KPSC AE Civil cheat sheet

A simply supported beam with central point load W and span L has maximum bending moment WL/4 at midspan; a cantilever with end load W has WL at the fixed end.

Section modulus Z = I/y_max; bending stress σ = M/Z. For a rectangle Z = bd²/6, for a circle Z = πd³/32.

Shear stress in a rectangular section varies parabolically, zero at extreme fibres and τ_max = 1.5 V/A at the neutral axis; for a circle τ_max = 4V/3A.

Euler buckling load P_cr = π²EI/L_eff²; effective length factors: both ends pinned 1.0, fixed-fixed 0.5, fixed-pinned 0.7, fixed-free 2.0.

Torsion of a solid circular shaft: τ = Tr/J with J = πd⁴/32, and angle of twist θ = TL/GJ in radians.

Principal stresses: σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τ²]; maximum shear = radius of Mohr's circle.

Strain energy under axial load U = P²L/2AE; proof resilience is the strain energy at the elastic limit per unit volume.

Moment of inertia by parallel axis theorem I = I_c + A·d²; for a rectangle about its base I = bd³/3.

Coefficient of restitution e = relative velocity of separation / relative velocity of approach; e = 1 elastic, e = 0 perfectly plastic.

Static indeterminacy of a plane frame = 3m + r − 3j; kinematic indeterminacy counts unknown joint displacements after ignoring axial deformations where appropriate.

A three-hinged arch is statically determinate; bending moment at any section equals the simple-beam moment minus H·y.

Influence line for shear at a section of a simply supported beam is two parallel lines with a jump equal to 1 at the section.

Slope-deflection equation: M_AB = M_FAB + (2EI/L)(2θ_A + θ_B − 3Δ/L).

Müller-Breslau principle: the influence line for a function equals the deflected shape when a unit displacement is introduced at the release corresponding to that function.

Castigliano's second theorem: partial derivative of total strain energy with respect to a load gives the displacement at that load.

For a fixed beam with central point load W, fixed-end moments are WL/8 (hogging) and the free bending moment WL/4 gives midspan moment WL/8.

Degree of freedom of a truss member: axial force only; a plane truss needs m = 2j − 3 members for internal determinacy.

Reciprocal theorem (Maxwell-Betti): deflection at A due to unit load at B equals deflection at B due to unit load at A.

Limit state of collapse in flexure (IS 456): Mu = 0.36 f_ck b x_u (d − 0.42 x_u), with x_u/d limited to 0.53 for Fe 250, 0.48 for Fe 415 and 0.46 for Fe 500.

Minimum tension steel in a beam = 0.85 b d / f_y; maximum tension steel = 0.04 b D; minimum shear reinforcement governs when τ_v < 0.5 τ_c.

Design shear strength of concrete τ_c depends on percentage steel and grade; maximum shear stress τ_c,max is 2.5 MPa for M20 and 3.1 MPa for M25.

Development length L_d = φ σ_s /(4 τ_bd); anchorage value of a standard 90° bend is 8φ and of a 45° bend is 4φ.

Nominal mix volumes: M20 = 1:1.5:3, M15 = 1:2:4, M10 = 1:3:6; design mix is required for M25 and above.

Limit state of serviceability: vertical deflection limited to span/250 overall and span/350 for partitions; crack width limited to 0.3 mm in mild exposure.

Steel design (IS 800:2007) is by limit state method; partial safety factor for material γ_m0 = 1.10 for yielding and γ_m1 = 1.25 for ultimate resistance.

Design bending strength of a laterally supported plastic section M_d = β_b Z_p f_y / γ_m0, with β_b = 1.0 for plastic and compact sections.

Bolted connection: nominal shear capacity of a bearing bolt V_nsb = (f_ub/√3)(n_n A_nb + n_s A_sb); block shear and bearing must also be checked.

Bernoulli's equation for steady incompressible flow along a streamline: p/γ + v²/2g + z = constant.

Reynolds number Re = ρVD/μ; laminar pipe flow below about 2000, turbulent above about 4000, transition in between.

Darcy-Weisbach head loss h_f = fLV²/(2gD); for laminar flow f = 64/Re.

Discharge over a sharp-crested rectangular weir Q = (2/3)C_d√(2g)·b·H^1.5; over a V-notch Q = (8/15)C_d√(2g)·tan(θ/2)·H^2.5.

Orifice discharge Q = C_d·A√(2gH); coefficients satisfy C_d = C_v × C_c, with C_c = area of vena contracta / orifice area.

Hydraulic jump conjugate depths satisfy y2/y1 = 0.5(√(1+8F1²) − 1); the jump is an energy-dissipating transition from supercritical to subcritical flow.

Most efficient trapezoidal channel has half the top width equal to the sloping side length, i.e. hydraulic radius = y/2.

Manning's equation V = (1/n)R^(2/3)S^(1/2); n is about 0.013 for finished concrete and 0.025 for earth channels.

Centre of pressure on a vertical submerged plane is at depth h̄ + I_G/(A·h̄) below the free surface.

