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Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load
International Journal of Non-Linear Mechanics 35 (2000) 467}480
Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load M. Ganapathi!,*, B.P. Patel!, P. Boisse", M. Touratier" !Institute of Armament Technology, Girinagar, Pune-411 025, India "LMMS, E! cole Nationale Supe& rieure d'Arts et Metiers, Paris-75013, France
Abstract Using a C1 QUAD-8 shear-#exible plate element, based on a new kind of kinematics which allows one to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers in the laminates, the non-linear instability behaviour of plates subjected to periodic in-plane load has been studied. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia e!ects. Primarily, an attempt is made here to understand the geometrically non-linear parametric instability characteristics of isotropic and composite plates through a "nite element formulation with dynamic response analysis. The non-linear governing equations obtained here are solved using the Newmark integration scheme coupled with a modi"ed Newton}Raphson iteration procedure. The analysis brings out various characteristic features of the phenomenon, which are known from experiments, i.e. existence of beats, their dependency on the forcing frequency, the in#uence of initial conditions and load amplitudes, and the typical character of vibrations in the di!erent regions. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic instability; Finite element; Non-linear response; Beats; Plates; Composite
1. Introduction Structural components made of plate/shell elements often "nd applications in the construction of aerospace, mechanical, nuclear and o!shore structures. They are, in general, subjected to various types of dynamic loads. Knowledge pertaining to the stability behaviour of such plates/shells for different types of system parameters is essential for the assessment of the structural failures and optimal design.
* Corresponding author. Fax: 00-91-020-592509. E-mail address: [email protected] (M. Ganapathi)
Dynamic instability analysis of isotropic beams and plates, subjected to periodic in-plane loads, has received considerable attention in the literature and has been reviewed by Bolotin [1] whereas the study related to composite laminates has been fairly attempted by many investigators [2}7]. All these works are based on linear analysis in which periodic solutions in the form of Fourier series are employed, and then, the boundaries of the instability regions are obtained using eigenvalue approach. Furthermore, the primary instability region that occurs in the vicinity of 2u (u * lowest natural 1 1 frequency) has been given more importance in all the work [2}7]. However, study using non-linear theory has been sparsely treated in Refs. [8}13].
0020-7462/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 3 4 - 7
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M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Ref. [8] has dealt with the non-linear dynamic instability of laminates by adopting eigenvalue analysis coupled with Fourier series solutions whereas asymptotic stability criteria are derived in Ref. [9] using Liapunov's direct method. Harmonic balance method is used in Ref. [10] for solving the equation of motion of plate, which includes the geometric non-linearities. Some works related to non-linear dynamic stability analysis of shells based on analytical methods are available in Refs. [11}13]. Ref. [1] has emphasised the existence of beats in experimentally obtained responses for the postcritical motion of non-linear systems. Phenomena like beats, e!ect of initial conditions, etc., are not brought out on the dynamic stability region in the earlier studies [8}10] due to the limitation of the solution approach. Using the method of formal expansion into trigonometric series [2}8] in conjunction with eigenvalue analysis in practice cannot bring out such phenomena because of the di$culties that arise in taking into account the initial conditions. Therefore, to predict the actual dynamic characteristics of the plates under periodic in-plane excitation load, a study concerning the non-linear dynamic response of the plates is necessary. In this paper, an attempt is made to study the non-linear behaviour of the isotropic plates subjected to periodic in-plane loads through a "nite element formulation coupled with dynamic response analysis. Here, an eight-noded shear-#exible plate element, developed recently based on C1 continuity requirement for transverse displacement as outlined in Refs. [14,15], is extended to analyse parametric resonance pertaining to plates with periodic loading. Further, this element is derived using the layer-wise theory based on the principle given in Refs. [16}19]. The present model includes the effects of in-plane and rotary inertia. Since the transverse shear deformation is represented by cosine functions, which is of a higher order, no shear correction factor is introduced in the analysis. A detailed parametric study to highlight the in#uence of initial conditions and dynamic load parameter on parametric resonance is carried out considering a simply supported square thin elastic plate.
2. Formulation A laminated composite plate is considered with the co-ordinates x, y along the in-plane directions and z along the thickness direction, as shown in Fig. 1. Using formulations based on shear #exible theory, the displacements in kth layer u(k), v(k) and w(k) at a point (x, y, z) from the median surface are expressed as functions of mid-plane displacement u, v, w and independent rotation h and h of normal x y in xz and yz planes, respectively, as, u(k)(x, y, z, t)"u(x, y, t)!zLw/Lx #[f (z)#g(k)(z)]MLw/Lx#h N 1 x 1 #g(k)(z)MLw/Ly#h N, 2 y v(k)(x, y, z, t)"v(x, y, t)!zLw/Ly #g(k)(z)[Lw/Lx#h ] 3 x #[f (z)#g(k)(z)]MLw/Ly#h N, 4 y 2 w(k)(x, y, z, t)"w(x, y, t),
(1)
where t is the time. The functions involved in Eq. (1) for de"ning the kinematics are as follows: f (z)"h/n sin(nz/h)!h/n b cos(nz/h), 1 55 f (z)"h/n sin(nz/h)!h/n b cos(nz/h), 2 44 g(k)"a(k)z#d(k), i"1, 2, 3, 4, k"1, 2, 3, 2, N, i i i (2)
Fig. 1. Plate Geometry and loading.
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
where N is the number of layers of the multilayered structure, h is the total thickness of the laminate, n is equal to 3.141592, and b , b , 44 55 a(k), d(k) are coe$cients to be determined from i i contact conditions for displacements and stresses between the layers and from the boundary conditions on the top and bottom surfaces of the plate. The details of the derivations of these coe$cients can be found from Refs. [14,15]. Von Karman's assumptions for moderately large deformation allows Green's strains to be written in terms of mid-plane deformation of Eq. (1) for a plate as, MeN"Me N#Me N. L NL
(3)
The linear strains in terms of mid-plane deformation can be written as
GH e0 s
Me N" . L -
(4a)
c0
The mid-plane strains Me0N, bending strains (due to lower- and higher-order terms involved in de"ning the kinematics, Eq. (1)), MsN,M-N and shear strains Mc0N in Eq. (3) are written as
G
Me0N"
H G H Lu/Lx Lv/Ly
,
(4b)
Lu/Ly#Lv/Lx
MsN"!
L2w/Ly2
,
(4c)
2L2w/Lx Ly
G H
G
MNN"[A]Me0N#[B]MsN#[E]M-N,
(5a)
MMN"[B]TMe0N#[D]MsN#[BI ]M-N,
(5b)
MM I N"[E]TMe0N#[B]TMsN#[DI ]M-N.
(5c)
Similarly, the transverse shear stress resultants MQN representing the quantities (Q , Q ) are related to xz yz the transverse strains Mc0N through the constitutive relation as MQN"[AI ]Mc0N.
