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Dynamic instability of composite plates subjected to non-uniform in-plane loads
Journal of Sound and Vibration 331 (2012) 53–65
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Dynamic instability of composite plates subjected to non-uniform in-plane loads L.S. Ramachandra n, Sarat Kumar Panda Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, India
a r t i c l e i n f o
abstract
Article history: Received 20 March 2009 Received in revised form 12 August 2011 Accepted 15 August 2011 Handling Editor: M.P. Cartmell Available online 9 September 2011
In this paper, the dynamic instability of a shear deformable composite plate subjected to periodic non-uniform in-plane loading is studied for four sets of boundary conditions. The static component and the dynamic component of the applied periodic inplane loading are assumed to vary according to either parabolic or linear distributions. Initially, the plate membrane problem is solved using the Ritz method to evaluate the plate in-plane stress distributions within the prebuckling range due to the applied nonuniform in-plane edge loading. Subsequently using the evaluated stress distribution within the plate, the equations governing the plate instability boundaries are formulated via Hamilton’s variational principle. Employing Galerkin’s method, these partial differential equations are reduced into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability behaviour. Following Bolotin’s method, the instability regions are determined from the boundaries of instability, which represents the periodic solution of the differential equations with period T and 2T to the Mathieu equations. The instability regions are determined for uniform, linear and parabolic dynamic in-plane loads using first-order and second-order approximations. Numerical results are also presented to bring out the effects of span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Composite materials are increasingly used in aerospace, mechanical, civil and naval structures for its high strength to weight ratio, stiffness and inherent directional properties. Structural components (beams, plates and shells) are often subjected to periodic in-plane loads and become dynamically unstable for certain combination of load amplitude, disturbing frequency and frequency of transverse vibration of the structure. The knowledge of static and dynamic stability behaviour of composite structural members subjected to various types of loads are essential in the design of these components. Bolotin [1] has reviewed the literature on the dynamic instability of isotropic beams, plates and shells in his book. Buckling and dynamic instability of composite plates subjected to uniform in-plane compressive loading have been extensively studied by various authors [2–11]. Often, plates are a part of a complex structural system and hence the load coming on it may not be always uniform. For example, in the case of I-beam or wide flanged beam subjected to bending moment at the ends or lateral loads on the flange, the web of the beam is subjected to non-uniform in-plane loads. The load exerted on the aircraft wings, or on the stiffened plate in the ship structures or on the slabs of a multi-storey building
n
Corresponding author. Tel.: þ91 03222 83445; fax: þ 91 03222 82254. E-mail address: [email protected] (L.S. Ramachandra).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.08.010
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by the adjoining structures usually is non-uniform. The type of distribution in an actual structure depends on the relative stiffnesses of the adjoining elements. A few studies have been reported on the static buckling analysis of plates subjected to non-uniform in-plane compressive loading [12–15]. Srinivasan and Chellapandi [5] investigated the dynamic instability of rectangular laminated composite plates subjected to uniform dynamic loading using finite strip method in conjunction with the Bolotin’s method. Authors have neglected the transverse shear deformation, in-plane inertia and rotary inertia of the plate while estimating the width of instability regions. The effect of shear deformation on the dynamic instability of antisymmetric simply supported angle ply plates was reported by Bert and Birman [6]. Srinivasan et al. [7] have obtained dynamic instability regions of orthotropic circular and annular plates subjected to time varying in-plane loads using finite element method. The parametric instability of a laminated composite plate under uniaxial, harmonically varying in-plane load was studied by Moorthy et al. [8] adopting first-order shear deformation theory within the framework of finite element method. The dynamic instability of simply supported isotropic cylindrical panels under combined static and periodic axial forces was reported by Ng et al. [9]. Authors have modelled the shell panel using the Donnel’s shell theory in conjunction with the first-order shear deformation theory and have obtained a system of Mathieu–Hill equations through normal mode expansion. The dynamic instability regions are obtained based on the Bolotin’s method. Recently, Chattopadhyay and Radu [10] studied instability associated with composite plates subjected to dynamic uniform in-plane load considering both the transverse shear deformation and the rotary inertia. They have presented two instability regions considering second-order approximation within the framework of finite element method. Dey and Singha [11] studied dynamic instability of skew plates subjected to periodic uniform in-plane loading using finite element approach. In their formulation, authors have included effects of transverse shear deformation, in-plane and rotary inertia. So far, to the best of authors’ knowledge, there is no work available in the published literature on the dynamic instability analysis of composite plates subjected to dynamic non-uniform in-plane loads. In this investigation, critical buckling loads and dynamic instability regions of a rectangular isotropic and composite plate, subjected to static and dynamic (Nxx ¼ Ns þ Nt cosðptÞ, p is the excitation frequency and Nxx is the applied edge loading) non-uniform (linearly varying and parabolically varying) in-plane loads, respectively, are studied considering higher-order shear deformation theory (HSDT). In the present study both static (Ns) and dynamic (Nt) component of the load is assumed to vary in a similar manner. Since the applied in-plane edge load is non-uniform, in the first step, considering only the static component (Ns) of the in-plane load the plane elasticity problem is solved by minimizing the membrane strain energy to evaluate the stress distribution sxx, syy and txy within the plate. Finally, superposing the stress distribution due to static (Ns) and dynamic loads (Nt cosðptÞ), the stress distributions within the plate are determined. Using the above stress distributions and through Hamilton’s principle, the governing partial differential equations of plate motion are derived. Adopting Galerkin’s approximation, the governing partial differential equations are converted into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability. Adopting Fourier series method, periodic solutions to Mathieu equations are sought and the instability regions are determined by solving the associated eigenvalue problem. Finally, the influence of boundary conditions, span to thickness ratios, shear deformation and aspect ratios on the dynamic instability regions of the plates are reported in the paper. 2. Formulation A composite rectangular plate of length a, width b and composed of n layers of equal thickness is considered with the co-ordinate axes xy in the in-plane directions and the z-axis in the thickness direction of the plate as shown in Fig. 1. The equations of motion of the plate subjected to dynamic in-plane loads in terms of force and moment resultants can be written as [16]: ^ xx,x þ N ^ xy,y ¼ r u þ r ðf w, x Þ 4 r f N 1 0 1 3 1 2 3h ,tt 4 ^ xy,x þ N ^ yy,y ¼ r v þ r ðf w, y Þ N r f 2 0 1 3h2 3 2 ,tt
Fig. 1. Geometry and loading of the plate.
L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
^ xx w, x þ N ^ xy w, y Þ, x þ ðN ^ xy w, x þ N ^ yy w, y Þ, y Mxx,xx þ2Mxy,xy þMyy,yy þ ðN 4 ¼ r0 wþ r1 ðu,x þ v, y Þ þ r2 ðf1,x þ f2,y w, xx w, yy Þ þ 2 r4 ðf1,x þ f2,y Þ 3h ,tt 4 4 4 4 a Pxx,x þPxy,y Qxx ¼ r1 2 r3 u þ r2 2 r4 ðf1 w, x Þ 2 r4 2 r6 f1 3h 3h 3h 3h ,tt 4 4 4 4 a Pxy,x þ Pyy,y Qyy ¼ r1 2 r3 v þ r2 2 r4 ðf2 w, y Þ 2 r4 2 r6 f2 3h 3h 3h 3h ,tt where the inertia terms are defined according to rm ¼
R h=2 h=2
55
(1)
rzm dz ðm ¼ 0,1,. . .,6Þ and u, v and w are the displacement
components, respectively, along the x, y and z directions; f1 and f2 are, respectively, the total rotations of the crosssections perpendicular to x-axis and y-axis, respectively; ( ),x and ( ),y denote, respectively, partial differentiation with ^ xx , N ^ yy , N ^ xy are the plate internal stress resultants; Mxx, respect to x and y co-ordinates; r is the mass density of the plate; N Myy, Mxy are the moment resultants; Pxx, Pyy, Pxy are the additional moment resultants due to additional changes of ^ xx , N ^ yy , N ^ xy are related curvature f1,x, f2,y, (f2,x þ f1,y). Q a ,Q a are the additional transverse shear force resultants. Now, N xx
yy
to Nxx, Nyy, Nxy and nxx, nyy, nxy through, ^ xx ¼ ½Nxx nxx N ^ N xy ¼ ½Nxy nxy ^ yy ¼ ½Nyy nyy N
(2)
where nxx, nyy, nxy are the plate internal stress resultants due to applied end non-uniform in-plane loading and Nxx, Nyy, Nxy n o are the plate stress resultants because of large deflection. The force ðNT ¼ Nxx Nyy Nxy Þ and moment n o n o n o 0 0 0 ðMT ¼ Mxx Myy Mxy , PT ¼ Pxx Pyy Pxy Þ, resultants are related to membrane strains ðe0T ¼ exx eyy exy Þ, and n o n o 0 0 0 T bending strains ðwT ¼ w, xx w, yy 2w, xy , f ¼ f1,x f2,y f1,y þ f2,x Þ, respectively, through the constitutive relationship defined as N ¼ Ae0 þ Bw þ Cf
(3)
M ¼ BT e0 þDw þ Ef
(4)
P ¼ CT e0 þ Ew þ Ff
(5)
In the present work, the bold upright letters are used to denote matrices and vectors. The additional transverse shear force n o n o a a T resultants ðQ aT ¼ Qyy Qxx Þ are related to shear strains ðf ¼ f2 f1 Þ by Q a ¼ Hf
(6)
where the elements of matrices A, B, C, E, F and H of Eqs. (3)–(6) are defined as Z h=2 fAij ,Bij ,Dg ¼ Qij ð1,z,z2 Þ dz fCij ,Eij ,Fij g ¼
Z Z
h=2 h=2
Qij ð1,z,f ðzÞÞf ðzÞdz
for i,j ¼ 1,2,6
h=2 h=2
Qij ðf 0 ðzÞÞf 0 ðzÞ dz
fHij g ¼
for i,j ¼ 4,5
(7)
h=2
here, Qij (i,j ¼1,2,6) are the transformed plane-stress reduced stiffness coefficients and Qij (i,j ¼4,5) are the transformed through-thickness shear stiffness coefficients; eox , eoy and goxy are the reference surface strains and are defined as 1 2
eox ¼ uo,x þ ðwo,x Þ2 ;
1 2
eoy ¼ vo,y þ ðwo,y Þ2 ;
goxy ¼ uo,y þ vo,x þ wo,x wo,y
(8)
where uo, vo and wo are, respectively, the middle surface displacement components along the x, y and z directions, respectively. The von Ka´rma´n strain–displacements relations, Eq. (8), are used to account for the deformed configuration of the plate with moderate displacements and small rotations. Using Eqs. (2)–(8), the governing Eq. (1) are then expressed in terms of displacement components and are given in Appendix A. 3. Plate prebuckling analysis In the present investigation, parabolically and linearly varying in-plane compressive dynamic loads are considered. In the case of linearly varying in-plane load, the stress distributions within the plate coincide with the applied edge loading.
