Advanced Civil Engineering Handbook › Chapter 4

Chapter 4 of 14 · 42 formulas · 81 one-liners

Reinforced Concrete Design (IS 456:2000) — formulas & one-liners

Limit state and working stress design of RCC beams, slabs, columns and footings as per IS 456:2000, with prestressed concrete basics from IS 1343.

4.1 Limit state basics, materials and load combinations

Design compressive strength of concrete (flexure)

f_{c d} = \frac{0.67 f _{c k}}{γ _{m}} = \frac{0.67 f _{c k}}{1.5} = 0.446 f_{c k}

f_ck = characteristic cube strength (MPa); γm = 1.5 for concrete

Peak stress of the IS 456 design stress block (cl. 38.1)

Design yield strength of steel

f_{y d} = \frac{f _{y}}{γ _{m}} = \frac{f _{y}}{1.15} = 0.87 f_{y}

f_y = characteristic yield strength (MPa); γm = 1.15 for steel

IS 456 cl. 36.4.2

Short-term modulus of elasticity of concrete

E_{c} = 5000 f_{c k}

E_c in MPa; f_ck in MPa

IS 456 cl. 6.2.3.1; e.g. M20 → about 22 360 MPa

Flexural strength (modulus of rupture) of concrete

f_{cr} = 0.7 f_{c k}

f_cr in MPa; f_ck in MPa

IS 456 cl. 6.2.2; e.g. M20 → 3.13 MPa

Partial safety factor for material strength at the ultimate limit state is 1.5 for concrete and 1.15 for steel (IS 456 cl. 36.4.2).
Ultimate load combinations (IS 456 Table 18): 1.5(DL + LL), 1.5(DL + WL) or 0.9DL + 1.5WL when dead load is favourable, and 1.2(DL + LL + WL).
Serviceability load combinations (IS 456 Table 18): 1.0(DL + LL), 1.0(DL + WL) and 1.0DL + 0.8LL + 0.8WL.
Characteristic strength is the value below which not more than 5% of test results are expected to fall (IS 456 cl. 36.1).
Maximum compressive strain in concrete is 0.0035 in bending and 0.002 in axial compression (IS 456 cl. 38.1 and 39.1).
Modulus of elasticity of steel Es is taken as 2 × 10⁵ N/mm² (200 kN/mm²) in IS 456.
Creep coefficient as per IS 456 cl. 6.2.5.1 is 2.2 for loading at 7 days, 1.6 at 28 days and 1.1 at 1 year.
In the absence of test data, total shrinkage strain of concrete may be taken as 0.0003 (IS 456 cl. 6.2.4.1).
Unit weight of plain concrete is 24 kN/m³ and of reinforced concrete 25 kN/m³ (IS 875 Part 1).

4.2 Flexure: stress block and singly reinforced beams

Total compressive force in concrete

C = 0.36 f_{c k} b x_{u}

b = width; x_u = depth of neutral axis

Resultant acts at 0.42 x_u from the extreme compression fibre

Depth of neutral axis (from C = T)

x_{u} = \frac{0.87 f _{y} A _{s t}}{0.36 f _{c k} b}

A_st = area of tension steel; b = width

Valid only while x_u ≤ x_u,max (under-reinforced)

Limiting neutral axis depth ratio

\frac{x _{u, m a x}}{d} = \frac{700}{1100 + 0.87 f _{y}}

f_y in MPa; d = effective depth

From ε_cu = 0.0035 and steel strain f_y/(1.15E_s) + 0.002 at failure, cl. 38.1(f)

Moment of resistance of under-reinforced section

M_{u} = 0.87 f_{y} A_{s t} (d - 0.42 x_{u})

d = effective depth; x_u = actual NA depth

IS 456 Annex G-1.1

Moment of resistance in terms of steel only

M_{u} = 0.87 f_{y} A_{s t} d (1 - \frac{A _{s t} f _{y}}{b d f _{c k}})

A_st = tension steel area; b, d in mm

IS 456 Annex G-1.1(b)

Limiting moment of resistance

M_{u, l i m} = 0.36 f_{c k} b x_{u, m a x} (d - 0.42 x_{u, m a x})

x_u,max from the limiting ratio

Written as Q b d²: Q = 0.149 f_ck (Fe250), 0.138 f_ck (Fe415), 0.133 f_ck (Fe500)

