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Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering
journal homepage: www.elsevier.com/locate/soildyn
Vibration analysis of concrete foundations retrofit with NFRP layer resting
on soil medium using sinusoidal shear deformation theory
Abdollah Zamania, Mahmood Rabani Bidgolib,
a
b
MARK
⁎
Department of Civil Engineering, Khomein Branch, Islamic Azad University, Khomein, Iran
Department of Civil Engineering, Jasb Branch, Islamic Azad University, Jasb, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords:
Vibration
Concrete foundation
NFRP layer
CNTs
Soil medium
Sinusoidal shear deformation theory
After decades of service, concrete foundations may be in need of strengthening and retrofitting due to continuous
aging. However, the main objective of this work is presenting a mathematical model for vibration of concrete
foundation with nano fiber reinforced polymer (NFRP) layer resting on soil medium. The nano fibers of NFRP
layer are made from carbon nanotubes (CNT) where the effective material properties of nano-composite
structure are obtained by Mori-Tanaka model considering agglomeration effects. The soil medium is simulated
with spring constants of Winker foundation model. Based on the sinusoidal shear deformation theory (SSDT),
energy method and Hamilton's principle, the motion equations are derived. Applying Navier method, the frequency of the structure is calculated analytically so that the effects of volume percent and agglomeration of
CNTs, soil medium, structural damping and geometrical parameters of structure are shown on the frequency of
system. Results show that considering NFRP layer leads to higher frequency for NFRP thickness to concrete
foundation thickness ratio of higher than 0.005.
1. Introduction
The concrete foundations resting on soil substrates are one of the
most commonly used structural elements in civil engineering applications such as pavement of highways, footing of buildings and bases of
machines. Since vibration of foundations cause many crucial problems
in civil engineering field especially for instruments with high-level
safety and functionality, studying this topic is essential and therefore
the vibrations may need to be reduced.
There are several ways to reduce vibration of concrete foundations
such as improving the properties of them with additive materials, retrofitting with fiber reinforced polymer (FRP) layers and so on. Persson
et al. [1] done a numerical research on reducing building vibrations
through foundation improvement. Investigations in recent years have
clearly demonstrated that confinement with a FRP cover leads to a
substantial improvement in the strength, ductility and dynamical behavior of concrete elements. Wei and Wu [2] presented a unified
stress–strain model of concrete for circular, square, and rectangular
columns confined by FRP jackets. Finite element analysis of masonry
panels strengthened with FRPs was studied by Grande et al. [3]. They
discussed some strategies, in the framework of the nonlinear finite
element analysis, regarding the elements choice and the models to
adopt for reliable nonlinear analyses of masonry structures reinforced
⁎
with FRP strips. Hemmatnezhad et al. [4] made an experimental, numerical and analytical investigation on free vibrational behavior of
glass fiber reinforced polymer (GFRP)-stiffened composite cylindrical
shells. Static and free vibration of reinforced concrete beams with
carbon fiber reinforced polymer (CFRP) rectangular rods were analyzed
by Capozucca and Bossoletti [5]. Pan and Wu [6] proposed an analytical modeling of bond behavior between FRP plate and concrete. CFRP
composite retrofitting effect on the dynamic characteristics of arch
dams was investigated by Altunisik et al. [7]. The influence of using
CFRP wraps on performance of buried steel pipelines under permanent
ground deformations was studied by Mokhtari and Alai Nia [8]. Bond
behavior of FRP carbon plates externally bonded over steel and concrete elements was considered experimentally and numerically by
Ceroni et al. [9]. Seismic strengthening of infilled reinforced concrete
frames by CFRP was studied by Erol and Karadogan [10].
On the other hand, recent progresses in innovative and advanced
engineering materials have given more emphasis on utilization of nano
scale reinforcing elements. One of perfect candidates for using in the
reinforcement phase of composite materials is CNTs. Because their
Young's modulus and tensile strength are about 1 TPa and 150 GPa,
respectively [11]. Many researchers studied the mechanical behaviors
of CNTs reinforced composite plates. The nonlinear vibration behaviors
of a reinforced composite plate with the CNTs under combined the
Corresponding author.
E-mail address: m.rabanibidgoli@gmail.com (M.R. Bidgoli).
http://dx.doi.org/10.1016/j.soildyn.2017.09.018
Received 24 July 2017; Received in revised form 21 August 2017; Accepted 23 September 2017
0267-7261/ © 2017 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
graded plates using higher-order shear deformation theory was studied
by Tran et al. [36]. Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory was studied
by Wang and Wu [37].
Dynamic analyses of concrete structures improved by nanoparticles
based on mathematical models are the novel topics in civil engineering.
Research works undertaken in relation to mathematical modeling of
concrete structures are very limited in the literature. One of these investigations is the research of Jafarian Arani and Kolahchi [38]. They
studied buckling analysis of concrete columns retrofit with CNT-reinforced polymer layer by using Euler-Bernoulli and Timoshenko beam
models. To the best of the authors’ knowledge, the mathematical
modeling of concrete foundations strengthened with NFRP layer has not
been reported yet. However, due to lack of this subject in the literature,
vibration analysis of concrete foundations retrofitted by NFRP layer
resting on soil medium is presented in this work. The soil medium is
simulated by spring constants. In order to obtain the equivalent material properties of nano-composite structure, the Mori-Tanaka model is
used considering agglomeration effects of nano fibers. Applying SSDT,
the motion equations are obtained based on Hamilton's principal. Also,
Navier method is applied for obtaining the frequency of the system.
NFRP layers have the short and long-term effects on the strength and
lifetime of concrete structures. Hence, the aim of this research is
studying the effects of NFRP layer on the vibration response of the
concrete foundations as well as showing the effects of volume percent
and agglomeration of CNTs, soil medium, structural damping and
geometrical parameters of structure on the frequency of system.
parametric and forcing excitations were studied by Guo and Zhang
[12]. An element-free analysis of CNT-reinforced composite plates with
column supports and elastically restrained edges under large deformation was considered by Zhang et al. [13]. Vibration analysis of CNTsreinforced thick laminated composite plates based on Reddy's higherorder shear deformation theory was done by Zhang and Selim [14].
