Dynamic instability of thin rectangular plates subjected to uniform in-plane harmonic compressive load applied along two opposite edges are investigated in this paper. The dynamic stiffness method (DSM), as a consequence the dynamic stiffness matrices, is used to analyze the free vibration, the static stability, and dynamic instability of thin plates under different boundary conditions.
Trang 1DYNAMIC INSTABILITY OF THIN PLATES BY THE DYNAMIC
STIFFNESS METHOD
Master Hung Quoc Huynh
Faculty of Civil Engineering, Central University of Construction
Abtract: Dynamic instability of thin rectangular plates subjected to uniform in-plane
harmonic compressive load applied alon
g two opposite edges are investigated in this paper The dynamic stiffness method (DSM), as
a consequence the dynamic stiffness matrices, is used to analyze the free vibration, the static stability, and dynamic instability of thin plates under different boundary conditions The boundaries of the dynamic instability principal regions are obtained using Bolotin’s method Results obtained such as free vibration frequencies, static buckling critical load and dynamic instability principal regions are compared with the results previously published to ascertain the validity of the method
Keywords: Dynamic stability; static stability; dynamic stiffness method; plate
1 Introduction
Various plate structures are widely used in
aircraft, ship, bridge, building, and some
circumstances, these structures are exposed
to dynamic loading Plate structures are often
designed to withstand a considerable in-plane
load along with the transverse loads The
dynamic instability of thin rectangular plates
under periodic in-plane loads has been
investigated by a number of researchers
The dynamic stability of rectangular plates
under various in-plane periodic forces was
studied by Bolotin [1], as well as by Yamaki
and Nagai [2] Hutt and Salama [3]
demonstrated the application of the finite
element method to the dynamic stability of
plates subjected to uniform harmonic loads
Takahasi and Konishi [4] studied the
dynamic stability of a rectangular plate
subjected to a linearly distributed load such
as pure bending or a triangularly distributed
load applied along the two opposite edges using harmonic balance method Nguyen and Ostiguy [5] considered the influence of the aspect ratio and boundary conditions on the dynamic instability and non-linear response
of rectangular plates Guan-Yuan Wu and Yan-shin Shih [6] investigated the effects of various system parameters on the regions of instability and the non-linear response characteristics of rectangular cracked plates using incremental harmonic balance (IHB) method The dynamic instability behaviour
of rectangular plates under periodic in-plane normal and shear loadings was studied by Singh and Dey [7] using energy-based finite difference method Srivastava et al [8] employed the nine-noded isoparametric quadratic element with five degree-of-freedom method to investigate the dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading
Trang 2The dynamic instability analysis of
composite laminated rectangular plates and
prismatic plate structures was determined by
Wang and Dawe [9] using the finite strip
method Wu Lanhe et al [10] analyzed the
dynamic stability of thick functionally graded
material plates subjected to
aero-thermo-mechanical loads, using a novel numerical
solution technique, the moving least squares
differential quadrature method The dynamic
instability of laminated sandwich plates
subjected to in-plane edge loading was
studied by Anupam Chakrabarti and Abdul
Hamid Sheikh [11] using the proposed finite
element plate model based on refined higher
order shear deformation theory Dynamic
stability analysis of composite plates
including delaminations were performed by
Adrian G Radu and Aditi Chattopadhyay
[12] using a higher order theory and
transformation matrix approach
In this paper, the problem of dynamic
stability of plates subjected to periodic
in-plate load along two opposite edges is
studied by the dynamic stiffness method The
problem is solved by the dynamic stiffness
method in order to investigate the efficiency
and the reliability of this method for solving
above-mentioned problems The boundaries
of the dynamic instability principal regions
are obtained using Bolotin’s method The
dynamic stability equation is solved to plot
the relationship of the parameters of load,
natural frequency, frequency of excitation
from the computational program by Matlab
Results obtained, such as free vibration
