The Ultimate GATE CE Cheatsheet

# Matrix Operations
  Rank: max linearly independent rows/cols
  det(A) = 0 → singular (no inverse)
  A⁻¹ = adj(A)/det(A)

  Eigenvalues: det(A − λI) = 0
  |A| = product of eigenvalues, tr(A) = sum of eigenvalues

  Ax = b:
    rank(A) = rank([A|b]) = n → unique solution
    rank(A) = rank([A|b]) < n → infinite solutions
    rank(A) < rank([A|b]) → no solution

  Cramer's Rule: xᵢ = det(Aᵢ)/det(A)

# Key Derivatives & Integrals
  d/dx(xⁿ) = nx^(n−1),  d/dx(eˣ) = eˣ,  d/dx(ln x) = 1/x
  Product: (uv)' = u'v + uv'
  Chain: (f∘g)' = f'(g)·g'(x)

  ∫ xⁿ dx = x^(n+1)/(n+1),  ∫ eˣ dx = eˣ
  ∫₀^∞ e^(−ax) dx = 1/a

# Partial Derivatives & Vector Calculus
  Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
  Divergence: ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
  Curl: ∇×F = determinant matrix
  Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²

# First Order ODE: y' + Py = Q
  IF = e^(∫P dx)
  y × IF = ∫(Q × IF)dx + C

# Second Order ODE: ay'' + by' + cy = 0
  ar² + br + c = 0
  D > 0: y = C₁e^(r₁x) + C₂e^(r₂x)
  D = 0: y = (C₁ + C₂x)e^(rx)
  D < 0: y = e^(αx)(C₁cosβx + C₂sinβx)

# Probability
  P(A∪B) = P(A) + P(B) − P(A∩B)
  P(A|B) = P(A∩B)/P(B)
  Bayes: P(A|B) = P(B|A)P(A)/P(B)

# Random Variables
  E[X] = μ, Var(X) = σ² = E[X²]−μ²
  Binomial(n,p): E = np, Var = np(1−p)
  Poisson(λ): E = λ, Var = λ
  Normal(μ,σ²): Z = (X−μ)/σ ~ N(0,1)
  Exponential(λ): E = 1/λ, Var = 1/λ²

# Correlation & Regression
  r = Σ(xᵢ−x̄)(yᵢ−ȳ) / √[Σ(xᵢ−x̄)²·Σ(yᵢ−ȳ)²]
  y = a + bx,  b = r(Sᵧ/Sₓ)

# Numerical Integration
  Trapezoidal: I ≈ (h/2)[y₀ + 2Σyᵢ + yₙ]
  Simpson's 1/3: I ≈ (h/3)[y₀ + 4y₁ + 2y₂ + 4y₃ + ... + yₙ]

# Root Finding
  Newton-Raphson: x_{n+1} = x_n − f(x_n)/f'(x_n)  (Order 2)
  Bisection: x = (a+b)/2, Error ≤ (b−a)/2ⁿ

  Regula-Falsi: x = (af(b)−bf(a))/(f(b)−f(a))

# Numerical ODE
  Euler's: y_{n+1} = y_n + h·f(x_n, y_n)     (Order 1)
  Runge-Kutta 4th: y_{n+1} = y_n + (k₁+2k₂+2k₃+k₄)/6  (Order 4)

# Curve Fitting
  Least squares: minimize Σ(yᵢ − (a+bxᵢ))²
  b = Σ(xᵢ−x̄)(yᵢ−ȳ)/Σ(xᵢ−x̄)²,  a = ȳ − bx̄

# Degree of Static Indeterminacy
  Beams: DSI = (m + r) − 2j
  Trusses: DSI = (m + r) − 2j  (for simple truss)
    m = members, r = reactions, j = joints
  Frames: DSI = 3m + r − 3j (with internal hinges: subtract hinges)

  Kinematic Indeterminacy (DKI):
  D_k = Total DOFs − Available equilibrium equations

# Influence Lines
  IL diagram = variation of response function (reaction, SF, BM)
  as a unit load moves across the structure

  Max BM under a wheel load:
    Place load such that the load and resultant are equidistant from center

  Max SF at a section: load just to the right of section

# Maxwell-Betti Reciprocal Theorem
  Work done by load system 1 through deflection due to load system 2
  = Work done by load system 2 through deflection due to load system 1
  f₁₂ = f₂₁

