Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load
The dynamic response of pavement plates to a localized Friedlander load based on the threeparameter foundation model with the account of soil inertia is analyzed.
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2018
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load / S.W. Alisjahbana, Safrilah, J.C.P. Putra, A. Asmi, I. Alisjahbana, S. Kiryu, B.S. Gan// Проблемы прочности. — 2018. — № 5. — С. 22-30. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1739892020-12-29T01:26:24Z Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load Alisjahbana, S.W. Safrilah Putra, J.C.P. Asmi, A. Alisjahbana, I. Kiryu, S. Gan, B.S. Научно-технический раздел The dynamic response of pavement plates to a localized Friedlander load based on the threeparameter foundation model with the account of soil inertia is analyzed. Выполнен анализ динамической реакции мостовых плит на локализованную нагрузку Фридлендера на базе трехпараметрической модели основы с учетом инерции грунта. 2018 Article Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load / S.W. Alisjahbana, Safrilah, J.C.P. Putra, A. Asmi, I. Alisjahbana, S. Kiryu, B.S. Gan// Проблемы прочности. — 2018. — № 5. — С. 22-30. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173989 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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English |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Alisjahbana, S.W. Safrilah Putra, J.C.P. Asmi, A. Alisjahbana, I. Kiryu, S. Gan, B.S. Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load Проблемы прочности |
| description |
The dynamic response of pavement plates to a localized Friedlander load based on the threeparameter foundation model with the account of soil inertia is analyzed. |
| format |
Article |
| author |
Alisjahbana, S.W. Safrilah Putra, J.C.P. Asmi, A. Alisjahbana, I. Kiryu, S. Gan, B.S. |
| author_facet |
Alisjahbana, S.W. Safrilah Putra, J.C.P. Asmi, A. Alisjahbana, I. Kiryu, S. Gan, B.S. |
| author_sort |
Alisjahbana, S.W. |
| title |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load |
| title_short |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load |
| title_full |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load |
| title_fullStr |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load |
| title_full_unstemmed |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load |
| title_sort |
dynamic response of pavement plates to the positive and negative phases of the friedlander load |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2018 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173989 |
| citation_txt |
Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load / S.W. Alisjahbana, Safrilah, J.C.P. Putra, A. Asmi, I. Alisjahbana, S. Kiryu, B.S. Gan// Проблемы прочности. — 2018. — № 5. — С. 22-30. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-15T10:51:43Z |
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| fulltext |
UDC 539.4
Dynamic Response of Pavement Plates to the Positive and Negative Phases of
the Friedlander Load
S. W. Alisjahbana,
a
Safrilah,
a
J. C. P. Putra,
a
A. Asmi,
a
I. Alisjahbana,
b
S. Kiryu,
c
and
B. S. Gan
d
a Civil Engineering Department, Bakrie University, Jakarta, Indonesia
b Civil Engineering Department, Stanford University, Stanford, USA
c Department of Architecture, Graduate School of Engineering, Nihon University, Koriyama, Japan
d Department of Architecture, College of Engineering, Nihon University, Koriyama, Japan
The dynamic response of pavement plates to a localized Friedlander load based on the three-
parameter foundation model with the account of soil inertia is analyzed. The pavement plate is
represented by a thin orthotropic plate of finite dimensions, which can rotate and transfer
deformation along the contour. The subgrade is simulated with the Pasternak foundation model,
including the inertia soil factor, the localized dynamic load is simulated with the Friedlander decay
function allowing for the positive and negative phases; with the time distribution described by the
Dirac function. The governing equation of the problem is solved with the modified Bolotin method for
determining the natural frequencies and mode numbers of the system. The Mathematica program is
used to define the natural frequencies of the system from the transcendental equations. Analysis
results for several parameters related to the dynamic response of plates to a localized dynamic load,
which includes both positive and negative phases, are presented. The impact of the Friedlander load
with the negative phase added on the response of the pavement plate is numerically simulated.
Keywords: pavement plate, Friedlander load, Pasternak foundation, positive and negative
phases, modified Bolotin method.
