WBPSC AE Civil cheat sheet

Hooke's law: σ = E·ε, valid within the proportional limit; E for steel ≈ 200–210 GPa, concrete ≈ 25–30 GPa.

Poisson's ratio μ = –lateral strain / longitudinal strain; typical values: steel 0.25–0.30, concrete 0.15–0.20, cork ~0.

Bending stress formula: σ = M·y / I, with y measured from the neutral axis; maximum stress occurs at the extreme fibre.

Shear stress in a beam: τ = V·Q / (I·b), where Q = first moment of area about the neutral axis of the portion above the level considered.

Torsion of a solid circular shaft: τ_max = T·r / J, angle of twist θ = T·L / (G·J), J = π·d⁴/32.

Euler buckling load: P_cr = π²·E·I / (L_e)²; effective length L_e = L (pinned-pinned), 0.5L (fixed-fixed), 0.7L (fixed-pinned), 2L (fixed-free).

Principal stresses: σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x – σ_y)/2)² + τ_xy²]; maximum shear τ_max = (σ₁ – σ₂)/2.

Strain energy under axial load U = P²L/(2AE); under bending U = ∫ M²/(2EI) dx; under torsion U = T²L/(2GJ).

Moment of inertia: rectangle I_xx = b·h³/12; circle I = π·d⁴/64; parallel-axis theorem I = I_c + A·d².

Static indeterminacy = (number of reactions + internal member forces) – (3 × number of rigid bodies); for a plane frame, D_s = 3m + r – 3j.

Kinematic indeterminacy = number of independent joint displacements (rotations + translations) not restrained.

Castigliano's theorem: ∂U/∂P = deflection at the point of application in the direction of P.

Slope-deflection equation: M_AB = M_AB^F + (2EI/L)(2θ_A + θ_B – 3Δ/L).

Three-moment equation for continuous beams: M_A·L₁/I₁ + 2M_B(L₁/I₁ + L₂/I₂) + M_C·L₂/I₂ = –6(A₁x₁/(I₁L₁) + A₂x₂/(I₂L₂)).

Influence line for a simply supported beam: reaction at A = 1 – x/L; shear at section = –x/L (left) or 1 – x/L (right); moment at section = x(L – a)/L for unit load at x < a.

Müller-Breslau principle: the influence line for a function is the deflected shape when the corresponding restraint is released by a unit displacement.

Stiffness method: K·Δ = F; member stiffness for axial bar k = AE/L; for beam element k = EI/L³ × [12, 6L, –12, 6L; …].

Limit state of collapse (IS 456): design strength of concrete in bending = 0.36·f_ck·b·x_u; steel stress = 0.87·f_y; limiting x_u/d = 0.53 for Fe415, 0.48 for Fe500.

Minimum reinforcement in RCC beams: 0.85·b·d/f_y (IS 456); maximum tension steel = 0.04·b·D.

Shear design: τ_v = V_u/(b·d) must not exceed τ_c,max (3.1 MPa for M25); nominal shear reinforcement ≥ 0.4·b·s/f_y.

Development length L_d = φ·σ_s/(4·τ_bd); for deformed bars in tension τ_bd = 1.6 MPa (M25) per IS 456.

Steel bolt value in shear: V_dsb = (f_ub·A_nb/√3)/γ_mb; in bearing: V_dpb = 2.5·k_b·d·t·f_u/γ_mb, with γ_mb = 1.25.

Design tensile strength of a plate: T_dg = f_y·A_g/γ_m0 (γ_m0 = 1.10); T_dn = 0.9·f_u·A_n/γ_m1 (γ_m1 = 1.25).

Effective length of compression members (IS 800): l_eff = K·L; K = 1.0 (pinned), 0.65 (fixed-pinned), 0.5 (fixed-fixed), 2.0 (cantilever).

Slenderness ratio λ = l_eff/r; λ ≤ 180 for compression members carrying design loads, ≤ 250 for secondary members.

Lacing and battening: lacing slenderness ≤ 50 or 0.7 × member slenderness; batten plates designed for transverse shear V_t = V/(N_s·N_b).

Continuity: A₁V₁ = A₂V₂; Bernoulli (ideal): p/γ + V²/2g + z = constant; with head loss h_f between sections.

Reynolds number Re = ρVD/μ = VD/ν; laminar flow in pipe Re < 2000; turbulent Re > 4000.

Darcy-Weisbach: h_f = f·L·V²/(2gD); f = 64/Re for laminar flow; Colebrook-White for turbulent.