Zhang et al. [15] conducted the first-known vibration analysis of CNT
reinforced functionally graded composite triangular plates subjected to
in-plane stresses. Ahmadi et al. [11] analyzed the multi-scale bending,
buckling and vibration of carbon fiber/carbon nanotube-reinforced
polymer nanocomposite plates with various shapes. Shen and Wang
[16] investigated the small- and large-amplitude vibrations of compressed and thermally postbuckled carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations. Kumar and Srinivas [17] calculated the vibration, buckling and bending behavior of
functionally graded multi-walled carbon nanotube reinforced polymer
composite plates using the layer-wise formulation.
Also, several works have been made in the field of vibration analysis
of plates on elastic foundations in recent years. Xiang et al. [18] studied
analytically the vibration of rectangular Mindlin plates with simply
supported boundary conditions on Pasternak foundation. The finite
element method was used by Omurtag et al. [19] to study the free vibration of thin plates resting on Pasternak foundation. Lam et al. [20]
employed the Green's functions to obtain canonical exact solutions of
elastic bending, buckling and vibration for Levy plates resting on twoparameter elastic foundations. By employing the Rayleigh-Ritz method,
the three dimensional vibration of rectangular thick plates on elastic
foundations was analyzed by Zhou et al. [21]. By using the differential
quadrature method (DQM), developing a three dimensional layerwisefinite element method and coupling a three-dimensional (3D) discrete
layer approach with DQM, Malekzadeh et al. studied the free vibration
analysis of rectangular [22,23] plates resting on elastic foundation.
Ferreira et al. [24] used the radial basis function collocation method to
study the static deformation and free vibration of plates on Pasternak
foundation. Kumar and Lal [25] studied the vibration analysis of nonhomogeneous orthotropic rectangular plates with bilinear thickness
variation resting on Winkler foundation. Element Free Galerkin method
was used by Bahmyari and Rahbar-Ranji [26] for free vibration analysis
of orthotropic plate with variable thickness and resting on nonuniform
elastic foundation. Bahmyari and khedmati [27] considered the vibration analysis of nonhomogeneous moderately thick plates with point
supports resting on Pasternak elastic foundation using element free
Galerkin method. Vibrational analysis of advanced composite plates
resting on elastic foundation was studied by Mantari et al. [28]. They
derived the governing equations of a type of functionally graded plates
resting on elastic foundation by employing the Hamilton's principal.
Nguyen-Thoi et al. [29] presented an edge-based smoothed three-node
Mindlin plate element (ES-MIN3) for static and free vibration analyses
of plates. Uğurlu [30] analyzed the vibration of elastic bottom plates of
fluid storage tanks resting on Pasternak foundation based on boundary
element method. An original first shear deformation theory to study
advanced composites on the elastic foundation was presented by
Mantari and Granados [31].
Recently, higher-order shear deformation theories (HSDTs) are used
for mathematical modeling of structures. Because these theories can be
employed for thick plates without any necessity to use shear correction
factors. One of HSDTs is sinusoidal shear deformation theory. Neves
et al. [32,33] investigated a quasi-3D sinusoidal shear deformation
theory for the static and free vibration analysis of functionally graded
plates. A simple quasi-3D sinusoidal shear deformation theory was
considered for functionally graded plates by Thai and Kim [34]. Their
studies showed that the obtained results with this theory are more accurate than those obtained by other higher-order shear deformation
theories. A new sinusoidal shear deformation theory (SSDT) for
bending, buckling, and vibration of functionally graded plates was investigated by Thai and Vo [35]. Isogeometric analysis of functionally
2. Mathematical modeling
As shown in Fig. 1, a concrete foundation including NFRP layer
resting on soil medium with length a, width b, concrete thickness 2h
and NFRP layer thickness hf is considered.
2.1. Sinusoidal Shear Deformation Theory
Based on SSDT, the displacement field can be expressed as [39]
U1 (x , y, z , t ) = U (x , y, t ) − z
∂Wb
h
πz ⎞ ∂Ws
− ⎛z − ( sin
)
,
∂x
π
h ⎠ ∂x
⎝
(1)
U2 (x , y, z , t ) = V (x , y, t ) − z
∂Wb
h
πz ⎞ ∂Ws
− ⎛z − ( sin
,
)
∂y
π
h ⎠ ∂y
⎝
(2)
U3 (x , y, z , t ) = Wb (x , y, t ) + Ws (x , y, t ),
(3)
where (U , V , Wb, Ws ) denote the displacement components. Based on
above relations, the strain-displacement equations may be written as
0
b
s
⎛ k xx ⎞
⎛ εxx ⎞
ε
⎛ k xx ⎞
⎛ xx ⎞
h
πz ⎞ k s
0
b ⎟
εyy = ⎜ ε yy ⎟ + z ⎜ k yy
⎛
⎜ yy ⎟,
+ z − sin
⎜ ⎟
π
h ⎠⎜ s ⎟
⎜⎜ b ⎟⎟ ⎝
⎜ 0 ⎟⎟
γ
k
⎝ xy ⎠ ⎜ γxy
k xy
⎝ xy ⎠
⎝ ⎠
⎝ ⎠
(4)
s
γ
⎛⎜ γyz ⎞⎟ = cos πz ⎛ yz ⎞,
⎜γ s ⎟
γxz
h
⎝ ⎠
⎝ xz ⎠
Fig. 1. A schematic figure for concrete foundation with NFRP layer.
142
(5)
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
Vr = Vrinclusion + Vrm
where
∂U
0
⎛ ∂x
⎛ εxx ⎞
⎜ ∂V
0
⎜ ε yy ⎟ =
⎜ ∂y
⎜⎜ 0 ⎟⎟
γxy
⎜ ∂U +
⎝ ⎠
⎝ ∂y
∂2Wb
∂x 2
⎞
⎛−
b
⎛ k xx ⎞ ⎜
2W ⎟
∂
b
b
⎜ k yy ⎟ = ⎜ −
⎟,
∂y 2
⎜⎜ b ⎟⎟ ⎜
⎟
2
k
∂ W
⎝ xy ⎠ ⎜− 2 ∂x ∂yb ⎟
⎠
⎝
⎞
⎟
⎟ ,
∂V ⎟
∂x ⎠
Vrinclusion
∂2Ws
∂x 2
⎛−
⎞
s
⎛ k xx ⎞ ⎜
⎟
∂2Ws
s
⎜ k yy ⎟ = ⎜ − 2 ⎟
∂y
⎜k s ⎟ ⎜
2 ⎟
⎝ xy ⎠ ⎜− 2 ∂ Ws ⎟
∂x ∂y ⎠
⎝
and
are the volumes of nanofibers dispersed in the
where
spherical inclusions and in the matrix, respectively. Introduce two
parameters ξ and ζ describe the agglomeration of nanofibers
(6a)
ξ=
Vinclusion
,
V
(11)
ζ=
Vrinclusion
.