Introduction. Dynamic response of a thin orthotropic plate subjected to a dynamic
load is important, not only for pavement design but also for many other applications [1].
For instance, Uzan and Lytton [2] used measured pavement dynamic response information
to investigate pavement non-destructive evaluation. Another application can be found in
military countermine detection, where an unmanned automatic vehicle equipped with
landmine detection sensors transverses minefield to identify locations of potential landmine
[3]. A significant difference between static theory and its dynamic counterpart is that
inertial effect, ignored in the former, is taken into account in the latter [4]. To better
understand the dynamic response of a thin orthotropic plate subjected to a dynamic load, it
is necessary to analyze the dynamic effects of a dynamic load on a concrete pavement. In
1954, Kenney [5] studied the steady-state response of a moving load on a beam on elastic
foundation. Some studies using the finite element method to solve the response of a thin
plate subjected to dynamic loads with applications in pavement design and non-destructive
evaluation have been developed earlier [6–8]. More recently, finite element procedures
have been developed to carry out the response of continuous pavements of finite
dimensions to moving vehicle loads [6, 9]. In the studies mentioned above, the rigid
pavements are modeled as orthotropic plates sitting on Winkler soil type. However, in
engineering practice, the effect of foundation inertia on the central deflection of the plate
has significant effects on the dynamic response of the plate modeling the pavement [10].
For this reason, Gibigaye et al. [11] take into account the inertia of the foundation soil on
the dynamic response of civil engineering structures.
© S. W. ALISJAHBANA, SAFRILAH, J. C. P. PUTRA, A. ASMI, I. ALISJAHBANA, S. KIRYU, B. S. GAN,
2018
22 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
This work investigates the dynamic analytical response of pavement plate subjected
to localized Friedlander dynamic loads sitting on an inertial soil. The rigid pavement is
modeled as a thin plate, allowing the rotations and the vertical deformations along its
edges. To take into account its inertia, the soil is modeled as a three-parameter type of soil
[11]. The free vibration solution of the problem is solved by the method of separation of
variables so that the superposition gives a solution satisfying the boundary conditions. The
pavement plate deflection is expressed as eigenfunction products, that is obtained based on
the orthogonality properties of eigenfunctions [12]. The localized Friedlander dynamic load
is expressed as the Dirac delta function [13]. Particular emphasis is focused on the
evaluation of midpoint displacements of the concrete pavement plates subjected to the
Friedlander decaying function which includes both positive and negative phase of loading
with different thickness and the varied position of the localized load.
1. Description of the Pavement Plates. Consider a rectangular pavement plate of
thickness h sitting on the three-parameter type foundation model (k , Gs , and m0) within
the linear elastic region. The origin of the Cartesian coordinate system is located in the
mid-plane with the z-axis perpendicular to the plane as shown in Fig. 1. The a and b are
dimensions along the x- and y-axes, respectively.
According to the classic theory of thin plates and taking into account the reduced
mass of soil, the deflection of the concrete pavement plate is governed by the following
partial differential equation [11]:
D
w x y t
t
B
w x y t
x y
D
w x y t
y
x y
�
�
�
� �
�
�
4
4
4
2 2
4
4
2
( , , ) ( , , ) ( , , )
� � � � �kw x y t G w x y ts( , , ) ( , , ),2
(1a)
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 23
Dynamic Response of Pavement Plates ...
Fig. 1. The geometry of the pavement concrete plate supported by a three-parameter type foundation
model.
( )
( , , ) ( , , )
( , , ),�
�
�
�
�
�
� � �m h
w x y t
t
h
w x y t
t
p x y t0
2
2
(1b)
where w x y t( , , ) is the vertical deflection of the orthotropic plate at point (x y, ) and time t,
Dx is the flexural rigidity of plate in the x-direction, B is the torsional rigidity, Dy is the
flexural rigidity of the plate in the y-direction, k is the stiffness coefficient of the elastic
foundation, Gs is the shear modulus of the shear layer of the elastic foundation, � is the
mass density of the plate, m0 is a linear reduced mass of the foundation stiffness
coefficient, h is the thickness of the plate, and � is the critical damping ratio of the plate.
