Stresses in Pavements

Introduction to Vehicle Pavement Interaction (VPI)

Vehicle pavement interaction (VPI) is a system in which mutual forces develop between the

vehicle tyre and the pavement surface. The VPI describes all the components related to vehicle

and pavement and their effect on each other including their influence on expected output. The

basic elements involved with their associated characterises as inputs to the VPI may be broadly

classified as three types, which are (a) pavement profile or roughness, (b) vehicle

characteristics including its operating conditions and (c) pavement layered structures with

their stiffness values. Applications of dynamic loads through tyre rolling cause transient

pavement responses that differ from the static type of loading effects. The incorporation of

this transient nature of loading effect into the design of pavements and vehicles is the most

important issue associated with VPI.

2.1 Components of Vehicle Pavement Interaction (VPI)

There are different types of models based on different parametric components and their

defined interface conditions. These models are analysed to determine the resulting pavement

stresses, strains and deflections.

The details of the main components with their representing interface conditions considered

for modelling VPI may be categorised as given below (Collop and Cebon, 1995).

(a) Longitudinal profile of pavement: - The surface profile characteristics in terms of unevenness

(BI), roughness (IRI or PI), texture and surface friction (SN) are considered to be the main causes

for dynamic tyre loads

(b) The vehicle characteristics: - The Vehicle/ tyre loading history (i.e., time versus loading

intensity) under the influence of pavement surface is modelled with vehicular characteristics

such as suspension system, axle loads with configuration, tyre contact area, speed, vehicle

dimension and acceleration/deceleration.

(c) Stiffness or resilience characteristics of pavement layered structure: - The resulting response

(i.e., stress/strain/deflection) due to the above input conditions of loading is analysed with

defined boundary conditions of pavement layered structure. With reference to the calculated

and measured responses in terms of stresses or strains or deflections (or also in combination)

are evaluated to evolve failure criteria of pavement and such analysis results are ultimately

used for design of mixes and thickness.

(d) Criteria considered for evaluation of VPI The type of analysis to be carried out and the

decision criteria depend on the type of analysis method used such as linear, elastic or a

combination of methods such as non-linear elastic or finite element methods. Based on these

results, a suitable type of pavement/structure will be designed with appropriate economic

models (i.e., life cycle cost analysis-LCCA), as final decision for construction.

The nature of VPI largely depends on special relationship between vehicle tyre dynamic loads,

pavement profile characteristics, pavement structural strength variation and the response of

vehicle suspension system (Divine, 1997). The following methods are widely used to consider

the above components (Styen et al., 2012)

Stresses in Flexible Pavements

3.1 Engineering Properties of Bituminous Materials

The behaviour of bitumen is determined with reference to anticipated climatic conditions and

intensity of traffic loads. These input parameters play a vital role during selection of suitable

binders for construction of pavements. Thus, engineering properties of bitumen should be

determined considering the following factors.

• Temperature (maximum, minimum and average values)

• Loading time

• Intensity of load Bitumen behaves differently in different combinations of the above values.

For example, bitumen behaves elastic under short duration of loading at normal temperature.

But, in case of prolonged loading and as the temperature increases, it loses elasticity and flows

like a fluid. This is the reason why bituminous pavements at higher temperature tend to deform

without cracking.

3.1.1 Stiffness Modulus of Bitumen

In 1954, van der Pod developed the concept of a stiffness modulus for thermoplastic polymers

which was useful for determining of the engineering properties of bitumen. The concept NOS

modified by Heukelom and Klomp (1964) and Heukelom (1973). In this method, the bitumen

is characterised by its stiffness modulus, (ail), the ratio between applied stress to the observed

strain at a given time of loading (t in s) and temperature (T in °C). The value of the stiffness

modulus of bitumen may be expressed as follows.

Ebit (t,T)= stress/ strain at (t,T)……………….3.1

The above equation holds good and is suitable for in service pavements if the assumed strain

is not more than 1 per cent. If the loading time changes, preferably under variable

temperatures similar to seasonal variations, the value of stiffness of bitumen also changes. To

account for this variability of the stiffness modulus of bitumen, a stiffness nomograph was

developed by van der Poel and modified by Heukelom based on three parameters, viz. time of

loading (t), temperature difference (T-T800 pen) and penetration index (PI). The equation

developed by Heukelom and Klomp (1964) or a method prescribed by Bonnaure et al. (1977)

may be used to determine the stiffness values of bituminous mixes. Measured stiffness values

greatly agree with the values predicted by Bonnaure's method and it is adopted by the Shell

method (1977) of the asphalt pavement design (Ullidtz and Peattie 1980). To understand the

implication of the true engineering properties of bitumen at different ranges of temperatures

from cold to hot under cyclic loading similar to field conditions, it is necessary to know its visco-

elastic properties.