Vr
(12)
∂W
s
s
−4 ⎛ ∂y ⎞
⎛ γyz ⎞
=
⎜ ⎟
⎜γ s ⎟
h3 ⎜ ∂Ws ⎟
⎝ xz ⎠
⎝ ∂x ⎠
(6b)
However, the average volume fraction Cr of nanofibers in the
composite is
2.2. Stress-strain relations
Based on Hook's law, the stress-strain relation of the concrete
foundation can be written as
k+m
l
Q⏟
12
n
Q⏟
22
k−m

0
0
0
0
0
0
0
Q13
⎡
K = K out ⎢1 +
⎢
⎣
(7)
l
Q⏟
23
l 
k+m
Q⏟
32
Q
33
0
0
0
0
p
Q⏟
44
0
0
0
0
m
Q⏟
55
0
n=
Em2 cm (1 + cr − cm νm) + 2cm cr (kr nr − lr2)(1 + νm)2 (1 − 2νm)
(1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )]
Em [2cm2 kr (1 − νm) + cr nr (1 + cr − 2νm) − 4cm lr νm]
+
Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )
Em [Em cm + 2pr (1 + νm)(1 + cr )]
p=
2(1 + νm)[Em (1 + cr ) + 2cm pr (1 + νm)]
Em [Em cm + 2mr (1 + νm)(3 + cr − 4νm)]
m=
2(1 + νm){Em [cm + 4cr (1 − νm)] + 2cm mr (3 − νm − 4νm2 )}
(14)
)
G
(
)
(15)
where
(δr − 3Km χr ) Cr ζ
,
3(ξ − Cr ζ + Cr ζχr )
Kin = Km +
K out = Km +
Gin = Gm +
(8)
Gout = Gm +
Cr (δr − 3Km χr )(1 − ζ )
,
3[1 − ξ − Cr (1 − ζ ) + Cr χr (1 − ζ )]
(ηr − 3Gm βr ) Cr ζ
,
2(ξ − Cr ζ + Cr ζβr )
Cr (ηr − 3Gm βr )(1 − ζ )
,
2[1 − ξ − Cr (1 − ζ ) + Cr βr (1 − ζ )]
(16)
(17)
(18)
(19)
where χr , βr , δr , ηr may be calculated as
Em {Em cm + 2kr (1 + νm)[1 + cr (1 − 2νm)]}
k=
2(1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )]
Em {cm νm [Em + 2kr (1 + νm)] + 2cr lr (1 − νm2 )]}
(1 + νm)[Em (1 + cr − 2νm) + 2cm kr (1 − νm − 2νm2 )]
⎤
⎥ ,
K
1 + α (1 − ξ ) ⎛ in − 1⎞ ⎥
⎝ K out
⎠⎦
(
where σij , εij, γij, k , m , n, l, p are the stress components, the strain
components and the stiffness coefficients respectively. According to the
Mori-Tanaka method, the stiffness coefficients are given by [40]
l=
K
ξ ⎛ in − 1⎞
⎝ K out
⎠
⎡
⎤
ξ G in − 1
out
⎥,
G = Gout ⎢1 +
Gin
⎢
1 + β (1 − ξ ) G − 1 ⎥
out
⎣
⎦
0
⎤
⎥
0
⎥ ε11
⎧ ⎫
0 ⎥ ⎪ ε22 ⎪
⎥ ⎪ ε33 ⎪
0 ⎥ ⎨ γ23 ⎬
⎥ ⎪ γ13 ⎪
⎪γ ⎪
0 ⎥ ⎩ 12 ⎭
⎥
p⎥
Q⏟
66 ⎥
⎦
(13)
Assume that all the orientations of the nanofibers are completely
random. Hence, the effective bulk modulus (K) and effective shear
modulus (G) may be written as
where Cij are elastic constants. In addition, the stress-strain relation of
NFRP layer is
⎡ 
Q
⎢ 11l
⎢ Q⏟12
σ
⎧ σ11 ⎫ ⎢ k − m
⎪ 22 ⎪ ⎢ 
⎪ σ33 ⎪
Q31
=
⎨ σ23 ⎬ ⎢ 0
⎪ σ13 ⎪ ⎢
⎪ σ12 ⎪
⎩ ⎭ ⎢ 0
⎢
⎢ 0
⎢
⎣
Vr
.
