CHAPTER 5 CONCRETE PAVEMENTS - TU Delft

CHAPTER 5
CONCRETE PAVEMENTS
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5.1 Introduction:
Concrete pavements always require technical provisions to prevent uncontrolled
cracking due to hardening shrinkage of the concrete and due to a decrease of the
temperature. The possible measures are:
- in plain (unreinforced) concrete pavements every 3 to 6 m a transverse joint is
made, and in wide pavements also longitudinal joints are made; this means that
the pavement is divided into concrete slabs
- in reinforced concrete pavements such an amount of reinforcement (0.6 to 0.75%)
is applied, that every 1.5 to 3 m a very narrow crack appears
- in prestressed concrete pavements by prestressing such compressive stresses
are introduced that the resulting flexural tensile stresses in the concrete due to
shrinkage, prestressing, temperature and traffic loadings stay within acceptable
values.
The application of the ‘zero-maintenance’ but very expensive prestressed concrete
pavements is limited to extremely heavily loaded pavements, especially airport
platforms (for instance at Amsterdam Airport Schiphol). Reinforced concrete
pavements, with a noise-reducing and permeable wearing course of porous asphalt
(‘ZOAB’), are nowadays sometimes applied on Dutch motorways. In all other cases,
however, plain concrete pavements are applied and for that reason in this lecture
note only attention is paid to this type of concrete pavement.
More extensive information about the design (and construction) of concrete
pavements can be found in (1,2,3,4,5).
5.2 Structure of plain concrete pavements:
5.2.1 General:
Figure 5.1 shows in general terms the plain concrete pavement structure.
Plain concrete toplayer
Base
Sub-base
Subgrade
Substructure
Figure 5.1: Plain concrete pavement structure.
The toplayer consists of cement concrete, that exhibits an elastic behavior until the
moment of failure. The Young’s modulus of elasticity of the concrete toplayer is much
higher than that of the underlying layers, which results in a great load spreading in
the toplayer and hence in low stresses in the underlying substructure (base plus subbase
plus subgrade).
Because of the great load spreading in the concrete toplayer, for reasons of strength
a base is not (always) necessary. Nevertheless generally a base (with a high
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esistance to erosion) is applied to prevent as much as possible the loss of support of
the concrete toplayer, which could result in unevenness and/or early cracking of the
concrete. A base is also necessary for carrying the modern, heavy construction
equipment (slipformpaver) for the concrete toplayer.
The sub-base is especially needed to protect the concrete pavement structure
against frost/thaw damage. The minimum thickness of the sub-base is thus
dependent on both the thickness of the overlaying layers and the frost penetration
depth.
5.2.2 Subgrade:
Because of the high Young’s modulus of elasticity of the concrete toplayer the
bearing capacity of the subgrade has only a small effect on the stresses in the
concrete layer due to a traffic loading.
Due to this small effect it is common use in the design of concrete pavement
structures to simply schematize the subgrade into a system of independent vertical
linear-elastic springs with a stiffness (‘modulus of subgrade reaction’) k0 (see 5.3.2).
The bearing capacity of the subgrade (modulus of subgrade reaction) does have a
great effect on the vertical displacements (deflections) of a concrete pavement
structure due to a traffic loading.
Because of the characteristic behavior of a concrete pavement structure and the high
repair costs in case of failure, it is important to use in the design of the concrete
pavement structure a relative low modulus of subgrade reaction, for instance the
value that has a 95% probability of exceeding.
Besides the bearing capacity also the settlement behavior of the subgrade is
important.
Except the connection to bridges, founded on piles, equal settlements of the
subgrade generally are not a problem.
However, by unequal subgrade settlements extra flexural stresses are introduced in
the concrete pavement structure. The magnitude of these stresses is dependent on
the wavelength and amplitude of the settlement pattern (related to the dimensions of
the concrete pavement) and on the velocity of the settlement process (because of
stress relaxation in the cement concrete).
In practice (plain) concrete pavements are only applied on subgrades that do not
exhibit settlements (such as the sand subgrade in the southern and eastern parts of
The Netherlands) or on subgrades with rather limited and uniform settlements (for
instance at Amsterdam Airport Schiphol).
5.2.3 Sub-base:
The sub-base may have the following functions:
- raising the road surface above the level of the surrounding subsoil
- preventing damage of the pavement structure due to frost/thaw action
- temporary storage of rainwater that penetrated into the pavement structure
- soil improvement, i.e. replacement of unsuitable subgrade material
- platform for the construction of the overlaying pavement layers
- spreading the traffic loadings.
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The thickness of the sub-base is thus dependent on the designed height level of the
road surface, the frost penetration depth (in cold climates), the permeability and
bearing capacity of the subgrade, the traffic loadings (especially during the
construction of the road) and the properties of the sub-base material.
Generally an unbound granular material (like gravel, crusher run, blast furnace slags,
sand etc.) is applied for the sub-base. The grading of the sub-base material should
meet the filter laws at the boundary with the subgrade material and at the boundary
with the unbound base material (if any).
In The Netherlands between the subgrade and the base (nearly) always sand is
applied (because of its availability and the lack of other natural road building
materials), where a distinction is made between ‘sand for sub-base’ and ‘sand for fill’.