3.1.2 Visco-elastic Properties of Bituminous Materials

Bitumen becomes hard and soft by cooling and heating respectively. These states of bitumen

may be reversed as and when required. This property of bitumen is known as thermos-

plasticity. Based on temperature and rate of loading, the bitumen may behave like a dual

characteristic material. Elastic material at a given temperature, the material undergoes

immediate deformation or strain under loading and immediately recovers upon unloading. In

other words, if the material is linear elastic, the stress is proportional to the strain according to

Hooke's law, i.e., stress = Young's modulus value x strain (Wilhelm Flugge 1975). Viscous

material At a given temperature, the material undergoes delayed deformation or strain under

loading but it may or may not recover its initial conditions. A Newtonian material will recover

completely. But in Bingham materials, power law fluids and pseudo-plastic materials, complete

recovery of strain will not take place due to plastic straining during loading. In other words, if

the material is viscous, the stress is proportional to the strain rate (Wilhelm Flugge 1975).

Bitumen exhibits intermediate characteristics ranging from elastic to viscous and vice versa

due to change in temperature and rate of application of load. Because of this, bituminous mixes

are known as visco-elastic materials. The behaviour of visco-elastic material depends on

temperature, time and frequency of loading.

Figure 3.1 Phase difference between stress and strain (Yang 2004; Relent 2001)

When a visco-elastic material is subjected to sinusoidal loading (Figure 3.1), the occurrence of

strain (Co) is delayed due to the phase difference of the applied stress (0-0) at a given time (t).

This difference of response is known as phase angle (y). As the material transforms from being

elastic to visco-elastic during change of temperature, the value of the phase angle (9) changes

from 0 °C to 90 °C. The energy dissipated by the material is determined by using the phase

angle value. The values of complex modulus and dynamic complex modulus of bitumen are

determined from the following popularly known relationships (Huang 2004; Di Benedetto et

al. 2001 and Kailas 1970). Complex modulus (E*) = Real part of modulus value + Imaginary part

of modulus value

……………….3.2 𝐸∗(𝑓, 𝑡) = (𝜎𝑜/𝜀𝑜) sin (𝜔𝑡) +i. (𝜎𝑜/𝜀𝑜) sin(𝜔𝑡 − ∅)

Dynamic modulus = Absolute value of complex modulus = [E]*. ……………… (3.3)

If the viscous effect is ignored, the ratio between the maximum values of stress to strain is

termed as the elastic modulus. The values of complex bulk modulus (K*) and shear modulus

(G*) may be determined from the following popularly known relationships.

K* = E*/ (3 — 6µ) ……….. (3.4) G* = E/ [2(1+ µ)] …………… (3.5)

Where, µ = Poisson's ratio

To characterise visco-elastic properties of binders at lower temperatures, another parameter

called the creep compliance is used. The creep compliance may be calculated using the

following relationship.

D (t, T) = 1 / E bit (t, T)…………. (3.6)

Shear compliance is also used to characterise the visco-elasticity of bituminous materials. The

following relationship is used to calculate shear compliance (J (t, T)).

J (t, T) = 3/ E bit (t, T) (3.7) …………….3.7

The above equations holds good if the material is assumed to be linear visco-elastic and

isotropic. The values of dynamic modulus, bulk modulus, shear modulus, creep compliance and

shear compliance provide important information to explain the visco-elastic behaviour of the

bituminous mix over a range of temperature and frequency of loading.

STRESS ANALYSIS OF FLEXIBLE PAVEMENTS

3.2.1 BOUSSINESQ'S THEORY (SINGLE LAYER THEORY)

In 1885, Boussinesq presented a theory for calculating the stresses in soil mass based on the

following assumptions.

1. The soil is homogenous (without any stratification) and isotropically linear elastic which

means that it abides by Hook's law.

2. The soil mass boundary is a semi-infinite elastic half-space.

3. The load is applied on to a level surface.

4. The soil mass is weightless.

The following equations are based on Boussinesq's theory and are widely used to calculate

stresses in soil mass.

CASE 1: Point load

Figure 3.2 Stress due to point load

Consider a point load on the soil mass as depicted in Figure 3.2.

Vertical stress due to point load 𝜎𝑧 =3𝑃

2𝜋𝑧2

1

[+(𝑟𝑧⁄ )2]5/2………….3.8

CASE 2:

Uniformly distributed load over a circular area generally, for pavement analysis, the equivalent

circular contact area of a tyre on pavement surface is taken.