# Methods of Analysis
  Slope Deflection Method:
    M_AB = (2EI/L)(2θ_A + θ_B − 3δ/L) + MF_AB

  Moment Distribution Method (Hardy Cross):
    Distribution factor: DF = (EI/L) / Σ(EI/L) at joint
    Carry-over factor = 1/2 (prismatic members)
    Iterate until convergence

  Virtual Work Method:
    External: W_ext = ΣP_i × δ_i
    Internal: W_int = ∫M·m/(EI) dx

  Castigliano's Theorem:
    δᵢ = ∂U/∂Pᵢ  (deflection at i due to load Pᵢ)

# Simply Supported Beams (Length L)
  Central point load P:
    M_max = PL/4 (center),  V_max = P/2

  UDL w:
    M_max = wL²/8 (center),  V_max = wL/2

  Moment at supports: M = 0
  Deflection (center, point load): δ = PL³/(48EI)
  Deflection (center, UDL): δ = 5wL⁴/(384EI)

# Cantilever (L, tip load P)
  M_max = PL (fixed end)
  Deflection (tip): δ = PL³/(3EI)
  UDL w: M_max = wL²/2, δ = wL⁴/(8EI)

# Fixed Beam (Length L)
  Central point load P:
    M_max = PL/8 (at supports, hogging)
    M_center = PL/8 (sagging)
    δ_center = PL³/(192EI)

  UDL w:
    M_supports = wL²/12, M_center = wL²/24

# Propped Cantilever (L, UDL w)
  M_fixed = wL²/2, M_prop = wL²/8
  Reaction at prop = 3wL/8

# Truss Analysis (Method of Joints)
  At each joint: ΣFₓ = 0, ΣFᵧ = 0
  Zero-force members:
    1. If two non-collinear members meet at unloaded joint → both zero
    2. If three members meet, two collinear, no load → third is zero

  Method of Sections: cut truss, apply equilibrium to one part

# Stress Block Parameters (Limit State)
  Limiting depth of neutral axis:
    x_u,max / d = 0.48 (Fe415), 0.46 (Fe500), 0.42 (Fe250)

  Moment of Resistance (singly reinforced):
    M_u = Q·b·d²   where Q depends on steel grade
    Fe250: Q = 0.149,  Fe415: Q = 0.138,  Fe500: Q = 0.133

  Area of steel:
    A_st = 0.5·(f_ck/f_y)·[1 − √(1 − 4.6·M_u/(f_ck·b·d²))]·b·d

  Minimum steel (tension):
    A_st,min = 0.26·(f_ck/f_y)·b·d (but ≥ 0.13% bd)

  Maximum steel (tension):
    A_st,max = 0.04·b·D (4% of gross cross-section)

# Design of Beams
  Effective span: leff = clear span + d (simply supported)
  Min width: b ≥ L/25 (simply supported), L/20 (continuous)
  Min depth: d ≥ L/20 (simply supported), L/26 (continuous)

  Nominal cover: 25mm (mild), 40mm (moderate), 50mm (severe)

# Shear
  τ_v = V_u/(b·d)  (nominal shear stress)
  τ_c = from table (depends on % steel and grade)
  If τ_v > τ_c → design shear reinforcement
  V_us = V_u − τ_c·b·d
  Asv/sv = V_us/(0.87·f_y·d)  (stirrup design)

# Column Design
  Short column: le/D ≤ 12
  Slender: le/D > 12

  Pu = 0.4·f_ck·Ac + 0.67·f_y·Asc  (short axially loaded)
  Min eccentricity: e_min = L/500 + D/30 ≥ 20mm

  Effective length: le = K·L (K depends on end conditions)
  Min steel: 0.8% of Ag, Max: 6% (4% preferred)
  Min bars: 4 for rectangular, 6 for circular

# Slab Design
  One-way: ly/lx > 2
  Two-way: ly/lx ≤ 2
  Effective depth: d = span/(basic ratio × modification factor)

  BM coefficients (simply supported one-way):
    M = w·l²/8

  Two-way (simply supported): Mx = αx·w·lx², My = αy·w·lx²

# Tension Members
  T_d = min(T_dg, T_db)
  T_dg = A_g·f_y/γ_m0  (yielding of gross section)
  T_db = 0.9·A_n·f_u/γ_m1  (rupture at net section)
  γ_m0 = 1.10,  γ_m1 = 1.25