‘Sand for fill’ is used at a depth of more than 1 m below the road surface, so below
the depth of frost penetration. It must consist of mineral material in which particles
smaller than 2 µm are present up to a maximum of 8% by mass. The amount of
particles passing the 63 µm sieve must be not more than 50% by mass. The degree
of compaction should be at least 93% MPD (Maximum Proctor Density) and the
average degree of compaction should be at least 98% MPD.
‘Sand for sub-base’ is applied at a depth of less than 1 m below the road surface,
which means that it may be within the frost penetration depth and that it clearly is
subjected to traffic load stresses. Of the particles passing the 2 mm sieve, the
amount of particles passing the 63 µm sieve must be not more than 15% by mass. If
this amount is between 10 and 15% by mass, then, of the particles passing the 2 mm
sieve, the amount of particles passing the 20 µm sieve must be not more than 3% by
mass. The degree of compaction should be at least 95% MPD and the average
degree of compaction should be at least 100% MPD.
Similar to the subgrade, also the bearing capacity of the sub-base has a limited effect
on the stresses in the concrete layer due to a traffic loading and a considerable effect
on the deflections of the concrete pavement structure due to a traffic loading.
In the design of concrete pavement structures the effect of a sub-base generally is
taken into account by means of a certain increase (dependent on the thickness and
Young’s modulus of elasticity of the sub-base) of the modulus of subgrade reaction
(see 5.3.2).
5.2.4 Base:
In The Netherlands mostly a cement-bound base is applied in (plain) concrete
pavements. Like cement concrete also a cement-bound base material is subjected to
shrinkage due to the hardening process and due to a decrease of temperature. Due
to the friction with the underlying layer this shrinkage results in cracking. The more
cement in the base material, the less but wider cracks will occur.
Without measures there will grow an uncontrolled crack pattern in the cement-bound
base, with variable crack distances and crack widths. The major cracks give the risk
of reflection cracking, which means the growth of cracks from the base into the
concrete toplayer.
There are some measures to prevent this reflection cracking:
1. Preventing the adhesion between the cement-bound base and the concrete
toplayer by the application of a ‘frictionless’ layer (plastic or asphalt layer) on top
of the base.
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2. Not preventing the adhesion between the cement-bound base and the concrete
toplayer, but controlling the cracking in the base by means of:
a. distribution of the construction traffic for the concrete top layer, in such a way
that there will grow a regular pattern of fine cracks in the cement-bound base;
of course the cement-bound base should not be totally destructed by the
construction traffic
b. in case of unreinforced concrete pavements, by weakening the cement-bound
base at regular distances, so that the location of the cracks is fixed (similar to
the joints in the plain cement concrete); the cracks in the cement-bound base
have to be exactly below the joints in the concrete toplayer.
In the first instance there will be a substantial adhesion between the concrete
toplayer and a cement-bound base. However, due to a different displacement
behavior (caused by the temperature variations and the traffic loadings) of the
concrete layer and the base, during time this adhesion will disappear to a great
extent. For reasons of safety therefore in general in the structural concrete
pavement design it is assumed that there exists no adhesion between the
concrete layer and a cement-bound base.
In the design of concrete pavement structures the effect of a cement-bound base in
general is also taken into account by means of an increase (dependent on the
thickness and Young’s modulus of elasticity of the base) of the ‘modulus of subgrade
reaction’ of the subgrade plus the sub-base (see 5.2.3 and 5.3.2).
Because of its high resistance to erosion and reasonable costs usually lean concrete
is applied for the cement-bound base. Lean concrete is a mixture of gravel and sand
(in a ratio of about 2:1), cement (80 to 100 kg/m 3 ) and water. Lean concrete is
manufactured in a concrete plant and then spread and compacted by means of a
slipformpaver. The Young’s modulus of elasticity, after 28 days, of uncracked lean
concrete is 15,000 to 20,000 N/mm 2 .
In plain concrete pavements the thickness of the lean concrete base varies between
150 to 200 mm (roads) and 600 mm (airport platforms). To enable an adequate
support of the slipformpaver for the overlaying concrete pavement at either side of
the pavement the base has to be 0.5 m wider than the concrete pavement.
The evenness of the lean concrete base has to be good (maximum deviation of 15
mm under a 3 m long straightedge) to obtain an overlaying concrete pavement of
rather uniform thickness.
5.2.5 Concrete toplayer:
The concrete mixture of a (plain) concrete pavement is manufactured in a concrete
plant; for big projects a mobile plant close to the works-site is used. The concrete
mixture consists of gravel or crushed stone and sand (in a ratio of about 2:1), cement
(300 to 400 kg/m 3 ) and water; sometimes one or more additives and/or pigments (for
coloring) are added to the mix.
Generally concrete for pavements must fulfil the requirements for environment class
3 (moist environment together with thaw salts) of the VBT 1986. This means that the
water/cement-ratio of the concrete mix may not exceed the value 0.55 (if an airentraining
admixture is used in the mix) or 0.45 (if no air-entraining admixture is
used).