H
(4e)
(6)
The di!erent matrices involved in Eqs. (5a), (5b), (5c) and (6) are de"ned as follows [15]:
P P P P P P
[A]"
[D3 ]"
c0 Lw/Lx#h x . Mc0N" 1 " c0 Lw/Ly#h 2 y
(4f )
If MNN represents the membrane stress resultants (N , N , N ) and MMN, MM I N represent the bending xx yy xy stress resultants due to lower- and higher-order terms involved in de"ning the kinematics [(M , xx M , M ), (M I ,M I ,M I )], one can relate these yy xy xx yy xy to membrane strains Me0N and bending strains (MsN, M-N) through the constitutive relations as
[B3 ]" (4d)
H
(1)(Lw/Lx)2 2 Me N" (1)(Lw/Ly)2 . 2 NL (Lw/Lx)(Lw/Ly)
[D]"
Lc0/Lx 1 Lc0/Ly M-N" 2 , Lc0/Ly 1 Lc0/Lx 2
GH G
The non-linear components of the in-plane strains are
[E]"
L2w/Lx2
469
[A3 ]"
h@2
P
h@2 [Q ] dz, [B]" z[Q ] dz, 1 1 ~h@2 ~h@2 h@2
[Z(z)]T[Q ] dz, 1 ~h@2 h@2
z2[Q ] dz, 1 ~h@2 h@2
z[Z(z)][Q ] dz, 1 ~h@2 h@2
[Z(z)]T[Q ][Z(z)] dz, 1 ~h@2 h@2
[Y(z)]T[Q ][Y(z)] dz. 5 ~h@2
(7a)
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M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
free vibration of plate as
The matrices Y(z) and Z(z) are given as
C
Y(z)"
Lf /Lz#Lg(k)/Lz 1 1 Lg(k)/Lz 3
D
Lg(k)/Lz 2 , Lf /Lz#Lg(k)/Lz 2 4
[M]Md$ N#([K]#(1)[N ]#(1)[N ]#[N ] 2 3 1 2 3 #N0[K ])MdN"M0N, x G
Z(z)
C
f #g(k) 1 1 0 " g(k) 3
0
0
f #g(k) 4 2 g(k) 2
g(k) 3 f #g(k) 1 1
g(k) 2 0
D
.
f #g(k) 4 2
(7b)
For a composite laminate of thickness h consisting of N layers with stacking angle / (k"1, 2, 3,2, N), the layer thickness h , the nek k cessary expressions for computing the reduced sti!ness coe$cients of ([Q ], [Q ]) available in Ref. 1 5 [20], are used here. Following the procedure given in Ref. [21] and using Eqs. (4a), (4b), (4c), (4d), (4e), (4f), (5a), (5b) and (5c), the potential energy functional ; is written as ;(d)"MdNT[(1)[K]#[(1)[N (d)]#( 1 )[N (d)] 2 6 12 1 2 #(1)[N ]]MdN, (8) 2 3 where d, [K] are the vector of the degrees of freedom associated to the displacement "eld in a "nite element discretisation, and the linear sti!ness matrix of the laminate. [N ] and [N ] are the 1 2 non-linear sti!ness matrices linearly and quadratically dependent on the "eld variables, respectively. [N ] is the transverse shear sti!ness matrix of the 3 plate. The kinetic energy of the plate is written as
PAP
1 ¹(d)" 2
h@2
B
o[u5 (k)v5 (k)w5 (k)]T[u5 (k)v5 (k)w5 (k)] dz dA, ~h@2 (9)
where the dot over the variable denotes the partial derivative with respect to time and o is the mass density. The potential energy due to external in-plane force, N0 in x-direction, is written as, x 1 =(d)" N0[Lw/Lx]2 dA. (10) 2 x
P
Substituting Eqs. (8)}(10) in Lagrange's equation of motion, one obtains the governing equation for the
(11)
where [M] is the consistent mass matrix, [K ] is G the geometric sti!ness matrices, respectively. MdK N is the acceleration vector. The coe$cient in the sti!ness and mass matrices can be rewritten as the product of term having thickness co-ordinate z alone and the term containing x and y. In the present study, while performing the integration for the evaluation of the sti!ness and mass coe$cients, terms having thickness coordinate z are explicitly integrated whereas the terms containing x and y are evaluated using full integration with 4]4 points Gauss integration rule.
3. Parametric instability analysis The state of periodic load is the uniform pulsating axial compressive force N0, which may be x de"ned as N0"(N #N cos )t)"(a#b cos )t)N0 x 0 1 #3
(12)
where a"N /N0 , b"N /N0 . N0 , ) are static 1 #3 #3 0 #3 bucking load of the plate and the frequency of the dynamic in-plane load, respectively. From the Eqs. (11) and (12), we have the governing equation of the form [M]Md$ N#[[K]#(1)[N ]#(1)[N ]#[N ] 2 3 1 2 3 #(aN0 #bN0 cos )t)[K ]]MdN"M0N. #3 #3 G
(13)
Eq. (13) represents the dynamic stability problem of a system subjected to a periodic in-plane axial force. The dynamic instability region is determined by solving Eq. (13), by employing the implicit method [22]. In the implicit method, equilibrium conditions are considered at the same time step for which solution is sought. If the solution is known at time t and one wished to obtain the displacements, etc., at time t#*t, then the equilibrium equations
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
considered at time t#*t are given as M]Md$ N #[[N(d)]MdN] "M0N, (14) t`*t t`*t where Md$ N and MdN are the vectors of the t`*t t`*t nodal accelerations and displacement at time t#*t, respectively. [[N(d)]MdN] is the internal t`*t force vectors at time t#*t and is given by [[N(d)]MdN] t`*t "([K]#(1)[N (d)]#(1)[N (d)] 3 2 1 2 . (15) #[N ]#N0 (a#b cos )t)[K ])MdN #3 G t`*t 3 In developing equations for the implicit integration, the internal forces [N(d)]MdN at the time t#*t is written in terms of the internal forces at time t, by using the tangent sti!ness approach, as [[N(d)]MdN] "[[N(d)]MdN] #[K (d)] M*dN, t`*t t T t (16) where [K (d)] "[[K]#[N ]#[N ]#[N ]# T t 1 2 3 N0 (a#b cos )t)[K ]] is the tangent sti!ness #3 G matrix and M*dN"MdN !MdN . Substituting Eq. t`*t t (16) into Eq. (14), one obtains the governing equation at t#*t as [M]Md$ N
#[K (d)] M*dN"![[N(d)]MdN] . (17) t`*t T t t To improve the solution accuracy and to avoid numerical instabilities, it is necessary to employ iteration within each time step in order to achieve equilibrium. The resulting non-linear equations obtained by the above procedure have been solved using the Newmark direct integration method, corresponding to the constant average acceleration method. Equilibrium is achieved for each time step through a modi"ed Newton}Raphson iteration scheme until the required convergence criteria [23] is satis"ed within the speci"c tolerance limit of less than 1%. The problem is now reduced to that of "nding the dynamic response of the systems for the given values of a, b, ), and the initial conditions. The variation of the amplitudes of vibrations of the plate (w) with respect to ), for the chosen initial conditions, and the value of static and dynamic part of the in-plane loads (a, b), can be obtained. From the plot of such variation in w}) plane, one
471
can demonstrate the instability regions associated with the given plate subjected to harmonically excited in-plane load.
4. Elemental description The eight-noded element used here is based on Hermite cubic function for transverse displacement, w according to the C@ continuity requirement, Serendipity quadratic function for the in-plane displacements u, v and rotations h , h . Further, the x y element needs eight nodal degrees of freedom (u, v, w, Lw/Lx, Lw/Ly, L2w/Lx Ly, h , h ) for all corner x y nodes and four degrees of freedom (u, v, h , h ) for x y the mid-node of all the four sides. The element is developed based on new kinematics as given in Eq. (1), which accounts for interlayer continuity for displacements and transverse shear stresses of the laminate. The element behaves very well for both thick and thin situations. It has no spurious mode and is represented by correct rigid body modes.
