Composite Structures 102 (2013) 306–314

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Composite Structures
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Corrigendum

Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically
stiffened FGM double curved shallow shells on elastic foundation’’
[Compos. Struct. 99 (2013) 88–96]
Nguyen Dinh Duc ⇑
University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 10 April 2013
Keywords:
Nonlinear dynamic
Eccentrically stiffened FGM double curved
shallow shells
Imperfection
Elastic foundation

a b s t r a c t
This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundations and being subjected
to axial compressive load and transverse load. The formulations are based on the classical shell theory

under periodic axial loading [4]. The group of Ng and Lam also published results on generalized differential quadrate for free vibration
of rotating composite laminated conical shell with various boundary conditions in 2003 [5]. In the same year, Yang and Shen [6]
published the nonlinear analysis of FGM plates under transverse
and in-plane loads.
In 2004, Zhao et al. studied the free vibration of two-side simply-supported laminated cylindrical panel via the mesh-free kpRitz method [7]. About vibration of FGM plates Vel and Batra [8]
gave three dimensional exact solution for the vibration of FGM
rectangular plates. Also in this year, Sofiyev and Schnack investigated the stability of functionally graded cylindrical shells under
linearly increasing dynamic tensional loading in [9] and obtained
the result for the stability of functionally graded truncated conical
shells subjected to a periodic impulsive loading [10], and they published the result of the stability of functionally graded ceramic–
metal cylindrical shells under a periodic axial impulsive loading
in 2005 [11]. Ferreira et al. received natural frequencies of FGM
plates by meshless method [12], 2006. In [13], Zhao et al. used
the element-free kp-Ritz method for free vibration analysis of conical shell panels. Liew et al. studied the nonlinear vibration of a
coating-FGM-substrate cylindrical panel subjected to a temperature gradient [14] and dynamic stability of rotating cylindrical
shells subjected to periodic axial loads [15]. Woo et al. investigated
the non linear free vibration behavior of functionally graded plates
[16]. Kadoli and Ganesan studied the buckling and free vibration
analysis of functionally graded cylindrical shells subjected to a
temperature-specified boundary condition [17]. Also in this year,


307

N.D. Duc / Composite Structures 102 (2013) 306–314

Wu et al. published their results of nonlinear static and dynamic
analysis of functionally graded plates [18]. Sofiyev has considered
the buckling of functionally graded truncated conical shells under
dynamic axial loading [19]. Prakash et al. studied the nonlinear axisymmetric dynamic buckling behavior of clamped functionally

transients response of doubly curved shallow shells subjected to
excited external forces obtained the dynamic critical buckling
loads are evaluated based on the displacement response using
the criterion suggested by Budiansky–Roth. Obtained results show
effects of material, geometrical properties, eccentrically stiffened,
elastic foundation and imperfection on the dynamical response of
FGM shallow shells.
2. Eccentrically stiffened double curved FGM shallow shell on
elastic foundations
Consider a ceramic–metal stiffened FGM double curved shallow
shell of radii of curvature Rx, Ry length of edges a, b and uniform
thickness h resting on an elastic foundation.
A coordinate system (x, y, z) is established in which (x, y) plane
on the middle surface of the panel and z is thickness direction (Àh/
2 6 z 6 h/2) as shown in Fig. 1.

z
b

a

h

y

x

Ry
Rx



½EðzÞ; qðzފ ¼ ðEm ; qm Þ þ ðEcm ; qcm Þ


N
2z þ h
2h

ð3Þ

where

Ecm ¼ Ec À Em ; qcm ¼ qc À qm ;

mðzÞ ¼ const ¼ m

ð4Þ

It is evident from Eqs. (3), (4) that the upper surface of the panel
(z = Àh/2) is ceramic-rich, while the lower surface (z = h/2) is metal-rich, and the percentage of ceramic constituent in the panel is
enhanced when N increases.
The panel–foundation interaction is represented by Pasternak
model as

qe ¼ k1 w À k2 r2 w
2

2

2

B e C B e0 C
B
C
@ y A ¼ @ y A À z@ ky A
0
cxy
2kxy
cxy

ð6Þ

where e0x ; e0x and c0xy are normal and shear strain at the middle surface of the shell, and kx, ky,kxy are the curvatures. The nonlinear
strain–displacement relationship based upon the von Karman theory for moderately large deflection and small strain are:


308

0

N.D. Duc / Composite Structures 102 (2013) 306–314

1

0

1

0

1

h

z

z2
x1
2

O

h=2

ri ð1; zÞ dz i ¼ x; y; xy

ð8Þ

Àh=2

The constitutive stress–strain equations by Hooke law for the
shell material are omitted here for brevity. The shell reinforced
by eccentrically longitudinal and transversal stiffeners is shown
in Fig. 1. The shallow shell is assumed to have a relative small rise
as compared with its span. The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique.
Then intergrading the stress–strain equations and their moments
through the thickness of the shell, the expressions for force and
moment resultants of an eccentrically stiffened FGM shallow shell
are obtained [32]:




