Investigation of influence of heavy traffic loads on asphalt pavement response by SAFEM

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491
Bearing Capacity of Roads, Railways and Airfields – Loizos et al. (Eds)
© 2017 Taylor & Francis Group, London, ISBN 978-1-138-29595-7
Investigation of influence of heavy traffic loads on asphalt pavement
response by SAFEM
P. Liu, D. Wang & M. Oeser
Institute of Highway Engineering, RWTH Aachen University, Germany
ABSTRACT: A computational model which is able to calculate the response of the pavement fast and
precisely is greatly important to accurately assess the impact of heavy traffic loads on asphalt pavements.
A specific program SAFEM was developed based on a semi-analytical finite element method for this
objective. It is a three-dimensional FE program that requires only a two-dimensional mesh by incorporat-
ing the semi-analytical method using Fourier series in the third dimension. The impact of heavy traffic
loads was analyzed in terms of stress distribution, surface deflection and fatigue life. The results indicate
that the SAFEM-program is an efficient and fast tool that is capable of accurately predicting the struc-
tural response of pavements to traffic loads.
In the following sections, the mathematical basis
of the SAFEM will be shortly introduced, and the
impact of heavy traffic loads on the asphalt pave-
ment is analyzed. A brief summary and conclu-
sions are provided at the end of this paper.
2 DESCRIPTION OF SEMI-ANALYTICAL
FINITE ELEMENT METHOD
The first step in the FE formulation is to express the
element coordinates and element displacements in
the form of interpolations using the natural coor-
dinate system of the element. By using SAFEM,
the general form of the shape functions defining
the variation of displacements can be written as a
Fourier series in which z ranges between zero and a.
The pavement is assumed to be held at z= 0 and
z= a in a manner preventing all displacements in
the XY plane but permitting unrestricted motion
in the z-direction, as shown in Figure2 (Fritz 2002,
1 INTRODUCTION
In the past decades the Finite Element (FE) method
has been developed rapidly and was increasingly
used in many industrial fields as well as in the rou-
tine pavement design and assessment process. But
several limitations exist in the conventional FE
packages such as ABAQUS, e.g. the complexity
and hence the time-consuming user training proc-
ess often renders it impractical to be used by a road
engineer. The specifically developed FE tools for
pavement analysis usually offer the results quickly
but sometimes the accuracy is relatively low due
to oversimplifications of the modelling, e.g. from
a Three-Dimensional (3D) condition to a Two-
Dimensional (2D) plane strain or axisymmetric one.
Therefore, it is necessary to find means that both
improve the computational speed without increas-
ing the resource requirement and still offer the com-
putational accuracy. The Semi-Analytical Finite
Element Method (SAFEM) is one of the methods
which can meet the requirements. For a typical
pavement structure problem as seen in Figure 1,
the geometry and material properties usually do
not vary in one of the coordinate direction (out-
of-plane direction, for this case the z-direction),
but the boundary conditions, e.g., the load terms,
exhibit a significant variation in that direction. As
a result, the pavement structure problem could not
be simplified as a 2D plane strain case. However, by
assuming that the displacements in the geometrical
out-of-plane direction can be represented using a
Fourier series and exploiting its orthogonal proper-
ties, the problem of such a class can be numerically
solved by a series of 2D FE-meshes which will be
shown later (Fritz 2002, Hu et al. 2008, Liu etal.
2013, Liu etal. 2014, Zienkiewicz & Taylor 2005).
Figure 1. Pavement structure geometry and load
mode.