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In some case gravel may be used as coarse aggregate in the concrete mix. For
heavily trafficked pavements however crushed stone shall be used to improve the
skid resistance. Because of the difference in costs between gravel and crushed stone
in those cases the concrete pavement is sometimes constructed in two sub-layers
(‘wet in wet’, i.e. by means of two slipformpavers closely behind each other), i.e. a
bottom layer of gravel concrete and a 60 to 80 mm thick upper layer of stone
concrete. The alternative of course is to use stone concrete over the whole concrete
pavement thickness, in this case only one slipformpaver is required.
The most important properties (with respect to the design of concrete pavement
structures) of some Dutch concrete qualities are shown in table 5.1.
Generally on heavily loaded concrete pavements, such as motorways and airport
platforms, the concrete quality B45 is used. On lightly loaded pavements (bicycle
tracks, rural roads, etc.) mostly concrete quality B35 and sometimes B45 is applied.
The concrete quality B55 is used for precast elements such as concrete blocks, tiles
and edge restraints (kerbs).
concrete quality
Property B25 B35 B45 B55
Mean cube compressive strength
(N/mm²) after 28 days
33 43 53 63
Characteristic* cube compressive
strength (N/mm²) after 28 days
25 35 45 55
Characteristic* compressive
strength (N/mm²) of cylinders taken
from the pavement after 28 days
20 28 36 44
Characteristic* splitting strength
(N/mm²) after 28 days
Characteristic* flexural tensile
strength (N/mm²):
2.25 2.75 3.25 3.75
after 28 days
3.5 4.2 4.9 5.6
after 90 days
3.9 4.6 5.3 6.0
Dynamic modulus of elasticity
(N/mm²)
30500 32500 34500 36500
Density (kg/m 3 ) 2300 – 2400
Poisson’s ratio 0.15 – 0.20
Coefficient of linear thermal
expansion (°C -1 )
1 x 10 -5 – 1.2 x 10 -5
* 95% probability of exceeding
Table 5.1: Mechanical properties of (Dutch) cement concrete for concrete pavement
structures (2,3).
Extensive information about the manufacturing of the concrete mix and the
construction of concrete pavements can be found in (4,5,6).
Immediately after construction the fresh concrete has to be protected against drying
through a curing compound, wet jute bags or a roof structure.
In plain concrete pavements furthermore joints have to be realized within 12 to 24
hours after construction of the pavement. This is necessary to prevent uncontrolled
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(‘wild’) cracking, due to the cooling down of the fresh concrete in the first night after
construction or due to a strong temperature decrease of the hardened elastic
concrete, because of friction with the underlying layer. One distinguishes:
- in the transverse direction: contraction joints, expansion joints (at the end of
the concrete pavement, for instance in front of bridges) and construction joints
(at the end of a daily production)
- in the longitudinal direction: contraction joints and construction joints (between
two lanes of concrete placement).
Through the transverse and longitudinal contraction joints a plain concrete pavement
is divided into concrete slabs. To limit the temperature gradient stresses (see 5.3.3)
the slabs should be more or less square with a maximum horizontal dimension
smaller than about 5 m (on roads) and 7.5 m (on airports) respectively.
Figure 5.2 shows an example of a plain concrete pavement for a two-lane industrial
road.
Figure 5.2: Concrete slab configuration, with dowel bars and ty bars, of a two-lane
industrial road.
For a better load transfer, dowel bars are applied in the transverse contraction joints
of heavier loaded concrete pavements at mid-height of the concrete slab. A dowel
bar is a steel bar with a diameter of about 10% of the concrete pavement thickness
(normally 25 mm in road pavements and 32 mm in airport pavements) and a length of
500 to 600 mm. The distance between the dowel bars is 300 to 500 mm. The dowel
bars should by no means obstruct the horizontal movements of the concrete slabs
due to the variation of the absolute temperature and therefore they have a
bituminous or plastic coating to prevent adhesion to the concrete (figure 5.3).
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Figure 5.3: Transverse contraction joint with dowel bar.
In longitudinal contraction joints so-called ty bars are applied to prevent two adjacent
rows of concrete slabs to float away from each other due to variation of the absolute
temperature. The ty bars are located at mid-height, or even somewhat higher, of the
concrete slab. The profiled steel ty bars have a diameter of 16 mm and a length of at
least 600 mm. At both ends the ty bars are fixed into the concrete, however the
central one-third part of the ty bar has a coating (which prevents bond to the
concrete) to distribute the occurring movements of the concrete slabs due to varying
absolute temperatures over a sufficient length so that no flow of the ty bar steel
occurs (figure 5.4). In longitudinal contraction joints normally 3 ty bars per concrete
slab length are applied (see figure 5.2).
Figure 5.4: Longitudinal contraction joint with ty bar.
Contraction joints are made by sawing a 3 mm wide cut into the hardening concrete.
This sawing has to be done as soon as possible and certainly within 24 hours after
the placement of the concrete. The depth of the saw cut for longitudinal contraction
joints should be 40 to 45% of the concrete thickness and for transverse contraction
joints about 35% of the concrete thickness. By these saw cuts the concrete is
weakened to such an extent that the inevitable cracks (due to shrinkage of the
hardening concrete or a decrease of the absolute temperature of the hardened
concrete) will appear below the saw cuts.
The joints may remain open (which is usually done at rural roads) or they may be
filled. In this latter case by further sawing the joints have to be widened (e.g. to 8 mm)
to a certain depth to enable filling of the joints (with a bituminous material or with
special hollow plastic profiles) and to limit the strains in the joint-filling material at
changing joint widths due to temperature variations.