5. Results and discussion The study, here, has been focussed mainly on understanding the various characteristic features of the phenomenon concerning the non-linear dynamic stability behaviour of plates. Although the formulation is valid for a thin/moderately thick multi-layered anisotropic plate, the numerical study is carried out only for a thin square isotropic plate. Based on progressive mesh re"nement, 8]8 meshes idealisation is found to be adequate to model the full plate for the present analysis. Before proceeding to the detailed analysis on non-linear instability of plate, the formulation developed herein is validated by considering the free vibration and buckling analyses of isotropic, orthotropic, and laminated plates. Table 1 shows the natural frequencies for fairly thick isotropic and orthotropic plates along with the exact solutions of three-dimensional elasticity theory [24,25]. For anti-symmetric cross-ply laminates, the fundamental frequencies and critical buckling loads obtained are given in Table 2 along with the exact analytical
472
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Table 1 Comparison of natural frequencies of isotropic and orthotropic square plates (a/h"10, all sides are simply supported, orthotropic material properties of Aragonite crystal Ref. [25]) Isotropic [uH "u h/(o/G)1@2] ./ ./
Orthotropic [uHH"u h/(o/C )1@2] ./ ./ 11
Mode (m, n)
Present
Ref. [24]
Mode (m, n)
Present
Ref. [25]
(1, 1) (1, 2) (2, 2)
0.09303 0.22201 0.34074
0.0932 0.0226 0.3421
(1, 1) (1, 2) (2, 1)
0.04740 0.10325 0.11880
0.0474 0.1033 0.1188
Table 2 Comparison of frequency and buckling load (due to N in the y direction) of antisymmetric cross-ply square plates (FSDT, TSDT-"rsty and third-order shear deformation theories) (all sides are simply s, material properties: E /E "40, G /E " L T LT T 0.6, G /E "0.5, l "0.25) TT T LT a/h
No. of layers
Ref. [26]
5 10 100
2 10 2 10 2 10
Buckl. load [NH "N a2/(E h3)1@2] #3 : T
Freq. [)H"(ua2/h)(o/E )1@2] T Present
FSDT
TSDT
8.833 11.644 10.473 15.779 } (11.3366)! } (18.6716)!
9.087 11.673 10.568 15.771 } } } }
9.1899 11.6361 10.6066 15.7418 11.3011 } 18.6099 }
Ref. [26]
Present
FSDT
TSDT
8.277 11.494 11.353 25.45 } (13.0136)! } (35.2982)!
8.769 12.109 11.562 25.423 } } } }
8.9744 12.0857 11.6488 25.3306 12.9439 } 35.0962 }
!FSDT using eight-noded C0 serendipity element.
results based on third-order shear deformation theory [26]. It is observed from these tables that the present results agree very well with the existing literature. Next, a linear dynamic instability analysis of two- and four-layered angle-ply square laminated plates is carried out for the simply supported end conditions. The plot of primary instability region (in terms of non-dimensional frequencies, )1 (")Joa4/E h) versus dynamic in-plane load T (b) is shown along with those of Ref. [27] in Fig. 2. They are found to be in very good agreement. For the dynamic response analysis, the initial amplitudes are assumed to be proportional to the normalised linear #exural mode vectors [28]. By varying the initial amplitude vectors, the vibration responses are evaluated and then, the instability
regions are analysed. In the absence of any estimate of the time step for the non-linear dynamic analysis in the literature, the critical time step of a conditionally stable "nite di!erence scheme [29,30] is introduced as a guide. Subsequently, a convergence study is conducted to select a time step, which yields a stable and accurate solution. The material properties and the geometrical properties used in the present analysis are Isotropic case: E "E "7.031]105 N/cm2, c "0.25, 1 2 12 o"2.547]10~6 N s2/cm4, a"b"243.8 cm, h"0.635 cm.
(18)
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
473
Fig. 2. Comparison of primary instability regions versus dynamic in-plane load parameters bM ("N /E h) plot for square angle-ply L T laminates (a/h"10, a"0.03; Material properties: E /E "40.0, G /E "0.6, G /E "0.5, c "0.25). L T LT T LT T LT
Composite laminate case [31]: E "172.7 GPa, E "7.2 GPa, L T G "G "3.7 GPa, c "0.3, o"1566 Kg/m3, LT TT LT a"b"0.2 m, h"0.00032 m, (19) where E and c are Young's modulus and Poisson's ratio. L and T are the longitudinal and transverse directions, respectively with respect to "bres. All the layers are of equal thickness and the ply-angle is measured with respect to the x-axis. a, b are the length and breadth of the plates. The simply supported boundary conditions, considered here, are: v"w"h "Lw/Ly"0 at x"0, a, y (20) u"w"h "Lw/Lx"0 at y"0, b. x Firstly, a dynamic instability analysis of thin isotropic square plates, based on the linear analysis wherein periodic solution introduced in the form of Fourier series in conjunction with eigenvalue approach, is carried out. Furthermore, the problem is focused mainly on the determination of boundaries of the primary instability region that occurs in the vicinity of simple resonance of "rst o!er, 2u (u is 1 1 the lowest natural frequency calculated using linear theory). This is by far the largest one compared to
Fig. 3. Instability regions versus dynamic in-plane load parameters b for plate (a"0).
the neighbourhoods of combination resonance of "rst order, and has been viewed as the most critical/dangerous zone having the greatest practical importance [1}5]. The plot of primary instability region obtained through the eigenvalue approach, in terms of excitation frequencies ) in the neighbourhood of 2u versus the dynamic in-plane load 1 parameter b is shown in Fig. 3. The width of the
474
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Fig. 4. Linear (left-hand side) and non-linear (right-hand side) dynamic responses of plate for di!erent values of in-plane forcing frequency (a"0, b"0.5, w /h"0.4). 0
primary instability *) is the separation of the boundaries of the primary instability region for the given plate. This is normally used as an instabil-
ity measure to study the in#uence of other parameters. For instance, for the dynamic load parameter b"0.5, the width of the instability region is
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
depicted by ) ()() (63.7()(69.2 rad/s; 1 2 u "33 rad/s is the fundamental #exural frequency 1 of the plate). Next, the dynamic response (linear and non-linear) of the thin isotropic plate, for the chosen value of dynamic load parameter (b"0.5) and initial amplitudes w /h"0.4, is evaluated for various 0 values of the excitation frequency and is presented in Fig. 4. One can observe the appearance of beats from this "gure, as highlighted in the work of Bolotin [1]. It can also be seen from Fig. 4 that the beats frequency obtained from the non-linear analysis is di!erent from those of the linear output. Furthermore, it can be noticed from the linear response analysis that the amplitude of vibrations of the plate is more when the in-plane excitation frequency is just in the upper primary stability zone ()') ) in comparison with those of the lower 2
475
stability region () ()). The non-linear response 1 corresponding to the resonance frequency ()"33 rad/s) is of a limit cycle oscillation but the amplitude is more than the initial amplitudes. Further, it is interesting to note from the non-linear responses that, in the vicinity of upper stability region, there exists an excitation frequency where considerable amount of increase in the amplitudes above the initial values is seen while the linear output remains almost constant, see for instance, Figs. 4c and d. It can be inferred from the comparison of the non-linear output given in Figs. 4b and c that in a way the upper stability region predicted by the linear analysis is more critical/dangerous than the primary instability ) ()() . Fig. 5 1 2 describes the variation of the maximum amplitude of vibrations of the plate with the in-plane excitation frequency for the selected initial amplitudes,
Fig. 5. Non-dimensional amplitudes of vibrations of plate versus in-plane forcing frequencies for di!erent values for initial amplitudes and dynamic load parameter (a"0; b"0.3, 0.5 and 0.7; w /h"0.0025, 0.2, 0.4 and 0.6). 0
476
