Full text of DOI:10.1061/(ASCE)0733-947X(2000)126:1(46)

FLEXIBLE
P
V
AVEMENT
ARIABLE
T
T
HERMAL
S
TRESSES WITH
EMPERATURE
By Wing-Gun Wong1 and Yang Zhong2
(Reviewed by the Highway Division)
A
BSTRACT
:
A comprehensive analytical treatment of flexible pavement thermal stresses with variable temper-
ature is presented in this paper. General solutions for equations of equilibrium expressed in terms of displacement
and variable temperature are derived by Laplace transformation, Hankel transformation, and Laplace transfor-
mation with respect to the time and radial and vertical coordinates, respectively. For multi-layered problems,
the transfer matrix method is utilized to obtain the general solutions. The calculated results confirm the impor-
tance and the need to account for the thermal stresses in design and analysis of flexible pavement.
INTRODUCTION
The thermal transfer equation for the flexible pavement can
be expressed as follows:
It has been recognized that critical stresses in flexible pave-
ments result from traffic load. The analysis of such stresses
has conventionally been performed based on the theory of an
elastically multilayered half-space problem, which can be ob-
tained by closed analytical solutions. It has also been recog-
nized that the effect of temperature is important; for example,
low temperature cracking of flexible pavements in one of the
open problems in pavement engineering. Many research works
have considered the calculation of thermal stresses in concrete
pavements [e.g., Harik (1992) and Choubane and Tia (1993)].
However, very little effort has dealt with the field of pavement
stresses caused by temperature.
This paper has three objectives. The first objective is to
present general solutions of thermal stress for a single layer.
The Laplace transformation, Hankel transformation, and La-
place transformation with respect to time and radial and ver-
tical coordinates, respectively, are used to obtain explicit gen-
eral solutions. The second objective is to derive a set of
fundamental solutions for the elastic multilayered half-space
problem with variable temperature. Complete explicit solu-
tions are presented in the domain of Laplace transformation
and Hankel transformation. The final objective is to show the
importance of thermal stress in the design and analysis of flex-
ible pavements by the calculated results.
ѨT
␭ٌ2T =
(2)
Ѩt
The constitutive relations can be expressed as
␮
ѨU
ET
␴ = 2G
e ϩ
Ϫ ␣
Ϫ ␣
Ϫ ␣
(3a)
(3b)
(3c)
(3d)
r
ͩ ͪ
1 Ϫ 2␮
Ѩr
1 Ϫ 2␮
␮
U
ET
␴ = 2G
␪
e ϩ
ͩ ͪ
1 Ϫ 2␮
r
1 Ϫ 2␮
␮
ѨW
Ѩz
ET
␴ = 2G
z
e ϩ
ͩ
ͪ
1 Ϫ 2␮
1 Ϫ 2␮
ѨU ѨW
␶ = G
ϩ
zr
ͩ ͪ
Ѩz
Ѩr
where
ѨU
U
r
ѨW
Ѩ2
Ѩr2
1 Ѩ
r Ѩr Ѩz2
Ѩ2
E
2
e =
ϩ
ϩ
;
ٌ =
ϩ
ϩ
; G =
Ѩr
Ѩz
2(1 ϩ ␮)
where E and G = modulus and shear modulus of pavement
materials, respectively; ␮ and ␣ = Poisson’s ratio and expand-
ing parameter, respectively; U and W represent the radial and
vertical displacements of pavement, respectively; T = function
of temperature; and ␭ = parameter of thermal transfer. To ob-
tain the solutions, the following equations should be solved:
GOVERNING EQUATIONS AND GENERAL SOLUTION
FOR SINGLE LAYER
The calculation model of a flexible pavement consists of a
multilayer and is treated as an axial symmetric elastic layered
half-space problem as shown in Fig. 1. The governing equa-
tions can be expressed by displacements and the variation of
temperature as a basic unknown as follows:
1
Ѩe
U
r2
2(1 Ϫ ␮) ѨT
ϩ ٌ2U Ϫ
=
␣
(1a)
(1b)
1 Ϫ 2␮ Ѩr
1 Ϫ 2␮
2(1 Ϫ ␮) ѨT
Ѩr
1
Ѩe
ϩ ٌ2W =
␣
1 Ϫ 2␮ Ѩz
1 Ϫ 2␮
Ѩz
1Assoc. Prof., Dept. of Civ. and Struct. Engrg., Hong Kong Polytechnic
Univ., Hung Hom, Kowloon, Hong Kong. E-mail: cewgwong@
polyu.edu.hk
2Res. Assoc., Dept. of Civ. and Struct. Engrg., Hong Kong Polytechnic
Univ., Hung Hom, Kowloon, Hong Kong.