For this purpose, a uniformly loaded circular area is considered for calculating the stresses in

the soil mass. To calculate stresses under a uniformly loaded area using flexible plates (similar

to a rubber tyre), equation (3.8) may be integrated over the circular area (Figure 3.3).

3

Figure 3.3 Stress under a uniformly loaded circular area

∫ ∫ (3𝑃((𝑟𝑑𝜃)𝑑𝑟)

2𝜋𝑧2

𝑎

0[

2𝜋

0

1

1+(𝑟/𝑧)2]…………1

The above equation can be simplified as

Vertical stress under the centre of the circular area (𝜎𝑧)= 𝑝(1 − [1

1+(𝑟/𝑧)2]3/2…. 2

Horizontal or radial stress beneath the centre of circular area,(𝜎𝑟)

(𝜎𝑟)= 0.5𝑃[(1 + 2𝜇) − (2𝑧(1+𝜇)

√(𝑎2+𝑧2)+ (

𝑧3

(𝑎2+𝑧2)32

)]………… 3

Elastic vertical displacement at z=0, under a circular flexible plate ∆𝑠=2𝑃𝑎(1−𝜇2)

𝐸………… 3

If 𝜇 = 0.5 the above equation can be written as ∆𝑠=1.5𝑃𝑎

𝐸 …….. 4

Where, 𝑃= uniformly distributed pressure

𝑎= radius of circularly loaded area

𝐸= Elastic modulus of soil

𝜇= Poisson’s ratio of soil

The stress distribution under the flexible plate is uniform since the flexible plate bends (or

flexes) under loading, similar to a rubber sheet. Therefore; the elastic surface deflection under

the flexible plate is non-uniform. If a circular rigid plate (or steel plate) is used to apply load on

to the soil, the elastic surface displacement is uniform but the stress' distribution is non-

uniform (Ullidtz 1987). Analysis of plate load test data is one of the best examples of an

application of Boussinesq's theory. The following formula may be used to calculate the elastic

surface deflection under a rigid plate.

Vertical displacement at z=0, under a circular rigid plate (∆𝑠) =𝜋𝑃𝑎(1−𝜇2)

𝐸….. 5

If 𝜇 = 0.5 the above equation can be written as ∆𝑠=1.18𝑃𝑎

𝐸…… 6

Q. A semi-infinite soil mass is subjected to stress under a circular plate having a 15 cm radius.

The load intensity over the plate is 4000 kg. Calculate the vertical stress in the soil under the

axis of the circular plate at 2 m depth.

Solution: - Intensity of pressure over plate = 4000/ (n x 152) = 5.66 kg/cm2.

Use equation 2

(𝜎𝑧)= 5.66(1 − [1

1+(15/2)2]3/2

= 5.64kg/cm2

Q. Calculate the rebound surface deflection on single layer pavement under a wheel load of

40KN with a tyre pressure of 0.8 Mpa. The effective elastic modulus of sub-grade may be taken

as 40MPa and Poisson’s ratio of the soil as 0.5

Solution 𝑡𝑦𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑊𝐻𝐸𝐸𝐿 𝐿𝑂𝐴𝐷

𝑇𝑌𝑅𝐸 𝐶𝑂𝑁𝑇𝐴𝐶𝑇 𝐴𝑅𝐸𝐴

𝑹𝑨𝑫𝑰𝑼𝑺 𝑶𝑭 𝑻𝒀𝑹𝑬 𝑪𝑶𝑵𝑻𝑨𝑪𝑻 𝑨𝑹𝑬𝑨 = 𝒂 √𝑾𝑯𝑬𝑬𝑳 𝑳𝑶𝑨𝑫

𝝅𝑿 𝑻𝒀𝑹𝑬 𝑷𝑹𝑬𝑺𝑺𝑼𝑹𝑬

𝒂 √𝟒𝟎𝑿𝟏𝟎𝟑

𝝅𝑿 𝟎.𝟖𝑿𝟏𝟎𝟔= 0.126m = 12.6 cm

Intensity of pressure =𝒑 =𝒍𝒐𝒂𝒅

𝒂𝒓𝒆𝒂=

𝟒𝟎𝒙𝟏𝟎𝟑

𝝅𝒙𝟎.𝟏𝟐𝟔𝟐= 80990.14N/m2

If a flexible pavement is considered, rebound surface deflection= ∆𝑠=1.5𝑃𝑎

𝐸

∆𝑠=1.580990.14𝑥0.126

40𝑥106= 0.00378m= 3.78mm

TWO LAYER THEORY

3.2.2 Burmister's Theory

In 1943, Donald M. Burmister developed a method of analysing a two-layered soil system which

resembled a flexible pavement having its top layer stiffer than its bottom layer. Later, ' in 1945,

he extended the method to include three-layered system. As an important outcome of

Burmister's numerical solutions, the flexible pavement thickness design method was

developed for airfield pavements (ASCE vol.115, 1950).