frequencies, static buckling critical load, and principal regions of dynamic instability, are
published to ascertain the validity of the method
2 Dynamic stability analysis
Assume that a rectangular plate with
length a, width b, and thickness h is
subjected to uniform harmonic in-plane loads
boundaries Both unloaded edges can be simply supported (SS) or clamped (C) A
Cartesian co-ordinate system (x, y, z) is
introduced as shown in Fig 1
SS
a
b
N x
h
y v
x, u z,w
O
SS
Edge a
Edge b
Bu ck
g in
on
ha lf-w
ave
Buckling in several half-waves
N = N + N cos x s t t
Fig 1 Rectangular plate subjected to dynamic inplane loads
The equations of motion for generally isotropic plates are given by Timoshenko [13], and can be reduced to the following set
of equations
4
2 x 2
in which
4
w
where w is the displacement at mid-surface in
Trang 3coordinates, t is the time, and is the mass
density per unit volume The flexural rigidity
is defined as D = Eh 3 /12(1- 2 ) in which E is
Young’s modulus and is Poisson ratio
In the above equation, the in-plane load
factor N x is periodic and can be expressed in
the form:
N xN sN cosΩt t (3)
where N s is the static portion of N x , N t is the
is the frequency of excitation The lowest
critical static buckling load N cr may be
expressed interns of N s and N t as follows:
N s s N cr,N t d N cr (4)
where s and d are static and dynamic load
factors, respectively
The transverse deflection function w,
satisfying the geometric boundary conditions,
can be written as
1
( , , ) ( ) ( )
N
m m
m x
w x y t Y y sin f t
a
where m is the number of half-waves (normal
spatial mode in x-direction), a is the length of
plate in x-direction, f(t) are unknown
functions of time, and Y m (y) are functions to
be determined in order to satisfy the equation
of motion (1)
By substituting Eq (5) into Eq (1), the
following fourth order ordinary differential
equations are obtained
2
( ) 0
IV
m m
h
D
k Y f t D
(6)
where k mm /a (7)
Equations (6) represent a system of
second-order differential equations for the time
functions with periodic coefficients of the standard Mathieu-Hill equations, describing the instability behavior of the plate subjected
to a periodic in-plane compressive load
The analysis of a given structural system
determination of boundaries between the stable and unstable regions The dynamic instability boundaries are determined using the method suggested by Bolotin [1] The stability and instability of their solution depends on the parameters of the system The boundaries between stable and unstable regions in the parameter space are formed by
periodic solutions of period T and 2T, where
T = 2/ The principal instability region
(first instability region) is usually the most important in dynamic stability analysis, because of its width as well as due to structural damping, which often neutralize higher regions
The boundaries of the principal instability
region with period of 2T are of practical
importance and their solution can be achieved in the form of Fourier series
1,3,5,
k
where a k and b k are vectors independent of time
equations (6) leads to an eigenvalue system for the dynamic stability boundary
0 0
0 0
(9)
Trang 4where
2 ''
1
2
IV
2
cr m
Y 2k Y
N
2
IV
2
Y 2k Y
N 9Ω h
3
IV
2
Y 2k Y
N 25Ω h
2
N
k Y
2 D
It has been shown by Bolotin [l] that
solutions with period 2T are the ones of
greatest practical importance, and that as a
first approximation the boundaries of the
principal regions of dynamic instability can
be determined from element (1, 1) of
determinant (9)
2 ''
2
IV
2
cr m
Y 2k Y
N
(10)
3 Dynamic stiffness method
The general solution of differential equations
(10) has the form
( ) ( ) ( )
( ) ( )
m
Y y C sinh c y C cosh c y
C sin d y C cos d y
where
1/2
1/2
1
2
1
2
2
cr
2
cr
N
h Ω
N
h Ω
(12)
where C1, C2, C3 and C4 are the coefficients
to be determined from the four boundary
conditions, edge a at y = 0, and edge b at y = b
3.1 Generalized displacements
a
b
N x
N x h
y, v
x, u
z, w
O
W m1
W m2
W m1 '
W m2 '
Q ym1
M ym1
Q ym2
M ym2
Fig 2 Generalized displacements and generalized forces of plate
Generalized displacement vector can be expressed as
'
'
( , 0) ( ,0) u
( , ) ( , )
T
W x b W x b
(13)
then
( , 0) (0); ( ,0) (0);
( , ) ( ); ( , ) ( )
W x b Y b W x b Y b
(14)
The generalized displacement vector {u} can
be determined by substituting Eqs (14) into Eqs (13) taking into account (11) and
evaluating it at y=0 and y=b, then Eq (13)
can be rewritten in matrix form u K 1 C (15)
where C T C1 C2 C3 C4 and
1
K
( ) ( ) ( ) ( ) ( ) ( ) os( ) ( )
sinh bc cosh bc sin bd cos bd
c cosh bc c sinh bc d c bd d sin bd
(16)
where [K1] is the shape function
