ISRO Scientist Civil cheat sheet

Euler's method: y(n+1) = y(n) + h·f(x(n), y(n)); error is O(h), so halving h roughly halves the error.

Modified Euler (Heun): predictor yp = y(n) + h·f(x(n),y(n)), then corrector y(n+1) = y(n) + (h/2)[f(x(n),y(n)) + f(x(n+1),yp)]; error O(h^2).

Newton-Raphson iteration x(n+1) = x(n) − f(x(n))/f'(x(n)) converges quadratically near a simple root; fails where f'(x) = 0.

Secant method: x(n+1) = x(n) − f(x(n))·(x(n) − x(n−1))/(f(x(n)) − f(x(n−1))); order ≈1.618, needs no derivative.

Trapezoidal rule error = −(b−a)h^2·f''(ξ)/12; Simpson's 1/3 rule needs an even number of intervals and has error −(b−a)h^4·f⁗(ξ)/180.

Gauss elimination with partial pivoting selects the largest-magnitude pivot in the column; a zero pivot without pivoting signals a singular or ill-conditioned system.

Jacobi and Gauss-Seidel iteration converge if the coefficient matrix is strictly diagonally dominant (|a(ii)| > Σ|a(ij)|, j≠i); Gauss-Seidel usually converges faster.

Laplace transform: L{e^(at)} = 1/(s−a), L{sin at} = a/(s²+a²), L{t^n} = n!/s^(n+1); L{f'} = sF(s) − f(0).

For a square matrix, eigenvalues satisfy det(A − λI) = 0; Cayley-Hamilton states every matrix satisfies its own characteristic equation.

Simple bending: M/I = σ/y = E/R; maximum bending stress occurs at the extreme fibre, σ = My/I.

Shear stress in a beam section: τ = VQ/(Ib), where Q is the first moment of area of the portion beyond the fibre about the neutral axis.

Torsion of a circular shaft: τ/r = T/J = Gθ/L; for a solid shaft J = πd⁴/32, polar section modulus Zp = πd³/16.

Euler buckling load Pcr = π²EI/(Leff)²; effective length Leff = L (pinned-pinned), 0.5L (fixed-fixed), 0.7L (fixed-pinned), 2L (cantilever).

Fixed-end moments for a UDL w over span L: both ends wL²/12; for a central point load P: both ends PL/8.

Slope-deflection equation: M(AB) = M(FAB) + (2EI/L)(2θA + θB − 3Δ/L), with clockwise moments positive by convention.

Limit state of collapse (IS 456): Mu,lim = 0.36·fck·b·xu,lim·(d − 0.42·xu,lim) with xu,lim/d = 0.53 for Fe 415 and 0.48 for Fe 500.

Minimum tension steel in RCC beams (IS 456): Ast/bd = 0.85/fy; minimum shear reinforcement governs when τv < 0.5τc.

Design shear strength of concrete τc depends on percentage steel pt = 100Ast/bd; maximum shear stress τc,max depends on concrete grade only.

Steel design (IS 800:2007) is limit-state based; design compressive strength uses buckling class a, b, c or d, with imperfection factor α from 0.21 to 0.76.

Three-phase relations: e = wG/s (saturated), γ = (G + e·S)·γw/(1+e), γd = γ/(1+w), and γsat = (G+e)γw/(1+e).

Relative density Dr = (e(max) − e)/(e(max) − e(min)); a well-graded, angular sand has a higher φ and higher Dr.

Terzaghi bearing capacity (strip footing): qu = c·Nc + q·Nq + 0.5·γ·B·Nγ; for square and circular footings the last term uses 0.4γB·Nγ.

Effective stress principle: σ' = σ − u; the effective stress controls shear strength, consolidation and bearing capacity.

Consolidation: coefficient of consolidation Cv = T·H²/t; for 50% consolidation Tv = 0.197, for 90% Tv = 0.848.

Compression index Cc from e-log σ' curve; settlement of normally consolidated clay Sc = Cc·H·log((σ'0+Δσ)/σ'0)/(1+e0).

Shear strength by Mohr-Coulomb: τf = c' + σ'·tan φ'; for a UU test on saturated clay φu ≈ 0 and τ = cu.

Liquefaction susceptibility rises with loose, saturated, uniformly graded fine sand and high water table; standard penetration N < 15 is a warning sign.

Pile capacity: Qu = Qp + Qs; for a driven pile in sand Qp ≈ q'·Nq·Ap with Nq about 8 to 12 for φ ≈ 30–35°.

Continuity, momentum and Bernoulli equations: p/γ + z + v²/2g = constant along a streamline for steady, frictionless, incompressible flow.

Reynolds number Re = ρVD/μ; pipe flow is laminar below 2000 and turbulent above 4000 for circular pipes.

Darcy-Weisbach head loss hf = fLV²/(2gD); Hagen-Poiseuille for laminar pipe flow gives f = 64/Re.

Manning's equation: V = (1/n)·R^(2/3)·S^(1/2); for a rectangular channel hydraulic radius R = A/P.

Critical depth in a rectangular channel: yc = (q²/g)^(1/3), where q is discharge per unit width; specific energy is minimum at critical depth.

Hydraulic jump: the sequent-depth relation y2/y1 = 0.5[(1 + 8F1²)^0.5 − 1]; energy loss increases with the upstream Froude number.

Rational method peak flow Qp = C·i·A/360 (SI, i in mm/h, A in ha); time of concentration controls the design intensity.

Unit hydrograph: a 1 cm (or 1 mm) direct runoff hydrograph from uniform rainfall over a unit duration; S-curve converts between durations.

Penman-Monteith is the reference evapotranspiration method; crop water requirement ETc = Kc·ETo.

Duty and delta relation: delta (m) = 8.64B/D, where B is the base period in days and D the duty in ha/cumec.