492
Hu et al. 2008, Liu et al. 2013, Liu et al. 2014,
Zienkiewicz & Taylor 2005):
U
u
v
w
N
lz
a
lz
a
lz
a
k
N
k
l
L
=
⎧
⎨
⎪
⎧
⎧
⎨
⎨
⎩
⎪
⎨
⎨
⎩
⎩
⎫
⎬
⎪
⎫
⎫
⎬
⎬
⎭
⎪
⎬
⎬
⎭
⎭
=
⎡
⎣
⎢
⎡
⎡
=
=
∑
∑
1
6
1
00
0
lz
0
00
si
n
si
co
s
π
z
z
zz
π
z
z
⎢
⎢
⎢
⎢⎢⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎣
⎢
⎢
⎤
⎦
⎥
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎦
⎥
⎥
⎧
⎨
⎪
⎧
⎧
⎨
⎨
⎩
⎪
⎨
⎨
⎩
⎩
⎫
⎬
⎪
⎫
⎫
⎬
⎬
⎭
⎪
⎬
⎬
⎭
⎭
=
=
∑
u
v
w
NU
⋅
k
l
k
l
k
l
l
l
L
l
1
(1)
where
uv
k
l
k
l
,
and
w
k
l
are the displacements of the
node at the term of the Fourier series along x-, y-
and z-directions, respectively. l identifies the term
of the Fourier series and L is the number of terms
considered; Nk is the shape functions of the node
in the XY plane which are the same as those for
displacement approximation used in 2D prob-
lems, and their definition in detail can be found in
(Zienkiewicz & Taylor 2005).
The loading function defining the variation of
load along the z-direction is given a similar form as
the displacements (Hu etal. 2008):
lz
a
l
L
l
l
L
{}
p
=
∑∑
p
lz
a
()
xy
=
)
y
11
a
l
=
a
l
z
z
z
(2)
p
P
l
l
a
Z
l
a
Z
t
P
P
tt
Z
t
n
c
l
t
os
co
s
()
x
y
,
y
⎛
⎝
⎛
⎛
⎝
⎝
⎞
⎠
⎞
⎞
⎠
⎠
−
⎡
⎣
⎢
⎡
⎡
⎣
⎣
⎤
⎦
⎥
⎤
⎤
⎦
⎦
=
∑
2
12
Z
t
Z
co
s
1
π
ππ
Z
l
(3)
where Pt is the tire load pressure; Zt1 is the z coor-
dinate where the tire load starts; Zt2 is the z coordi-
nate where the tire load ends.
After determining the element displacement,
geometrical and physical equations can be used
to obtain the strain and stress of one element.
The strain-displacement matrix
k
B
l
is defined as
follows:
B
N
x
lz
a
N
y
lz
a
l
a
N
lz
a
N
y
lz
k
B
l
k
N
k
N
k
N
k
N
=
∂
∂
∂
∂
−
∂
∂
s
i
n
si
s
i
n
s
i
n
π
z
z
π
z
z
ππ
lz
z
i
π
z
z
00
0
0
k
si
n
00
a
a
N
x
lz
a
l
a
N
lz
a
N
y
lz
a
l
a
N
lz
a
N
x
k
N
k
N
k
N
k
N
k
N
∂
∂
∂
∂
∂
∂
si
n
co
sc
k
os
co
s
π
z
z
ππ
lz
z
π
z
z
ππ
lzz
0
0
0
co
c
c
s
lz
a
π
z
z
⎡
⎣
⎢
⎡⎡
⎢
⎢
⎢
⎢
⎢⎢
⎢
⎢⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎢
⎢
⎢⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎢
⎢
⎢
⎢
⎢
⎣⎣
⎢
⎢
⎤
⎦
⎥
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎦
⎥
⎥
(4)
By using the principle of minimum potential
energy, a typical sub-matrix of the element stiff-
ness matrix (Klm)e is (Hu etal. 2008):
DB
d
xd
y
d
z
e
vo
l
T
m
()
K
l
()
B
l
∫∫∫
v
(5)
A typical term for the force vector becomes:
d
xdydz
d
d
e
vo
l
T
l
()
F
l
()
N
l
{}
p
∫∫∫
v
(6)
From Equations4 and 5, the stiffness matrix of
one element includes (Zienkiewicz & Taylor 2005):
I
lz
a
z
a
d
z
I
lz
a
z
a
dz
I
lz
a
a
a
1
I
I
0
2
I
I
0
3
II
=
⋅
=
⋅
=
∫
0
∫
0
∫
si
nc
⋅
os
si
ns
⋅
in
co
s
ππ
z
z
mzz
ππ
z
z
mzz
i
π
z
z
00
a
mz
a
dz
∫
00
⋅
⋅
co
s
π
zz
(7)
The integrals exhibit orthogonal properties
which ensure that (Zienkiewicz & Taylor 2005):
II
af
or
f
f
lm
f
or
l
m
23
II
II
1
2
0
=
I
3
I
≠
⎧
⎨
⎪
⎧
⎧
⎨
⎨
⎩
⎪
⎨
⎨
⎩
⎩
,
(8)
Only when l and m are both odd or even num-
bers, the first integral I1 is zero. Due to the special
structure of the Bl matrix, all terms that include
I1 become zero. This means that the matrix (Klm)e
becomes diagonal. Thus, the stiffness matrix can
be reduced and the final assembled equations have
the following form:
Figure 2. Schematic representation of an SAFEM
situation.