2. Localized Friedlander Loading. Figure 2 shows a localized Friedlander decaying
loading function which includes the positive and the negative phases, P t( ), that can be
expressed by the following equations [14, 15]:
P t p
t
t
er
d
t
td( ) , max� �
�
�
�
�
�
�
1
�
for 0� �t t d , (2)
P t p
t t
t
t t
t
r
d
d
d
d
( )
. ( ) ( )
, min��
�
�
�
�
�
� �
�
�
�
�
� �
675
1 �
�
2
for t t t td d d� � �
�
. (3)
Here, pr, max is the peak pressure, t d is positive phase duration, � is a waveform
parameter, pr, min is the negative peak pressure, and t d�
is negative phase duration.
The localized dynamic loading is a function of position and time, p x y t( , , ), that can
be expressed by using the Dirac delta function, �[ ],� as follows:
p x y t P t x x t y y t P t x x y y( , , ) ( ) [ ( )] [ ( )] ( ) [ ] [ ]� � � � � �� � � �0 0 , (4)
where P t( ) is the localized dynamic loading, which can be expressed by Eqs. (2) and (3),
x t( ) is the position of the localized dynamic loading in the x-direction, y t( ) is the position
of the localized dynamic loading in the y-direction, x0 is the initial position of the dynamic
loading in the x-direction, and y0 is the initial position of the localized dynamic loading in
the y-direction.
24 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
S. W. Alisjahbana, Safrilah, J. C. P. Putra, et al.
Fig. 2. The localized Friedlander decaying load.
3. Vibration Analysis. In the first part of this paper, the free vibration of pavement
concrete plate on a Pasternak foundation, including the m0 factor as a soil parameter, is
studied using Levy’s solution. The free vibration solution of the system is set as
w x y t W x y t X x Y y t( , , ) ( , )sin ( ) ( )sin ,� �� � (5)
where W x y( , ) is only a function of the position coordinates and � is the circular
frequency of the system.
The eigenfrequencies of the systems can be solved from the undamped free vibration
of Eq. (1) by postulating that the eigenfrequencies are analogous to the case of a plate with
simple support at all edges, which can be expressed as
�
�
� � �
mn x y s
h m
D
p
a
B
p q
a b
D
q
b
k G
p2
0
4 4
4
2 2 4
2 2
4 4
4
2
1
2�
�
� � � �
( )
� �
2
2
2 2
2a
q
b
�
�
�
�
�
�
�
�
�
�
�
, (6)
where p and q are real numbers to be solved from a system of two transcendental
equations, obtained from the solution of two auxiliary Levy’s type problems, also known as
the modified Bolotin method [16].
3.1. First and Second Auxiliary Levy-Type Problems. The solution of the free
vibration problem in Eq. (6) can be obtained from
W x y X x
q y
b
mn( , ) ( )sin ,�
�
�
�
�
(7)
which satisfies the elastic vertical translation (ksx1, ksx2) and rotational (krx1, krx2)
boundary conditions [12]. The solutions of the characteristic equation become
X x A
p
a
x A
p
a
x A
ab
xmn ( ) cos sin cosh�
�
�
��
�
�
��1 2 3
� � ��
�
�
��
�
�
�A
ab
x4 sinh ,
��
(8)
where
�
�
� � �
2 2 2
2 2
2 2
2
Bq a
D
p b
G a b
Dx
s
x
.
Equation (8) gives the general form of the eigenmode of the pavement plate in the
x-direction. Substituting the eigenmode according to Eq. (8) into the boundary conditions
along x� 0 and x a� , results in the linear algebra equation in terms of Ai , as follows:
[ ]{ } { } ,a A 0
T T
� (9)
where aij are the coefficients of [ ].a
To obtain a non-trivial solution, it is necessary to propose that the determinant of
Eq. (9) is zero. The determinant of Eq. (9) is called the transcendental equation of the first
auxiliary Levy’s type problem. The second auxiliary Levy-type problem in the y-axis can
be determined analogously to the above formulations.
