Strain Accumulation Due to Cyclic

Loadings

Mamdouh Mohamad

Civil Engineering, master's level (120 credits)

2018

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering

Preface

The master’s thesis is the final part of my two years master’s programme in civil engineering

with specialization in mining and geotechnical engineering at Luleå University of Technology,

corresponding to 30 credits.

I would like to thank my supervisors Jan Laue and Tommy Edeskär from Luleå University of

Technology for their guidance during these months. In fact I could not have achieved my master’s

thesis without their strong support and patient advices that were very valuable to complete its

various chapters. I would like also to send my appreciation to my parents who supported me with

love and blessings. Thank you all for your absolute support.

Mamdouh Mohamad,

Luleå, March 2018

Abstract

The formation of plastic strains in non-cohesive soils due to large number of loading cycles is

a phenomenon of great importance in geotechnical and civil engineering. It constitutes a

considerable cause for failures and deformations in various types of engineering applications

including pavements. Strain accumulation due to cyclic loading has been studied for years

through different models. This thesis reviews various models and focuses on the Bochum model

through which, the most contributing soil and traffic parameters on permanent strains formation

in pavement subgrades can be figured out. This represents the base for studying the serviceability

of increasing the gross weights of vehicles that affect the behavior and size of cyclic loading. This

was discussed through investigating the efficacy of increasing the number of vehicle axles and

through increasing the vehicle gross weight while keeping the number of axles to check their

impacts at the levels of strain formation in soil and consequently on its deformation. The results

showed a considerable difference in settlements after changing the axle configurations of vehicles

through increasing its number of axles. The work is expected to open a new area of scientific

research in pavement designs seeking for ideal configurations of vehicle axles and to provide an

advanced approach for studying soil deformations due to higher cyclic loadings.

Sammanfattning

Ackumulering av plastiska deformationer orsakade av upprepade lastcykler är viktigt

fenomen att ta hänsyn till inom geoteknik. Det bidrar till brott och deformationer i en rad

tillämpningsområden inom byggsektorn, framför allt inom väg- och järnvägsbyggnad.

Ackumulering av plastiska deformationer har studerats länge och flera modeller har föreslagits. I

detta arbete granskas flera av dessa modeller. Bochums modell har utvärderats närmare då de

ingående variablerna har varit möjliga att skatta. Modellen och de skattade variablerna har

använts för att studera brukstillstånd för en typisk vägkonstruktion för olika laster och

axelkonfigurationer med avseende på töjningar och sättningar för terrass och undergrund.

Resultaten visar att axelkonfiguration och last har en stor inverkan på ackumulerade plastiska

deformationer i undergrunden. Resultaten visar att axelkonfigurationen och last är viktig att ta

hänsyn till i vägdimensionering för att minimera sättningar i terrassen. Mer forskning behövs för

att undersöka olika jordtypers egenskaper för denna typ av modeller för att kunna generalisera

resultaten.

Table of Contents

Preface ..................................................................................................................................2

Abstract ................................................................................................................................3

Sammanfattning ....................................................................................................................4

1. Introduction ..................................................................................................................1

1.1. Background ..................................................................................................................... 1

1.2. Purpose/Aim .................................................................................................................... 3

1.3. Limitations ....................................................................................................................... 4

2. Literature Review ..........................................................................................................4

2.1. Soil Parameters ................................................................................................................ 4

2.1.1. Void Ratio ............................................................................................................... 4

2.1.2. Initial Density .......................................................................................................... 6

2.1.3. Particles shape (Roundness and Sphericity) ............................................................ 7

2.1.4. Water Content ........................................................................................................ 10

2.1.5. Particle size ............................................................................................................ 11

2.1.6. Summary................................................................................................................ 11

2.2. Traffic and Vehicle Parameters ..................................................................................... 12

2.2.1. Vehicle Speed ........................................................................................................ 12

2.2.2. Tire’s Contact Areas and Inflation pressure .......................................................... 13

2.2.3. Tire’s shape and Axle Configurations ................................................................... 13

2.3. Stress State .................................................................................................................... 14

2.3.1. Boussinesq’s solution ............................................................................................ 14

2.3.2. Odemark ................................................................................................................ 17

2.4. Accumulation Models for Soils ..................................................................................... 19

2.4.1. Hysteretic Versus Accumulation Constitutive Models for Soil ............................ 20

2.4.2. Comparison between Various Accumulation Models ........................................... 21

2.4.3. Bochum Accumulation Model for Sand under Cyclic Loading ............................ 22

2.4.4. Polarization and Shape of Strain Loop’s Effect on Strain Accumulation ............. 30

3. Analysis ....................................................................................................................... 33

3.1. Soil Parameters .............................................................................................................. 33

3.2. Superstructure Parameters ............................................................................................. 34

3.3. Stress Parameters ........................................................................................................... 35

3.4. Vehicle Parameters ........................................................................................................ 36

3.4.1. Case 1 .................................................................................................................... 37

3.4.2. Case 2 .................................................................................................................... 38

3.4.3. Case 3 .................................................................................................................... 38

3.5. Bochum Model .............................................................................................................. 39

4. Results ......................................................................................................................... 41

5. Conclusions ................................................................................................................. 59

6. References ................................................................................................................... 60

7. Table of abbreviations ................................................................................................. 64

Appendix A ......................................................................................................................... 66

Case 1 .................................................................................................................................... 66

Case 2 .................................................................................................................................... 82

Case 3 .................................................................................................................................... 98

Appendix B ....................................................................................................................... 114

Case 1 .................................................................................................................................. 114

Case 2 .................................................................................................................................. 132

Case 3 .................................................................................................................................. 151

Appendix C ....................................................................................................................... 174

Case 1 .................................................................................................................................. 174

Case 2 .................................................................................................................................. 175

Case 3 .................................................................................................................................. 176

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1. Introduction

1.1. Background

The residual settlement in soil increases with the number of loading cycles. This is due to the fact that,

with each cycle, irreversible deformation stays in soil. This phenomenon is of great importance in

geotechnical and civil engineering where there were many accidents relating the damages of structures to

cyclic loading of their foundations (Wichtmann, 2005). Examples on structures that can be affected by

cyclic loadings are:

Water tanks due to the change in water levels and fillings.

Coastal structures due to waves

Pavements due to crossing vehicles which is the case under study in this paper.

The current pavement design criterion takes into consideration all physical factors that contribute in the

permanent strain formation in soil due to cyclic loadings. The criterion is being used in Europe to achieve

stability, durability and persistence of pavements, specifies the maximum gross weight of vehicles to be 65

tons. The aim of the paper is to study the possibility of increasing the gross weight of vehicles without

affecting the safety of pavements by setting suitable proposals. The effect of increasing the cyclic loadings

and the applicability of the proposals to handle this increase were studied based on the variations that would

take place in the values of soil permanent deformations at the level of subgrade. Different models and

scientific articles were presented and described in a simplified manner in the literature to give an overview

about experimental studies performed on soil samples and to conclude their behavior under repeating loads.

The review handled all essential influencing parameters on strain formation in soils subgrade including

soil properties, stress states and loading factors. In road structures subjected to repeated traffic loadings,

subgrade plays a significant role in supporting the asphalt, base layers and traffic loadings. Therefore, the

subgrade should have sufficient bearing capacity to achieve its function properly. Practical observations

showed that, the assumption that subgrade soils are elastic is not consistent. Instead, it is more reasonable to

treat subgrade as elastic plastic materials. However, subgrades show elastic behavior only at small strains,

nonlinear behavior at larger strains and plastic behavior above yield strength (Salgado & Kim, 2002).

It has been found that soil subgrade has a crucial role in the initiation and propagation of permanent

deformation that can directly influence the pavement performance (Huang, 1993). Under heavy traffic loads,

soil subgrade may deform and that leads to distress in the pavement structure and consequently to rutting and

cracking (Elliott et al., 1998). Rutting is a phenomenon of concern because it is as a result of permanent

deflection of pavement surface due to accumulation of the plastic deformation of all pavement layers

including the subgrade.

Researchers were primarily using triaxial shear tests and direct shear tests to investigate the influence of

different soil properties on strains formation. Some of these properties were found to have direct and explicit

impact like void ratio and initial density. The triaxial tests performed by Wichtmann (2005) on soil samples

with distinct void ratios confirmed that the accumulation rate of residual strain increases with the increase in

void ratio . The direct shear tests done by Silver and Seed (1971) on soil samples with various initial

densities showed that, the residual strains are increasing significantly when there is a slight decrease in the

initial density. On the other hand, the impact of water content and soil particles shape was controversial,

where different researchers had different findings. However, they agreed that, these two factors must be

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taken into consideration in any analysis on strain formations. Thus, their effect was considered minor and it

was handled with respect to its impact on void ratio of soil.

In addition to soil characteristics, the thesis examined vehicles and traffic parameters that have primary

influence on accumulation of plastic strains. The ones were considered are: vehicle speeds, axle loads, the

number of load repetitions, and tire-contact areas.

Vehicle speed influences the values of obtained stresses and strains under cyclic loadings. In general,

the greater the vehicles speed, the smaller the strains in the pavement (Huang, 2004). This was explained in

more details in the literature.

Axles are integral components of wheeled vehicles. They transmit driving torque to the wheel and keep

the location of the wheels relative to each other and to the vehicle body. In order to determine the wheel/axle

loads that preserve the safety of pavement during its life time, the value of Load Equivalency Factor (LEF)

has to be determined. This value converts the damage from wheel loads of various magnitudes and

repetitions to damage obtained from an equivalent number of standard loads. The standard load is the

equivalent single axle load (ESAL) whose most commonly used value is 10 tons (18000 lb). The value of

LEF is important to understand the extent of damage pavements would have due to repeated loading cycles

of vehicles. It is calculated according to what is called “fourth power law” which is expressed

mathematically as follows: (Hjort et al., 2008)

, such that:

= vehicle axle load

= equivalent single axle load= 18000 lb

There are various axle configurations for different vehicles due to the difference in their sizes and their

intended bearing loading capacities. These configurations influence soil differently because their loading

cycles produce different values of stresses and strains and thus, different deformations. The three main

configurations examined in the review are summarized with their properties in table 1: (Hjort et al., 2008).

Table 1. Different wheel configurations (Hjort et al., 2008)

Single axle A single axle with more than 1.8 m spacing from other axles.

Tandem axle A configuration of two axles, with less than 1.8 m spacing between the axles.

The suspension of a tandem axle is such that the load on the tandem axle is

shared rather equally between the constituent axles.

The maximum load is dependent on axle spacing and suspension.

A bogie: two axles with shared suspension and spacing less than 1.3 m.

Triple axle A configuration of three axles, with relatively short longitudinal distance

between the axles.

The suspension of the tri-axle is such that the load on the tri-axle is shared

rather equally between the constituent axles.

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As already mentioned, the aim is to study the influence of the increment of vehicle gross weights at the

level of soil subgrades. A proposal to compensate this increase would be by distributing the excess loads on

additional axles which consequently means, modifying the wheel configurations. This proposal and its

corresponding effects had further investigation and discussion in the analysis part.

The traffic pressure and its distribution in soil are related to the structure of the loading surface and to

the wheel soil interaction or what is called the wheel footprint. Tire contact pressure, or nominal ground

pressure is the nominal tire load divided by the tire contact area, and tire contact area is tire diameter

multiplied by tire width. Usually, the contact pressure is assumed to be equal to the tire pressure and the tire

contact pressure is assumed to be uniformly distributed over a circular area (Pezo et. al, 1989). This is due to

the fact that, heavier axle loads have higher tire pressures and thus they are more destructive to pavements,

so the use of tire pressure as the contact pressure is a safe assumption (Huang, 2004). Generally, models that

give evaluation for soil damages are based on maximum pressure evaluation rather than on contact area.

In order to figure out the development of stresses in soil layers due to cyclic loading, different

approaches were studied. The wheel firstly gives compressive stresses in its front, then when the wheel is

passing, the stresses change to tensile, and when it leaves, the stresses are recovered to compression. This

behavior is related to different vehicle and soil effects and some of them are usually difficult to be

implemented in a model. So some approaches make simplifications that lead to underestimating the stresses

or strains in the model.

The mechanistic-empirical (ME) is one of these approaches through which loads and responses are

covered by a mechanistic model, in general an elastic or visco-elastic model. This approach serves the

calculation of horizontal strains in the lower edge of the pavement and vertical strains at the subgrade. It

assumes that, the pavement has an elastic load response. The calculation of stresses, strains and deflections in

this approach is done by using Bousinissque’s and Odemark’s approaches under certain assumptions.

In fact, the analysis of cyclic loadings must not be restricted to the assumptions set by Odemark and

Bousinissque’s. There is a need for a moving load model that takes into considerations a more realistic

behavior of pavements. This model must consider the rotation of principal stresses of the crossing wheel

loads, the time dependency of stresses development and by all means, the non static state of loads. When

dealing with flexible pavements, there are complicated calculation methods to analyze the load distribution

in soil. These methods demand larger resources like: adequate computer programs, high mathematical and

programming skills, and good knowledge in soil properties and models. However, simpler methodologies

can be used for developing acceptable models for wheel soil interaction. Some of them focus on maximum

values of tire contact pressure and others on contact area.

1.2. Purpose/Aim

The aim of the thesis is to check the prospect and reliability of increasing the gross weights of vehicles

crossing the pavements to limits higher than the ones under usage nowadays. This is to be investigated in

light of the possible fallouts would be obtained with respect to plastic strain formation in soil subgrade which

is due to combined effects of soil and traffic.

The goals to be achieved in the paper can be summarized as follows:

1. Reviewing different models and scientific articles that studied the effect of soil parameters on plastic

strain formation and dividing them into major and minor factors.

2. Looking on different models discussing traffic and vehicle parameters affecting soil deformation.

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3. Calculating the soil deformation through accumulated strain rates and settlements formation in

subgrade. This was implemented through Bochum model’s methodology.