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In the case of parabolic in-plane load, initially the static component (Ns) of the in-plane loading is applied at the plate edge and stress fields within the plate are obtained by solving the plate membrane problem. The correct stress distribution within the plate is the one which minimizes the membrane strain energy of the plate and satisfies the boundary conditions [17] of the problem. The membrane strain energy of a composite plate of thickness h is given by 8 9T 2 9 31 8 n n > ZZ > < xx > = A11 A12 A16 < xx > = 1 6 7 nyy nyy dx dy V¼ (9) 4 A12 A22 A26 5 > 2 A> :n > ; A :n > ; A A xy xy 16 26 66 where, nyy ¼
@2 F , @x2
nxx ¼
@2 F , @y2
nxy ¼
@2 F , @x@y
Aij ¼
Z
h=2
Qij dz
(10)
h=2
where Qij is the transformed reduced stiffness coefficients and F is the stress function. Following Timoshenko and Goodier [17], Ritz method is adopted to minimize the membrane strain energy of the plate in this study. The boundary conditions [17] of the plate membrane problem are given here for the parabolically varying uniaxial in-plane edge loading (see Fig. 1) y y 1 x ¼ 0,a, N xy ¼ 0, Nxx ¼ 4N0 b b y ¼ 0,b, N xy ¼ 0, Nyy ¼ 0 (11) The stress function is assumed in the form of a series as [17]
F ¼ F0 þ a1 F1 þ a2 F2 þ a3 F3 þ a4 F4 þ ,
(12)
where, a1,a2,a3,y are constants to be determined such that the membrane strain energy is minimized and boundary conditions are satisfied. In the present investigation, only four terms are considered in Eq. (12). Thus stress function for the parabolic in-plane edge loading is assumed as y2 y y2 2 þðx2 axÞ2 ðy2 ybÞ2 ða1 þ a2 x þ a3 yÞ F ¼ 2N0 (13) 3 b 2b where
F0 ¼ 2N 0
y2 y y2 2 3 b 2b
which gives Nyy ¼
@2 F0 ¼0 , @x2
N xx ¼
N xy ¼
@2 F0 ¼0 @x@y
@2 F0 y y 1 ¼ 4N0 2 b b @y
In the present study, only four terms in the series are considered. The functions F1, F2, F3 in Eq. (12) are chosen such that the stresses corresponding to them vanish at the boundary. Substituting the Eq. (13) into Eq. (9) and carrying out integration, a second degree polynomial in a1, a2 and a3 is obtained. Then the constants a1, a2 and a3 are evaluated from the 3 algebraic equations resulting from the condition, @V=@a1 ¼ 0, @V=@a2 ¼ 0, @V=@a3 ¼ 0. The explicit expression so obtained for constants ak (k¼1,2,3) for 3-layered cross-ply composite plate [01/901/01] are
a1 ¼
N0 ð0:0934a8 þ 16:7313a6 b2 þ 65:6356a4 b4 þ32:2675a2 b6 þ0:3475b8 Þ ð0:1359a12 b2 þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(14)
a2 ¼
N0 ð0:6073a6 þ 28:3450a4 b2 þ88:7251a2 b4 þ 0:9738b6 Þ ð0:1359a12 b2 þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(15)
a3 ¼
N 0 ð2:2589b6 þ 54:6644a2 b4 þ 88:7235a4 b2 þ 0:5049a6 Þ 12 2 ð0:1359a b þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(16)
Superposing the distribution due to static (Ns) and dynamic loads (Nt cos(pt)), the stress distributions within the plate are determined. Using these plate stress distributions and from Hamilton’s principle, the governing partial diffrential equations (Eq. (1)) of the plate dynamic instability are obtained. 4. Plate buckling analyses Neglecting the inertia terms in Eq. (1) we obtain the governing partial differential equations for the buckling analysis of plates. The critical buckling load of the composite rectangular plate subjected to parabolically varying in-plane
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57
compressive load is obtained by using Galerkin’s method. In the present investigation following four sets of boundary conditions are used: SSSS, CSCS, SCSC and CCCC. Where S stands for the simply supported edge and C for the clamped edge. The letters indicate the boundary conditions on the edge of the plate in the anti clockwise fashion starting from the left hand corner. In the Galerkin’s method, the out-of-plane displacement field w(x,y) satisfying the boundary conditions of the plate is expressed as the product of beam functions [18] as wðx,yÞ ¼
1 X 1 X
Xm ðxÞYn ðyÞ
(17)
m¼1n¼1
where Xm(x) and Yn(y) are the eigenfunctions of the beam having the same boundary conditions as that of two opposite edges of the plate. In the present case following beam functions are assumed for different edge conditions: (a) Simply supported at x¼0 and x ¼a ss ðxÞ ¼ sin Xm
mpx a
ðm ¼ 1,2,3:::Þ
(18)
(b) Clamped support along two opposite edges, i.e., at x ¼0 and x ¼a, according to [18–20] cc ðxÞ ¼ cos xm Xm
x 1 sinðxm =2Þ x 1 þ cosh xm a 2 sinhðxm =2Þ a 2
ðm ¼ 2,4,6. . .Þ
(19)
where xm are the roots of the equation tanðxm =2Þ þ tanhðxm =2Þ ¼ 0
(20)
and cc ðxÞ ¼ sin xm Xm
x 1 sinðxm =2Þ x 1 cosh xm a 2 sinhðxm =2Þ a 2
ðm ¼ 3,5,7. . .Þ
(21)
where xn are obtained as roots of the equation, tanðxm =2Þtanhðxm =2Þ ¼ 0
(22)
The functions Yn(y) are similarly chosen based on the condition at y¼0 and y¼b by replacing x by y and a by b and m by n in the above equations. m and n are, respectively, the number of nodal lines along x and y directions. In the case of simple support only normal in-plane displacements are allowed and in-plane tangential displacements and out of plane displacements are prevented. (a) Boundary conditions at the simply supported edges are o
vo ¼ wo ¼ Pxx ¼ f2 ¼ Mxx ¼ 0
nxx Nxx ¼ Nxx ,
o
nyy Nyy ¼ Nyy ,
o
o
u ¼ w ¼ f1 ¼ Pyy ¼ Myy ¼ 0
at
x ¼ 0,a
at
y ¼ 0,b
(23)
Following displacement fields satisfy the above boundary conditions (Eq. (23)): u~ o ¼
mpx npy sin Umn cos a b m¼1n¼1 j i X X
j i X X
v~ o ¼
Vmn sin
m¼1n¼1
~o¼ w
j i X X
mpx npy cos a b
ss Wmn Xm ðxÞXnss ðyÞ
m¼1n¼1 o
f~ 1 ¼ o
f~ 2 ¼
mpx npy sin Kmn cos a b m¼1n¼1 j i X X
mpx npy cos Lmn sin a b m¼1n¼1 j i X X
(24)
(b) The boundary conditions for CCCC plate are
o
o
nxx Nxx ¼ N xx ,
w0,x ¼ vo ¼ wo ¼ f1 ¼ f2 ¼ 0
nyy Nyy ¼ Nyy ,
w0,y ¼ uo ¼ wo ¼ f1 ¼ f2 ¼ 0
o
o
at x ¼ 0,a at y ¼ 0,b
(25)
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L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
Following displacement fields satisfy the boundary conditions defined in (Eq. 25): u~ o ¼
mpx npy sin Umn cos a b m¼1n¼1 j i X X
j i X X
v~ o ¼
Vmn sin
mpx
m¼1n¼1
~o¼ w
i X
j X
a
cos
npy b
cc Wmn Xm ðxÞXncc ðyÞ
m¼1n¼1 0
f~ 1 ¼
j i X X
Kmn sin
m¼1n¼1
mpx a
sin
npy b
mpx npy 0 f~ 2 ¼ Lmn sin sin a b m¼1n¼1 j i X X
(26)
Using Eqs. (24) and (26), and employing Galerkin’s method the Eq. (1) (without inertia terms) in displacement terms are converted into a set of homogeneous algebraic equations for simply supported and clamped boundary conditions, respectively. Solving the associated eigenvalue problem critical buckling loads are evaluated numerically in Section 6.