Tension steel for a given moment (singly reinforced)

A_{s t} = \frac{0.5 f _{c k}}{f _{y}} [1 - 1 - \frac{4.6 M _{u}}{f _{c k} b d ^{2}}] b d

M_u in N·mm; b, d in mm

Use only when M_u ≤ M_u,lim

Limiting percentage of tension steel

p_{t, l i m} = \frac{0.36 f _{c k}}{0.87 f _{y}} \cdot \frac{x _{u, m a x}}{d} \times 100

p_t,lim in percent

M20: 1.76% (Fe250), 0.96% (Fe415), 0.76% (Fe500)

Minimum tension reinforcement in beams

\frac{A _{s}}{b d} = \frac{0.85}{f _{y}}

A_s = minimum tension steel; f_y in MPa

IS 456 cl. 26.5.1.1(a)

The IS 456 design stress block is parabolic up to a strain of 0.002 and rectangular from 0.002 to 0.0035, with a maximum design stress of 0.446 fck (cl. 38.1).
Limiting xu,max/d is 0.53 for Fe250, 0.48 for Fe415 and 0.46 for Fe500 (IS 456 Note under cl. 38.1).
For M20 concrete with Fe415 steel, Mu,lim = 2.76 b d² (N·mm, with b and d in mm).
A section with xu < xu,max is under-reinforced: steel yields first, giving a ductile failure with warning.
If xu exceeds xu,max the section is over-reinforced; IS 456 requires a larger depth or compression steel (doubly reinforced design) instead.
Maximum tension reinforcement in a beam is 4% of bD, and maximum compression reinforcement is also 4% of bD (IS 456 cl. 26.5.1.1 and 26.5.1.2).
Effective span of a simply supported beam is the lesser of (clear span + effective depth) and the centre-to-centre distance between supports (IS 456 cl. 22.2).
For lateral stability, a simply supported beam's clear distance between lateral restraints must not exceed 60b or 250b²/d, whichever is less (IS 456 cl. 23.3).
For a cantilever, the clear distance from the free end to the lateral restraint must not exceed 25b or 100b²/d, whichever is less (IS 456 cl. 23.3).
Side-face reinforcement of 0.1% of the web area is needed when the web depth exceeds 750 mm, split equally on both faces at spacing not more than 300 mm or the web thickness (IS 456 cl. 26.5.1.3).

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4.3 Doubly reinforced and flanged (T and L) beams

Extra moment carried by compression steel

M_{u} - M_{u, l i m} = (f_{sc} - 0.446 f_{c k}) A_{sc} (d - d^{'})

f_sc = stress in compression steel; A_sc = compression steel; d' = cover to its centre

The 0.446 f_ck term (displaced concrete) is often neglected

Additional tension steel to balance compression steel

A_{s t 2} = \frac{( f _{sc} - 0.446 f _{c k} ) A _{sc}}{0.87 f _{y}}

Total A_st = A_st,lim + A_st2

A_st,lim is the steel for M_u,lim

Effective flange width of T-beam

b_{f} = \frac{l _{0}}{6} + b_{w} + 6 D_{f}

l₀ = distance between points of zero moment; b_w = web width; D_f = flange thickness

IS 456 cl. 23.1.2(a); not more than actual width

Effective flange width of L-beam

b_{f} = \frac{l _{0}}{12} + b_{w} + 3 D_{f}

Symbols as for T-beam

IS 456 cl. 23.1.2(b); for an isolated T-beam b_f = l₀/(l₀/b + 4) + b_w

Moment of resistance of T-beam when x_u > D_f (D_f/d ≤ 0.2)

M_{u} = 0.36 f_{c k} b_{w} x_{u} (d - 0.42 x_{u}) + 0.45 f_{c k} (b_{f} - b_{w}) D_{f} (d - \frac{D _{f}}{2})

b_f = flange width; b_w = web width

IS 456 Annex G-2.2, with x_u,max replaced by x_u as per G-2.2.2

Equivalent flange depth when D_f/d > 0.2

y_{f} = 0.15 x_{u} + 0.65 D_{f} \leq D_{f}

Replace D_f by y_f in the T-beam formula

IS 456 Annex G-2.2.1

When the neutral axis lies within the flange (xu ≤ Df), a flanged beam is designed as a rectangular beam of width bf.
For continuous beams and frames, l0 for the flange width may be taken as 0.7 times the effective span (IS 456 cl. 23.1.2 Note).
Where the slab's main bars run parallel to a T-beam, transverse flange reinforcement of at least 60% of the slab's mid-span main steel must be provided over the beam (IS 456 cl. 23.1.1(b)).

