non-linear dynamic instability of laminated composite

STD Structural Dynamics

NON-LINEAR DYNAMIC INSTABILITY OF LAMINATED COMPOSITE PLATES ON WINKLER FOUNDATION USING DYNAMIC STIFFNESS METHOD

Hung Q. Huynh1, Hien Luong T. Nguyen2 and Hai Nguyen3 1

Faculty of Civil Engineering, MienTrung University of Civil Engineering, Tuyhoa, Vietnam 2 Faculty of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam 3 Faculty of Applied Sciences, Ho Chi Minh City University of Technology, Vietnam

ABSTRACT The analysis of dynamic instability due to parametric vibration of structures subjected to dynamic loadings is of both theoretical and practical importance. In this study, the dynamic instability and non-linear parametric vibrations of composite laminated rectangular plates supported on elastic foundations, subjected to periodic in-plane loads are theoretically analyzed by using the dynamic stiffness method. The authors present how to establish the dynamic stiffness matrices of rectangular plates subjected to static and periodic in-plane forces based on von Karman’s large deflection plate theory. A set of second-order ordinary differential non-linear equations of extended Mathieu–Hill type with periodic coefficients is formed to determine the regions of dynamic instability and non-linear responses based on Bolotin’s method. The nonlinear temporal response of the system is determined. The influences of various parameters such as static loading factor, dynamic loading factor, aspect ratio, and elastic foundation stiffness on the dynamic stability and non-linear response characteristics are investigated and discussed in this study. Keywords: Dynamic instability, non-linear response, dynamic stiffness method, elastic foundation.

1. INTRODUCTION The dynamic instability of laminated rectangular plates under periodic in-plane loads has been investigated by a number of researchers in recent years. As is well known, when a flat plate sustains a periodic in-plane load of the form N(t)=Ns +Ndcosθt, it may become laterally unstable over certain regions of the (excitating frequency, the in-plane load or the amplitude of system, etc. ) parameter space, and this phenomenon is referred to as parametric or dynamic instability. The dynamic stability of isotropic rectangular plates under various in-plane periodic forces was studied by Bolotin (1964). Nguyen and Ostiguy (1989) considered the influence of the aspect ratio and boundary conditions on the dynamic instability and nonlinear response of rectangular plates theoretically and experimentally. The dynamic 1985

instability behaviour of rectangular plates was studied by Singh and Dey (1992) using energy-based finite difference method. Wu and Shih (2005,2006) investigated the effects of various system parameters on the regions of instability and the non-linear response characteristics of plates using Galerkin’s method. Srivastava et al. (2003, 2010) used the finite element model to study the dynamic instability of stiffened plates subjected to nonuniform harmonic in-plane edge loading. The dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures was determined by Wang and Dawe (2002) using the finite strip method. Ramachandra and Panda (2012), and Chen et al. (2013) have studied the dynamic instability analysis of composite plates used the Galerkin’s method. Noh and Lee (2014) used the finite element analysis to study the dynamic stability of delaminated composite skew plates. For elastic foundation, Saha et al. (1997) used the Galerkin’s method to study the dynamic stability of a rectangular plate on non-homogeneous foundation, subjected to uniform compressive in-plane bi-axial dynamic loads. The dynamic instability of laminated composite plates supported on elastic foundations is investigated by Patel et al. (1999) used the finite element model. In this paper, the problem of dynamic instability and non-linear response of composite laminated rectangular plates supported on elastic foundations is studied by the dynamic stiffness method (DSM). A detailed parametric study to bring out the influences of static loading factor, dynamic loading factor, aspect ratio, and elastic foundation stiffness on the dynamic instability region and non-linear response are investigated and discussed. 2. Governing equations Assume that a laminated composite plates supported on elastic foundation with length a, width b, and thickness h is subjected to uniform harmonic in-plane loads Nx. The coordinates x, y along the in-plane directions and z along the thickness direction is introduced as shown in Figure 1.

Nx = Ns + Nd cosθ t

x, u

O

Nx

z, w y v Nx

K

b

h a h

Nx K

Figure 1: Geometry and load configuration of the plate.