5. Dynamic instability analysis The applied periodic non-uniform in-plane load N xx ¼ Ns þNt cos pt has a static and a dynamic component. The static component (Ns) and the dynamic component (Nt) of the applied loading are assumed to vary accordingly to either parabolic or linear distributions. Both Ns and Nt are assumed to vary in a similar manner. Adopting Galerkin’s approximation and using approximate displacement fields (Eqs. (24) and (26)), the governing partial differential Eq. (1) of the plate subjected to dynamic non-uniform in-plane loads are reduced to a set of ordinary differential equations (Mathieu type of equation) describing the plate dynamic instability behaviour as € þ ½KNs KG Nt KG cos ptX ¼ 0 MX
(27)
where M, K and KG are, respectively, the mass, stiffness and geometric stiffness matrices. In the present investigation, only the linear dynamic instability of rectangular composite plates is studied and hence the nonlinear terms in the Eq. (1) are not considered. The Eq. (27) is a second-order differential equation with periodic coefficients. For evaluation of the buckling load, employing Galerkin’s procedure to Eq. (1) without inertia and harmonic terms, the following set of homogeneous algebraic equations are derived ½KNcr KG W ¼ 0
(28)
For non-trivial solution, the determinant of the coefficient matrix must vanish. Hence 9KNcr KG 9 ¼ 0
(29)
where Ncr is the static buckling load. The critical buckling load is obtained from the solution of linear eigenvalue problem given by Eq. (29). Similarly the solutions of the eigenvalue problem associated with the Eq. (27) neglecting terms containing Ns and Nt gives the natural frequencies. The eigenvalue problem for evaluating free vibration frequencies are stated as 9Ko2 M9 ¼ 0
(30)
In the Eq. (27), Ns and Nt are varied as Ns ¼ aNcr and Nt ¼ bNcr, where a and b are static and dynamic load factors, respectively. The regions of instability are located by boundaries of the instability having periodic solutions with period T and 2T to the Eq. (27). These are determined by the method suggested by Bolotin [1] in this investigation. The solutions to Eq. (27) with periods T and 2T, respectively, are assumed in the form of Fourier series as 1 X lpt lpt xðtÞ ¼ b0 þ þ bl cos (31) al sin 2 2 l ¼ 2,4,6 xðtÞ ¼
1 X l ¼ 1,3,5
al sin
lpt lpt þbl cos 2 2
(32)
where al and bl are the arbitrary constants. Here, it may be noted that, two solutions of identical periods bound the region of instability and, two solutions of different periods bounds the region of stability. Substituting Eq. (31) or (32) into Eq. (27) and equating the coefficients of identical sinðlpt=2Þ and cosðlpt=2Þ leads to a system of homogeneous algebraic equations in al and bl. For non-trivial solution the determinant of the coefficient matrix of al and bl must vanish. The size of the above determinant is infinite as we have assumed the solution in the form of infinite series. The determinants are shown to be
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59
belonging to a class of converging determinant known as normal determinant [1]. The first-order (from Eq. (33)) and second-order (from Eq. (34)) approximation to boundaries of principal regions of instability corresponding to period 2T are obtained by solving following two eigenvalue problems, respectively 9K n 70:5bNcr KG 0:25Mp21 9 ¼ 0 n K 7 0:5bNcr KG bNcr KG
0:5bNcr KG 0:25M 2 p K n 2:25Mp21 2 0
(33) 0 ¼0 0
(34)
Secondary zone of instability with first-order approximation (from Eqs. (35) and (36)) and second-order approximations (from Eqs. (37) and (38)) corresponding to period T are determined from 9K n Mp21 9 ¼ 0 Kn bNcr KG Kn 0:5bNcr KG Kn bN K cr G 0
0:5bNcr KG 2 0 p Kn 1 0
(35) 0 ¼0 M
0:5bNcr KG M 2 n 2 p2 K 4Mp1 0
0:5bNcr KG Kn 0:5bNcr KG
0 ¼0 0
0 0:5bNcr KG p22 0 0 K n 4Mp21 0
(36)
0 M 0
(37) 0 0¼0 0
(38)
where K* ¼K NsKG. Similarly third and fourth instability region can be computed by following the same procedure. However they are not given here for the sake of brevity. 6. Buckling numerical results The following mechanical properties are assumed, E11 ¼ E22, G23 ¼G13 ¼G12 ¼ E22/2.5, n12 ¼0.25 for isotropic plate and E1/E2 ¼40, G12 ¼G13 ¼0.5E2, G23 ¼0.6E2, n12 ¼0.25 for composite lamina in the analysis. The natural frequencies obtained from the present method for various boundary conditions are compared with those obtained by Leissa [21] in Table 1 for a square isotropic plate. It is observed that the two results agree with each other exactly for a very thin plate. To validate the present formulation, the dimensionless buckling load ðki ¼ Ncr ð2bÞ2 =p2 DÞobtained from the present method for an isotropic plate subjected to uniform and parabolic uniaxial in-plane loads are compared with those of Timoshenko and Gere [22] and Wang et al.[15] in Table 2. The buckling loads obtained from the commercial finite element software ANSYS are also given in the Table 2. Converged buckling loads were obtained in the present method using 4 terms in the displacement field approximation [(m ¼1, n¼ 1), (m¼1, n ¼3), (m¼ 3, n¼1), (m¼3, n ¼3)]. The present results are in close agreement with the published and the commercial software ANSYS results. Hence, all subsequent analyses were carried out with 4 terms in the displacement field approximations. 7. Numerical results for Instability regions The material properties of the plate are the same as used in the case of buckling analysis. The dynamic instability pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffi regions are presented as a plot of dimensionless excitation frequency (pð rh=DÞ for isotropic plate and pð r=E22 h2 Þ for composite plate) or frequency ratio (p/o) against dimensionless dynamic load factor (b), where o is the natural frequency of the plate. The principal dynamic instability regions (with period 2T) of isotropic simply supported rectangular plate Table 1 Dimensionless fundamental frequency for a square simply supported isotropic plate. a/h
Dimensionless natural frequency (l)
10 20 50 100 1000 Ref. [21] For isotropic plate l ¼ oa2
pffiffiffiffiffiffiffiffiffiffiffiffi rh=D.
SSSS
SCSC
CSCS
CCCC
19.10 19.57 19.71 19.73 19.74 19.74
27.12 28.45 28.87 28.94 28.95 28.95
27.12 28.45 28.87 28.94 28.95 28.95
33.59 35.34 35.91 35.98 35.99 35.99
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L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
Table 2 Critical buckling coefficient (ki) of isotropic plate for various aspect ratios. Type of in-plane load distribution
Aspect ratio (a/b)
Dimensionless buckling coefficient, kia Present solution
ANSYS
Others
Uniform load
0.5 1 1.5
6.25 3.99 4.33
6.25 4.00 4.33
6.25 [22] 4.00 [22] 4.34 [22]
Parabolic load
0.5 1 3
7.27 5.24 5.62
7.28 5.27 5.62
7.27 [15] 5.24 [15] 5.69 [15]
a
ki ¼ Ncr ð2bÞ2 =p2 D .
Fig. 2. First and second approximation to principal instability region for a SSSS square isotropic plate subjected to uniform edge loading: (a) a ¼0 and (b) a ¼0.6.
(a/b¼ 1, a/h¼100) obtained from the present method are compared with that of Hutt and Salam [23] in Fig. 2(a) and (b), respectively, for two static load factors a ¼0 and 0. It is observed from figures that the present results agree well with the reference values. It is observed from Fig. 2(a) and (b) that the width of the instability region increases with the increase in static and dynamic load factors. Dynamic instability regions of a simply supported square symmetric four layered cross-ply [01/901/01] composite plate (a/b¼ 1, a/h ¼25, a ¼0) obtained from the present approach are compared with those of Moorthy et al. [8] in Fig. 3. It is observed that the present results agree well with the reference values. The first four instability zones of an isotropic simply supported square plate (a/b¼1, a/h¼100, a ¼ 0) under parabolic loading is shown in Fig. 4(a). In this figure, the principal zone and the third zone of instability corresponds to period 2T, whereas, secondary zone and the fourth zone of instability corresponds period T. At a dynamic load factor of 0.8, the width (Dp) of instability is 0.827o for principal instability zone, 0.145o for second zone of instability, 0.077o and 0.037o for third and fourth zone of instability, respectively. Fig. 4(b) shows the first four instability regions of a simply supported (SSSS) cross-ply [01/901/01] laminate (a/b¼1, a/h ¼100 and a ¼0) subjected to parabolic in-plane loading. The width of instability for all four zones increases with the increase of dynamic load factor. At a dynamic load factor of 0.8, width of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi instability (Dp) zones, respectively, are 15:428ð E22 h2 =rÞ for principal or first zone, 2:910ð E22 h2 =rÞ for second zone, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:796ð E22 h2 =rÞ for third zone and 1:676ð E22 h2 =rÞ for fourth zone. The width of the principal instability zone is the maximum compared to all other instability zones and hence of greater practical importance. It occurs in the vicinity of 2o (o is the lowest natural frequency of the plate). The width of instability zones decreases marginally for higher-order approximation as shown in Fig. 4(a) and (b). At a dynamic load factor, b ¼0.8, the width of principal zone is 0.827o from first-order approximation and 0.781o from second-order approximation for isotropic plate. The width of the instability zone for period 2T is more compared to the width for period T. Hence, only instability zones results are presented with period 2T with second-order approximation in subsequent figures. The effect of shear deformation for a square 3-layered cross-ply [01/901/01] composite SSSS plate (a/b¼ 1, a ¼ 0) subjected to parabolic in-plane loading is highlighted in Fig. 5. As the ratio of side to thickness decreases the shear deformation becomes pronounced, resulting in the decrease of natural frequency. It is also observed from the figure that as
L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
61
Fig. 3. First- and second-order approximation to first two dynamic instability regions of a SSSS square cross-ply composite plate subjected to uniform edge loading.
Fig. 4. Four instability zones of SSSS square plate subjected to parabolic in-plane edge loading: (a) isotropic plate and (b) cross-ply composite plate.
the side to thickness ratio decreases, the width of the instability zone decreases. The width of instability zone (excitation frequency range), Dp, is compared at a dynamic load factor, b ¼0.8, and for various a/h ratios. For a/h¼100, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp ¼ 17:375ð E22 h2 =rÞ; for a/h ¼50, Dp ¼ 15:117ð E22 h2 =rÞ; for a/h¼20, Dp ¼ 14:572ð E22 h2 =rÞ; and for a/h¼10, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width of instability zone is, Dp ¼ 11:994 E22 h2 =r. The influence of aspect ratio (a/b) on the primary instability zone of an isotropic plate (a/h ¼100 and a ¼0) subjected to parabolic in-plane loading is shown in Fig. 6(a). For a/b¼2, 1.5, 1, 0.5 and b ¼0.8, the instability zone widths are pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 22:72ð D=rhÞ, 21:61ð D=rhÞ, 16:06ð D=rhÞ and 10:03ð D=rhÞ, respectively. Fig. 6(b) shows the influence of aspect ratio (a/b) on instability zone of simply supported (SSSS) 3-layered cross-ply [01/901/01] laminate (a/h¼100 and a ¼0) subjected to parabolic in-plane load. Here a is kept constant and b is varied. It is observed that as the ratio of width of unloaded edge to the width of loaded edge decreases the width of instability zone decreases. For a/b¼2, 1.5, 1 and 0.5 and b ¼0.8, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values of instability zones are, 22:01ð E22 h2 =rÞ, 17:53ð E22 h2 =rÞ, 15:29ð E22 h2 =rÞ and 14:17ð E22 h2 =rÞ, respectively. The influence of aspect ratio is more in the case of composite plate as compared to isotropic plate. In the present case, the static and dynamic parts of linearly varying in-plane loads are denoted by Ns ¼ aN cr ð1Zðy=bÞÞ and Nt ¼ bNcr ð1Zðy=bÞÞ where aNcr and bN cr are the intensity of static and dynamic component of compressive load at the edge y ¼0. By taking various values of Z, different in-plane load distribution (uniform (Z ¼0), trapezoidal (Z ¼0.5), triangular (Z ¼1), partial tension (Z ¼ 1.5) and pure bending (Z ¼2.0)) are obtained. For parabolic in-plane load the static and dynamic component of compressive load at the edge y¼ 0 are Ns ¼ aN cr ð1Zðy2 =b2 ÞÞ and Nt ¼ bNcr ð1Zðy2 =b2 ÞÞ,
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Fig. 5. Influence of length to thickness (a/h) ratio on the principal instability zone of square cross-ply composite SSSS plates subjected to parabolic in-plane loading.