4.4 Shear and torsion

Nominal shear stress

τ_{v} = \frac{V _{u}}{b d}

V_u = factored shear; b = width (web width for flanged); d = effective depth

IS 456 cl. 40.1

Shear to be carried by shear reinforcement

V_{u s} = V_{u} - τ_{c} b d

τ_c = design shear strength of concrete (Table 19)

Required only when τ_v > τ_c

Shear capacity of vertical stirrups

V_{u s} = \frac{0.87 f _{y} A _{s v} d}{s _{v}}

A_sv = total area of stirrup legs; s_v = spacing

IS 456 cl. 40.4(a)

Minimum shear reinforcement

\frac{A _{s v}}{b s _{v}} \geq \frac{0.4}{0.87 f _{y}}

A_sv = stirrup leg area; s_v = spacing; f_y ≤ 415 MPa in this formula

IS 456 cl. 26.5.1.6

Equivalent shear due to torsion

V_{e} = V_{u} + 1.6 \frac{T _{u}}{b}

T_u = factored torsional moment; b = breadth of beam

IS 456 cl. 41.3.1

Equivalent bending moment due to torsion

M_{e 1} = M_{u} + \frac{T _{u}}{1.7} (1 + \frac{D}{b})

D = overall depth; b = breadth

IS 456 cl. 41.4.2

Maximum shear stress τc,max (IS 456 Table 20) is 2.5, 2.8, 3.1, 3.5, 3.7 and 4.0 N/mm² for M15, M20, M25, M30, M35 and M40 and above.
For M20 concrete, τc (IS 456 Table 19) is 0.28, 0.36, 0.48, 0.62 and 0.82 N/mm² at 100As/bd of 0.15, 0.25, 0.50, 1.00 and 3.00 or more.
If τv exceeds τc,max, the section must be redesigned; stirrups cannot make up the deficit (IS 456 cl. 40.2.3).
In solid slabs τv must not exceed half of τc,max (IS 456 cl. 40.2.3.1).
Shear strength of solid slabs is kτc, with k from 1.00 for slabs 300 mm or thicker up to 1.30 for slabs 150 mm or thinner (IS 456 cl. 40.2.1.1).
Maximum spacing of vertical stirrups is 0.75d, and of 45° inclined stirrups d, but never more than 300 mm (IS 456 cl. 26.5.1.5).
When τv < τc, minimum shear reinforcement is still provided in beams, except in members of minor importance such as lintels (IS 456 cl. 26.5.1.6).
Near a support, the shear strength of concrete may be enhanced to 2dτc/av, where av is the distance of the section from the support face (IS 456 cl. 40.5).
Spacing of torsion stirrups must not exceed x1, (x1 + y1)/4 or 300 mm, where x1 and y1 are the short and long centre-to-centre dimensions of the stirrup (IS 456 cl. 26.5.1.7).

4.5 Bond, development length, cover and detailing

Development length

L_{d} = \frac{ϕ σ _{s}}{4 τ _{b d}}

φ = bar diameter; σ_s = 0.87 f_y at design load; τ_bd = design bond stress

IS 456 cl. 26.2.1

Development length check at simple supports and points of inflection

L_{d} \leq 1.3 \frac{M _{1}}{V} + L_{0}

M₁ = moment of resistance of bars continuing into the support; V = shear force; L₀ = embedment beyond the support centre

IS 456 cl. 26.2.3.3(c); factor 1.3 when bar ends are confined by a compressive reaction (otherwise 1.0)