On roads the thickness of plain concrete pavements varies between 180 mm (bicycle
tracks) and 300 mm (motorways). On airports and other very heavily loaded
pavements a plain concrete pavement with a thickness up to 450 mm is applied.
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5.3 Stresses and displacements in plain concrete pavements:
5.3.1 Introduction:
The structural design of plain concrete pavements mainly concerns the analysis of
the occurring (flexural) tensile stresses in the concrete pavement and the allowable
(flexural) tensile stresses, taking into account fatigue of the concrete. The occurring
(flexural) tensile stresses are mainly due to the traffic loadings and to the temperature
gradients.
In this chapter some analytical models for the calculation of the occurring flexural
tensile stresses in plain concrete pavements are presented. However first the
modulus of subgrade reaction ko is discussed and the increase of this ko-value
through a sub-base and a base.
5.3.2 Modulus of substructure reaction:
One of the input parameters in the design of concrete pavement structures is the
bearing capacity of the substructure (base plus sub-base plus subgrade). Generally
the complete substructure is modeled as a dense liquid, which means that in the
substructure no shear stresses can occur. The bearing capacity of the substructure
thus is expressed as the ‘modulus of substructure reaction’ k, which is defined as
(figure 5.5):
k = p / w
(5.1)
where: k = modulus of substructure reaction (N/mm 3 )
p = vertical stress (N/mm 2 ) at the top of the substructure
w = vertical displacement (deflection) (mm) at the top of the
substructure
Figure 5.5: Definition of the ‘modulus of substructure reaction’ k.
In principal the modulus of substructure reaction k has to be determined in situ by
means of a plate bearing test. However, for reasons of costs, plate bearing tests are
not always done. Then the modulus of substructure reaction has to be determined in
an indirect way, with an increasing possibility of inaccuracy.
Table 5.2 gives an indication of the value of the modulus of subgrade reaction ko. For
a certain subgrade that ko-value should be chosen that represents the bearing
capacity for the actual moisture content, degree of compaction and grading.
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Subgrade k0 (N/mm 3 )
Well graded gravel and gravel/sand-mixtures, hardly any fine material
Poor graded gravel, hardly any fine material
Gravel/sand/clay-mixtures
Well graded sand and sand with gravel, hardly any fine material
Poor graded sand, hardly any fine material
Sand/clay-mixtures
Very fine sand, sand with loam
Vast clay
Weak clay and peat
Table 5.2: Rough k0-values for various types of subgrade (2).
0.08 – 0.13
0.08 – 0.13
0.05 – 0.13
0.05 – 0.10
0.04 – 0.10
0.03 – 0.08
0.03 – 0.05
0.01 – 0.03
0.00 – 0.01
When the CBR-value of the subgrade is known, then an indication of the ko-value can
also be obtained by means of figure 5.6.
Figure 5.6: Rough relationship between ko and CBR for various types of subgrade (2)
As already mentioned in paragraph 5.2, generally a sub-base and/or a base are
constructed over the subgrade. The effect of these layers can be estimated by means
of figure 5.7.
The k-value at the top of a layer is found from the k-value at the top of the underlying
layer and the thickness hf (mm) and the dynamic modulus of elasticity Ef (N/mm²)
(table 5.3) of the layer under consideration. This procedure has to be repeated for
each (sub-)base layer, so at the end the ‘modulus of substructure reaction’ k on top
of the substructure, i.e. directly beneath the concrete top layer, is found.
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Example
Sand subgrade: ko = 0.045 N/mm 3
Sand sub-base, hf = 500 mm, Ef = 100 N/mm²: at top of sub-base k = 0.055 N/mm 3
Lean concrete base, hf = 150 mm, Ef = 8000 N/mm²: at top of base k = 0.105 N/mm 3
The k-value of 0.105 N/mm 3 is used by the Dutch State Highway Authorities in the
structural design of concrete pavements for motorways.
Figure 5.7: Nomograph for the determination of the k-value on top of a
(sub-)base layer (1).
Material Ef (N/mm²)
Sand
Gravel/sand
Crusher run
Crushed masonry
Crushed concrete
Mix granulate
Blast furnace slags: granular
hydraulic
Sandcement*
Lean concrete*
* uncracked
100 - 200
200 - 400
300 - 600
200 - 300
400 - 1000
300 - 500
400 - 800
2000 - 3000
6000 - 12000
15000 - 20000
Table 5.3: Rough Ef-values for some (sub-)base materials.
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5.3.3 Stresses due to temperature variations:
Temperature variations lead to stresses in the concrete top layer. These stresses can
be distinguished into (figure 5.8):
1. stresses due to a temperature change ∆T which is constant over the thickness
of the concrete layer
2. stresses due to a temperature gradient ∆t which is constant over the thickness
of the concrete layer
3. stresses due to an irregular temperature over the thickness of the concrete
layer.
Figure 5.8: Temperature in the concrete toplayer in case of heating at the surface.
A regular temperature increase or decrease ∆T leads to compressive and tensile
stresses respectively in the concrete toplayer due to friction over the underlying layer.