Fig. 6. Principal instability zone of SSSS square plate subjected to parabolic in-plane edge loading for different length to width (a/b) ratios: (a) isotropic plate and (b) cross-ply composite plate.
respectively. The principal dynamic instability regions are evaluated for the 3-layered symmetric cross-ply [01/901/01] square composite plate (a/b ¼1, a/h ¼100 and a ¼0.5) subjected to time dependent linearly varying in-plane loads with Z ¼0, 0.5, 1, 1.5, 2 and time dependent parabolically varying in-plane load and are represented as a plot of dimensionless pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi excitation frequency pð r=E22 h2 Þ against dimensionless dynamic load factor (b) of uniform loading in Fig. 7. It may be observed that the width of the dynamic instability region is the maximum for the uniform in-plane load and minimum for the pure bending case. This is due to the fact that the buckling load is minimum for uniform loaded plate and maximum for the pure in-plane bending case. The widths of the primary instability (range of excitation frequency) zones are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10:85ð E22 h2 =rÞ, 7:25ð E22 h2 =rÞ, 4:64ð E22 h2 =rÞ, 2:59ð E22 h2 =rÞ and 0:92ð E22 h2 =rÞ for values of Z ¼0, 0.5, 1, 1.5 and 2, respectively, for a dynamic load factor of 0.4 (note that the dynamic load factor corresponds to that of uniform pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi loading).The width for parabolic loading is 8:59ð E22 h2 =rÞ. Next the effect of boundary conditions on the principal instability region of plate (a/b¼1, a/h ¼100 and a ¼0) is studied by taking four different boundary conditions: all edges simply supported (SSSS); all edges clamped (CCCC); opposite loaded edges are clamped and the other two edges are simply supported (CSCS) and opposite loaded edges are simply supported and the other two edges are clamped (SCSC). The boundaries of instability are given in Fig. 8(a) for a square isotropic plate subjected to parabolic in-plane dynamic loading for four different boundary conditions. The principal instability region shifts to lower frequencies for SSSS case as compared to plate with other boundary conditions at the same dynamic load factor (b ¼ Nt =Ncr of SSSS plate) as observed from the figure. It is also observed from the Fig. 8(a) that
L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
63
Fig. 7. Principal instability zones for parabolic in-plane edge load and different linearly varying loads for a SSSS square cross-ply composite plate.
Fig. 8. Effect of boundary conditions on principal instability region of a square plate subjected to parabolic in-plane edge loading: (a) isotropic plate and (b) cross-ply composite plate.
the width of instability zone for CCCC plate is the minimum among all boundary conditions due to its higher natural frequency. The width of instability zone (excitation frequency range) increases with decrease of end restraint. pffiffiffiffiffiffiffiffiffiffiffiffi The widths of instability zones for a dynamic load factor of 0.8 of a square isotropic plate are 15:25ð D=rhÞ, pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 12:87ð D=rhÞ, 10:61ð D=rhÞ and 10:10ð D=rhÞ for SSSS, CSCS, SCSC and CCCC boundary conditions, respectively. The widths of instability zones are calculated for 3-layered symmetric cross-ply [01/901/01] square composite plates in Fig. 8(b) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a dynamic load factor of 0.8 and the values are 15:58ð E22 h2 =rÞ, 13:89ð E22 h2 =rÞ, 8:92ð E22 h2 =rÞ, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8:38ð E22 h2 =rÞ for SSSS, SCSC, CSCS, and CCCC, respectively. Finally the influence of static in-plane parabolic load on the dynamic instability regions of SSSS isotropic plate (a/b¼ 1, a/h¼ 100) is shown in Fig. 9. As the static in-plane load increases, the natural frequency of the loaded plate decreases and the instability regions occur at relatively lower disturbing frequency. The width of both principal instability zone and secondary instability zone increases with the increase in static in-plane loading. 8. Conclusions In this paper, buckling and dynamic instability of composite plates subjected to dynamic non-uniform in-plane loading is studied. The static buckling load of non-uniformly (parabolically) loaded plate is evaluated in two steps. The instability regions depend on the applied in-plane load variation (uniform, parabolic or nonlinear), the natural frequency of unloaded structure and frequency of excitation. The width of instability region for period 2T is more compared to the width for period T and hence has greater practical importance. The first-order approximation predicts accurately the instability
64
L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
Fig. 9. The effect of static in-plane parabolic edge load on the dynamic instability region of a SSSS square isotropic plate.
region if b ¼0.4, even in the presence of static load, beyond which second-order approximation must be used to calculate accurately the instability regions. Because of the shear deformation, the width of instability narrows down. As the aspect ratio (a/b) increases the width of instability zone becomes wider and this influence is more in the case of composite plate as compared to isotropic plate. The boundary conditions of the plate have a significant influence on the dynamic instability regions. The width of the instability zone increases with the decrease in end restraint and increases with the increase of static and dynamic loads.
Acknowledgements The authors sincerely thank the reviewers for their comments which have helped in improving the manuscript. Appendix A Nonlinear governing partial differential equations of cross-ply composite plate in displacement variables are A11 uo,xx þ A66 uo,yy þðA12 þA66 Þvo,xy þðA11 wo,xx þ A66 wo,yy Þwo,x þðA12 þ A66 Þwo,y wo,xy ¼ 0
(A-1)
ðA12 þ A66 Þuo,xy þ A66 vo,xx þA22 vo,yy þ ðA66 wo,xx þ A22 wo,yy Þwo,y þ ðA12 þ A66 Þðwo,x wo,xy Þ ¼ 0
(A-2)
o
o
o
D11 wo,xxxx 2D12 wo,xxyy D22 wo,yyyy 4D66 wo,xxyy þE11 f1,xxx þE12 ðf1,xyy þ f2,xxy Þ o o o þ E22 f2,yyy þ 2E66 ðf1,xyy þ f2,xxy Þ þ wo,xx ðA11 p1 þ A12 q1 Þ þ wo,x ðA11 p1,x þA12 q1,x Þ o o o nxx w,xx nxx,x w,x þ 2w,xy ðA66 r1 Þ þwo,x ðA66 r1,y Þ þ wo,y ðA66 r1,x Þ2nxy wo,xy nxy,y wo,x nxy,x wo,y þwo,yy ðA12 p1 þ A22 q1 Þ þ wo,y ðA12 p1,y þ A22 q1,y Þnyy wo,yy nyy,y wo,y ¼ 0
(A-3)
o
o
o
o
(A-4)
o
o
o
o
(A-5)
E22 wo,yyy ðE12 þ 2E66 Þwo,xxy þF66 f2,xx þ F22 f2,yy þ ðF12 þ F66 Þf1,xy H44 f2 ¼ 0 E11 wo,xxx ðE12 þ 2E66 Þwo,xyy þF11 f1,xx þ F66 f1,yy þ ðF12 þ F66 Þf2,xy H55 f1 ¼ 0 where, p1 ¼ uo,x þ 0:5ðwo,x Þ2 ;
q1 ¼ vo,x þ 0:5ðwo,y Þ2 ;
p2 ¼ wo,xx ; q2 ¼ wo,yy ; o o p3 ¼ f1,x ; q3 ¼ f2,y ; r3
r1 ¼ uo,x þ vo,x þ wo,x wo,y ; r2 ¼ 2wo,xy ; o
o
¼ f1,y þ f2,x ;
References [1] V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day Publication, San Francisco, 1964. [2] M.S.S. Prabhu, S. Durvasula, Thermal buckling of restrained skew plates, ASCE Journal of Engineering Mechanics 100 (1974) 1292–1295.