3.2. Determination of Mode Numbers. The mode numbers p and q, respectively, in
the x- and y-directions of the system can be solved by solving the transcendental equations
simultaneously. The solutions cannot be determined analytically. Hence the Mathematica
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 25
Dynamic Response of Pavement Plates ...
software is used to obtain solutions numerically. The eigenfunction of the orthotropic plate
is therefore given by
W x y X x Y ymn mn
nm
( , ) ( ) ( ).�
�
�
�
�
��
11
(10)
3.3. Determination of the Time Function T tmn ( ). The general solution in the time
domain is given by
T t
X x dx Y y dy
hQ t
p y ymn
mn mn
ba
mn mn
( )
( ) ( )
( )
( ( ,�
�
��
00
21� � �
, ) sin ( )) ,
( )
� � � � �
�� �
e t dmn t
mn
t
� �
� �� 1 2
0
(11)
where p x y t( , , ) is the localized Friedlander decaying function dynamic loading,
respectively, for positive phase according to Eq. (3) and for negative phase according to
Eq. (4). Finally, the dynamic deflection of the pavement plate on Pasternak foundation
models with the inclusive of the inertial soil factor is found by multiplying Eqs. (10) and
(11).
4. Numerical Results and Discussion. Numerical calculations are carried out for
different parameters of the problem. The number of modes in the x- and y-directions is
taken as m� 1 2 5, , ... , and n� 1 2 5, , ... , by considering the convergence of the
eigenvalues. In this work, a finite rectangular pavement plate sitting on the three-type
parameter foundation model is considered as shown in Fig. 1. The size of the pavement
plate is 5 3 5� . m. For comparison sake, two values of the plate thicknesses are considered:
h� 0.18 and 0.20 m. The physical characteristics of the pavement plate are: density
�� 2500 kg/m3, Poisson’s ratios �x � 0.2 and � y � 0.3, the elastic moduli (Ex and E y )
of the concrete plate is 27 8 106. � and 30 106
� Pa, respectively. The boundary conditions
properties of the pavement plate are: ks ks ks ksx x y y1 2 1 2 250� � � � MN/m and krx1 �
� � � �kr kr krx y y2 1 2 1.0 N m� /rad [17]. It is also assumed that the damping ratio of the
system equals � � 5%. Two values of soil parameters are considered: Case 1, Hs � 0.5 m,
k � 166.2 MN/m, Gs � 2.596 MN/m2, m0 � 252.326 kg m� and Case 2, Hs � 2.5 m,
k � 33 25. MN/m, Gs � 12.98 MN/m2, m0 � 1261.63 kg m� [11]. The localized
Friedlander dynamic load parameters which include the positive and the negative phase
according to Eqs. (2) and (3) are: pr, max � 28.906 kN/m2, t d � 0.0018 s, �� 0.35,
pr, min .��7 2265 kN/m2 [14]. Four different locations of the localized Friedlander
dynamic loading are studied in detail to determine the mid-point deflection of the rigid
pavement subjected to localized Friedlander with the emphasis in the effect of the
negative phase of the load.
4.1. Variation of Deflection during the Negative Phase of the Load. Figure 3 shows
the variation of deflection under load as a function of time for three different values of plate
thickness for Case 1. It is found that the dynamic deflection of the plate decreases as the
thickness of the plate increases. Increasing the thickness of the pavement plates, from 0.18
to 0.2 m for a damped system (5%) result in a decrease of the mid-point displacement by
7.9%. It can be seen from Fig. 3, the maximum dynamic deflection of the pavement plate
occurs just after the negative phase occurs of the Friedlander load, t t td d� �
�
, for all
different values of thickness considered in this work. In conclusion, the negative phase of
the load contributes to increasing the maximum dynamic deflection of the system.
