrups in a beam are provided mainly to resist

    1. bending moment
    2. shear
    3. temperature
    4. axial load
    Answer

    B. shear

    Stirrups resist diagonal tension from shear.

28. The basic span ÷ effective depth ratio for a simply supported beam is

    1. 10
    2. 7
    3. 20
    4. 26
    Answer

    C. 20

    Basic values are 7 for cantilever, 20 for simply supported and 26 for continuous.

29. The minimum steel in slabs with mild steel (Fe250) is

    1. 0.15% of the gross area
    2. 1% of the gross area
    3. 0.12% of the gross area
    4. 0.8% of the gross area
    Answer

    A. 0.15% of the gross area

    0.12% applies to HYSD bars.

30. Torsion steel in two-way slabs is provided at the

    1. centre of the span
    2. mid-point of each edge
    3. corners
    4. top of the beam only
    Answer

    C. corners

    Restrained corners tend to lift and twist.

31. The minimum period of curing for ordinary Portland cement concrete is

    1. 1 day
    2. 7 days
    3. 3 hours
    4. 28 hours
    Answer

    B. 7 days

    Concrete should be kept moist for at least 7 days with ordinary Portland cement.

32. Which of the following statements is/are correct? 1. An under-reinforced section gives a ductile failure. 2. An over-reinforced section is preferred in design.

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

    A. 1 only

    Over-reinforced sections fail suddenly and are avoided.

33. Which of the following statements is/are correct? 1. The partial safety factor for concrete is 1.5. 2. The partial safety factor for steel is also 1.5.

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

    A. 1 only

    Steel uses 1.15.

34. Which of the following statements is/are correct? 1. A slab with ly ÷ lx more than 2 acts as a one-way slab. 2. In a one-way slab the main steel is provided in both directions.

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

    A. 1 only

    In a one-way slab main steel is in the short span only; distribution steel is across it.

35. Which of the following statements is/are correct? 1. A column with le ÷ D less than 12 is a short column. 2. Longitudinal steel in a column should be between 0.8% and 6%.

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

    C. Both 1 and 2

    Both are IS 456 rules.

36. Which of the following statements is/are correct? 1. In a T-beam, the slab acts as the flange. 2. The flange of a T-beam resists compression.

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

    C. Both 1 and 2

    The slab forms the flange which resists compression in sagging moment.

37. Which of the following statements is/are correct? 1. The modulus of elasticity of concrete is taken as 5000 √fck. 2. The flexural strength of concrete is taken as 0.7 √fck.

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

    C. Both 1 and 2

    Both are IS 456 expressions in N/mm².

38. Which of the following statements is/are correct? 1. Stirrups resist shear force. 2. Main bars in a beam are placed in the compression zone only.

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

    A. 1 only

    Main bars are placed in the tension zone.

39. Which of the following statements is/are correct? 1. The diameter of a slab bar should not exceed one-eighth of the slab thickness. 2. The maximum spacing of main slab bars is the lesser of 3d and 300 mm.

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

    C. Both 1 and 2

    Both are IS 456 slab rules.

40. Which of the following statements is/are correct? 1. Development length increases with bar diameter. 2. Development length decreases when the design bond stress is higher.

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

    C. Both 1 and 2

    Ld = φ σs ÷ (4 τbd).

41. Which of the following statements is/are correct? 1. Doubly reinforced beams have steel only in the tension zone. 2. Doubly reinforced beams are used when the depth of the beam is restricted.

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

    B. 2 only

    They have steel in both tension and compression zones.

42. Which of the following statements is/are correct? 1. A raft footing covers the whole area under the structure. 2. An isolated footing always supports two or more columns.

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

    A. 1 only

    An isolated footing carries a single column; a combined footing carries two or more.

43. Match the member type with the basic span ÷ effective depth ratio: (a) Cantilever (b) Simply supported (c) Continuous. Values: (1) 7 (2) 20 (3) 26

    1. a-3, b-2, c-1
    2. a-2, b-1, c-3
    3. a-1, b-3, c-2
    4. a-1, b-2, c-3
    Answer

    D. a-1, b-2, c-3

    Basic values are 7, 20 and 26 respectively.

44. Match the steel with the limiting xu,max ÷ d: (a) Fe250 (b) Fe415 (c) Fe500. Values: (1) 0.46 (2) 0.48 (3) 0.53

    1. a-1, b-2, c-3
    2. a-2, b-3, c-1
    3. a-3, b-1, c-2
    4. a-3, b-2, c-1
    Answer

    D. a-3, b-2, c-1

    Fe250 0.53, Fe415 0.48, Fe500 0.46.

45. The cover to reinforcement is provided mainly to

    1. reduce the self weight
    2. protect steel from corrosion and fire
    3. speed up curing
    4. increase the span
    Answer

    B. protect steel from corrosion and fire

    Concrete cover shields steel from moisture, air and heat.