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Fig. 6. The non-linear dynamic responses of plate for di!erent values of dynamic load parameter, b (a"0, )"72, 75 rad/s and w /h"0.4). 0
w /h ("0.0025, 0.2, 0.4, 0.6) with di!erent values 0 for the dynamic load parameter, b ("0.3, 0.5, 0.7). It is inferred from the frequency responses shown in Fig. 5 that the unstable zone predicted by the linear
analysis based on eigenvalue approach almost coincides with those obtained from the response study (i.e. unstable region is around )"2u ). This 1 can be con"rmed by comparing the frequency
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
477
Fig. 7. The non-linear dynamic responses of plate for di!erent values of initial amplitude, w /h (a"0, )"68, 71 rad/s and b"0.3). 0
values given in Fig. 3 for various values of the load parameter with those evaluated from Fig. 5b. From Fig. 5a, one can understand that for very low values of initial amplitudes, the non-linear response char-
acteristics are almost the same as those of the linear analysis (i.e. the frequency range in which amplitudes starts building up is almost the same by linear and non-linear studies), as expected. However, the
478
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
critical/dangerous zone predicted through the non-linear response characteristics is di!erent from those of linear analysis (around )"2u ) and also, 1 does not fall around )"2u (u is the natural /- /frequency predicted by non-linear theory), especially with the increase in the initial amplitudes and dynamic load parameter. This zone is almost in between the upper primary instability region and the frequency curve )"2u . It may be further /concluded that there is a de"nite increase in the amplitudes of the vibration level and instability width when there is an increase in the values of initial amplitudes and load parameters. The in#uence of initial conditions and dynamic load parameter on the beat phenomenon is demonstrated in Figs. 6 and 7 considering two di!erent values for the forcing frequency. It is evident from Fig. 6 that, with an increase in the value of the load amplitude, the existence of the forcing frequency in the vicinity of upper primary stability zone wherein signi"cant increase in the vibration level noticed shifts to higher one. Also, it is seen from these "gures that the beats of the non-linear output are
more pronounced at higher dynamic load parameters and at lesser initial amplitudes. Further, in general, this parametric study reveals that the strength of the instability region and its features depend highly on the combination of chosen initial amplitudes and load parameters. Next, non-linear dynamic stability analysis of eight-layered cross-ply composite square plate (03/903/03/903) is carried out. Fig. 8 describes the 4 variation of the maximum amplitude of vibrations with the in-plane excitation frequency for the selected initial amplitudes (w /h"0.4) and dy0 namic load parameter (b"0.5). It may be concluded from the frequency responses shown in Fig. 8 that the critical/dangerous zone obtained through the non-linear response characteristics are qualitatively similar to those of isotropic case. A similar study is made for angle-ply laminate (453/!453/453/!453) and the results for the load 4 parameter b equal to 0.5 are presented in Fig. 9 for two values of initial conditions (w /h"0.2, 0.4). 0 The general behaviour of angle-ply laminates is qualitatively similar to that of cross-ply plate. It is
Fig. 8. Non-dimensional amplitudes of vibrations of eight-layered cross-ply laminate versus in-plane forcing frequencies (a"0, b"0.5, w /h"0.4 (03/903/03/903) ). 0 4
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
479
Fig. 9. Non-dimensional amplitudes of vibrations of eight-layered angle-ply laminate versus in-plane forcing frequencies (a"0, b"0.5, w /h"0.2, 0.4 and (453/!453/453/!453) ). 0 4
also evident from Fig. 9 that with an increase in the initial amplitudes, the actual instability region predicted through the non-linear analysis deviates from the linear one (primary instability region) and shifts to higher-frequency range (i.e. the frequency range in which amplitudes starts building up). Furthermore, it may also be concluded from the study of composite laminates that an increase in the initial amplitudes increases not only the level of amplitudes of vibration but also the instability width. It is further inferred from Figs. 8 and 9 that the angle-ply laminate has high stability strength and the occurrence of critical zone is noticed at much higher excitation frequencies compared to those of the cross-ply case.
6. Conclusions Non-linear parametric instability of elastic square plates under periodic in-plane load has been carried out considering a shear #exible plate element along with dynamic response approach. The
Newmark numerical integration procedure in conjunction with modi"ed Newton}Raphson iterations scheme has been employed to obtain the responses. This study enables one to obtain all the characteristic features of the physical phenomenon concerning the dynamic instability of plates subjected to periodic in-plane load. From the detailed analysis, some of the observations made here are as follows: (i) The appearance of beats phenomenon has been established through the dynamic response. (ii) The amplitudes of the vibrations of plate are more than the initial amplitudes when the excitation frequency falls in the upper part of instability region, as per the linear study. (iii) The beats of the non-linear output are more predominant at higher excitation load parameters and at lower initial amplitudes. (iv) The actual critical/dangerous zone is just above the primary instability region (upper stability region) predicted by the linear analysis.
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M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
(v)
The e!ect of initial amplitude is to increase the instability width and the level of amplitudes of vibrations. (vi) In general, the instability region and its features depend mostly on the combination of initial amplitudes and load parameters. (vii) The non-linear characteristics of laminated composite plates are qualitatively similar to those of isotropic cases. (viii) The angle-ply laminate has high-stability strength and the occurrence of critical zone is noticed at much higher excitation frequencies compared to those of the cross-ply case. References
[1] V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964. [2] V. Birman, Dynamic stability of unsymmetrically laminated rectangular plates, Mech. Res. Commun. 12 (1985) 81}86. [3] R.S. Srinivasan, P. Chellapandi, Dynamic stability of rectangular laminated composite plates, Comput. Struct. 24 (1986) 233}238. [4] C.W. Bert, V. Birman, Dynamic stability of shear deformable antisymmetric angle-ply plates, Int. J. Solids Struct. 23 (1987) 1053}1061. [5] J. Moorthy, J.N. Reddy, Parametric instability of laminated composite plates with transverse shear deformation, Int. J. Solids Struct. 26 (1990) 801}811. [6] L.W. Chen, J.Y. Yang, Dynamic stability of laminated composite plates by the "nite element method, Comp. Struct. 36 (1990) 845}851. [7] M. Ganapathi, T.K. Varadan, V. Balamurugan, Dynamic instability of laminated composite curved panels using "nite element method, Comp. Struct. 53 (1994) 335}342. [8] V. Balamurugan, M. Ganapathi, T.K. Varadan, Nonlinear dynamic instability of laminated composite plates using "nite element method, Comp. Struct. 60 (1996) 125}130. [9] A. Tylikowski, Dynamic stability of non-linear antisymmetrically laminated angle-ply plates, Int. J. Non-linear Mech. 28 (1993) 291}300. [10] J.R. Marin, N.C. Perkins, W.S. Vorus, Non-linear response of predeformed plates subjected to harmonic in-plane edge loading, J. Sound Vib. 176 (1994) 515}529. [11] H. Ray, C.W. Bert, Dynamic instability of suddenly heated, thick, composite shells, Int. J. Engng Sci. 22 (1984) 1259}1268. [12] V. Birman, Thermal dynamic problems of reinforced composite cylinders, ASME, J. Appl. Mech. 57 (1990) 941}947.