  Net area: A_n = A_g − n·d₀·t + Σ(p²/4g)·t
  n = number of holes in diagonal, d₀ = hole diameter

# Compression Members
  Euler buckling: P_cr = π²EI/(Le)²
  Effective length: depends on end conditions
  Design compressive stress: f_cd from IS 800 Table 9
    Based on λ = KL/r (slenderness ratio)
    f_cd = f_y / [0.6 + 0.21(λ/λ̄)²] approx

  Max slenderness ratio:
    Main member: 180, Secondary: 200, Tension: 350

# Beams (Laterally Supported)
  M_d = Z_p·f_y/γ_m0  (plastic section capacity)
  Z_p = plastic section modulus
  Laterally unsupported: use LTB equations

  Shear: V_d = 0.6·f_y·A_v/√3 / γ_m0
  A_v = depth × web thickness (for rolled sections)

# Bolted Connections
  Bearing type: V_dsb = 2.5·k_b·d·t·f_u/γ_mb
    k_b = smaller of (3.0, e/(3d₀), p/(3d₀) − 0.25, f_ub/f_u + 0.5)
  Friction grip (HSFG): V_dsf = μ_f·n_e·K_h·F₀
    F₀ = 0.9·f_₀·A_b  (proof load)

# Welded Connections
  Fillet weld effective throat: t_e = 0.7·s (s = leg length)
  Design strength: f_wd = f_u/(√3·γ_mw)
  γ_mw = 1.25
  Min weld size: depends on thinner part (3mm to 8mm)

# Phase Relations
  Void ratio: e = V_v/V_s
  Porosity: n = V_v/V = e/(1+e)
  Water content: w = W_w/W_s × 100%
  Degree of saturation: S = V_w/V_v × 100%
  Specific gravity: G = ρ_s/ρ_w

  Dry density: ρ_d = G·ρ_w/(1+e) = ρ/(1+w)
  Bulk density: ρ = G(1+w)ρ_w/(1+e)
  Saturated density: ρ_sat = (G+e)ρ_w/(1+e)
  Submerged density: ρ' = (G−1)ρ_w/(1+e) = ρ_sat − ρ_w

  Useful: e = wG (when S = 100%)

# Atterberg Limits
  Liquid Limit (LL): water content at transition to liquid
  Plastic Limit (PL): water content at transition to plastic
  Shrinkage Limit (SL): water content below which no volume change

  Plasticity Index: PI = LL − PL
  Liquidity Index: LI = (w−PL)/PI

# Soil Classification (IS)
  Coarse: Sand (0.075–4.75mm), Gravel (4.75–80mm)
  Fine: Silt (0.002–0.075mm), Clay (<0.002mm)

  Casagrande's chart: PI vs LL
  Above A-line → Clay, Below → Silt
  A-line: PI = 0.73(LL − 20)

  Unified (USCS): GW, GP, GM, GC, SW, SP, SM, SC, CL, ML, OL, CH, MH, OH, Pt
  IS Classification: Similar but uses group names with specific % criteria

# Standard Proctor (IS:2720)
  Mold: 1000cc, Hammer: 2.6kg, Drop: 310mm, Layers: 3, Blows: 25/layer

  Modified Proctor (Heavy):
  Hammer: 4.89kg, Drop: 450mm, Layers: 5, Blows: 25/layer
  Higher compactive effort → higher ρ_d(max), lower OMC

  Zero Air Void Line: ρ_d = G/(1+wG) × ρ_w  (theoretical max)
  Actual compaction curve always lies below ZAV line

# Permeability (Darcy's Law)
  v = k·i   (seepage velocity)
  q = k·i·A  (discharge)
  i = hydraulic gradient = h/L

  k (coefficient of permeability)
  Typical values:
    Gravel: 10⁻¹–10 cm/s
    Sand: 10⁻³–10⁻¹ cm/s
    Silt: 10⁻⁵–10⁻³ cm/s
    Clay: <10⁻⁶ cm/s

  Allen-Hazens: k = C·D₁₀² (for clean sand)
  Kozeny-Carman: k = (γ/μ)·(e³/(1+e))·(1/S₀²)

