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Flexible pavements consist of one or more asphalt layers and usually also a base. Mostly the base is composed of unbound (granular) materials but also bound bases (obtained by stabilizing the base material with e.g. cement) are applied. In The Netherlands the asphalt layer(s) plus the (un)bound base are normally resting on sand, either the natural sand subgrade or a constructed sand sub-base. Figure 7.1 is an example of a flexible pavement structure for a motorway.
50 mm porous asphalt wearing course
210 mm stone asphalt concrete,
4 layers: 3 x 50 mm + 1 x 60 mm
300 mm unbound base of e.g. concrete granulate or mix granulate sand sub-base or sand subgrade
Figure 7.1: Example of a flexible pavement structure for a heavily loaded motorway.
Nowadays the design of the thickness of pavements for roads, airports, industrial yards etc. is based on the calculation of stresses and strains, occurring within the structure due to the traffic loadings, and the comparison with the allowable stresses and strains. In this respect the thickness design of a pavement is essentially the same as for e.g. a concrete beam.
Usually the linear elastic multi-layer theory is used to calculate the occurring stresses and strains. This however implies that the actual material behavior is simplified to a great extent because most road building materials don’t behave linear elastic (see chapter 4). Unbound materials behave strongly stress dependent and asphalt mixes are visco-elastic materials. Nevertheless, the assumption of linear elastic material behavior is in most cases justified and that is certainly the case if the occurring stresses and strains in the structure are rather limited.
Of course the traffic loading has to be known to enable the thickness design of the pavement structure. Furthermore the elastic modulus of the various pavement layers must be known as the amount of traffic load spreading strongly depends on the bending stiffness of the subsequent layers. From basic applied mechanics it is known that the bending stiffness is related to the product E.h
3 , where E is the elastic modulus and h the layer thickness.
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A pavement structure is a three-dimensional structure and for that reason also the Poisson ratio of the various layers is relevant.
Finally one should know whether the subsequent pavement layers are fully bonded (which implies that the horizontal displacements just above and just below the interface are equal) or that they can move relatively to each other in the horizontal direction.
In this chapter it will be explained how the occurring stresses and strains in a flexible pavement can be calculated. The mathematical backgrounds are however not discussed as they are rather complicated. Instead use will be made of available graphs and computer programs.
First the occurring stresses and strains in a half-space will be discussed.
Although Boussinesq’s theory already has been explained in the course on
Soil Mechanics, in this course it will be demonstrated how this theory can be applied in the structural design of earth and gravel roads.
Then the occurring stresses and strains in a two-layer system are discussed.
An asphalt pavement laid directly on top of a sand subgrade (so without a base) is an example of a two-layer system.
Next attention is paid to three-layer systems and multi-layer systems. The occurring stresses and strains in this type of structures can be calculated by means of a computer program that is added to this lecture note.
Finally it is demonstrated how all this information can be used in the thickness design of an asphalt pavement structure.
When a load, uniformly distributed over a circular contact area (e.g. a truck wheel load) is placed on a homogeneous soil then normal and shear stresses occur at any soil element. This is schematically shown in figure 7.2.
Logically the stresses are dependent on the magnitude of the wheel load, the radius of the circular contact area and the distance to the center of the load.
Boussinesq has developed equations to determine the vertical stress and the radial stress on a vertical line through the load center (the shear stresses are zero because of symmetry). These equations are:
σ z
= p ( 1 – z 3 / {( a 2 + z 2 ) 1.5
})
σ r
= ( p/2 ) {( 1 + 2 ν ) – 2 ( 1 + ν ) z / [( a 2 + z 2 ) 1.5
] + z 3 / [( a 2 + z 3 ) 1.5
]}
σ t
= σ r
τ rz
= τ zr
= 0; τ rt
= τ tr
= 0; τ zt
= τ tz
= 0 where: p = a = radius of the load contact area, z = depth below the surface, r = radial distance to the load centre,
ν =
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Figure 7.2: Stresses in a half-space due to a circular load (1).
Figure 7.3 gives in a graphical way the vertical, radial, tangential and shear stresses as a function of the depth z, the distance to the load center z and
Poisson’s ratio ν .