V
Cr =
0
0 ⎤ εxx
C11 C12 C13 0
σ
⎡ ⎤
⎡ σxx ⎤ ⎡
0
0
0 ⎥ ⎢ εyy ⎥
C
C
C
⎢
yy
21
22
23
⎢ ⎥
0
0 ⎥ ⎢ εzz ⎥
⎢ σzz ⎥ ⎢C31 C32 C33 0
⎥ ⎢ γ ⎥,
⎢ σzy ⎥ = ⎢ 0
0
0
0
0
C
44
⎥ ⎢ zy ⎥
⎢σ ⎥ ⎢
xz
⎢
0
0
0
0 C55 0 ⎥ ⎢ γxz ⎥
⎢ ⎥
⎥ γzy ⎥
⎢
⎣ σzy ⎦ ⎣ 0
0
0
0
0 C66 ⎦ ⎢
⎣ ⎦
(10)
Vrm
χr =
3(Km + Gm) + kr − lr
,
3(kr + Gm)
βr =
1 ⎧ 4Gm + 2kr + lr
4Gm
+
k
G
p
+
5⎨
3(
)
(
r
m
r + Gm )
⎩
2[Gm (3Km + Gm) + Gm (3Km + 7Gm)] ⎫
,
Gm (3Km + Gm) + mr (3Km + 7Gm) ⎬
⎭
(21)
δr =
(2kr − lr )(3Km + 2Gm − lr ) ⎤
1⎡
n + 2lr +
,
⎥
⎢ r
3⎣
kr + Gm
⎦
(22)
ηr =
4Gm pr
8Gm mr (3Km + 4Gm)
1 ⎡2
+
(nr − lr ) +
5⎢
(pr + Gm)
3Km (mr + Gm) + Gm (7mr + Gm)
⎣3
+
(9)
+
where the subscripts m and r stand for matrix and reinforcement respectively. Cm and Cr are the volume fractions of the matrix and the
nanofibers respectively and kr ، lr ، nr ، pr, mr are the Hills elastic
modulus for the nanofibers [40]. The experimental results show that the
assumption of uniform dispersion for nanofibers in the matrix is not
correct and the most of nanofibers are bent and centralized in one area
of the matrix. These regions with concentrated nanofibers are assumed
to have spherical shapes, and are considered as ‘‘inclusions’’ with different elastic properties from the surrounding material. The total volume Vr of nanofibers can be divided into the following two parts [41]
(20)
2(kr − lr )(2Gm + lr ) ⎤
.
⎥
3(kr + Gm)
⎦
(23)
where, Km and Gm are the bulk and shear moduli of the matrix which
can be written as
Km =
Em
3(1 − 2υm)
Gm =
Em
.
2(1 + υm)
,
Furthermore, β , α can be obtained from
143
(24)
(25)
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
α=
(1 + υout )
,
3(1 − υout )
β=
2(4 − 5υout )
,
15(1 − υout )
υout =
(26a)
Qx = A55g
∂
∂2
Ws + GA55g
Ws,
∂x
∂x ∂t
(39)
Q y = A66g
∂2
∂
Ws + GA66g
Ws,
∂y ∂t
∂y
(40)
(26b)
3K out − 2Gout
.
6K out + 2Gout
∂
∂2
∂2
∂
∂2
U − B11 2 Wb − A11zf 2 Ws + A12z V − B12 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− A12zf 2 Ws,
∂y
(41)
MxxB = A11z
(27)
Finally, the elastic modulus (E) and poison's ratio (υ) can be calculated as
E=
9KG
3K + G
υ=
3K − 2G
.
6K + 2G
,
∂
∂2
∂2
∂
∂2
U − A11zf 2 Wb − E11 2 Ws + A12f V − A12zf 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− E12 2 Ws,
∂y
(42)
MxxS = A11f
(28)
(29)
∂
∂2
∂2
∂
∂2
U − B21 2 Wb − A21zf 2 Ws + A22z V − B22 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− A22zf 2 Ws,
∂y
(43)
MyyB = A21z
2.3. Energy method
The potential energy can be written as
U=
1
2
h
∫A ∫− h2 (σxx εxx + σyy εyy + σxy γxy + σxz γxz+σyz γyz ) dzdA
∂
∂2
∂2
∂
∂2
U − A21zf 2 Wb − E21 2 Ws + A22f V − A22zf 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− E22 2 Ws,
∂y
(44)
MyyS = A21f
(30)
2
Combining Eqs. (1)–(3) and (30) yields
U=
∂U
∂U
∂V
∂V
∂Ws
1
+ Nxy
+ Nxy
+ Nyy
+ Qx
∫ ⎜⎛Nxx
∂x
∂y
∂x
∂y
∂x
2 A⎝
∂Ws
∂2Ws
∂2Ws
∂2Ws
+ Qy
− MxxS
− MyyS
− 2MxyS
∂y
∂x 2
∂y 2
∂y ∂x
− MxxB
∂2Wb
∂2Wb
∂2Wb ⎞
− MyyB
− 2MxyB
⎟ dA ,
∂x 2
∂y 2
∂y ∂x ⎠
h
∫−h
⎡ MxxB ⎤