493
K
K
K
U
U
U
F
F
F
LL
LL
L
11
22
1
2
1
2


⎡
⎣
⎢
⎡
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎣
⎢
⎢
⎤
⎦
⎥
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎦
⎧
⎨
⎪
⎧
⎧
⎪
⎨
⎨
⎪
⎪
⎩
⎪
⎪
⎩
⎩
⎪
⎪
⎫
⎬
⎪
⎫
⎫
⎪
⎬
⎬
⎪
⎪
⎭
⎪
⎬
⎬
⎪
⎭
⎭
⎪
⎪
+
⎧
⎨
⎪
⎧
⎧
⎪
⎨
⎨
⎪
⎪
⎩
⎪
⎨
⎨
⎪
⎪
⎩
⎩⎩⎩
⎪
⎪⎪⎪
⎫
⎬
⎪
⎫
⎫
⎪
⎬
⎬
⎪
⎪
⎭
⎪
⎬
⎬
⎪
⎭
⎭
⎪
⎪
=
0
(9)
The Equation9shows that the large system of
equations splits up into L separate problems, i.e.,
the equilibrium equations are fully decoupled for
each harmonic of Fourier series, which can be well
adapted to the individual processors of a parallel
computer, and thus reduce the computational time
significantly compared to the sequential solving
procedure.
The accuracy and efficiency of SAFEM have
been validated by former studies (Liu et al. 2017a,
2017b).
3 ANALYSIS OF THE IMPACT OF HEAVY
TRAFFIC LOADS ON THE ASPHALT
PAVEMENT
3.1 The definition of models
The thicknesses and the material properties of the
asphalt pavement layers are listed in Table1. The
excess length in both ends of the traffic direction
was defined as 20 times of the loading radius to
limit the computational time. The width of the
pavement was defined as 3750 mm (Gohl 2006,
Rabe 2004, 2007, 2014). The upper three asphalt
layers were totally bound. The two contact layers
among the asphalt base course, gravel base layer,
frost protection layer and sub-grade were defined
as being partially bound, which means same dis-
placement happens in vertical direction but differ-
ent displacements in horizontal directions.
A standard single axle with dual wheels was uti-
lized and its configuration can be seen in Figure3.
Assuming that the influence of one pair of dual
wheels on the other pair on the other end is small
enough it can be neglected to simplify the model,
which is proved by former study (Liu et al. 2017a),
therefore only one pair was loaded in the center of
the pavement surface. Considering different work-
ing conditions, five different levels of axle loads
were applied. The corresponding loading param-
eters are shown in Table2.