3.2 Generalized forces Generalized force vector can be expressed as
Q ( , 0) ( ,0) ( , ) ( , )
T
(17)
The Kirchhoff shear force Q y (x,y) and the bending moment M y (x,y) of the plate along the line y=constant are [15]
Trang 5
( , )
( , )
y
y
y x
(18)
The generalized force which are determined
to Eqs (18) can be written
''' 2 ' '' 2 ( , )
( , )
Q x y D Y k Y
M x y D Y k Y
(19)
The generalized force vector {Q} can be
determined by substituting Eqs (19) into Eq
(17) taking into account (11) and evaluating
it at y=0 and y=b, then Eq (17) can be
rewritten in matrix form
Q K 2 C (20)
where K 2 is the generalized stiffness matrix
2
K
D
(21)
Explicit expressions of the elements k ij of
the generalized stiffness matrix [K2] are as
follows:
2 3
);
m
m
31
32
33
34
m m m m
(22)
m m
42
43
4
4
4
)
m m m m
c sinh b c k v sinh b c
k c cosh b c k v cosh b c
k d sin b d k v sin b d
k
k
d cos b d k v cos b d
By substituting Eq (15) into Eq (20), the generalized nodal displacements and nodal forces are related,
Q K 2 K 1 1 u
Therefore, Q D u (23) Where
D K 2 K 1 1 (24) Matrix [D] in equation (24) is the required
dynamic stiffness matrix With the dynamic
vibration, static stability and dynamic stability problems of the plate structures can
be solved
3.3 Static stability and vibration of the plate
Two parameters c and d of the dynamic
stiffness matrix [D] for solving the static
determined as follows :
r
r
(25)
where ra b/ is aspect ratio of plate, N m
represents the static critical load of plate for
non-dimensional static critical loading factor of
plate for the m mode, which is defined as
/
(26) The non-dimensional natural frequency parameter (natural frequency factor) m of plate is defined as
where m is the natural frequency for the m
mode of plate
Trang 63.4 Dynamic instability of the plate
For analyzing the dynamic stability, two
parameters c and d of the dynamic stiffness
matrix [D] are determined as in Eq (12)
The non-dimensional static critical loading
factor cr of plate is defined as
/
(28)
determined as
*
/ 2(1 )
(29)
The natural frequency of lateral free
vibration of a rectangular plate loaded by a
uniform in-plane force is defined as
*
1
(30)
The non-dimensional frequency of excitation
parameter is as follows
Λ Ωa 2 h D/ (31)
3.5 Dynamic instability of thin plates by the
dynamic stiffness method
Step 1 The motion equation (23) of plate
would be:
Q D u (32)
Step 2 Apply the constraints as dictated by
the boundary conditions Apply boundary
conditions of the problem to eliminate
degeneracy of the dynamic stiffness matrix
Equation (32) has the form:
Q D u
(33)
Step 3 Derive the dynamic stability equation
infinitely large, [D*] must vanish and this
displacemant in the plate must also tend to
infinity Therefore, for dynamic instability
D
det 0
Step 4 Solve dynamic stability equation
D
det 0
(34)
4 Numerical results and discussions
4.1 Static stability and vibration problems 4.1.1 Problem 1 An example is investigated
for the static stability and natural vibration
analysis of a thin square plate P1 (a=b) with
all four edges simply supported and compressed by uniformly distributed
in-plane forces along its opposite edges (Fig 3)
N x
y
x SS
SS
a=b
b
Buckling in one half-wave (m = 1)
P.1
(a)
(b)
N x
Fig 3 Thin square plate P1 (SS-SS-SS-SS) The dynamic stability equation (34) is solved by plotting the relationship m - m
using Matlab program, which determines the static critical loading factors m and the free vibration frequency factors m
Natural frequency factor
0 2 4 6 8
4
2
Fig 4 Relation m - m (plate P1, mode m=1)
Trang 7Natural frequency factor
0
2
4
6
8
6.2499
5
Fig 5 Relation m - m (plate P1, mode m=2)
Natural frequency factor
0
2
4
6
8
10
12
10 11.111
Fig 6 Relation m - m (plate P1, mode m=3)
It is observed from Fig 4-6 that the lowest
static critical loading factor and the free
vibration frequency factors are determined
cr 4,1 2;2 5;3 10
The lowest static critical buckling load
N cr 42D b/ 2
The free vibration frequencies
2 2
1 2( /a ) D/ h
2 2
2 5( /a ) D/ h
2 2
3 10( /a ) D/ h
Table 1 Comparison of cr and m of square
plate P1
Ref
[13,14]
m
Results obtained in the present analysis are compared with those of Yamaki and Nagai [2] and Timoshenko [13,14] in Table 1, which shows a good agreement
4.1.2 Problem 2 This problem considers a
thin square plate P3 (a=b) with two edges
simply supported and two edges clamped and compressed by uniformly distributed in-plane forces along its opposite edges for the static stability and free vibration frequency (Fig 7)
N x
y
x SS
a=b
b
)
(a)
(b)
SS C
C
P.3
N x
Fig 7 Thin square plate P3 (SS-C-SS-C)
Natural frequency factor
0 2 4 6 8