The use of the graphs is illustrated by means of a practical example that deals with the evaluation of an earth road in a tropical African country. The trucks on the road transport cacao, trees, cement etc. and in general they are overloaded: axle loads of 150 kN frequently occur. The number of trucks is however low, say a few trucks per day. The unpaved road has a top layer of laterite (a red-colored tropical weathered material) and for reason of simplicity it is assumed that this may be considered as a half-space. The question now is whether damage will occur on this road, while it is known that the cohesion and the angle of internal friction of the applied laterite have the following values.
Cohesion c [kPa] Angle of internal friction ϕ [ 0 ] season
Wet
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Figure 7.3: Stresses in a half-space due to a circular load (1).
Assume that wide base tyres are mounted on all the truck axles; this means that at either side of any axle there is one tyre with a load of 75 kN. It is further assumed that the tyre pressure in all cases is 850 kPa. As stated earlier, as a first approximation the contact pressure between the tyre and the road surface can be taken equal to the tyre pressure. This implies that p = 850 kPa.
The radius a of the circular contact area then follows from: a = √ ( 75 / [ 850 x π ] ) = 0.168 m
The Poisson’s ratio is taken as 0.5.
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In this specific example only the stresses in the load center (r = 0) are taken into account.
It follows from figure 7.3 that the occurring deviatoric stress σ dev a depth of 0.168 m (z = a):
is greatest at
σ z
= 0.6 x p = 510 kPa, σ r
= σ t
= 0.1 x p = 85 kPa, σ dev
= σ z
- σ t
= 425 kPa
The Mohr’s circle of occurring stresses now can be drawn, see figure 7.4. This figure learns that in the dry season the stress circle remains very much below
Coulomb’s failure envelope. To a smaller extent this is also valid for the (most critical) wet season. The conclusion from this analysis is that the laterite road is strong enough to carry the limited number of 150 kN axle loads.
But then another transport-firm starts to use the road and that firm places such a great amount of products on its trucks that it results in extreme heavy axle loads of 225 kN. In such a case also the tyre pressure must increase, say to 1275 kPa. So both the axle load and the tyre pressure increase with a factor of 1.5. This means that the radius of the contact area remains the same: a = 0.168 m. The occurring stresses at the depth z = 0.168 m thus also increase with a factor of 1.5. Figure 7.4 shows that the Mohr’s circle for these occurring stresses just touches the Coulomb’s failure envelope for the wet season. This means that the road immediately fails (shear failure) due to the passage of only one such heavily overloaded truck in the wet season! cirkels van Mohr en faalomhullenden
1200
1000
800
600
400
200 faalomhullende natte seizoen cirkel van Mohr 150 kN as cirkel van Mohr 225 kN as faalomhullende droge seizoen
0
0
-200
200 400 600 800 1000 1200 1400 1600 1800 2000 spanning [kPa]
Figure 7.4: Mohr’s circles and Coulomb’s failure envelopes for the laterite road.
In The Netherlands earth and gravel roads form only a very small part of the road network. However, still today the great majority of the world road network
(around 70%) consists of earth and gravel roads!
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In this course emphasis is however laid to flexible pavement structures that are relevant for The Netherlands. As already mentioned these structures nearly always consist of asphalt layers and a base on top of sand (sub-base or subgrade). In some cases a base is however not applied and the asphalt layers are directly laid on the subgrade. In such a case a two-layer system is present and in the next paragraph it is discussed how the occurring stresses due to traffic loadings can be calculated in such a system.
Burmister was the first person that developed mathematical solutions for the calculation of the stresses due to traffic loadings in a two-layer system. These mathematical solutions are also transformed into graphs and the most important ones are presented in the figures 7.5, 7.6 en 7.7. Figure 7.5 enables the determination of the radial stress at the bottom of the top-layer in the load center. The vertical stress at the top of the subgrade in the load centre can be determined with figure 7.6. Finally figure 7.7 allows the determination of the vertical displacement (deflection) at the pavement surface in the load center.