Note. Discussion open until July 1, 2000. To extend the closing date
one month, a written request must be filed with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review and
possible publication on September 9, 1998. This paper is part of the
Journal of Transportation Engineering, Vol. 126, No. 1, January/Feb-
ruary, 2000. ᭧ASCE, ISSN 0733-947X/00/0001-0046–0049/$8.00 ϩ
$.50 per page. Paper No. 19223.
FIG. 1. Calculated Model of Flexible Pavement
46 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000
J. Transp. Eng. 2000.126:46-49.

iable of z and the inverse integral transformation, which are
shown, respectively, by (11) and (12)
ϩϱ
1
Ѩe
U
r2
2(1 Ϫ ␮) ␣T
ϩ ٌ2U Ϫ
=
␣
(4a)
1 Ϫ 2␮ Ѩr
1 Ϫ 2␮
␣r
1
Ѩe
2(1 Ϫ ␮) ␣T
ѨT
Ѩt
Ϫpz
ϩ ٌ2W =
␣
;
␭ٌ2T =
(4b,c)
͵
˜
¯
F(r, p, s) =
F(r, z, t)e dz
(11)
(12)
1 Ϫ 2␮ Ѩz
1 Ϫ 2␮
Ѩz
Ϫϱ
␣ϩiϱ
To omit the variable of time, the Laplace integral transfor-
mation is utilized. Let
pz
˜
F(r, p, s)e dp
¯
F(r, z, s) =
͵
␣Ϫiϱ
ϱ
Thus (10) can be written as follows:
F(r, z, t)eϪst dt
(5)
ˆ
F(r, z, s)
͵
Ϫϱ
2(1 Ϫ ␮)
1 Ϫ 2␮
␨
2(1 Ϫ ␮)
1 Ϫ 2␮
2
p2 Ϫ
␨
U Ϫ
pW ϩ
␣T
˜
˜
˜
ͩ
ͪ
1 Ϫ 2␮
and the inverse transformation is
␣ϩiϱ
1
¯
¯
¯
␨W(␨, 0, s)
= pU(␨, 0, s) ϩ UЈ(␨, 0, s) Ϫ
st
ˆ
F(r, z, t) =
F(r, z, s)e ds
(6)
1 Ϫ 2␮
(13a)
͵
␣Ϫiϱ
␨
2(1 Ϫ ␮)
1 Ϫ 2␮
2(1 Ϫ ␮)
1 Ϫ 2␮
2
˜
˜
˜
pU ϩ
p2 Ϫ ␨ W Ϫ
␣pT
ͩ
ͪ
From the integral transformation of (5), (4) becomes
1 Ϫ 2␮
ˆ
r2
ˆ
1
Ѩeˆ
U
2(1 Ϫ ␮)␣ ѨT
2
␨
2(1 Ϫ ␮)
ˆ
ϩ ٌ U Ϫ
=
(7a)
¯
¯
WЈ(␨, 0, s)
=
U(␨, 0, s) ϩ
ͫ
1 Ϫ 2␮ Ѩr
1 Ϫ 2␮ Ѩr
1 Ϫ 2␮
1 Ϫ 2␮
ˆ
1
Ѩeˆ
2(1 Ϫ ␮)␣ ѨT
1 Ϫ 2␮ Ѩz
2
2
2
2(1 Ϫ ␮)
1 Ϫ 2␮
2
2(1 Ϫ ␮)
ˆ
ˆ
; ␭ٌ T = sT (7b,c)
2
ϩ ٌ W =
¯
¯
␣T
ϩ
p W(␨, 0, s) Ϫ
ͬ
1 Ϫ 2␮ Ѩz
1 Ϫ 2␮
(13b)
2
2
where the initial temperature is assumed as zero. Now (7) be-
comes a coupled ordinary differential equation group of the
second order. To solve (7) the Hankel integral transformation,
as follows, should be used:
˜
¯
¯
[␭(p Ϫ ␨ ) Ϫ s ]T = ␭pT(␨, 0, s) ϩ ␭TЈ(␨, 0, s) (13c)
Eq. (3) can be written as follows:
␮
1 Ϫ ␮
¯
¯
WЈ(␨, z, s)
␴¯z(␨, z, s) = 2G
␨U(␨, z, s) ϩ
ͫ
ͬ
ϱ
1 Ϫ 2␮
1 Ϫ 2␮
¯
ˆ
U(␨, z, s) =
rU(r, z, s)J1(␨r) dr
(8a)