þ C 2 ky
1 À m2 x
1 À m2
s2
1Àm
1 À m2




E2
E2 m 0
E3
EI1
E3 m
kx À
Mx ¼
þ C 1 e0x þ
e À
þ
ky
1 À m2
1 À m2 y
1 À m2 s1
1 À m2




E2 m 0

s1

s1

s1

s2

a

Fig. 2. Configuration of an eccentrically stiffened shallow shells

Mx;xx þ 2Mxy;xy þ My;yy þ

@2w
@t 2

ð11Þ

where


 


Z h
2
A
A
q

h
E1 ¼ Em þ
Nþ1

@2w
@t 2

ð13Þ

Calculating from Eq. (9), obtained:

2

E2 ¼

ð10Þ

In above relations (9) and (10), the quantities A1, A2 are the cross
section areas of the stiffeners and I1, I2, z1, z2 are the second moments of cross section areas and eccentricities of the stiffeners with
respect to the middle surface of the shell respectively, E is elasticity
modulus in the axial direction of the corresponding stiffener witch
is assumed identical for both types of stiffeners (Fig. 2). In order to
provide continuity between the shell and stiffeners, suppose that
stiffeners are made of full metal (E = Em) if putting them at the metal-rich side of the shell, and conversely full ceramic stiffeners
(E = Ec) at the ceramic-rich side of the shell [32].
The nonlinear dynamic equations of a FGM shallow shells based
on the classical shell theory are [33]

N xy;x þ Ny;y


Ecm Nh
2ðN þ 1ÞðN þ 2Þ

!
Em
1
1
1
3
E3 ¼
À
þ
h
þ Ecm
N þ 3 N þ 2 4N þ 4
12
EA1 z1
EA2 z2
C1 ¼
; C2 ¼
s1
s2

s2

0
x
0
y


D 1 À m2


 
2
EA1
E1
EA2
E1
E1 m
D¼
þ
þ
À
2
2
2
s1
1Àm
s2
1Àm
1Àm




E2
E2 m
E2
E2 m

2
2
1Àm
1Àm
1Àm
1Àm
E2
B66 ¼
E1
ð15Þ
A11 ¼

Substituting once again Eq. (14) into the expression of Mij in (9)
leads to:

M x ¼ B11 Nþx B21 NÀy D11 kx À D12 ky
M x ¼ B12 Nþx B22 NÀy D21 kx À D22 ky
M xy ¼

B66 NÀxy 2D66 kxy

ð16Þ


309

N.D. Duc / Composite Structures 102 (2013) 306–314

The approximate solutions of w, w⁄ and f satisfying boundary
conditions (23) are assumed to be [27–31]

E2
E2 m
¼
þ
À C2 þ
B12
B22 À
s2 1 À m 2
1 À m2
1 À m2


E3 m
E2
E2 m
¼
À C1 þ
B22
B12 À
1 À m2
1 À m2
1 À m2


E3 m
E2
E2 m
B21 À
¼
À C2 þ

gh00 ðxÞ ¼ p0 h gx00 ðyÞ ¼ r0 h

Then Mij into Eq. (13) and f(x, y) is stress function defined by

Nx ¼ f;yy ;

w ¼ WðtÞ sin km x sin dn y

ð18Þ

Subsequently, substitution of Eqs. (24a and b) into Eqs. (22) and
(24c) into Eq. (19) and applying the Galerkin procedure for the
resulting equation yield leads to:

Â

g A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4

þ

þ f;xx w;yy þ

þ ðD12 þ D21 þ

þ f;yy w;xx À 2f ;xy w;xy

f;yy f;xx
@2w
þ
þ q À k1 w þ k2 r2 w ¼ q 2

2

A11 f;xxxx þ ðA66 À 2A12 Þf;xxyy þ A22 f;yyyy þ B21 w;xxxx

ð22Þ

@2W
@t2

ð27Þ

where m,n are odd numbers, and k ¼ ab.
Eliminating g from two obtained equations leads to non-linear
second-order ordinary differential equation for f(t):
"

ð21Þ

Ã

32
k2 p2 hW 2
W gmnp2 4 þ
ðm r 0 þ n2 p0 k2 Þ
3
a ! a2


p2 m2 n2 k2
16h r 0 p0

þ

Ã
Ã
e0x;yy þ e0y;xx À c0xy;xy ¼ w2;xy À w;xx w;yy À wÃ2
;xy þ w;xx w;yy

From the constitutive relations (18) in conjunction with Eq. (14)
one can write

ð26Þ

Ã
B21 m4 þ ðB11 þ B22 À B66 Þn2 m2 k2 þ B12 n4 k4 À ðW

À W0Þ

in which w (x, y) is a known function representing initial small
imperfection of the eccentrically stiffened shallow shells. The geometrical compatibility equation for an imperfect shallow shells is
written

w;yy À wÃ;yy w;xx À wÃ;xx
À
:
Rx
Ry

n2 k2 m2
ðW À W 0 Þ
þ


a2

À

For an imperfect FGM curved panel, Eq. (13) are modified into
form

D22 wÃ;yyyy

ð25Þ

!

