Advanced Civil Engineering Handbook › Chapter 10

Chapter 10 of 14 · 43 formulas · 82 one-liners

Highway & Transportation Engineering — formulas & one-liners

Road planning, IRC geometric design, traffic engineering, pavement materials and design (IRC:73, IRC:37-2018, IRC:58-2015, IS 73), drainage, maintenance, and airport and bridge basics.

10.1 Road development, classification & cross-section (IRC:73)

Ordinate of parabolic camber

y = \frac{2 x ^{2}}{nW}

y = drop below crown at distance x from crown (m); camber = 1 in n; W = carriageway width (m)

Parabolic (elliptical) camber suits fast vehicles

Jayakar Committee (1927) led to the Central Road Fund (1929) and the formation of the Indian Roads Congress (1934).
Nagpur Road Plan (1943–63) set a target road density of 16 km per 100 km² and adopted the star-and-grid pattern.
Bombay Road Plan (1961–81) set a target road density of 32 km per 100 km².
Lucknow Road Plan (1981–2001) set a target road density of 82 km per 100 km².
The Nagpur plan classified roads into National Highways, State Highways, Major District Roads, Other District Roads and Village Roads.
Carriageway width per IRC:73: single lane 3.75 m, two-lane without raised kerbs 7.0 m, two-lane with raised kerbs 7.5 m, intermediate lane 5.5 m, multi-lane 3.5 m per lane.
Camber (IRC:73) for high-type bituminous or cement concrete surface is 1 in 50 (2.0%) in heavy rainfall and 1 in 60 (1.7%) in light rainfall areas.
IRC recommends a minimum shoulder width of 2.5 m; roadway width for a two-lane NH/SH in plain terrain is 12 m.
Design speed for NH and SH in plain terrain is 100 km/h (ruling) and 80 km/h (minimum); in mountainous terrain 50 km/h and 40 km/h.

10.2 Sight distance

Stopping sight distance (level road)

S S D = 0.278 V t + \frac{V ^{2}}{254 f}

V = design speed (km/h); t = reaction (PIEV) time = 2.5 s; f = longitudinal friction coefficient (0.35–0.40)

First term = lag distance; second = braking distance

SSD on a gradient

S S D = 0.278 V t + \frac{V ^{2}}{254 ( f \pm 0.01 n )}

n = gradient (%); + for ascending, − for descending

Overtaking sight distance (two-way road)

O S D = d_{1} + d_{2} + d_{3} = v_{b} t + (v_{b} T + 2 s) + v T

v_b = speed of overtaken vehicle (m/s); v = design speed (m/s); t = 2 s; s = 0.7v_b + 6 (m); T = √(4s/a) (s); a = acceleration (m/s²)

On one-way roads OSD = d₁ + d₂; v_b may be taken as v − 4.5 m/s (V − 16 km/h)

IRC takes total reaction (PIEV) time as 2.5 s for SSD and 2 s as the reaction time of the overtaking driver.
Height of driver's eye is taken as 1.2 m; object height is 0.15 m for SSD and 1.2 m for OSD.
IRC:73 SSD values: 120 m at 80 km/h and 180 m at 100 km/h.
Intermediate sight distance is twice the SSD and is provided where OSD cannot be provided.
Minimum length of an overtaking zone is 3 × OSD; desirable length is 5 × OSD.

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10.3 Horizontal & vertical alignment

Super-elevation + friction balance

e + f = \frac{V ^{2}}{127 R}

e = super-elevation (m/m); f = lateral friction (0.15); V = speed (km/h); R = radius (m)

IRC design super-elevation (mixed traffic)

e = \frac{( 0.75 V ) ^{2}}{127 R}

e = super-elevation for 75% of design speed, friction neglected

Limit e to 7% (plain/rolling, snow-bound hills) or 10% (hills, not snow-bound); then check f ≤ 0.15 at full speed

Extra widening on curves

W_{e} = \frac{n l ^{2}}{2 R} + \frac{V}{9.5 R}

n = number of lanes; l = wheelbase (6.1 m); R = radius (m); V = speed (km/h)