(A-6)
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[3] A.W. Leissa, E.F. Ayoub, Vibration and buckling of simply supported rectangular plate subjected to pair of in-plane concentrated forces, Journal of Sound and Vibration 127 (1988) 155–171. [4] P. Sundaresan, G. Singh, V.G. Rao, Buckling of moderately thick rectangular composite plate subjected to partial edge compression, International Journal of Mechanical Sciences 40 (1998) 1105–1117. [5] R.S. Srinivasan, P. Chellapandi, Dynamic stability of rectangular laminated composite plates, Computers & Structures 24 (1986) 233–238. [6] C.W. Bert, V. Birman, Dynamic instability of shear deformable antisymmetric angle-ply plates, International Journal of Solids and Structures 23 (1987) 1053–1061. [7] R.S. Srinivasan, P. Muralidharan, L.S. Ramachandra, Axisymmetric parametric stability of composite annular plates, Journal of Sound and Vibration 140 (1990) 175–180. [8] J. Moorthy, J.N. Reddy, R.H. Plaut, Parametric instability of laminated composite plates with transverse shear deformation, International Journal of Solids and Structures 26 (1990) 801–811. [9] Y.T. Ng, Y.K. Lam, J.N. Reddy, Dynamic stability of cylindrical panels with transverse shear effects, International Journal of Solids and Structures 36 (1999) 3483–3496. [10] A. Chattopadhyay, A.G. Radu, Dynamic instability of composite laminates using higher order theory, Computers & Structures 77 (2000) 453–460. [11] P. Dey, M.K. Singha, Dynamic stability analysis of composite skew plates subjected to periodic in-plane load, Thin Walled Structures 44 (2006) 937–942. [12] A.W. Leissa, J.H. Kang, Exact solution for vibration and buckling of an SS–C–SS–C rectangular plate loaded by linearly varying inplane stresses, International Journal of Mechanical Sciences 44 (2002) 1925–1945. [13] P. Jana, K. Bhaskar, Stability analysis of simply-supported rectangular plates under non-uniform uniaxial compression using rigorous and approximate plane stress solutions, Thin Walled Structures 44 (2006) 507–516. [14] H. Zhong, C. Gu, Buckling of simply supported rectangular Reissner–Mindlin plates subjected to linearly varying inplane loading, ASCE Journal of Engineering Mechanics 132 (2006) 578–581. [15] X. Wang, Xinfeng Wang, X. Shi, Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method, International Journal of Mechanical Sciences 49 (2006) 447–453. [16] K.P. Soldatos, A refined laminated plate and shell theory with applications, International Journal of Sound and Vibration 144 (1) (1991) 109–129. [17] S.P. Timoshenko, J.N. Goodier, Theory of Elastic Stability, McGraw-Hill Publication, New York, 1970. [18] G.B. Warburton, The vibration of rectangular plates, Proceedings of the Institutes of Mechanical Engineers, Series A 1 (68, 1954) 371–384. [19] D. Young, Vibration of rectangular plates by the Ritz method, Journal of Applied Mechanics 17 (4) (1950) 448–453. [20] R.K. Kaul, Y. Cadambe, The natural frequencies of thin skew plates, Aeronautical Quarter 7 (1956) 337–352. [21] A.W. Leissa, The free vibration of rectangular plates, Journal of Sound and Vibration 31 (1973) 257–293. [22] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, McGraw-Hill Publication, New York, 1961. [23] J.M. Hutt, A.E. Salam, Dynamic instability of plates by finite element, ASCE Journal of Engineering Mechanics 3 (1971) 879–899.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Dynamic instability of composite plates subjected to non-uniform in-plane loads L.S. Ramachandra n, Sarat Kumar Panda Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, India
a r t i c l e i n f o
abstract
Article history: Received 20 March 2009 Received in revised form 12 August 2011 Accepted 15 August 2011 Handling Editor: M.P. Cartmell Available online 9 September 2011
In this paper, the dynamic instability of a shear deformable composite plate subjected to periodic non-uniform in-plane loading is studied for four sets of boundary conditions. The static component and the dynamic component of the applied periodic inplane loading are assumed to vary according to either parabolic or linear distributions. Initially, the plate membrane problem is solved using the Ritz method to evaluate the plate in-plane stress distributions within the prebuckling range due to the applied nonuniform in-plane edge loading. Subsequently using the evaluated stress distribution within the plate, the equations governing the plate instability boundaries are formulated via Hamilton’s variational principle. Employing Galerkin’s method, these partial differential equations are reduced into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability behaviour. Following Bolotin’s method, the instability regions are determined from the boundaries of instability, which represents the periodic solution of the differential equations with period T and 2T to the Mathieu equations. The instability regions are determined for uniform, linear and parabolic dynamic in-plane loads using first-order and second-order approximations. Numerical results are also presented to bring out the effects of span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Composite materials are increasingly used in aerospace, mechanical, civil and naval structures for its high strength to weight ratio, stiffness and inherent directional properties. Structural components (beams, plates and shells) are often subjected to periodic in-plane loads and become dynamically unstable for certain combination of load amplitude, disturbing frequency and frequency of transverse vibration of the structure. The knowledge of static and dynamic stability behaviour of composite structural members subjected to various types of loads are essential in the design of these components. Bolotin [1] has reviewed the literature on the dynamic instability of isotropic beams, plates and shells in his book. Buckling and dynamic instability of composite plates subjected to uniform in-plane compressive loading have been extensively studied by various authors [2–11]. Often, plates are a part of a complex structural system and hence the load coming on it may not be always uniform. For example, in the case of I-beam or wide flanged beam subjected to bending moment at the ends or lateral loads on the flange, the web of the beam is subjected to non-uniform in-plane loads. The load exerted on the aircraft wings, or on the stiffened plate in the ship structures or on the slabs of a multi-storey building
n
Corresponding author. Tel.: þ91 03222 83445; fax: þ 91 03222 82254. E-mail address: [email protected] (L.S. Ramachandra).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.08.010
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by the adjoining structures usually is non-uniform. The type of distribution in an actual structure depends on the relative stiffnesses of the adjoining elements. A few studies have been reported on the static buckling analysis of plates subjected to non-uniform in-plane compressive loading [12–15]. Srinivasan and Chellapandi [5] investigated the dynamic instability of rectangular laminated composite plates subjected to uniform dynamic loading using finite strip method in conjunction with the Bolotin’s method. Authors have neglected the transverse shear deformation, in-plane inertia and rotary inertia of the plate while estimating the width of instability regions. The effect of shear deformation on the dynamic instability of antisymmetric simply supported angle ply plates was reported by Bert and Birman [6]. Srinivasan et al. [7] have obtained dynamic instability regions of orthotropic circular and annular plates subjected to time varying in-plane loads using finite element method. The parametric instability of a laminated composite plate under uniaxial, harmonically varying in-plane load was studied by Moorthy et al. [8] adopting first-order shear deformation theory within the framework of finite element method. The dynamic instability of simply supported isotropic cylindrical panels under combined static and periodic axial forces was reported by Ng et al. [9]. Authors have modelled the shell panel using the Donnel’s shell theory in conjunction with the first-order shear deformation theory and have obtained a system of Mathieu–Hill equations through normal mode expansion. The dynamic instability regions are obtained based on the Bolotin’s method. Recently, Chattopadhyay and Radu [10] studied instability associated with composite plates subjected to dynamic uniform in-plane load considering both the transverse shear deformation and the rotary inertia. They have presented two instability regions considering second-order approximation within the framework of finite element method. Dey and Singha [11] studied dynamic instability of skew plates subjected to periodic uniform in-plane loading using finite element approach. In their formulation, authors have included effects of transverse shear deformation, in-plane and rotary inertia. So far, to the best of authors’ knowledge, there is no work available in the published literature on the dynamic instability analysis of composite plates subjected to dynamic non-uniform in-plane loads. In this investigation, critical buckling loads and dynamic instability regions of a rectangular isotropic and composite plate, subjected to static and dynamic (Nxx ¼ Ns þ Nt cosðptÞ, p is the excitation frequency and Nxx is the applied edge loading) non-uniform (linearly varying and parabolically varying) in-plane loads, respectively, are studied considering higher-order shear deformation theory (HSDT). In the present study both static (Ns) and dynamic (Nt) component of the load is assumed to vary in a similar manner. Since the applied in-plane edge load is non-uniform, in the first step, considering only the static component (Ns) of the in-plane load the plane elasticity problem is solved by minimizing the membrane strain energy to evaluate the stress distribution sxx, syy and txy within the plate. Finally, superposing the stress distribution due to static (Ns) and dynamic loads (Nt cosðptÞ), the stress distributions within the plate are determined. Using the above stress distributions and through Hamilton’s principle, the governing partial differential equations of plate motion are derived. Adopting Galerkin’s approximation, the governing partial differential equations are converted into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability. Adopting Fourier series method, periodic solutions to Mathieu equations are sought and the instability regions are determined by solving the associated eigenvalue problem. Finally, the influence of boundary conditions, span to thickness ratios, shear deformation and aspect ratios on the dynamic instability regions of the plates are reported in the paper. 2. Formulation A composite rectangular plate of length a, width b and composed of n layers of equal thickness is considered with the co-ordinate axes xy in the in-plane directions and the z-axis in the thickness direction of the plate as shown in Fig. 1. The equations of motion of the plate subjected to dynamic in-plane loads in terms of force and moment resultants can be written as [16]: ^ xx,x þ N ^ xy,y ¼ r u þ r ðf w, x Þ 4 r f N 1 0 1 3 1 2 3h ,tt 4 ^ xy,x þ N ^ yy,y ¼ r v þ r ðf w, y Þ N r f 2 0 1 3h2 3 2 ,tt
Fig. 1. Geometry and loading of the plate.