a2

ðm2 þ k2 n2 Þ À

p4 P2
a4

P1

þ

2 2

p2 m2
a2


Ry
Rx


512m2 n2 k 1
16h r 0 p0
16q
@2W
þ
À WðW 2 À W 20 Þ
À
þ
¼q 2
4
2
2
9a
P1 mnp Rx Ry
mnp
@t
þ ðW À W 0 Þ4

p

2 2

p2

2


¼ 0. The
nonlinear Eq. (28) can be solved by the Newmark’s numerical integration method or Runge–Kutta method.

In the present study, suppose that the stiffened FGM shallow
shell is simply supported at its all edges and subjected to a transverse load q(t), compressive edge loads r0(t) and p0(t). The boundary conditions are

w ¼ N xy ¼ M x ¼ 0;

Nx ¼ Àr 0 h at x ¼ 0; a

w ¼ N xy ¼ M y ¼ 0;

Ny ¼ p0 h at y ¼ 0; b:

ð23Þ

where a and b are the lengths of in-plane edges of the shallow shell.

P1 ¼ A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4
P2 ¼ B21 m4 þ ðB11 þ B22 À 2B66 Þm2 n2 k2 þ B12 n4 k4
4

2 2 2

ð29Þ

4 4

P3 ¼ D11 m þ ðD12 þ D21 þ 4D66 Þm n k þ D22 n k



32mnp2 k2 P2
3a4
P1

p4 P2
a4 P1

À

p2 m2
a2

Ry

!
p4 P2 p2 m2 P2
2 2
2
ðm
þ
k
n
Þ
À
þ
a2
a2
a4 P1
a2 RP 1


4.2.1. Imperfect eccentrically stiffened FGM cylindrical panel acted on
by axial compressive load
Eq. (28) in this case Rx ? 1, Ry = R, p0 = q = 0;r0 – 0 can be
rewritten as:

þ

m2 r 0 À k1 À k2

32mnp2 k2 P2
32mnk2 m2 1
þ WðW À W 0 Þ
4
3a
P1
3a2
R P1
2

 512m2 n2 k 1
@
W
À W W 2 À W 20
¼q 2
9a4
P1
@t
Â



where denoting

H1 ¼ x2L ¼
H2 ¼
H3 ¼

1

q

Q4 À Q3 À Q5

q

xNL

m2 r0 ¼ k1 þ k2

ð34Þ

q

ð35Þ

where xNL is the nonlinear vibration frequency and s is the amplitude of nonlinear vibration.
4.2. Nonlinear dynamic buckling analysis of imperfect eccentrically
stiffened FGM shallow shell
The aim of considered problems is to search the critical dynamic
buckling loads. They can be evaluated based on the displacement

P1

ð37Þ

p2
a2

ðm2 þ k2 n2 Þ þ
"

p4 P2
a4 P1

À

p2 m2 P2
a2 RP 1

À

p2 m 2 P 2
a2 RP 1

1 16m3 nk2 16mnp2 k2 P2
À
2
a
3P 1 R
a4
P1

Taking of W – 0, i.e. considering the shell after the loss of stability we obtain

a2

Seeking solution as W(t) = scos xt and applying procedure like
Garlerkin method to Eq (33), the frequency-amplitude relation of
nonlinear free vibration is obtained

a2

2

þ W3

p2 h

ðQ 1 þ Q 2 Þ

p2

a2 RP 1
a2 RP 1
"
1 16m3 nk2 16mnp2 k2 P2
À W2 2
À
a
3P1 R
a4
P1


Q6 ¼

xL ¼

p2

p4
a4

m4

P3 þ

2

ÀW

ð38Þ
From Eq. (38) the upper buckling load can be determined by
W=0
rupper ¼

a2
m2 h

p

2


m4

!

2

R P1
ð39Þ

dr0
And the lower buckling load is found using the condition dW
¼ 0,
it follows:

rlower ¼

a2
2

k1 þ k2

p2 hm
p2 m2 P2 p4

À
"

a2 RP 1

þ

a2 3P 1 R
a4
P1
3a4
P1
3a2
R P1
"


2
512m2 n2 k 1
1 16m3 nk2 16mnp2 k2 P 2
þþ4
À
9a4
P1
a2 3P 1 R
a4
P1
##
32mnp2 k2 P 2 32mnk2 m2 1
þ
À
ð40Þ
3a4
P1
3a2
R P1


1
1
"
#

 1 16m3 nk2 16mnp2 k2 P
2
2
2
À
þ W À W0
a2 3RP 1
a4
P1
2

Table 1
The dependence of the fundamental frequencies of nature vibration of spherical FGM
double curved shallow shell on volume ratio N.