First term = mechanical widening; second = psychological widening

Transition length – rate of change of centrifugal acceleration

L_{s} = \frac{0.0215 V ^{3}}{C R}, C = \frac{80}{75 + V}

L_s (m); V (km/h); R (m); C = allowable rate (m/s³), limited to 0.5 ≤ C ≤ 0.8

Transition length – rate of introducing super-elevation (rotation about centre line)

L_{s} = \frac{e N ( W + W _{e} )}{2}

N = rate of change (1 in N): 150 for plain/rolling, 60 for hilly; W = carriageway width; W_e = extra widening

If rotated about inner edge, L_s = e N (W + W_e)

Transition length – IRC empirical (plain/rolling)

L_{s} = \frac{2.7 V ^{2}}{R}

V (km/h); R (m)

Mountainous/steep terrain: L_s = V²/R; adopt the largest of the three criteria

Shift of circular curve

S = \frac{L _{s}^{2}}{24 R}

L_s = transition length (m); R = radius (m)

Grade compensation on curves

GC = \frac{30 + R}{R} %, G C_{m a x} = \frac{75}{R} %

R = radius of curve (m)

Not needed where the gradient is flatter than 4%

Summit curve length for SSD (L > S)

L = \frac{N S ^{2}}{4.4}

N = deviation angle (algebraic difference of grades, as a ratio); S = SSD (m)

General: L = NS²/[2(√H + √h)²], H = 1.2 m, h = 0.15 m

Summit curve length for SSD (L < S)

L = 2 S - \frac{4.4}{N}

S = SSD (m); N = deviation angle

Valley curve length – comfort criterion

L = 2 \frac{N v ^{3}}{C}

L = total length (m); v = speed (m/s); C = allowable rate of change of centrifugal acceleration (0.6 m/s³)

Valley curve length – headlight sight distance (L > S)

L = \frac{N S ^{2}}{1.5 + 0.035 S}

S = SSD (m); headlight height 0.75 m, beam inclined 1° upward

For L < S: L = 2S − (1.5 + 0.035S)/N

Maximum super-elevation is 7% for plain and rolling terrain and snow-bound hill roads, 10% for hill roads not bound by snow, and 4% for urban roads with frequent intersections.
Minimum super-elevation equals the camber of the road, provided for drainage.
IRC adopts a lateral friction coefficient of 0.15 for horizontal curve design and 0.35–0.40 longitudinal friction for SSD.
The ideal transition curve is the spiral (clothoid), in which radius × length is constant; IRC recommends the spiral.
Rate of introduction of super-elevation is 1 in 150 for plain/rolling and 1 in 60 for mountainous terrain (IRC).
Gradients per IRC for plain/rolling terrain: ruling 3.3% (1 in 30), limiting 5% (1 in 20), exceptional 6.7% (1 in 15).
Gradients for mountainous terrain and steep terrain above 3000 m elevation: ruling 5%, limiting 6%, exceptional 7%; for steep terrain up to 3000 m: 6%, 7% and 8%.
Summit curves are generally designed as simple parabolas; valley curves use a cubic parabola (or spiral) transition.
Valley curve length is governed by the headlight sight distance at night and by comfort (centrifugal) criterion.

10.4 Traffic engineering

Fundamental relation of traffic flow

q = k v_{s}

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

Greenshields model: v = v_f(1 − k/k_j)

Maximum flow (Greenshields)

q_{m a x} = \frac{v _{f} k _{j}}{4}

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

Theoretical capacity of a lane

C = \frac{1000 V}{S}

C = veh/h; V = speed (km/h); S = minimum spacing (m), e.g. stopping distance + vehicle length

Time mean speed vs space mean speed (Wardrop)

v_{t} = v_{s} + \frac{σ _{s}^{2}}{v _{s}}

v_t = time mean speed; v_s = space mean speed; σ_s = standard deviation of space speeds

v_t ≥ v_s; space mean speed is the harmonic mean of spot speeds

Webster optimum cycle length

C_{0} = \frac{1.5 L + 5}{1 - Y}

C₀ (s); L = total lost time per cycle (s); Y = Σ y_i, y_i = q_i/s_i (critical flow ratio of each phase)

Saturation flow s ≈ 525w PCU/h for approach width w = 5.5–18 m

Effective green time of a phase (Webster)

g_{i} = \frac{y _{i}}{Y} (C_{0} - L)

y_i = flow ratio of phase i; Y = sum of critical flow ratios

Practical capacity of a weaving section (rotary, IRC:65)