⎢ MyyB ⎥ =
⎢M ⎥
⎣ xyB ⎦
σ
⎡ σxx ⎤
⎢ yy ⎥ dz +
⎥
⎢ σxy ⎦
⎣
h
∫−h
∂
∂
∂2
∂2
U + 2A 44z V − 2B44
Wb − 2A 44zf
Ws,
∂y
∂x
∂x ∂y
∂x ∂y
(45)
MxyS = 2A 44f
∂
∂
∂2
∂2
U + 2A 44f V − 2A 44zf
Wb − 2E44
Ws,
∂y
∂x
∂x ∂y
∂x ∂y
(46)
where
(31)
h
(A11 , A12 , A22 , A 44 ) = ∫−h (C11, C12, C22, C44 ) dz
where the stress resultant-displacement relations can be written as
⎡ Nxx ⎤
⎢ Nyy ⎥ =
⎢N ⎥
⎣ xy ⎦
MxyB = 2A 44z
h + hf
+ ∫h
f
∫h
σ
⎡ σxx ⎤
⎢ yy ⎥ zdz +
⎢
⎦
⎣ σxy ⎥
h + hf
∫h
⎡ σxx ⎤
⎢σ f ⎥
⎢ yy ⎥ dz
⎢σ f ⎥
⎣ xy ⎦
f
⎡ σxx ⎤
h + hf ⎢ f ⎥
⎢ σyy ⎥
(Q11, Q12, Q22, Q44 ) dz ,
(47)
h
(A11z , A12z , A22z , A 44z ) = ∫−h (C11, C12, C22, C44 ) zdz
(32)
h + hf
+ ∫h
(Q11, Q12, Q22, Q44 ) zdz ,
(48)
h
(A11f , A12f , A22f , A 44f ) = ∫−h (C11, C12, C22, C44 ) fdz
zdz
h + hf
⎢σ f ⎥
⎣ xy ⎦
+ ∫h
(33)
(Q11, Q12, Q22, Q44 ) fdz ,
(49)
h
⎡ MxxS ⎤
⎢ MyyS ⎥ =
⎢M ⎥
⎣ xyS ⎦
⎡Q x ⎤ =
⎢Q y ⎥
⎣ ⎦
h
∫−h
σ
⎡ σxx ⎤
⎢ yy ⎥ fdz +
⎢
⎦
⎣ σxy ⎥
σ
h
∫h
h + hf
h + hf
∫−h ⎡⎢ σxxyy ⎤⎥ pdz + ∫−h
⎣
⎦
(A11zf , A12zf , A22zf , A 44zf ) = ∫−h (C11, C12, C22, C44 ) zfdz
f
⎡ σxx ⎤
⎢σ f ⎥
⎢ yy ⎥ fdz
⎢σ f ⎥
⎣ xy ⎦
h + hf
+ ∫h
(34)
h
h + hf
(35)
∂2
∂
∂2
∂2
∂
∂2
Nyy = A21 U − A21z 2 Wb − A21f 2 Ws + A22 V − A22z 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− A22f 2 Ws,
∂y
Nxy = A 44
∂
∂
∂2
∂2
U + A 44 V − 2A 44z
Wb − 2A 44f
W,
∂y
∂x
∂x ∂y
∂x ∂y
(51)
(B11, B12, B22, B44 ) = ∫ (C11, C12, C22, C44 ) z 2dz
h + hf
+ ∫h
∂2
∂
∂
U − A11z 2 Wb − A11f 2 Ws + A12 V − A12z 2 Wb
∂x
∂x
∂x
∂y
∂y
∂2
− A12f 2 Ws,
∂y
Nxx = A11
(Q55, Q66) gdz ,
h
−h
Substituting stress-strain relation into Eqs. (32)–(35), the stress resultant-displacement relations can be obtained as follows
∂2
(50)
(A55g , A66g ) = ∫−h (C55, C66) gdz
+ ∫h
⎡ σxx ⎤ pdz,
⎢ σyy ⎥
⎣ ⎦
(Q11, Q12, Q22, Q44 ) zfdz ,
(Q11, Q12, Q22, Q44 ) z 2dz ,
(52)
h
(E11, E12, E22, E44 ) = ∫−h (C11, C12, C22, C44 ) f 2 dz
h + hf
+ ∫h
(36)
(Q11, Q12, Q22, Q44 ) f 2 dz ,
(53)
The kinetic energy of system may be written as
K=
1
ρ
2
h
2
∫A ∫− 2h ⎛ ⎛⎝ ∂∂ut ⎞⎠
⎜
2
⎝
2
2
∂v
∂w
+ ⎛ ⎞ + ⎛ ⎞ ⎟⎞ dz dA .
⎝ ∂t ⎠
⎝ ∂t ⎠ ⎠
(54)
The external work due to soil medium can be written as [42]
(37)
We =
(38)
∬ (−Kw w) wdA,
(55)
where Kw is Winkler's spring modulus. The governing equations can be
144
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
derived by Hamilton's principal as follows
∂2 ⎛
∂
∂2
∂2
∂
∂2
A11z U − B11 2 Wb − A11zf 2 Ws + A12z V − B12 2 Wb
2
∂x ⎝
∂x
∂x
∂x
∂y
∂y
⎜
∫0
t
(δU − δK − δWe ) dt = 0.
(56)
− A12zf
Substituting Eqs. (30), (54) and (55) into Eq. (56) yields the following governing equations
∂
∂
∂ 2U
∂3Wb
∂3Ws
Nxx +
Nxy − I0 2 + I1
+ J1
= 0,
2
∂x
∂y
∂t
∂x ∂t
∂x ∂t 2
+2
+
(57)
∂2
Ws ⎞
∂y 2 ⎠
∂2 ⎛
∂
∂
∂2
∂2
2A 44z U + 2A 44z V − 2B44
Wb − 2A 44zf
Ws ⎞
∂x ∂y ⎝
∂y
∂x
∂x ∂y
∂x ∂y ⎠
⎜
(58)
⎟
∂2 ⎛
∂
∂2
∂2
∂
∂2
A21z U − B21 2 Wb − A21zf 2 Ws + A22z V − B22 2 Wb
2
∂y ⎝
∂x
∂x
∂x
∂y
∂y
⎜
− A22zf
∂
∂
∂ 2V
∂3Wb
∂3Ws
Nxy +
Nyy − I0 2 + I1
+ J1
= 0,
2
∂x
∂y
∂t
∂y ∂t
∂y ∂t 2
⎟
∂2
Ws ⎞
