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Nonlinear dynamic analysis of a laminated hybrid composite plate subjected to time-dependent external pulses
Acta Mechanica Solida Sinica, Vol. 25, No. 6, December, 2012 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES Mehmet S¸enyer1
Zafer Kazancı2
1
( Turkish Air Force Academy, Aeronautics and Space Technologies Institute, 34149, Ye¸silyurt, Istanbul, Turkey) (2 Turkish Air Force Academy, Aerospace Engineering Department, 34149, Ye¸silyurt, Istanbul, Turkey)
Received 6 December 2010, revision received 26 December 2011
ABSTRACT Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von K´ arm´ an large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.
KEY WORDS nonlinear vibration, hybrid composite plate, Newmark method, dynamic loads
I. INTRODUCTION In recent years, requirements for high performance materials in aircraft and aerospace structures and development of numerical methods to solve structural problems have made it imperative to rapidly develop the technology of composite materials. With this improvement advanced laminated composites are playing their roles in various engineering applications such as space station structures, aircraft, automobiles and submarines. Conventional applications allow using the same composite materials in the structure of laminated plates. Hybrid composite laminated plates which should be exposed to timedependent pulses, such as blast loads occurring from fuel and nuclear explosions, gust and sonic boom pulses provide more advantages to the minimization of weight, cost and displacement. A fairly large amount of the nonlinear responses of laminated composite plates subjected to the exponentially decaying (blast load) pulses have been reported in several studies[1–12] . In addition, Dobyns[13] investigated the static and dynamic analysis of orthotropic plates with the transient loading conditions taken into consideration. Serge Abrate[14] examined the transient response of beams, plates, and shells to impulsive loads using the modal expansion technique for pulse shapes typically observed during impact and explosion. Kazancı and Mecitoˇglu[15] studied the nonlinear damped vibration of
Corresponding author. E-mail: [email protected]
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a laminated composite plate subjected to blast load. Birman and Bert[16] considered the response of simply supported anti-symmetrically laminated angle-ply plates to explosive blast loading. Kazancı[17] investigated the dynamic response of orthotropic sandwich composite plates impacted by time-dependent pulses. An analytical tool for the nonlinear large deflection response of simply supported laminated plates is presented by Tsouvalis and Papazoglou[18]. While the previous studies mentioned above are mainly on laminated composite plates, several are related to hybrid plates laminated with various materials as reported in the literature. Walker and Smith[19] used a methodology for using genetic algorithms with the finite element method to minimize a weighted sum of the mass and deflection of fiber-reinforced structures subjected to a uniformly distributed load over the entire surface. Kuan et al.[20] observed the properties of composite materials based on two types of self-reinforced polypropylene (SRPP) and a glass fiber-reinforced polypropylene under quasi-static and dynamic loading conditions. Sayer et al.[21] investigated the impact behavior of hybrid composite plates. Chen and Fung[22] derived non-linear partial differential equations based on the Reissner–Mindlin plate theory for large amplitude vibration of a hybrid composite plate in a general non-uniform initial stress state. Ibrahim et al.[23] studied a nonlinear finite element model for the aero-thermal buckling and free vibration behavior of an imperfect SMAHC panel under combined thermal and aerodynamic loads. Rahul et al.[24] used the genetic algorithm (GA) to solve an optimization problem related to the design of symmetric laminated composite plate subjected to transverse impact. Chen et al.[25] investigated the natural frequency and buckling loads of hybrid Al/GFRP/Al plates in a general non-uniform initial stress state. Chen et al.[26] tackled numerical solutions to hybrid laminate plates in a general non-uniform initial stress state based on various plate theories. In this study, the behaviors of hybrid composite laminated plates under the action of different types of time-dependent lateral loads are analyzed. The geometric nonlinearity effects are taken into account with the von K´ arm´ an large deflection theory of thin plates. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate displacement functions are assumed for the space domain by considering the nonlinear static deformation obtained using ANSYS software. They are substituted into the equations of motion and then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The Newmark Method is applied to solve the system of coupled nonlinear equations and a program code in MATLAB software is written for use in calculation. The transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thickness. The results of approximately numerical analysis are obtained and discussions are made about the displacement-time histories.