L.S. Ramachandra, S.K. Panda / Journal of Sound and Vibration 331 (2012) 53–65
^ xx w, x þ N ^ xy w, y Þ, x þ ðN ^ xy w, x þ N ^ yy w, y Þ, y Mxx,xx þ2Mxy,xy þMyy,yy þ ðN 4 ¼ r0 wþ r1 ðu,x þ v, y Þ þ r2 ðf1,x þ f2,y w, xx w, yy Þ þ 2 r4 ðf1,x þ f2,y Þ 3h ,tt 4 4 4 4 a Pxx,x þPxy,y Qxx ¼ r1 2 r3 u þ r2 2 r4 ðf1 w, x Þ 2 r4 2 r6 f1 3h 3h 3h 3h ,tt 4 4 4 4 a Pxy,x þ Pyy,y Qyy ¼ r1 2 r3 v þ r2 2 r4 ðf2 w, y Þ 2 r4 2 r6 f2 3h 3h 3h 3h ,tt where the inertia terms are defined according to rm ¼
R h=2 h=2
55
(1)
rzm dz ðm ¼ 0,1,. . .,6Þ and u, v and w are the displacement
components, respectively, along the x, y and z directions; f1 and f2 are, respectively, the total rotations of the crosssections perpendicular to x-axis and y-axis, respectively; ( ),x and ( ),y denote, respectively, partial differentiation with ^ xx , N ^ yy , N ^ xy are the plate internal stress resultants; Mxx, respect to x and y co-ordinates; r is the mass density of the plate; N Myy, Mxy are the moment resultants; Pxx, Pyy, Pxy are the additional moment resultants due to additional changes of ^ xx , N ^ yy , N ^ xy are related curvature f1,x, f2,y, (f2,x þ f1,y). Q a ,Q a are the additional transverse shear force resultants. Now, N xx
yy
to Nxx, Nyy, Nxy and nxx, nyy, nxy through, ^ xx ¼ ½Nxx nxx N ^ N xy ¼ ½Nxy nxy ^ yy ¼ ½Nyy nyy N
(2)
where nxx, nyy, nxy are the plate internal stress resultants due to applied end non-uniform in-plane loading and Nxx, Nyy, Nxy n o are the plate stress resultants because of large deflection. The force ðNT ¼ Nxx Nyy Nxy Þ and moment n o n o n o 0 0 0 ðMT ¼ Mxx Myy Mxy , PT ¼ Pxx Pyy Pxy Þ, resultants are related to membrane strains ðe0T ¼ exx eyy exy Þ, and n o n o 0 0 0 T bending strains ðwT ¼ w, xx w, yy 2w, xy , f ¼ f1,x f2,y f1,y þ f2,x Þ, respectively, through the constitutive relationship defined as N ¼ Ae0 þ Bw þ Cf
(3)
M ¼ BT e0 þDw þ Ef
(4)
P ¼ CT e0 þ Ew þ Ff
(5)
In the present work, the bold upright letters are used to denote matrices and vectors. The additional transverse shear force n o n o a a T resultants ðQ aT ¼ Qyy Qxx Þ are related to shear strains ðf ¼ f2 f1 Þ by Q a ¼ Hf
(6)
where the elements of matrices A, B, C, E, F and H of Eqs. (3)–(6) are defined as Z h=2 fAij ,Bij ,Dg ¼ Qij ð1,z,z2 Þ dz fCij ,Eij ,Fij g ¼
Z Z
h=2 h=2
Qij ð1,z,f ðzÞÞf ðzÞdz
for i,j ¼ 1,2,6
h=2 h=2
Qij ðf 0 ðzÞÞf 0 ðzÞ dz
fHij g ¼
for i,j ¼ 4,5
(7)
h=2
here, Qij (i,j ¼1,2,6) are the transformed plane-stress reduced stiffness coefficients and Qij (i,j ¼4,5) are the transformed through-thickness shear stiffness coefficients; eox , eoy and goxy are the reference surface strains and are defined as 1 2
eox ¼ uo,x þ ðwo,x Þ2 ;
1 2
eoy ¼ vo,y þ ðwo,y Þ2 ;
goxy ¼ uo,y þ vo,x þ wo,x wo,y
(8)
where uo, vo and wo are, respectively, the middle surface displacement components along the x, y and z directions, respectively. The von Ka´rma´n strain–displacements relations, Eq. (8), are used to account for the deformed configuration of the plate with moderate displacements and small rotations. Using Eqs. (2)–(8), the governing Eq. (1) are then expressed in terms of displacement components and are given in Appendix A. 3. Plate prebuckling analysis In the present investigation, parabolically and linearly varying in-plane compressive dynamic loads are considered. In the case of linearly varying in-plane load, the stress distributions within the plate coincide with the applied edge loading.
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In the case of parabolic in-plane load, initially the static component (Ns) of the in-plane loading is applied at the plate edge and stress fields within the plate are obtained by solving the plate membrane problem. The correct stress distribution within the plate is the one which minimizes the membrane strain energy of the plate and satisfies the boundary conditions [17] of the problem. The membrane strain energy of a composite plate of thickness h is given by 8 9T 2 9 31 8 n n > ZZ > < xx > = A11 A12 A16 < xx > = 1 6 7 nyy nyy dx dy V¼ (9) 4 A12 A22 A26 5 > 2 A> :n > ; A :n > ; A A xy xy 16 26 66 where, nyy ¼
@2 F , @x2
nxx ¼
@2 F , @y2
nxy ¼
@2 F , @x@y
Aij ¼
Z
h=2
Qij dz
(10)
h=2
where Qij is the transformed reduced stiffness coefficients and F is the stress function. Following Timoshenko and Goodier [17], Ritz method is adopted to minimize the membrane strain energy of the plate in this study. The boundary conditions [17] of the plate membrane problem are given here for the parabolically varying uniaxial in-plane edge loading (see Fig. 1) y y 1 x ¼ 0,a, N xy ¼ 0, Nxx ¼ 4N0 b b y ¼ 0,b, N xy ¼ 0, Nyy ¼ 0 (11) The stress function is assumed in the form of a series as [17]
F ¼ F0 þ a1 F1 þ a2 F2 þ a3 F3 þ a4 F4 þ ,
(12)
where, a1,a2,a3,y are constants to be determined such that the membrane strain energy is minimized and boundary conditions are satisfied. In the present investigation, only four terms are considered in Eq. (12). Thus stress function for the parabolic in-plane edge loading is assumed as y2 y y2 2 þðx2 axÞ2 ðy2 ybÞ2 ða1 þ a2 x þ a3 yÞ F ¼ 2N0 (13) 3 b 2b where
F0 ¼ 2N 0
y2 y y2 2 3 b 2b
which gives Nyy ¼
@2 F0 ¼0 , @x2
N xx ¼
N xy ¼
@2 F0 ¼0 @x@y
@2 F0 y y 1 ¼ 4N0 2 b b @y
In the present study, only four terms in the series are considered. The functions F1, F2, F3 in Eq. (12) are chosen such that the stresses corresponding to them vanish at the boundary. Substituting the Eq. (13) into Eq. (9) and carrying out integration, a second degree polynomial in a1, a2 and a3 is obtained. Then the constants a1, a2 and a3 are evaluated from the 3 algebraic equations resulting from the condition, @V=@a1 ¼ 0, @V=@a2 ¼ 0, @V=@a3 ¼ 0. The explicit expression so obtained for constants ak (k¼1,2,3) for 3-layered cross-ply composite plate [01/901/01] are
a1 ¼
N0 ð0:0934a8 þ 16:7313a6 b2 þ 65:6356a4 b4 þ32:2675a2 b6 þ0:3475b8 Þ ð0:1359a12 b2 þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(14)
a2 ¼
N0 ð0:6073a6 þ 28:3450a4 b2 þ88:7251a2 b4 þ 0:9738b6 Þ ð0:1359a12 b2 þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(15)
a3 ¼
N 0 ð2:2589b6 þ 54:6644a2 b4 þ 88:7235a4 b2 þ 0:5049a6 Þ 12 2 ð0:1359a b þ 190a10 b4 þ 3155a8 b6 þ 1000a6 b8 þ6084a4 b10 þ 708a2 b12 þ 9:7487b14 Þ
(16)
Superposing the distribution due to static (Ns) and dynamic loads (Nt cos(pt)), the stress distributions within the plate are determined. Using these plate stress distributions and from Hamilton’s principle, the governing partial diffrential equations (Eq. (1)) of the plate dynamic instability are obtained. 4. Plate buckling analyses Neglecting the inertia terms in Eq. (1) we obtain the governing partial differential equations for the buckling analysis of plates. The critical buckling load of the composite rectangular plate subjected to parabolically varying in-plane
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57
compressive load is obtained by using Galerkin’s method. In the present investigation following four sets of boundary conditions are used: SSSS, CSCS, SCSC and CCCC. Where S stands for the simply supported edge and C for the clamped edge. The letters indicate the boundary conditions on the edge of the plate in the anti clockwise fashion starting from the left hand corner. In the Galerkin’s method, the out-of-plane displacement field w(x,y) satisfying the boundary conditions of the plate is expressed as the product of beam functions [18] as wðx,yÞ ¼
1 X 1 X
Xm ðxÞYn ðyÞ
(17)
m¼1n¼1
where Xm(x) and Yn(y) are the eigenfunctions of the beam having the same boundary conditions as that of two opposite edges of the plate. In the present case following beam functions are assumed for different edge conditions: (a) Simply supported at x¼0 and x ¼a ss ðxÞ ¼ sin Xm
mpx a
ðm ¼ 1,2,3:::Þ
(18)
(b) Clamped support along two opposite edges, i.e., at x ¼0 and x ¼a, according to [18–20] cc ðxÞ ¼ cos xm Xm
x 1 sinðxm =2Þ x 1 þ cosh xm a 2 sinhðxm =2Þ a 2
ðm ¼ 2,4,6. . .Þ
(19)
where xm are the roots of the equation tanðxm =2Þ þ tanhðxm =2Þ ¼ 0
(20)
and cc ðxÞ ¼ sin xm Xm
x 1 sinðxm =2Þ x 1 cosh xm a 2 sinhðxm =2Þ a 2
ðm ¼ 3,5,7. . .Þ
(21)
where xn are obtained as roots of the equation, tanðxm =2Þtanhðxm =2Þ ¼ 0
(22)
The functions Yn(y) are similarly chosen based on the condition at y¼0 and y¼b by replacing x by y and a by b and m by n in the above equations. m and n are, respectively, the number of nodal lines along x and y directions. In the case of simple support only normal in-plane displacements are allowed and in-plane tangential displacements and out of plane displacements are prevented. (a) Boundary conditions at the simply supported edges are o
vo ¼ wo ¼ Pxx ¼ f2 ¼ Mxx ¼ 0
nxx Nxx ¼ Nxx ,
o
nyy Nyy ¼ Nyy ,
o
o
u ¼ w ¼ f1 ¼ Pyy ¼ Myy ¼ 0
at
x ¼ 0,a
at
y ¼ 0,b
(23)
Following displacement fields satisfy the above boundary conditions (Eq. (23)): u~ o ¼
mpx npy sin Umn cos a b m¼1n¼1 j i X X
j i X X
v~ o ¼
Vmn sin
m¼1n¼1
~o¼ w
j i X X
mpx npy cos a b
ss Wmn Xm ðxÞXnss ðyÞ
m¼1n¼1 o
f~ 1 ¼ o
f~ 2 ¼
mpx npy sin Kmn cos a b m¼1n¼1 j i X X
mpx npy cos Lmn sin a b m¼1n¼1 j i X X
(24)
(b) The boundary conditions for CCCC plate are
o
o
nxx Nxx ¼ N xx ,
w0,x ¼ vo ¼ wo ¼ f1 ¼ f2 ¼ 0
nyy Nyy ¼ Nyy ,
w0,y ¼ uo ¼ wo ¼ f1 ¼ f2 ¼ 0
o
o
at x ¼ 0,a at y ¼ 0,b
(25)
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Following displacement fields satisfy the boundary conditions defined in (Eq. 25): u~ o ¼
mpx npy sin Umn cos a b m¼1n¼1 j i X X
j i X X
v~ o ¼
Vmn sin
mpx
m¼1n¼1
~o¼ w
i X
j X
a
cos
npy b
cc Wmn Xm ðxÞXncc ðyÞ
m¼1n¼1 0
f~ 1 ¼
j i X X
Kmn sin
m¼1n¼1
mpx a
sin
npy b
mpx npy 0 f~ 2 ¼ Lmn sin sin a b m¼1n¼1 j i X X
(26)
Using Eqs. (24) and (26), and employing Galerkin’s method the Eq. (1) (without inertia terms) in displacement terms are converted into a set of homogeneous algebraic equations for simply supported and clamped boundary conditions, respectively. Solving the associated eigenvalue problem critical buckling loads are evaluated numerically in Section 6.
