TRANSPORTATION ENGINEERING: Traffic engineering and design of pavements Study Notes

Traffic Engineering

Traffic engineering is that phase of engineering which deals with planning and geometric design of streets, highways, abutting lands, and with traffic operation thereon, as their use is related to the safe, convenient and economical transportation of persons and goods.

Theoretical maximum capacity, (C)

where S = Minimum clear distance between two vehicles (m).

S = 0.2V + 6 where, v = Speed of vehicle in km/hr.

C = Theoretical maximum capacity in vehicle/hour

H_t = Time headway in ‘Sec’.

Where,

C = Traffic capacity or traffic volume in vehicle/hour

δ = Traffic density in vehicle/km

V = Traffic velocity in km/hr

Passenger Car Unit

The PCU may be considered as a measure of the relative space requirement of a vehicle class compared to that of passenger car under a specified set of roadway, traffic and other conditions.

Accident Studies

The problem of accident is very acute in highway transportation due to complex flow patterns of vehicular traffic presence of mixed traffic and pedestrians. Traffic accidents may involve property damages, personal injuries or even casualties.

Where, e = Coefficient of restitution

 Velocity of separation

 Velocity of approach.

V_A = Velocity of vehicle ‘A’ of mass m_A before collision.

V_B = Velocity of vehicle ‘B’ of mass m_B before collision.

 Velocity of vehicle A after collision

 Velocity of vehicle B after collision.

e = 1 for perfectly elastic collision.

e = 0 for perfectly inelastic collision or plastic collapse.

i.e., both vehicle move with same velocity after collision.

Momentum Equation

Types of Collision

Collision of moving vehicle with parked vehicle (assumption: collision is perfectly plastic)

Where, v₁ = Initial velocity of moving vehicle in km/hr.

v₂ = Velocity of moving vehicle after travelling distance ‘s₁’ (in meter)

v₃ = Common velocity of moving & parked vehicle at the time of collision.

s₂ = Distance travelled by both vehicle till both vehicle comes in rest finally.

Two Vehicle Approaching from Right Angle Collide at an Intersection

Where,

 are skid distance just before collision.  are skid distance after collision.

 are speed of vehicle A & B respectively after the collision.

 are speed of vehicle A & B respectively after skidding a distance 

 are speed of vehicle A & B respectively before skidding.

Design Cycle Metho

Trial Cycle Method

Where, X_A = Number of vehicle accumulated in one cycle time on Road A.

X_B = Number of vehicle accumulated in one cycle time on Road B.

T = Total cycle time in ‘sec’ (assumed)

G_A = 2.5 X_A η_A = Traffic count on road A in 15 minutes.

G_B = 2.5 X_B η_B = Traffic count on road B in 15 minutes.

Where, T’ = Total cycle time (Actual)

A_A = Amber time on road A

A_B = Amber time on road B

G_A & G_B are green time on road A & B respectively.

It T’ = T then O.K. otherwise repeat the process.

Approximate Method

Whey, R_A = Red time on road A

R_B = Red time on road B

G_AP = Green time on road A for pedestrians

G_BP = Green time on road B for pedestrians

W_A = Width of road A

W_B = Width of road B

1.2 m/s = Speed of pedestrians G_A = R_B-A_A

G_B = R_A-A_B

Where, G_A = Green time on road A

G_B = Green time on road B

A_A & A_B are Amber time on road A & B respectively.

Websters Method

Where, C_O = Optical cycle time

L = Total lost time

L = 2n+R

Where, n = number of phase

R = All red time

Where, q_A = Normal flow on road A

q_B = Normal flow on road B

S_A = Saturation flow on road A

S_B = Saturation flow on road B

Where, G_A & G_B are green time on road A & B respectively.

Annual Average Daily Traffic (AADT or ADT)

Space Mean Speed (V_s)

Where, V_s = Space mean speed in km/hr.

d = Length of road in meter

n = Number of individual vehicle observations

t_i = Observed travel time (sec) for i^th vehicle of travel distance ‘d’ meter.