However, for plain concrete pavements (that generally consist of slabs with both a
length and a width smaller than 5 m (roads) or 7.5 m (airports) these stresses are
such small that they can be neglected.
The irregular temperature results in internal concrete stresses, which are only
relevant for very thick concrete slabs. For normal concrete slab thicknesses they also
can be neglected.
On the contrary, the temperature gradients ∆t cause flexural stresses in the concrete
slab, which are for plain concrete pavements in the same order of magnitude as
those caused by the traffic loadings, and thus cannot be neglected at all.
The temperature gradient ∆t is defined as (figure 5.8):
T Tb
∆ t =
(5.2)
h
t −
where: Tt = temperature (°C) at the top of the concrete layer
Tb = temperature (°C) at the bottom of the concrete layer
h = thickness (mm) of the concrete layer
For the structural design of plain concrete pavements in The Netherlands only the
positive temperature gradients are relevant because:
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1. positive temperature gradients mainly occur during the day, together with the
(majority of the) heavy truck traffic
2. both positive temperature gradients and the traffic loading cause flexural tensile
stresses at the bottom of the concrete slab at the most critical locations of the
slab.
A positive temperature gradient causes warping of a concrete slab. Due to the dead
weight of the concrete slab there are flexural tensile stresses at the bottom of the
slab. These stresses are called ‘warping stresses’.
These exist several theories for the calculation of the flexural tensile stresses in
concrete slabs due to positive temperature gradients, for instance those of
Westergaard-Bradbury and Eisenmann (1,4).
In the Dutch VNC-method for the structural design of plain concrete pavements
however a somewhat different model is used, because one has realized that the most
critical point of the pavement structure is somewhere at an edge of the concrete slab.
At the edges there is by definition a uni-axial stress situation (only stresses parallel to
the edge and no stress perpendicular to the edge). For the calculation of the
temperature gradient stresses this means that only a concrete beam (with unit width)
needs to be taken into account and not an entire concrete slab (1,2,7).
In the case of a small positive temperature gradient ∆t the warping of the concrete
slab along the edge is smaller than the compression of the substructure
(characterized by the modulus of substructure reaction k) due to the deadweight of
the concrete slab. This implies that the concrete slab remains fully supported. The
flexural tensile stress σt at the bottom of the concrete slab in the center of a slab edge
can then be calculated by means of the equation:
h⋅∆t
σ t = α E
(5.3)
2
where: h = thickness (mm) of the concrete slab
∆t = small positive temperature gradient (°C/mm)
α = coefficient of linear thermal expansion (°C -1 )
E = Young’s modulus of elasticity (N/mm²) of concrete
At positive temperature gradients greater than a certain limit value l t ∆ the slab’s
edge looses contact with the substructure and is only supported at its ends over a
certain support length. In this case the flexural tensile stress σt at the bottom of the
concrete slab in the center of a slab edge follows from the equation:
σt = 1.8 * 10 -5 L ’2 /h (5.4)
where: h = thickness (mm) of the concrete slab
L ’ = slab span (mm) in longitudinal or transverse direction:
'
L = L − 3
h
k ∆t
with: L = slab length or width (mm)
k = modulus of substructure reaction (N/mm 3 )
∆t = great positive temperature gradient (°C/mm)
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The value of the limit temperature gradient ∆tl follows from equalizing the equations
5.3 and 5.4 for both the center of the longitudinal slab edge and the center of the
transverse slab edge.
5.3.4 Stresses due to traffic loadings:
The basic theory for the structural design of plain concrete pavements has been
developed by Westergaard in 1926 (8). Westergaard considers a single, fully
supported slab (without any load transfer to adjacent slabs) of the concrete
pavement. The slab is resting on an elastic foundation, i.e. vertical elastic springs
with a stiffness k (the modulus of substructure reaction k, see 5.3.2).
Westergaard developed equations for the maximum flexural tensile stress in the
concrete slab and the maximum vertical displacement (deflection) of the concrete
slab due to a single wheel load, located in the interior (middle), along the edge or in a
corner of the slab (figure 5.9). In the cases that the single wheel load is in the interior
or along the edge of the slab, the flexural tensile stress is maximal at the bottom of
the slab in the center of the wheel load. In the case that the single wheel load is in a
corner of the slab, the flexural tensile stress is maximum at the top of the slab at
some distance (around 1 m) of the corner.
Figure 5.9: Positions of the load in Westergaard’s theory.
The edge loading case is most important for the structural design of plain concrete
pavements. There exist numerous Westergaard-equations for this loading case, but
the most correct equations (which yield nearly the same result) are the following
‘new’ Westergaard-equations from 1948 (9,10):
circular loading area
( 1 + υ)
( 3 + υ)
3
3 P ⎧ ⎛ E h ⎞ 4 1 − υ
σ =
⎨l
n ⎜
⎟ + 1.
84 − υ + + 1.
18 2
4
π h ⎩ ⎝100
k a ⎠ 3 2
semi-circular loading area
( 1 + υ)
( 3 + υ)
( ) ⎬
⎭ ⎫ a
1 + υ
2 (5.5)
3
3 P ⎪⎧
⎛ E h ⎞ 4
a ⎪⎫
2
σ =
⎨l
n
+ − + ( + ) ⎬
⎪⎩
⎜
⎟ 3.
84 υ 0.