[13] V.B. Kovtunov, Dynamic stability and nonlinear parametric vibration of cylindrical shells, Comp. Struct. 46 (1993) 149}156. [14] M. Touratier, An e$cient standard plate theory, Int. J. Eng. Sci. 29 (1991) 901}916. [15] A. Beakou, M. Touratier, A rectangular plate "nite element for analysing composite multilayered shallow shell in statics, vibration and buckling, Int. J. Numer. Methods Eng. 36 (1993) 627}653. [16] J.N. Reddy, A generalization of Two-dimensional theories of laminated composite plates, Commun. Appl. Numer. Methods 3 (1987) 173}180. [17] K.N. Cho, C.W. Bert, A.G. Striz, Free vibrations of laminated rectangular plates analyzed by higher order theory individual layer theory, J. Sound Vib. 145 (1991) 429}442. [18] A. Nosier, R.K. Kapania, J.N. Reddy, Free vibration analysis of laminated plates using a layerwise theory, AIAA J. 31 (1993) 2335}2346. [19] M.R. Eslami, M. Shariyat, M. Shakeri, Layerwise theory for dynamic buckling and postbuckling of laminated composite cylindrical shells, AIAA J. 36 (1998) 1874}1882. [20] R.M. Jones, Mechanics of Composite Materials, McGraw-Hill, New York, 1975. [21] S. Rajasekaran, D.W. Murray, Incremental "nite element matrices, ASCE J. Struct. Div. 99 (1973) 2423}2438. [22] K. Subbaraj, M.A. Dokainish, A survery of direct timeintegration methods in computational structural dynamics II: implicit methods, J. Comput. Struct. 32 (1989) 1387}1401. [23] P.G. Bergan, R.W. Clough, Convergence criteria of iterative process, AIAA J. 10 (1972) 1107}1108. [24] S. Srinivas, C.V. Joga Rao, A.K. Rao, An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates, J. Sound Vib. 12 (1970) 187}199. [25] S. Srinivas, A.K. Rao, Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. J. Solids Struct. 6 (1970) 1463}1481. [26] J.N. Reddy, A.A. Khdeir, Buckling and vibration of laminated composite plates using various theories, AIAA J. 27 (1989) 1808}1817. [27] L.W. Chen, J.Y. Yang, Dynamic stability of laminated composite plates by the "nite element method, Comput. Struct. 36 (1990) 845}851. [28] M. Ganapathi, T.K. Varadan, Large amplitude vibrations of circular cylindrical shells, J. Sound Vib. 192 (1996) 1}14. [29] J.W. Leech, Stability of "nite di!erence equations for the transient response of a #at plate, AIAA J. 3 (1965) 1772}1773. [30] T.Y. Tsai, P. Tong, Stability of transient solution of moderately thick plate by "nite di!erence method, AIAA J. 9 (1971) 2603. [31] K.N. Koo, I. Lee, Vibration and damping analysis of composite laminates using shear deformable "nite element, AIAA J. 31 (1971) 728}735.
Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load M. Ganapathi!,*, B.P. Patel!, P. Boisse", M. Touratier" !Institute of Armament Technology, Girinagar, Pune-411 025, India "LMMS, E! cole Nationale Supe& rieure d'Arts et Metiers, Paris-75013, France
Abstract Using a C1 QUAD-8 shear-#exible plate element, based on a new kind of kinematics which allows one to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers in the laminates, the non-linear instability behaviour of plates subjected to periodic in-plane load has been studied. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia e!ects. Primarily, an attempt is made here to understand the geometrically non-linear parametric instability characteristics of isotropic and composite plates through a "nite element formulation with dynamic response analysis. The non-linear governing equations obtained here are solved using the Newmark integration scheme coupled with a modi"ed Newton}Raphson iteration procedure. The analysis brings out various characteristic features of the phenomenon, which are known from experiments, i.e. existence of beats, their dependency on the forcing frequency, the in#uence of initial conditions and load amplitudes, and the typical character of vibrations in the di!erent regions. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic instability; Finite element; Non-linear response; Beats; Plates; Composite
1. Introduction Structural components made of plate/shell elements often "nd applications in the construction of aerospace, mechanical, nuclear and o!shore structures. They are, in general, subjected to various types of dynamic loads. Knowledge pertaining to the stability behaviour of such plates/shells for different types of system parameters is essential for the assessment of the structural failures and optimal design.
* Corresponding author. Fax: 00-91-020-592509. E-mail address: [email protected] (M. Ganapathi)
Dynamic instability analysis of isotropic beams and plates, subjected to periodic in-plane loads, has received considerable attention in the literature and has been reviewed by Bolotin [1] whereas the study related to composite laminates has been fairly attempted by many investigators [2}7]. All these works are based on linear analysis in which periodic solutions in the form of Fourier series are employed, and then, the boundaries of the instability regions are obtained using eigenvalue approach. Furthermore, the primary instability region that occurs in the vicinity of 2u (u * lowest natural 1 1 frequency) has been given more importance in all the work [2}7]. However, study using non-linear theory has been sparsely treated in Refs. [8}13].
0020-7462/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 3 4 - 7
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Ref. [8] has dealt with the non-linear dynamic instability of laminates by adopting eigenvalue analysis coupled with Fourier series solutions whereas asymptotic stability criteria are derived in Ref. [9] using Liapunov's direct method. Harmonic balance method is used in Ref. [10] for solving the equation of motion of plate, which includes the geometric non-linearities. Some works related to non-linear dynamic stability analysis of shells based on analytical methods are available in Refs. [11}13]. Ref. [1] has emphasised the existence of beats in experimentally obtained responses for the postcritical motion of non-linear systems. Phenomena like beats, e!ect of initial conditions, etc., are not brought out on the dynamic stability region in the earlier studies [8}10] due to the limitation of the solution approach. Using the method of formal expansion into trigonometric series [2}8] in conjunction with eigenvalue analysis in practice cannot bring out such phenomena because of the di$culties that arise in taking into account the initial conditions. Therefore, to predict the actual dynamic characteristics of the plates under periodic in-plane excitation load, a study concerning the non-linear dynamic response of the plates is necessary. In this paper, an attempt is made to study the non-linear behaviour of the isotropic plates subjected to periodic in-plane loads through a "nite element formulation coupled with dynamic response analysis. Here, an eight-noded shear-#exible plate element, developed recently based on C1 continuity requirement for transverse displacement as outlined in Refs. [14,15], is extended to analyse parametric resonance pertaining to plates with periodic loading. Further, this element is derived using the layer-wise theory based on the principle given in Refs. [16}19]. The present model includes the effects of in-plane and rotary inertia. Since the transverse shear deformation is represented by cosine functions, which is of a higher order, no shear correction factor is introduced in the analysis. A detailed parametric study to highlight the in#uence of initial conditions and dynamic load parameter on parametric resonance is carried out considering a simply supported square thin elastic plate.
2. Formulation A laminated composite plate is considered with the co-ordinates x, y along the in-plane directions and z along the thickness direction, as shown in Fig. 1. Using formulations based on shear #exible theory, the displacements in kth layer u(k), v(k) and w(k) at a point (x, y, z) from the median surface are expressed as functions of mid-plane displacement u, v, w and independent rotation h and h of normal x y in xz and yz planes, respectively, as, u(k)(x, y, z, t)"u(x, y, t)!zLw/Lx #[f (z)#g(k)(z)]MLw/Lx#h N 1 x 1 #g(k)(z)MLw/Ly#h N, 2 y v(k)(x, y, z, t)"v(x, y, t)!zLw/Ly #g(k)(z)[Lw/Lx#h ] 3 x #[f (z)#g(k)(z)]MLw/Ly#h N, 4 y 2 w(k)(x, y, z, t)"w(x, y, t),
(1)
where t is the time. The functions involved in Eq. (1) for de"ning the kinematics are as follows: f (z)"h/n sin(nz/h)!h/n b cos(nz/h), 1 55 f (z)"h/n sin(nz/h)!h/n b cos(nz/h), 2 44 g(k)"a(k)z#d(k), i"1, 2, 3, 4, k"1, 2, 3, 2, N, i i i (2)
Fig. 1. Plate Geometry and loading.