Q_{w} = \frac{280 w ( 1 + \frac{e}{w} ) ( 1 - \frac{p}{3} )}{1 + \frac{w}{l}}

Q_w = PCU/h; w = weaving width (m); e = average entry width (m); l = weaving length (m); p = proportion of weaving traffic

PCU values (IRC:64, rural roads): car 1.0, two-wheeler 0.5, bicycle 0.5, bus/truck 3.0, bullock cart 8.0.
IRC:64 recommends Level of Service B for the design of rural highways.
Design service volume of a two-lane two-way rural road in plain terrain with low curvature is about 15,000 PCU/day (IRC:64).
The 30th highest hourly volume of the year is generally adopted as the design hourly volume.
The 85th percentile spot speed is used to fix the upper speed limit, the 15th percentile for the lower limit and the 98th percentile as the geometric design speed.
Time mean speed is the arithmetic mean of spot speeds at a point; it is always equal to or greater than space mean speed.
IRC:65 design speed for rotaries is 40 km/h in rural areas and 30 km/h in urban areas.
Rotaries are suitable when total entering traffic is roughly between 500 and 3000 vehicles per hour.
Lost time per phase covers start-up delay and the unused part of amber; pedestrian walking speed for signal timing is taken as 1.2 m/s (IRC:93).

10.5 Pavement materials & tests

Aggregate impact / crushing / abrasion value

Value (%) = \frac{W _{2}}{W _{1}} \times 100

W₁ = weight of oven-dry sample; W₂ = weight passing the specified sieve (2.36 mm for impact & crushing, 1.70 mm for Los Angeles)

Angularity number

A N = 67 - \frac{100 W}{C G _{a}}

W = mass of aggregate filling the cylinder (g); C = mass of water filling it (g); G_a = specific gravity of aggregate

Ranges from 0 (rounded) to about 11 (very angular)

California Bearing Ratio

C B R = \frac{P _{T}}{P _{S}} \times 100

P_T = test load at given penetration; P_S = standard load: 1370 kg at 2.5 mm, 2055 kg at 5.0 mm

Plunger 50 mm dia at 1.25 mm/min; soaked 4 days; higher of 2.5 mm and 5.0 mm values (repeat if 5 mm higher)

Aggregate impact test: hammer of 13.5–14 kg falling 380 mm, 15 blows, on aggregate passing 12.5 mm and retained on 10 mm sieve.
Aggregate impact value below 10% indicates exceptionally strong, 10–20% strong, 20–30% satisfactory for road surfacing, and above 35% weak aggregate.
Aggregate crushing test applies 40 t load at a uniform rate over 10 minutes; maximum crushing value is 30% for surface course and 45% for base course.
MoRTH limits the Los Angeles abrasion value to 30% for bituminous concrete and 35% for DBM.
Flaky particles have thickness less than 0.6 × mean size; elongated particles have length more than 1.8 × mean size.
MoRTH (5th revision) limits the combined flakiness and elongation index of bituminous-course aggregate to 35% (earlier 30%).
Penetration test: 100 g needle for 5 s at 25 °C; penetration is reported in units of 0.1 mm.
Ductility test: briquette of 10 mm × 10 mm cross-section pulled at 50 mm/min; IS 1208 test temperature is 27 °C.
IS 73:2013 grades paving bitumen by absolute viscosity at 60 °C: VG10 (800–1200 P), VG20 (1600–2400 P), VG30 (2400–3600 P), VG40 (3200–4800 P).
Minimum penetration (0.1 mm) at 25 °C per IS 73: VG10 80, VG20 60, VG30 45, VG40 35.
Minimum softening point per IS 73: VG10 40 °C, VG20 45 °C, VG30 47 °C, VG40 50 °C.
VG30 is the most widely used paving grade in India; VG40 suits heavy-traffic/hot areas and VG10 cold regions and surface dressing.
Marshall specimen is 101.6 mm dia × 63.5 mm high, compacted with a 4.54 kg hammer falling 457 mm, 75 blows per face for heavy traffic.
Marshall test is done at 60 °C with a loading rate of 50.8 mm/min; MoRTH requires minimum stability 9 kN, flow 2–4 mm, air voids 3–5% and VFB 65–75% for DBM/BC.