Time mean speed (v_t)

Where, V_t = Time mean speed (km/hr)

V_i = Observed instantaneous speed of i^th vehicles (km/hr)

n = number of vehicles observed.

Speed & delay study by floating car method

Average journey time (t) in minute

Where,

q = Flow of vehicles (volume per minutes) in one direction of the stream.

n_a = Average number of vehicles counted in the direction of the stream when the test vehicle travel in the opposite direction.

n_y = The average number of vehicles overtaking the test vehicle minus the number of vehicle overtaken when the test is in the direction of ‘q’.

t_w = Average journey time when the test vehicle is travelling with the stream q.

t_a = Average journey time, in minute when the test vehicle is running against the stream ‘q’.

Relationship between speed, travel time, volume, density & capacity

Travel time per unit length of road, 

Where, V = Speed in km/hr

q=kV_s

Where, q = Average volume of vehicle passing a point during a specified period of time (vehicle per hour).

k = Average density or number of vehicle occupying a unit length of roadway at a give instant (vehicles/km).

V_s = Space-mean speed of vehicles in a unit roadway length (km/hr)

Capacity flow or maximum flow, (q_max)

where, V_SF = Free mean speed i.e., maximum speed at zero density.

K_j = Jam density i.e., maximum density at zero speed

Where, s = Spacing between vehicles.

Rotary intersection

A rotary intersection or traffic rotary is an enlarged road intersection where all converging vehicles are forced to move round a large central island in one direction (clockwise direction) before they can weave out of traffic flow into their respective directions radiating from the central island.

Radius of rotary, (R)

Where, V = Design speed of vehicle (km/hr)

f = Coefficient of friction may be taken as 0.43 & 0.47 for the speed of 40 & 30 km/hr respectively after allowing a factor of safety of 1.5.

(R_min)_{Central Island} = 1.33 (R)_{Entry curve.}

(R_min)_{Central Island} = Minimum radius of central Island.

(R_min)_{Entry curve} = Radius of entry curve.

Where, w = Width of weaving section.

e₁ = Entry width,

e₂ = Width of non-weaving section.

Capacity of Rotary (Q_P)

Where, Q_P = Practical capacity of weaving section of a rotary in PCU per hour.

w = Width of weaving section (6 to 18 m).

e = Average width of entry ‘e₁’ & width of non-weaving section e₂ for the range e/w = 0.4 to 

L = Length of weaving section between the ends of channelizing islands in meter for the range of 

p = Proportion of weaving traffic or weaving ratio.

where, a = Left turning traffic moving along left extreme lane.

d = right turning moving along right extreme lane.

b = Crossing/weaving traffic turning towards right while entering the rotary.

c = Crossing/weaving traffic turning towards left while leaving the rotary.

Parking Facilities

Number of spaces (N)

 for parallel parking with equal spacing facing the same direction.

 for parallel parking when two cars placed closely.

 for 30° angle parking

 for 45° angle parking

 for 60° angle parking

 for 90° angle parking

Out of various angles used in angle parking, 45 degree angle is considered the best from all considerations discussed above.

Highway Lighting

Spacing between lighting units =

Trip Distribution

Where, T_ij = Number of trips from zone I to zone j.

G_i = Trips generated in zone i.

A_i = Trips attracted to zone j.

F_ij = Empirically derived ‘Friction Factor’ calculated on area wise basis.

n = Number of zones in the urban area.

Rigid pavements are those which posses note worthy flexural strength or flexural rigidity. The stresses are not transferred from grain to grain to the lower layers as in the ease of flexible pavement layers. The rigid pavements are made of Portland cement concrete-either plain, reinforced or prestressed concrete. The plain cement concrete slabs are expected to take-up about 40 kg/cm² flexural stress.