  Laboratory: Constant head (k > 10⁻⁴), Falling head (k < 10⁻⁴)
  Falling head: k = (aL/At)·ln(h₁/h₂)

# Consolidation (Terzaghi, 1D)
  Coefficient of compressibility: a_v = −Δe/Δσ
  Compression index: C_c = −Δe/Δ(log σ)
    (slope of e-log σ curve, virgin compression)

  C_c estimation:
    Skempton: C_c = 0.009(LL − 10)  (normally consolidated clay)
    C_s (recompression) ≈ C_c/5 to C_c/10

  Coefficient of volume compressibility: m_v = a_v/(1+e₀)
  Coefficient of consolidation: c_v = k/(m_v·γ_w)

  Settlement: S = C_c·H·log(σ_f/σ_i)/(1+e₀)

  Time rate: T_v = c_v·t/d²  (d = drainage path)
  U = f(T_v) from Taylor's or Casagrande's curve
  t₉₀ ≈ T₉₀·d²/c_v, T₅₀ = 0.197

# Shear Strength
  τ_f = c + σ·tan(φ)  (Mohr-Coulomb)

  Direct Shear Test: c, φ from τ vs σ plot
  Triaxial Test:
    UU (Unconsolidated Undrained): φ_u ≈ 0, c_u varies
    CU (Consolidated Undrained): c_cu, φ_cu, c', φ' (with pore pressure)
    CD (Consolidated Drained): c_d, φ_d = c', φ'

  Effective stress: σ' = σ − u
  τ_f = c' + σ'·tan(φ')  (effective stress criterion)

  Vane Shear Test: c_u = T/(πD²H/2 + πD³/6)
  Standard Penetration Test (SPT): N-value correlates to soil properties

# Lateral Earth Pressure
  Rankine (smooth wall, vertical backfill):
    Active: K_a = (1−sinφ)/(1+sinφ) = tan²(45−φ/2)
    Passive: K_p = (1+sinφ)/(1−sinφ) = tan²(45+φ/2)
    At rest: K₀ = 1−sinφ (normally consolidated)

  Pressure distribution:
    Active: triangular, P_a = ½K_a·γ·H²
    Passive: triangular, P_p = ½K_p·γ·H²
    With surcharge: P_a = ½K_a·γ·H² + K_a·q·H

  Coulomb (rough wall, inclined backfill):
    P_a = ½K_a·γ·H²
    K_a = sin²(α+φ)/[sin²α·sin(α−δ)·(1+√(sin(φ+δ)sin(φ−β)/sin(α−δ)sin(α+β)))²]

# Bearing Capacity (Terzaghi)
  q_ult = c·N_c + q·N_q + 0.5γ·B·N_γ
  c = cohesion, q = γD_f (surcharge), B = footing width
  N_c, N_q, N_γ = bearing capacity factors (function of φ)

  Strip footing: N_c = 5.14, N_q = 1, N_γ = 0 (φ=0)
  Square: multiply by 1.3
  Circular: multiply by 1.3

  Safe bearing: q_safe = q_ult/FOS + γ·D_f
  Net bearing: q_net = q_ult − γ·D_f

  Meyerhof correction for inclined/eccentric loads

# Pile Foundations
  Load capacity: Q_ult = Q_point + Q_friction
  Q_point = 9·c·A_p (clay),  A_p = πd²/4
  Q_friction = α·c̄·A_s (clay, α ≈ 0.5–1.0)
  A_s = πdL (surface area)

  Dynamic formula (Engineering News): R = Wh/(s+c)
  H pile capacity from SPT: Q = 400N × A_p (approx)

# Precipitation
  Mean rainfall (Thiessen): P_avg = ΣAᵢPᵢ/ΣAᵢ
  Isohyetal: weighted by area between isohyets

  Arithmetic mean: P_avg = ΣPᵢ/n (for uniform catchment)

# Evapotranspiration
  Penman (combines energy balance + mass transfer)
  Thornthwaite: based on temperature

  Evaporation: Dalton's law (wind + vapor pressure deficit)
  Pan coefficient: K_p = lake evaporation / pan evaporation (0.6–0.8)