10.6 Flexible pavement design (IRC:37-2018)

Cumulative design traffic

N_{d es} = \frac{365 [ ( 1 + r ) ^{n} - 1 ]}{r} \times A \times D \times F

N_des = standard axles (divide by 10⁶ for msa); r = annual growth rate (decimal); n = design period (years); A = initial CVPD; D = lane distribution factor; F = VDF; A = P(1 + r)^x, P = CVPD at last count, x = years to completion

Load equivalency (fourth power law)

E F = (\frac{W}{W _{s}})^{4}

W = actual axle load; W_s = standard axle load: 80 kN (single axle dual wheel), 148 kN (tandem), 224 kN (tridem)

Resilient modulus of subgrade (CBR ≤ 5%)

M_{R} = 10 \times C B R

M_R in MPa; CBR in %

Resilient modulus of subgrade (CBR > 5%)

M_{R} = 17.6 \times (C B R)^{0.64}

M_R in MPa; CBR in %

Subgrade rutting criterion (90% reliability)

N_{R} = 1.41 \times 1 0^{- 8} \times (\frac{1}{ε _{v}})^{4.5337}

N_R = allowable standard axles for 20 mm rut; ε_v = vertical compressive strain on top of subgrade

80% reliability: coefficient 4.1656 × 10⁻⁸

IRC:37-2018 applies to design traffic above 2 msa; low-volume roads (below 2 msa) are designed per IRC:SP:72.
Design period per IRC:37-2018 is 20 years for NH, SH and urban roads and 15 years for other roads.
Where reliable data are not available, an annual traffic growth rate of 5% may be adopted (IRC:37-2018).
Lane distribution factor: single-lane road 1.0, two-lane single carriageway 0.50, four-lane single carriageway 0.40 of total two-way traffic.
Lane distribution factor for dual carriageways: two-lane 0.75, three-lane 0.60 and four-lane 0.45 of the traffic in each direction.
Indicative VDF (IRC:37-2018) in plain/rolling terrain: 1.7 for 0–150 CVPD, 3.9 for 150–1500 CVPD and 5.0 above 1500 CVPD.
IRC:37-2018 adopts mechanistic-empirical design using the linear-elastic layered program IITPAVE with tyre contact pressure 0.56 MPa.
Fatigue cracking is controlled by the horizontal tensile strain at the bottom of the bituminous layer; rutting by the vertical compressive strain on top of the subgrade.
Top 500 mm of subgrade is compacted to at least 97% of the maximum dry density from modified (heavy) Proctor compaction.
Benkelman beam deflection (IRC:81) uses a truck with 8170 kg rear axle load and 5.6 kg/cm² tyre pressure; deflections are corrected to 35 °C.
Characteristic deflection is mean + 2σ for NH/SH and mean + σ for other roads (IRC:81); overlay design is now by FWD per IRC:115.

10.7 Rigid pavement design (IRC:58-2015)

Radius of relative stiffness

l = [\frac{E h ^{3}}{12 ( 1 - μ ^{2} ) k}]^{1/4}

E = modulus of concrete; h = slab thickness; μ = Poisson's ratio (0.15); k = modulus of subgrade reaction

Equivalent radius of resisting section (a < 1.724h)

b = 1.6 a^{2} + h^{2} - 0.675 h

a = radius of load contact area; h = slab thickness

If a ≥ 1.724h, b = a

Westergaard interior load stress

σ_{i} = \frac{0.316 P}{h ^{2}} [4 lo g_{10} \frac{l}{b} + 1.069]

P = wheel load (kg); h (cm); σ in kg/cm²

Westergaard edge load stress

σ_{e} = \frac{0.572 P}{h ^{2}} [4 lo g_{10} \frac{l}{b} + 0.359]

P = wheel load (kg); h (cm); σ in kg/cm²

Westergaard corner load stress

σ_{c} = \frac{3 P}{h ^{2}} 1 - (\frac{a 2}{l})^{0.6}

a = radius of contact area; l = radius of relative stiffness

Kelley's modification uses exponent 1.2 instead of 0.6

Edge warping stress (Bradbury)

σ_{t e} = \frac{C E α t}{2}

C = Bradbury coefficient (from L/l); α = coefficient of thermal expansion; t = temperature differential between top and bottom