4. Checking the effect of increasing the gross weight of vehicles and changing their axle configurations

on soil deformation.

1.3. Limitations

There were several limitations encountered through the different parts of the analysis. Determining the

real influence of the soil and traffic parameters is one of these limitations where the effects of some of these

parameters are controversial and questionable. Only the more logical and consistent hypotheses were

presented in the analysis. In addition to that, no lab tests were conducted in the thesis. Alternately, only

theoretical and logical analysis was used by using inputs from similar case studies and laboratory works.

Using assumed conservative values for some parameters might increase the uncertainties of the gotten

results. Furthermore, excel sheets were used in order to plot different relations and to deduce values of some

parameters, for example, they were used to deduce the total values of stresses and strains with respect to

distance using the “interpolation tool”. Adjustments needed after using this tool to give more accurate and

reasonable outputs.

2. Literature Review

In order to get better understanding for how the cyclic loading over pavements can affect the residual

strain formation and the corresponding types of failures, several parameters were analyzed. Some of these

parameters are related to soil properties, others are related to the stress state whereas other factors are linked

to loading functions. The disparate parameters are discussed in this paper in light of different researches and

analysis.

2.1. Soil Parameters

Non cohesive soil properties have a significant impact on determining the extent to which soil deforms

under cyclic loading. This paper handles the following soil properties that, after reviewing considerable

number of scientific papers, were found to be the most influential.

1. Void Ratio

2. Initial Density

3. Granular Shape (Roundness, Sphericity, Roughness)

4. Water Content

5. Granular Size

2.1.1. Void Ratio

The void ratio of the soil is one of the most effective parameters in soil mechanics and a primary feature

that determines soil properties and governs soil’s strength and its mechanical behavior when loading.

Wichtmann (2005) studied the influence of soil void ratio using cyclic triaxial tests. He used six samples

with different initial void ratios as follows: 0.581, 0.627, 0.65, 0.675, 0.717, and 0.806. The triaxial tests

were performed under the following conditions:

5

= 200 kPa,

=

=0.75

= 60 kPa

The relation between the deviatoric stresses ( ) and the deviatoric strains ( ) in the first cycle is shown

in figure 1, whereas the relation between residual strains, both deviatoric ( ) and volumetric ( ), as a

function of the initial void ratios at the end of the cycle is shown in figure 2.

Figure 1. q - relation for the first cycle (Wichtmann, 2005)

Figure 2. Residual strains with respect to initial void ratios (Wichtmann, 2005)

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During the first loading, the deviatoric strain increases as the initial void ratio increases. Similarly, the

residual volumetric and deviatoric strains at the end of the cycle show higher values with higher void ratios.

The strain accumulation with respect to void ratios through increasing number of cycles is shown in

figure 3. It shows that higher strains correspond to higher void ratios as the number of load cycles increases.

Figure 3. Strain accumulation with respect to void ratios and number of cycles (Wichtmann, 2005)

Thus, the performed triaxial tests under identical stresses and different initial void ratios confirm that,

the accumulation rate of residual strain increases with the initial void ratio.

2.1.2. Initial Density

To study the effect of soil initial density on strain formation, the results of simple shear tests done by

Silver and Seed (1971) were presented. They have done tests on different soil samples with different initial

relative densities: 0.45, 0.6 and 0.8. They studied how the accumulation of the residual axial strain of various

samples is affected as a function of shearing strains amplitude after 10 cycles.

The results of the shear tests showed a strong relation between the axial strain accumulation rates and

the initial densities. As shown in figure 4 below, after 10 cycles and with increasing shear strain amplitudes,

the residual strains are increasing significantly when there is a slight decrease in the initial density. The

recorded strains for the sample of initial density of 0.45 are two times greater than that of 0.6 and almost six

times greater than that of 0.8.

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Figure 4. Residual axial strains as a function of shear strain amplitudes for samples with different initial densities (Silver and

Seed, 1971)

2.1.3. Particles shape (Roundness and Sphericity)

The relation between the particle shape and the soil response was discussed through many researches

whose results were not so consistent. This makes the influence of granular shape a controversial issue. The

criteria selected to study the soil response was not the same too. Some of researchers used to look at the

shear modulus of different samples whereas others studied the response in light of volumetric or shear strains

formation that are obtained from triaxial or cyclic shear tests.

Barrett (1980) has determined the shape properties of the soil in three factors: form, roundness, and

roughness. According to Bui et al. (2007), the form represents the overall shape of the particle or the

uniformity of its dimensions (height, width and length), while the roundness and the roughness represent the

surface irregularities of the particles at large and small scales respectively.

Bui et al. (2007 and the references there in) mentioned that, the particle characteristics have an

important influence on soil strength and deformation when large strains (0.1%) are applied. Cho et al. (2006)

noticed that, the increase in irregularity leads to the increase of maximum and minimum void ratios

( and ) and to the decrease in soil stiffness. Santamarina and Cascante (1998) made a test using

steel spheres with different degrees of roughness, and they found out that, the small strain stiffness decreases

as the soil surface roughness increases. Cho et al. (2006) tried to build a relation between the particles shape

and the shear wave velocity and consequently with the maximum shear modulus. They found out that, the

maximum shear strain ( ) decreases with the irregularity of soil. On the other hand, their result was

contentious because some of the parameters used in the equation of shear wave velocity were related to void

ratio of the soil that they did not talk about in their paper. In other words, void ratio of the soil might be the

reason that was affecting the results of maximum shear modulus and accordingly, particle shape might have

a minor effect only.

The roundness and spherity, and the corresponding rate of strain formation under loading, was inspected

by Ali et al. (2013). Couple of triaxial and cyclic shear tests was carried out on four different soil types from

Turkey and Cyprus. The four soil samples have different shapes with respect to their roundness and spherity.

8

The roundness can be defined as the extent to which the soil particles’ corners and edges are rounded.

Whereas the spherity is to which level the soil particle is similar to the sphere in shape. The roundness and

the spherity of the soil particles are defined according to the following equations set by Wadell (1932) and

whose parameters are explained in figures 5 and 6:

Roundness:

Spherity:

With: Average diameter of the inscribed circles formed at each corner of the particle

Diameter of the greatest diameter of the soil particle

Smallest diameter of the sphere circumscribing the soil particle

In this study, three triaxial tests were carried out, one consolidated drained, one consolidated undrained

and one unconsolidated undrained. They were executed under different confining pressures of 450, 500 and

550 kPa and with a back pressure of 400 kPa. On the other hand the cyclic shear tests were performed with

three values of normal stresses: 50, 100 and 150 kPa. The tests studied the behavior of the soil particles

under monotonic and cyclic loading.

The four types of soil were Narli, Birecik and Trakya Sands obtained from Turkey and a crushed stone

produced commercially from Northern Cyprus. The four types were artificially graded to narrow size range

where almost 90% of the obtained sand particles were almost between 63 μm and 2.0 mm as described in

figure 7. The crushed stone soil particles had relatively angular shape whereas all the other soil samples were

relatively rounded.

Figure 5. Graphical representation of roundness R (redrawn from Muszynski and Stanley, 2012)

Figure 6. Graphical representation of sphericity S (redrawn from Muszynski and Stanley, 2012)

9

Figure 7. Particle size distribution of different soil samples (Ali et al., 2013)

The R value was estimated using visual methods, whereas the S value using manual/visual methods.

The visual method means by using binary images that are produced by Scanning Electron Micrography

(SEM) and scanner. The values of R and S for the four types of soil are summarized in table 2:

Table 2. Roundness and spherity of soil particles (Ali el at.,2013)

Soil type Roundness Spherity

Crushed Sand 0.19 0.56

Narli Sand 0.26 0.61

Birecik Sand 0.28 0.60

Trakya Sand 0.35 0.65

The results of the tests are presented in figures 8 and 9. For the three soil samples that have relatively

rounded shapes namely, Trakya, Narli and Birecik, it was obvious that, higher values of roundness and

spherity lead to higher strength and lower volumetric strains. On the other hand, the crushed sand recorded

higher strength in the cyclic shear test even though it had lower roundness and spherity than the other soil

samples. Ali et al. (2013) assumed that, this could be due to amount of fines and grading beside the particles

shape.

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Figure 8. Deviatoric stresses and axial strains under drained consolidation conditions (Ali et al., 2013)

Figure 9. Volumetric and axial strains in drained consolidation conditions and 150 kPa effective stress (Ali et al., 2013)

2.1.4. Water Content

Shear strength of the soil is believed to be highly affected by the moisture conditions (water content)

mainly when soil contains clay, where shear strength decreases with the increase of water content (Blahova

et al. 2013). For the purpose of understanding the influence of water content on soil strength, Blahova et al.

(2013) performed direct shear tests using three soil samples with different water contents as follows: 9%,

10% and 11%. The authors studied the response with respect to three factors: cohesion, friction angle and

maximum shear stresses obtained after being consolidated under three values of normal stresses: 25, 50 and

100 kPa. The results are shown in table 3.

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Table 3. Strength parameters as a function of water content, Blahova et al. (2013)

Water content c(kPa) Φ(⁰) τmax 50kN τmax 100kN τmax 200kN

9% 24,84 9,00 34,39 38,22 57,32

10% 21,00 13,40 35,70 40,80 70,10

11% 5,09 7,47 12,74 16,56 31,85

It was expected that, as water content increases, the strength parameters will decrease, which is not

consistent with the results shown in the table. Even though, the strength parameters were lowest for the

highest value of water content (11%), but the friction angle and the shear stresses recorded at water content

of 10% were higher than those for water content of 9%.

2.1.5. Particle size

One of the soil aspects that control its strength is the grain size particles. This is due to many researches

that have analyzed the relation between particle size of the soil and the obtained peak and residual strength.

Some of researchers got a conclusion that, the relationship is inverse whereas others claimed it is direct. The

shear strength of the soil can be described based on the values of friction angle obtained from direct shearing.

Alias et al. (2014) displayed the results obtained by different researchers who tried to figure out the relation

between the grain size and the friction angle of the soil (references there in). According to the authors, the

results of the different researchers were not compatible; where some found out that, larger grain size

decrease the friction angle, whereas others found the opposite.

Alias et al. (2014) also presented the tests implemented by Nakao and Fityus (2008). They revealed that

the peak and residual effective friction angles for soil grains of size less than 4.75 mm were 32.8° and 31.6°

respectively, whereas the peak and residual strengths for those of size less than 19 mm were much higher at

37.1° and 34.2° respectively.

In order to verify the real relation between the grains size of granular materials and the strength of the

soil, Alias et al. (2014) used small direct shear tests for particle size less than 2.36 mm and large direct shear

tests for grain size greater than 20mm. The tests were carried out with a shearing rate of 0.09 mm/min and

similar normal stresses of 100, 200 and 300 kPa. The results of Alias et al. (2014) proved the relationship set

by earlier researches that, when particle size increases, the peak and residual strength increase.

2.1.6. Summary

Based on the above, it can be concluded that, the outcome of some soil properties is different from the

others with respect to their contribution and effectiveness. The influence of soil void ratio, initial density and

particle size is more significant than particle shape and water content whose effects can be considered minor

or indirect.

It is accepted by many researchers that, particle shape affects the soil response by controlling the void

ratio of the soil. Cho et. al., (2006) concluded that, the minimum and maximum void ratios increase when

both spherity and roundness decrease. The study of Wichtmann (2005) as discussed earlier showed that, as

void ratio increases higher strain will be formed. Consequently, lower R and S values lead to higher strains

formation which fits with the results obtained by Ali et al. (2013)

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Regarding water content, Blahova et al. (2013) found out that, there is a relation between the strength of

soil and water content, but the number of samples he used in the analysis was not enough to provide high

degrees of credibility of the findings.

To sum up, all of these factors will be taken into consideration in the analysis and more focus will be

given to void ratio and initial density of the soil.

2.2. Traffic and Vehicle Parameters

Recently, considerable research efforts have been concentrated on the calculation and prediction of

cyclic wheel loads and their impacts on road deteriorations. In addition to considering the influence of

different soil properties, this has been done through investigating various vehicle aspects like its speed, its

tires pressure and its wheel configurations, to name some.

2.2.1. Vehicle Speed

Low vehicle speeds causes significant damage to the pavement structure. As vehicle speed increases, the

hot mix asphalt materials exhibit a higher strength due to increasing loading frequency (Ghauch and Abou

Jaoude, 2011). The effect of vehicle speed due to cyclic loadings on strain accumulation in soil subgrade has

been studied by Ghauch and Abou Jaoude (2011) and Daba et al. (2013). They have done it in different

ways. Ghauch and Abou Jaoude’s model (2011) simulated 50 loading cycles in order to examine the strain

formation in typical overlays on rigid pavements. Her model considered three vehicle speeds: 8, 32 and 48

km/h. Figures 10 and 11 show the development of strains under loading cycles with respect to the different

speeds.

Figure 10. Variation of maximum horizontal strains in

subgrade with respect to number of cycles and different

speeds (Ghauch and Abou Jaoude, 2011)

Figure 11. Variation of maximum shear strains in

subgrade with respect to number of cycles and

different speeds (Ghauch and Abou Jaoude, 2011)

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It is clear in figure 10 that the speed of 8 km/h compared to 48 km/h, creates not only a higher initial

maximum horizontal strain, but also a higher rate of horizontal strain accumulation, approximately 6 times

larger which is very similar to the behavior of the shear strain response observed in figure 11.

Gedafa et al. (2013) have done an experiment that involved construction of four thick pavement

structures in order to check the effect of traffic speed on stresses and strains. The pavements were arranged

with gauges to measure the tensile strains at the bottom of asphalt base layers and to measure the stress on

the top of subgrade. Five passes were performed by the truck with velocities of 30 km/h, 65 km/h, and 95

km/h. They found out that, as the traffic speed increases, the obtained stresses and strains are less.