# Infiltration
  Horton: f = f_c + (f₀ − f_c)e^(−kt)
  Philip: f = S·t^(−1/2) + K  (S = sorptivity, K = steady rate)

# Runoff
  Rational formula: Q = C·i·A/360
  C = runoff coefficient, i = intensity (mm/hr), A = area (ha)
  Q in m³/s

  Unit Hydrograph:
    D-hr UH: 1cm direct runoff from D-hr uniform rainfall
    Convolution: Q(t) = Σᵢ Iᵢ × UH(t−i+1)
    S-curve: continuous rainfall → cumulative runoff
    T-hr UH from D-hr: lag by T, subtract, divide by T/D

  Flood frequency: Gumbel, Log-Pearson Type III
  Gumbel: x_T = x̄ + K·σ  (K from Gumbel table vs T)

# Streamflow & Hydrographs
  Hydrograph: Q vs time plot
  Base flow: slow subsurface contribution
  Direct runoff: rapid surface contribution
  Unit hydrograph assumption: linear system, time invariance

# Manning's Equation
  V = (1/n)·R^(2/3)·S^(1/2)
  Q = A·V = (1/n)·A·R^(2/3)·S^(1/2)
  n = Manning's roughness coefficient
  R = A/P = hydraulic radius
  S = bed slope
  P = wetted perimeter

  n values:
    Concrete: 0.013–0.015
    Earth: 0.020–0.030
    Masonry: 0.025–0.035

# Specific Energy
  E = y + V²/(2g) = y + Q²/(2gA²)
  Critical depth (rectangular): y_c = (q²/g)^(1/3)
  Critical flow: Fr = 1,  Minimum specific energy

  Fr (Froude): Fr = V/√(gD)  (D = hydraulic depth = A/T)
  Fr < 1 → subcritical (tranquil)
  Fr > 1 → supercritical (rapid/torrential)

# Hydraulic Jump
  Conjugate depths (rectangular):
    y₂/y₁ = ½(√(1+8Fr₁²) − 1)
  Energy loss: ΔE = (y₂−y₁)³/(4y₁y₂)
  Length of jump: L_j ≈ 5(y₂−y₁)

# Weirs & Notches
  Rectangular weir: Q = (2/3)·C_d·L·√(2g)·H^(3/2)
  Triangular (V-notch): Q = (8/15)·C_d·tan(θ/2)·√(2g)·H^(5/2)
  C_d ≈ 0.62 (typical)

# Most Economical Sections (Maximum discharge for given area)
  Rectangular: B = 2y (width = 2 × depth)
  Trapezoidal: half of a regular hexagon
  Triangular: 90° vertex angle

# Crop Water Requirement
  Delta: Δ = B·D (duty × base period)
  Duty: area irrigated per unit discharge
  Delta: total depth of water required by crop in base period

# Canal Design
  Kennedy's Theory:
  V₀ = 0.55·m·D^0.64
  m = critical velocity ratio (1.0–1.2)

  Lacey's Theory:
  V = (Q·f²/140)^(1/6)
  f = silt factor = 1.76√(d_mm)
  R = 5V²/(2f)
  P = 4.75√Q  (wetted perimeter for regime channel)

  Discharge: Q = V × A
  Lined canals: design based on Manning's equation

# Water Logging & Drainage
  Causes: excessive irrigation, poor drainage, seepage
  Control: drainage pipes, tile drains, open drains
  Drainage coefficient = design discharge per unit area

# Water Quality Parameters
  BOD₅ = L₀(1−e^(−kt))
  k = 0.23/day (typical at 20°C)
  Ultimate BOD: L₀ = BOD₅/(1−e^(−kt))
  Temperature correction: k_T = k₂₀ × 1.047^(T−20)

  COD > BOD > ThOD
  COD/BOD ratio: 1.5–2.5 for domestic sewage

  DO = Dissolved Oxygen (saturation ≈ 9.2 mg/L at 20°C)
  Hardness: Ca²⁺, Mg²⁺ (temporary: carbonate, permanent: non-carbonate)

# Water Treatment Plant
  Screens → Aeration → Coagulation → Flocculation → Sedimentation
  → Filtration → Disinfection → Distribution

  Coagulation: Alum dose = Al₂(SO₄)₃ (removes turbidity)
  Filtration: Sand filter (slow 0.1–0.4 m/hr, rapid 4–8 m/hr)
  Disinfection: Chlorination (residual Cl₂ > 0.2 mg/L)