II. EQUATIONS OF MOTION In this section, a mathematical model for the laminated hybrid composite plate subjected to dynamic loads is presented. A rectangular plate with length a, width b, and thickness h is depicted in Fig.1. The Cartesian axes are used in the derivation. The displacement functions of a thin plate can be expanded in the series. If the first few terms in the series are taken, the displacement functions can be approximated as follows: ∂w0 ∂x 0 ∂w v = v0 − z ∂y 0 w=w
u = u0 − z
(1a) (1b) (1c)
where u, v and w are the displacement components in the x, y and z directions. ( )0 indicates the displacement components of reference surface. The strain-displacement relations for the von K´arm´ an plate can be written as εx = ε0x + zκx εy = ε0y + zκy
(2a) (2b)
εxy = ε0xy + zκxy
(2c)
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Fig. 1. Laminated hybrid composite plate.
where ε0x =
1 ∂u0 + ∂x 2
∂w0 ∂x
2
2 1 ∂w0 ∂v 0 + = ∂y 2 ∂y 0 ∂v 0 ∂w0 ∂w0 ∂u + + ε0xy = ∂y ∂x ∂x ∂y ε0y
(3a) (3b) (3c)
and ∂ 2 w0 (4a) ∂x2 ∂ 2 w0 (4b) κy = − ∂y 2 ∂ 2w0 (4c) κxy = −2 ∂x∂y The effective elastic constants are used for defining the constitutive model of the laminated composite. The constitutive equations can then be expressed as ⎧ ⎫ ⎡ ⎫ ⎤⎧ ¯ 11 Q ¯ 12 Q ¯ 16 ⎨ εx ⎬ Q ⎨ σx ⎬ ¯ 12 Q ¯ 22 Q ¯ 26 ⎦ εy σy = ⎣Q (5) ⎩ ⎭ ¯ ¯ ¯ 66 ⎩ εxy ⎭ σxy Q16 Q26 Q ¯ ij ’s are the elastic constants for a laminated composite. where σx , σy and σxy are stress components, Q The force and moment components of the plate can be written as
0 ε [A] [B] {N } = (6) [B] [D] {M } {κ} κx = −
The coefficients in the matrices are Aij = Bij = Dij =
n
¯ ij )k (hk − hk−1 ) (Q
k=1 n
1 2
1 3
k=1 n k=1
(7a)
¯ ij )k (h2 − h2 ) (Q k k−1
(7b)
¯ ij )k (h3 − h3 ) (Q k k−1
(7c)
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where Aij , Bij , and Dij are the extensional, coupling and bending matrices, respectively. Using the constitutive equations and the strain-displacement relations in the virtual work and applying the variational principles, nonlinear dynamic equations of a laminated composite plate can be obtained in terms of mid-plane displacements as follows: L11 u0 + L12 v 0 + L13 w0 + N1 (w0 ) + m¨ ¯ u0 − qx = 0
(8a)
L21 u0 + L22 v 0 + L23 w0 + N2 (w0 ) + m¨ ¯ v 0 − qy = 0
(8b)
L31 u0 + L32 v 0 + L33 w0 + N3 (u0 , v 0 , w0 ) + m ¯w ¨ 0 − qz = 0
(8c)
¯ is the mass of unit area of where Lij and Ni denote linear and nonlinear operators, respectively. m the mid-plane, qx , qy and qz are the load vectors in the axial direction. The explicit expressions of the operators can be found in Kazancı and Mecitoˇ glu[15] . The boundary conditions are in the following form for a clamped plate: u0 (0, y, t) = u0 (a, y, t) = u0 (x, 0, t) = u0 (x, b, t) = 0 ∂u0 ∂u0 ∂u0 ∂u0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y v 0 (0, y, t) = v 0 (a, y, t) = v 0 (x, 0, t) = v 0 (x, b, t) = 0 ∂v 0 ∂v 0 ∂v 0 ∂v 0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y w0 (0, y, t) = w0 (a, y, t) = w0 (x, 0, t) = w0 (x, b, t) = 0 ∂w0 ∂w0 ∂w0 ∂w0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y and initial conditions are given by u0 (x, y, 0) = 0, u˙ 0 (x, y, 0) = 0,
v 0 (x, y, 0) = 0, v˙ 0 (x, y, 0) = 0,
w0 (x, y, 0) = 0 w˙ 0 (x, y, 0) = 0