In 1962, Ahlvin and Ulery of the US Army Corps of Engineers (USACE) reported solutions for

calculating the stresses, strains and deflections on a homogenous half-space withstanding a

uniform circular load (Ahlvin and Ulery) 1962). Since then, the Burmister's theory with required

modifications is being used for various pavement engineering applications.

Many researchers have developed different methods for analysing multi-layered systems

based on elastic, non-linear, visco-elastic and finite element methods. The solution of various

problems related to a multi-layered system is made easy by the use of computers. Burmister's

two-layered system of analysis is briefly explained below.

Assumptions

• Every layer material is homogenous, isotropic and ideally elastic. They are characterised by a

unique elastic modulus and Poisson's ratio.

• A pavement consists of two layers. The elastic modulus value" of the top layer is more than

the bottom layer (i.e., El > E2).

• The layers are weightless and infinite in the horizontal direction.

• The top layer has finite thickness (h) but the bottom layer is infinitely thick.

• The top layer is in full contact with the bottom layer.

• The interface between these layers is rough and there is no loss of transfer of applied stress

from the top layer to the bottom layer.

• There will be no shear and normal stresses outside the loaded area.

• The applied stress is uniform over a circular area. The displacement. equations given by

Burmister for a two-layered system are as follows (Figure 3.4).

Surface deflection under the centre of the flexible plate= ∆=1.5𝑝𝑎

𝐸2x𝐹d

Surface deflection under the centre of the flexible plate∆=1.18𝑝𝑎

𝐸2x𝐹d

where, ∆ = Vertical deflection at surface (when z = 0)

• P = Uniform pressure on circular plate

a = Radius of the circular plate

E2 = Elastic modulus of sub-grade

Fd = Displacement factor, depends on ratios of E1/E2 and h/a Refer to Figure 3.5

E1 = Elastic modulus of the top layer

h = Thickness of the top layer

Fig- 3.5 Burmister’s two layered system

By introducing additional pavement layers over the sub-grade; the vertical stress induced on

the sub-grade, exactly under the centre of the loading plate, can be reduced from 68% to 30%.

To make this possible, the radius of the additional layer has to be equal to the radius of the

loading plate, the elastic modulus value of the additional layer has to be approximately equal

to ten times the elastic modulus value of the sub-grade (E1 ≈ 10E2) and Poisson's ratio of both

the layers has to be equal to 0.5. Due to this reason, construction of pavements is carried out

in multiple layers instead of a single thicker layer.

𝑸. 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒕𝒉𝒆 𝒓𝒆𝒒𝒖𝒊𝒓𝒆𝒅 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 𝒐𝒇 𝒂𝒏 𝒂𝒊𝒓𝒇𝒊𝒆𝒍𝒅 𝒇𝒍𝒆𝒙𝒊𝒃𝒍𝒆 𝒑𝒂𝒗𝒆𝒎𝒆𝒏𝒕

𝒃𝒂𝒔𝒆𝒅 𝒐𝒏 𝑩𝒖𝒓𝒎𝒊𝒔𝒕𝒆𝒓′𝒔 𝒕𝒉𝒆𝒐𝒓𝒚 𝒖𝒔𝒊𝒏𝒈 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝒑𝒍𝒂𝒕𝒆 𝒍𝒐𝒂𝒅 𝒕𝒆𝒔𝒕 (𝑷. 𝑳𝟕)𝒅𝒂𝒕𝒂

𝒂𝒏𝒅 𝒐𝒕𝒉𝒆𝒓 𝒊𝒏𝒑𝒖𝒕 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔:

𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 𝒐𝒇 𝒑𝒍𝒂𝒕𝒆 𝒖𝒔𝒆𝒅 − 𝟕𝟓𝒄𝒎

• 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒂𝒕 𝟏. 𝟐𝟓 𝒎𝒎 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒘𝒉𝒆𝒏 𝒕𝒉𝒆 𝒑𝒍𝒂𝒕𝒆 𝒍𝒐𝒂𝒅 𝒕𝒆𝒔𝒕 𝒊𝒔 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒆𝒅

𝒐𝒏 𝒕𝒉𝒆 𝒔𝒖𝒃𝒈𝒓𝒂𝒅𝒆 𝟎. 𝟖𝟐𝒌𝒈/𝒄𝒎𝟐. 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒂𝒕 𝟏. 𝟐𝟓 𝒎𝒎 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒘𝒉𝒆𝒏