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
where N is the number of layers of the multilayered structure, h is the total thickness of the laminate, n is equal to 3.141592, and b , b , 44 55 a(k), d(k) are coe$cients to be determined from i i contact conditions for displacements and stresses between the layers and from the boundary conditions on the top and bottom surfaces of the plate. The details of the derivations of these coe$cients can be found from Refs. [14,15]. Von Karman's assumptions for moderately large deformation allows Green's strains to be written in terms of mid-plane deformation of Eq. (1) for a plate as, MeN"Me N#Me N. L NL
(3)
The linear strains in terms of mid-plane deformation can be written as
GH e0 s
Me N" . L -
(4a)
c0
The mid-plane strains Me0N, bending strains (due to lower- and higher-order terms involved in de"ning the kinematics, Eq. (1)), MsN,M-N and shear strains Mc0N in Eq. (3) are written as
G
Me0N"
H G H Lu/Lx Lv/Ly
,
(4b)
Lu/Ly#Lv/Lx
MsN"!
L2w/Ly2
,
(4c)
2L2w/Lx Ly
G H
G
MNN"[A]Me0N#[B]MsN#[E]M-N,
(5a)
MMN"[B]TMe0N#[D]MsN#[BI ]M-N,
(5b)
MM I N"[E]TMe0N#[B]TMsN#[DI ]M-N.
(5c)
Similarly, the transverse shear stress resultants MQN representing the quantities (Q , Q ) are related to xz yz the transverse strains Mc0N through the constitutive relation as MQN"[AI ]Mc0N.
H
(4e)
(6)
The di!erent matrices involved in Eqs. (5a), (5b), (5c) and (6) are de"ned as follows [15]:
P P P P P P
[A]"
[D3 ]"
c0 Lw/Lx#h x . Mc0N" 1 " c0 Lw/Ly#h 2 y
(4f )
If MNN represents the membrane stress resultants (N , N , N ) and MMN, MM I N represent the bending xx yy xy stress resultants due to lower- and higher-order terms involved in de"ning the kinematics [(M , xx M , M ), (M I ,M I ,M I )], one can relate these yy xy xx yy xy to membrane strains Me0N and bending strains (MsN, M-N) through the constitutive relations as
[B3 ]" (4d)
H
(1)(Lw/Lx)2 2 Me N" (1)(Lw/Ly)2 . 2 NL (Lw/Lx)(Lw/Ly)
[D]"
Lc0/Lx 1 Lc0/Ly M-N" 2 , Lc0/Ly 1 Lc0/Lx 2
GH G
The non-linear components of the in-plane strains are
[E]"
L2w/Lx2
469
[A3 ]"
h@2
P
h@2 [Q ] dz, [B]" z[Q ] dz, 1 1 ~h@2 ~h@2 h@2
[Z(z)]T[Q ] dz, 1 ~h@2 h@2
z2[Q ] dz, 1 ~h@2 h@2
z[Z(z)][Q ] dz, 1 ~h@2 h@2
[Z(z)]T[Q ][Z(z)] dz, 1 ~h@2 h@2
[Y(z)]T[Q ][Y(z)] dz. 5 ~h@2
(7a)
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M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
free vibration of plate as
The matrices Y(z) and Z(z) are given as
C
Y(z)"
Lf /Lz#Lg(k)/Lz 1 1 Lg(k)/Lz 3
D
Lg(k)/Lz 2 , Lf /Lz#Lg(k)/Lz 2 4
[M]Md$ N#([K]#(1)[N ]#(1)[N ]#[N ] 2 3 1 2 3 #N0[K ])MdN"M0N, x G
Z(z)
C
f #g(k) 1 1 0 " g(k) 3
0
0
f #g(k) 4 2 g(k) 2
g(k) 3 f #g(k) 1 1
g(k) 2 0
D
.
f #g(k) 4 2
(7b)
For a composite laminate of thickness h consisting of N layers with stacking angle / (k"1, 2, 3,2, N), the layer thickness h , the nek k cessary expressions for computing the reduced sti!ness coe$cients of ([Q ], [Q ]) available in Ref. 1 5 [20], are used here. Following the procedure given in Ref. [21] and using Eqs. (4a), (4b), (4c), (4d), (4e), (4f), (5a), (5b) and (5c), the potential energy functional ; is written as ;(d)"MdNT[(1)[K]#[(1)[N (d)]#( 1 )[N (d)] 2 6 12 1 2 #(1)[N ]]MdN, (8) 2 3 where d, [K] are the vector of the degrees of freedom associated to the displacement "eld in a "nite element discretisation, and the linear sti!ness matrix of the laminate. [N ] and [N ] are the 1 2 non-linear sti!ness matrices linearly and quadratically dependent on the "eld variables, respectively. [N ] is the transverse shear sti!ness matrix of the 3 plate. The kinetic energy of the plate is written as
PAP
1 ¹(d)" 2
h@2
B
o[u5 (k)v5 (k)w5 (k)]T[u5 (k)v5 (k)w5 (k)] dz dA, ~h@2 (9)
where the dot over the variable denotes the partial derivative with respect to time and o is the mass density. The potential energy due to external in-plane force, N0 in x-direction, is written as, x 1 =(d)" N0[Lw/Lx]2 dA. (10) 2 x
P
Substituting Eqs. (8)}(10) in Lagrange's equation of motion, one obtains the governing equation for the
(11)
where [M] is the consistent mass matrix, [K ] is G the geometric sti!ness matrices, respectively. MdK N is the acceleration vector. The coe$cient in the sti!ness and mass matrices can be rewritten as the product of term having thickness co-ordinate z alone and the term containing x and y. In the present study, while performing the integration for the evaluation of the sti!ness and mass coe$cients, terms having thickness coordinate z are explicitly integrated whereas the terms containing x and y are evaluated using full integration with 4]4 points Gauss integration rule.
3. Parametric instability analysis The state of periodic load is the uniform pulsating axial compressive force N0, which may be x de"ned as N0"(N #N cos )t)"(a#b cos )t)N0 x 0 1 #3
(12)
where a"N /N0 , b"N /N0 . N0 , ) are static 1 #3 #3 0 #3 bucking load of the plate and the frequency of the dynamic in-plane load, respectively. From the Eqs. (11) and (12), we have the governing equation of the form [M]Md$ N#[[K]#(1)[N ]#(1)[N ]#[N ] 2 3 1 2 3 #(aN0 #bN0 cos )t)[K ]]MdN"M0N. #3 #3 G
(13)
Eq. (13) represents the dynamic stability problem of a system subjected to a periodic in-plane axial force. The dynamic instability region is determined by solving Eq. (13), by employing the implicit method [22]. In the implicit method, equilibrium conditions are considered at the same time step for which solution is sought. If the solution is known at time t and one wished to obtain the displacements, etc., at time t#*t, then the equilibrium equations
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
considered at time t#*t are given as M]Md$ N #[[N(d)]MdN] "M0N, (14) t`*t t`*t where Md$ N and MdN are the vectors of the t`*t t`*t nodal accelerations and displacement at time t#*t, respectively. [[N(d)]MdN] is the internal t`*t force vectors at time t#*t and is given by [[N(d)]MdN] t`*t "([K]#(1)[N (d)]#(1)[N (d)] 3 2 1 2 . (15) #[N ]#N0 (a#b cos )t)[K ])MdN #3 G t`*t 3 In developing equations for the implicit integration, the internal forces [N(d)]MdN at the time t#*t is written in terms of the internal forces at time t, by using the tangent sti!ness approach, as [[N(d)]MdN] "[[N(d)]MdN] #[K (d)] M*dN, t`*t t T t (16) where [K (d)] "[[K]#[N ]#[N ]#[N ]# T t 1 2 3 N0 (a#b cos )t)[K ]] is the tangent sti!ness #3 G matrix and M*dN"MdN !MdN . Substituting Eq. t`*t t (16) into Eq. (14), one obtains the governing equation at t#*t as [M]Md$ N
#[K (d)] M*dN"![[N(d)]MdN] . (17) t`*t T t t To improve the solution accuracy and to avoid numerical instabilities, it is necessary to employ iteration within each time step in order to achieve equilibrium. The resulting non-linear equations obtained by the above procedure have been solved using the Newmark direct integration method, corresponding to the constant average acceleration method. Equilibrium is achieved for each time step through a modi"ed Newton}Raphson iteration scheme until the required convergence criteria [23] is satis"ed within the speci"c tolerance limit of less than 1%. The problem is now reduced to that of "nding the dynamic response of the systems for the given values of a, b, ), and the initial conditions. The variation of the amplitudes of vibrations of the plate (w) with respect to ), for the chosen initial conditions, and the value of static and dynamic part of the in-plane loads (a, b), can be obtained. From the plot of such variation in w}) plane, one
471
can demonstrate the instability regions associated with the given plate subjected to harmonically excited in-plane load.