III. METHODS OF SOLUTION 3.1. Finite Element Solution The laminated composite plate is analyzed using ANSYS finite element software. The plate is discretized by using the eight-node laminated shell elements (SHELL 91) which have geometric nonlinear capability. Four hundred elements are used for the discretization. Large deformation static analyses and transient response analyses are performed for the laminated composite plate under the blast. Transient response analysis is based on the Newmark Method. 3.2. Present Solution (Approximate-Numerical) The equations of motion given as Eqs.(8) can be reduced to a time domain by choosing some approximation functions for the displacement field and applying the Galerkin method. The couplednonlinear equations in the time domain are solved by using the Newmark Method. The approximation functions are selected so as to satisfy the natural boundary conditions. I J 0 u = Uij (t)φij (x, y) (9a) i=1 j=1
v0 =
K L
Vkl (t)ψkl (x, y)
(9b)
k=1 l=1
w0 =
M N m=1 n=1
Wmn (t)χmn (x, y)
(9c)
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As even the simplest multi-term approximation results in hundreds of integral terms during the application of the Galerkin procedure, it is impractical. Therefore, one term approximation functions for the displacement components are used in this study. As mentioned by Strang[27] , choosing the approximation functions is the crucial point that should be most important for the one-term solutions. The approximation function should closely resemble the first mode of the plate. It can be determined by considering the results of static large deformation analysis of laminated composite plate under the uniform pressure load by using ANSYS software. The approximation functions are determined by examining the finite element results obtained from the static large deformation results as follows: 2πy a 0 2 2 u = U11 (t)x (x − a) x − 1 − cos (10a) 2 b b 2πx y 2 (y − b)2 y − (10b) v 0 = V11 (t) 1 − cos a 2 2πy 2πx 0 1 − cos (10c) w = W11 (t) 1 − cos a b Applying the Galerkin method to the equations of motion given in Eqs.(8), the time-dependent nonlinear differential equations can be obtained: ¨ + a 1 U + a2 V + a3 W + a4 W 2 + a 5 = 0 a0 U
(11a)
b0 V¨ + b1 V + b2 U + b3 W + b4 W 2 + b5 = 0
(11b)
¨ + c1 W + c 2 W 2 + c 3 W 3 + c 4 U + c5 V + c 6 U W + c 7 V W + c 8 = 0 c0 W
(11c)
The coefficients of the equations are given in the Appendix. The dot denotes the derivative with respect to time. The initial conditions can be expressed as U (0) = 0,
V (0) = 0,
W (0) = 0
U˙ (0) = 0,
V˙ (0) = 0,
˙ (0) = 0 W
The nonlinearly coupled equations of motion are solved by using the Newmark Method. We may arrange Eqs.(11) in the matrix format: ¨ + [KL ] {Q} + [KN L ] {Q} = {F } [M ] {Q}
(12)
¨ = {U ¨ V¨ W ¨ }T denote the displacement and acceleration where {Q} = {U V W }T and {Q} vectors, respectively. In Eq(12), [M ], [KL ], and [KN L ] matrices are ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ a1 a2 a3 0 0 a4 W a0 0 0 ⎦ 0 b4 W [M ] = ⎣ 0 b0 0 ⎦ , [KL ] = ⎣ b2 b1 b3 ⎦ , [KN L ] = ⎣ 0 (13a) 2 0 0 c0 c6 W c 7 W c 2 W + c 3 W c4 c5 c1 and {F } vector is {F } =
a5 b 5 c8
T
(13b)
The Newmark Method uses forward difference expansions in the time interval Δt, in which it is assumed that ¨ n} + δQ ¨ n+1 ]Δt {Q˙ n+1 } = {Q˙ n } + [(1 − δ){Q