2.2.2. Tire’s Contact Areas and Inflation pressure

The three main factors that are pertained to vehicle’s tires and affect the service life of pavements are:

tire inflation pressure, tire contact pressure and tire contact area.

Tire inflation pressure is the previously specified value of the pressure as agreed between the vehicle

manufacturer and the tire producers. It influences the response of the vehicles when it comes to driving

comfort, directional stability, cornering and the general handling behaviour. The tire contact area is the area

of contact between the tire and the ground, whereas the tire contact pressure is the wheel load over the tire

contact area. It is frequently assumed in pavement designs that the tire contact pressure is equal to the tire

inflation pressure, and the tire contact pressure is uniformly distributed over a circular area. This is based on

the idea that, if an inflated membrane is in contact with a flat surface, the contact pressure at each point is

equal to the membrane's inflation pressure and the contact area is circular. In fact, it has been explained

theoretically and experimentally that contact pressures are not uniform and contact areas are not circular.

(Pezo et al., 1989).

The rutting and fatigue failures of asphaltic concrete pavements were found to be directly affected by

the three interrelated factors. Eisenmann and Hilmer (1987) state that pavement rutting is caused by

mechanical abrasion. They related this mechanical process to the irreversible material deformations that are

as a result of high tire contact pressure concentrations, which are in turn generated by a change in tire

inflation pressure and/or wheel load. Papagianakis and Haas (1986) believe that the compressive strains at

the top of the asphalt surface are greatly influenced by high tire inflation and contact pressures. (Varin and

Saarenketo, 2014) also showed that, tire inflation pressure can have a significant effect on stresses and

strains in the upper parts of pavement structures, and causes fatigue and rutting. For example the pavement

lifetime of a road subjected to tire inflation pressures of 1000 kPa can be half that subjected to tire pressures

of 800 kPa.

2.2.3. Tire’s shape and Axle Configurations

Over the last decade there has been an intention in the transportation industry to use heavier trucks and

heavier axle configurations. Sweden has recently suggested raising the maximum gross weight of trucks to

74 tons. (Varin and Saarenketo, 2014).

It was believed that, increasing the total weight of heavy vehicles will not affect road structures if the

number of axles is increased while the axle weights are not raised. Although, increasing the number of axles

of a vehicle causes an increase in the pore water pressures in the road structure and/or in the subgrade. This

decreases the stiffness of the structural materials in the road.

14

The damaging risk to pavement structures increases as the axle weights of vehicles increase. The

estimated life time of pavements for example with 10 tons axles can be tens of percent shorter than its life

time when the axle weight is 8 tons when crossed by vehicles of same tire type and tire pressure. (Varin and

Saarenketo, 2014).

The tire type is also an important factor affecting the pavement structures. Tire type has however a

much greater impact on pavement lifetime than small increases in total weight. For instance, the stresses

induced by super single tires are much higher than the stresses induced by dual tires. In northern European

countries, the rutting speed can be 8-18 times higher with super single tires than with dual tires. (Varin and

Saarenketo, 2014).

In future the northern European countries will be facing the effects of increasing number of super single

tires and higher tire pressures, even in those countries where the gross truck weights or axle weights are not

being increased. In Scotland, the most critical issue will be the subgrade, and in Norway the fatigue of

pavement under super single loading will be important. The main challenge in Finland and Sweden will be

the new heavier trucks and the problems that arise from their use. (Varin and Saarenketo, 2014).

2.3. Stress State

There are several approaches used in order to calculate the stresses and strains in soil after being loaded

at various depths. One of the emerging approaches is the mechanistic-empirical (ME) approach that is

commonly used in pavement designs. In ME-design, the load and response are covered by a mechanistic

model, in general an elastic or visco-elastic model. The general principle of this approach is to measure the

horizontal strain in the lower edge of the pavement and the vertical strain at the subgrade. This is due to

assuming the pavement to have an elastic load response through which empirical relationships are set to

predict the lifetime of the pavement and the subgrade.

2.3.1. Boussinesq’s solution

The simplest way to analyze the stresses and strains in flexible pavement is to assume it as a

homogeneous half-space (Huang, 2014). Boussinesq has provided a methodology for calculating stresses,

strains and deflections in soil after being subjected to wheel loads. According to his method, the concentrated

load can be integrated into circular load area. Boussinesq’s solution is based on some assumptions as

follows:

The soil volume is a semi-infinite space; it is infinite in the horizontal plane and in depth from the

surface.

The soil volume is made up of elastic materials that are characterized by an elastic modulus E

(Young's modulus) and Poisson's ratio µ.

The material properties are considered homogenous and isotropic.

Figure 12 shows a homogeneous half-space of an elastic modulus E and a Poisson ratio ν subjected to a

circular load with a radius a and a uniform pressure q. In order to get better understanding for the way

Boussinesq used to calculate stresses in the soil body below the surface, a small cylindrical element with

center at a distance z below the surface and r from the axis is presented.

Due to symmetry, there are only three normal stresses: , and .

15

Figure 12. Homogenous half space of a circular load (Huang, 2004)

Based on Boussinesq’s assumptions, vertical ( ) and radial stresses ( ) induced at any depth in the

soil mass below the center of the circular loading plate are:

Vertical ( ) and radial strains ( ) can be determined using the following equations:

16

The deflection can be calculated according to this equation:

Regarding the values of the Poisson’s ratio, Dore and Zubeck (2009) have set typical values with

respect to different soils and pavement materials. They are summarized in the table 4 below:

Table 4. Poisson's ratios of soil materials (Dore and Zubeck 2009)

Soil/Material Range Typical value

Asphalt concrete 0.30-0.40 0.35

Portland Cement Concrete 0.15-0.20 0.15

Dense Graded Aggregates 0.30-0.40 0.35

Dense Sand 0.30-0.45 0.35

Loose Sand 0.20-0.40 0.30

Fine Grained Soils 0.20-0.50 0.40

Saturated Clays 0.40-0.50 0.45

On the other hand, in order to calculate the stresses and strains at points other than that are just below

the center of the loads, Boussinesq’s equations for a point load in polar coordinates were used as explained in

figure 13 and the following equations:

Figure 13. Boussinesq’s application in polar coordinates, (Per Ullidtz. 1987)

17

The equations used for this purpose are:

Vertical stress:

Radial Stress:

Vertical Strain

Radial Strain

2.3.2. Odemark

Odemark (1949), has developed a simple method to convert a system of layers with different moduli

into an equivalent system whose layers are of the same modulus and in which Boussinesq’s equations can be

still used. This allows the estimation of stresses and strains in soil layered systems by what is called the

method of equivalent thicknesses or MET that can be illustrated using figure 14 below: (Ullditz, 1987)

Figure 14. Odemark's transformation of a layered system (Edeskär, 2016)

18

This stiffness of a layer is proportional to:

This proportionality means that:

The assumptions set by Odemark for his equivalent thickness theory are:

The layer must have decreasing rigidity with depth with minimum ratio between two adjacent layers

equal to two ( / > 2)

The layers must have at least a thickness equal to the radius of the loading plate.

All layers above the subgrade soil are considered to have pure flexural behaviour and to have perfect

friction interfaces.

Figure 15 represents a schematic representation of Odemark principle;

Figure 15. Schematic illustration of the Odemark principle. (Edeskär, 2016)

The steps of calculating stresses in different soil layers according to Odemark are summarized as

follows:

When the analysis is conducted at a depth less than the depth of the first interface, stresses and strains

can be computed with the Boussinesq equations. In this case the properties of the first layer are used.

When the analysis is conducted at the first interface or between the first and the second interface, the

first layer is transformed in an equivalent thickness, h, of the material in the second layer according to the

following equation:

19

and Properties of the first layer

and Properties of the second layer

Different values of the correction factor depend on the point at which stresses are calculated such that:

= 1.0 for the first interface

= 1.1 for the first interface if the radius of the plate <

= 0.8 for all other cases

When two adjacent layers have the same Poisson's ratio, the equation can be written in a simplified form

such that:

If the Poisson's ratio can be assumed equal for all layers in the system, a general formula for all layers

can be written such that:

If the Poisson's ratio is not considered to be equal, the equation for a n-layer system becomes:

Deflections represent the summation of the values of deflections of the layers plus the deflection of the

subgrade. The compression of an individual layer is the difference between the deflection at the top and the

bottom of the layer in the transformed system.

2.4. Accumulation Models for Soils

Various types of models were implemented to analyze strain accumulation in soil. The main two

categories of these models are the hysteretic models and the accumulation constitutive models. This paper

compares between these two branches and discusses in details the Bochum accumulation model that reduced

the limitations and shortcomings of the other models and provided more reasonable and accurate approach

through considering more variables and inputs.

20

2.4.1. Hysteretic Versus Accumulation Constitutive Models for Soil

Soil has strongly non-linear behavior, and this leads to gradual reduction of secant shear modulus and to

increasing of hysteretic damping ratio as the shear strain amplitude increases. Hysteretic models are used in

soil mechanics applications in order to represent the loading and unloading behaviour of soil under certain

conditions that is the evolution of strain with respect to applied stresses as exemplified in figure 16:

Figure 16. Hysteretic strain accumulation (C. Karg, 2007)

In this model, the growth of stresses and strains is expressed as a function of increments on the

hysteresis loop. The major problem with this model is that, it is not applicable to describe strain

accumulation through hundreds or thousands of cycles where the limit for accurate prediction of plastic

deformations can be estimated at about only 50 to 100 load cycles. (C. Karg, 2007)

On the other hand, the accumulation models allow the prediction of accumulated stresses and strains due

to cyclic loading through very large number of cycles, possibly even millions of cycles. This represents the

main advantage of these models over the hysteretic models where the possible number of cycles to be

evaluated is in principle unlimited.

On contrast to the hysteretic models, in accumulation models stresses or strains accumulation are

described as a function of the applied number of cycles instead of stress and strain increments in each single

hysteresis loop. Hence, the accumulation curve can be drawn as indicated in figure 17: (C. Karg, 2007)

Figure 17. Strain accumulation in accumulation models (C. Karg, 2007)

21

2.4.2. Comparison between Various Accumulation Models

Numerous accumulation models have been implemented in order to evaluate the accumulated plastic

strains ( ) in soil exposed to repeated loads. A power law is used by several models for this aim such that:

= AN b

, with N being the number of cycles applied, A and b are empirical model parameters

depending on many factors like: soil type, soil properties and stress state. Another type of accumulation

models uses logarithmic formulations to describe the development of strain accumulation with respect to the

number of cycles. Also in these models, there are empirical determined parameters that must be considered.

(C. Karg, 2007)

All these models are based on empirical field and laboratory investigations and they use the number of

loading cycles as essential parameter. Depending on the amount of data available, the models cover several

relations like: stress dependencies, influence of the initial density and cyclic loading options which

consequently leads to better investigation for the problems.

Common problems with these models are:

1. The influences of important parameters as void ratio or state of stress are often neglected or cannot

be adequately taken into account.

2. Many models are based on very low numbers of cycles which leads to a wrong prediction of strain

accumulation obtained from large number of cycles.

Advanced accumulation models should allow forecasting of both the deviatoric and the volumetric

portions of accumulated strain. The models dealing with this request either suggest two separate empirical

equations for deviatoric and volumetric strains or use coupled formulations. For calculating the total

accumulated strain components, most models use incremental formulations based on the strain accumulation

rates of the deviatoric and the volumetric portions. The rate means the derivative of the accumulated strain

with respect to the number of load cycles instead of with respect to time.

Semi-explicit approaches are used sometimes to describe the strain accumulation. In this approach,

empirical based predictions are combined with advanced constitutive laws to achieve a tensorial formulation

of the accumulation rate. Bochum accumulation model (Niemunis et al., 2004; Wichtmann, 2005) is a good

example on the semi-explicit approach. (C. Karg, 2007).

Table 5 below shows the main properties of different accumulation models. It compares the main

features of the models including their input parameters and limitations. (C. Karg, 2007. References are in

there).

22

Table 5. Accumulation models and their properties

Model Type of

strain Equation

Material

constant

Input

parameter Limitations

Marr &

Christian

(1981)

Volumetric

and Vertical

Volumetric strain rate:

Cv, DV,

C1, D1

stress

amplitude

The large number of model parameters makes

it difficult to understand the respective

physical meaning and its determination.

Vertical strain rate :

Bouckoval

as et al.

(1984)

Volumetric

and

Deviatoric

Volumetric strain rate:

A, B, a, b

and c

cyclic shear

strain

amplitude

The influences of the average mean pressure

and the void ratio are not captured.

Deviatoric strain rate:

Sawicki &

Swidzinski

(1987 and

1989)

Volumetric

φ( )=C1 ln(1+C2 )

C1 and C2

cyclic shear

strain

amplitude

Only low numbers of cycles are investigated.

Influence of the average stress state on the

strain accumulation is not taken into account.

Li & Selig

(1996) Deviatoric

Li’s equation:

a,b,m and

n

Deviatoric

Stress

amplitude

Number of cycles investigated is small

(N=1,000).

Influences of the average mean pressure and

the void ratio are not captured.

Amended equation:

Gidel et al.

(2001) Vertical

,0, B,

m, n, and

s

Deviatoric

and mean

stresses

The indication of some parameters is not clear.

The influence of the void ratio and soil

structure is neglected.

2.4.3. Bochum Accumulation Model for Sand under Cyclic Loading

This model is an explicit accumulation model developed in Bochum since 1999 and originated from the

model of Sawicki and Swidzinski. The model is considered a good first approach for predicting the

settlement of granular soil with respect to low level vibrations. Many influencing parameters are investigated

and captured in the Bochum accumulation model some of them were not considered in the previous models

like the void ratio and the average stress. The material parameters included in the empirical equations of this

23

model can be determined by laboratory tests. The basic parameter of the model is the strain amplitude. It

considers the accumulation of the deviatoric strains and it uses the number of cycles instead of time t.