4. Elemental description The eight-noded element used here is based on Hermite cubic function for transverse displacement, w according to the C@ continuity requirement, Serendipity quadratic function for the in-plane displacements u, v and rotations h , h . Further, the x y element needs eight nodal degrees of freedom (u, v, w, Lw/Lx, Lw/Ly, L2w/Lx Ly, h , h ) for all corner x y nodes and four degrees of freedom (u, v, h , h ) for x y the mid-node of all the four sides. The element is developed based on new kinematics as given in Eq. (1), which accounts for interlayer continuity for displacements and transverse shear stresses of the laminate. The element behaves very well for both thick and thin situations. It has no spurious mode and is represented by correct rigid body modes.
5. Results and discussion The study, here, has been focussed mainly on understanding the various characteristic features of the phenomenon concerning the non-linear dynamic stability behaviour of plates. Although the formulation is valid for a thin/moderately thick multi-layered anisotropic plate, the numerical study is carried out only for a thin square isotropic plate. Based on progressive mesh re"nement, 8]8 meshes idealisation is found to be adequate to model the full plate for the present analysis. Before proceeding to the detailed analysis on non-linear instability of plate, the formulation developed herein is validated by considering the free vibration and buckling analyses of isotropic, orthotropic, and laminated plates. Table 1 shows the natural frequencies for fairly thick isotropic and orthotropic plates along with the exact solutions of three-dimensional elasticity theory [24,25]. For anti-symmetric cross-ply laminates, the fundamental frequencies and critical buckling loads obtained are given in Table 2 along with the exact analytical
472
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Table 1 Comparison of natural frequencies of isotropic and orthotropic square plates (a/h"10, all sides are simply supported, orthotropic material properties of Aragonite crystal Ref. [25]) Isotropic [uH "u h/(o/G)1@2] ./ ./
Orthotropic [uHH"u h/(o/C )1@2] ./ ./ 11
Mode (m, n)
Present
Ref. [24]
Mode (m, n)
Present
Ref. [25]
(1, 1) (1, 2) (2, 2)
0.09303 0.22201 0.34074
0.0932 0.0226 0.3421
(1, 1) (1, 2) (2, 1)
0.04740 0.10325 0.11880
0.0474 0.1033 0.1188
Table 2 Comparison of frequency and buckling load (due to N in the y direction) of antisymmetric cross-ply square plates (FSDT, TSDT-"rsty and third-order shear deformation theories) (all sides are simply s, material properties: E /E "40, G /E " L T LT T 0.6, G /E "0.5, l "0.25) TT T LT a/h
No. of layers
Ref. [26]
5 10 100
2 10 2 10 2 10
Buckl. load [NH "N a2/(E h3)1@2] #3 : T
Freq. [)H"(ua2/h)(o/E )1@2] T Present
FSDT
TSDT
8.833 11.644 10.473 15.779 } (11.3366)! } (18.6716)!
9.087 11.673 10.568 15.771 } } } }
9.1899 11.6361 10.6066 15.7418 11.3011 } 18.6099 }
Ref. [26]
Present
FSDT
TSDT
8.277 11.494 11.353 25.45 } (13.0136)! } (35.2982)!
8.769 12.109 11.562 25.423 } } } }
8.9744 12.0857 11.6488 25.3306 12.9439 } 35.0962 }
!FSDT using eight-noded C0 serendipity element.
results based on third-order shear deformation theory [26]. It is observed from these tables that the present results agree very well with the existing literature. Next, a linear dynamic instability analysis of two- and four-layered angle-ply square laminated plates is carried out for the simply supported end conditions. The plot of primary instability region (in terms of non-dimensional frequencies, )1 (")Joa4/E h) versus dynamic in-plane load T (b) is shown along with those of Ref. [27] in Fig. 2. They are found to be in very good agreement. For the dynamic response analysis, the initial amplitudes are assumed to be proportional to the normalised linear #exural mode vectors [28]. By varying the initial amplitude vectors, the vibration responses are evaluated and then, the instability
regions are analysed. In the absence of any estimate of the time step for the non-linear dynamic analysis in the literature, the critical time step of a conditionally stable "nite di!erence scheme [29,30] is introduced as a guide. Subsequently, a convergence study is conducted to select a time step, which yields a stable and accurate solution. The material properties and the geometrical properties used in the present analysis are Isotropic case: E "E "7.031]105 N/cm2, c "0.25, 1 2 12 o"2.547]10~6 N s2/cm4, a"b"243.8 cm, h"0.635 cm.
(18)
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
473
Fig. 2. Comparison of primary instability regions versus dynamic in-plane load parameters bM ("N /E h) plot for square angle-ply L T laminates (a/h"10, a"0.03; Material properties: E /E "40.0, G /E "0.6, G /E "0.5, c "0.25). L T LT T LT T LT
Composite laminate case [31]: E "172.7 GPa, E "7.2 GPa, L T G "G "3.7 GPa, c "0.3, o"1566 Kg/m3, LT TT LT a"b"0.2 m, h"0.00032 m, (19) where E and c are Young's modulus and Poisson's ratio. L and T are the longitudinal and transverse directions, respectively with respect to "bres. All the layers are of equal thickness and the ply-angle is measured with respect to the x-axis. a, b are the length and breadth of the plates. The simply supported boundary conditions, considered here, are: v"w"h "Lw/Ly"0 at x"0, a, y (20) u"w"h "Lw/Lx"0 at y"0, b. x Firstly, a dynamic instability analysis of thin isotropic square plates, based on the linear analysis wherein periodic solution introduced in the form of Fourier series in conjunction with eigenvalue approach, is carried out. Furthermore, the problem is focused mainly on the determination of boundaries of the primary instability region that occurs in the vicinity of simple resonance of "rst o!er, 2u (u is 1 1 the lowest natural frequency calculated using linear theory). This is by far the largest one compared to
Fig. 3. Instability regions versus dynamic in-plane load parameters b for plate (a"0).