(i) Modulus of subgrade reaction (k),

where, k = Modulus of subgrade reaction (kg/cm²/cm)

p = Pressure required for ‘Δ’ deflection (kg/cm²)

Δ = Deflection(cm)

For 75 cm dia plate, Δ = 1.25 mm

(ii) Radius of Relative Stiffness (l)

where, l = Radius of relative stiffness, cm

E = Modulus of elasticity of cement concrete (kg/cm²)

μ = Poisson’s ratio for concrete = 0.15

h = Slap thickness (cm)

k = Subgrade modulus or modulus of subgrade reaction (kg/cm³)

(iii) Equivalent Radius of Resisting Section (b)

when a < 1.724 h

(b) b=a when a > 1.724h

where, a = Radius of contact area (cm)

h = Slab thickness (cm)

(iv) Glodbeck’s Formula for Stress due to Corner Load

where, S_C = Stress due to corner load (kg/cm²)

P = Corner load assumed as a concentrated point load, (kg)

h = Thickness of slab (cm).

(v) Westergards Stress Equation

(a) Stress at Interior Loading (S_i)

(b) Stress at Edge Loading (S_e)

(c) Stress at Corner Loading (S_c)

where, h = Slab thickness (cm)

P = Wheel load (kg)

a = Radius of contact area (cm)

l = Radius of relative stiffness (cm)

b = Radius at resisting section (cm).

(vi) Warping Stresses

(a) Stress at Interior Region 

where,  Warping stress at interior region (kg/cm²)

E = Modulus of elasticity of concrete, (kg/cm²)

α = Coefficient of thermal expansion (/°c)

C_X = Coefficient based on  in desired direction.

C_y = Coefficient based on  in right angle to the above direction.

μ = Poissons’s ratio ∼ 0.15

L_X & L_y are the dimensions of the slab considering along X & y directions along the length & width of slab.

(b) Stress at Edge Region 

(c) Stress Corner Region 

where, a = Radius of contact area

l = Radius of relative stiffness

(vii) Frictional Stress (S_f)

where, S_F = Frictional stress (kg/cm²)

W = Unit weight of concrete, (kg/cm³)

f = Friction constant or coefficient of subgrade restraint

L = Slab length (m)

B = Slab width (m)

(viii) Combination of Stresses

1. Critical Combination During Summer

(a) Stress for edge/interior regions at Bottom = (+ load stress) + (warping stress of day time) – Frictional stress

(b) Stress for corner region at top = (+ load stress + warping stress at night)

     2. Critical Combination During Winter

(a) Stress for edge/interior at bottom = (+ load stress + warping stress at day time + Frictional stress)

(b) Stress for corner at top = (load stress + warping stress at night)

Design of Joints in Cement Concrete Pavements

(i) The spacing of expansion joints, (L_e)

Where, δ’ = Maximum expansion in slab (cm)

L_e = Spacing of expansion joint (m)

α = Coefficient of thermal expansion of concrete (/°c)

(ii) The spacing of the contraction joint, (L_c)

(a) When reinforcement is not provided

where, L_c = Spacing of contraction joint (m)

S_C = Allowable stress in tension in cement concrete.

f = Coefficient friction ∼ 1.5

w = Unit weight of cement concrete (kg/m³).

(b) When reinforcement is provided

where, S_S = Allowable tensile stress in steel (kg/cm²)

∼ 1400kg/cm²

A_S = Total area of steel in cm².

(iii) Longitudinal Joints

where, A_S = Area of steel required per meter length of joint (cm²)

b = Distance between the joint & nearest free edge (m)

h = Thickness of the pavement (cm)

f = Coeff. of friction ∼ 1.5

w = Unit wt. of concrete (kg/cm³)

S_s = Allowable working stress in tension for steel (kg/cm²)

where, L_t = Length of tie bar

S_S = Alloable stress in tension (kg/cm²) ∼ 1400

S_b = Allowable bond stress in concrete (kg/cm²)

= 24.6 kg/cm² for deformed bars

= 17.5 kg/cm² for plain tie bars

d = diameter of tie bar (cm).

IRC recommendations for design of cement concrete pavements

A_d = P’[1+r]⁽ⁿ⁺²⁰⁾

where, A_d = Number of commercial vehicles per day (laden weight > 3 tonnes)

P’ = Number of commercial vehicles per day at last count.

r = Annual rate of increase in traffic intensity.

n = Number of years between the last traffic count & the commissioning of new cement concrete pavement.