Rational method peak discharge Q = C·i·A/3.6 (Q in m³/s, i in mm/h, A in km²); runoff coefficient C is about 0.9 for roofs and 0.2 for sandy soil.

Dicken's formula for flood peak Q = C·A^(3/4); Ryves formula Q = C·A^(2/3); these are regional empirical methods.

Thiessen polygon method weights each rain gauge by its polygon area fraction; isohyetal method draws lines of equal rainfall.

Unit hydrograph: a 1 cm (or 1 mm) direct runoff hydrograph from uniform rainfall over a unit duration; ordinates scale linearly and the time base is unchanged for different rainfall depths.

Infiltration follows Horton's equation f = f_c + (f_0 − f_c)e^(−kt); φ-index is the constant loss rate giving rainfall excess equal to direct runoff.

Penman-Monteith is the standard reference evapotranspiration method; pan coefficient converts pan evaporation to lake evaporation.

S-curve method converts a unit hydrograph of one duration to another duration using the S-curve and a time shift.

Storage-yield analysis: mass curve of cumulative inflow versus demand gives required reservoir capacity from the maximum cumulative deficit.

Return period T and probability of exceedance relate as P = 1/T; risk of at least one exceedance in n years = 1 − (1 − 1/T)^n.

Terzaghi bearing capacity (strip footing): q_u = cN_c + qN_q + 0.5γBN_γ; use 0.4γBN_γ for square and 0.3γBN_γ for circular footings.

Effective stress σ' = σ − u; quick sand condition occurs when the upward seepage gradient equals critical gradient i_c = (G − 1)/(1 + e).

Coefficient of active earth pressure K_a = (1 − sinφ)/(1 + sinφ); passive K_p = 1/K_a for a dry cohesionless backfill.

Consolidation: C_v = T_v·d²/t; time factor T_v = 0.197 for 50% consolidation and 0.848 for 90%.

Compression index C_c = Δe/log(σ'2/σ'1); coefficient of consolidation from Casagrande's logarithm-of-time or Taylor's square-root-of-time construction.

Standard penetration test N value corrected for overburden; N ≈ 10 corresponds to loose sand, N > 30 to dense sand.

Liquid limit and plastic limit give plasticity index PI = LL − PL; PI > 30 indicates high plasticity clay.

Pile capacity = end bearing + skin friction; group efficiency by Converse-Labarre formula; negative skin friction develops when surrounding soil settles.

Slope stability: factor of safety = resisting shear strength / mobilized shear stress; Taylor's stability number S_n = c/(F·γ·H).

Levelling: height of instrument = reduced level of backsight station + backsight reading; reduced level of any station = HI − intermediate or foresight reading.

Rise and fall method checks: Σ backsight − Σ foresight = Σ rise − Σ fall = last RL − first RL.

Curvature correction = 0.0785 d² m (d in km), always negative; refraction correction is about one-seventh of curvature and positive.

Tacheometry: distance D = (f/i)·s + (f + d) for a horizontal line of sight; multiplying constant is usually 100.

Theodolite: included angle = difference of vernier readings; face-left and face-right readings should differ by 180° to detect collimation error.

Total station measures slope distance and zenith angle; horizontal distance = slope distance × sin(zenith), vertical difference = slope distance × cos(zenith).

Contour interval is constant for a map; contours crossing a stream form a V pointing upstream.

GPS requires at least four satellites for a 3D position fix; a total station with GNSS is a standard geomatics field setup.

Scale: representative fraction 1:1000 means 1 cm on the map equals 10 m on the ground.

Stopping sight distance SSD = v·t + v²/(2gf) with v in m/s; reaction time t is 2.5 s and f is about 0.35–0.4.

Superelevation e = v²/(127R) − f with v in km/h; maximum e is 7% for plain terrain and 10% for hilly roads without snow.

Extra widening on horizontal curves = n·L²/(2R) + V/(9.5√R), combining mechanical and psychological widening.

Transition curve length by rate of change of centrifugal acceleration L = 0.0215 V³/(CR), with C about 80/(75+V).

Los Angeles abrasion test limits wear of aggregates; aggregate impact value and crushing value should be below about 30% for wearing courses.

Marshall mix design: optimum bitumen content at maximum stability and bulk density with air voids of 3–5%.

CBR value is the ratio of penetration load to standard load, expressed as a percentage; it is used in flexible pavement thickness design.

Railway: cant e = GV²/(127R) in mm with G in mm; negative superelevation is provided on curves where a road and rail cross.

Traffic: PCU converts mixed traffic to passenger car units; a rotary needs weaving sections and entry capacity depends on weaving width.

BOD5 tests oxygen consumed at 20 °C in 5 days; BOD exerted at time t follows first-order kinetics L_t = L_0(1 − e^(−kt)).

Water demand: per capita demand in Indian cities is typically 135 lpcd; peak factor is 1.5 for hourly and 3 for daily maximum.

Hardness of water is expressed as mg/L CaCO3 equivalent; temporary hardness is removed by boiling, permanent by lime-soda or ion exchange.