Design bond stress τbd for plain bars in tension is 1.2, 1.4, 1.5, 1.7 and 1.9 N/mm² for M20, M25, M30, M35 and M40 and above (IS 456 cl. 26.2.1.1).
For deformed bars (IS 1786) the bond stress values are increased by 60%, and for bars in compression by a further 25% (IS 456 cl. 26.2.1.1).
Development length is about 47φ for Fe415 deformed bars in tension in M20 concrete.
The anchorage value of a bend is 4φ for every 45° of bend, up to a maximum of 16φ; so a standard U-hook counts as 16φ and a 90° bend as 8φ (IS 456 cl. 26.2.2.1).
Development length of bundled bars is increased by 10% for two bars, 20% for three bars and 33% for four bars in contact (IS 456 cl. 26.2.1.2).
Lap length in flexural tension is the greater of Ld and 30φ; in direct tension the greater of 2Ld and 30φ (IS 456 cl. 26.2.5.1(c)).
Lap length in compression is the greater of the development length in compression and 24φ (IS 456 cl. 26.2.5.1(c)).
Lap splices should not be used for bars larger than 36 mm; such bars are normally welded, and lapping is allowed only with additional spirals where welding is impracticable (IS 456 cl. 26.2.5.1(a)).
Horizontal clear distance between parallel main bars must not be less than the bar diameter or 5 mm more than the nominal maximum aggregate size (IS 456 cl. 26.3.2).
Nominal cover (IS 456 Table 16) is 20, 30, 45, 50 and 75 mm for mild, moderate, severe, very severe and extreme exposure.
Minimum cover is 40 mm for longitudinal bars in columns and 50 mm for footings (IS 456 cl. 26.4.2.1 and 26.4.2.2).
Actual cover may exceed nominal cover by up to 10 mm but must never be less (tolerance +10 mm, −0 mm; IS 456 cl. 26.4.2).
Minimum grade for RCC (IS 456 Table 5) is M20 for mild, M25 for moderate, M30 for severe, M35 for very severe and M40 for extreme exposure.
Maximum free water-cement ratio for RCC (IS 456 Table 5) is 0.55 for mild, 0.50 for moderate, 0.45 for severe and very severe, and 0.40 for extreme exposure.
Minimum cement content for RCC (IS 456 Table 5) is 300 kg/m³ for mild and moderate, 320 for severe, 340 for very severe and 360 kg/m³ for extreme exposure.
Cement content above 450 kg/m³ of OPC should not be used unless special care is taken (IS 456 cl. 8.2.4.2).

4.6 Slabs, deflection and cracking

Permissible span/effective-depth ratio

(\frac{L}{d})_{m a x} = (\frac{L}{d})_{basic} \times k_{t} \times k_{c} \times k_{f}

Basic = 7 (cantilever), 20 (simply supported), 26 (continuous); k_t, k_c, k_f = modification factors for tension steel, compression steel and flanges

IS 456 cl. 23.2.1; basic values for spans up to 10 m

Steel service stress for the tension modification factor

f_{s} = 0.58 f_{y} \frac{A _{s t, req}}{A _{s t, prov}}

A_st,req = steel required; A_st,prov = steel provided

Used to read k_t from IS 456 Fig. 4

Moment coefficients, simply supported two-way slab (corners free to lift)

M_{x} = α_{x} w l_{x}^{2}, α_{x} = \frac{r ^{4}}{8 ( 1 + r ^{4} )}, α_{y} = \frac{r ^{2}}{8 ( 1 + r ^{4} )}

