Transportation Engineering — Formula Sheet

⭐ Stopping Sight Distance (SSD):

SSD = 0.278 V t + V²/(254 f)    (V in km/h; t = 2.5 s; f = longitudinal friction)

On grade: SSD = 0.278 V t + V²/[254(f ± g)]    (+uphill; –downhill)

Lag distance = 0.278 V t;   Braking distance = V²/(254 f)

⭐ Overtaking Sight Distance (OSD):

OSD = d₁ + d₂ + d₃

d₁ = 0.278 V_b t₁ (reaction distance; t₁ = 2 s)

d₂ = 0.278 V_b t₂ + s (overtaking distance; s = safe spacing = 0.7V_b + 6 m)

d₃ = d₁ (opposing vehicle reaction)

ISD = 2 × SSD

⭐ Superelevation on horizontal curves:

e + f = V²/(127R)    (V in km/h; R in m; e fraction; f lateral friction ≈ 0.15)

Minimum radius: R_min = V²/[127(e_max + f)]

e_max = 7% (plain/rolling); 10% (mountainous); f ≈ 0.12–0.15

Extra widening on curves:

W_e = nl²/(2R) + V/(10√R)    (l = wheelbase ≈ 6 m; n = lanes; V in km/h)

Setback distance (S < L_c): m = S²/(8R)

Setback distance (S > L_c): m = L_c(2S – L_c)/(8R)

Grade compensation: GC = 75/R %    (R in metres)

⭐ Simple circular curve elements (Δ = intersection angle; R = radius):

Tangent length: T = R tan(Δ/2)

Arc length: L = πRΔ/180 = RΔ (Δ in radians)

Chord length: C = 2R sin(Δ/2)

Mid-ordinate: M = R[1 – cos(Δ/2)]

External distance: E = R[sec(Δ/2) – 1]

Chainage of PC = Chainage of PI – T; Chainage of PT = Chainage of PC + L

Degree of curve (arc definition, highway): R = 1719/D (D in degrees)

Degree of curve (chord definition, railway): R = 1746/D

⭐ Transition (spiral) curve:

Length (rate of change of centrifugal acc.): L_s = V³/(C × R)    (V in m/s; C ≈ 0.6 m/s³)

IRC simplified: L_s = 2.7V³/R    (V in m/s)

Length (superelevation): L_s = e × N × W    (N = 150 for V ≤ 80 km/h)

Spiral angle: φ_s = L_s/(2R) radians

Shift: S = L_s²/(24R)

Combined curve tangent length: T_L = (R + S)tan(Δ/2) + L_s/2

Central angle of circular arc: Δ_c = Δ – 2φ_s

Algebraic grade difference: N = |G₁ – G₂| (use decimal fraction, e.g., 0.06 for 6%)

Summit: G₁ > G₂ (uphill then downhill); Valley: G₁ < G₂ (downhill then uphill)

⭐ Summit curve length — SSD (H₁ = 1.2 m, H₂ = 0.15 m):

L > S: L = NS²/4.4    (N in decimal, S in m)

L < S: L = 2S – 4.4/N

⭐ Summit curve length — OSD (H₁ = H₂ = 1.2 m):

L > O: L = NO²/9.6

L < O: L = 2O – 9.6/N

⭐ Valley curve length — Headlight sight distance (h = 0.75 m, α = 1°):

L > S: L = NS²/(1.5 + 0.035S)

L < S: L = 2S – (1.5 + 0.035S)/N

Valley curve — Comfort criterion:

L = NV²/1300    (N = numerical % value, e.g., 7 for 7%; V in km/h)

Use larger L from both criteria.

K-value: K = L/N (metres per % grade change)

Highest/lowest point from BVC: x = G₁L/N (G₁, N as decimal fractions)

CBR test: CBR (%) = (Test load / Standard load) × 100

Standard loads: 2.5 mm penetration = 13.44 kN; 5.0 mm = 20.07 kN

Design CBR = soaked CBR at 95% Modified Proctor MDD (96 hours soaking)

⭐ Vehicle Damage Factor (VDF / LEF — 4th power law):

LEF = (W/W_std)⁴    (W_std = 80 kN = 8.16 tonnes)

VDF = Σ(n_i × LEF_i) / Σn_i

⭐ Design traffic (N) in Million Standard Axles (MSA):

N = 365 × A × [(1+r)ⁿ – 1]/r × VDF × LDF / 10⁶

A = initial CVPD; r = growth rate (fraction); n = design life (years)

LDF: single lane = 1.0; 2-lane undivided = 0.75; 4-lane divided = 0.40

Design life: 15 years (NH/SH new); 10 years (overlays)

⭐ IRC:37 — Total pavement thickness from CBR–MSA chart (selected values):

⭐ Radius of relative stiffness:

l = [Eh³/(12(1–μ²)k)]^1/4

E = 30,000 MPa (M40 concrete); μ = 0.15; k = modulus of subgrade reaction (MN/m³)

Equivalent radius b (for stress calculations):

If a < 1.724h: b = √(1.6a² + h²) – 0.675h

If a ≥ 1.724h: b = a

⭐ Westergaard’s stress equations:

Edge (critical): σ_e = (0.572P/h²)[4 log(l/b) + 0.359]

Interior: σ_i = (0.316P/h²)[4 log(l/b) + 1.069]

Corner: σ_c = (3P/h²)[1 – (a√2/l)⁰·⁶]

P = wheel load (N); h = slab thickness (m)

Edge stress > Interior stress; Corner stress varies.