1 ¨ n } + α{Q ¨ n+1 } (Δt)2 − α {Q {Qn+1 } = {Qn } + {Q˙ n }Δt + 2
(14a) (14b)
Here α and δ are Newmark integration parameters. Since the primary aim is the computation of displacement {Qn+1 }, the governing Eq.(12) is evaluated at time tn+1 as ¨ n+1 } + [KL ] {Qn+1 } + [KN L ] {Qn+1 } = {F } [M ] {Q
(15)
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The solution for the displacement at time tn+1 is obtained by rearranging Eq.(14), such that ¨ n+1 } = k0 ({Qn+1 } − {Qn }) − k2 {Q˙ n } − k3 {Q ¨ n} {Q
(16a)
¨ n } + k7 {Q ¨ n+1 } {Q˙ n+1 } = {Q˙ n } + k6 {Q
(16b)
where k0 = 1/[α(Δt)2 ], k2 = 1/[α(Δt)], k3 = 1/(2α) − 1, k6 = Δt(1 − δ), k7 = δΔt. The equations for ¨ n+1 } are then combined with Eq.(15) to form {Q˙ n+1 } and {Q
¨ n} (k0 [M ] + [KL ] + [KN L ]) {Qn+1 } = {F } + [M ] k0 {Qn } + k2 {Q˙ n } + k3 {Q (17) Once a solution is obtained for {Qn+1 }, the velocities and accelerations are updated as described in Eqs.(16). Finally, if the matrices and vectors given in Eqs.(13) are substituted into Eq.(17), the equations of motion are reduced to A1 U n+1 + A2 V n+1 + A3 W n+1 = A4 B1 U n+1 + B2 V n+1 + B3 W n+1 = B4
(18)
C1 U n+1 + C2 V n+1 + C3 W n+1 = C4 The coefficients in the equations are given in the Appendix. Linear equations in Eq.(18) are solved by the Gaussian elimination algorithm to calculate the displacement {Q} = {U V W }T .
IV. DYNAMIC LOADINGS Advanced supersonic/hypersonic flight vehicles are subjected to several dynamic loadings such as gust, sonic boom pulses and nuclear explosion. The analysis and design of structures subjected to blast loads require a detailed understanding of blast phenomena and the dynamic response of structures. In this section, various types of time-dependent external blast pulses are summarized and demonstrated in Figs.2-5.
Fig. 3. Sine load. Fig. 2. Exponentially decaying (blast) load.
4.1. Blast Loading If the blast source is distant enough from the plate, the blast pressure can be described in terms of the Friedl¨ ander exponential decay equation as[28] t e−αt/tp P (t) = Pm 1 − (19) tp where the negative phase of the blast is included. In this equation, Pm is the peak blast pressure, tp is positive phase duration, and α is waveform parameter.
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Fig. 4. Rectangular loading.
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Fig. 5. Triangular loading.
4.2. Sine Loading The sine pulse can be represented as ⎧ πt ⎨ Pm sin (0 < t < rtp ) P (t) = rtp ⎩ 0 (t < 0 and t > rtp )
(20)
where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. 4.3. Rectangular Loading For the rectangular pulse, the load function is given as Pm (0 < t < rtp ) P (t) = 0 (t < 0 and t > rtp )
(21)
where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. 4.4. Triangular Loading Sonic boom effect could be modeled as an N-shaped pressure pulse and may be described as ⎧ t ⎨ (0 < t < rtp ) Pm 1 − (22) P (t) = tp ⎩ 0 (t < 0 and t > rtp ) where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. For r = 1 the sonic boom transformed into a triangular pulse (air-blast), for r = 2 the pulse corresponds to a symmetric N-shaped pulse (sonic boom), and for r = 2 the pulse corresponds to a non-symmetric N-pulse.