∂y 2 ⎠
⎟
∂ 2W
∂2Ws ⎞
∂3U
∂3V ⎞
− Kw W − Kw Ws = I0 ⎛ 2 b +
+ I1 ⎛
+
2
2
t
t
x
t
y ∂t 2 ⎠
∂
∂
∂
∂
∂
⎝
⎠
⎝
⎜
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎜
⎟
∂ 4W
∂ 4W
∂ 4W
∂ 4W
− I2 ⎛ 2 b2 + 2 b2 ⎞ − J2 ⎛ 2 s2 + 2 s2 ⎞ ⎤,
∂y ∂t ⎠ ⎥
∂y ∂t ⎠
⎝ ∂x ∂t
⎝ ∂x ∂t
⎦
∂2
∂2
∂2
∂ 2W
∂2Ws ⎞
MxxB + 2
MxyB + 2 MyyB − Kw w − I0 ⎛ 2 b +
2
∂x
∂x ∂y
∂y
∂t 2 ⎠
⎝ ∂t
3
3
4
4
4
∂U
∂V ⎞
∂W
∂W
∂W
∂ 4W
− I1 ⎛
+
+ I2 ⎛ 2 b2 + 2 b2 ⎞ + J2 ⎛ 2 s2 + 2 s2 ⎞
2
2
∂y ∂t ⎠
∂y ∂t ⎠
∂y ∂t ⎠
⎝ ∂x ∂t
⎝ ∂x ∂t
⎝ ∂x ∂t
⎜
⎟
⎟
⎜
⎟
(64)
⎟
∂2 ⎛
∂
∂2
∂2
∂
∂2
A11f U − A11zf 2 Wb − E11 2 Ws + A12f V − A12zf 2 Wb
∂x 2 ⎝
∂x
∂x
∂x
∂y
∂y
= 0,
⎜
(59)
− E12
∂2
∂2
∂2
∂
∂
MxxS + 2
MxyS + 2 MyyS +
Qx +
Q y − Kw w
2
∂x
∂x ∂y
∂y
∂x
∂y
+2
∂ 2W
∂2Ws ⎞
− I0 ⎛ 2 b +
∂t 2 ⎠
⎝ ∂t
3
3
∂U
∂V ⎞
∂ 4W
∂ 4W
∂ 4W
∂ 4W
− J1 ⎛
+
+ J2 ⎛ 2 b2 + 2 b2 ⎞ + K2 ⎛ 2 s2 + 2 s2 ⎞
2
2
∂y ∂t ⎠
∂y ∂t ⎠
∂y ∂t ⎠
⎝ ∂x ∂t
⎝ ∂x ∂t
⎝ ∂x ∂t
= 0,
⎜
⎟
⎜
⎟
⎜
⎟
⎜
+
∂2
Ws ⎞
∂y 2 ⎠
⎟
∂2 ⎛
∂
∂
∂2
∂2
Wb − 2E44
W⎞
2A 44f U + 2A 44f V − 2A 44zf
∂x ∂y ⎝
∂y
∂x
∂x ∂y
∂x ∂y ⎠
⎜
⎟
∂2 ⎛
∂
∂2
∂2
∂
∂2
A21f U − A21zf 2 Wb − E21 2 Ws + A22f V − A22zf 2 Wb
2
∂y ⎝
∂x
∂x
∂x
∂y
∂y
⎜
⎟
− E22
∂2
Ws+⎞
∂y 2
⎠
⎟
∂ ⎛
∂
∂ ⎛
∂
A55g Wss ⎞ +
A66g Ws ⎞+
∂x ⎝
∂x
∂y ⎝
∂y ⎠
⎠
∂3V ⎞
∂ 2W
∂2Ws ⎞
∂3U
+ J1 ⎛
+
− Kw W − Kw Ws = I0 ⎛ 2 b +
2
2
∂y ∂t 2 ⎠
∂t ⎠
⎝ ∂t
⎝ ∂x ∂t
(60)
+
where
⎜
⎟
⎜
h
−h
(I0, I1, I2, J1, J2 , K2) = ∫ ρ (1, z , f , zf ,
h + hf
+ ∫h
z 2,
f 2)
⎜
⎟
∂ 4W
∂ 4W
∂ 4W
∂ 4W
− J2 ⎛ 2 b2 + 2 b2 ⎞ − K2 ⎛ 2 s2 + 2 s2 ⎞ ⎤,
x
t
y
t
x
t
y ∂t ⎠ ⎥
∂
∂
∂
∂
∂
∂
∂
⎠
⎝
⎝
⎦
⎜
ρf (1, z , f , zf , z 2, f 2 ) dz.
⎟
(61)
⎟
⎜
⎟
(65)
Substituting Eqs. (36)–(46) into Eqs. (57)–(60), the governing
equations can be written as follows
3. Solution procedure
∂ ⎛
∂
∂2
∂2
∂
∂2
A11 U − A11z 2 Wb − A11f 2 Ws + A12 V − A12z 2 Wb
∂x ⎝ ∂x
∂x
∂x
∂y
∂y
⎜
− A12f
+
∂2
∂y 2
Steady state solutions to the governing equations of the system
which relate to the simply supported boundary conditions can be assumed as
Ws ⎞
⎠
⎟
∂ ⎛
∂
∂
∂2
∂2
A 44 U + A 44 V − 2A 44z
Wb − 2A 44f
Ws ⎞
∂y ⎝
∂y
∂x
∂x ∂y
∂x ∂y ⎠
⎜
∂ 2U
∂3Wb
∂3Ws ⎤
= ⎡I0 2 − I1
− J1
,
2
⎢ ∂t
⎥
∂x ∂t
∂x ∂t 2 ⎦
⎣
nπy iωt
mπx
)sin(
)e ,
L
b
(66)
nπy iωt
mπx
)cos(
)e ,
L
b
(67)
U (x , y, t ) = u 0 cos(
⎟
V (x , y, t ) = v0 sin(
(62)
nπy iωt
mπx
)sin(
)e ,
L
b
(68)
nπy iωt
mπx
)sin(
)e ,
L
b
(69)
Wb (x , y, t ) = wb0 sin(
∂2
∂2
∂ ⎛
∂
∂
A 44 U + A 44 V − 2A 44z
Wb − 2A 44f
Ws ⎞
∂x ⎝
∂y
∂x
∂x ∂y
∂x ∂y ⎠
⎜
⎟
Ws (x , y, t ) = ws0 sin(
∂ ⎛
∂
∂2
∂2
∂
∂2
+
A21 U − A21z 2 Wb − A21f 2 Ws + A22 V − A22z 2 Wb
∂y ⎝
∂x
∂x
∂x
∂y
∂y
⎜
− A22f
Substituting Eqs. (66)–(69) into Eqs. (62)–(65) yields
∂2
Ws ⎞
∂y 2 ⎠
⎡ K11
⎢ K21
⎢ K31
⎢
⎣ K 41
⎟
∂ 2V
∂3Wb
∂3Ws ⎤
= ⎡I0 2 − I1
− J1
,
2
⎢
∂y ∂t
∂y ∂t 2 ⎥
⎣ ∂t
⎦
(63)
K12
K22
K32
K 42
K13
K23
K33
K 43
K14 ⎤ u 0
⎤
K24 ⎥ ⎡
⎢ v0 ⎥ = 0,
K34 ⎥ ⎢ wb0 ⎥
⎥ ws0 ⎥
⎦
⎣
K 44 ⎦ ⎢
(70)
where Kij are defined in Appendix A. Finally, for calculating the frequency of the system (ω ), the determinant of matrix in Eq. (70) should
be equal to zero.