(Wichtmann, 2005)

Bochum accumulation model describes sand behavior under cyclic loading by a multiplicative approach

with the functions: , , , , and , such that:

describe the influence of the strain amplitude (scalar or absolute value)

describes the average mean pressure

describes the average stress ratio

describes the void ratio

describes the change of the polarization of the strain loop

describes the number of cycles N and it is divided into two components:

decays with number of cycles

independent of the number of cycles

In order to get better understanding for the meaning of each of these parameters, the tests done by

different researchers who implemented this model are discussed.

T. Wichtmann, A. Niemunis and Th. Triantafyllidis have studied the strain accumulation in sand due to

drained uniaxial cyclic loading. The empirical formulas were derived on the basis of numerous cyclic triaxial

tests and cyclic multiaxial direct simple shear (CMDSS) tests.

In these tests, only the vertical stress component was varied and the corresponding influence of strain

amplitude, average stress and initial density were figured out. The purpose of the cyclic triaxial tests was to

provide data for an explicit constitutive model which describes the strain cyclic accumulation that was

calculated such that:

The value of the rate of strain is proposed to be such that:

acc * * * * * * (Wichtmann et al. 2004)

acc rate of strain accumulation

The cyclic triaxial tests done by Wichtmann et al. (2004) allowed separating the above mentioned six

parameters, providing experimental evidence for the assumed dependencies and determining the material

constants.

Table 6 below summarizes the functions and the material constants of the proposed model that

were derived on the basis of cyclic triaxial test results and experimental findings.

24

Table 6. Functions and parameters of the proposed model, Wichtmann et al. (2004b)

The soil used in the tests is uniform medium sand whose properties are summarized in figure 18

showing the grain size distribution, friction angle and the maximum and minimum void ratios:

25

Figure 18, Grain size distribution for a tested sand (Wichtmann et al. 2004b)

Influence of the strain amplitude ( )

Soil strain obtained in the axial direction is denoted with and the ones in the lateral direction are

denoted as and that correspond to radial and tangential strains. These values are obtained using axial

and radial strains transducers that are fixed directly on the triaxial specimen as shown in figure 19 and

provide the opportunity to measure the soil deformation with high degree of accuracy.

Figure 19. Transducer (gdsinstruments.com)

The volumetric and deviatoric strains are derived from the obtained axial and lateral values such that:

Volumetric strain:

Deviatoric strain:

In case of cyclic loading the strain is made up of two portions: an accumulated or residual portion and an elastic or resilient portion as explained in figure 20:

26

Figure 20. Evolution of total strain in a cyclic triaxial test (Wichtmann, 2005)

The strain amplitude can be described by the volumetric and the deviatoric components

and

, respectively.

Shear strain amplitude can be used as an alternative to the deviatoric component

.

The influence of the shear strain amplitude on strain accumulation was studied by Wichtmann et al.

(2004) in a series of cyclic triaxial tests with:

12 kPa ≤

≤ 94 kPa (varying stress amplitudes)

= 200kPa (constant average stress)

=

0.75 (in all tests)

0.55 ≤ ≤ 0.64 (initial relative density of the dry specimens)

The development of the values of shear strain amplitudes with the number of cycles is shown in

figure 21:

27

Figure 21. Shear strain amplitude as a function of the number of cycles (Wichtmann et al., 2004b)

A relation between the values of accumulated strains with the number of cycles N through

different shear strain amplitudes obtained from figure 21 is plotted and shown in figure 22. It is clear

that the accumulated strain increases with the number of cycles and higher shear strain amplitudes

cause larger accumulated strains.

Figure 22. Accumulated strain as a function of the number of cycles in tests with different shear strain amplitudes

(Wichtmann et al., 2004b)

Influence of the mean pressure ( )

The mean pressure or the average effective pressure is the term that describes the state of stress in the

static state. In order to study the influence of mean pressure on the development of accumulated strain, a

series of six compression tests were performed by Wichtmann et al. (2004) with the following properties:

28

(Different average mean pressure)

=

0.75 (Identical average stress ratio)

ζ=

0.3 (amplitude ratio); where

is the stress amplitude and p is the mean

effective stress.

Figure 23 presents the accumulated strain as a function of the average mean pressure for different

numbers of cycles. In this comparison the residual strain has been normalized, i.e. divided by the amplitude

function.

Figure 23. Accumulated strain divided by the in dependence on the average mean pressure for different number of

cycles (Wichtmann et al., 2004b)

The figure showed that, the accumulated strain significantly increased with decreasing the average mean

pressure. This is reasonable since the soil stiffness increases proportional to the effective mean pressure that

leads to decreasing the strain accumulation rates.

Influence of the average stress ratio ( =

Wichtmann et al. (2004) performed eleven tests to figure out the effect of the average stress ratio on

strain accumulation and found out that the strain amplitudes slightly decreased with increasing average stress

ratio as shown in figure 24:

29

Figure 24. Strain amplitudes in tests with different average stress ratios (Wichtmann et al., 2004b)

Influence of the void ratio

The influence of void ratio on strain formation is included in Bochum model unlike other accumulation

models. Wichtmann et al. (2004) did tests on specimens with different void ratios. The specimens that were

saturated have the following properties:

0.580 ≤ ≤ 0.688

0.63 ≤ ≤ 0.99

= 200 kPa

= 0.75

ζ = 0.30

The obtained relation between the accumulated strains and the void ratios is shown in figure 25 that

shows that, higher accumulated strain is greater for looser materials.

30

Figure 25. Accumulated strain divided by the amplitude function in dependence on the actual void ratio for different

numbers of cycles (Wichtmann et al., 2004b)

2.4.4. Polarization and Shape of Strain Loop’s Effect on Strain Accumulation

In soil the cyclic stress and strain paths can be either in-phase or out-of-phase. The in-phase loops are

one dimensional loops that are usually caused by quasi-static fixed sources like in Watergates, tanks, etc.

Whereas the stress and strain loops are out-of-phase in traffic loads like a passing wheel as described in

figure 26 that shows how the principal stress axes in soil are varying as wheels move:

Figure 26. Out-of-phase stress loops in soil due to traffic loads on the surface (Wichtmann el al. 2007)

31

The two types of cycles are different with respect to their effect on strain formation in soil. Out of-

phase loops produce larger accumulation rates than in-phase cycles, so out-of-phase behavior of the cycles

must not be ignored because this will under-estimate the residual deformations.

The polarization of the cycles is a concept describes the orientation of the loop in the stress/strain space

and its change leads to variation in the level of strain development in soil.

Any explicit model has to describe the strain accumulation due to in-phase and out-of-phase strain

loops, the influence of the polarization of the cycles and the cycle’s shape. (Wichtmann et al., 2004).

Influence of polarization changes

In order to study the influence of polarization changes, Yamada and Ishihara (1982), have done drained

triaxial tests on loose saturated sand. After isotropic consolidation four cycles were tested.

In the first cycle: octahedral shear stress amplitude was applied at p = constant.

In the second cycle: the loading was rotated by an angle θ in the octahedral plane but the amplitude was

kept constant.

In the third cycle: the loading was applied in the same direction as the first cycle, but the amplitude was

increased.

In the fourth cycle: the specimen was sheared again in the direction of the second cycle and the

amplitude of the third cycle was used.

The results of the cases are shown in figure 27:

Figure 27. True triaxial tests after Yamada & Ishihara [9]: Influence of a rotation of the stress path by θ = 0⁰, 90⁰ and

150⁰ on the accumulation of volumetric strain (Wichtmann el al. 2007)

It is clear that, the residual strain increased with increasing the rotation of shearing direction with θ as in

the second and the fourth cycles compared to the situation in the first and the third cases.

32

Another test was made using a cyclic multidimensional simple shear (CMDSS) test through which a

sudden 90⁰ change of the direction of cyclic shearing was applied.

Figure 28 makes a comparison between tests with and without a change of the polarization after 1,000

cycles. It is obvious that, the change in polarization caused an increase of the accumulation rate.

(Wichtmann et al., 2004)

Figure 28. Temporary increase of the accumulation rate due to a sudden 90⁰ change in polarization (Wichtmann el al. 2007)

Influence of the shape of the cycles

Pyke et al. (1975) used two shaking tables in order to test dry sand sample under multiaxial cyclic

loading. 2-D shearing was allowed through mounting transversely one of the tables on the other. They found

out that, when the applied stress path was approximately circular i.e. elliptic shearing, the settlement was

twice larger than that of a uniaxial stress path with the same maximum shear stress as described in figure 29:

(Wichtmann et al., 2004).

33

Figure 29. Shaking table tests after Pyke et al.: comparison of uniaxial and circular stress cycles (Wichtmann el al. 2007)

3. Analysis

Building on the diverse parameters discussed in the review, the stress/strain response of a soil specimen

under uniaxial cyclic loading of 100000 cycles was investigated. This was handled through several excel

sheets that drew the mathematical relations between the various input parameters and the corresponding

outputs: stresses, strains, accumulated strains and settlements.

3.1. Soil Parameters

The soil sample used for analysis to refer to the subgrade layer has similar properties as the one used by

Wichtmann et al. (2005) who performed similar analysis to inspect strain formation based on the Bochum

Model Criteria. The advantage of using the same soil type was the possibility of assuming some factors of

Bochum model whose values depend on laboratory tests. The soil sample was uniform medium course to

course sand whose grain size distribution and properties are displayed in the figure 30 and table 7.

34

Figure 30. Tested grain size distribution curves of a quartz sand (Wichtmann, 2005)

Table 7. Soil sample properties (Wichtmann, 2005)

d50 0.55 mm

U 1.8

C 1.2

emax 0.874

emin 0.577

c 31.2⁰

3.2. Superstructure Parameters

The superstructure was assumed to be of high performance (BBÖ) (TRV, 2005) that is crushed rock

based with bitumen bound superstructure. Some of the features of this type of superstructures are:

improvement of freeze-thaw durability, reduction of permeability, and increasing of wear resistance. The

layers’ properties and dimensions of the superstructure are illustrated in figure 31 and table 8:

35

Figure 31. Layers of the superstructure

Table 8. Layers' properties (TRV, 2005 - The Austroad Pavement Design Guide,1992)

Layer Thickness (m) Stiffness Modulus, E, (Pa)

Wearing course 0.04 3.5E9

Bound Base Layer 0.08 3E9

Unbound Base Layer 0.08 2E8

Subbase 0.46 1.5E8

Subgrade - 7.5E7

3.3. Stress Parameters

The stresses and strains were calculated at different depths below the ground surface level and the

deepest depth comprised in the analysis was 10 m. The equivalent thickness method set by Odemark was

36

used in order to convert the actual layers of the superstructure into ones with equivalent thicknesses. Figure

32 represents the actual thicknesses of the ground layers and figure 33 represents the equivalent layers.

The stresses and strains were calculated at disparate distances from the load center by handling the

wheel loads as point loads and calculated just below the load center by considering the load as a strip load.

The distances were of 10 m strips and the total distance was approximately 120 m.

3.4. Vehicle Parameters

The analysis encompassed three different types of vehicles varied with respect to either their load values

or their wheel configuration whereas the vehicle speed and wheel contact pressure were kept constant for the

three cases and assumed to be 60 km/h and 2.5 bars respectively.

Wheel loads were selected in light of the permissible maximum weights of trucks in Sweden set by the

international transport forum. The truck used in the first case had 6 axles as shown in figure 34.

Figure 33. Equivalent thicknesses of the ground

layers

Figure 32. Actual thicknesses of the ground

layers

37

Figure 34. 6 axles truck (packer3d.com)

Case 2 handled a truck with same axle configuration as that in case 1 but with higher axle loads,

whereas the case 3 handled a truck with 2 additional axles.

Evaluation of stresses and strains was performed with respect to the effect of each truck wheel and its

contact area with the ground. The contact areas were derived from the assumed wheel loads and their contact

pressures such that:

Radius of the area of contact m

Wheel Load kN

Contact Pressure kPa

Contact Area

3.4.1. Case 1

The features of the first truck such as the axle loadings and their dimensions are described in tables 9

and 10:

Table 9. Truck features, case 1

Front Single

Axle

Tandem

Axle

Triple Axle Minimum

Distance ‘a’

Minimum

Distance ‘d’

Gross Vehicle

Weight

10 ton 18 ton 26 ton 10.25 m 1.5 m 54 ton

38

Table 10. Truck wheels properties, case 1

Wheel Load

(kN)

Radius (m) Contact Area

(m2)

Contact Pressure

(kPa)

Wheel of the front axle 49 0.25 0.2 250

Wheels of the tandem axle 44.2 0.237 0.18 250

Wheels of the triple axle 42.5 0.233 0.17 250

3.4.2. Case 2

Same steps were followed for the second case and the only difference was to make amendments on the

wheel loads such that the maximum gross weight of the new vehicle was increased from 54 tons to 74 ton.