r = l_y/l_x; w = factored load per unit area; l_x = shorter span

IS 456 Annex D-2 (Table 27); M_y = α_y w l_x²

A slab supported on all four edges is designed as one-way when ly/lx exceeds 2 and as two-way when ly/lx ≤ 2.
For spans above 10 m, the basic span/depth ratio is multiplied by 10/span (in metres), except for cantilevers, whose deflection must be calculated (IS 456 cl. 23.2.1(b)).
For two-way slabs up to 3.5 m span with live load up to 3 kN/m², span/overall depth may be 35 (simply supported) and 40 (continuous) for mild steel, multiplied by 0.8 for Fe415, i.e. 28 and 32 (IS 456 cl. 24.1 Note 2).
Minimum reinforcement in slabs is 0.12% of the gross section for HYSD bars and 0.15% for mild steel bars (IS 456 cl. 26.5.2.1).
Diameter of slab bars must not exceed one-eighth of the total slab thickness (IS 456 cl. 26.5.2.2).
Main bars in slabs are spaced at not more than 3d or 300 mm, and distribution bars at not more than 5d or 450 mm, whichever is smaller (IS 456 cl. 26.3.3(b)).
Torsion steel at a corner where both edges are discontinuous is 3/4 of the mid-span steel in four layers over a length of one-fifth of the shorter span; with one edge discontinuous it is half of this (IS 456 Annex D-1.8, D-1.9).
Final deflection under all loads, including creep and shrinkage, should not exceed span/250 (IS 456 cl. 23.2(a)).
Deflection after construction of partitions and finishes should not exceed span/350 or 20 mm, whichever is less (IS 456 cl. 23.2(b)).
Surface crack width should generally not exceed 0.3 mm, and 0.004 times the nominal cover in aggressive environments (IS 456 cl. 35.3.2).
Table 12 bending moment coefficients for continuous beams and slabs: +1/12 (DL) and +1/10 (LL) near the middle of the end span, and −1/10 (DL) and −1/9 (LL) at the support next to the end support.

4.7 Columns and footings

Axial load capacity of short column (with minimum eccentricity)

P_{u} = 0.4 f_{c k} A_{c} + 0.67 f_{y} A_{sc}

A_c = net concrete area; A_sc = area of longitudinal steel

IS 456 cl. 39.3; valid when e_min ≤ 0.05D; multiply by 1.05 for helically tied columns (cl. 39.4)

Minimum eccentricity

e_{m i n} = \frac{l}{500} + \frac{D}{30} \geq 20 mm

l = unsupported length; D = lateral dimension in the direction considered

IS 456 cl. 25.4

Additional moment in slender column

M_{a} = \frac{P _{u} D}{2000} (\frac{l _{e x}}{D})^{2}

l_ex = effective length; D = depth in the plane of bending

IS 456 cl. 39.7.1

Biaxial bending interaction

(\frac{M _{ux}}{M _{ux 1}})^{α_{n}} + (\frac{M _{u y}}{M _{u y 1}})^{α_{n}} \leq 1

M_ux1, M_uy1 = uniaxial capacities at load P_u; α_n = 1.0 for P_u/P_uz ≤ 0.2 and 2.0 for P_u/P_uz ≥ 0.8 (linear between); P_uz = 0.45 f_ck A_c + 0.75 f_y A_sc

IS 456 cl. 39.6

Punching (two-way) shear strength of slab or footing

τ_{c, punch} = k_{s} \cdot 0.25 f_{c k}, k_{s} = 0.5 + β_{c} \leq 1

β_c = short side/long side of column

IS 456 cl. 31.6.3.1; critical perimeter at d/2 from the column face

A column is short when both lex/D and lex/b are less than 12; otherwise it is slender (IS 456 cl. 25.1.2).
Unsupported length of a column must not exceed 60 times its least lateral dimension (IS 456 cl. 25.3.1).
Effective length factors (IS 456 Table 28, recommended values): both ends fixed 0.65L, fixed–hinged 0.80L, both hinged 1.0L, fixed–free 2.0L.
Longitudinal steel in columns must be at least 0.8% and not more than 6% of the gross area, with 4% recommended as a practical maximum (IS 456 cl. 26.5.3.1).
A column needs at least 4 longitudinal bars if rectangular and 6 if circular, each at least 12 mm in diameter (IS 456 cl. 26.5.3.1).
Lateral ties must be at least one-quarter of the largest longitudinal bar diameter and not less than 6 mm (IS 456 cl. 26.5.3.2(c)).
Tie pitch must not exceed the least lateral dimension, 16 times the smallest longitudinal bar diameter, or 300 mm (IS 456 cl. 26.5.3.2(c)).
Helix pitch must not exceed 75 mm or one-sixth of the core diameter, and must not be less than 25 mm or three times the helix bar diameter (IS 456 cl. 26.5.3.2(d)).
Thickness of a footing at its edge must be at least 150 mm on soil and at least 300 mm above the tops of piles (IS 456 cl. 34.1.2).
In footings, bending moment is checked at the face of the column and one-way shear at distance d from the face (IS 456 cl. 34.2.3.2 and 34.2.4.1).
In a rectangular footing, a fraction 2/(β + 1) of the short-direction steel is placed in a central band of width equal to the short side, where β = long side/short side (IS 456 cl. 34.3.1(c)).
Permissible bearing stress at a column base is 0.45fck√(A1/A2), where √(A1/A2) is limited to 2 (IS 456 cl. 34.4).