It is important to realize that the magnitude of the occurring traffic load stresses is dependent on the magnitude and the geometry of the load, the ratio of the thickness of the top-layer and the radius of the circular contact area, and the ratio of the elastic modulus values of the top-layer and the bottom layer (subgrade).
When using the graphs it should be realized that they are all valid for a
Poisson’s ratio of 0.5 for both layers and that full bond between the top-layer and the subgrade has been assumed.
The use of the graphs is illustrated with an example for a motorway pavement structure that consists of 300 mm asphalt (h) directly laid on the sand subgrade. The elastic modulus E elastic modulus E
2
1
of the asphalt amounts 5000 MPa and the
of the sand subgrade is 100 MPa. The pavement structure is subjected to wheel loadings of 50 kN and the tyre pressure (contact pressure) is 700 kPa. We want to know the radial stress at the bottom of the asphalt top-layer in the load center as well as the vertical stress at the top of the sand subgrade in the load center.
It can be calculated from the magnitude of the wheel load and the contact pressure that the radius of the circular contact area a = 150 mm.
So we find:
E
1
/ E
2
= 50, h / a = 2, p = 700 kPa.
To determine the radial stress at the bottom of the asphalt the bottom graph of figure 7.5 is the easiest one to use. It is read from this graph:
σ r
/ p = 1
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Figure 7.5: Graphs for determination of the radial stress in the load center at the bottom of the top-layer of a two-layer system (1).
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Figure 7.6: Graph for determination of the vertical stress in the load center at the top of the bottom layer of a two-layer system (1).
Figure 7.7: Graph for determination of the vertical displacement (deflection) in the load center at the surface of a two-layer system (1).
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The minus sign means that the radial stress is a flexural tensile stress because the contact pressure is a compressive stress. In the remaining part of this calculation example tensile stresses are however given a positive sign and compressive stresses a negative sign, which results in:
σ r
= -1 x p = -1 x -700 = 700 kPa
It appears from figure 7.6 that:
σ z
/ p = 0.043
In this case σ z
and p have the same sign and that means that σ compressive stress. This leads to: z
is a
σ z
= 0.043 x p = 0.043 x -700 = -30 kPa
In chapter 4 it has been explained that knowledge about the fatigue behavior of asphalt is important because a (truck) wheel load does not pass only one time over the pavement but millions of times. It was also discussed in chapter
4 that usually the occurring strain instead of the stress is used as input in the asphalt fatigue relationship. This implies that the occurring strain at the bottom of the asphalt layer must be known for the determination of the allowable number of load repetitions until fatigue damage (cracking) occurs.
This strain cannot be calculated with the equation ε = σ /E because at the bottom of the asphalt layer there is not a one-dimensional but a threedimensional stress situation.
In the load centre at the bottom of the asphalt layer there is not only a radial stress σ r
but also a tangential stress σ t
(see also figure 7.2). The vertical line through the load center is the axis of symmetry, therefore is valid σ t
= σ r and the shear stresses are zero
.
Furthermore there is a vertical stress at the bottom of the asphalt layer.
Because of the required balance of vertical stresses the vertical stress at the bottom of the asphalt layer is equal to the vertical stress at the top of the subgrade, and this has already been determined above.
At the bottom of the asphalt layer in the load center thus the following stresses are present:
σ r
= σ t
= 700 kPa, σ z
= -30 kPa
The radial strain at the bottom of the asphalt layer can now be calculated with the equation:
ε r
= [ σ r
- νσ t
- νσ z
] / E
1
= [0.7 – 0.5 x 0.7 – 0.5 x (-0.03)] / 5000 = 7.3 x 10 -5
Be aware of the fact that the stresses were calculated in kPa while the elastic modulus E
1
of the asphalt was given in MPa. For the calculation of the asphalt strain all values are given in MPa.
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To enable the calculation of the vertical strain ε z
at the top of the subgrade the radial stress σ r
and the tangential stress σ t
at that location must be known.
These stresses are however absolutely not equal to σ r
and σ t
at the bottom of the asphalt layer.
Another question is whether it is also possible to calculate the stresses σ z
, σ r and σ t
at the surface of the top-layer in the load center. This is not possible through the given graphs but reasonable estimates can nevertheless be made. Because of the balance of vertical stresses, the vertical stress at the surface of the top-layer must be equal to the contact pressure, so in that point is valid:
σ z
= -700 kPa.