5 1 2 υ
(5.6)
2
4
π h ⎝100
k a2
⎠ 3
l ⎪⎭
l
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where:
σ = flexural tensile stress (N/mm²)
P = single wheel load (N)
p = contact pressure (N/mm²)
a =
P
= radius (mm) of circular loading area
π p
a2
=
2 P
π p
= radius (mm) of semi-circular loading area
E = Young’s modulus of elasticity (N/mm²) of concrete
υ = Poisson’s ratio of concrete
h = thickness (mm) of concrete layer
k = modulus of substructure reaction (N/mm 3 )
3
E h
l = 4
2
12( 1−
υ ) k
= radius (mm) of relative stiffness of concrete layer
It follows from the equations 5.5 and 5.6 that in the case of an edge loading on a
single concrete slab the maximum flexural tensile stress is mainly dependent on the
magnitude of the single wheel load P and the thickness h of the concrete slab, the
other factors are of minor importance.
It is remarked that the size (length and width) of the concrete slab is not included in
the equations 5.5 and 5.6 because they are only valid for rather big concrete slabs,
with horizontal dimensions of at least 8ℓ * 8ℓ (ℓ = radius of relative stiffness of the
concrete slab). The concrete slabs of modern plain concrete pavements do not fulfil
this requirement, nevertheless the equations 5.5 and 5.6 are used in most current
design methods to calculate the traffic load stresses.
Pickett and Ray have transformed the Westergaard-equations into influence charts.
These charts also allow the determination of the flexural tensile stress and the
deflection due to complex load systems, such as dual wheel tyres, tandem and triple
axles, and airplane gears.
In figure 5.10 the influence chart of Pickett and Ray for the bending moment along
the slab edge is shown. In this influence chart the wheel load contact area has to be
drawn on scale; to this end the radius of relative stiffness (l) of the concrete toplayer
is drawn as a reference.
The flexural tensile stress σ at the bottom of the concrete layer due to a wheel load P
is found from the influence chart for the bending moment M by means of the
equation:
2 ( N/
mm )
2
M 6 p l N
σ = =
(5.7)
2
1 2
h
10000 h
6
where: p = contact pressure (N/mm 2 ) of wheel load P
l = radius (mm) of relative stiffness of concrete layer
h = thickness (mm) of concrete layer
N = number of blocks at the chart, covered by wheel load P
174

Figure 5.10: Influence chart of Pickett and Ray for the bending moment along the
slab edge.
175

The equations 5.5 and 5.6 and figure 5.10 are valid for the calculation of flexural
tensile stresses due to traffic loadings in a single concrete slab. However, in reality a
plain concrete pavement consists of a number of concrete slabs with joints between
them. The load transfer in these joints is dependent on the joint width (which
depends on the slab length), the amount of traffic and the type of joint construction
(which means: aggregate interlock, ty bars and dowel bars).
Teller and Sutherland have defined the total load transfer in a joint as follows (11):
W
=
2w
u
100 (5.8)
wl
+ wu
where: W = joint efficiency (%) related to deflections
wl = deflection (mm) at the joint edge of the loaded concrete slab
wu = deflection (mm) at the joint edge of the unloaded concrete slab
In joints of plain concrete pavements without dowel bars the joint efficiency W
decreases from 70-90% just after construction to 10-30% at long term due to the
polishing effect of the repeated traffic loadings on the concrete of the joint sides. In
structural design calculations for these joints without dowel bars the safe value W =
0% (i.e. a free edge) generally is used.
In joints with dowel bars at long term there remains a joint efficiency W of 50-60%.
Similar to equation 5.8, however, the joint efficiency can also be defined as (12):
W
'
= 100
σ
l
2 σ
u
+ σ
u
Where: W ‘ = joint efficiency (%) related to flexural stresses in the concrete slab
σl = flexural tensile stress (N/mm 2 ) at the joint edge of the loaded
concrete slab
σu = flexural tensile stress (N/mm 2 ) at the joint edge of the unloaded
concrete slab
The joint efficiency W ‘ with respect to flexural stresses appears to be (much) smaller
than the joint efficiency W with respect to deflections, as is illustrated in figure 5.11.
(5.9)
Figure 5.11: Flexural stresses related to deflections for doweled concrete slabs (12).
176

The load transfer in joints can be incorporated in the design of plain concrete
pavement structures by means of a reduction of the actual wheel load Pact to the
wheel load P (to be used in equation 5.5 or 5.6) according to:
⎛ W ⎞
P = ( 1 − 1/
2 W / 100)
Pact
= ⎜1
− ⎟ Pact
(5.10)
⎝ 200 ⎠
respectively
'
⎛ W ' ⎞
( 1 − 1/
2 W / 100)
Pact
= ⎜1
− Pact
P =
⎟
⎝ 200 ⎠
(5.11)
In analytical design methods, which are primarily based on a correlation between
occurring and allowable flexural tensile stresses in the concrete pavement, equation
5.11 should be used (instead of equation 5.10).
Example
transverse contraction joint with dowel bars: say W = 60%
equation 5.8: wu = 0.43 wl
figure 5.11: σu = 0.17 σl
equation 5.9: W’= 30%
equation 5.11: P = 0.85 Pac
5.4 VNC Design method:
5.4.1 General:
In this chapter the current Dutch design method for plain concrete pavements is
briefly described. This analytical design method has been developed by the
Netherlands Cement Industry Association (2).
