Limit State Design of Beams in Flexure

IS 456 designs RCC members by the limit state method: strength at collapse (limit state of collapse) and behaviour in service (deflection and cracking).

Partial safety factors

• Loads: 1.5 (DL + LL); 1.2 (DL + LL + WL/EL); 1.5 or 0.9 for DL with WL/EL.
• Materials: γ_m = 1.5 for concrete, 1.15 for steel. Design strengths: concrete 0.67 f_ck/1.5 = 0.446 f_ck; steel f_y/1.15 = 0.87 f_y.

The stress block

Assumptions: plane sections stay plane; tensile strength of concrete is ignored; maximum compressive strain in concrete 0.0035 in bending; parabolic–rectangular stress block.

• Total compression C = 0.36 f_ck b x_u, acting at 0.42 x_u from the top.
• Tension T = 0.87 f_y A_st.
• From C = T: x_u = 0.87 f_y A_st/(0.36 f_ck b).

Limiting neutral axis and balanced sections

• x_u < x_u,max: under-reinforced (steel yields first: ductile, preferred).
• x_u = x_u,max: balanced. x_u > x_u,max: over-reinforced (concrete crushes suddenly; not allowed). If it happens, take x_u = x_u,max.

Moment of resistance (under-reinforced): M_u = 0.87 f_y A_st (d − 0.42 x_u).

Doubly reinforced and flanged beams

• If the design moment exceeds M_u,lim, add compression steel: M_u − M_u,lim = f_sc A_sc (d − d').
• T-beams: effective flange width b_f = l₀/6 + b_w + 6D_f (for T-beams). If the neutral axis lies within the flange, design it as a rectangular beam of width b_f.

Serviceability

• Span/effective depth limits: 7 (cantilever), 20 (simply supported), 26 (continuous), for spans up to 10 m, modified for steel stress and percentage.
• Minimum tension steel: A_s/(bd) = 0.85/f_y. Maximum: 4% of bD.

Worked example

b = 230 mm, d = 450 mm, M20, Fe 415, A_st = 3 bars of 16 mm (603 mm²).

• x_u = 0.87 × 415 × 603/(0.36 × 20 × 230) = 131.5 mm; x_u,max = 0.48 × 450 = 216 mm → under-reinforced.
• M_u = 0.87 × 415 × 603 × (450 − 0.42 × 131.5) = 85.9 kN·m.

Practice