It is furthermore known that the asphalt top-layer behaves as a bending beam under the wheel loading and that its neutral line will be somewhat below the middle of the top-layer. When the ratio E
1
/ E
2
increases the neutral line moves into the direction of the middle of the top-layer. The horizontal stresses at the top of the layer therefore will be about equal to the horizontal stresses at the bottom of the layer. The sign is however opposite as through the bending flexural compressive stresses are present in the upper part of the asphalt layer and flexural tensile stresses in the lower part. At the surface of the top-layer in the load center the stresses are thus:
σ r
= σ t
≈ -700 kPa
Figure 7.8 presents the radial stresses σ r
in a two-layer system. The figure makes clear that the top-layer indeed acts as a bending beam: in the case of a ratio E
1
/ E top-layer.
2
of 10 and higher the neutral line is about in the middle of the
Figure 7.8: Radial stresses in the load center as a function of depth in a twolayer system (1).
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Graphs are also available to determine the occurring stresses, strains and displacements in three-layer systems. The use of these graphs is however rather complicated and therefore no attention is given to them. Another reason to do so is that the analyses can also be done fast and easy with one of the available linear-elastic multi-layer computer programs. In this paragraph therefore the computer program WESLEA is discussed that is added to these lecture notes on a CD-ROM. Appendix I gives a short description how the input for this program has to be prepared and how the output is obtained. The use of the WESLEA program is further explained here by discussing a small example problem.
The example problem concerns the calculation, for the three-layer system depicted in figure 7.9, of the stresses and strains at the bottom of the asphalt layer and at the top of the subgrade, in both cases in the load centre. The required input parameters are all given in figure 7.9. Full bond between the various layers is assumed. The location at the bottom of the asphalt layer is referred to as ‘position 1’ and the location at the top of the subgrade as
‘position 2’.
After having prepared the input as explained in Appendix I and having done the calculation, the results given in table 7.1 are obtained.
Remark! The sign convention used in WESLEA is different from the one used until now. WESLEA uses the so-called soil mechanics convention; in this convention a tensile stress or tensile strain gets the – sign, while a compressive stress or compressive strain gets the + sign.
50 kN wheel load tyre pressure 700 kPa
200 mm asphalt, E = 5000 MPa, ν = 0.35
300 mm unbound base, E = 400 MPa, ν = 0.35
ε z
, σ z
ε r
, σ r subgrade (sand), E = 150 MPa, ν = 0.35
Figure 7.9: Input for the calculation example with WESLEA.
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Position 1
X Y Z
Normal stress [kPa] -798.02 -798.02 118.31
Normal strain [ µ m/m] -112.02 -112.02 135.38
Displacement [ µ 240.54
X Y Z
Normal stress [kPa]
Normal strain [ µ m/m]
-1.96
-81.64
-1.96
-81.64
31.35
218.14
Displacement [ µ m]
Table 7.1: The stresses and strains calculated with WESLEA in the two positions indicated in figure 7.9.
In figure 7.9 the stress and the strain at the bottom of the asphalt layer are indicated as σ r
and ε r
respectively, while WESLEA gives the stresses in
Cartesian coordinates. However, for an axial symmetric load (such as the one in this example) in the vertical line through the load center is valid: σ r
= σ t
= σ x
= σ y
.
So it is very easy to calculate the occurring stresses and strains in any point of a certain asphalt pavement structure by means of the WESLEA program.
The obtained output allows a pavement life analysis that is discussed in the following paragraph.
7.5.1 Introduction:
In this paragraph the principles of the structural design of an asphalt pavement and the determination of its life are discussed. Prior to that however attention is paid to the various types of damage that may occur on asphalt pavements and that in principle should be taken into account in the structural design. It will appear from the overview of damage types that in this course only a limited number of damage types is addressed and that only a limited number of design criteria is taken into account.