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Flexible Pavements

Flexible pavements are those, which on the whole have low or negligible flexural strength and are rather flexible in their structural action under the loads.

A typical flexible pavement consists of four components: 1. soil subgrade 2. sub-base course 3. base course 4. surface course.

(i) Stress Under Road Surface as per Boussineq’s Equation,

where,

σ_z = vertical stress at depth z.

q = surface pressure.

z = depth at which σ_z is computed.

a = radius loaded area.

(ii) As per IRC

Maximum legal axle load = 8170 kg

Equivalent single wheel load = 4085 kg.

(v) Equivalent Single Wheel Load (ESWL) 

Methods of Flexible Pavement Design

(i) Group Index Method

G.I = 0.2a + 0.005ac + 0.01bd

(ii) C.B.R Method

(a) 

(b) The thickness of Pavement, (T)

where, P = Wheel load in kg.

CBR = California bearing ratio in percent

p = Tyre pressure in kg/cm²

A = Area of contact in cm².

A=πa²

a = Radius of contact area.

(c) Number of a heavy vehicle per day for design (A),

A=P[1+r]⁽ⁿ⁺¹⁰⁾

where, A = No. of vehicles at the end of design period.

P = Number of heavy vehicles per day at least count.

r = Annual rate of increase of heavy vehicles

n = Number of years between the last count & the year of completion of construction.

(d) CBR Method of pavement design by cumulative standard axle load,

where,

N_s = Cumulative number of standard axle load

A’ = Number of the commercial vehicle per day when construction is completed considering the number of lanes.

n = Design life of the pavement, taken as 10 to 15 years.

F = Vehicle damage factor.

D = Lane distribution factor

(iii) California Resistance Value Method

where, T = Total thickness of pavement, (cm)

k = Numerical constant = 0.166

T.I = Traffic Index

T.I = 1.35(EWL)^0.11

R = Stabilometer resistance value

C = Choesiometer value.

where, T₁ & T₂ are the thickness values of any two pavement layers & C₁ & C₂ are their corresponding Cohesiometer values.

(iv) Triaxial Method

(a) Thickness of pavement required for single layer, (T_S)

where, T_S = Thickness in cm

P = Wind load in kg

X = Traffic coefficient

Y = Rainfall coefficient

E_S = Modulus of elasticity of subgrade soil (kg/cm²)

a = Radius of contact area (cm)

Δ = Design deflection (0.25 cm)

(b) Thickness of Pavement Consist of Two layer system,

where, E_P = Modulus of elasticity of pavement material

(v) MC Load Method

where, T = Required thickness of gravel base (cm)

P = Gross wheel load, (kg)

k = Base course constant.

(vi) Burmister Method (Layered System)

Displacement equations given by Burmister are,

where,  are Poisons ratio for soil subgrade & pavement.

For single layer, F₂ = 1

P = Yielded pressure

E_S = Subgrade modulus

a = Radius of loaded area

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Group Index of Soils (G.I)

In order to classify the fine-grained soils within one group and for judging their suitability as subgrade material, an indexing system has been introduced in HRB classification which is termed as group index. Soils are thus assigned arbitrary numerical numbers known as group index (GI). Group index is function of percentage and is given by the equation G.I = 0.2a + 0.005ac + 0.01bd

Where,

Where, W_L = Liquid limit, I_P = Plasticity Index.

P = Percentage fines passing from 0.074 mm sieve.

0≤G.I≤20 Lower the group → best quality.

Plate Bearing Test

(i) 

Here,

k = Modulus of subgrade reaction

P = Pressure Corresponding to settlement of 0.125 cm.

(ii)

Where,

k_s = Modulus of subgrade reaction for soaked condition.

P_s = Pressure required in the soaked condition to produce same deformation as deformation Produce by pressure ‘P’ in consolidated condition.

k = Modulus of subgrade reaction for the consolidated stage.