(8b)
(8c)
(8d)
͵
␣E
0
¯
T(␨, z, s)
Ϫ
1 Ϫ 2␮
(14a)
ϱ
¯
¯
¯
ˆ
␶¯ (␨, z, s) = G[UЈ(␨, z, s) Ϫ ␨W(␨, z, s)]
(14b)
zr
W(␨, z, s) =
rW(r, z, s)J0(␨r) dr
͵
0
In (14) let z = 0 and put it into (13); then omit UЈ(␨, 0, s)
and WЈ(␨, 0, s). The following equations can be obtained:
ϱ
␶¯ (␨, z, s) =
r␶ˆzr(r, z, s)J1(␨r) dr
r␴ˆz(r, z, s)J0(␨r) dr
zr
͵
2(1 Ϫ ␮)
1 Ϫ 2␮
␨
2(1 Ϫ ␮)
1 Ϫ 2␮
2
p2 Ϫ
␨
U Ϫ
pW = Ϫ
␣T
˜
˜
˜
0
ͩ
ͪ
1 Ϫ 2␮
ϱ
2␮
1 Ϫ 2␮
2(1 Ϫ ␮)
␴¯z(␨, z, s) =
͵
¯
¯
ϩ pU(␨, 0, s) Ϫ ␨
W(␨, 0, s) ϩ
␶¯ (␨, 0, s)
zr
0
E
(15a)
The inverse transformation is as follows:
␨
2(1 Ϫ ␮)
2(1 Ϫ ␮)
1 Ϫ 2␮
ϱ
2
p2 Ϫ ␨ W =
˜
˜
˜
␣pT
pU ϩ
ͩ
ͪ
ˆ
¯
1 Ϫ 2␮
1 Ϫ 2␮
U(␨, z, s) =
␨U(r, z, s)J1(␨r) d␨
(9a)
(9b)
(9c)
(9d)
͵
0
2(1 Ϫ ␮)
2(1 Ϫ ␮)
¯
¯
ϱ
ϱ
ϱ
ϩ ␨U(␨, 0, s) ϩ
pW(␨, 0, s) ϩ
␴¯z(␨, 0, s)
(15b)
1 Ϫ 2␮
E
ˆ
¯
W(␨, z, s) =
␨W(r, z, s)J0(␨r) d␨
͵
0
ˆ
␶zr(␨, z, s) =
␨␶¯ (r, z, s)J1(␨r) d␨
zr
͵
0
␴ˆz(␨, z, s) =
␨␴¯z(r, z, s)J0(␨r) d␨
͵
0
After using the integral transformation [(8)], (7) becomes
2
¯
¯
Ѩ U 2(1 Ϫ ␮)
1
ѨW
2(1 Ϫ ␮)
1 Ϫ 2␮
2
2
␨
¯
¯
␣␨T
Ϫ
␨ U Ϫ
= Ϫ
Ѩz2
1 Ϫ 2␮
1 Ϫ 2␮
Ѩz
(10a)
2
¯
¯
¯
2(1 Ϫ ␮) Ѩ W
1 Ϫ 2␮ Ѩz2
␨
ѨU 2(1 Ϫ ␮) ѨT
2
¯
Ϫ ␨ W ϩ
=
␣
(10b)
(10c)
1 Ϫ 2␮ Ѩz
1 Ϫ 2␮
Ѩz
2
¯
Ѩ T
Ѩz2
2
¯ ¯
Ϫ ␭␨ T = sT
␭
Again, using the Laplace integral transformation to the var-
FIG. 2. Computing Model of Practical Pavement
JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 47
J. Transp. Eng. 2000.126:46-49.

2
2
2
where
˜
¯
¯
[␭(p Ϫ ␨ ) Ϫ s ]T = ␭pT(␨, 0, s) ϩ ␭TЈ(␨, 0, s) (15c)
¯
¯
¯
Y(␨, z, s) = [U(␨, z, s), W(␨, z, s), ␶¯ (␨, z0, s), ␴¯z(␨, z, s),
zr
Having solved (15) and by using the inverse integral trans-
formation [(12)], the solutions of (15) can be written in the
form of a matrix as follows:
T
¯
¯
T(␨, z, s), TЈ(␨, z, s)]
¯
¯
¯
Y(␨, 0, s) = [U(␨, 0, s), W(␨, 0, s), ␶¯ (␨, 0, s), ␴¯z(␨, 0, s),
zr
¯
U(␨, z, s)
T
¯
¯
¯
W(␨, z, s)
T(␨, 0, s), TЈ(␨, 0, s)]
␶¯ (␨, z, s)
zr
where [⌽] = transfer matrix; and Y(␰, 0, s) = boundary con-
dition of top layer. Eq. (16) establishes the relationship be-
tween the quantities of the upper surface and lower surface of
the j layer by the transfer matrix [⌽].