7.5.2 Damage types on asphalt roads and design criteria to be used:
When determining the required thickness of an asphalt pavement structure two design criteria should be taken into account, i.e. cracking and permanent deformation. It already has been explained that horizontal tensile stresses and horizontal tensile strains occur at the bottom of an asphalt layer, laid directly on the subgrade or on an unbound base, due to bending of the structure under the traffic load. After many load repetitions these flexural tensile stresses/strains may lead to fatigue cracking. This fatigue cracking starts at the bottom of the asphalt layer, gradually propagates upward and finally
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appears at the road surface as so-called alligator cracking in the wheel tracks.
Figure 7.10 shows an example of this particular type of cracking. An asphalt pavement structure must be designed in such a way that this type of serious damage does not occur too early.
Figure 7.10: Example of alligator cracking on an asphalt pavement.
Besides of alligator cracking in the wheel tracks also frequently longitudinal cracks are observed. These longitudinal cracks mostly penetrate to a depth of not more than about 50 mm. The cause and propagation of this type of cracking is not yet fully understood. It is however clear that they occur due to the complex distribution of stresses in the contact area between the tyre and the road surface. In the contact area not only vertical stresses occur but also horizontal shear stresses. In regular asphalt pavement design calculations these shear stresses are however not taken into account (in the proceeding examples also a uniform vertical contact pressure over a circular contact area was assumed) and by consequence the development of this surface cracking cannot be analyzed. The propagation of these surface cracks is most probably the result of traffic and climatic influences. To a great extent surface cracking can be prevented by a correct asphalt mix composition. This course
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is not the right place to extensively discuss the occurrence and propagation of surface cracking; reference is made to the course CT4860 ‘Structural design of pavements’. One should however realize that surface cracking is a major reason for maintenance of asphalt wearing courses.
Asphalt layers are not only applied on an unbound base but also frequently on a cement-bound base. For instance, on Amsterdam Airport Schiphol the pavement structure on a runway consists of 200 mm polymer-modified asphalt layers on 600 mm lean concrete base. Although a linear elastic multilayer calculation reveals that no tensile stresses or tensile strains occur at the bottom of the asphalt layer, there are however cracks present in the asphalt.
The causes of these cracks are the following.
Each cement-bound material will try to shrink due to the hardening process and due to a decrease of temperature. The shrinkage is however to a great extent obstructed because of the friction with the underlying layer and this results in tensile stresses in the cement-bound material. If these tensile stresses become too great (shrinkage) cracks occur. This type of cracking is thus strongly dependent on the climatic conditions and on the properties of the cement-bound material. The shrinkage cracks remain not exclusively within the cement-bound base but they want to propagate into the bonded asphalt layers. This mechanism is schematically shown in figure 7.11a.
The material properties of the cement-bound base exhibit quite some variation and as a result also the distance between the (transverse) cracks varies. The greater the strength of the cement-bound base material, the greater both the crack distance and the crack width and the movements around the crack due to temperature variations. So the greater the crack distance the greater the movements at the crack and the more heavily loaded the bonded asphalt. asphalt originally closed crack opens because of shrinkage cement-bound base a: Shrinkage in the cement-bound b: The traffic wheel loadings base results in tensile stresses result in great shear stresses in the bonded asphalt layer in the asphalt layer
Figure 7.11: Propagation of cracks from the cement-bound base into the bonded asphalt layer.
The effects of the temperature movements can be reduced by regulation of the crack distance in the cement-bound base. On Amsterdam Airport Schiphol this has been done by creating notches, to a depth of 1/3 of the base
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thickness, at regular distances (about 7 m). Through these notches the base weakens to such an extent that the shrinkage cracks will occur there. The limited crack distance results in smaller movements around the crack and as a result the asphalt layer is less heavily loaded. The principle of a notch is similar to that of a contraction joint in plain concrete pavements (see chapter
5).
But even a narrow crack always is a weak point in the pavement structure. At such a crack bending moments cannot be transmitted, load transfer is only possible through cross-forces. As indicated in figure 7.11b, during the passage of a wheel load not only substantial shear stresses occur in the asphalt layer above the crack but also an extra large bending moment, and as a result the crack wants to propagate from the base into the asphalt layer. The asphalt layer also has to be designed to resist this type of cracking. This subject is however outside the scope of this course; reference is made to the course CT4860 ‘Structural design of pavements’.
