Construction Materials, Concrete and Steel Structures

Syllabus

• Construction materials and management: cement, aggregates, concrete, bricks, steel, timber; concrete mix design; project planning and network analysis (PERT, CPM); cost estimation.
• Concrete structures: working stress and limit state design; beams, slabs, columns and footings; bond and development length; prestressed concrete.
• Steel structures: working stress and limit state design; tension and compression members, beams, beam-columns, column bases; bolted and welded connections; plastic analysis.

Construction materials and management

• Abrams' law: concrete strength falls as the water-cement ratio rises.
• Characteristic strength f_ck is the strength below which not more than 5% of results fall: f_target = f_ck + 1.65s.
• PERT: expected time t_e = (t_o + 4t_m + t_p)/6 and standard deviation σ = (t_p − t_o)/6. The project time is approximately normal, with mean and variance summed along the critical path.
• CPM: total float = LS − ES = LF − EF. Activities with zero total float form the critical path, the longest path through the network.

Reinforced concrete (IS 456, limit state method)

• Design strength of concrete in flexure: 0.67f_ck/1.5 = 0.446f_ck; the stress block gives a compressive force of 0.36f_ck b x_u acting at 0.42x_u from the top.
• Design strength of steel: f_y/1.15 = 0.87f_y.
• Limiting neutral axis depth x_u,max/d: 0.53 (Fe 250), 0.48 (Fe 415), 0.46 (Fe 500).
• Limiting moment M_u,lim = 0.36f_ck b x_u,max (d − 0.42x_u,max), which gives 0.148, 0.138 and 0.133 f_ckbd² for Fe 250, Fe 415 and Fe 500.
• Neutral axis of a singly reinforced section: x_u = 0.87f_yA_st/(0.36f_ckb).
• Minimum tension steel: A_s/(bd) = 0.85/f_y; maximum 4% of bD.
• Short axially loaded column: P_u = 0.4f_ckA_c + 0.67f_yA_sc.
• Development length L_d = φσ_s/(4τ_bd).
• Working stress method: modular ratio m = 280/(3σ_cbc).
• Prestressed concrete: losses come from elastic shortening, shrinkage, creep, relaxation of steel, friction and anchorage slip. For a concentric prestress P on area A, the uniform stress is P/A; an eccentricity e adds ± Pe/Z.

Steel structures (IS 800, limit state method)

• Partial safety factors: γ_m0 = 1.10 (yielding) and γ_m1 = 1.25 (rupture).
• Tension member: design strength in gross-section yielding T_dg = A_gf_y/γ_m0; in net-section rupture T_dn = 0.9A_nf_u/γ_m1; the smaller value (with block shear) governs.
• Fillet weld: effective throat = 0.7 × size.
• Plastic analysis: plastic moment M_p = f_yZ_p; shape factor = Z_p/Z_e = 1.5 for a rectangle, 16/(3π) ≈ 1.70 for a solid circle, about 1.12 to 1.14 for I-sections.
• Collapse loads: simply supported beam with a central load W_c = 4M_p/L; fixed beam with a central load 8M_p/L; fixed beam with a UDL w_c = 16M_p/L².

Worked example

A singly reinforced beam 300 mm wide with effective depth 500 mm uses M25 concrete and Fe 415 steel with A_st = 1200 mm². Is it under-reinforced, and what is its moment of resistance?

b, d, fck, fy, Ast = 300, 500, 25, 415, 1200
xu = 0.87 * fy * Ast / (0.36 * fck * b)
xu_max = 0.48 * d
print(f"xu = {xu:.2f} mm, xu,max = {xu_max:.0f} mm -> {'under' if xu < xu_max else 'over'}-reinforced")
Mu = 0.87 * fy * Ast * (d - 0.42 * xu)
print(f"Mu = {Mu / 1e6:.2f} kN m")
Mlim = 0.36 * fck * b * xu_max * (d - 0.42 * xu_max)
print(f"Mu,lim = {Mlim / 1e6:.2f} kN m = {Mlim / (fck * b * d * d):.4f} fck b d^2")

# PERT for one activity: optimistic 4, most likely 6, pessimistic 14 days
to, tm, tp = 4, 6, 14
print(f"te = {(to + 4 * tm + tp) / 6:.2f} days, sigma = {(tp - to) / 6:.3f} days")

xu = 160.47 mm, xu,max = 240 mm -> under-reinforced
Mu = 187.43 kN m
Mu,lim = 258.68 kN m = 0.1380 fck b d^2
te = 7.00 days, sigma = 1.667 days

Practice questions