The number of truck axles and its axle configuration were not changed. The features of the truck of the

second case and the modified loading properties are shown in tables 11 and 12 below:

Table 11. Truck features, case 2

Front Single

Axle

Tandem

Axle

Triple Axle Minimum

Distance ‘a’

Minimum

Distance ‘d’

Gross Vehicle

Weight

10 ton 26 ton 38 ton 10.25 m 1.5 m 74 ton

Table 12. Truck wheels properties, case 2

3.4.3. Case 3

In the third case, the influence of changing the wheel configuration of the truck was detected. This was

done by adding two axles to the truck and keeping its gross weight at 74 tons. One axle was added to the

tandem axle group and one to the triple axle group. The distances ‘a’ and ‘d’ between the axles were adjusted

and decreased to 9 m and 1.2 m respectively. The features of the truck of the third case and the modified

loading properties are shown in tables 13 and 14 below:

Wheel Load

(kN)

Radius (m) Contact Area

(m2)

Contact Pressure

(kPa)

Wheel of the front axle 49 0.25 0.2 250

Wheels of the tandem axle 63.7 0.285 0.25 250

Wheels of the triple axle 63.7 0.285 0.25 250

39

Table 13. Truck features, case 3

Front Single

Axle

Triple

Axle

Quadrant

Axle

Minimum

Distance ‘a’

Minimum

Distance ‘d’

Gross Vehicle

Weight

10 ton 30 ton 34 ton 9 m 1.2 m 74 ton

Table 14. Truck wheels properties, case 3

Wheel Load

(kN)

Radius (m) Contact Area

(m2)

Contact Pressure

(kPa)

Wheel of the front axle 49 0.25 0.2 250

Wheels of the triple axle 49 0.25 0.19 250

Wheels of the quad. axle 41.7 0.23 0.17 250

3.5. Bochum Model

The stresses and strains obtained were used later in the accumulation model to investigate strain

accumulation with respect to applied loading cycles. The accumulation model used for this purpose was

Bochum Model that takes into consideration high number of parameters and therefore leads to more accurate

results. Bochum model is composed of various factors some of them are related to stresses and strains values

and others are obtained from laboratory testing.

Since this paper does not include any laboratory experiments, the parameters related to lab work were

taken from similar tests done by Wichtmann et al. (2005) for the same soil sample. The parameters that

represent the components of the multiplicative factors of the Bochum model are summarized in table 15

below:

40

Table 15. Bochum models' parameters (Wichtmann et al. (2005)

Parameter Value Parameter Value

0.00034 4

0.55 200

0.00006 31.2

0.43 11.93

100 0.577

2.05 0.874

0.54 1

0.874

0.0001

Strain accumulation was calculated with respect to 1, 10, 100, 1000, 10000 and 100000 cycles. The

value of the strain accumulation was also calculated at 73000 cycles for the first and the second case that is

equal to:

Max. number of cycles (Gross weight of the first truck Gross weight of the second truck).

This is in order to notice how increasing the loading of the truck will affect the results while keeping the

axle configuration.

The values of strain rates obtained and the corresponding strain values were plotted with respect to the

number of cycles and the curves of the three cases were compared.

The impact of increasing the loading cycle and the axle configuration was also discussed through

comparing the values of settlements obtained from each case. The settlement was calculated for the various

numbers of cycles and the total settlements that are the summation of settlements at different depths of each

case were compared.

Through analysis, some of the assumptions have taken into account to facilitate using Bochum Model

that is complicated and includes lots of factors and laboratory work:

1. The analysis was carried out once using the maximum value of the void ratio of the soil sample and

once using the minimum value.

2. The deviatoric and volumetric stresses calculated using Boussinesq’s/Odemark’s equations were

handled as average values in the model.

3. The factor of the model that describes the change of polarization of the strain loop ( ) was assumed

to be 1.

41

4. Results

The calculation of stresses and strains were done in Excel sheets through mathematical equations whose

factors are the aforementioned soil, vehicle and superstructure parameters. Truck wheels were given numbers

based on the order of crossing. The sheets, named according to the wheel number, present the calculation of

the equivalent thicknesses of the soil layers based on Odemark’s equations. They also present the derivation

of vertical stresses and strains and the horizontal stresses and strains below center of the wheel as well as at

various distances from the center based on Boussinesq’s equations. The vehicle path was assumed to be of

approximately 120 m length. The points of interest were 50 m before a wheel crossing a specific point

location and 70 m after crossing it. The stresses and strains values obtained at the each equivalent layer for

the different distances were summarized in tables and used in other excel sheets for the three cases under

study such that:

Case 1: 6 wheels truck, 54 ton gross weight

Case 2: 6 wheels truck, 74 tons gross weight

Case 3: 8 wheels truck, 74 tons gross weight

The relations between the vertical and radial stresses with respect to time were also studied. Excel

sheets were made for each soil depth where time was deducted by dividing distances from a certain point in

the vehicle path over an assumed vehicle velocity of 60 km/h. The assumed vehicle path was 100 m long.

The total length of the path was divided into positive and negative values. This means that, the path has an

assumed origin in its middle such that the negative values refer to the distances from the wheel center before

crossing the origin and the positive values refer to the distances from the wheel center after crossing the

origin as indicated in figures: 35, 36 and 37.

These plots were performed to see how stresses were developed in the soil subgrade with respect to time

as vehicle passes. The results obtained at a depth of 10.24 m for the three cases are shown in the graphs 35,

36 and 37. Plots at the other depths were included in appendix B that includes also the results for the radial

stresses.

42

Figure 35. Vertical Stress, v=60 km/h, z= 10.24 m, Case 1

Figure 36. Vertical Stress, v=60 km/h, z= 10.24 m, Case 2

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

43

Figure 37. Vertical Stress, v=60 km/h, z= 10.24 m, Case 3

Other Excel sheets were performed to plot the relation between stresses and strains with respect to

distance throughout the vehicle path at the different depths. The Excel sheets were given names that

correspond to the equivalent soil layers. The stresses and strains values obtained from the different passing

wheels were performed in the same plots and thus the curves are superposed. Thus, the total values of

stresses and strains, which are the summation of the overlapped curves, with respect to the distance from the

wheel center were then deducted.

The figures below (39-50) represent the obtained relations at the deepest point under study, which is

10.24 m, for the three cases. The x-axis in these figures corresponds to the distance from the load center of

wheel 1, on which stress values of the other consecutive wheels, that will cross the same distance, were

represented as well. Figure 38 is a schematic representation for the wheels path.

These plots were performed to inspect the development of stresses and strains at the level of subgrade

from which the accumulated strains would then be calculated and consequently the total settlements.

As shown in figures 39, 40 and 41, vertical stresses of each wheel increase as the distance from the

center of the wheel decreases till reaching the maximum when the distance is 0 that is just below the wheel

center, then stresses start to decrease as the wheel crosses and moves away. This scenario applies in the three

cases. However, the total stresses obtained in second case are the highest, and stresses of case 3 are higher

than those of case 1.

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

44

Figure 38. Schematic representation of the wheels, their positions and their paths

Vertical stress, case 1

Figure 39. Vertical stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 1.

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress VS. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

Wheel 1 Wheel 2 Wheel 3 Wheel 4 Wheel 5 Wheel 6

45

Vertical stress, case 2

Figure 40. Vertical stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 2

Vertical stress, case 3

Figure 41. Vertical stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 3

Radial stresses with respect to the distance from the load center for the three cases are plotted in figures

42, 43 and 44. The development scenario of radial stresses in the three cases is similar to that of vertical

stresses, whereas case 3 shows higher values than the ones obtained in case1 and 2 whose results are close to

each other.

7.00E-02

7.00E-01

7.00E+00

7.00E+01

7.00E+02

7.00E+03

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ver

tica

l Str

ess

(Pa)

Distance

Vertical Stress VS. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ver

tica

l Str

ess

(P

a)

Distance (m)

Vertical Stress VS. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

46

Radial stress, case 1

Figure 42. Radial stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 1

Radial stress, case 2

Figure 43. Radial stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 2

-5.00E+04

5.00E+04

1.50E+05

2.50E+05

3.50E+05

4.50E+05

5.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-4.00E+04

6.00E+04

1.60E+05

2.60E+05

3.60E+05

4.60E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

total

47

Radial stress, case 3

Figure 44. Radial stresses vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 3

The values of vertical strains obtained in case 1 are less than those of case 2 and 3 even though. On the

other hand, case 2 and case 3 show approximately equal values. This is described in figures 45, 46 and 47.

Vertical Strain, case 1

Figure 45. Vertical strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 1

-1.00E+05

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

7.00E+05

8.00E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-7.21E-08

9.28E-07

1.93E-06

2.93E-06

3.93E-06

4.93E-06

5.93E-06

6.93E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

48

Vertical Strain, case 2

Figure 46. Vertical strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 2

Vertical Strain, case 3

Figure 47. Vertical strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 3

-8.31E-07

1.17E-06

3.17E-06

5.17E-06

7.17E-06

9.17E-06

1.12E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-1.00E-06

1.00E-06

3.00E-06

5.00E-06

7.00E-06

9.00E-06

1.10E-05

1.30E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

49

With respect to radial strains, the results obtained in the three cases were very close as shown in figures

48, 49 and 50.

Radial Strain, case 1

Figure 48. Radial strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 1

Radial Strain, case 2

Figure 49. Radial strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 2

-2.50E-06

-2.00E-06

-1.50E-06

-1.00E-06

-5.00E-07

-8.00E-20

5.00E-07

1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

-2.50E-06

-2.00E-06

-1.50E-06

-1.00E-06

-5.00E-07

-8.00E-20

5.00E-07

1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

50

Radial Strain, case 3

Figure 50. Radial strains vs. distance from wheel centers corresponding to the path of wheel 1, z= 10 m, case 3

The relations between stresses/strains and the distance from the center of load at the other depths under

study are included in appendix A. Tables (16-18) below summarize the results of the plots of the three cases.

The values of stresses/strains included in the tables correspond to the peak total values of the curves.

From table 16, it is remarkable that for the three cases, going deeper in the soil layers (subgrade) leads

to less vertical stresses. This is reasonable where the highest values of stresses must be taken by the

superstructure that has higher stiffness than the subgrades. It is also clear that, increasing the axle loads lead

to formation of higher stresses (case 2) whereas increasing the number of axles while keeping the same

loadings did not make noticeable changes (case 3).

Table 16. Peak vertical stresses at various depths

Vertical Stress (Pa)

Depth (m) 0.66 1 2 3 4 6 10

Case 1 2.1E5 2.5E4 1.25E4 5.92E3 4.97E3 1.37E3 1.09E3

Case 2 2.18E5 3.25E4 1.6E4 6.3E3 7E3 3.18E3 1.47E3

Case 3 2.36E5 5.41E4 1.6E4 6.31E3 7.11E3 3.19E3 1.44E3

Table 17 shows that, higher radial stresses were formed when going deeper in the soil layers. Increasing

the loadings did not affect the values of radial stresses explicitly but the significant effect came from

changing the wheel configuration. Higher number of axles generated more radial stresses (case 3).

-2.60E-06

-2.10E-06

-1.60E-06

-1.10E-06

-6.00E-07

-1.00E-07

4.00E-07

9.00E-07

1.40E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

51

Table 17. Peak radial stresses at various depths

Radial Stress (Pa)

Depth (m) 0.66 1 2 3 4 6 10

Case 1 1.7E5 4.4E5 4.4E5 4.42E5 4.56E5 4.7E5 4.82E5

Case 2 1.7E5 4.4E5 4.4E5 4.4E5 4.51E5 4.67E5 4.8E5

Case 3 2.25E5 5.9E5 6.8E5 7.4E5 7.6E5 7.4E5 7.8E5

Table 18 clarifies that, the vertical strains decrease significantly with depths. Increasing the loadings

(case 2) as well as increasing the number of axles (case 3) raises the formation of vertical strains.

Table 18. Peak vertical strains at various depths

Vertical Strain

Depth (m) 0.66 1 2 3 4 6 10

Case 1 2.48E-3 3.93E-4 1.41E-4 6.76E-5 3.95E-5 1.84E-5 7.37E-6

Case 2 2.86E-3 4.84E-4 1.74E-4 8.39E-5 4.91E-5 2.29E-5 1.15E-5

Case 3 3.12E-3 6.38E-4 1.8E-4 8.64E-5 6.44E-5 2.58E-5 1.12E-5

Table 19 explains that, increasing axle loads (case 2) obviously increases the radial strain formations

that develop further with depth. On the other hand, increasing the number of axles (case 3) does not make

considerable difference.

Table 19. Peak radial strains at various depths

Radial Strain

Depth (m) 0.66 1 2 3 4 6 10

Case 1 -5.79E-8 -3.16E-8 6.3E-7 1.15E-6 1.44E-6 1.52E-6 1.5E-6

Case 2 -8.46E-8 -6.51E-8 7.75E-7 1.43E-6 1.82E-6 1.94E-6 2.4E-6

Case 3 -5.56E-8 -4.44E-8 8.04E-7 1.49E-6 1.89E-6 2.02E-6 2.6E-6

The maximum values of stresses and strains obtained were substituted in the Bochum Model’s

equations. Excel sheets were used for this aim for the three cases and for the different depths. The material

constants and the model’s factors used in the sheets were taken from Wichtmann et al. (2005) who did

similar analysis using the same soil sample. The factors and material constants are summarized in table 20

below.

52

Table 20. Summary of factors and constants of tested sand (Niemunis et al. 2005)

First, strain rates were calculated for different number of cycles by multiplying all factors of Bochum’s

equation such that:

= * * * * *

The excel sheets allow the deduction of accumulated strains at the different depths by summing the

values of strain rates obtained from the different number of cycles. Plots were drawn to describe the relation

between the strain rates and accumulation strains with respect to the number of cycles for the three cases.

The plots are included in appendix C.

Remarks on the plots:

The values of the strain rate that are in concept the derivation of the accumulation strain with respect to

the number of load applications decrease with the increment of number of cycles. Higher values of strain

rates were obtained at closest points to the ground surface and the values decrease when going deeper in the

subgrade layer. It is also clear that, increasing the loading values (case 2) leads to higher strain rates. This is

explained in figure 51 that compares between the strain rates of case 1 and case 2 at three selected depths:

1.32 m, 3.24 m and 10.24 m.

53

Figure 51. Strain rates vs. number of cycles, cases 1 and 2

On contrary, changing the axle configuration by increasing the number of axles while keeping the same

axle loads leads to less strain formation as proved in figure 52 that compares between cases 2 and 3.

Figure 52. Strain rates vs. number of cycles, cases 2 and 3

Additionally, the values of accumulated strains increase with the number of cycles. Higher values refer

to the points closer to the ground surface and they decrease as going deeper in the soil layers. The

accumulated strains witness remarkable increase within the first few number of cycles (10 cycles), and then

the increment starts to be minor and slight.