the neighbourhoods of combination resonance of "rst order, and has been viewed as the most critical/dangerous zone having the greatest practical importance [1}5]. The plot of primary instability region obtained through the eigenvalue approach, in terms of excitation frequencies ) in the neighbourhood of 2u versus the dynamic in-plane load 1 parameter b is shown in Fig. 3. The width of the
474
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
Fig. 4. Linear (left-hand side) and non-linear (right-hand side) dynamic responses of plate for di!erent values of in-plane forcing frequency (a"0, b"0.5, w /h"0.4). 0
primary instability *) is the separation of the boundaries of the primary instability region for the given plate. This is normally used as an instabil-
ity measure to study the in#uence of other parameters. For instance, for the dynamic load parameter b"0.5, the width of the instability region is
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depicted by ) ()() (63.7()(69.2 rad/s; 1 2 u "33 rad/s is the fundamental #exural frequency 1 of the plate). Next, the dynamic response (linear and non-linear) of the thin isotropic plate, for the chosen value of dynamic load parameter (b"0.5) and initial amplitudes w /h"0.4, is evaluated for various 0 values of the excitation frequency and is presented in Fig. 4. One can observe the appearance of beats from this "gure, as highlighted in the work of Bolotin [1]. It can also be seen from Fig. 4 that the beats frequency obtained from the non-linear analysis is di!erent from those of the linear output. Furthermore, it can be noticed from the linear response analysis that the amplitude of vibrations of the plate is more when the in-plane excitation frequency is just in the upper primary stability zone ()') ) in comparison with those of the lower 2
475
stability region () ()). The non-linear response 1 corresponding to the resonance frequency ()"33 rad/s) is of a limit cycle oscillation but the amplitude is more than the initial amplitudes. Further, it is interesting to note from the non-linear responses that, in the vicinity of upper stability region, there exists an excitation frequency where considerable amount of increase in the amplitudes above the initial values is seen while the linear output remains almost constant, see for instance, Figs. 4c and d. It can be inferred from the comparison of the non-linear output given in Figs. 4b and c that in a way the upper stability region predicted by the linear analysis is more critical/dangerous than the primary instability ) ()() . Fig. 5 1 2 describes the variation of the maximum amplitude of vibrations of the plate with the in-plane excitation frequency for the selected initial amplitudes,
Fig. 5. Non-dimensional amplitudes of vibrations of plate versus in-plane forcing frequencies for di!erent values for initial amplitudes and dynamic load parameter (a"0; b"0.3, 0.5 and 0.7; w /h"0.0025, 0.2, 0.4 and 0.6). 0
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Fig. 6. The non-linear dynamic responses of plate for di!erent values of dynamic load parameter, b (a"0, )"72, 75 rad/s and w /h"0.4). 0
w /h ("0.0025, 0.2, 0.4, 0.6) with di!erent values 0 for the dynamic load parameter, b ("0.3, 0.5, 0.7). It is inferred from the frequency responses shown in Fig. 5 that the unstable zone predicted by the linear
analysis based on eigenvalue approach almost coincides with those obtained from the response study (i.e. unstable region is around )"2u ). This 1 can be con"rmed by comparing the frequency
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Fig. 7. The non-linear dynamic responses of plate for di!erent values of initial amplitude, w /h (a"0, )"68, 71 rad/s and b"0.3). 0
values given in Fig. 3 for various values of the load parameter with those evaluated from Fig. 5b. From Fig. 5a, one can understand that for very low values of initial amplitudes, the non-linear response char-
acteristics are almost the same as those of the linear analysis (i.e. the frequency range in which amplitudes starts building up is almost the same by linear and non-linear studies), as expected. However, the
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critical/dangerous zone predicted through the non-linear response characteristics is di!erent from those of linear analysis (around )"2u ) and also, 1 does not fall around )"2u (u is the natural /- /frequency predicted by non-linear theory), especially with the increase in the initial amplitudes and dynamic load parameter. This zone is almost in between the upper primary instability region and the frequency curve )"2u . It may be further /concluded that there is a de"nite increase in the amplitudes of the vibration level and instability width when there is an increase in the values of initial amplitudes and load parameters. The in#uence of initial conditions and dynamic load parameter on the beat phenomenon is demonstrated in Figs. 6 and 7 considering two di!erent values for the forcing frequency. It is evident from Fig. 6 that, with an increase in the value of the load amplitude, the existence of the forcing frequency in the vicinity of upper primary stability zone wherein signi"cant increase in the vibration level noticed shifts to higher one. Also, it is seen from these "gures that the beats of the non-linear output are
more pronounced at higher dynamic load parameters and at lesser initial amplitudes. Further, in general, this parametric study reveals that the strength of the instability region and its features depend highly on the combination of chosen initial amplitudes and load parameters. Next, non-linear dynamic stability analysis of eight-layered cross-ply composite square plate (03/903/03/903) is carried out. Fig. 8 describes the 4 variation of the maximum amplitude of vibrations with the in-plane excitation frequency for the selected initial amplitudes (w /h"0.4) and dy0 namic load parameter (b"0.5). It may be concluded from the frequency responses shown in Fig. 8 that the critical/dangerous zone obtained through the non-linear response characteristics are qualitatively similar to those of isotropic case. A similar study is made for angle-ply laminate (453/!453/453/!453) and the results for the load 4 parameter b equal to 0.5 are presented in Fig. 9 for two values of initial conditions (w /h"0.2, 0.4). 0 The general behaviour of angle-ply laminates is qualitatively similar to that of cross-ply plate. It is
Fig. 8. Non-dimensional amplitudes of vibrations of eight-layered cross-ply laminate versus in-plane forcing frequencies (a"0, b"0.5, w /h"0.4 (03/903/03/903) ). 0 4
M. Ganapathi et al. / International Journal of Non-Linear Mechanics 35 (2000) 467}480
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Fig. 9. Non-dimensional amplitudes of vibrations of eight-layered angle-ply laminate versus in-plane forcing frequencies (a"0, b"0.5, w /h"0.2, 0.4 and (453/!453/453/!453) ). 0 4
also evident from Fig. 9 that with an increase in the initial amplitudes, the actual instability region predicted through the non-linear analysis deviates from the linear one (primary instability region) and shifts to higher-frequency range (i.e. the frequency range in which amplitudes starts building up). Furthermore, it may also be concluded from the study of composite laminates that an increase in the initial amplitudes increases not only the level of amplitudes of vibration but also the instability width. It is further inferred from Figs. 8 and 9 that the angle-ply laminate has high stability strength and the occurrence of critical zone is noticed at much higher excitation frequencies compared to those of the cross-ply case.
6. Conclusions Non-linear parametric instability of elastic square plates under periodic in-plane load has been carried out considering a shear #exible plate element along with dynamic response approach. The
Newmark numerical integration procedure in conjunction with modi"ed Newton}Raphson iterations scheme has been employed to obtain the responses. This study enables one to obtain all the characteristic features of the physical phenomenon concerning the dynamic instability of plates subjected to periodic in-plane load. From the detailed analysis, some of the observations made here are as follows: (i) The appearance of beats phenomenon has been established through the dynamic response. (ii) The amplitudes of the vibrations of plate are more than the initial amplitudes when the excitation frequency falls in the upper part of instability region, as per the linear study. (iii) The beats of the non-linear output are more predominant at higher excitation load parameters and at lower initial amplitudes. (iv) The actual critical/dangerous zone is just above the primary instability region (upper stability region) predicted by the linear analysis.
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(v)
The e!ect of initial amplitude is to increase the instability width and the level of amplitudes of vibrations. (vi) In general, the instability region and its features depend mostly on the combination of initial amplitudes and load parameters. (vii) The non-linear characteristics of laminated composite plates are qualitatively similar to those of isotropic cases. (viii) The angle-ply laminate has high-stability strength and the occurrence of critical zone is noticed at much higher excitation frequencies compared to those of the cross-ply case. References
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