3.2 The response of the asphalt pavement
structure under the heavy traffic loads
With an increase of the axle load, the contact pres-
sure between the tire and pavement increases. As a
result the vertical compressive stress in the asphalt
pavement structure will increase, which will cause
rutting. The variation tendencies of the compres-
sive stress distribution along the depth of the
asphalt pavement subjected to the different axle
loads are similar, seen from Figure 4. The com-
pressive stress caused by larger axle load is obvi-
ously larger than that caused by smaller one, which
means an overweight load will lead to more serious
destruction of the asphalt pavement.
The values of the compressive stresses on the
top of each pavement layer are listed in Table 3.
The values decrease as the depth from the pave-
ment surface increases. The upper three asphalt
layers bear much larger compressive stress than
those lower ones. The compressive stress on the
top of the surface course caused by the axle load
of 20000kg is 1.11 times that caused by a stand-
ard axle load of 10000kg. The compressive stress
Figure 3. The configuration of single axle with dual
wheels.
Table2. Loading parameters of the asphalt pavement
(Yu etal. 2013).
Axle
load Contact
pressure Contact
area Wheel
width Wheel
length
Distance
between
wheels
Pp A WLD
kg MPa mm2mm mm mm
10000 0.70 35256 156 226 320
12500 0.76 40172 166 242 320
15000 0.82 45056 176 256 320
17500 0.85 50220 186 270 320
20000 0.89 55185 195 283 320
Table 1. Thicknesses and material properties of the
pavement layers.
Layer
Thickness
μ
E
Mm MPa
Surface course 40 0.35 11150
Binder course 50 0.35 10435
Asphalt base course 110 0.35 6893
Gravel base layer 150 0.49 157.8
Frost protection layer 570 0.49 125.7
Sub-grade 2000 0.49 98.9

494
a significantly increased compressive stress in
the asphalt pavement structure. The compressive
stress in asphalt binder course is relatively large
and increases more significantly; the service life of
the asphalt pavement will decrease if this phenom-
enon is not considered enough.
Similarly, the surface deflection of the asphalt
pavement will increase as the axle load increases.
The deflection distribution along the traffic direc-
tion offset from the midpoint between the dual
wheels centers is shown in Figure5.
The maximum surface deflections due to differ-
ent axle loads are listed in Table4. The deflection
increases to 159% when the axle load of 10000kg
increases to 20000kg, i.e., the surface deflection
is sensitive to the change of the axle load. It indi-
cates that when the asphalt pavement is designed
for a region with many heavy traffic loads, it
should be considered that the thicknesses and stiff-
ness of the pavement structural layers should be
increased adequately in order to support the sur-
face deflection.
The fatigue life of the asphalt pavement sub-
jected to the different axle loads can be calculated
according to the Equation10 (FGSV 2009):
NN
in
N
N
si
tu
f
N
⋅
N
f
N
⋅
−
2
8
2
8
3
4
19
4
.
.
ε
−
−
(10)
where Ninsitu is the number of load cycles until the
macro-crack initiates; Nf is the shift factor, for this
case Nf=1540; ε is the computational tensile strain
at the bottom of the asphalt base course.
Figure4. Comparison of the compressive stress distri-
bution along the depth of the pavement derived from the
different axle loads.
Table3. The compressive stresses (MPa) on the top of
each pavement structure layer.
Axle load
kg Surface
course Binder course Asphalt
base course
10000 0.801 0.663 0.410
12500 0.845 0.720 0.451
15000 0.871 0.776 0.490
17500 0.874 0.803 0.511
20000 0.891 0.840 0.538
Axle load
kg Gravel
base layer Frost
protection layer Sub-grade
10000 0.088 0.076 0.037
12500 0.101 0.088 0.042
15000 0.115 0.099 0.048
17500 0.125 0.109 0.053
20000 0.136 0.119 0.058
Figure 5. The deflection distribution offset from the
midpoint between the centers of the dual wheels.
Table 4. The maximum surface deflections due to
different axle loads.