145
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
4. Numerical results and discussion
Here, a concrete foundation with Young's modulus of Ec = 20 GPa
and Poisson's ratio of νc = 0.2 is considered. The NFRP layer is made
from a polymer with Young's modulus of Em = 18 GPa and Poisson's
ratio of νm = 0.3 which is reinforced by CNTs by Young's modulus of
Er = 1 TPa and Poisson's ratio of νr = 0.3. The effects of different
parameters are shown on the frequency (i.e. eigenfrequency) of the
structure. The motivation for the choice of NFRP thickness, CNT volume percent and agglomeration of CNTs parameters is discussing about
the NFRP layer effect (main goal of this work) on the frequency of the
concrete foundation. In addition, the effects of different soil mediums
and geometrical parameters of concrete foundation are considered on
the frequency of the structure.
4.1. Validation
Fig. 2. Effects of structural damping on the dimensionless frequency versus NFRP layer
thickness to foundation thickness ratio.
In this paper, to validate the results, the frequency (i.e. eigenfrequency) of the structure is obtained by neglecting the soil medium
(Kw = 0 ) and NFRP layer. Therefore, all the mechanical properties and
type of loading are the same as reference [43]. So the non-dimensional
frequency
is
considered
as
Ω=
ρhω2a4
D0
in
which
D0 = E1 h3/(12(1 − ν12 ν21)) . The results are compared with five references which have used different solution methods. Whitney [43] is used
exact solution while Secgin and Sarigul [44] applied discrete singular
convolution approach. The numerical solution methods of Dai et al.
[45], Chen et al. [46] and Chow et al. [47] are mesh-free, finite element
and Ritz, respectively. As can be observed from Table 1, the results of
present work are in accordance with the mentioned references.
4.2. Effects of different parameters
Considering structural damping is crucial in most engineering problems. For this purpose, based on Kelvin-Voigt model [48], the stiffness
coefficient (Cij ) in Eq. (7) can be replaced by Cij (1 + g ∂/ ∂t ) where g is
the structural damping parameter. Using the solution procedure outlined
in
Section
3,
Eq.
(70)
can
be
written
as
where
and
[M ][d ]̈ + [C ][d]̇ + [K ][d] = [0]
[M ],
[C ],
[K ]
[d] = [u 0 , v0, wb0, ws0] are mass matrix, damp matrix, stiffness matrix
and dynamic vector, respectively. Hence, utilizing the eigenvalue problem, the frequency of the structure may be calculated. In order to show
the effect of structural damping on the dimensionless frequency
(Ω = ωL ρm / Em ) versus NFRP thickness to concrete foundation thickness ratio, Fig. 2 is plotted. As can be seen, considering structural
damping leads to lower frequency. Since the exact value of the structural damping parameter (g) is unclear for concrete foundation with
NFRP layer, we chose g = 0 in the following figures.
Fig. 3 illustrates the effect of the CNT volume fraction on the dimensionless frequency of structure versus NFRP thickness to concrete
foundation thickness ratio. It can be seen that with increasing the values of CNT volume fraction, the frequency of the system is increased.
This is due to the fact that the increase of CNT leads to a harder
structure. In addition, increasing the NFRP thickness to concrete
Fig. 3. Effects of CNT volume percent on the dimensionless frequency versus NFRP layer
thickness to foundation thickness ratio.
foundation thickness ratio, the frequency is decreased for hf / h < 0.005
and increased for hf / h > 0.005. It is perhaps due to this fact that for
hf / h < 0.005, the destructive effect of CNTs agglomeration is higher
than the effect of NFRP layer while for hf / h > 0.005, the NFRP layer
shows the positive effect on the frequency of the structure.
Fig. 4 presents the agglomeration of CNTs on the dimensionless
Table 1
Validation of present work with the other references.
Method
Ref. [43]
Ref. [44]
Ref. [45]
Ref. [46]
Ref. [47]
Present
Mode number
1
2
3
4
15.171
15.171
15.17
15.18
15.19
15.169
33.248
33.248
33.32
33.34
33.31
33.241
44.387
44.387
44.51
44.51
44.52
44.382
60.682
60.682
60.78
60.78
60.79
60.674
Fig. 4. Effects of CNT agglomeration on the dimensionless frequency versus the NFRP
layer thickness to foundation thickness ratio.
146
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
Fig. 5. Effects of soil medium on the dimensionless frequencyversus NFRP layer thickness
to foundation thickness ratio.
Fig. 6. Effects of length of concrete foundation on the dimensionless frequencyversus
NFRP layer thickness to foundation thickness ratio.
frequency versus the NFRP thickness to concrete foundation thickness
ratio. It is shown that considering agglomeration of CNTs leads to lower
frequency in the structure since in this state, the stiffness of the structure decreases. In addition, the effect of agglomeration of CNTs on the
frequency increases with enhancing the NFRP thickness to concrete
foundation thickness ratio.
The dimensionless frequency of the nano-composite concrete foundation versus NFRP thickness to concrete foundation thickness ratio is
demonstrated in Fig. 5 for different soil mediums. In this figure, four
cases of loose sand, dense sand, Clayey medium dense sand and Clayey
soil are considered with the spring constants stated in Table 2. As can be
seen, considering soil medium increases the frequency of the structure.
It is due to the fact that considering soil medium leads to stiffer
structure. Furthermore, the frequency of the dense sand medium is
higher than the other cases since the spring constant of this medium is
maximum.
The effect of the length of concrete foundation on the dimensionless
frequency of the system versus NFRP thickness to concrete foundation
thickness ratio is depicted in Fig. 6. As can be seen, the frequency of the
structure decreases with increasing the length of concrete foundation. It
is because increasing the length leads to softer structure. In addition,
the decreasing rate of frequency is reduced with increasing the concrete
foundation length.
Fig. 7 shows the dimensionless frequency of the structure versus
NFRP thickness to concrete foundation thickness ratio for different
concrete foundation widths. It can be also found that the frequency of
the structure decrease with increasing the width which is due to the
higher stiffness of system with lower width of the concrete foundation.
The effect of length to thickness ratio of concrete foundation on the
dimensionless frequency versus NFRP thickness to concrete foundation
thickness ratio is shown in Fig. 8. It can be found that with increasing
the length to thickness ratio, the frequency of the structure is decreased.
It is because with rising the length to thickness ratio, the stiffness of the
structure will be decreased.
The effect of three different modes on the dimensionless frequency
versus NFRP thickness to concrete foundation thickness ratio is plotted
in Fig. 9. As can be seen, with increasing the vibrational mode, the
Fig. 7. Effects of width of concrete foundation on the dimensionless frequency versus
NFRP layer thickness to foundation thickness ratio.