𝒕𝒉𝒆 𝒑𝒍𝒂𝒕𝒆 𝒍𝒐𝒂𝒅 𝒕𝒆𝒔𝒕 𝒊𝒔 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒆𝒅 𝒐𝒏 𝒂 𝒃𝒂𝒔𝒆 𝒄𝒐𝒖𝒓𝒔𝒆 𝒐𝒇 𝟏𝟔𝟎 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔

= 𝟐. 𝟏 𝒌𝒈/𝒄𝒎𝟐. 𝑫𝒆𝒔𝒊𝒈𝒏 𝒘𝒉𝒆𝒆𝒍 𝒍𝒐𝒂𝒅 = 𝟐𝟑𝟎𝟎𝟎 𝒌𝒈; 𝑻𝒊𝒓𝒆 𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆

= 𝟏𝟓 𝒌𝒈/𝒄𝒎𝟐; (𝒂) 𝑰𝒇 𝒂𝒍𝒍𝒐𝒘𝒂𝒃𝒍𝒆 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏

= 𝟎. 𝟏𝟐𝟓 𝒄𝒎 𝒂𝒏𝒅 (𝒃). 𝑰𝒇 𝒂𝒍𝒍𝒐𝒘𝒂𝒃𝒍𝒆 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏. = 𝟎. 𝟓𝟎 𝒄𝒎

Solution:- (𝒊)𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒆𝒍𝒂𝒔𝒕𝒊𝒄 𝒎𝒐𝒅𝒖𝒍𝒖𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒖𝒃

− 𝒈𝒓𝒂𝒅𝒆 𝒇𝒓𝒐𝒎 𝒂 𝒑𝒍𝒂𝒕𝒆 𝒍𝒐𝒂𝒅 𝒕𝒆𝒔𝒕 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒆𝒅. 𝒐𝒏 𝒕𝒉𝒆 𝒔𝒖𝒃 − 𝒈𝒓𝒂𝒅𝒆

𝑭𝒓𝒐𝒎 𝒕𝒉𝒆 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏𝒔𝒉𝒊𝒑, 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 ∆=1.18𝑝𝑎

𝐸2

𝒘𝒉𝒆𝒓𝒆, 𝑬𝟐 𝒊𝒔 𝒕𝒉𝒆 𝒆𝒍𝒂𝒔𝒕𝒊𝒄 𝒎𝒐𝒅𝒖𝒍𝒖𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒖𝒃 − 𝒈𝒓𝒂𝒅𝒆 𝒂𝒏𝒅

𝒂 =𝒅𝒊𝒂𝒎𝒆𝒕𝒆𝒓 𝒐𝒇 𝒑𝒍𝒂𝒕𝒆

𝟐=

𝟕𝟓

𝟐= 𝟑𝟕. 𝟓𝒄𝒎

𝑶𝒏 𝒔𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒊𝒐𝒏 𝒐𝒇 𝒗𝒂𝒍𝒖𝒆𝒔

𝟎. 𝟏𝟐𝟓 𝒄𝒎 = 𝟏. 𝟏𝟖𝑿

𝟎. 𝟖𝟐𝒌𝒈𝒄𝒎𝟐 𝑿𝟑𝟕. 𝟓𝒄𝒎)

𝑬𝟐

𝑬𝟐 = 𝟐𝟗𝟎. 𝟐𝟖𝟏𝒈/𝒄𝒎𝟐

(ii) Determination of elastic modulus of granular layer from the results of a plate load test conducted

on a granular layer

Use this equation for deflection ∆=

1.18𝑝𝑎

𝐸2x𝐹𝑑

𝐹𝑑 𝑖𝑠 𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜𝑠 𝑜𝑓 ℎ

𝑎 𝑎𝑛𝑑

𝐸1

𝐸1 𝑎𝑛𝑑 𝐸2 𝑖𝑠 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏𝑔𝑟𝑎𝑑𝑒

𝐹𝑑 = 0.39 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 ℎ

𝑎=

16𝑐𝑚

37.5 𝑐𝑚= 0.426

𝑁𝑜𝑤 𝑟𝑒𝑎𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸1

𝐸2 𝑟𝑎𝑡𝑖𝑜 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒 3.5 ∴

𝐸1

𝐸1 = 200 ∴ 𝐸1 = 200𝐸2; 𝐸1 = 200𝑥290.28 = 58056𝑘𝑔/𝑐𝑚2

(III) DESIGN OF FLEXIBLE PAVEMENT:-

𝑡𝑦𝑟𝑒 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑎𝑟𝑒𝑎 𝑜𝑛 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 = 𝜋𝑋𝑎2 𝑤ℎ𝑒𝑟𝑒 𝑎 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡𝑦𝑟𝑒 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑜𝑛 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡

𝜋𝑋𝑎2 =𝑊𝐻𝐸𝐸𝐿 𝐿𝑂𝐴𝐷

𝑇𝑌𝑅𝐸 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒=

23000𝑘𝑔

15𝑘𝑔/𝑐𝑚2 ∴ 𝑎 = 22.09𝑐𝑚

𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑎 𝑓𝑙𝑒𝑥𝑖𝑏𝑙𝑒 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒 𝑖𝑠 ∆𝑎= 1.5𝑝. 𝑎

𝐸2𝐹𝑑

(a)if allowable deflection = 0.125cm therfore on substituting the values, we get

0.125 = 1.515𝑘𝑔/𝑐𝑚2𝑋22.09𝑐𝑚

290.28𝑘𝑔/𝑐𝑚2𝐹𝑑

Therefore 𝐹𝑑 = 0.073

𝑛𝑜𝑤 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 ℎ

𝑎𝑏𝑎𝑠𝑒𝑑 𝑜𝑛

𝐸1

𝐸2= 200

ℎ

𝑎= 2.79, ⇒ ℎ = 2.79𝑋𝑎 = 2.79𝑋22.09 = 61.63𝑐𝑚

Therefore for the base course provide a thickness of 62cm

(b)if allowable deflection = 0.50cm

0.50 = 1.515𝑘𝑔/𝑐𝑚2𝑋22.09𝑐𝑚

290.28𝑘𝑔/𝑐𝑚2 𝐹𝑑 𝐹𝑑 = 0.438

𝑛𝑜𝑤 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 ℎ

𝑎𝑏𝑎𝑠𝑒𝑑 𝑜𝑛

𝐸1

𝐸2= 200

ℎ

𝑎= 0.36, ⇒ ℎ = 0.36𝑋𝑎 = 0.36𝑋22.09 = 7.95𝑐𝑚

Therefore for the base course provide a thickness of 8cm

3.2.3 Linear Elastic Multi-Layered Pavement System

𝐼𝑛 𝑟𝑒𝑎𝑙𝑖𝑡𝑦, 𝑎 𝑓𝑙𝑒𝑥𝑖𝑏𝑙𝑒 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑙𝑎𝑦𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑒𝑎𝑐ℎ 𝑙𝑎𝑦𝑒𝑟 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑒𝑑 𝑤𝑖𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 (ℎ), 𝑒𝑎𝑐ℎ ℎ𝑎𝑣𝑖𝑛𝑔 𝑎 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠 𝑠𝑢𝑐ℎ 𝑎𝑠 𝑡ℎ𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 (𝐸)𝑎𝑛𝑑 𝑃𝑜𝑖𝑠𝑠𝑜𝑛′𝑠 𝑟𝑎𝑡𝑖𝑜 (µ). 𝐴 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑓𝑖𝑔𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑒𝑑 𝑡𝑜 𝑎 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑝) 𝑝𝑙𝑎𝑐𝑒𝑑 𝑜𝑛 𝑎 𝑚𝑢𝑙𝑡𝑖− 𝑙𝑎𝑦𝑒𝑟𝑒𝑑 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚 𝑖𝑠 𝑑𝑒𝑝𝑖𝑐𝑡𝑒𝑑 𝑖𝑛 𝐹𝑖𝑔𝑢𝑟𝑒 3.9. 𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑙𝑦 𝑚𝑎𝑑𝑒 𝑓𝑜𝑟 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑜𝑓 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠, 𝑠𝑡𝑟𝑎𝑖𝑛𝑠 𝑎𝑛𝑑 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑎 𝑚𝑢𝑙𝑡𝑖 − 𝑙𝑎𝑦𝑒𝑟𝑒𝑑 𝑓𝑙𝑒𝑥𝑖𝑏𝑙𝑒 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚. • 𝑇ℎ𝑒 𝑏𝑜𝑡𝑡𝑜𝑚 (𝑛𝑡ℎ) 𝑙𝑎𝑣𝑒𝑟 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑢𝑏 − 𝑔𝑟𝑎𝑑𝑒 𝑎𝑛𝑑 ℎ𝑎𝑠 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠. • 𝐴𝑏𝑜𝑣𝑒 𝑡ℎ𝑒 𝑠𝑢𝑏𝑔𝑟𝑎𝑑𝑒, 𝑒𝑎𝑐ℎ 𝑙𝑎𝑣𝑒𝑟 ℎ𝑎𝑠 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑎𝑛𝑑 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛.