# Wastewater Treatment
  Primary: Screens → Grit chamber → Primary sedimentation
    Removes: 30–40% SS, 25–35% BOD

  Secondary (Biological):
    Activated Sludge: F/M ratio = 0.2–0.5/day
    SRT = V·X/(Q·X_e)  (Sludge Retention Time)
    HRT = V/Q  (Hydraulic Retention Time)

    Trickling Filter: organic loading rate = BOD/(volume)
    Sludge age = underflow concentration / yield

  Tertiary: Nitrification, Phosphorus removal, Filtration

# Sludge
  Sludge Volume Index (SVI) = (settled volume × 1000) / (MLSS × sample vol)
  SVI < 100 → good settling, SVI > 200 → bulking

  Digester gas: CH₄ + CO₂ (methane ≈ 60–65%)

# Air Pollution Standards
  NAAQS (India): PM₂.₅ < 40 μg/m³ (annual), PM₁₀ < 60
  CO < 2 mg/m³ (1hr), SO₂ < 50 μg/m³ (annual)
  NO₂ < 40 μg/m³ (annual), O₃ < 100 μg/m³ (8hr)

  Air Quality Index (AQI): 0–500
    0–50: Good, 51–100: Satisfactory, 101–200: Moderate
    201–300: Poor, 301–400: Very Poor, 401–500: Severe

# Stack Dispersion (Gaussian Plume)
  C(x,y,z) = Q/(2πuσᵧσ_z) × exp(−y²/2σᵧ²)
    × [exp(−(z−H)²/2σ_z²) + exp(−(z+H)²/2σ_z²)]
  H = stack height + plume rise
  σᵧ, σ_z = dispersion coefficients (function of x)

# Solid Waste Management
  Waste generation: 0.3–0.6 kg/capita/day (India)
  Composition: Organic ~50%, Paper ~15%, Plastic ~5%

  Landfill: area = (W × ρᵢ)/(ρ_c × D)
  Leachate: contaminated water from landfill

  Incineration: reduces volume 90%, weight 75%
  Composting: organic waste → manure

  3R: Reduce, Reuse, Recycle
  4R: Reduce, Reuse, Recycle, Recover

# Design Speed (IRC)
  NH/SH: 80–100 km/h
  MDR: 60–80 km/h
  ODR: 40–50 km/h
  VR: 30 km/h

# Sight Distance
  SSD (Stopping Sight Distance):
    SSD = 0.278Vt + V²/(254f)
    V = speed (kmph), t = reaction time (2.5s), f = coefficient of friction

  OSD (Overtaking Sight Distance):
    OSD = 0.278Vb × T  where T = √(4S + 2d₁)/Vb
    d₁ = 0.278Vb × t₁  (t₁ = reaction time for overtaking)

  ISD (Intermediate Sight Distance) = 2 × SSD

# Horizontal Curves
  R = V²/(127(e+f))
  e = superelevation, f = lateral friction (0.15 max)

  Max superelevation: e_max = 0.07 (plain), 0.10 (hilly)
  Extra widening: W_e = nl²/(2R) + V/(9.5√R)
  Transition curve: L = V³/(46.5 × R × C)  (C = rate of change of centrifugal acc)
  Min length: L = V³/(36 × R × C) for 2-lane

# Vertical Curves
  Summit: convex, L = NS²/(√(2H₁ + √2H₂))²  (H = headlight height)
  Valley: concave, comfort: L = NV²/(46.5)  (N = deviation rate)
  L = NV²/(C) where C = comfort factor

  Gradient: Ruling max: 3.3% (NH), 5% (MDR)
  Length at summit: L > 60m (mechanical), L > 15m (non-mechanical)

# Flexible Pavement (Bituminous)
  CBR Method (IRC 37):
    Thickness based on CBR value and cumulative traffic (msa)
    t = √(P/(0.3×π×CBR×p))

  Traffic: CSA = A × (365×L×DF×N) / (10⁶)
  A = vehicles/day, L = design life, DF = vehicle damage factor, N = lanes

  Components: Subgrade → Sub-base → Base → Wearing course

# Rigid Pavement (Concrete)
  Design: based on wheel load, modulus of subgrade reaction (k)
  Westergaard equations:
    Stress: σ = 3P/(2πh²) × [log(l/b) + 0.6159]
    l = (Eh³/(12k))^(1/4) (radius of relative stiffness)