32mnp2 k2 P2
32m3 nk2 1
þ
WðW
À
W
Þ
0
3a4
P1


þ ÀW 2

ð41Þ

The static critical load can be determined by the equation to be re€ ¼ 0; W 0 ¼ 0 and using condition
duced from Eq. (41) by putting W
dq
¼ 0.
dW
4.2.3. Imperfect eccentrically stiffened FGM shallow spherical panel
under transverse load
Eq. (28) in this case Rx = Ry = R, p0 = r0 = 0 can be rewritten as:

"

W Àk1 À k2

p2

2

2 2

p4 P2

!

p2 m2 þ n2 k2 P2


À P3 À
P1
a
R
P1 a4
R
!
"
#

 1 m2 þ n2 k2 16mnk2 16mnp2 k2 P
2
þ W 2 À W 20
À
a2
R
3P1
a4
P1
!
2
2 2
2
2 2
32mnk m þ n k
1
2 32mnp k P 2
þ ÀW
þ WðW À W 0 Þ
P1

K1, K2

xL (rad/s)
Reinforced

Unreinforced

K1 = 200, K2 = 0
K1 = 200, K2 = 10
K1 = 200, K2 = 20
K1 = 200, K2 = 30
K1 = 0, K2 = 10
K1 = 100, K2 = 10
K1 = 150, K2 = 10
K1 = 200, K2 = 10

33.574 Â 10 5
39.034 Â 10 5
44.079 Â 10 5
48.535 Â 10 5
26.734 Â 105
31.534 Â 10 5
35.585 Â 10 5
39.034 Â 10 5

32.865 Â 105
38.515 Â 105
43.273 Â 105
46.371 Â 105
25.646 Â 105

0.0385
0.0304

0.0597
0.0506
0.0456
0.0396
0.0381

0.0597
0.0506
0.0456
0.0396
0.0380

0.0577
0.0490
0.0442
0.0383
0.0366

0.0588
0.0492
0.0403
0.0381
0.0364

FGM cylindrical panel
(0, 0.5)
0

0.0485
0.0413
0.0390

FGM plate
(0, 0)

a ¼ b ¼ 2 m; h ¼ 0:01 m;
Ec ¼ 380 Â 109 N=m2 ;

3

qm ¼ 2702 kg=m ; qc ¼ 3800 kg=m3 ;
s1 ¼ s2 ¼ 0:4;

z1 ¼ z2 ¼ 0:0225 ðmÞ;

ð43Þ

4

5.7419

x 10

Reinforced, Rx =Ry =3(m), N=5

m ¼ 0:3

Table 1 presents the dependence of the fundamental frequencies


5.7415

5.7414

5.7413

0

0.05

τ

0.1

Fig. 3. Frequency-amplitude relation.

0.15


312

N.D. Duc / Composite Structures 102 (2013) 306–314

Fig. 4. Effect of eccentrically stiffeners on nonlinear dynamic response of the
shallow spherical FGM shell.

Fig. 6. Influence of elastic foundations on nonlinear dynamic response of the
eccentrically stiffened shallow spherical FGM shell.


Fig. 4 shows the effect of eccentrically stiffeners on nonlinear
respond of the FGM shallow shell on elastic foundation. The FGM


N.D. Duc / Composite Structures 102 (2013) 306–314

313

Fig. 8. Effect of dynamic loads on nonlinear response.
Fig. 10. Influence of initial imperfection on nonlinear dynamic response of the
eccentrically stiffened spherical panel.

Fig. 9. Effect of Rx on nonlinear dynamic response.

shell considered here is spherical panel Rx = Ry = 5 m. Clearly, the
stiffeners played positive role in reducing amplitude of maximum
deflection. Relation of maximum deflection and velocity for spherical shallow shell is expressed in Fig. 5.
Fig. 6 shows influence of elastic foundations on nonlinear dynamic response of spherical panel. Obviously, elastic foundations
played negative role on dynamic response of the shell: the larger
k1 and k2 coefficients are, the larger amplitude of deflections is.
Fig. 7 shows effect of volume metal-ceramic on nonlinear dynamic response of the eccentrically stiffened shallow spherical
FGM shell.
Figs. 8 and 9 show effect of dynamic loads and Rx on nonlinear
dynamic response of the eccentrically stiffened shallow spherical
FGM shell.

Fig. 11. Nonlinear dynamic response of eccentrically stiffened spherical and
cylindrical FGM panel.

Fig. 10 shows influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel. The

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