• 𝑇ℎ𝑒 𝑠𝑒𝑙𝑓/𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏 − 𝑔𝑟𝑎𝑑𝑒 𝑎𝑛𝑑 𝑜𝑡ℎ𝑒𝑟 𝑙𝑎𝑦𝑒𝑟𝑠 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑 𝑎𝑠 𝑤𝑒𝑖𝑔ℎ𝑡𝑙𝑒𝑠𝑠 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑙𝑜𝑠𝑠 𝑜𝑓 𝑠𝑡𝑟𝑒𝑠𝑠 𝑎𝑡 𝑒𝑎𝑐ℎ 𝑙𝑎𝑦𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒. 𝑇ℎ𝑒 𝑙𝑎𝑦𝑒𝑟𝑠 𝑎𝑟𝑒 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑙𝑦 𝑖𝑛 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑎𝑛𝑑. 𝑏𝑜𝑡𝑡𝑜𝑚 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠. 𝐵𝑢𝑡 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑚𝑒𝑡ℎ𝑜𝑑𝑠, 𝑎 𝑠𝑚𝑜𝑜𝑡ℎ 𝑙𝑎𝑦𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒 𝑚𝑎𝑦 𝑏𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑. • 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑟 𝑟𝑎𝑑𝑖𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑙𝑎𝑦𝑒𝑟. • 𝑇ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑙𝑎𝑦𝑒𝑟 𝑖𝑠 𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑙𝑦 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑛𝑑 ℎ𝑎𝑠 𝑛𝑜 𝑢𝑛𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝑠. • 𝑂𝑛𝑙𝑦 𝑠𝑡𝑎𝑡𝑖𝑐 𝑙𝑜𝑎𝑑𝑠 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑. • 𝐸𝑎𝑐ℎ 𝑙𝑎𝑦𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑠𝑦𝑠𝑡𝑒𝑚 𝑖𝑠 𝑙𝑖𝑛𝑒𝑎𝑟𝑙𝑦 − 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑜𝑢𝑠 𝑎𝑛𝑑 𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑖𝑐. • 𝑇ℎ𝑒𝑟𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑛𝑜 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠, 𝑠𝑡𝑟𝑎𝑖𝑛𝑠 𝑎𝑛𝑑 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑡 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑑𝑒𝑝𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏 − 𝑔𝑟𝑎𝑑𝑒.

Figure 3.9.- Multi-layered pavement system

Φ(r, z) = [{𝐴 + 𝐵𝑧}]𝑒−𝑚𝑧 + (𝐶 + 𝐷𝑧)𝑒𝑚𝑧}]𝐽𝑂(𝑚𝑟) 𝑊ℎ𝑒𝑟𝑒, 𝑟 = 𝑟𝑎𝑑𝑖𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑍 = 𝑑𝑒𝑝𝑡ℎ 𝐴, 𝐵, 𝐶, 𝐷 = 𝐸𝑎𝑐ℎ 𝑙𝑎𝑦𝑒𝑟 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝐽𝑂 = 𝐵𝑒𝑠𝑠𝑒𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑘𝑖𝑛𝑑 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 𝑧𝑒𝑟𝑜

𝑊𝑖𝑡ℎ 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠 (𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠)

𝑡ℎ𝑎𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑦 𝑡ℎ𝑒 𝑔𝑜𝑣𝑒𝑟𝑛𝑖𝑛𝑔 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛,

𝑡ℎ𝑒 𝐻𝑒𝑛𝑘𝑒𝑙 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑡𝑜 𝑜𝑏𝑡𝑎𝑖𝑛

𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠 𝑎𝑛𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙𝑠..

𝑇ℎ𝑒 𝑑𝑒𝑡𝑎𝑖𝑙𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑚𝑎𝑦 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑

𝑓𝑟𝑜𝑚 𝐽𝑒𝑢𝑓𝑓𝑟𝑜𝑦 𝑎𝑛𝑑 𝐵𝑎𝑐ℎ𝑒𝑙𝑒𝑧 (1962), 𝑉𝑒𝑟𝑠𝑡𝑟𝑎𝑒𝑡𝑒𝑛 (1967; 1972)𝑎𝑛𝑑

𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 (𝑆𝑐ℎ𝑖𝑓𝑓𝑚𝑎𝑛 1962; 𝐴𝑠ℎ𝑡𝑜𝑛 𝑎𝑛𝑑 𝑀𝑜𝑎𝑣𝑒𝑛𝑧𝑎𝑑𝑒ℎ 1967).

