GATE CE cheat sheet

Civil Engineering (CE)

• BOD5 exerted at 20°C: BOD5 = (D1 − D2)(1 − f) / P, where D1, D2 are initial and final DO (mg/L), f = volume of seed in sample/total volume, P = volume of sample/total volume; standard 300 mL bottle.

• Reynolds number Re = ρVD/μ = VD/ν; laminar pipe flow Re < 2000, transitional 2000–4000, turbulent > 4000; Darcy–Weisbach head loss hf = fLV²/(2gD).

• Moment–area theorems: θ_B − θ_A = area of M/EI diagram between A and B; Δ_B − Δ_A − θ_A·L = moment of M/EI diagram about B.

• Slope deflection equation (no support settlement): M_AB = (2EI/L)(2θ_A + θ_B − 3ψ) + FEM_AB, with ψ = chord rotation; FEM for fixed-end beam under UDL w: wL²/12.

• IS 456:2000: minimum grade M20 for RC; minimum clear cover: slab 15 mm, beam 25 mm, column 40 mm, footing 50 mm; development length Ld = φσs/(4τbd).

• Effective stress principle: σ' = σ − u; for capillary rise hc, u = −γw·hc above water table; hc = 2T cosα/(d·γw) with T ≈ 0.073 N/m.

• Terzaghi bearing capacity (strip footing, general shear): q_u = cNc + qNq + 0.5γBNγ; Nq = e^(π tanφ) tan²(45° + φ/2).

• Horizontal curve: degree of curve D = 5730/R (arc definition, R in m); superelevation e + f = V²/(127R) for V in km/h.

• Traffic flow: q = k·v; Greenshields model v = vf(1 − k/kj), giving q = vf·k − (vf/kj)k², capacity qmax = vf·kj/4 at k = kj/2.

• Hydrology: rational method Q = (1/3.6)·C·i·A (Q in m³/s, i in mm/h, A in km²); SCS-CN: Q = (P − 0.2S)²/(P + 0.8S), S = (25400/CN) − 254 (mm).