Highway & Transportation Engineering Formulas

🛣️ Horizontal Alignment & Curves

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Superelevation (IRC Method)

e + f = V² / (127 R)

Equilibrium equation for lateral friction and superelevation. IRC recommends considering only 75% of design speed for e.

e = Superelevation (tan θ ≈ sin θ)

f = Coefficient of lateral friction (typically 0.15)

V = Design speed (km/h)

R = Radius of curve (m)

Limit: e ≤ 0.07 (7%)

Extra Widening (Horizontal Curve)

W_e = W_m + W_p = (n L²) / (2R) + V / (9.5 √R)

Total extra widening required for a curved road. Sum of mechanical and psychological widening.

n = Number of lanes

L = Wheelbase of longest vehicle (typically 6.1m)

V = Speed (km/h)

Mechanical term accounts for off-tracking. Psychological term for safety clearance.

Transition Curve Length (IRC)

L_s = (2.7 × V²) / R

Length of transition curve based on rate of change of acceleration (IRC empirical formula).

V = Speed (km/h)

R = Radius (m)

Also check based on rate of introduction of superelevation: L_s = N × e × B, where B = road width.

Setback Distance (Sight Distance on Curve)

m = R [1 - cos(90 S / πR)]

Distance from centerline to obstacle clearance for Stopping Sight Distance (SSD).

m = Setback distance

S = Required sight distance

If S > Length of curve, different formula applies. For Overtaking: S must be compared with length of curve.

👁️ Sight Distance Calculations

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Stopping Sight Distance (SSD) IRC 38

SSD = 0.278 V t + (V² / (254 (f ± n)))

Minimum distance required to stop safely without collision. Sum of lag distance and braking distance.

V = Speed (km/h)

t = Perception-reaction time (2.5 sec)

f = Longitudinal friction coefficient (0.35–0.4)

n = Grade (decimal, +ve uphill, -ve downhill)

Overtaking Sight Distance (OSD)

OSD = d₁ + d₂ + d₃

Total distance required to safely overtake.

d₁ = Distance traveled during reaction time (before maneuver)

d₂ = Distance during actual overtaking (critical zone)

d₃ = Clearance distance (approx 30–60m)

d₂ = 2√(2 a T³) or derived from speeds. Approx: OSD ≈ 3 × SSD for single lane.

Intermediate Sight Distance (ISD)

ISD = 2 × SSD

Used where OSD cannot be provided fully. Halfway between SSD and OSD.

Headlight Sight Distance

HSD = 0.278 V t + V² / (254(f ± tan α))

Sight distance available at night, limited by headlight beam throw.

α = Beam inclination (usually 1°)

Critical in valleys (summit curves) at night.

⬆️⬇️ Vertical Curves

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Length of Summit Curve (Sight Distance Control)

L_s = (N S²) / [4.4] (if L > S)

Parabolic curve length to provide required stopping sight distance over a hill.

N = Deviation angle (n₁ − n₂)

S = Required sight distance

If L < S, use L = 2S − 4.4/N. Formula depends on eye height (1.2m) and object height (0.15m).

Length of Valley Curve (Comfort)

L_v = 2 × [ (N V³) / C ]^0.5

Length based on comfort criteria to minimize centrifugal jerk.

C = Rate of change of acceleration (0.6 m/s³ typical)

V = Speed (m/s)

Also checked for Headlight Sight Distance. Max(L_v,comfort, L_v,sight) is used.

Gradient Calculation

Grade (%) = (Δh / L) × 100

Slope expressed as percentage.

Δh = Elevation difference

L = Horizontal distance

IRC limits: Ruling (max 1 in 30), Limiting (1 in 20), Minimum (1 in 400 for drainage).

🚗 Traffic Flow & Analysis

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Fundamental Relationship

q = k × v

Basic relation between flow, density, and speed.

q = Flow (vehicles/hr)

k = Density (veh/km)

v = Space mean speed (km/hr)

Time Mean vs Space Mean Speed

v_s = (Σt) / Σ(t/v_i) | v_t = (Σv_i) / n

v_s (Space Mean) is always less than or equal to v_t (Time Mean). Used in q=kv.

t = Travel time

Greenberg's Model (Density-Speed)

v = v_f ln(k_j/k)

Logarithmic relationship between speed and density. Valid for high densities.

v_f = Free flow speed

k_j = Jam density

Capacity (Greenshields Linear Model)

q_max = (k_j × v_f) / 4

Maximum flow capacity occurs at half jam density and half free flow speed.

k_j = Jam density

v_f = Free flow speed

Occurs at k = k_j/2 and v = v_f/2

Traffic Density from Spot Speed

k = 1 / h (where h = headway in km/veh)

Inverse of average time headway converted to distance.