Manning's equation: V = (1/n)·R^(2/3)·S^(1/2); for a circular pipe flowing full, R = D/4.

Hydrostatic force on a plane surface: F = γ·h̄·A, where h̄ is the depth to the centroid; centre of pressure y_cp = ȳ + I_G/(ȳ·A).

Orifice discharge: Q = C_d·A·√(2gH); C_d ≈ 0.61–0.65 for sharp-edged orifice; C_v ≈ 0.98, C_c ≈ 0.62.

Venturi meter: Q = C_d·A₁A₂√(2gΔh)/√(A₁² – A₂²); typical C_d = 0.98.

Hydraulic jump: conjugate depths satisfy y₂/y₁ = ½(√(1 + 8F₁²) – 1); energy loss ΔE = (y₂ – y₁)³/(4y₁y₂).

Specific speed of a turbine: N_s = N√P/H^(5/4); for a pump N_s = N√Q/H^(3/4).

Water balance: P – Q – E – ΔS = 0 over a catchment for a given period.

Rational method: Q_p = C·i·A/3.6 (Q in m³/s, i in mm/h, A in km²); C values: 0.9 impervious, 0.3–0.5 agricultural.

Time of concentration Kirpich: t_c = 0.01947·L^0.77·S^(–0.385) (minutes, L in m, S dimensionless).

Unit hydrograph: volume of direct runoff = 1 cm over the catchment; base width ≈ 5.5·t_p; t_p ≈ 0.6·t_c.

Infiltration Horton: f(t) = f_c + (f_0 – f_c)·e^(–kt); φ-index = (P – R)/t_e.

Return period T = 1/(1 – P(X ≤ x)); Gumbel reduced variate y = –ln[ln(T/(T – 1))].

Penman-Monteith reference evapotranspiration: ET₀ = [0.408Δ(R_n – G) + γ·900/(T+273)·u₂·(e_s – e_a)]/[Δ + γ(1 + 0.34u₂)].

Reservoir capacity by mass curve: maximum cumulative difference between inflow and demand ordinates.

Well hydraulics: Dupuit steady flow to a well Q = πK(H² – h²)/ln(R/r); Thiem equation for confined aquifer Q = 2πKb(H – h)/ln(R/r).

Water demand: per capita 135 lpcd (urban domestic); peak factor 1.5–3.0; fire demand (Kuichling) Q = 3182√P (L/min, P in thousands).

Surface overflow rate for settling tanks: SOR = Q/A; for plain sedimentation 15–30 m³/m²·d; detention time 2–4 h.

Rapid sand filter: rate 100–150 L/m²·min (or 4–6 m³/m²·h); backwash rate 600–900 L/m²·min.

Chlorine demand: dose – residual = demand; breakpoint chlorination requires residual free Cl₂ ≥ 0.2 mg/L after 30 min contact.

BOD₅ = BOD_ultimate × (1 – e^(–k·5)); k₂₀ = 0.1–0.3 per day; temperature correction k_T = k₂₀·1.047^(T–20).

Sewage flow: 80% of water supply; peak factor = 1.5 + 2.5/√P (P in thousands).

Activated sludge: F/M = Q·S₀/(V·X); typical F/M 0.2–0.4 d⁻¹; sludge age θ_c = V·X/(Q_w·X_r + Q_e·X_e).

Trickling filter (NRC): efficiency E = 100/(1 + 0.44√(W/(V·F))); organic loading 0.08–0.32 kg BOD/m³·d.

Septic tank: detention 24 h; sludge storage 30–60 L/person/year; soak pit design based on soil percolation rate.

Effective stress: σ' = σ – u; capillary rise h_c = 2T cosθ/(r·γ_w) ≈ 0.3/d (cm, d in mm).

Terzaghi bearing capacity (strip): q_u = c·N_c + q·N_q + 0.5·γ·B·N_γ; N_q = e^(π tanφ)·tan²(45+φ/2).

Standard penetration test: N₆₀ = N_measured × (C_E·C_S·C_R·C_B); correction for overburden N₁ = N·√(100/σ'_v) (kPa).

Consolidation: C_c = Δe / log(σ'₂/σ'₁); coefficient of consolidation c_v = T_v·H_dr²/t; T_v = 0.197 for 50% consolidation.

Shear strength: τ = c' + σ'·tanφ'; for normally consolidated clay c' ≈ 0; for overconsolidated clay c' > 0.