Rapid sand filters have a filtration rate of about 100–150 L/m²/min and need backwashing; slow sand filters operate at 100–200 L/m²/day.

Chlorination: breakpoint is the dose beyond which free residual chlorine appears; a residual of 0.2 mg/L is maintained in distribution.

Sewage: BOD of domestic sewage is about 200–300 mg/L; standard rate trickling filters and activated sludge are aerobic biological treatments.

Sedimentation tank design uses overflow rate = Q/A; for an ideal settling basin, particles with settling velocity above the overflow rate are removed.

Air pollution: PM2.5 refers to particles below 2.5 µm; National Ambient Air Quality Standards prescribe 24-hour limits for PM10 and PM2.5.

Noise is measured on a logarithmic decibel scale; Leq is the equivalent continuous sound level over a period.

Ordinary Portland cement: initial setting time must not be less than 30 minutes and final setting time not more than 600 minutes.

Le Chatelier apparatus measures soundness of cement; Vicat apparatus measures setting time and standard consistency.

Concrete workability is measured by slump (true, shear, collapse), compaction factor and Vee-Bee consistometer; slump is not suitable for very dry mixes.

Cube strength of concrete is tested on 150 mm cubes at 28 days; cylinder strength is about 0.8 times cube strength.

Bulking of sand increases with moisture up to about 4–6% and must be corrected for volume batching.

Bricks: first-class bricks have crushing strength above 10 N/mm² and water absorption below 20%; efflorescence indicates soluble salts.

Steel: Fe 415 has yield strength 415 N/mm²; HYSD bars have higher strength and better bond than mild steel.

Timber: seasoning reduces moisture; the fibre saturation point is about 25–30% moisture content.

Bitumen is graded by penetration or viscosity; the penetration test is at 25 °C with a 100 g needle for 5 seconds.

PERT expected time t_e = (t_o + 4t_m + t_p)/6; variance = ((t_p − t_o)/6)²; the critical path has zero total float.

CPM total float = latest start − earliest start = latest finish − earliest finish; free float = total float − head event slack.

Earned value: cost variance CV = EV − AC and schedule variance SV = EV − PV; CPI = EV/AC and SPI = EV/PV.

Rate analysis includes materials, labour, plant, overheads and contractor profit; lead and lift are added for earthwork.

Trapezoidal rule for area of an irregular plot: A = h[(O1 + On)/2 + O2 + O3 + … + O(n−1)].

Prismoidal rule: V = h/3[(A1 + An) + 4(sum of odd areas) + 2(sum of even areas)].

Centre line method suits walls meeting at junctions; long wall–short wall method is used for load-bearing buildings.

Depreciation by straight line: D = (C − S)/n; sinking fund method sets aside an annual amount that accumulates with interest.

Safety: a permit-to-work system is required for confined-space entry and hot work; scaffolding must be tied to the structure.

The Constitution of India was adopted on 26 November 1949 and came into force on 26 January 1950; Dr B. R. Ambedkar chaired the Drafting Committee.

Fundamental Rights are in Part III (Articles 12–35); Directive Principles in Part IV (Articles 36–51); Fundamental Duties in Article 51A added by the 42nd Amendment.

Article 356 provides President's Rule when constitutional machinery fails in a state; Article 360 covers financial emergency.

The Fundamental Right to Property was removed by the 44th Amendment (1978) and made a legal right under Article 300A.

Panchayati Raj was constitutionalised by the 73rd Amendment (1992); urban local bodies by the 74th Amendment.

Indian Standard Time is UTC+05:30, based on the 82.5° E longitude passing near Mirzapur.

Basic physics: work = force × displacement; power = work/time; the SI unit of pressure is the pascal (N/m²).

Environmental law: the Environment (Protection) Act 1986 is umbrella legislation; the Water Act 1974 and Air Act 1981 preceded it.

Karnataka was formed on 1 November 1956 by the States Reorganisation Act, initially named Mysore State and renamed Karnataka in 1973.

Bengaluru is the capital; the state has 31 districts, and Vijayanagara was carved out as the 31st district in 2021.

Karnataka's major rivers are the Krishna, Cauvery, Tungabhadra, Sharavathi and Netravathi; the Krishna and Cauvery disputes involve inter-state tribunals.

The Western Ghats run along Karnataka's western edge; Kudremukh, Bandipur, Nagarahole, Anshi and Bhadra are prominent protected areas.

Jog Falls on the Sharavathi is one of India's highest plunge waterfalls; the Sharavathi hydro project and Linganamakki dam are on the same river.

Karnataka's economy is led by IT and biotechnology in Bengaluru; the state is a major producer of coffee, arecanut, silk and sandalwood.

Karnataka Public Service Commission (KPSC) conducts the AE and other recruitment examinations from its office in Bengaluru.

Heritage: Hampi (Vijayanagara), Pattadakal and Belur–Halebidu are UNESCO-listed or protected sites; Mysore Dasara is a state festival.

Karnataka's coastline is about 320 km along the Arabian Sea, with New Mangalore Port at Panambur as the major port.