5. Dynamic instability analysis The applied periodic non-uniform in-plane load N xx ¼ Ns þNt cos pt has a static and a dynamic component. The static component (Ns) and the dynamic component (Nt) of the applied loading are assumed to vary accordingly to either parabolic or linear distributions. Both Ns and Nt are assumed to vary in a similar manner. Adopting Galerkin’s approximation and using approximate displacement fields (Eqs. (24) and (26)), the governing partial differential Eq. (1) of the plate subjected to dynamic non-uniform in-plane loads are reduced to a set of ordinary differential equations (Mathieu type of equation) describing the plate dynamic instability behaviour as € þ ½KNs KG Nt KG cos ptX ¼ 0 MX
(27)
where M, K and KG are, respectively, the mass, stiffness and geometric stiffness matrices. In the present investigation, only the linear dynamic instability of rectangular composite plates is studied and hence the nonlinear terms in the Eq. (1) are not considered. The Eq. (27) is a second-order differential equation with periodic coefficients. For evaluation of the buckling load, employing Galerkin’s procedure to Eq. (1) without inertia and harmonic terms, the following set of homogeneous algebraic equations are derived ½KNcr KG W ¼ 0
(28)
For non-trivial solution, the determinant of the coefficient matrix must vanish. Hence 9KNcr KG 9 ¼ 0
(29)
where Ncr is the static buckling load. The critical buckling load is obtained from the solution of linear eigenvalue problem given by Eq. (29). Similarly the solutions of the eigenvalue problem associated with the Eq. (27) neglecting terms containing Ns and Nt gives the natural frequencies. The eigenvalue problem for evaluating free vibration frequencies are stated as 9Ko2 M9 ¼ 0
(30)
In the Eq. (27), Ns and Nt are varied as Ns ¼ aNcr and Nt ¼ bNcr, where a and b are static and dynamic load factors, respectively. The regions of instability are located by boundaries of the instability having periodic solutions with period T and 2T to the Eq. (27). These are determined by the method suggested by Bolotin [1] in this investigation. The solutions to Eq. (27) with periods T and 2T, respectively, are assumed in the form of Fourier series as 1 X lpt lpt xðtÞ ¼ b0 þ þ bl cos (31) al sin 2 2 l ¼ 2,4,6 xðtÞ ¼
1 X l ¼ 1,3,5
al sin
lpt lpt þbl cos 2 2
(32)
where al and bl are the arbitrary constants. Here, it may be noted that, two solutions of identical periods bound the region of instability and, two solutions of different periods bounds the region of stability. Substituting Eq. (31) or (32) into Eq. (27) and equating the coefficients of identical sinðlpt=2Þ and cosðlpt=2Þ leads to a system of homogeneous algebraic equations in al and bl. For non-trivial solution the determinant of the coefficient matrix of al and bl must vanish. The size of the above determinant is infinite as we have assumed the solution in the form of infinite series. The determinants are shown to be
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belonging to a class of converging determinant known as normal determinant [1]. The first-order (from Eq. (33)) and second-order (from Eq. (34)) approximation to boundaries of principal regions of instability corresponding to period 2T are obtained by solving following two eigenvalue problems, respectively 9K n 70:5bNcr KG 0:25Mp21 9 ¼ 0 n K 7 0:5bNcr KG bNcr KG
0:5bNcr KG 0:25M 2 p K n 2:25Mp21 2 0
(33) 0 ¼0 0
(34)
Secondary zone of instability with first-order approximation (from Eqs. (35) and (36)) and second-order approximations (from Eqs. (37) and (38)) corresponding to period T are determined from 9K n Mp21 9 ¼ 0 Kn bNcr KG Kn 0:5bNcr KG Kn bN K cr G 0
0:5bNcr KG 2 0 p Kn 1 0
(35) 0 ¼0 M
0:5bNcr KG M 2 n 2 p2 K 4Mp1 0
0:5bNcr KG Kn 0:5bNcr KG
0 ¼0 0
0 0:5bNcr KG p22 0 0 K n 4Mp21 0
(36)
0 M 0
(37) 0 0¼0 0
(38)
where K* ¼K NsKG. Similarly third and fourth instability region can be computed by following the same procedure. However they are not given here for the sake of brevity. 6. Buckling numerical results The following mechanical properties are assumed, E11 ¼ E22, G23 ¼G13 ¼G12 ¼ E22/2.5, n12 ¼0.25 for isotropic plate and E1/E2 ¼40, G12 ¼G13 ¼0.5E2, G23 ¼0.6E2, n12 ¼0.25 for composite lamina in the analysis. The natural frequencies obtained from the present method for various boundary conditions are compared with those obtained by Leissa [21] in Table 1 for a square isotropic plate. It is observed that the two results agree with each other exactly for a very thin plate. To validate the present formulation, the dimensionless buckling load ðki ¼ Ncr ð2bÞ2 =p2 DÞobtained from the present method for an isotropic plate subjected to uniform and parabolic uniaxial in-plane loads are compared with those of Timoshenko and Gere [22] and Wang et al.[15] in Table 2. The buckling loads obtained from the commercial finite element software ANSYS are also given in the Table 2. Converged buckling loads were obtained in the present method using 4 terms in the displacement field approximation [(m ¼1, n¼ 1), (m¼1, n ¼3), (m¼ 3, n¼1), (m¼3, n ¼3)]. The present results are in close agreement with the published and the commercial software ANSYS results. Hence, all subsequent analyses were carried out with 4 terms in the displacement field approximations. 7. Numerical results for Instability regions The material properties of the plate are the same as used in the case of buckling analysis. The dynamic instability pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffi regions are presented as a plot of dimensionless excitation frequency (pð rh=DÞ for isotropic plate and pð r=E22 h2 Þ for composite plate) or frequency ratio (p/o) against dimensionless dynamic load factor (b), where o is the natural frequency of the plate. The principal dynamic instability regions (with period 2T) of isotropic simply supported rectangular plate Table 1 Dimensionless fundamental frequency for a square simply supported isotropic plate. a/h
Dimensionless natural frequency (l)
10 20 50 100 1000 Ref. [21] For isotropic plate l ¼ oa2
pffiffiffiffiffiffiffiffiffiffiffiffi rh=D.
SSSS
SCSC
CSCS
CCCC
19.10 19.57 19.71 19.73 19.74 19.74
27.12 28.45 28.87 28.94 28.95 28.95
27.12 28.45 28.87 28.94 28.95 28.95
33.59 35.34 35.91 35.98 35.99 35.99
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Table 2 Critical buckling coefficient (ki) of isotropic plate for various aspect ratios. Type of in-plane load distribution
Aspect ratio (a/b)
Dimensionless buckling coefficient, kia Present solution
ANSYS
Others
Uniform load
0.5 1 1.5
6.25 3.99 4.33
6.25 4.00 4.33
6.25 [22] 4.00 [22] 4.34 [22]
Parabolic load
0.5 1 3
7.27 5.24 5.62
7.28 5.27 5.62
7.27 [15] 5.24 [15] 5.69 [15]
a
ki ¼ Ncr ð2bÞ2 =p2 D .
Fig. 2. First and second approximation to principal instability region for a SSSS square isotropic plate subjected to uniform edge loading: (a) a ¼0 and (b) a ¼0.6.
(a/b¼ 1, a/h¼100) obtained from the present method are compared with that of Hutt and Salam [23] in Fig. 2(a) and (b), respectively, for two static load factors a ¼0 and 0. It is observed from figures that the present results agree well with the reference values. It is observed from Fig. 2(a) and (b) that the width of the instability region increases with the increase in static and dynamic load factors. Dynamic instability regions of a simply supported square symmetric four layered cross-ply [01/901/01] composite plate (a/b¼ 1, a/h ¼25, a ¼0) obtained from the present approach are compared with those of Moorthy et al. [8] in Fig. 3. It is observed that the present results agree well with the reference values. The first four instability zones of an isotropic simply supported square plate (a/b¼1, a/h¼100, a ¼ 0) under parabolic loading is shown in Fig. 4(a). In this figure, the principal zone and the third zone of instability corresponds to period 2T, whereas, secondary zone and the fourth zone of instability corresponds period T. At a dynamic load factor of 0.8, the width (Dp) of instability is 0.827o for principal instability zone, 0.145o for second zone of instability, 0.077o and 0.037o for third and fourth zone of instability, respectively. Fig. 4(b) shows the first four instability regions of a simply supported (SSSS) cross-ply [01/901/01] laminate (a/b¼1, a/h ¼100 and a ¼0) subjected to parabolic in-plane loading. The width of instability for all four zones increases with the increase of dynamic load factor. At a dynamic load factor of 0.8, width of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi instability (Dp) zones, respectively, are 15:428ð E22 h2 =rÞ for principal or first zone, 2:910ð E22 h2 =rÞ for second zone, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:796ð E22 h2 =rÞ for third zone and 1:676ð E22 h2 =rÞ for fourth zone. The width of the principal instability zone is the maximum compared to all other instability zones and hence of greater practical importance. It occurs in the vicinity of 2o (o is the lowest natural frequency of the plate). The width of instability zones decreases marginally for higher-order approximation as shown in Fig. 4(a) and (b). At a dynamic load factor, b ¼0.8, the width of principal zone is 0.827o from first-order approximation and 0.781o from second-order approximation for isotropic plate. The width of the instability zone for period 2T is more compared to the width for period T. Hence, only instability zones results are presented with period 2T with second-order approximation in subsequent figures. The effect of shear deformation for a square 3-layered cross-ply [01/901/01] composite SSSS plate (a/b¼ 1, a ¼ 0) subjected to parabolic in-plane loading is highlighted in Fig. 5. As the ratio of side to thickness decreases the shear deformation becomes pronounced, resulting in the decrease of natural frequency. It is also observed from the figure that as
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Fig. 3. First- and second-order approximation to first two dynamic instability regions of a SSSS square cross-ply composite plate subjected to uniform edge loading.