Increasing truck loads lead to higher accumulation strains as shown in figure 53 that compares between

cases 1 and 2, whereas increasing the number of axles while keeping the loads reduce the accumulation as

described in figure 54 that compares between cases 2 and 3.

5.00E-11

5.00E-10

5.00E-09

5.00E-08

5.00E-07

5.00E-06

5.00E-05

5.00E-04

5.00E-03

1 10 100 1000 10000 100000 1000000

ε'

Log N

ε' vs. N

z= 1.32 m (case 1)

z= 3.24 m (case 1)

z= 10.24 m (case 1)

z=1.32 m (case 2)

z= 3.24 m (case 2)

z= 10.24 m (case 2)

5.00E-11

5.00E-10

5.00E-09

5.00E-08

5.00E-07

5.00E-06

5.00E-05

5.00E-04

5.00E-03

1 10 100 1000 10000 100000 1000000

ε'

Log N

ε' vs. N

z=1.32 m (case 2)

z= 3.24 m (case 2)

z= 10.24 m (case 2)

z=1.32 m (case 3)

z= 3.24 m (case 3)

z= 10.24 m (case 3)

54

Figure 53. Accumulated strains vs. number of cycles, cases 1 and 2

Figure 54. Accumulated strains vs. number of cycles, cases 2 and 3

Another comparison was implemented with respect to settlement attained due to cyclic loading. Each

depth under study was assumed to witness settlement and thus affect a certain thickness of soil. The assumed

influenced thickness at each depth is displayed in table 21. The accumulated strain at each depth was

multiplied by the assumed layer thickness to get the settlement. After calculating the settlements they were

summed up together with respect to the depth. The resultants of each case were afterwards compared.

Table 21. Affected soil thicknesses by stresses used for estimation of settlements.

Depth (m) 0.66 1 2 3 4 6 10

Thickness (m) 0.9 0.84 1 1 1 3 5

1.20E-07

1.20E-06

1.20E-05

1.20E-04

1.20E-03

1.20E-02

1 10 100 1000 10000 100000 1000000

ε

Log N

ε vs N

z= 1.32 m (case 1)

z= 3.24 m (case 1)

z=10.24 m (case 1)

z= 1.32 m (case 2)

z= 3.24 m (case 2)

z= 10.24 m (case 2)

2.00E-07

2.00E-06

2.00E-05

2.00E-04

2.00E-03

2.00E-02

1 10 100 1000 10000 100000 1000000

ε

Log N

ε vs N

z= 1.32 m (case 2)

z= 3.24 m (case 2)

z= 10.24 m (case 2)

z= 1.32 m (case 3)

z= 3.24 m (case 3)

z= 10.24 m (case 3)

55

Tables 22, 23 and 24 below present the results of the three cases:

Case 1:

Table 22. Case 1, total settlement for different number of load cycles (N) at different depths (0.66-10 m)

Total Settlement (m)

N Depth (m)

0.66 1 2 3 4 6 10

1 8.4157E-03 8.2590E-04 1.5236E-04 4.5486E-05 1.9263E-05 1.0405E-05 2.4043E-06

10 1.0424E-02 1.0230E-03 1.8871E-04 5.6338E-05 2.3859E-05 1.2887E-05 2.9780E-06

100 1.0658E-02 1.0460E-03 1.9295E-04 5.7605E-05 2.4396E-05 1.3177E-05 3.0449E-06

1000 1.0683E-02 1.0484E-03 1.9341E-04 5.7740E-05 2.4453E-05 1.3208E-05 3.0521E-06

10000 1.0687E-02 1.0488E-03 1.9348E-04 5.7761E-05 2.4462E-05 1.3213E-05 3.0532E-06

73000 1.0689E-02 1.0490E-03 1.9351E-04 5.7770E-05 2.4466E-05 1.3215E-05 3.0537E-06

100000 1.0689E-02 1.0490E-03 1.9351E-04 5.7770E-05 2.4466E-05 1.3215E-05 3.0537E-06

Case 2:

Table 23. Case 2, total settlements for different number of load cycles (N) at different depths (0.66-10 m)

Total Settlement (m)

N Depth (m)

0.66 1 2 3 4 6 10

1 8.7369E-03 1.2314E-03 2.2951E-04 6.8082E-05 2.7758E-05 1.4457E-05 4.9704E-06

10 1.0821E-02 1.5252E-03 2.8427E-04 8.4325E-05 3.4381E-05 1.7906E-05 6.1563E-06

100 1.1065E-02 1.5595E-03 2.9066E-04 8.6221E-05 3.5154E-05 1.8308E-05 6.2947E-06

1000 1.1091E-02 1.5632E-03 2.9134E-04 8.6424E-05 3.5236E-05 1.8351E-05 6.3095E-06

10000 1.1095E-02 1.5638E-03 2.9144E-04 8.6455E-05 3.5249E-05 1.8358E-05 6.3117E-06

73000 1.1097E-02 1.5640E-03 2.9149E-04 8.6469E-05 3.5255E-05 1.8361E-05 6.3128E-06

100000 1.1098E-02 1.5643E-03 2.9154E-04 8.6482E-05 3.5260E-05 1.8364E-05 6.3137E-06

56

Case 3:

Table 24. Case 3, total settlements for different number of load cycles (N) at different depths (0.66-10 m)

Total Settlement (m) (m)

N Depth (m)

0.66 1 2 3 4 6 10

1 7.4691E-03 1.2229E-03 1.2630E-04 3.8180E-05 1.8866E-05 8.8805E-06 2.7025E-06

10 9.2512E-03 1.5147E-03 1.5643E-04 4.7289E-05 2.3367E-05 1.0999E-05 3.3473E-06

100 9.4592E-03 1.5487E-03 1.5995E-04 4.8352E-05 2.3893E-05 1.1247E-05 3.4226E-06

1000 9.4814E-03 1.5524E-03 1.6032E-04 4.8466E-05 2.3949E-05 1.1273E-05 3.4306E-06

10000 9.4848E-03 1.5529E-03 1.6038E-04 4.8483E-05 2.3957E-05 1.1277E-05 3.4318E-06

100000 9.4863E-03 1.5532E-03 1.6040E-04 4.8491E-05 2.3961E-05 1.1279E-05 3.4324E-06

The results are plotted in graphs 55, 56 and 57.

Case 1:

Figure 55. Settlements (m) vs. number of load cycles (N), case 1

1.50E-06

1.50E-05

1.50E-04

1.50E-03

1.50E-02

1 10 100 1000 10000 100000 1000000

ε acc

Log N

Settlement VS N

z= 0.45 m

z= 1.32 m

z=2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

57

Case 2:

Figure 56. Settlements (m) vs. number of load cycles (N), case 2

Case 3:

Figure 57. Settlements (m) vs. number of load cycles (N), case 3

The figures in all cases show the proportional relation between the number of cycles and the value of

produced settlements. As the number of cycles increases the settlement increases. The remarkable

deformation of soil at various depths was generated after little number of cycles (10 to 50), and then it started

to have a very slight increase with higher number of cycles.

It is also remarkable that, increasing the wheel loads leads to higher deformation in all soil layers. The

deformation increases by almost 25 % in case 2 after increasing the loads without changing the axle

configuration. On the other hand, case 3 shows fewer deformations than that of case 2. The values of

2.00E-06

2.00E-05

2.00E-04

2.00E-03

2.00E-02

1 10 100 1000 10000 100000 1000000

ε acc

Log N

Settlement VS N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

1.80E-06

1.80E-05

1.80E-04

1.80E-03

1.80E-02

1 10 100 1000 10000 100000 1000000

ε acc

Log N

Settlement VS N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

58

settlements obtained at 73000 cycles in case 2 are higher than that obtained after 100000 cycles in case 3.

This indicates that, increasing the number of vehicle axles is an efficient factor in reducing soil settlements.

The total settlement is the value obtained by summing up the settlements at the different depths. Table

25 and figure 58 summarize the results obtained for the three cases:

Table 25. Total settlement (m) for analyzed cases at different load cycles (N)

N Case 1 Case 2 Case 3

1 9.4715E-03 1.0313E-02 8.8754E-03

10 1.1731E-02 1.2774E-02 1.0993E-02

100 1.1995E-02 1.3061E-02 1.1240E-02

1000 1.2023E-02 1.3092E-02 1.1267E-02

10000 1.2028E-02 1.3096E-02 1.1271E-02

100000 1.2030E-02 1.3100E-02 1.1272E-02

Figure 58. Total settlements vs. number of cycles, all cases

The curves clearly show that, increasing the values of loads leads to higher soil settlements and

increasing the number of axles decreases the settlements.

8.80E-03

9.80E-03

1.08E-02

1.18E-02

1.28E-02

1.38E-02

1 10 100 1000 10000 100000 1000000

ε acc

Log N

Settlement VS N

Case 3

Case 2

Case 1

59

5. Conclusions

The conclusions that can be drawn handle in particular the difference between the impact of increasing

vehicle loads and vehicle number of axles with respect to: stresses, accumulation strains and soil settlement.

The most contributing soil parameters in the strain formation according to the review are: soil void ratio,

initial density and the particle shape, and they were taken into consideration in the analysis. They were given

legitimate values in light of Wichtmann et al. (2005) laboratory work that was also helpful to assume some

of the parameters used in Bochum model. The vehicle parameters like the speed, wheel loads and tire

pressure were also given suitable values that fit with the Swedish transportation guidelines. This selection of

parameters gave reasonable outputs to a certain level. However, the limitations related to the way of

calculating stresses and strains must be taken into further consideration. Assuming for example the uniform

load as a point load in the calculations can dispute the credibility of the findings. Nevertheless, the general

behavior of the stresses and strains formation in the subgrade layer with respect to time and with respect to

the number of loading cycles was justifiable and admissible.

The comparison between the three different study cases proved the ultimate effects of axle loads and

axle configurations in the strain formation and consequently on subgrade deformation due to settlement.

Increasing the axle loads plays significant role in the development of vertical stresses and radial strains in the

soil layers. Changing the axle configuration can imminently influence the values of radial stresses in soil

layers where higher number of axles generates more radial stresses.

The overall impact of obtained vertical and radial stresses and strains can be understood by looking on

the values of settlement they produce in the soil layers. Comparing the settlements from the different cases

shows that, increasing the wheel loads leads to much higher settlements in the subgrade. On contrary,

changing the axle configuration by increasing them can relieve the deformation in soil.

According to these findings, it is reasonable to say that, increasing the gross weight of vehicles to 74

tons in Sweden might be acceptable through regulating their axle configurations. This can lead to mitigating

the impact on roads to some extent. This requires further studies for the different types of trucks to determine

the ideal configuration of their axles through which the deformation of soil subgrades can be relieved to

minimum limits.

60

6. References

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Blahova, K., Ševelova, L. & Pilařova, P. (2013). Influence of Water Content on the Shear Strength

Parameters of Clayey Soil in Relation to Stability Analysis of A Hillside in Brno Region. ACTA

Universitatis Agriculturae Et Silviculturae Mendelianae Brunensis, Czech Republic.

Bui, M.T., Clayton, C.R.I. & Priest, J.A. (2007). Effects of Particle Shape on Gmax of Geomaterials. Pitilakis,

K. 4th International Conference on Earthquake Geotechnical Engineering, Paper No. 1536. Aristotle

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Cabalar, A. F, Dulundu, K. & Tuncay, K. (2013). Strength of Various Sands in Triaxial and Cyclic Direct

Shear Tests. Elsevier, Amsterdam.

Cho, G.C., Dodds, J.S. & Santamarina, J.C. (2006). Particle Shape Effects on Packing Density, Stiffness and

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602.

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Structural Design of Asphalt Pavements. University of Michigan, Ann Arbor, Michigan. pp. 392-403.

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Library, Fayetteville, North Carolina.

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GDS Hall Effect Local Strain Transducers. (October, 2017). Retrieved from

http://www.apastyle.org/learn/faqs/web-page-no-author.aspx .

Gedafa, D.S., Hossain, M. & Romanoschi, S.A. (2013). Effect of Traffic Speed on Stresses and Strains in

Asphalt Perpetual Pavement. Journal of Civil Engineering and Architecture, Volume 7, No. 8 (Serial No.

69), pp. 956-963.

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Bottom-Up Reflective Cracking. Civil-Comp Press, Stirlingshire, Scotland.

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Karg, C. (2007). Modelling of Strain Accumulation Due to Low Level Vibrations in Granular Soils. Faculty

of Engineering Ghent University. Ghent, Belgium.

Muszynski, M.R. & Stanley, J.V. (2012). Particle Shape Estimates of Uniform Sands: Visual and Automated

Methods Comparison. Journal of Materials in Civil Engineering 24 (2), 194–206.

Nakao, T. & Fityus, S. (2008). Direct Shear Testing of A Marginal Material Using A Large Shear Box.

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Niemunis, A., Wichtmann, T. and Triantafyllidis, Th. (2005). A High-Cycle Accumulation Model for Sand.