IRC:58-2015 adopts a design period of 30 years for concrete pavements.
Pavement quality concrete should have 28-day flexural strength of at least 4.5 MPa (about M40).
IRC:58 takes E of concrete as 30,000 MPa, Poisson's ratio 0.15 and coefficient of thermal expansion 10 × 10⁻⁶ per °C.
IRC:58-2015 checks both bottom-up cracking (load at edge, day-time positive temperature gradient) and top-down cracking (load near corner/transverse joint, night-time negative gradient).
Fatigue is checked by cumulative fatigue damage (Miner's hypothesis), which must not exceed 1.0.
For a two-lane two-way highway, 25% of the total two-way commercial traffic is taken for rigid pavement design.
Dowel bars (plain round bars) transfer load across transverse contraction/expansion joints; tie bars (deformed bars) hold longitudinal joints together.
Contraction joints in plain jointed concrete pavement are typically spaced at 4.5 m.

10.8 Highway drainage & pavement maintenance

Rational formula for surface run-off

Q = \frac{C I A}{360}

Q = run-off (m³/s); C = run-off coefficient; I = rainfall intensity for duration = time of concentration (mm/h); A = catchment area (ha)

Manning's formula for side drains

V = \frac{1}{n} R^{2/3} S^{1/2}

V = velocity (m/s); n = Manning's roughness; R = hydraulic radius (m); S = longitudinal slope

Filter criteria for sub-surface drains: D15 of filter / D85 of soil < 5 (to prevent piping) and D15 of filter / D15 of soil > 5 (for permeability).
Catch-water drains are provided on the hill side above a cut slope to intercept surface water before it reaches the road.
Cross-drainage works: culvert for total span up to 6 m, minor bridge for 6–60 m and major bridge above 60 m.
Bleeding (flushing) is the upward movement of excess bitumen to the surface, caused by high binder content or low air voids.
Alligator (fatigue) cracking indicates repeated overloading or a weak base/subgrade.
Routine maintenance includes pothole patching, crack sealing and drain cleaning; periodic renewal uses surface dressing, slurry seal, micro-surfacing or thin overlays.
Road roughness is measured with a bump integrator (mm/km) and expressed as International Roughness Index (m/km).

10.9 Airport & bridge basics

Airport reference temperature

A R T = T_{a} + \frac{T _{m} - T _{a}}{3}

T_a = monthly mean of average daily temperature of the hottest month; T_m = monthly mean of maximum daily temperature of that month

Standard temperature at elevation H (m): 15 − 0.0065H °C

Impact factor for IRC Class A loading (RCC bridges)

I = \frac{4.5}{6 + L}

L = span (m)

Steel bridges: I = 9/(13.5 + L)

Lacey's regime waterway

P = 4.8 Q

P = wetted perimeter ≈ regime width (m); Q = design discharge (m³/s)

Lacey's normal scour depth (regime conditions)

d_{s m} = 0.473 (\frac{Q}{f})^{1/3}

d_sm = mean depth of scour below HFL (m); f = silt factor = 1.76√d_m (d_m in mm)

Basic runway length is increased by 7% for every 300 m of elevation above mean sea level.
Runway length is further increased by 1% for every 1 °C by which the airport reference temperature exceeds the standard atmospheric temperature at that elevation.
If the combined elevation and temperature correction exceeds 35%, a specific study is required (ICAO).
Runway length (after elevation and temperature correction) is increased by 20% for every 1% of effective gradient.
Runways are oriented along the prevailing wind using the wind rose (Type I: direction and duration; Type II: with wind intensity).
ICAO requires runway usability (wind coverage) of at least 95%, with permissible cross-wind 37 km/h for reference field length ≥ 1500 m, 24 km/h for 1200–1500 m and 19 km/h below 1200 m.
IRC:6 live loads are Class 70R, Class A and Class B; Class A train of vehicles totals 554 kN and Class B 332 kN.
Class 70R loading comprises a 700 kN tracked vehicle or a 1000 kN wheeled vehicle.
Maximum scour depth for design is taken as 2.0 d_sm at piers and 1.27 d_sm at abutments with approach retained (IRC:78).
The economical span of a multi-span bridge is the one at which the cost of superstructure of one span about equals the cost of one substructure (pier + foundation).

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.