Fig. 4. Four instability zones of SSSS square plate subjected to parabolic in-plane edge loading: (a) isotropic plate and (b) cross-ply composite plate.
the side to thickness ratio decreases, the width of the instability zone decreases. The width of instability zone (excitation frequency range), Dp, is compared at a dynamic load factor, b ¼0.8, and for various a/h ratios. For a/h¼100, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp ¼ 17:375ð E22 h2 =rÞ; for a/h ¼50, Dp ¼ 15:117ð E22 h2 =rÞ; for a/h¼20, Dp ¼ 14:572ð E22 h2 =rÞ; and for a/h¼10, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width of instability zone is, Dp ¼ 11:994 E22 h2 =r. The influence of aspect ratio (a/b) on the primary instability zone of an isotropic plate (a/h ¼100 and a ¼0) subjected to parabolic in-plane loading is shown in Fig. 6(a). For a/b¼2, 1.5, 1, 0.5 and b ¼0.8, the instability zone widths are pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 22:72ð D=rhÞ, 21:61ð D=rhÞ, 16:06ð D=rhÞ and 10:03ð D=rhÞ, respectively. Fig. 6(b) shows the influence of aspect ratio (a/b) on instability zone of simply supported (SSSS) 3-layered cross-ply [01/901/01] laminate (a/h¼100 and a ¼0) subjected to parabolic in-plane load. Here a is kept constant and b is varied. It is observed that as the ratio of width of unloaded edge to the width of loaded edge decreases the width of instability zone decreases. For a/b¼2, 1.5, 1 and 0.5 and b ¼0.8, the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values of instability zones are, 22:01ð E22 h2 =rÞ, 17:53ð E22 h2 =rÞ, 15:29ð E22 h2 =rÞ and 14:17ð E22 h2 =rÞ, respectively. The influence of aspect ratio is more in the case of composite plate as compared to isotropic plate. In the present case, the static and dynamic parts of linearly varying in-plane loads are denoted by Ns ¼ aN cr ð1Zðy=bÞÞ and Nt ¼ bNcr ð1Zðy=bÞÞ where aNcr and bN cr are the intensity of static and dynamic component of compressive load at the edge y ¼0. By taking various values of Z, different in-plane load distribution (uniform (Z ¼0), trapezoidal (Z ¼0.5), triangular (Z ¼1), partial tension (Z ¼ 1.5) and pure bending (Z ¼2.0)) are obtained. For parabolic in-plane load the static and dynamic component of compressive load at the edge y¼ 0 are Ns ¼ aN cr ð1Zðy2 =b2 ÞÞ and Nt ¼ bNcr ð1Zðy2 =b2 ÞÞ,
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Fig. 5. Influence of length to thickness (a/h) ratio on the principal instability zone of square cross-ply composite SSSS plates subjected to parabolic in-plane loading.
Fig. 6. Principal instability zone of SSSS square plate subjected to parabolic in-plane edge loading for different length to width (a/b) ratios: (a) isotropic plate and (b) cross-ply composite plate.
respectively. The principal dynamic instability regions are evaluated for the 3-layered symmetric cross-ply [01/901/01] square composite plate (a/b ¼1, a/h ¼100 and a ¼0.5) subjected to time dependent linearly varying in-plane loads with Z ¼0, 0.5, 1, 1.5, 2 and time dependent parabolically varying in-plane load and are represented as a plot of dimensionless pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi excitation frequency pð r=E22 h2 Þ against dimensionless dynamic load factor (b) of uniform loading in Fig. 7. It may be observed that the width of the dynamic instability region is the maximum for the uniform in-plane load and minimum for the pure bending case. This is due to the fact that the buckling load is minimum for uniform loaded plate and maximum for the pure in-plane bending case. The widths of the primary instability (range of excitation frequency) zones are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10:85ð E22 h2 =rÞ, 7:25ð E22 h2 =rÞ, 4:64ð E22 h2 =rÞ, 2:59ð E22 h2 =rÞ and 0:92ð E22 h2 =rÞ for values of Z ¼0, 0.5, 1, 1.5 and 2, respectively, for a dynamic load factor of 0.4 (note that the dynamic load factor corresponds to that of uniform pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi loading).The width for parabolic loading is 8:59ð E22 h2 =rÞ. Next the effect of boundary conditions on the principal instability region of plate (a/b¼1, a/h ¼100 and a ¼0) is studied by taking four different boundary conditions: all edges simply supported (SSSS); all edges clamped (CCCC); opposite loaded edges are clamped and the other two edges are simply supported (CSCS) and opposite loaded edges are simply supported and the other two edges are clamped (SCSC). The boundaries of instability are given in Fig. 8(a) for a square isotropic plate subjected to parabolic in-plane dynamic loading for four different boundary conditions. The principal instability region shifts to lower frequencies for SSSS case as compared to plate with other boundary conditions at the same dynamic load factor (b ¼ Nt =Ncr of SSSS plate) as observed from the figure. It is also observed from the Fig. 8(a) that
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63
Fig. 7. Principal instability zones for parabolic in-plane edge load and different linearly varying loads for a SSSS square cross-ply composite plate.
Fig. 8. Effect of boundary conditions on principal instability region of a square plate subjected to parabolic in-plane edge loading: (a) isotropic plate and (b) cross-ply composite plate.
the width of instability zone for CCCC plate is the minimum among all boundary conditions due to its higher natural frequency. The width of instability zone (excitation frequency range) increases with decrease of end restraint. pffiffiffiffiffiffiffiffiffiffiffiffi The widths of instability zones for a dynamic load factor of 0.8 of a square isotropic plate are 15:25ð D=rhÞ, pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 12:87ð D=rhÞ, 10:61ð D=rhÞ and 10:10ð D=rhÞ for SSSS, CSCS, SCSC and CCCC boundary conditions, respectively. The widths of instability zones are calculated for 3-layered symmetric cross-ply [01/901/01] square composite plates in Fig. 8(b) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a dynamic load factor of 0.8 and the values are 15:58ð E22 h2 =rÞ, 13:89ð E22 h2 =rÞ, 8:92ð E22 h2 =rÞ, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8:38ð E22 h2 =rÞ for SSSS, SCSC, CSCS, and CCCC, respectively. Finally the influence of static in-plane parabolic load on the dynamic instability regions of SSSS isotropic plate (a/b¼ 1, a/h¼ 100) is shown in Fig. 9. As the static in-plane load increases, the natural frequency of the loaded plate decreases and the instability regions occur at relatively lower disturbing frequency. The width of both principal instability zone and secondary instability zone increases with the increase in static in-plane loading. 8. Conclusions In this paper, buckling and dynamic instability of composite plates subjected to dynamic non-uniform in-plane loading is studied. The static buckling load of non-uniformly (parabolically) loaded plate is evaluated in two steps. The instability regions depend on the applied in-plane load variation (uniform, parabolic or nonlinear), the natural frequency of unloaded structure and frequency of excitation. The width of instability region for period 2T is more compared to the width for period T and hence has greater practical importance. The first-order approximation predicts accurately the instability
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Fig. 9. The effect of static in-plane parabolic edge load on the dynamic instability region of a SSSS square isotropic plate.
region if b ¼0.4, even in the presence of static load, beyond which second-order approximation must be used to calculate accurately the instability regions. Because of the shear deformation, the width of instability narrows down. As the aspect ratio (a/b) increases the width of instability zone becomes wider and this influence is more in the case of composite plate as compared to isotropic plate. The boundary conditions of the plate have a significant influence on the dynamic instability regions. The width of the instability zone increases with the decrease in end restraint and increases with the increase of static and dynamic loads.
Acknowledgements The authors sincerely thank the reviewers for their comments which have helped in improving the manuscript. Appendix A Nonlinear governing partial differential equations of cross-ply composite plate in displacement variables are A11 uo,xx þ A66 uo,yy þðA12 þA66 Þvo,xy þðA11 wo,xx þ A66 wo,yy Þwo,x þðA12 þ A66 Þwo,y wo,xy ¼ 0
(A-1)
ðA12 þ A66 Þuo,xy þ A66 vo,xx þA22 vo,yy þ ðA66 wo,xx þ A22 wo,yy Þwo,y þ ðA12 þ A66 Þðwo,x wo,xy Þ ¼ 0
(A-2)
o
o
o
D11 wo,xxxx 2D12 wo,xxyy D22 wo,yyyy 4D66 wo,xxyy þE11 f1,xxx þE12 ðf1,xyy þ f2,xxy Þ o o o þ E22 f2,yyy þ 2E66 ðf1,xyy þ f2,xxy Þ þ wo,xx ðA11 p1 þ A12 q1 Þ þ wo,x ðA11 p1,x þA12 q1,x Þ o o o nxx w,xx nxx,x w,x þ 2w,xy ðA66 r1 Þ þwo,x ðA66 r1,y Þ þ wo,y ðA66 r1,x Þ2nxy wo,xy nxy,y wo,x nxy,x wo,y þwo,yy ðA12 p1 þ A22 q1 Þ þ wo,y ðA12 p1,y þ A22 q1,y Þnyy wo,yy nyy,y wo,y ¼ 0
(A-3)
o
o
o
o
(A-4)
o
o
o
o
(A-5)
E22 wo,yyy ðE12 þ 2E66 Þwo,xxy þF66 f2,xx þ F22 f2,yy þ ðF12 þ F66 Þf1,xy H44 f2 ¼ 0 E11 wo,xxx ðE12 þ 2E66 Þwo,xyy þF11 f1,xx þ F66 f1,yy þ ðF12 þ F66 Þf2,xy H55 f1 ¼ 0 where, p1 ¼ uo,x þ 0:5ðwo,x Þ2 ;
q1 ¼ vo,x þ 0:5ðwo,y Þ2 ;
p2 ¼ wo,xx ; q2 ¼ wo,yy ; o o p3 ¼ f1,x ; q3 ¼ f2,y ; r3
r1 ¼ uo,x þ vo,x þ wo,x wo,y ; r2 ¼ 2wo,xy ; o
o
¼ f1,y þ f2,x ;
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(A-6)
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