␴¯z(␨, z, s)
ͩ ͪ
¯
T(␨, z, s)
¯
TЈ(␨, z, s)
¯
␾
11
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
U(␨, 0, s)
12
22
32
42
13
23
33
43
14
24
34
44
15
25
35
45
16
26
36
46
¯
⌽
21
W(␨, 0, s)
GENERAL SOLUTION FOR MULTILAYER
⌽
␶¯ (␨, 0, s)
31
zr
=
Eq. (17) is suitable to any layer, particularly for the case of
z = zj when equation becomes
⌽
41
␴¯z(␨, 0, s)
ͫ ͬͩ ͪ
¯
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
⌽
T(␨, 0, s)
51
52
53
54
55
56
¯
TЈ(␨, 0, s)
(16)
¯
¯
61
62
63
64
65
66
Y(␨, zj, s) = [⌽j]Y(␨, zjϪ1, s)
(18)
where ⌽ij (i = 1, 2, . . . , 6; j = 1, 2, . . . , 6) = transfer function.
Letting j in (18) be equal to 1 and n, respectively, (18) can
be written as follows:
Eq. (16) can be expressed as follows:
¯
¯
¯
¯
Y(␨, z, s) = [⌽]Y(␨, 0, s), z ʚ [0, hj]
(17)
Y(␨, z1s) = [⌽1]Y(␨, z0, s)
(19a)
FIG. 3. Calculated Results of Pavement Temperature
FIG. 4. Calculated Results of Pavement Thermal Stress
48 / JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000
J. Transp. Eng. 2000.126:46-49.

¯
¯
SUMMARY AND CONCLUSIONS
Y(␨, zns) = [⌽n]Y(␨, znϪ1, s)
(19b)
At the interfaces, the compatibility conditions can be written
as follows:
A set of general solutions are presented in the Laplace and
Hankel transformation spaces for equations governing sym-
metric deformations of flexible pavements with variable tem-
perature. These general solutions are used to calculate the
stresses and displacements of flexible pavements caused by
temperature. Through the analysis and calculation, the main
findings in this paper are summarized as follows:
¯
¯
Y(␨, zjs) = Y(␨, zjϩ1, s)
(20)
Eq. (20) shows the equality of the interface stresses and the
continuity of the interface displacements. Using (19) and (20)
repeatedly, the following formula for the multilayered problem
can be obtained:
¯
¯
• Through the theoretical analysis, it proves that maximum
stresses of flexible pavements caused by temperature often
appears at the top surface of the pavement.
Y(␨, zn, s) = [⌽nϪ1]и[⌽nϪ2]иии[⌽2]и[⌽1]Y(␨, 0, s)
nϪ1
¯
[⌽j] Y(␨, 0, s)
=
ͩ͹ ͪ
• The magnitude and direction of stress caused by temper-
ature changes with time; for example, at some time the
stress is positive, whereas at other times the stress is neg-
ative. The reversal of stress direction can cause the fatigue
damage of pavement and also is one of the reasons for
the cracking of flexible pavement in cold regions.
• The temperature effects have been considered in the ma-
terial selection. The results in the present paper have
clearly demonstrated that temperature effects should also
be considered in the analysis and design for flexible pave-
ments.
j=1
(21)
The quantities of the top and bottom layers of flexible pave-
ments have been linked together by (21). Usually, ␶ and ␴z,
zr
of top and bottom layers, are known. Actually, (21) is a set of
simultaneous equations in terms of layer displacements. All of
the calculations in the above procedure are multiplication of
matrices, which can be programmed.
NUMERICAL EXAMPLES
To prove the correctness of the formulations in this paper,
a practical flexible pavement, shown in Fig. 2, is calculated.
The material properties are indicated below:
ACKNOWLEDGMENTS
This research was made possible by a grant from the Research Grant
Council of the Hong Kong government.
• Asphalt surface: E3 = 2,500 MPa; h3 = 0.15 m; ␭ = 1.0;
␣3 = 0.0022
3
• Cement stabilized sand base: E2 = 1,500 MPa; h2 = 0.20
APPENDIX. REFERENCES
m; ␭ = 1.2; ␣2 = 0.0028
2
• Lime stabilized soil subbase: E1 = 700 MPa; h1 = 0.30 m;
Choubane, B., and Tia, M. (1995). ‘‘Analysis and verification of effects
on concrete pavement.’’ J. Transp. Engrg., ASCE, 12(1), 75–81.
Harik, I. E., Jianping, P., Southgate, H., and Allen, D. (1994). ‘‘Temper-
ature effects on rigid pavement.’’ J. Transp. Engrg., ASCE, 120(1),
127–143.
␭ = 1.1; ␣1 = 0.0026
1
• Soil subbase: E0 = 500 MPa; ␭ = 1.0; ␣0 = 0.0030
0
The calculation results are shown in Figs. 3 and 4.
JOURNAL OF TRANSPORTATION ENGINEERING / JANUARY/FEBRUARY 2000 / 49
J. Transp. Eng. 2000.126:46-49.