Slope stability: factor of safety F = resisting moment / driving moment; infinite slope F = (c' + γ'·z·cos²β·tanφ')/(γ_sat·z·sinβ·cosβ).

Pile capacity: Q_u = Q_p + Q_s = q_p·A_p + Σ f_s·A_s; for driven piles in sand q_p = 40N (kPa) ≤ 400 kPa.

Liquefaction susceptibility: loose saturated sands with N₁₆₀ < 15 and fines < 15%; factor of safety = CRR/CSR.

Compaction: standard Proctor energy = 594 kJ/m³ (25 blows, 3 layers); modified Proctor = 2703 kJ/m³ (56 blows, 5 layers).

Highway design: stopping sight distance SSD = 0.278·V·t + V²/(254·(f ± g)); V in km/h, t = 2.5 s, f ≈ 0.35–0.40.

Superelevation: e + f = V²/(127·R); maximum e = 7% (plain terrain), 10% (hilly, not snow-bound).

Transition curve length: L_s = V³/(C·R) (C = 80 for plain, 60 for mountainous); or L_s = 0.0215·V³/R based on rate of change of centrifugal acceleration.

Extra widening on curves: W_e = n·l²/(2R) + V/(9.5√R) (metres).

Pavement thickness (IRC 37): CBR-based; for CBR ≤ 5, use design traffic in msa and catalogue charts; subgrade CBR ≥ 8% for good design.

Traffic flow: q = k·v (flow = density × speed); capacity of a 2-lane road ≈ 2000 PCU/h; PCU: car 1.0, bus/truck 3.0, motorcycle 0.5.

Webster's optimum cycle: C_o = (1.5L + 5)/(1 – Y); Y = Σ y_i (critical flow ratios); L = total lost time.

Bridge: IRC Class AA loading (tracked 700 kN, wheeled 400 kN); Class A (wheeled) and Class B (wheeled, lighter).

Flexible pavement: group index GI = 0.2a + 0.005ac + 0.01bd (a, b, c, d from sieve and LL/PI).

Chain correction: C_a = L·(l'/l – 1) for tape length error; C_t = L·α·(T_m – T_0).

Compass: true bearing = magnetic bearing ± declination (east +, west –); interior angle = FB of next line – BB of previous line.

Levelling: HI = RL + BS; RL of new station = HI – FS; rise = previous staff – current staff; fall = current – previous.

Theodolite traverse: latitudes = l·cosθ, departures = l·sinθ; closure error e = √(ΣL² + ΣD²); Bowditch correction: C_L = –ΣL·(l/Σl).

Tacheometry: D = K·s·cos²θ + C·cosθ; V = (K·s·sin2θ)/2 + C·sinθ; K = 100, C = 0 for anallatic lens.

Curves: degree of curve D = 1719/R (20 m chord); tangent length T = R·tan(Δ/2); length of curve L = π·R·Δ/180.

Triangulation: well-conditioned triangle has angles between 30° and 120°; strength of figure R = (number of conditions)/(number of observations).

GPS: position from ≥ 4 satellites; PDOP < 6 for good geometry; DGPS improves accuracy to sub-metre.

Remote sensing: spatial resolution of Landsat 8 OLI = 30 m (multispectral), 15 m (panchromatic).

Cement: initial setting time ≥ 30 min, final ≤ 600 min (IS 4031); soundness by Le Chatelier ≤ 10 mm; specific gravity 3.10–3.15.

Concrete mix: water-cement ratio by Abrams' law; strength = A/B^w/c; for M25 w/c ≈ 0.50; minimum cement 300 kg/m³ for moderate exposure.

Workability: slump 25–75 mm (normal), 75–125 mm (high); compaction factor 0.75–0.80 (low), 0.85–0.92 (medium), 0.92–0.96 (high).

Aggregate: coarse aggregate size 20 mm (standard), 40 mm (mass concrete); flakiness index ≤ 30%, elongation ≤ 30% for good aggregate.

Bricks: compressive strength ≥ 3.5 MPa (common), ≥ 10 MPa (class A); water absorption ≤ 20% (common), ≤ 15% (class A).

Steel: Fe415, Fe500, Fe550; yield strength 415, 500, 550 MPa; elongation ≥ 14.5% (Fe415), ≥ 12% (Fe500).

Timber: moisture content ≤ 12% for structural use; permissible stress in bending ≈ 8–10 MPa for teak.