26 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
S. W. Alisjahbana, Safrilah, J. C. P. Putra, et al.
4.2. Influence of the Soil Parameters on the Dynamic Response. Figure 4 shows the
dynamic response of the system subjected to a Friedlander decaying dynamic loading for
three types of soil conditions. At h� 0.18 m and t � 0.004 s, in Case 1, the maximum
dynamic deflection of the pavement plate is 6 48092 10 5. �
� m, while Case 2 it is
6 60447 10 5. �
� m. It is noticed that the maximum dynamic deflection increases linearly with
the increase in dynamically activated depth parameter, Hs . This result agrees very well
with the previous work done by Gibigaye et al. [11].
4.3. Influence of the Inertia of the Soil Parameters on the Dynamic Response. To
study the influence of the inertia of the soil factor, m0, on the dynamic response of the
pavement plate, the dynamic deflection of pavement plate during the period of
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 27
Dynamic Response of Pavement Plates ...
Fig. 3. Variation of the deflection directly under the localized dynamic loading versus time for
different values of pavement’s thickness.
Fig. 4. Variation of the deflection directly under the localized dynamic loading versus time for
different values of soil parameters.
t t t td d d� � � is computed as shown in Fig. 5. It is noticed that the mid-point deflection
of the pavement plate on Pasternak foundation is 25.7% higher compared to the deflection
values for the three-parameter soil. In conclusion, inertial soil dramatically reduces the
dynamic response of the pavement plate when the dynamic load is at the center of the plate.
Figure 6 shows the influence of the internal bending moment during the negative
phase of the load for two different type of soil conditions. It is shown that the inertial soil
factor (m0) is an essential factor in reducing the maximum value of the internal moment in
the x-direction for as much as 29.3%.
4.4. Effect of the Location of the Dynamic Load along the x-Axis. Figure 7 shows
the time history of the mid-point deflection of the pavement plate subjected to localized
Friedlander loading for different values of x0 (x a0 0 125� . , 0 25. ,a 0 375. ,a and 0 5. a). The
results show that the dynamic deflection at mid-point increases as the position of the load
closes to the middle of the pavement plate.
28 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
S. W. Alisjahbana, Safrilah, J. C. P. Putra, et al.
Fig. 5. The influence of the dynamic deflection of the pavement plate over the plate region due to the
Friedlander decaying for two type of soil parameters conditions.
Fig. 6. Variation of the internal bending moment (Mx) as a function of x position computed during
the negative phase of the load.
Conclusions. This paper dealt with some significant results from an analytical study
of pavement concrete plate sitting on a three-type soil parameter foundation subjected to
localized Friedlander loading which includes positive and negative phases. The pavement
model consists of discrete orthotropic plates joined at the discontinuities by vertical springs
representing the load transfer devices. The foundation model used in this work is the
well-known Pasternak model which takes into account the interaction between soil layers,
and the inertial factor of soil. The main conclusions of this study are the following:
1. The negative phase of the localized dynamic load plays an important factor in
increasing the maximum dynamic deflection as well as the internal bending moment of the
system.
2. The soil inertia factor (m0) influences the maximum dynamic deflection of the
pavement plate. By including the m0 factor into the analysis, the maximum dynamic
deviation of the pavement plate was reduced by as much as 25.7%.
3. The effect of the plate thickness is significant because it affects the overall behavior
of the concrete plate.
4. The mid-span dynamic deflection depends on the position of the blast loading. The
mid-span dynamic deflection increases as the load near the middle of the plate.
5. This study covers the pavement plates interconnected by the conditions that allow
rotational and translational deformation along the boundaries. These conditions can be
extended to another type of boundary conditions.
While this paper deals mainly with computational results, Yuen and Nurick [18] and
Gibigaye et al. [11] reported that the dynamic response of the pavement plates is influenced
significantly by the thickness of the plate and the soil parameters. That approach provides a
satisfactory correlation and creates a better understanding of dynamic responses of the
pavement plate on three-type of soil parameters.
Acknowledgments. The research has been supported by the Director General of
Research and Development Reinforcement, Ministry of Research Technology and the
Higher Education Republic of Indonesia, contract No. 035/KM/PNT/2018.
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ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 29
Dynamic Response of Pavement Plates ...
Fig. 7. Dynamic deflection time history of pavement plate subjected to localized Friedlander load for
different value of x0 (Case 2).
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