⭐ Temperature warping stress (Bradbury):

σ_t = EαΔT C/[2(1–μ)]

α = 10 × 10⁻⁶/°C; 2(1–μ) = 1.70; C from Bradbury’s chart vs L/l

Combined design stress: σ = σ_e + σ_t

⭐ Flexural strength of concrete: f_r = 0.7√f_ck (MPa)

M40: f_r = 4.43 MPa; M45: f_r = 4.70 MPa

Fatigue (Teller-Sutherland, SR > 0.55): log N = (0.97 – SR)/0.0828

Unlimited fatigue life: SR = σ/f_r ≤ 0.45

CFD = Σ(n_i/N_i) ≤ 1.0 (Miner’s rule; design satisfied)

⭐ Fundamental flow equation: q = u_s × k

q = flow (veh/hr); u_s = space mean speed (km/h); k = density (veh/km)

Headway: h = 3600/q (s); Spacing: s = 1000/k (m)

⭐ Speed types:

TMS (arithmetic mean): u_t = Σu_i/n

SMS (harmonic mean): u_s = n/Σ(1/u_i) = L/t̄

u_t ≥ u_s always; u_t = u_s + σ²_s/u_s

Use SMS in q = uk (not TMS).

⭐ PCU conversion: Q_PCU = Σ(n_i × PCU_i)

Car = 1.0; Two-wheeler = 0.5; Auto-rickshaw = 0.5; Bus/truck = 2.2; Tractor-trailer = 4.5; Bullock cart = 8.0

Road capacity (IRC:64): 2-lane undivided = 1500 PCU/hr (both ways)

4-lane divided: 2000 PCU/hr per direction; 6-lane: 3000 PCU/hr per direction

Peak Hour Factor: PHF = PHV/(4 × V_15,max)

Design flow = PHV/PHF

Moving observer method: q = (m_a + m_w)/(t_a + t_w) veh/min

Journey time: t̄ = t_a – m_a/q

⭐ Webster’s optimal signal cycle:

C₀ = (1.5L + 5)/(1 – Y)

L = total lost time; Y = Σy_i; y_i = q_i/S_i (flow ratio per phase)

Green per phase: g_i = (y_i/Y)(C₀ – L)

Saturation flow S ≈ 1800 PCU/hr/lane (straight through)

⭐ Greenshields model:

u = u_f(1 – k/k_j)    (linear u–k)

q = u_f k(1 – k/k_j)    (parabolic q–k)

q = k_j u(1 – u/u_f)    (parabolic q–u)

⭐ Critical (optimum) conditions (Greenshields):

k_o = k_j/2;   u_o = u_f/2;   q_max = u_fk_j/4

Find k at given q: solve k² – k_jk + q/u_f × k_j = 0 (quadratic — 2 roots)

Greenberg model: u = u_o ln(k_j/k); k_o = k_j/e; q_max = u_ok_j/e

Underwood model: u = u_f e^–k/k_o; q_max = u_fk_o/e

⭐ Shockwave speed:

w = (q₂ – q₁)/(k₂ – k₁)

w < 0 → propagates upstream (back of queue growing); w > 0 → propagates downstream

⭐ Cant (superelevation):

e_th = GV²/(127R)    (G in mm; V in km/h; R in m; e in mm)

BG: G = 1676 mm; MG: G = 1000 mm

Maximum cant: BG = 165 mm; MG = 90 mm; NG = 65 mm

⭐ Equilibrium speed:

V_eq = √(127 R e / G)    (e in mm; R in m; G in mm; V in km/h)

⭐ Cant deficiency and excess:

CD = e_th,fast – e_actual    (max CD: BG = 75 mm)

CE = e_actual – e_th,slow    (max CE: BG = 75 mm)

Negative cant (max): 75 mm (BG, permitted only at crossovers/yards)

⭐ Grade compensation (BG):

GC = 70/R ‰    (R in metres) [MG: 52.5/R; NG: 35/R]

Gradient on curve = Ruling gradient – GC

Degree of curve (railway, chord definition):

R = 1746/D    (R in m; D in degrees)

Sleeper density: N_sleepers/km = (sleepers per rail length/rail length) × 1000

Sleeper spacing: s = Rail length / Sleepers per rail length

Maximum station gradient: 1 in 400 (2.5‰) for BG; level preferred