The plate theory used in this analysis may be described as the dynamic analog of von Karman’s large-deflection theory. The two coupled dimensionless differential equations governing the non-linear flexural vibration of the thin composite laminated plates are derived as follows: 1986

L2 ( F ) − L3 (W ) + (W , XX W ,YY − W , 2XY ) r 2 = 0

(1)

L1 (W ) + L3 ( F ) − L( F ,W ) + W ,TT +κW = 0

(2)

in which a comma denotes partial differentiation with respect to the corresponding coordinates, r=a/b is the plate aspect ratio, and where X=x/a, Y=y/b, W=w/h, F=ψ/E2h3, Λ=θ[ρa4/E2h2]1/2, τ =t[E2h2/ρa4]1/2, κ=Ka4/E2h3

(3)

In equations (3), w(x,y,t) is the lateral displacement, ψ is stress function, ρ is mass density per unit volume of plate, θ the frequency of excitation force, Ei are the Young’s moduli along the i principal direction of elasticity, t the time, K is linear stiffness of foundation, and where

(

)

* * * * L1(W ) = [D11 (W ,XXXX ) + 4D16 (W ,XXXY )r + 2 D12 (W ,XXYY )r 2 + 2D66

+ 4D*26 (W ,XYYY )r 3 + D*22 (W ,YYYY )r 4 ]/E2 h 3

(

)

* * L2 ( F ) = [A*22 ( F ,XXXX ) − 2 A*26 ( F ,XXXY )r + 2 A12 ( F ,XXYY )r 2 + A66 * * ( F ,XYYY )r 3 + A11 ( F ,YYYY )r 4 ]E2 h − 2 A16

(

)

(4)

(

)

* * * L3 ( ) = [B*21 (( ),XXXX ) + 2B*26 − B61 + B*22 − 2B66 (( ),XXXY ) r + B11 (( ),XXYY ) r 2

(

)

* * * + 2B16 − B62 (( ),XYYY ) r 3 + B12 (( ),YYYY ) r 4 ] / h

L ( F ,W ) = ( F ,YY W ,XX + F ,XX W ,YY −2F ,XY W ,XY )r 2 In equation (4), Aij, Bij and Dij are membrane stiffness, coupling stiffness and flexural stiffness of the plate, and in which A* = A-1, B* = -A-1B, D* = D-B A-1B. 3. Analysis 3.1. Method of solution An approximate solution of the governing equations (1) and (2) is, in the case of standing flexural waves, sought in the form of a double series in terms of separate space and time variables. The non-dimensional lateral displacement is expressed as

W ( X ,Y ,τ ) = ∑∑Wmn ( τ )X m ( X )Yn ( Y )

(5)

m n

and the dimensionless force function as

F = ∑∑ Fpq Z p ( X )Sq ( Y ) − Y 2 N X / 2

(6)

p q

where Wmn and Fpq are undetermined functions of the dimensionless time τ and where Xm, Yn, Zp and Sq are beam eigenfunctions, and where NX (=NXs+NXdcosΛτ ) is dimensionless in-plane loading. 1987

3.2. Parametric instability analysis Substituting equations (5) and (6) into equations (1) and (2) leads to a system of general non- linear ordinary differential equations for the time functions as follows

W ,ττ +( α0 +α1cosΛτ )W + β W 2 + γ W 3 = 0

(7)

where α0, α1, β and γ being given in the appendix. Equations (7) constitute the final form assumed by the equations of motion. They represent a system of second-order non-linear differential equations with periodic coefficients, which may be considered as extensions of the generalized standard MathieuHill equation. The boundaries of dynamic instability can be constructed by the periodic solution of period T and 2T; where T = 2π/θ. According to Bolotin’s method, the boundaries of the principal instability region with period of 2T are of practical importance and their solution can be achieved in the form of Fourier series

W (τ ) =

∞

k Λτ k Λτ ⎞ ⎛ + bk cos ⎜ ak sin ⎟ 2 2 ⎠ k =1,3,5,... ⎝

∑

(8)

where the coefficients ak and bk are vectors which are independent of time and are defined as the dimensionless amplitude of the system. Substituting Eq. (8) into Eq. (7) and consider the case k = 1, we obtain the following system of equations