  Dowel bars: transfer load across joints (steel, 25–40mm dia)
  Tie bars: hold slabs together at longitudinal joints

  Joints: Expansion, Contraction (shrinkage), Warping, Construction

# Highway Materials
  Aggregate tests:
    Crushing value: <30% (wearing), <45% (base)
    Impact value: <30% (wearing), <35% (base)
    Los Angeles Abrasion: <30%
    Flakiness Index: <25–30%

  Bitumen tests:
    Penetration: harder grade = lower penetration
    Softening point (Ring & Ball): min 60°C
    Ductility: min 40 cm
    Viscosity: flow property
    Flash & Fire point: safety

# Traffic Volume Studies
  ADT = Average Daily Traffic = Σtraffic / 365
  AADT = Annual Average Daily Traffic
  PCU (Passenger Car Unit):
    Car = 1, Bus = 3, Truck = 3, Motorcycle = 0.5, Cycle = 0.2

# Speed Studies
  Time mean speed: V̄_t = ΣVᵢ/n (at a point)
  Space mean speed: V̄_s = n/Σ(1/Vᵢ) (harmonic mean)
  V̄_t ≥ V̄_s always

# Density & Flow
  q = k × v  (flow = density × speed)
  Jam density: k_j (maximum)
  Free flow speed: v_f (maximum at zero density)
  Capacity: q_max = k_j × v_f / 4 (at k = k_j/2, v = v_f/2)

# Intersection Design
  Signal timing:
    Cycle length: C = (1.5L + 5)/(1 − Y)  (Webster)
    Y = Σyᵢ = Σ(q/s)_ᵢ  (flow ratio sum)
    L = lost time per cycle
    Green time: gᵢ = (yᵢ/Y) × (C − L)

  Weaving length: based on weaving volume ratio
  Island types: Divisional, Channelizing, Refuge

# Errors & Accuracy
  Precision: repeatability of measurements
  Accuracy: closeness to true value

  Systematic Error: consistent, follows pattern → can be corrected
  Random Error: statistical, follows normal distribution
  Gross Error: blunders

  Error propagation: if Z = aA + bB, then σ_Z = √(a²σ_A² + b²σ_B²)

# Traversing
  Sum of interior angles = (2n − 4) × 90°  (closed traverse)
  Sum of exterior angles = (2n + 4) × 90°

  Latitude (L) = l × cos(θ)  [North-South component]
  Departure (D) = l × sin(θ)  [East-West component]

  Bowditch correction:
    Latitude: L_c = L × (Σ|L|/Σ|total|)
    Departure: D_c = D × (Σ|D|/Σ|total|)

  Area: by coordinate method:
    A = ½|Σ(XᵢY_{i+1} − X_{i+1}Yᵢ)|

# Leveling
  Height of instrument: HI = RL + BS
  RL = HI − FS = HI − IS
  BS = backsight (on known point), FS = foresight

  Booking methods: Rise & Fall, Height of Instrument

  Curvature correction: C_c = 0.0785d² (d in km, in meters)
  Combined (curvature + refraction): C = 0.0673d²
  Refraction reduces curvature effect by 1/7

  Reciprocal leveling: eliminates curvature & refraction
  d/√D for distance measurement with stadia (tacheometry)

# Stadia Tacheometry
  D = K × s × cos²θ + C × cosθ  (inclined sight)
  V = K × s × sin2θ / 2 + C × sinθ
  RL = HI + V − h (h = height of target)
  K = 100 (stadia constant), C = 0 (anallactic lens)

# Horizontal Curves (Circular)
  Tangent length: T = R × tan(Δ/2)
  Curve length: L = πRΔ/180
  Mid-ordinate: M = R(1 − cos(Δ/2))
  External distance: E = R(sec(Δ/2) − 1)

  Transition curve (spiral):
    L = V³/(CR) where C = rate of change of centrifugal acceleration
    C = 0.3 m/s³ to 0.9 m/s³ (comfort)
    Shift = L²/(24R)

  Compound curve: two curves of different radii, same tangent
  Reverse curve: two curves with opposite curvature

# Vertical Curves
  Summit (crest): grade changes from +g₁ to −g₂
  Valley (sag): grade changes from −g₁ to +g₂

  Elevation at any point:
    y = y_BVC + g₁x + (g₂−g₁)x²/(2L)
  L = minimum length for comfort or sight distance