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63

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64

7. Table of abbreviations

Parameter Meaning

Radial stress

Tangential stress

Vertical stress

Stress uniformly applied on a plate of radius a at the

surface of the soil mass

Depth

, Poisson's ratio

Vertical strain

Radial strain

Young's modulus

d Deflection

average mean pressure

average deviatoric pressure

average stress ratio

deviatoric stress amplitude

Volumetric Strain Rate

Vertical Strain Rate

Isotropic Average Stress

Weighted Number of Cycles

Shear strain Amplitude

State Variable

φ State Variable Compaction

Accumulated Strain

Static Failure Deviatoric Stress

ζ amplitude ratio

65

Parameter Meaning

stress amplitude

Point Load

Horizontal coordinate of the point of consideration

(Boussinesq’s equations)

Vertical coordinate of the point of consideration

(Boussinesq’s equations)

Radial distance to the point of consideration

(Boussinesq’s equations)

Layer thickness

Equivalent layer thickness

Correction factor

N Number of loading cycles

Volumetric strain

Deviatoric strain

Initial relative density

Mean grain diameter

U Non uniformity index

Maximum void ratio

Minimum void ratio

Friction angle

Vehicle axle load

Equivalent single axle load= 18000 lb

Average diameter of the inscribed circles formed at

each corner of the particle

Diameter of the greatest diameter of the soil particle

Smallest diameter of the sphere circumscribing the soil

particle

c Cohesion

τmax Maximum shear stress

R Soil particle roundness

S Soil particle spherity

66

Appendix A

Case 1

Table A.1. Truck features, case 1

Speed 60 km/h

Wheel load 1 49 kN

Wheel load 2, 3 44.2 kN

Wheel load 4, 5, 6 42.5 kN

Figure A.1. z= 0.66 m

5.00E-06

5.00E-04

5.00E-02

5.00E+00

5.00E+02

5.00E+04

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

67

Figure A.2. z= 1 m

Figure A.3. z = 2 m

1.30E-04

1.30E-02

1.30E+00

1.30E+02

1.30E+04

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ver

tica

l Str

ess

(P

a)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

6.50E-04

6.50E-02

6.50E+00

6.50E+02

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

68

Figure A.4. z= 3 m

Figure A.5. z= 4 m

2.10E-03

2.10E-02

2.10E-01

2.10E+00

2.10E+01

2.10E+02

2.10E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 6

Total

Wheel 5

4.86E-03

4.86E-02

4.86E-01

4.86E+00

4.86E+01

4.86E+02

4.86E+03

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

69

Figure A.6. z= 6 m

Figure A.7. z= 10 m

1.50E-02

1.50E-01

1.50E+00

1.50E+01

1.50E+02

1.50E+03

-60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertica Stress VS. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

70

Figure A.8. z= 0.66 m

Figure A.9. z= 1 m

-1.00E+04

1.00E+04

3.00E+04

5.00E+04

7.00E+04

9.00E+04

1.10E+05

1.30E+05

1.50E+05

1.70E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-3.50E+04

1.50E+04

6.50E+04

1.15E+05

1.65E+05

2.15E+05

2.65E+05

3.15E+05

3.65E+05

4.15E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

71

Figure A.10. z= 2 m

Figure A.11. z= 3 m

-2.00E+04

3.00E+04

8.00E+04

1.30E+05

1.80E+05

2.30E+05

2.80E+05

3.30E+05

3.80E+05

4.30E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-2.50E+04

7.50E+04

1.75E+05

2.75E+05

3.75E+05

4.75E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

72

Figure A.12. z= 4 m

Figure A.13. z= 6 m

-3.00E+04

7.00E+04

1.70E+05

2.70E+05

3.70E+05

4.70E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-5.00E+04

5.00E+04

1.50E+05

2.50E+05

3.50E+05

4.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

73

Figure A.14. z= 10 m

-5.00E+04

5.00E+04

1.50E+05

2.50E+05

3.50E+05

4.50E+05

5.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

74

Figure A.15. z= 0.66 m

Figure A.16. z= 1 m

-2.00E-04

3.00E-04

8.00E-04

1.30E-03

1.80E-03

2.30E-03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-2.00E-05

3.00E-05

8.00E-05

1.30E-04

1.80E-04

2.30E-04

2.80E-04

3.30E-04

3.80E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

75

Figure A.17. z= 2 m

Figure A.18. z= 3 m

-1.00E-05

1.00E-05

3.00E-05

5.00E-05

7.00E-05

9.00E-05

1.10E-04

1.30E-04

1.50E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-3.00E-06

7.00E-06

1.70E-05

2.70E-05

3.70E-05

4.70E-05

5.70E-05

6.70E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

76

Figure A.19. z= 4 m

Figure A.20. z= 6 m

-4.00E-06

1.00E-06

6.00E-06

1.10E-05

1.60E-05

2.10E-05

2.60E-05

3.10E-05

3.60E-05

4.10E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-1.00E-06

4.00E-06

9.00E-06

1.40E-05

1.90E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

77

Figure A.21. z= 10 m

-7.21E-08

9.28E-07

1.93E-06

2.93E-06

3.93E-06

4.93E-06

5.93E-06

6.93E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

78

Figure A.22. z = 0.66 m

Figure A.23. z= 1 m

-9.00E-04

-8.00E-04

-7.00E-04

-6.00E-04

-5.00E-04

-4.00E-04

-3.00E-04

-2.00E-04

-1.00E-04

-2.40E-18

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

-1.55E-04

-1.35E-04

-1.15E-04

-9.50E-05

-7.50E-05

-5.50E-05

-3.50E-05

-1.50E-05

5.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Series6

Total

79

Figure A.24. z= 2 m

Figure A.25. z= 3 m

-5.50E-05

-4.50E-05

-3.50E-05

-2.50E-05

-1.50E-05

-5.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

-2.60E-05

-2.10E-05

-1.60E-05

-1.10E-05

-6.00E-06

-1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

80

Figure A.26. z= 4 m

Figure A.27. z= 6 m

-1.50E-05

-1.30E-05

-1.10E-05

-9.00E-06

-7.00E-06

-5.00E-06

-3.00E-06

-1.00E-06

1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-6.00E-06

-5.00E-06

-4.00E-06

-3.00E-06

-2.00E-06

-1.00E-06

-1.90E-19

1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

81

Figure A.28. z= 10 m

-2.00E-06

-1.50E-06

-1.00E-06

-5.00E-07

-7.00E-20

5.00E-07

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

82

Case 2

Table A.2. Truck features, case 2

Figure A.29. z= 0.66 m

6.80E-06

6.80E-04

6.80E-02

6.80E+00

6.80E+02

6.80E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

Speed 60 km/h

Wheel load 1 49 kN

Wheel load 2, 3 63.7 kN

Wheel load 4, 5, 6 63.7 kN

83

Figure A.30. z= 1 m

Figure A.31. z= 2 m

1.70E-04

1.70E-03

1.70E-02

1.70E-01

1.70E+00

1.70E+01

1.70E+02

1.70E+03

1.70E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

8.00E-04

8.00E-03

8.00E-02

8.00E-01

8.00E+00

8.00E+01

8.00E+02

8.00E+03

8.00E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(P

a)

Distance (m)

Vertical Stress vs. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

84

Figure A.32. z= 3 m

Figure A.33. z= 4 m

2.52E-03

2.52E-02

2.52E-01

2.52E+00

2.52E+01

2.52E+02

2.52E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

5.60E-03

5.60E-02

5.60E-01

5.60E+00

5.60E+01

5.60E+02

5.60E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

Toatl

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

85

Figure A.34. z= 6 m

Figure A.35. z= 10 m

1.75E-02

1.75E-01

1.75E+00

1.75E+01

1.75E+02

1.75E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(P

a)

Distance (m)

Vertical Stress vs. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

7.00E-02

7.00E-01

7.00E+00

7.00E+01

7.00E+02

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

Total

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

86

Figure A.36. z= 0.66 m

Figure A.37.z=1m

-3.00E+04

2.00E+04

7.00E+04

1.20E+05

1.70E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-5.00E+04

0.00E+00

5.00E+04

1.00E+05

1.50E+05

2.00E+05

2.50E+05

3.00E+05

3.50E+05

4.00E+05

4.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

87

Figure A.38. z= 2 m

Figure A.39. z= 3 m

-3.00E+04

2.00E+04

7.00E+04

1.20E+05

1.70E+05

2.20E+05

2.70E+05

3.20E+05

3.70E+05

4.20E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-3.00E+04

2.00E+04

7.00E+04

1.20E+05

1.70E+05

2.20E+05

2.70E+05

3.20E+05

3.70E+05

4.20E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

88

Figure A.40. z= 4 m

Figure A.41. z= 6 m

-4.00E+04

6.00E+04

1.60E+05

2.60E+05

3.60E+05

4.60E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-4.00E+04

6.00E+04

1.60E+05

2.60E+05

3.60E+05

4.60E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

89

Figure A.42. z= 10 m

-4.00E+04

6.00E+04

1.60E+05

2.60E+05

3.60E+05

4.60E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

total

90

Figure A.43. z= 0.66 m

Figure A.44. z= 1 m

-3.00E-04

2.00E-04

7.00E-04

1.20E-03

1.70E-03

2.20E-03

2.70E-03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-4.00E-05

6.00E-05

1.60E-04

2.60E-04

3.60E-04

4.60E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

91

Figure A.45. z= 2 m

Figure A.46. z= 3 m

-1.50E-05

3.50E-05

8.50E-05

1.35E-04

1.85E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-8.00E-06

2.00E-06

1.20E-05

2.20E-05

3.20E-05

4.20E-05

5.20E-05

6.20E-05

7.20E-05

8.20E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

92

Figure A.47. z= 4 m

Figure A.48. z= 6 m

-5.00E-06

5.00E-06

1.50E-05

2.50E-05

3.50E-05

4.50E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

-2.28E-06

2.72E-06

7.72E-06

1.27E-05

1.77E-05

2.27E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

total

93

Figure A.49. z= 10 m

-8.31E-07

1.17E-06

3.17E-06

5.17E-06

7.17E-06

9.17E-06

1.12E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

Total

94

Figure A.50. z= 0.66 m

Figure A.51. z= 1 m

-1.10E-03

-9.00E-04

-7.00E-04

-5.00E-04

-3.00E-04

-1.00E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

-1.90E-04

-1.40E-04

-9.00E-05

-4.00E-05

1.00E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

total

95

Figure A.52. z= 2 m

Figure A.53. z= 3 m

-6.60E-05

-5.60E-05

-4.60E-05

-3.60E-05

-2.60E-05

-1.60E-05

-6.00E-06

4.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

-3.10E-05

-2.60E-05

-2.10E-05

-1.60E-05

-1.10E-05

-6.00E-06

-1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

96

Figure A.54. z= 4 m

Figure A.55. z= 6 m

-1.80E-05

-1.30E-05

-8.00E-06

-3.00E-06

2.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

total

-8.00E-06

-6.00E-06

-4.00E-06

-2.00E-06

-1.30E-19

2.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

97

Figure A.56. z= 10 m

-2.50E-06

-2.00E-06

-1.50E-06

-1.00E-06

-5.00E-07

-8.00E-20

5.00E-07

1.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Wheel 6

Total

98

Case 3

Table A.3. Truck features, case 3

Figure A.57. z= 0.66 m

5.00E-06

5.00E-05

5.00E-04

5.00E-03

5.00E-02

5.00E-01

5.00E+00

5.00E+01

5.00E+02

5.00E+03

5.00E+04

5.00E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(P

a)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

Speed 60 km/h

Wheel load 1 49

Wheel load 2, 3, 4 49

Wheel load 5, 6, 7, 41.7

99

Figure.A.58…z=1m

Figure A.59. z= 2 m

1.45E-04

1.45E-03

1.45E-02

1.45E-01

1.45E+00

1.45E+01

1.45E+02

1.45E+03

1.45E+04

1.45E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

wheel1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

5.13E-04

5.13E-03

5.13E-02

5.13E-01

5.13E+00

5.13E+01

5.13E+02

5.13E+03

5.13E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

100

Figure A.60. z= 3 m

Figure A.61. z= 4 m

1.30E-03

1.30E-02

1.30E-01

1.30E+00

1.30E+01

1.30E+02

1.30E+03

1.30E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

4.77E-03

4.77E-02

4.77E-01

4.77E+00

4.77E+01

4.77E+02

4.77E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

101

Figure A.62. z= 6 m

Figure..A.63.z=10m

1.20E-02

1.20E-01

1.20E+00

1.20E+01

1.20E+02

1.20E+03

1.20E+04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ess

(Pa)

Distance (m)

Vertical Stress vs. Distance

wheel1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

tre

ss (

Pa)

Distance (m)

Vertical Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

102

Figure A.64. z= 0.66 m

Figure A.65. z= 1 m

-2.00E+04

3.00E+04

8.00E+04

1.30E+05

1.80E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-5.00E+04

5.00E+04

1.50E+05

2.50E+05

3.50E+05

4.50E+05

5.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

103

Figure A.66. z= 2 m

Figure A.67. z= 3 m

-8.00E+04

2.00E+04

1.20E+05

2.20E+05

3.20E+05

4.20E+05

5.20E+05

6.20E+05

7.20E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-5.00E+04

5.00E+04

1.50E+05

2.50E+05

3.50E+05

4.50E+05

5.50E+05

6.50E+05

7.50E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

104

Figure A.68. z= 4 m

Figure A.69. z= 6 m

-6.00E+04

4.00E+04

1.40E+05

2.40E+05

3.40E+05

4.40E+05

5.40E+05

6.40E+05

7.40E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-6.00E+04

4.00E+04

1.40E+05

2.40E+05

3.40E+05

4.40E+05

5.40E+05

6.40E+05

7.40E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tre

ss (

Pa)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

105

Figure A.70. z= 10 m

-1.00E+05

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

7.00E+05

8.00E+05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

tres

s (P

a)

Distance (m)

Radial Stress vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

106

Figure A.71. z= 0.66 m

Figure A.72. z= 1 m

-3.00E-04

2.00E-04

7.00E-04

1.20E-03

1.70E-03

2.20E-03

2.70E-03

3.20E-03

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-5.00E-05

5.00E-05

1.50E-04

2.50E-04

3.50E-04

4.50E-04

5.50E-04

6.50E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

107

Figure A.73. z= 2 m

Figure A.74. z= 3 m

-2.00E-05

3.00E-05

8.00E-05

1.30E-04

1.80E-04

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

whee4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-8.00E-06

1.20E-05

3.20E-05

5.20E-05

7.20E-05

9.20E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

108

Figure A.75. z= 4 m

Figure A.76. z= 6 m

-8.00E-06

2.00E-06

1.20E-05

2.20E-05

3.20E-05

4.20E-05

5.20E-05

6.20E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-1.70E-06

3.30E-06

8.30E-06

1.33E-05

1.83E-05

2.33E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ve

rtic

al S

trai

n

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

109

Figure A.77. z= 10 m

-1.00E-06

1.00E-06

3.00E-06

5.00E-06

7.00E-06

9.00E-06

1.10E-05

1.30E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Ver

tica

l Str

ain

Distance (m)