⎛ Λ2 3γ + A2 ⎜− 4 ⎝ 4

⎞ ⎛ α1 ⎞ ⎟ + ⎜ α0 ∓ ⎟=0 2 ⎝ ⎠ ⎠

(9)

where the amplitude dynamic of steady-state vibrations is denoted by A=(a2 +b2)1/2. 4. Dynamic stiffness formulations The boundary conditions for generalized displacements vector (Figure 2) are: δ = AC

(10)

where δ=[Yn1 ; φn1 ; Yn2 ; φn2], A=[1 0 1 0 ; 0 r1 0 r2 ; cosh(r1b) sinh(r1b) cos(r2b) sin(r2b); r1sinh(r1b) r1cosh(r1b) -r2sin(r2b) r2cos(r2b)], C=[C1 ; C2 ; C3 ; C4], and elements of the equation being given in the appendix. The boundary conditions for generalized forces vector (Figure 2) are: P = RC

(11)

where P=[QY1 ; MY1 ; QY2 ; MY2], R=[0 t1 0 -t2 ; t3 0 -t4 0 ; -t1sinh(r1b) -t1cosh(r1b) t2sin(r2b) t2cos(r2b); -t3cosh(r1b) -t3sinh(r1b) t4cos(r2b) t4sin(r2b)], and elements of the equation being given in the appendix. Using Eqs. (10) and (11) the dynamic stiffness matrix K (=RA-1) for the plate element can be obtained by eliminating the constant vector C to give: P = Kδ

(12) 1988

Once the DS matrix of a laminate element has been developed, it can be rotated and/or offset if required and thus can be assembled to form the global DS matrix of the final structure. The application of boundary conditions in DSM is also similar to that of FEM.

b Arbitrary B.C.

QY1 , Yn1

QY2 , Yn2

Line node 1

x

MY1, φ n1

MY2 , φ n2

y z

a

Line node 2

h

Arbitrary B.C.

Figure 2: Boundary conditions for displacements and forces for a laminated plate element.

5. Numerical results and discussions In the present study, the material constants for laminate composite material are considered as E1/E2=40; E2=6.96 GPa; G12 = G13 = 0.6E2; G23 = 0.5E2; ν12 = 0.25; ρ = 1580 kg/m3; a/b = varied; a/h=50. All the laminates are assumed to be of the same thickness and material properties. In this paper, the rectangular laminated plates have been calculated for [00/900/900/00] with different boundary conditions. Dynamic loading factor - βd

κ=100

0.6 0.5 0.4 0.3 0.2

DSM Patel 1999

0.1 0

0

Dynamic loading factor - βd

b) 0.7

a) 0.7

0.5 0.4 0.3 0.2

DSM Patel 1999

0.1 0

10 20 30 40 50 60 70 Non-dimensional excitation frequency - χ

κ=500

0.6

0

10 20 30 40 50 60 70 Non-dimensional excitation frequency - χ

Fig. 3: The comparison of the instability regions for the simply supported square orthotropic plate with elastic foundation parameter: E1/E2=25, G12/E2=0.5, ν12=0.25. a) κ=100; b) κ=500.

In Figure 3, the ordinate βd (=NXd/Ncr) denotes the dynamic component NXd of the periodic in-plane force normalized to the lowest critical load Ncr, and is called the ratio of dynamic critical loading, while the abscissa χ (=θa2(ρh/D22)1/2), and is called the nondimensional excitation frequencies. The dynamic instability region associated with a simply supported square orthotropic plate as different elastic foundation parameter has been calculated and shown in Figure 3 wherein the comparison with the region obtained 1989

by Patel et al. (1999) using the finite element method is made. These two regions of dynamic instability are closed. These results are found to be in good agreement. b)

1

1 0.8

0.6

0.6

βd

0.8

βd

a)