Permanent deformation of the various pavement layers due to the repeated traffic loadings is another important type of damage that should be taken into account in the structural pavement design. Such permanent deformations manifest themselves as rutting in the wheel tracks. Figure 7.12 is an example of this type of damage.
Figure 7.12: Example of rutting on an asphalt pavement.
The rutting observed at the road surface results from visco-plastic deformations of the asphalt layers and from plastic deformations of the
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unbound base, sub-base and subgrade. In chapter 4 these permanent deformations already have been discussed.
It is important to design the asphalt pavement structure in such a way that in all the layers the occurring stress levels remain sufficiently low to prevent these permanent deformations. The various layers also should possess sufficient resistance against permanent deformation. This is directly related with the choice of the type of asphalt mix, the type of base and sub-base material and the compaction. In this course not much attention is given to the permanent deformation of the asphalt layers and the unbound base and subbase. Reference is made to the courses CT4850 ‘Road building materials’ and CT4860 ‘Structural design of pavements’.
In this course only some rules of thumb are given and used to limit the permanent deformation in the subgrade. In chapter 4 the subgrade criterion already has been introduced. The meaning of that criterion is that the permanent deformation in the subgrade remains limited if the vertical elastic deformation at the top of the subgrade, which is calculated with WESLEA, remains below a certain value. In chapter 4 it also has been explained how, according to the Shell method, the permanent deformation in the asphalt layers can be calculated.
All the above-mentioned implies that in this course only attention is paid to the structural design of asphalt pavements laid directly on the subgrade or with an unbound (sub-)base. Furthermore, the determination of the layer thicknesses is only based on fatigue of the asphalt layer and permanent deformation within the subgrade. The relevant design parameters are thus the occurring horizontal strain at the bottom of the asphalt layer and the vertical compressive strain at the top of the subgrade.
A last important type of damage is raveling that frequently occurs in practice, especially on porous asphalt (‘zoac’) wearing courses. Raveling is the loss of aggregate at the road surface, resulting in a raw appearance of it. The occurring traffic loading, the climate and the properties of the wearing course asphalt material are the most important factors influencing raveling. Raveling is one of the most important causes of maintenance on motorways. Also this type of damage is not further discussed here, reference again is made to the courses CT4850 ‘Road building materials’ and CT4860 ‘Structural design of pavements’.
7.5.3 Steps in the structural design of an asphalt pavement:
For the structural design of an asphalt pavement the following steps have to be made.
Traffic loading
A traffic forecast is the basis for the determination of the traffic loading. This traffic forecast should not only describe the growth of the total amount of traffic but also the share of the truck traffic. In chapter 4, table 4.3 is given how the traffic class is determined in The Netherlands; this information is relevant for the structural design of an asphalt pavement.
The (truck) traffic loading is usually given as an axle load frequency distribution (see chapter 3, figure 3.3). On the basis of this frequency
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distribution the cumulative number of equivalent standard axle loads can be calculated. The magnitude of the standard axle load is normally taken as 80 or 100 kN. In The Netherlands mostly a standard axle load of 100 kN is used in the structural design of an asphalt pavement.
The transformation of the axle load frequency distribution into a number of equivalent standard axle loads is done by means of the load equivalency equation that already has been discussed in chapter 3, paragraph 3.4. For reasons of simplicity the exponent m in the equivalency equation is usually assigned the value of 4.
After having calculated the cumulative number of equivalent standard axle loads, it must be determined how the axle load is transmitted to the pavement.
Usually dual tyre wheel configurations are taken into account. This means that there are two tyres at either side of the axle. In such a dual tyre wheel the center-to-center distance between the two tyres is some 320 mm.
Furthermore a tyre pressure of 700 kPa is normally taken into account.
If one wants to analyze the effects of wide base (‘super single’) tyres (only one wide tyre at each side of the axle), a tyre pressure of 850 kPa should be used.