Bitumen: penetration grade 30/40, 60/70, 80/100; viscosity grade VG-10, VG-30, VG-40; softening point of VG-30 ≈ 47–54 °C.

PERT: expected time t_e = (t_o + 4t_m + t_p)/6; variance σ² = ((t_p – t_o)/6)²; project duration = sum of critical path t_e.

CPM: critical path has zero total float; total float = LFT – EFT = LST – EST.

Earned value: CPI = EV/AC; SPI = EV/PV; EAC = BAC/CPI (if CPI remains constant).

Rate analysis: overheads and contractor's profit typically 10–15% added to material and labour cost.

Earthwork: prismoidal formula V = (L/6)(A₁ + 4A_m + A₂); trapezoidal V = L·(A₁ + A₂)/2.

Building estimate: plinth area rate method (cost/m²); cube rate method (cost/m³); approximate quantity of cement ≈ 0.4 m³ per m³ of concrete (1:2:4).

Labour constants: mason 0.5 day per m³ of brickwork; unskilled labour 1.2 days per m³ of concrete.

Project network: AOA diagram uses dummy activities to maintain logic; AON diagram uses nodes for activities.

Duty (ha/cumec) and delta (m): delta = 8.64·B/D (B in days, D in ha/cumec); base period B = 8.64·D·Δ.

Consumptive use: Blaney-Criddle f = p·(0.46T + 8.13); Penman and Hargreaves alternatives.

Canal design: Kennedy's equation V₀ = 0.55·m·y^0.64; Lacey's regime: V = (Q·f²/140)^(1/6), R = 0.47(Q/f)^(1/3).

Silt factor f = 1.76√d_mm; critical velocity ratio m = V/V₀.

Weir: Broad-crested weir Q = 1.71·C_d·L·H^(3/2); sharp-crested Q = (2/3)·C_d·L·√(2g)·H^(3/2).

Hydraulic jump for energy dissipation: F₁ = V₁/√(g·y₁); jump length L_j ≈ 6.9(y₂ – y₁).

Gravity dam: factor of safety against overturning ≥ 1.5 (usual), sliding ≥ 1.0 (usual), 1.5 (extreme); no tension at heel.

Spillway: ogee profile Q = C·L·H_e^(3/2); C ≈ 2.2 (SI) for free overflow; energy dissipation by ski-jump or stilling basin.

Well irrigation: specific capacity = Q/s (m³/h per m drawdown); radius of influence R = 3000·s·√K (m).

Indian Constitution: Article 21 (right to life and personal liberty), Article 32 (writs), Article 356 (President's rule), Article 370 (special status to J&K, abrogated 2019).

Fundamental duties: Article 51A, added by 42nd Amendment (1976) on Swaran Singh Committee recommendation.

West Bengal: capital Kolkata; Sundarbans mangrove (UNESCO World Heritage); Damodar Valley Corporation (1948) is India's first multipurpose river valley project.

Indian geography: Tropic of Cancer passes through 8 states; Western Ghats are a biodiversity hotspot; Deccan Plateau is basaltic.

Economy: GDP growth measured by CSO; inflation measured by CPI (base 2012) and WPI (base 2011-12); RBI controls monetary policy.

Science: pH of pure water 7; speed of light 3×10⁸ m/s; SI unit of force newton; atomic number of carbon 6.

History: Indus Valley Civilization (2500–1500 BCE); Mauryan Empire (321–185 BCE); Bengal Renaissance (19th century).

Environment: Kyoto Protocol (1997), Paris Agreement (2015) target 1.5–2 °C; India's INDC: 33–35% emission intensity reduction by 2030 over 2005.

Percentage: x% of y = (x/100)·y; percentage increase = (new – old)/old × 100.

Profit and loss: SP = CP·(1 + profit%/100); discount = MP – SP; discount% = (MP – SP)/MP × 100.

Time and work: if A does a job in x days and B in y days, together 1/x + 1/y per day; time = xy/(x + y).

Simple interest SI = P·R·T/100; compound interest CI = P[(1 + R/100)^T – 1].

Average speed for equal distances = 2xy/(x + y); for equal times = (x + y)/2.

Engineering ethics: paramountcy of public safety, competence, honesty, confidentiality, conflict of interest disclosure; whistle-blowing when public safety is at risk.

Professional practice: IEI (Institution of Engineers India) code of ethics; registration with state council; continuing professional development.

Contract: offer + acceptance + consideration; FIDIC conditions for international projects; arbitration clause for dispute resolution.