0.4

βs =0

0.4

βs=0

0.2

βs =0.2 βs =0.5

0.2

βs=0.2 βs =0.5

0

0

20

40

60

0

80

0

20

40

χ

60

80

χ

Figure 4: The effect of static in-plane load NXs on dynamic instability regions for simply supported square cross-ply laminates plates. a) κ=100; b) κ=500.

a)

0.5 0.4

m=1

m=2

0.3 0.2 0.1 0

0

1 2 3 4 Normalized frequency parameter S

Dynamic loading factor βd

Dynamic loading factor βd

Figure 4 shows the effect of varying static preload and the extent of foundation on the regions of dynamic instability. It is observed that the natural frequencies and the critical load increase with the increase of the foundation stiffness. The results reveal that, with a relatively large increase of the foundation stiffness, consequently, augments the possibility of stability the system.

5

b)

0.5 0.4

m=1

0.3

m=2

m=3

m=4

0.2 0.1 0

0

1 2 3 Normalized frequency parameter S

4

Figure 5: The effect of aspect ratio r=(a/b) on the principal instability regions for the clamped rectangular cross-ply laminated plates, βs=0.4, κ=100. a) r=1; b) r=2.

The regions of parametric instability for two different values of the aspect ratio r are illustrated in Figure 5. In these figures, the ordinate βd, while the abscissa S(=θ/2Ωm) denotes the excitation frequency θ normalized to twice the natural frequency Ωm associated with the prevalent buckling mode mc, and is called the frequency parameter. Reference to this figure shows that an increase of the aspect ratio does bring the instability zones closer together. This means that an increase in r has a significant destabilizing effect on the system. Typical stationary frequency-response curves associated with various principal parametric resonances are shown in Figures 6. In the figures, the ordinate A(h), represents the steady state amplitude, corresponding to each natural mode of vibration m, as a function of the plate thickness, while the abscissa θ denotes the exciting frequency (in Hz). In those figures, solid and broken lines represent the stable and unstable solutions, respectively. 1990

3

m=1

a)

m=2

2

1

0 200

400 600 800 1000 Excitation frequency - θ(Hz)

Dynamic amplitude - A(h)

Dynamic amplitude - A(h)

The results illustrated in Figures 6 show that all of the frequency response curves exhibit a right-hand overhang which is typical of a hard spring effect, generally due to large deflections. It has been known that the overhang of the stationary response curves depends on the coefficient of non-linear terms; a positively cubic non-linearity bends the frequency-response curves to the right for hard spring effect while a negative one bends the curves to the left for soft spring influence. In the current work, the right-hand overhang is dependent upon the vibratory mode and for increasing m shows an increasing harder spring effect.

1200

3

b) m=1

m=2

2

1

0 200

400 600 800 1000 Excitation frequency - θ(Hz)

1200

Figure 6: Effect of varying dynamic in-plane load NXd on the parametric response of clamped square cross-ply laminated plates when βs=0.4 and κ=100. a) βd=0.2; a) βd=0.5.

6. Concluding remarks Based on the dynamic analog of the von Karman’s equations and extended dynamic stiffness method, the region of dynamic instability and non-linear parametric vibrations is determined for composite laminated plates supported on elastic foundations. The rectangular plates with simply supported and clamped boundary conditions are considered in this study. Various system parameters also play important roles in determining the instability and response characteristics of rectangular plates. An increase of the aspect ratio significantly destabilizes the system; elongated plates are much more susceptible to various parametric resonances than square plates. An increase in the static or dynamic component of the inplane load is usually destabilizing; it can render a stable plate unstable. An increase in the foundation stiffness make the system more sensitive to periodic forces. The origin of the principal dynamic instability region shifts to higher excitation frequencies with the increase in the value of foundation stiffness of the plates. 7. REFERENCES Bolotin, V.V.. (1964). The dynamic stability of elastic system, Holden-Day, San Francisco. Chen, W.R., et al. (2013). “Stability of parametric vibrations of laminated composite plates.” Applied Mathematics and Computation, 223, 127–138. Noh, M. H., and Lee, S. Y. (2014). “Dynamic instability of delaminated composite skew plates subjected to combined static and dynamic loads based on HSDT.” Composites: Part B, 58, 113– 121.