PCU Equivalent Factors (IRC)

Passenger Car Unit (PCU) converts mixed traffic to standard units.
PCU = 1.0 (Car/Jay)
PCU = 2.5 (Truck/Bus)
PCU = 1.0-3.0 (Motorcycle depending on condition)

Used to calculate total equivalent traffic volume Q = Σ(n_i × PCU_i)

🛣️ Flexible Pavement Design

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CBR Thickness (Group Index)

T = f(CBR)

Empirical thickness design using CBR charts (IRC:37). Total thickness above subgrade.

Depends on CBR value and expected wheel load.

Design Traffic (ESALs)

N = (365 × A × ((1+r)ⁿ − 1) × DF) / r

Cumulative number of standard axles (in millions) over design life.

A = Initial traffic (vpd)

r = Annual growth rate

n = Design life (years)

DF = Vehicle damage factor

Thicknes Ratio (Flexible Layers)

t₁ / t₂ = (C₁/C₂)^0.2 (Approx)

Rough equivalence between layers based on CBR values.

California Bearing Ratio (CBR)

CBR = (Load Test / Standard Load) × 100%

Ratio of force per unit area required to penetrate soil with standard plunger to that of standard crushed stone.

Standard loads: 1370 kg at 2.5mm, 2055 kg at 5.0mm penetration.

Use higher of the two % values. If correction needed due to convexity of curve.

Dynamic Cone Penetrometer (DCP) Correlation

CBR = 292 / DCP^1.12

Empirical correlation to estimate CBR from field DCP index.

DCP = mm/blow

🧱 Rigid Pavement Design

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Modulus of Subgrade Reaction (k)

k = p / Δ

Pressure per unit settlement (kg/cm³). Key parameter for Westergaard analysis.

p = Pressure applied by 75cm square plate

Δ = Settlement (0.125 cm)

Radius of Relative Stiffness (l)

l = [ (E h³) / (12 k (1 − μ²)) ]^0.25

Measure of slab stiffness relative to subgrade. Critical length for stress calculation.

E = Modulus of elasticity of concrete (MPa)

h = Slab thickness (cm)

k = Modulus of subgrade reaction

μ = Poisson's ratio (0.15)

Westergaard Interior Stress (Critical)

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

Stress at interior critical point under wheel load P.

P = Wheel load

b = Equivalent radius of resisting section

Bradley's Edge Stress (Critical)

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

Higher stress occurs at the edge of the slab, usually the design criterion.

l = Radius of relative stiffness

Corner Stress

σ_c = (3 P / h²) [1 − (a/l)^0.6]

Stress at the corner. Usually lower than edge but critical if warping exists.

a = Radius of contact area

Temperature Differential Stresses

σ_T = (E α ΔT) / 2

Stress induced by temperature gradient through slab depth (warping).

α = Coeff of thermal expansion

ΔT = Temp diff across depth

🚦 Intersections & Signals

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Weaving Width & Ratio

Weaving Ratio = (P+W) / (P+Q+W+R)

Proportion of weaving traffic to total traffic in rotary intersections.

P, Q, R, W = Traffic streams entering/exiting

Saturation Flow (Webster)

S = 2070 / (1 + (w/3.65)) (approx)

Max flow rate of vehicles passing a signal when green is given. Depends on lane width w.

Optimum Signal Cycle (Webster)

C_o = (1.5L + 5) / (1 - Y)

Cycle length that minimizes delay.

L = Total lost time per cycle

Y = Sum of critical flow ratios (y₁ + y₂...)

y_i = q_i / S_i

Green Time Allocation

G_i = (C - L) × (y_i / Y)

Effective green time distributed proportionally to critical flow ratios.

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Highway & Transportation
Formula Reference

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Highway & TransportationFormula Reference

Highway & Transportation
Formula Reference