4.8 Working stress method (IS 456 Annex B)

Modular ratio

m = \frac{280}{3 σ _{c b c}}

σ_cbc = permissible compressive stress in bending (MPa)

IS 456 Annex B-1.3; M20 → m = 13.33; allows for long-term creep

Critical (balanced) neutral axis depth factor

k = \frac{m σ _{c b c}}{m σ _{c b c} + σ _{s t}}, j = 1 - \frac{k}{3}

σ_st = permissible tensile stress in steel; x = kd; lever arm = jd

Balanced section reaches σ_cbc and σ_st together

Moment of resistance of balanced section

M = \frac{1}{2} σ_{c b c} k j b d^{2} = R b d^{2}

R = ½ σ_cbc k j

M20 with Fe415: k = 0.289, j = 0.904, R ≈ 0.91 MPa

Permissible stress in bending compression σcbc (IS 456 Table 21) is 5.0, 7.0, 8.5, 10.0, 11.5 and 13.0 N/mm² for M15, M20, M25, M30, M35 and M40.
Permissible direct compressive stress σcc (IS 456 Table 21) is 4.0, 5.0, 6.0, 8.0, 9.0 and 10.0 N/mm² for M15, M20, M25, M30, M35 and M40.
Permissible tensile stress in steel (IS 456 Table 22) is 140 N/mm² for Fe250 bars up to 20 mm (130 above 20 mm), 230 N/mm² for Fe415 and 275 N/mm² for Fe500.
Maximum shear stress τc,max in working stress design (IS 456 Table 24) is 1.6, 1.8, 1.9, 2.2, 2.3 and 2.5 N/mm² for M15 to M40.

4.9 Prestressed concrete basics (IS 1343)

Fibre stress in a prestressed beam

f = \frac{P}{A} \pm \frac{P e}{Z} \pm \frac{M}{Z}

P = prestress force; e = tendon eccentricity; Z = section modulus; M = external moment

Sign depends on the fibre (top or bottom) and the sense of P·e and M

Balancing load of a parabolic tendon (load balancing)

w_{b} = \frac{8 P h}{L ^{2}}

h = central sag of tendon; L = span

Upward UDL exerted by the tendon

Loss due to friction (post-tensioning)

P_{x} = P_{0} e^{- (μα + k x)}

μ = coefficient of friction; α = cumulative angle change (rad); k = wobble coefficient per m; x = distance from jack

Curvature plus wave (wobble) effect

Loss due to elastic shortening (pre-tensioned)

Δ f_{p} = m f_{c}

m = E_s/E_c; f_c = concrete stress at the level of the tendon

Zero for a single post-tensioned tendon

Losses common to pre- and post-tensioning are elastic shortening, creep, shrinkage and relaxation of steel; friction and anchorage slip losses occur only in post-tensioning.
Minimum grade of concrete is M30 for post-tensioned and M40 for pre-tensioned members (IS 1343:2012, unchanged from IS 1343:1980 cl. 4.2.1); higher grades are needed for more severe exposure.
High-tensile steel is essential in prestressing because losses of the order of 200–300 N/mm² would wipe out most of the prestress in mild steel.
The kern distance of a rectangular section is h/6 on either side of the centroid, so the middle-third rule applies.
The pressure line in a prestressed beam is displaced from the cable line by M/P, where M is the external moment.
A concordant cable produces no secondary (parasitic) moments in a continuous prestressed beam.
Hoyer's long-line method is a pre-tensioning system; Freyssinet, Magnel–Blaton, Gifford–Udall and Lee–McCall are post-tensioning anchorage systems.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.