1991

Patel, B. P., et al. (1999). “Dynamic instability of layered anisotropic composite plates on elastic foundations.” Engineering Structures, 21, 988-995. Ramachandra, L.S., and Panda, S.K. (2012). “Dynamic instability of composite plates subjected to non-uniform in-plane loads.” J. of Sound and Vibration, 331, 53-65. Saha, K. N., et al. (1997). “Dynamic stability of a rectangular plate on non-homogeneous Winkler foundation.” Computers & Structures, 63(6), 1213-1222. Singh, J.P., and Dey, S.S. (1992). “Parametric instability of rectangular plates by the energy based finite difference method.” Comput. Methods in Appl. Mech. and Eng., 97, 1 – 21. Srivastava, A.K.L., et al. (2003). “Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading.” J. of Sound and Vib., 262,1171-1189. Srivastava, A.K.L., et al. (2010). “Dynamic stability of stiffened plates with cutout subjected to harmonic in-plane partial edge loading.” Int. J. of Crashworthiness, 10(4), 403-417. Wang, S., and Dawe, D.J. (2002). “Dynamic instability of composite laminated rectangular plates and prismatic plate structures.” Comput. methods appl. Mech. and eng, 191, 1791–1826. Wu, G.Y., and Shih, Y.S. (2005). “Dynamic instability of rectangular plate with an edge crack.” Computers & Structures, 84, 1 -10. Wu, G.Y., and Shih, Y.S. (2006). “Analysis of dynamic instability for arbitrarily laminated skew plates.” J. of Sound and Vibration, 292, 315-340.

APPENDIX Elements of the equation (9) are

(

)

α0 = [X mYn ] −1 L1 ( X mYn ) + [L2 ( Z p Sq )] −1 L3 ( Z p Sq )L3 ( X mYn )+N Xs r 2 ( X m ,XX Yn ) + κ ( X mYn ) ;α1 = N Xd r 2 ( X γ = [X mYn ] −1 [L2 ( Z p Sq )] −1 ( Z p Sq ,YY X m ,XX Yn + Z p ,XX Sq X mYn ,YY −2Z p ,X Sq ,Y X m ,X Yn ,Y ) (X m ,XX Yn X rYs ,YY − X m ,X Yn ,Y X r ,X Ys ,Y )r 4

β = - [X mYn ] −1 [L2 ( Z p Sq )] −1 L3 ( Z p Sq )(( X m ,XX Yn X rYs ,YY − X m ,X Yn ,Y X r ,X Ys ,Y )r 2 + L3 ( X mYn )( Z p Sq ,YY X m ,XX Yn + Z p ,XX Sq X mYn ,YY −2Z p ,X Sq ,Y X m ,X Yn ,Y )r 2 )

Elements of the equation (10) are 1 r1 = − ( − J 2 + J 22 − 4J 1 J 3 );r2 = 2J 1

1 ( J 2 + J 22 − 4J 1 J 3 ); 2J 1

D12 + 2D66 r 4 D22 1 J 2 = ∫ 2r ( )X m ,XX X m dX ; J 1 = X X dX ; 3 3 ∫ m m E h E h 0 0 2 2 1

1⎛

2

N D ⎛ J 3 = ∫ ⎜ 113 X m ,XXXX + ⎜ N Xs ∓ Xd ⎜ 2 ⎝ 0 ⎝ E2 h Elements of the equation (11) are

⎞ ⎛ Λ 2 3γ A2 ⎞ ⎞ 2 κ r X , X X + + − + ⎟ m ⎟⎟ X m dX . m XX m ⎜ ⎟ 4 4 ⎠ ⎝ ⎠ ⎠

t1 = g1r13 + g 2 r1 ; t2 = g1r23 − g 2 r2 ; t3 = g1r12 + g3 ; t4 = g1r22 − g3 . g1 =

D12 + 4D66 1 D22 1 D12 1 X X dX ;g = X , X dX ;g = 2 3 ∫ m m ∫ m XX m ∫ X m ,XX X m dX E2 h 3 0 r 2 E2 h 3 0 r 2 E2 h 3 0 1992