The design model used in the VNC method for plain concrete pavements is shown in
figure 5.12. In the model two possibly critical locations of the most heavily loaded
traffic lane are indicated:
• ZR = center of the longitudinal edge of the concrete slab (free edge or longitudinal
joint)
• VR = center of the wheel track at the transverse joint with load transfer.
Two design criteria are used in the structural design of a plain concrete pavement:
1. a strength criterion, i.e. preventing the concrete pavement for cracking; the
required thickness of the concrete pavement is found from a strength criterion
which on one hand is determined by the occurring flexural tensile stresses
under traffic and temperature gradient loadings and on the other hand by the
fatigue strength of the concrete.
2. a stiffness criterion, i.e. preventing the development of longitudinal
unevenness at the transverse joints (so-called joint-faulting): the required
thickness of the concrete pavement is found from a stiffness criterion which on
one hand is determined by the occurring deflection at the transverse joints
under traffic loading and on the other hand by the allowable deflection.
177

Figure 5.12: Design model for plain concrete pavements in the VNC method.
First of all a plain concrete pavement structure has to be assumed, which means that
the length, width and thickness of the concrete slabs, the concrete quality, the type of
joints, the thickness and modulus of elasticity of the base and sand sub-base, the
modulus of subgrade reaction etc. have to be chosen. If it appears after the
calculation that the assumed pavement structure does not fulfil the technical and/or
economical requirements, then the analysis has to be repeated for a modified
pavement structure. The flow diagram of the VNC design procedure is shown in
figure 5.13.
This paragraph only deals with the technical aspects of the VNC design procedure,
the economical aspects will not be discussed.
5.4.2 Traffic loading:
First the cumulative number of heavy (truck) axle load repetitions on the most heavily
loaded traffic lane (the design lane) during the desired pavement life (20 tot 40 years)
has to be determined.
Only part of this cumulative number of heavy axle load repetitions is driving in the
center of the wheel track, point VR in figure 5.12 (on roads 40% to 50%) or exactly
over the longitudinal edge or longitudinal joint, point ZR in figure 5.12 (0% to 15%,
depending on the geometry of the road).
Furthermore the axle load or wheel load frequency distribution of the heavy (truck)
traffic has to be known or assumed. This wheel load frequency distribution has to be
as realistic as possible: underestimation of the actual wheel loadings may lead to
early and serious structural damage (cracking) of the plain concrete pavement
because of the susceptibility of concrete for overloading (brittle material behavior).
For illustration some theoretical axle/wheel-load frequency distributions, deducted
from actual axle load measurements on Dutch motorways, are given in table 5.4.
178

Figure 5.13: Flow diagram of the VNC design procedure for plain concrete
pavements (2).
179

Axle load
group
(kN)
Wheel load
group
(kN)
Frequency distribution (%)
Light traffic Medium traffic Heavy traffic
0-20 0-10 7.60 5.40 4.00
20-40 10-20 25.00 22.00 15.00
40-60 20-30 30.00 29.00 26.00
60-80 30-40 18.00 20.00 27.00
80-100 40-50 11.00 12.00 14.00
100-120 50-60 6.10 7.70 8.40
120-140 60-70 1.80 3.00 4.40
140-160 70-80 0.41 0.75 1.00
160-180 80-90 0.07 0.10 0.12
>180 >90 0.02 0.05 0.08
Table 5.4: Representative axle/wheel-load distributions of trucks on Dutch motorways
(3).
5.4.3 Temperature gradients:
The VNC design method includes a standard frequency distribution for the (positive)
temperature gradients within the concrete pavement. This distribution (table 5.5) can
be used in the design of plain concrete pavements for roads, bus bays etc. On
heavily loaded airports and industrial yards the temperature gradients will be
somewhat smaller because of the thicker concrete slabs.
Temperature gradient ∆t
(°C/mm)
0.00
0.01
0.02
0.03
0.04
0.05
≥ 0.06
Frequency
distribution (%)
71
17
6
3
2
1
0
Table 5.5: Standard temperature gradient frequency distribution according to the
VNC method (2).
5.4.4 Strength criterion:
In the two possibly critical locations of the concrete slab (VR and ZR, see figure 5.12)
the temperature gradient stress σti has to be calculated for every positive temperature
gradient ∆ti according to the procedure described in paragraph 5.3.3.
Also in the locations VR and ZR the traffic load stress σvi has to be calculated for
every wheel load Pi by means of the appropriate Westergaard equation (eq. 5.5 or
5.6), taking into account the load transfer (if any) at the slab edge under
consideration (joint efficiency W ’ , eq. 5.11).
180

Next a fatigue damage analysis is carried out for the locations VR and ZR by
calculating the allowable number of load repetitions Ni for each combination of wheel
load Pi and temperature gradient ∆ti. In the damage analysis the following concrete
fatigue relationship (50% fatigue curve, so an average relationship) is used:
12.
903 ( 0.
995 − σ max / fbtg)
i
= with 0.
5 ≤ σ / fbtg ≤
1.
000 − 0.
7525 σ / fbtg
log N i
max
min
where:
i
0.