Axle load
kg 10000 12500 15000 17500 20000
Maximum
deflection
mm −0.511 −0.594 −0.678 −0.742 −0.814
on the top of the binder course caused by an axle
load of 17500kg is already larger than that on the
top of the surface course caused by a 10000kg
axle load. As a result, overweight loads will cause
0.2 0.4 0 .6 0.8
'E
.s
Surface
cours
·
Q)
u
40
~
:::J
f/)
I I I I I I
l.oo!l
Binder
course
~
1/
~
....-
c
Q) 80
E
Q)
>
ro
~
Asphalt
base
course
a.
~
120 -Axle load = 10000 kg
E -Axle load = 12500 kg
.g
160
.c
a.
Q)
0
r-
h
~
-Axle load = 15000 kg
"" Axle load = 17500 kg
II' A -Axle
load=
20000 kg
200
Stress
[MPa]
(a) The depth from surface course to asphalt base course
'E
2oo
.s
Q)
~
600
i)l
c
~
1000
~
ro
a.
Q) 1400
£
E
e
~ 1800
a.
Q)
0
2200
0.03
I
I""
.
0.
06
0.
09
0.
12
0.
15
'1'..
I
"'-
1
}/
Gravel
base
layer
L...L.....L.:
Frost
protection
layer
Sub-grade
I I I I I
-Axle
load=
10000 kg
-Axle
load=
12500 kg
-Axle
load=
15000 kg
Axle
load=
17500 kg
-Axle load = 20000 kg
Stress
[MPa]
(b) The depth from gravel base layer to sub-grade.
0.00
-0
.
15
-0
.
30
'E
E
'-;-0.45
0
u
~-0
.
60
0
-0
.
75
-0
.
90
Axle
load
10000
kg
~
Axle
load
12500
kg
~
Axle
load
15000
kg
r-
r-
r-
Axle
load
17500
kg
r-
r-
Axle
load
20000
kg
~
500
1000 1500
2000
Offset
from
midpoint
between
centers of the wheels
[mm)

495
The tensile strain and fatigue life of the asphalt
pavement is shown in Table5. The theoretical value
seems large enough and thus the fatigue may be not
a big deal. However, considering the other influenc-
ing factors such as climate, the fatigue life would be
much less than the theoretical value. And according
to the computation, the fatigue life decreases to 14%
when the axle load is from 10000kg to 20000kg,
i.e., the increase of the axle load will significantly
speed up the destruction of the asphalt pavement.
4 SUMMARY AND CONCLUSION
This paper proposes to use the SAFEM for pre-
dicting the asphalt pavement structural responses
under heavy traffic loads to different extents. The
compressive stress distribution, surface deflection
and fatigue life derived from the SAFEM is ana-
lysed to investigate the impact of the heavy traffic
load on the asphalt pavement. The results indicate
that the overweight load will have a serious impact
on the destruction of asphalt pavement structure.
For further investigation, the SAFEM allows the
application of dynamic analysis and various mate-
rial properties, such as viscoelasticity for asphalt
and nonlinear elasticity for the sub-base of the
pavement. In order to further reduce the compu-
tational time, infinite elements should be coupled
with the finite element model. With these improve-
ments, the SAFEM should be more appropriate to
predict the impact of the heavy traffic load on the
asphalt pavement.
ACKNOWLEDGEMENTS
This paper is based on parts of the research projects
carried out at the request of the Federal Ministry
of Transport and Digital Infrastructure, requested
by the Federal Highway Research Institute, under
research projects No. 04.0259/2012/NGB and FE
88.0137/FE88.0138, as well as parts of the research
project carried out at the request of the German
Research Foundation, under research projects No.
FOR 2089. The authors are solely responsible for
the content.
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Table 5. The tensile strain and the fatigue life of the
asphalt pavement subjected to different axle loads.
Axle load
kg 10000 12500 15000 17500 20000
Tensile
strain
μm/m 193 224 255 279 306
Fatigue life 1.67e19 8.86e18 5.09e18 3.51e18 2.38e18