Fig. 8. Effects of length to thickness of concrete foundation on the dimensionless frequency versus NFRP layer thickness to foundation thickness ratio.
Table 2
Spring constants of soil medium under concrete foundation.
frequency increases.
Soil
Kw
Loose sand
Dense sand
Clayey medium dense sand
Clayey soil
4800–16,000
64,000–128,000
32,000–80,000
12,000–24,000
5. Conclusion
Innovative strengthening methods based on FRP material for extending the lifetimes of concrete structures (old industrial buildings,
147
Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
SSDT, considering NFRP with agglomerated nano fibers and simulating
the soil medium by Winkler model were the main contributions of this
work. The effective material properties of the NFRP layer were obtained
using the Mori-Tanaka model considering agglomeration effects. Based
on SSDT, the motion equations were derived using energy method and
Hamilton's principle. Exact solution was applied for obtaining the frequency of system so that the effects of the volume percent and agglomeration of CNTs, soil medium, structural damping and geometrical
parameters of concrete foundation were considered. The results show
that considering structural damping leads to lower frequency.
Increasing the NFRP thickness to concrete foundation thickness ratio,
the frequency is decreased for hf / h < 0.005 and increased for
hf / h > 0.005. It was shown that considering agglomeration of CNTs
leads to lower frequency in the structure. It can be seen that with increasing the values of CNT volume fraction, the frequency of the system
was increased. Considering soil medium increases the frequency of the
structure. The frequency of the structure decreases with increasing the
length and width of the concrete foundation. In addition, the decreasing
rate of frequency was reduced with rising the concrete foundation
length. Present results were in good agreement with those reported by
[43–47]. Finally, it is expected that the results presented in this paper
would be helpful for further works in mathematical modeling of concrete foundations retrofitted with nano fiber reinforced polymers.
Fig. 9. Effects of three different modes on the dimensionless frequency versus NFRP layer
thickness to foundation thickness ratio.
bridges, off-shore platforms, etc.) were gaining more and more interest
in the practical applications. Hence, in this article, vibration response of
the concrete foundations retrofitted with NFRP layer resting on soil
medium was presented. Mathematical modeling of the structure using
Appendix A
K11 = −
A11 m2π 2
A n2π 2
+ I0 ω2 − 44 2
2
a
b
(A1)
K12 = −
A12 m π 2 n
ab
(A2)
A12z m π 3 n2
A11z m3 π 3
I m π ω2
+
− 1
2
3
ab
a
a
2A 44z m π 3 n2
+
ab2
(A3)
K13 = −
2A 44f m π 3n2
K14 =
+
ab2
A11f m3 π 3
a3
+
A12f m π 3 n2
ab2
J1 m π ω2
−
a
(A4)
K21 = −
A 44 m π 2 n
A m π2 n
− 21
ab
ab
(A5)
K22 = −
A22 n2 π 2
+ I0 ω2
b2
(A6)
K23 =
K24 =
K31 =
A22z n3 π 3
A21z m2 π 3 n
+
3
b
a2 b
2A 44z m2 π 3 n
I n π ω2
+
− 1
a2 b
b
A22f n3 π 3
+
(A7)
A21f m2 π 3 n
b3
a2 b
2A 44f m2 π 3 n
J n π ω2
+
− 1
2
a b
b
(A8)
A21z m π 3 n2
I m π ω2
− 1
2
ab
a
4A 44z m π 3n2
A11z m3 π 3
+
+
a b2
a3
(A9)
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Soil Dynamics and Earthquake Engineering 103 (2017) 141–150
A. Zamani, M.R. Bidgoli
K32 =
4A 44z m2 π 3 n
I n π ω2
− 1
2
a b
b
A12z m2 π 3n
A22z n3 π 3
+
+
a2 b
b3
(A10)
B12 m2 π 4 n2
B n4 π 4
− 22 4
−K
2
2
a b
b
B m2 π 4 n2
4B44 m2 π 4 n2
−
− 21 2 2
a2 b2
a b
2 π 2 ω2
2 π 2 ω2
m
n
⎞ + I0 ω2
+ I2 ⎛
+
a2
b2 ⎠
⎝
K33 = −
⎜
−
⎟
B11 m4 π 4
a4
(A11)
m2 π 2 ω2
n2 π 2 ω2 ⎞
K34 = I0 ω2 + J2 ⎛
+
2
a
b2 ⎠
⎝
A22zf n4 π 4
A11zf m4 π 4
−
−
−K
b4
a4
2
4
2
2
A21zf m π n
A12zf m π 4 n2
−
−
2
2
a b
a2 b2
2
4
2
4A 44zf m π n
−
a2 b2
⎜
K 41 =
A11f m3 π 3
a3
+
K 42 = −
+
⎟
+
4A 44f m π 3 n2
a
b2
−
(A12)
J1 m π ω2
a
A21f m π 3 n2
a b2
(A13)
A12f m2 π 3 n
A22f n3 π 3
J1 n π ω2
+
+
b
a2 b
b3
4A 44f m2 π 3 n
a2 b
K 43 = J2 ⎛
⎝
⎜
(A14)
m2 π 2 ω2
n2 π 2 ω2 ⎞
+
a2
b2 ⎠
⎟
n2 π 2 ω2 ⎞
m2 π 2 ω2
+ K2 J ⎛
+
a2
b2 ⎠
⎝
2
4
2
A21zf m π n
A12zf m2 π 4 n2
−
−
2
2
a b
a2 b2
4
4
A11zf m π
A22zf n4 π 4
−
−K−
4
a
b4
4A 44zf m2 π 4 n2
−
+ I0 ω2
a2 b2
⎜
A66g n2 π 2
⎟
(A15)
E12 m2 π 4 n2
E m4 π 4
− 11 4
a2 b2
a
E21 m2 π 4 n2
−
−
a2
a2 b2
4A 44zf m2 π 4 n2
E n4 π 4
−
− 22 4
+ I0 ω2 − K
a2 b2
b
K 44 = −
b2
A55g m2 π 2
−
(A16)
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