833
(5.12)
Ni = allowable number of repetitions of wheel load Pi i.e. the traffic load
stress σvi until failure when a temperature gradient stress σti is present
σmin i = minimum occurring flexural tensile stress (= σti)
σmaxi = maximum occurring flexural tensile stress (= σvi + σti)
fbtg = average flexural tensile strength (N/mm 2 ) of unreinforced (plain)
concrete under loading of short duration:
fbtg = 1.4 (1.6 – h/1000) (1.05 + 0.05B) (5.13)
where: h = thickness (mm) of the concrete slab
B = characteristic cube compressive strength (N/mm 2 ) after 28
days (see table 5.1)
The design criterion (i.e. cracking occurs) is the cumulative damage law of Palmgren-
Miner:
n
i
∑ = 1.0 (5.14)
i N i
where:
ni = occurring number of repetitions of wheel load Pi i.e. the traffic load
stress σvi during the pavement life when a temperature gradient stress
σti is present
NI = allowable number of repetitions of wheel load Pi i.e. the traffic load
stress σvi until failure when a temperature gradient stress σti is present
5.4.5 Stiffness criterion:
To prevent longitudinal unevenness (joint faulting) at the transverse joints in the plain
concrete pavement the deflection of the transverse edge in the wheel track (location
VR in figure 5.12) due to the traffic loading should be limited. According to
Westergaard the deflection of the transverse edge is (2,13):
⎛ W ⎞ ⎛ P ⎞
wl = λ ⎜1
− ⎟ ⎜ 2 ⎟
(5.15)
⎝ 200 ⎠ ⎝ kl ⎠
181

where:
wl = deflection (mm) of the transverse edge of the loaded concrete slab
λ = parameter (-); in the case of one single wheel load on the concrete slab
and a Poisson’s ratio υ of concrete = 0.15 the λ-value is 0.431
W = joint efficiency (%) with respect to deflections (eq. 5.8 and 5.10)
P = wheel load (N); in the VNC method P = 50 kN = 50,000 N has to be
used
k = modulus of substructure reaction (N/mm 3 )
l = radius of relative stiffness (mm) of the concrete slab
The allowable deflection is dependent on the traffic loading:
w l = 4.8 e -0.35 log neq . (5.16)
where:
w l = allowable deflection (mm) of the transverse edge of the loaded concrete
neq
slab
= cumulative number of equivalent 50 kN wheel load repetitions during the
pavement life, calculated with the equation:
n
eq
∑
= i
where:
( P 50)
i
4
/ n
(5.17)
i
Pi = wheel load (kN)
ni = number of repetitions of wheel load Pi during the pavement life
The design criterion with respect to the stiffness of the plain concrete pavement is:
w l ≤ l
5.5 References:
w (5.18)
1. Houben, L.J.M.
Structural Design of Pavements – Part IV: Design of Concrete Pavements
Lecture Notes CT4860, Faculty of Civil Engineering and Geosciences, TU
Delft; Delft - 2003
2. Manual for Concrete Roads (in Dutch)
Cement Industry Association (VNC); ‘s-Hertogenbosch - 1993
3. Manual for Road Construction – Pavement Design (in Dutch), 4 th edition
Ministry of Transport, Public Works and Water Management, Road and
Hydraulic Engineering Division; Delft - 1998
182

4. Eisenmann, J.
Concrete Pavements – Design and Construction (in German)
Wilhelm Ernst & Sohn; Berlin/Munich/Dϋsseldorf - 1979
5. Vollpracht, A., H. Eifert, O. Hersel and W. Knopp
Road Construction Today – Concrete Pavements (in German), 4 th edition
Bundesverband der Deutschen Zementindustrie; Köln - 1995
6. Concrete Roads in Practice (in Dutch)
Association of Concrete Road Contractors; VCW/CROW; Ede - 1994
7. Leewis, M.
Theoretical knowledge leads to practical results (in Dutch)
Journal ‘BetonwegenNieuws’ no. 89, September 1992, pp. 20-22
8. Westergaard, H.M.
Stresses in Concrete Pavements Computed by Theoretical Analysis
Public Roads, Vol. 7, no. 2, 1926
9. Westergaard, H.M.
New Formulas for Stresses in Concrete Pavements of Airfields
ASCE, Transactions, Vol. 113, 1948
10. Ioannides, A.M., M.R. Thompson and E.J. Barenberg
The Westergaard Solutions Reconsidered
Workshop on Theoretical Design of Concrete Pavements, 5-6 June 1986,
Epen, The Netherlands
Record 1; CROW; Ede - 1987
11. Teller, L.W. and E.J. Sutherland
A Study of Action of Several Types of Transverse and Longitudinal Joint
Design
Public Roads, Vol. 17, no. 7, 1936
12. Barenberg, E.J. and D.M. Arntzen
Design of Airport Pavements as Affected by Load Transfer and Support
Conditions
Proceedings 2 nd International Conference on Concrete Pavement Design,
Purdue University, West Lafayette, Indiana, USA, 1981, pp. 161-170
13. Leewis, M. and H.E. van der Most
The stiffness criterion for plain concrete pavements (in Dutch)
Journal ‘BetonwegenNieuws’ no. 58, December 1984, pp. 12-16
183