Design period for water supply components is usually 30 years for treatment units and 20–30 years for distribution mains.

Per capita demand for Indian cities is typically 135 lpcd; fire demand uses the Kuichling formula Q = 3182√P (L/min, P in thousands).

Surface overflow rate (SOR) governs settling-tank design; for an ideal settling basin, SOR = Q/A equals the settling velocity of the critical particle.

Rapid sand filters: filtration rate about 100–150 L/m²/min; backwashing at 600–900 L/m²/min expands the bed by about 50%.

Breakpoint chlorination adds chlorine until the chlorine demand of ammonia and organics is satisfied; free residual appears beyond the breakpoint.

BOD5 = (D1 − D2)·(1 − f)·P, where f is the dilution factor correction; standard test is 20 °C, 5 days, with nitrification suppressed.

Activated sludge: F/M ratio, sludge age (SRT) and MLSS control the process; F/M ≈ 0.3–0.6 d⁻¹ gives conventional aeration.

Trickling filter efficiency by the NRC equation: E = 100/(1 + 0.44·(W/(V·F))^0.5), with W the BOD load and V the filter volume.

Air quality: PM2.5 and PM10 are size fractions; the National Ambient Air Quality Standards set 24-hour PM10 at 100 µg/m³ and PM2.5 at 60 µg/m³.

Sanitary landfill siting criteria: at least 200 m from habitation, above the water table, with a clay or HDPE liner and leachate collection.

Stopping sight distance SSD = v·t + v²/(2g(f ± n)), with v in m/s, t reaction time ≈ 2.5 s, and gradient n in decimals.

Transition curve length by rate of change of centrifugal acceleration: Ls = 0.0215·V³/(C·R) (V in km/h, C ≈ 80/(75+V)).

Superelevation: e + f = V²/(127R); maximum e is 7% on plain terrain and 10% on hilly roads without snow.

Flexible pavement design (IRC 37) uses the CBR of subgrade; for a soaked CBR of 5%, the design traffic and layer thicknesses come from the catalogue charts.

Railway sleeper density = number of sleepers per rail length (e.g., M+4 for a 13 m rail means 17 sleepers); higher density on sharper curves.

Cant for a railway curve: equilibrium cant e = GV²/(127R) with G in mm; maximum cant on broad gauge is 165 mm, cant deficiency limited to 75 mm.

Airport runway length corrections: elevation correction 7% per 300 m above MSL, temperature correction 1% per 1 °C above standard.

Traffic flow relation q = k·v; Greenshields' model gives v = vf(1 − k/kj) and capacity at k = kj/2.

Webster's optimum cycle length Co = (1.5L + 5)/(1 − ΣY), where L is total lost time and Y the sum of critical flow ratios.

Levelling: the difference in elevation is backsight minus foresight; for a closed loop the sum of BS minus sum of FS must equal zero.

Tacheometry: distance D = k·s + c for a horizontal line of sight, with k = 100 and c = 0 for an anallatic lens.

Theodolite traverse adjustment by the Bowditch rule distributes error proportional to the length of each leg; the transit rule distributes it proportional to latitude or departure.

Curve setting: for a simple circular curve, tangent length T = R·tan(Δ/2) and curve length L = R·Δ (Δ in radians).

Remote sensing: spatial resolution is the ground area of one pixel; Landsat multispectral bands are 30 m, panchromatic 15 m.

NDVI = (NIR − Red)/(NIR + Red); values range from −1 to +1 and high values indicate healthy vegetation.

GPS positioning needs at least four satellites to solve three coordinates plus receiver clock bias.

GIS raster overlay uses map algebra; vector overlay includes union, intersection and identity operations.

Total station combines electronic distance measurement with electronic theodolite angles; accuracy is typically ±(2 mm + 2 ppm).

Concrete mix design (IS 10262) targets a characteristic strength fck with a target mean strength fck + 1.65σ; M25 means a 28-day cube strength of 25 N/mm².

Water-cement ratio controls strength and durability; IS 456 caps w/c at 0.50 for moderate exposure and 0.45 for severe exposure.

Slump test measures workability of concrete; a slump of 25–75 mm suits normal RCC work, and the test is unsuitable for very dry mixes.

Bricks: a first-class brick has compressive strength above 10 N/mm², water absorption below 20% and a sharp, ringing sound.

PERT uses three time estimates: te = (to + 4tm + tp)/6, with variance ((tp − to)/6)².

CPM: total float = latest finish − earliest finish; a critical path has zero total float and determines project duration.

Earned value: CPI = EV/AC and SPI = EV/PV; CPI below 1 means the project is over budget.

Estimating: for a wall, volume of brickwork deducts openings above 0.1 m²; plaster area is measured on the finished surface.

Rate analysis includes materials, labour, plant, overheads and contractor profit; lead and lift affect the conveyance cost of materials.

Percentage change: successive changes of a% and b% give a net change of a + b + ab/100 percent.

Compound interest A = P(1 + r/n)^(nt); for simple interest A = P(1 + rt).

Time and work: if A does a job in a days and B in b days, together they take ab/(a+b) days.

Speed, time and distance: average speed for equal distances at v1 and v2 is 2v1v2/(v1+v2), not the arithmetic mean.

Permutations nPr = n!/(n−r)!; combinations nCr = n!/(r!(n−r)!); nC0 = nCn = 1.

Ratio and proportion: if a:b = c:d, then ad = bc; for a mixture replaced repeatedly, final quantity = initial·(1 − k/V)^n.

Number series and coding-decoding: look for constant differences, ratios, squares/cubes, or alternating patterns before assuming a rule.

Data interpretation: compute percentage share as (part/total)×100 and always check whether the question asks for a ratio or a percentage point difference.