Material data
The strength and stiffness characteristics of the various layers have to be known to enable a structural design calculation with the WESLEA multi-layer computer program. At least the following data has to be available:
the CBR-value of the subgrade,
the composition of the applied asphalt mixes,
the representative (most frequently occurring) speed of the trucks,
the temperature (in The Netherlands an asphalt temperature of 20 0 normally used for the structural design of asphalt pavements).
C is
The rules of thumb given in chapter 4, paragraph 4.2 then allow to reasonably estimate the E-value of the subgrade as well as the E-value of the unbound
(sub-)base.
The stiffness of the bitumen and next the stiffness of the asphalt mix can be determined on the basis of the mix composition by mass, the applied type of bitumen, the asphalt temperature and the loading time. The information given in chapter 4, paragraph 4.11.2 enables to determine the asphalt fatigue relationship as well as the healing factor of the asphalt.
The obtained E-values should always be checked on consistency. For instance, it is impossible that an asphalt mix has an E-value that is greater than that of cement concrete, quality B45.
Structural design calculations
The occurring strain at the bottom of the asphalt layer and the vertical compressive strain at the top of the subgrade can now be calculated by means of WESLEA.
In the calculations you may assume that all the pavement layers are fully bonded to each other.
Although the Poisson’s ratio is dependent on a number of factors, the following guidelines can be given for it:
asphalt at moderate temperatures and loading times, sand, nonsaturated clay, unbound sub-base and base materials: 0.35,
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asphalt at high temperatures and long loading times, saturated clay:
0.5,
cement-bound base materials: 0.2,
concrete:
In the calculations much care must be taken that the correct units are used because ‘nonsense in = nonsense out’.
Also realistic layer thicknesses should be used! The minimum thickness of a layer is about 2.5 to 3 times the maximum grain size.
Probably a number of calculations, with different layer thicknesses, are required to obtain the desired pavement life. Adapting the layer thicknesses has to be done in a systematic way and care must be taken that the stresses and strains are calculated at the correct positions within the (modified) pavement structure. It is recalled that WESLEA uses the soil mechanics sign convention, so the – sign means tension and the + sign means compression.
Pavement life
The fatigue life of the asphalt can be determined on the basis of the calculated strain at the bottom of the asphalt layer. The fatigue life resulting from the (laboratory) fatigue relationship has to be multiplied with the factors for healing (see chapter 4, paragraph 4.11.2) and for lateral wander (as stated earlier, a value of 2.5 is a reasonable assumption for the lateral wander factor).
The pavement life based on the subgrade criterion is found by inputting the calculated vertical compressive strain in the subgrade criterion given in chapter 4, paragraph 4.3. This found pavement life of course should not be multiplied with the healing factor and the lateral wander factor (you should be able to explain why this should not be done).
1. Meier,
Effects of traffic loadings on pavement structures (in German)
Forschungsarbeiten aus dem Strassenwesen. Kirschbaum Verlag;
Bonn/Bad Godesberg – 1968
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Introduction:
The WESLEA program has been developed for the American Waterways
Experiment Station (WES) of the US Army Corps of Engineers. It is a linear elastic multi-layer program that enables the analysis of a pavement structure consisting of maximum 5 layers (the subgrade counts as one layer). The number of circular loads is maximum 20. This is a very useful option because it enables to analyze the effects of complex load systems such as the landing gears of a Boeing 747 aircraft.
There are two options with respect to the bond between the layers: a. the subsequent layers are fully bonded to each other (this is the most commonly used option), b. the subsequent layers are not bonded to each other, so they can slip along each other without any friction (this option is only used for very special cases).
The starting point in the following description of the input and output of the program is the example given in figure 7.9.
The input:
On the main screen you first click “units” and then “SI”.
You have to realize that the WESLEA program has originally been developed for the American system of units. Your input in SI-units therefore is converted into American units and then of course some round-off errors occur. Also in the output you will notice this.
Next you click “input” and then “structure”. You input that the ‘number of layers” is 3.
The next step is the input of the properties of the various pavement layers. As
“material” for layer 1 you chose “asphalt” and for the elastic modulus you fill in
5000 MPa. As “material” for the layers 2 and 3 you chose “other”. The reason for doing so is that the choice for “GB” (= granular base) yields a confrontation with a maximum value for the elastic modulus that is hidden in the program.