Vertical Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

110

Figure A.87. z= 0.66 m

Figure A.79. z= 1 m

-1.20E-03

-1.00E-03

-8.00E-04

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

Series6

wheel 7

wheel 8

Total

-3.80E-04

-3.30E-04

-2.80E-04

-2.30E-04

-1.80E-04

-1.30E-04

-8.00E-05

-3.00E-05

2.00E-05

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

111

Figure A.80. z= 2 m

Figure A.81. z= 3 m

-9.00E-05

-8.00E-05

-7.00E-05

-6.00E-05

-5.00E-05

-4.00E-05

-3.00E-05

-2.00E-05

-1.00E-05

-1.26E-18

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-3.50E-05

-3.00E-05

-2.50E-05

-2.00E-05

-1.50E-05

-1.00E-05

-5.00E-06

-1.02E-18

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

112

Figure A.82. z= 4 m

Figure A.83. z= 6 m

-2.80E-05

-2.30E-05

-1.80E-05

-1.30E-05

-8.00E-06

-3.00E-06

2.00E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

-7.50E-06

-5.50E-06

-3.50E-06

-1.50E-06

5.00E-07

2.50E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

113

Figure A.84. z= 10 m

-2.60E-06

-2.10E-06

-1.60E-06

-1.10E-06

-6.00E-07

-1.00E-07

4.00E-07

9.00E-07

1.40E-06

-70.00 -50.00 -30.00 -10.00 10.00 30.00 50.00 70.00

Rad

ial S

trai

n

Distance (m)

Radial Strain vs. Distance

wheel 1

wheel 2

wheel 3

wheel 4

wheel 5

wheel 6

wheel 7

wheel 8

Total

114

Appendix B

Case 1

Figure B.1. Wheel 1, v=60 km/h, vertical stress, z= 0.45 m

Figure B.2. Wheel 1, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.83E-06

6.83E-05

6.83E-04

6.83E-03

6.83E-02

6.83E-01

6.83E+00

6.83E+01

6.83E+02

6.83E+03

6.83E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

115

Figure B.3. Wheel 2, v=60 km/h, vertical stress, z= 0.45 m

Figure B.4. Wheel 2, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

2.00E-02

2.00E-01

2.00E+00

2.00E+01

2.00E+02

2.00E+03

2.00E+04

2.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

116

Figure B.5. Wheel 3, v=60 km/h, vertical stress, z= 0.45 m

Figure B.6. Wheel 3, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

2.00E-02

2.00E-01

2.00E+00

2.00E+01

2.00E+02

2.00E+03

2.00E+04

2.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

117

Figure B.7. Wheel 4, v=60 km/h, vertical stress, z= 0.45 m

Figure B.8. Wheel 4, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

7.00E-01

7.00E+00

7.00E+01

7.00E+02

7.00E+03

7.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

118

Figure B.9. Wheel 5, v=60 km/h, vertical stress, z= 0.45 m

Figure B.10. Wheel 5, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-07

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

7.00E-01

7.00E+00

7.00E+01

7.00E+02

7.00E+03

7.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

119

Figure B.11. Wheel 6, v=60 km/h, vertical stress, z= 0.45 m

Figure B.12. Wheel 6, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time …

-60 -40 -20 0 20 40 60

7.00E-01

7.00E+00

7.00E+01

7.00E+02

7.00E+03

7.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

120

Figure B.13. Wheel 1, v=60 km/h, vertical stress, z= 6.24 m

Figure B.14. Wheel 1, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

121

Figure B.15. Wheel 2, v=60 km/h, vertical stress, z= 6.24 m

Figure B.16. Wheel 2, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

122

Figure B.17. Wheel 3, v=60 km/h, vertical stress, z= 6.24 m

Figure B.18. Wheel 3, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

123

Figure B.19. Wheel 4, v=60 km/h, vertical stress, z= 6.24 m

Figure B.20. Wheel 4, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

124

Figure B.21. Wheel 5, v=60 km/h, vertical stress, z= 6.24 m

Figure B.22. Wheel 5, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

125

Figure B.23. Wheel 6, v=60 km/h, vertical stress, z= 6.24 m

Figure B.24. Wheel 6, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

126

Figure B.25. Wheel 1, v=60 km/h, vertical stress, z= 10.24

Figure B.26. Wheel 1, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

127

Figure B.27. Wheel 2, v=60 km/h, vertical stress, z= 10.24 m

Figure B.28. Wheel 2, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

128

Figure B.29. Wheel 3, v=60 km/h, vertical stress, z= 10.24 m

Figure B.30. Wheel 3, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

129

Figure B.31. Wheel 4, v=60 km/h, vertical stress, z= 10.24 m

Figure B.32. Wheel 4, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

130

Figure B.33. Wheel 5, v=60 km/h, vertical stress, z= 10.24 m

Figure B.34. Wheel 5, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

131

Figure B.35. Wheel 6, v=60 km/h, vertical stress, z= 10.24 m

Figure B.36. Wheel 6, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

132

Case 2

Figure B.37. Wheel 1, v=60 km/h, vertical stress, z = 0.45 m

Figure B.38. Wheel 1, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

133

Figure B.39. Wheel 2, v=60 km/h, vertical stress, z = 0.45 m

Figure B.40. Wheel 2, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

134

Figure B.41. Wheel 3, v=60 km/h, vertical stress, z = 0.45 m

Figure B.42. Wheel 3, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

135

Figure B.43. Wheel 4, v=60 km/h, vertical stress, z = 0.45 m

Figure B.44. Wheel 4, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

136

Figure B.45. Wheel 4, v=60 km/h, vertical stress, z = 0.45 m

Figure B.46. Wheel 4, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

137

Figure B.47. Wheel 5, v=60 km/h, vertical stress, z = 0.45 m

Figure B.48. Wheel 5, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

138

Figure B.49. Wheel 6, v=60 km/h, vertical stress, z = 0.45 m

Figure B.50. Wheel 6, v=60 km/h, radial stress, z = 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

139

Figure B.51. Wheel 1, v=60 km/h, vertical stress, z= 6.24 m

Figure B.52. Wheel 1, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

140

Figure B.53. Wheel 2, v=60 km/h, vertical stress, z= 6.24 m

Figure B.54. Wheel 2, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

141

Figure B.55. Wheel 3, v=60 km/h, vertical stress, z= 6.24 m

Figure B.56. Wheel 3, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

142

Figure B.57. Wheel 4, v=60 km/h, vertical stress, z= 6.24 m

Figure B.58. Wheel 4, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

143

Figure B.59. Wheel 5, v=60 km/h, vertical stress, z= 6.24 m

Figure B.60. Wheel 5, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

144

Figure B.61. Wheel 6, v=60 km/h, vertical stress, z= 6.24 m

Figure B.62. Wheel 6, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

145

Figure B.63. Wheel 1, v=60 km/h, vertical stress, z= 10.24 m

Figure B.64. Wheel 1, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

146

Figure B.65. Wheel 2, v=60 km/h, vertical stress, z= 10.24 m

Figure B.66. Wheel 2, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σr

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

147

Figure B.67. Wheel 3, v=60 km/h, vertical stress, z= 10.24 m

Figure B.68. Wheel 3, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

148

Figure B.69. Wheel 4, v=60 km/h, vertical stress, z= 10.24 m

Figure B.70. Wheel 4, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

149

Figure B.71. Wheel 5, v=60 km/h, vertical stress, z= 10.24 m

Figure B.72. Wheel 5, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

0.006

0.06

0.6

6

60

600

6000

60000

600000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

150

Figure B.73. Wheel 6, v=60 km/h, vertical stress, z= 10.24 m

Figure B.74. Wheel 6, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

151

Case 3

Figure B.75. Wheel 1, v=60 km/h, vertical stress, z= 0.45 m

Figure B.76. Wheel 1, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

152

Figure B.77. Wheel 2, v=60 km/h, vertical stress, z= 0.45 m

Figure B.78. Wheel 2, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

153

Figure B.79. Wheel 3, v=60 km/h, vertical stress, z= 0.45 m

Figure B.80. Wheel 3, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

154

Figure B.81. Wheel 4, v=60 km/h, vertical stress, z= 0.45 m

Figure B.82. Wheel 4, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

155

Figure B.83. Wheel 5, v=60 km/h, vertical stress, z= 0.45 m

Figure B.84. Wheel 5, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

156

Figure B.85. Wheel 6, v=60 km/h, vertical stress, z= 0.45 m

Figure B.86. Wheel 6, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

157

Figure B.87. Wheel 7, v=60 km/h, vertical stress, z= 0.45 m

Figure B.88. Wheel 7, v=60 km/h, radial stress, z= 0.45 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

158

Figure B.89. Wheel 1, v=60 km/h, vertical stress, z= 6.24 m

Figure B.90. Wheel 1, v=60 km/h, radial stress, z= 6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

159

Figure B.91. Wheel 2, v=60 km/h, vertical stress, z=6.24 m

Figure B.92. Wheel 2, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

160

Figure B.93. Wheel 3, v=60 km/h, vertical stress, z=6.24 m

Figure B.94. Wheel 3, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

161

Figure B.95. Wheel 4, v=60 km/h, vertical stress, z=6.24 m

Figure B.96. Wheel 4, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

162

Figure B.97. Wheel 5, v=60 km/h, vertical stress, z=6.24 m

Figure B.98. Wheel 5, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

163

Figure B.99. Wheel 6, v=60 km/h, vertical stress, z=6.24 m

Figure B.100. Wheel 6, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

164

Figure B.101. Wheel 7, v=60 km/h, vertical stress, z=6.24 m

Figure B.102. Wheel 7, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

165

Figure B.103. Wheel 8, v=60 km/h, vertical stress, z=6.24 m

Figure B.104. Wheel 8, v=60 km/h, radial stress, z=6.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

166

Figure B.105. Wheel 1, v=60 km/h, vertical stress, z= 10.24 m

Figure B.106. Wheel 1, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

167

Figure B.107. Wheel 2, v=60 km/h, vertical stress, z= 10.24 m

Figure B.108. Wheel 2, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

168

Figure B.109. Wheel 3, v=60 km/h, vertical stress, z= 10.24 m

Figure B.110. Wheel 3, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

169

Figure B.111. Wheel 4, v=60 km/h, vertical stress, z= 10.24 m

Figure B.112. Wheel 4, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

170

Figure B.113. Wheel 5, v=60 km/h, vertical stress, z= 10.24 m

Figure B.114. Wheel 5, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

171

Figure B.115. Wheel 6, v=60 km/h, vertical stress, z= 10.24 m

Figure B.116. Wheel 6, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

172

Figure B.117. Wheel 7, v=60 km/h, vertical stress, z= 10.24 m

Figure B.118. Wheel 7, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

173

Figure B.119. Wheel 8, v=60 km/h, vertical stress, z= 10.24 m

Figure B.120. Wheel 8, v=60 km/h, radial stress, z= 10.24 m

-60 -40 -20 0 20 40 60

6.00E-06

6.00E-05

6.00E-04

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σz

(Pa)

Time (sec)

Distance (m)

Time vs σz

-60 -40 -20 0 20 40 60

6.00E-03

6.00E-02

6.00E-01

6.00E+00

6.00E+01

6.00E+02

6.00E+03

6.00E+04

6.00E+05

0.00 1.00 2.00 3.00 4.00 5.00 6.00

σr(

Pa)

Time (sec)

Distance (m)

Time vs σr

174

Appendix C

Case 1

Figure C.1. Strain rate vs. number of cycles, case 1

Figure C.2. Accumulated strains vs. number of cycles, case 1

5.00E-11

5.00E-10

5.00E-09

5.00E-08

5.00E-07

5.00E-06

5.00E-05

5.00E-04

5.00E-03

1 10 100 1000 10000 100000 1000000

ε'

Log N

ε' vs N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

1.20E-07

1.20E-06

1.20E-05

1.20E-04

1.20E-03

1.20E-02

1 10 100 1000 10000 100000 1000000

ε

Log N

ε vs N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z=10.24 m

175

Case 2

Figure C.3. Strain rate vs. number of cycles, case 2

Figure C. 4. Accumulated strains vs. number of cycles, case 2

1.50E-10

1.50E-09

1.50E-08

1.50E-07

1.50E-06

1.50E-05

1.50E-04

1.50E-03

1 10 100 1000 10000 100000 1000000

ε'

Log N

ε' vs N

z=0.45 m

z=1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

8.00E-07

8.00E-06

8.00E-05

8.00E-04

8.00E-03

1 10 100 1000 10000 100000 1000000

ε

Log N

ε vs. N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

176

Case 3

Figure C.5. Strain rates vs. number of cycles, case 3

Figure C.6. Accumulated strains vs. number of cycles, case 3

5.00E-11

5.00E-10

5.00E-09

5.00E-08

5.00E-07

5.00E-06

5.00E-05

5.00E-04

5.00E-03

1 10 100 1000 10000 100000 1000000

ε'

Log N

ε' vs N

z=0.45 m

z=1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m

2.00E-07

2.00E-06

2.00E-05

2.00E-04

2.00E-03

2.00E-02

1 10 100 1000 10000 100000 1000000

ε

Log N

ε vs N

z= 0.45 m

z= 1.32 m

z= 2.24 m

z= 3.24 m

z= 4.24 m

z= 6.24 m

z= 10.24 m