𝑇ℎ𝑒 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑚𝑒𝑡ℎ𝑜𝑑𝑠 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒

𝑎𝑏𝑜𝑣𝑒 𝑚𝑒𝑛𝑡𝑖𝑜𝑛𝑒𝑑 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑢𝑛𝑟𝑒𝑎𝑙𝑖𝑠𝑡𝑖𝑐 𝑖𝑛 𝑝𝑟𝑎𝑐𝑡𝑖𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑟𝑒𝑎𝑠𝑜𝑛𝑠. • 𝐸𝑎𝑐ℎ 𝑙𝑎𝑦𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑛𝑜𝑛 𝑒𝑙𝑎𝑠𝑡𝑖𝑐, 𝑖𝑛ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑎𝑛𝑑 𝑎𝑛 𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑖𝑐 𝑠𝑜𝑙𝑖𝑑.

𝑇ℎ𝑒𝑦 𝑛𝑒𝑣𝑒𝑟 𝑡𝑟𝑢𝑙𝑦 𝑜𝑏𝑒𝑦 𝐻𝑜𝑜𝑘′𝑠 𝑙𝑎𝑤.• • 𝑇ℎ𝑒′𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑢𝑏𝑏𝑎𝑠𝑒 𝑎𝑛𝑑 𝑏𝑎𝑠𝑒 𝑙𝑎𝑦𝑒𝑟𝑠 𝑎𝑟𝑒 𝑔𝑟𝑎𝑛𝑢𝑙𝑎𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠 𝑤ℎ𝑖𝑐ℎ 𝑠ℎ𝑜𝑤

𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑏𝑒ℎ𝑎𝑣𝑖𝑜𝑢𝑟 𝑜𝑛 𝑙𝑜𝑎𝑑𝑖𝑛𝑔. • 𝑇ℎ𝑒 𝑠𝑢𝑏 − 𝑔𝑟𝑎𝑑𝑒 𝑚𝑎𝑦 𝑐𝑜𝑛𝑠𝑖𝑠𝑡 𝑜𝑓 𝑛𝑜𝑛 − 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑤𝑖𝑡ℎ 𝑑𝑒𝑝𝑡ℎ. • 𝑇ℎ𝑒 𝑏𝑖𝑡𝑢𝑚𝑖𝑛𝑜𝑢𝑠 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑙𝑎𝑦𝑒𝑟 𝑠ℎ𝑜𝑤𝑠 𝑣𝑖𝑠𝑐𝑜𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝐵𝑒ℎ𝑎𝑣𝑖𝑜𝑢𝑟

𝑤ℎ𝑖𝑐ℎ 𝑣𝑎𝑟𝑖𝑒𝑠 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑎𝑛𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒. • 𝑇ℎ𝑒 𝑃𝑜𝑖𝑠𝑠𝑜𝑛′𝑠 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 ′𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠 𝑖𝑠 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑎𝑛𝑑 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑖𝑛 𝑠𝑖𝑡𝑢 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 • 𝑇ℎ𝑒 𝑤ℎ𝑒𝑒𝑙 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝑖𝑠 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑙𝑦 𝑑𝑦𝑛𝑎𝑚𝑖𝑐 𝑎𝑛𝑑 𝑡𝑟𝑎𝑛𝑠𝑖𝑒𝑛𝑡′ 𝑖𝑛 𝑛𝑎𝑡𝑢𝑟𝑒. • 𝑇ℎ𝑒 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 . 𝑜𝑓 𝑏𝑒𝑖𝑛𝑔 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑛 𝑡ℎ𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑎 𝑡𝑟𝑢𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 • 𝑆ℎ𝑒𝑎𝑟 𝑎𝑛𝑑 𝑟𝑎𝑑𝑖𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑙𝑎𝑦𝑒𝑟 𝑑𝑢𝑟𝑖𝑛𝑔 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑤ℎ𝑒𝑒𝑙 𝑙𝑜𝑎𝑑𝑠. • 𝐶𝑟𝑎𝑐𝑘𝑒𝑑 𝑏𝑜𝑢𝑛𝑑 𝑙𝑎𝑦𝑒𝑟𝑠 𝑐𝑎𝑢𝑠𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑒𝑑 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒.

ASSIGNMENT-1

Q1. Write a short note on stiffness modulus of bitumen.

Q2. What is the limitation of Boussinesq's theory of stress analysis?

Q3. Derive an expression for determination of vertical stress under a uniformly loaded circular area

with a neat sketch.

Q4. Explain why the pressure distribution is different under flexible and rigid plates.

Q5. What is the significance of multi-layered flexible pavement analysis? Explain the

considerations/ assumptions made for analysis of stresses, strains and deflections in a multi-

layered flexible pavement system.

Q6. Define how the assumptions made in a multi-layered flexible pavement system are unrealistic

in practice or in service conditions of pavement.

Q7. List out a few programs with salient details of the analytical solution/programs of multi-layered

flexible pavements.