Δ = Deformation in ‘cm’.

a = Radius of rigid plate in ‘cm’.

E = Modulus of elasticity of soil subgrade in kg/cm².

k.a = constant

a₁ = Radius of smaller plate (other plate)

k₁ = Modulus of subgrade reaction of other plate of radius ‘a₁’ cm.

California Bearing Ratio (CBR) Test

Test for Road Aggregate

(i) Aggregate crushing test

Aggregate crushing value 

Where, w₁ = Weight of the test sample in ‘gm’

w₂ = weight of the crushed material in ‘gm’ passed through 2.36 mm sieve.

(ii) Shape Tests

Where, Ga = specific gravity of aggregate

W = mass of mould containing aggregate

C = mass of mould containing water

(iii) Abrasion Test

(a) Los Angeles Abrasion Test

Bituminous Material

1. Product of fractional distillation of Petroleum: Gasoline, Naptha, Kerosene, Lubricating oil and Residue – Petroleum Bitumen.
2. Cutback Bitumen: Reduced Viscosity Bitumen

N – Numeral [0, 1, 2, 3, 4, 5]

Show progressive thickening from 0 to 5

1. Specific Gravity:

Bituminous → 0.97 – 1.02

Tar → 1.1 – 1.5

Bituminous Mixes

(i) Determination of Specific Gravity

Where, G_a = Average specific gravity of blended aggregate mix.

w₁, w₂, w₃, w₄ are % by weight of aggregate 1, 2, 3 & 4 respectively. G₁, G₂, G₃ & G₄ are specific gravities of the aggregate 1, 2, 3 & 4 respectively.

(ii) Specific Gravity of Compacted Specimen

(a) 

Where, G_t = Theoretical maximum specific gravity of the mix.

W_b = % by weight of bitumen.

G_b = Specific gravity of bitumen.

G_a = Average specific gravity of aggregates.

(b) Theoretical density γ_t, per-cent solids by volume

Where, G = Actual specific gravity of test specimen

G_t = Theoretical maximum specific gravity.

(c) Voids in the Mineral Aggregate (VMA) 

Where, V_b = % of bitumen

W_a = Aggregate content percent by weight

V_v = % air voids in the specimen.

(d) % Voids Filled with Bitumen (VFB)

Marshall Method Bituminous Mix Design

• Percentage Air Voids 

Where, G_m = Bulk density or mass density of the specimen

G_t = Theoretical specific gravity of mixture

Where,

W₁ = Percent by weight of coarse aggregate in total mix

W₂ = Percent by weight of fine aggregate in total mix

W₃ = Percent by weight of filler in total mix

W₄ = Percent by weight of bitumen in total mix

G₁ = Apparent specific gravity of coarse aggregate

G₂ = Apparent specific gravity of fine aggregate

G₃ = Apparent specific gravity of filter

G₄ = Specific gravity of bitumen

Percent Voids in Mineral Aggregate (VMA)

VMA = V_v + V_b

Here, V_v = Volume of air voids, %

V_b = Volume of bitumen, 

Per-cent Voids Filled with Bitumen (VFB)

HveeM Method of Bituminous mix Design Stabilimeter Value, (s)

Where, P_v = Vertical pressure at 28 kg/cm² or at a total load of 2268 kg.

P_h = Horizontal pressure corresponding to P_v = 28 kg/cm².

D₂ = Displacement on specimen represented as number of turns of pump handle to raise P_h from 0.35 to 7 kg/cm².

Cohesiometer Value, (c)

Where, L = Weight of shots in cm.

w = Diameter of width of specimen in cm

H = Height of specimen in cm

Stabilometer Resistance R-value

where, P_v = Vertical pressure applied (11.2 kg/cm²)

P_h = Horizontal pressure transmitted at P_v = 11.2 kg/cm².

D₂ = Displacement of stabilometer fluid necessary to increase the horizontal pressure from 0.35 to 7kg/cm², measured in number of revolutions of the calibrated pump handle. 

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