10 8.6044
2.9332
Fig 8 Relation m - m (plate P3, mode m=1)
Natural frequency factor
0 2 4 6 8 10
5.5466 7.6913
Fig 9 Relation m - m (plate P3, mode m=2)
Trang 8Natural frequency factor
0
2
4
6
8
10
12
14
11.9178
10.3566
Fig 10 Relation m - m (plate P3, mode m=3)
It is observed from Fig 7-10 that the lowest
static critical buckling load factor and the
determined
7.6913
cr
;1 2.9332;2 5.5466;
3 10.3566
The lowest static critical loading
N cr 7.69132D b/ 2
The free vibration frequency
2 2
2 2
2 2
Table 2 Comparison of cr and m of square
plate P3
Ref
[15]
m
Results obtained in the present analysis are
compared with those of Yamaki and Nagai
[2] and Timoshenko [15] in Table 2, which
shows a good agreement
4.2 Dynamic instability problems 4.2.1 Problem 1 This problem concerns the
dynamic stability of a thin square plate P1
(a=b) with all four edges simply supported
and compressed by uniformly distributed in-plane periodic forces along its opposite edges
(Fig 11)
SS
a=b
b
P.1
N = N + N cost x s cr d cr
N x
y
x
SS
Fig 11 Thin square plate P1 (SS-SS-SS-SS)
By solving the dynamic stability Eq (34),
we obtain the boundaries of the principal dynamic instability regions, which are presented in the non-dimensional frequency
of excitation parameter () versus dynamic
of the static load factor s , i.e., 0 and 0.6, are
considered
Case 1: the static load factor S = 0
0
Unstable
Dimensionless excitation frequency:
0 0.2 0.4 0.6 0.8 1 1.2
Fig 12 Principal instability region for the
square plate P1 (case 1, S = 0)
Case 2: the static load factor S = 0.6
Trang 9Dimensionless excitation frequency:
Principal region of dynamic instability for simply supported plate P.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Unstable
Fig 13 Principal instability region for the
square plate P1 (case 2, S = 0.6).
0)
d
0.6)
d
Results obtained in the present analysis are
compared with those of Hutt and Salam [3],
Srivastava, Datta and Sheikh [8], and
Chakrabarti and Sheikh [11] in Table 3 and
Table 4, which show a good agreement
4.2.2 Problem 2 An example is investigated
for the dynamic stability of a thin rectangular
plate P4 with two edges simply supported
and two edges clamped and compressed by
uniformly distributed in-plane periodic forces
along its opposite edges (Fig 14)
N x
y
x C
C
a = 1.667b
b
P.4
N = x s N + cr dN cos t cr
Trang 10Fig 14 Thin rectangular plate P4
(SS-C-SS-C)
(mode1,2,3)
Normalized frequency parameter: *
0.1
0.2
0.3
0.4
0
m=3 m=2
m=1
Fig.15.Principal instability regions for the
rectangular plate P4(modes m=1,2,3) for S = 0.5
Fig 16 Principal instability regions for the
0.5 of Ref [5]
The plots of the principal region of dynamic
instability for the rectangular plate P4 for
three modes (m=1,2,3) in Fig 15 are
compared and found to be in a very good
agreement with the results of Nguyen and
Ostiguy [5] in Fig 16
5 Conclussion
In the paper, the dynamic stiffness method
has been developed to analyze the thin plates
and to consider the effect of in-plane
dynamic forces on static stability, vibration
and dynamic stability of such plates
The dynamic stiffness matrices of thin
plates subjected to uniformly distributed
static plane edge loading and dynamic
in-plane edge loading are established On that
basis, the dynamic stability equation is established to analyze the problem of static stability, vibration and dynamic stability of thin plates by the dynamic stiffness method Research results obtained such as free vibration frequencies, static critical buckling load and principal regions of dynamic instability for the plates by the dynamic stiffness method are compared with the results previously published to be in a good agreement Thus in the analysis of plates structural one can use the dynamic stiffness method as a reliable and efficient tool
References
[1] Bolotin V.V 1964 The dynamic stability
of elastic system, San Francisco, Holden-Day [2] Yamaki N., Nagai K.1975 Dynamic stability of rectangular plates under periodic compressive forces, Report No 288 of the Institute of high speed mechanics, Tohoku University 32 103-127
[3] Hutt J.M., Salam A.E 1971 Dynamic instability of plates by finite element method, ASCE J of Eng Mech 3 879-899
[4] Takahashi K., Konishi Y 1988 Dynamic stability of a rectangular plate subjected to distributed in-plane dynamic force, J of Sound Vib 123 115-127
[5] Nguyen H., Ostiguy G.L 1989 Effect of
instability and non-linear response of rectangular plates, part I, theory, J of Sound and Vib 133 381-400
[6] Guan-Yuan W., Shih Y.S 2005 Dynamic instability of rectangular plate with an edge crack, Comput and Struct 84 1 -10