This limitation is by-passed through the choice of the material “other”. Next you input the values for the elastic modulus of the unbound base and the subgrade in MPa. Also the values of the Poisson’s ratio have to be input but you will observe that the value 0.35 is set as default value.
Next the thickness (in cm!) of the asphalt layer and the base has to be input.
Then you have to input whether the asphalt layer is fully bonded to the base and whether the base is fully bonded to the subgrade. As stated earlier this is a very reasonable assumption for most of the cases.
You now click the button “ok”.
You click again on “input” and then on “loads”.
There is the possibility to analyze various load configurations, which are: a. a single axle with dual tyre wheels, b. a tandem axle with dual tyre wheels, c. a triple axle with dual tyre wheels, d. a single axle with wide base tyre wheels (“steer”),
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e. your own load configuration.
Because we want to simulate the effects of a wide base tyre we click “steer”.
You may wonder if the combined effect of both wide base tyres (at both sides of the axle) is now analyzed. This is not the case, only the effects of one wheel at one side of the axle are analyzed; this is also the case for the other axle configurations. The reason to do so is that the wheel at the other side of the axle has hardly any effect on the stresses and strains under the wheel under consideration.
The program now asks for the “number of load repetitions”. Here you can input any number. The program only uses this number of load repetitions if you want to analyze the “standard locations” (see next step “evaluation”). In the program fatigue relations for the asphalt and the subgrade have been built in that are based on the greatest occurring horizontal tensile strain in the asphalt and the greatest occurring vertical compressive strain in the subgrade. These relations are however not universal applicable. When using the “standard locations” the pavement life is calculated on the basis of these calculated strain values.
Because as a matter of fact we are interested in such a fatigue calculation we input as ‘number of load repetitions” the expected number of repetitions of the considered wheel load, e.g. 1000000.
For “load” (wheel load) you fill in 50 kN and for “pressure” (tyre pressure) 700 kPa.
You now click the “ok” button.
Next you click again “input” and then “evaluation”. You see a grey screen with a “v” on “standard locations”. You observe that there are two locations. By means of clicking on “next location” or “previous location” you can see which locations they are. The first location appears to be in the load center at the bottom of the asphalt layer (z = 19.99 cm) and the second location is also in the load center at the top of layer 3, the subgrade (z = 49.99 cm). Of course it is strange that the program yields the depth z = 49.99 cm while the top of the subgrade is on the depth z = 50 cm, but this is the result of rounding-offs.
Therefore check whether the “locations” are in the correct layer.
If you remove the “v” on “standard locations” you can input your own locations but the program then does not perform a pavement life calculation.
If you have completed this part of the input you click on “ok”.
The output:
You have prepared all the required input and now you click on the main screen first “output” and then “view output”. You now see a grey table containing the stresses, strains and displacements that already have been given in table 7.1. You also get the pavement life based on “fatigue” (fatigue of the asphalt) and “rutting” (permanent deformation of the subgrade).
However, the program calculates these pavement lifes on the basis of the greatest horizontal tensile strain and the vertical compressive strain that have been calculated. So you have to be very careful in interpreting these numbers!
For example, the “rutting” life is completely irrelevant for position 1, at the bottom of the asphalt, and the “fatigue” life is completely irrelevant for position
3, the top of the subgrade.
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The calculated “damage factor” is the ratio between the “applied” (the occurring) and the “allowable” “number of load repetitions”.
Finally you can have a look to the fatigue relationship for the asphalt
(“fatigue”) and the criterion for the allowable permanent deformation in the subgrade (“rutting”) by means of the button “view transferfunctions”. It is stressed again that these functions are not universal applicable. The relations that normally used in The Netherlands already have been given in chapter 4.
Final remark:
The WESLEA program generates numerical solutions. The accuracy of the obtained calculation results depends among other things on the magnitude of the integration steps. In this calculation process errors may be introduced.
Simple checks are possible to investigate whether these errors have occurred and whether the program has functioned well. This is however beyond the scope of this course; reference is made to the course CT4860 ‘Structural design of pavements’. In that course also other programs will be discussed.
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