V. NUMERICAL RESULTS The solution of nonlinear-coupled equations given by Eqs.(11) is obtained by writing a MATLAB computer code based on the Newmark Method. First of all, the results obtained from the present method are validated with ANSYS results. The plate is discretized using the eight-node laminated shell elements (SHELL 91) with the geometric nonlinear capability. A six-layered hybrid composite laminated plate is chosen for comparison. The ply material properties used in the validation are given in Table 1. Unless otherwise stated, every ply has the same thickness which is 0.4×10−3 m. The stacking sequence of lamination is demonstrated in Fig.1. Fiber orientation is taken as [45/ − 45/45/ − 45/45/ − 45]. The dimensions of the plate are a = 0.3 m, b = 0.3 m, and h = 2.4 × 10−3 m. Friedl¨ander exponential decay equation given in Eq.(19) is used as the blast load. The maximum blast pressure Pm is taken to be 30 kPa. Parameters are chosen as α = 0.35 and tp = 0.0018 s. Comparison of the results for the non-dimensional deflection at the center of six-layered square plate with ANSYS is shown in Fig.6. The numerical results presented here have considered different loading conditions. In Fig.7, the dimensionless central deflections of the plate subjected to exponentially decaying (blast load) for different peak values are shown. 15 kPa, 30 kPa and 45 kPa are selected as blast peak pressure Pm for the clamped
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Table 1. Ply material properties
Ply material Woven-glass/epoxy Kevlar/epoxy(Aramid) E-glass/epoxy
E1 (GPa) 29.7 87 39
E2 (GPa) 29.7 5.5 8.6
Fig. 6. Comparison of the dimensionless deflection timehistories of the plate center.
G12 (GPa) 5.3 2.2 3.8
γ12 (Poisson’s ratio) 0.17 0.34 0.28
Density (kg/m3 ) 2200 1380 2100
Fig. 7. Time history of dimensionless (w/h) central deflection under blast load.
plate, respectively. Increasing the peak pressure value increases the amplitude and frequency, as expected. Figure 8 displays the effect of aspect ratio on dimensionless central deflection that corresponds to a blast load for selected aspect ratios. The mid-plane area of the plate is preserved as a constant value for all the aspect ratios. It was demonstrated that decreasing the aspect ratio results in lower amplitude and higher frequencies. In Fig.9, the effects of fiber orientation are depicted. Three lamination schemes are investigated. However, there is no significant discrepancy as seen in the chosen orientation and loading condition.
Fig. 8. Effect of aspect ratio on dimensionless (w/h) central deflection under blast load.
Fig. 9. Effect of fiber-orientation on dimensionless (w/h) central deflection under blast load.
In Fig.10, the dynamic response of the plate by changing the plate thickness is presented. The dimensionless center deflections of the plates for various ply thicknesses (0.3, 0.4 and 0.5 mm) are investigated. Decreasing the plate thickness results in higher deflection and frequencies. The frequency is significantly affected by the layer thickness. Figure 11 displays the effect of the shock pulse length factor (r) on the dynamic response of hybrid plate subjected to a sine load. The shape of the pulse is determined by the change of r as can be seen in Fig.4. As stated above, tp is positive phase duration which defines the extension in time of the forced motion in this case. Maximum deflection amplitudes are nearly the same for the first peak
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Fig. 10. Effect of thickness on dimensionless (w/h) central deflection under blast load.
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Fig. 11. Time history of dimensionless (w/h) central deflection under sine load.
values. Meanwhile, there is a special case for r = 2. The deflection amplitude for r = 2 is significantly lower than the others since the strong blast effect ends when the amplitude is close to the equilibrium condition. Thus, the hybrid plate oscillation is close to the equilibrium condition when the strong blast effect ends. In Fig.12, the dynamic response to a rectangular pulse for different r values is highlighted. Similarly, as previously mentioned for Fig.11, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the deflection amplitude. In the forced motion regime, the deflection amplitudes are almost the same for the first peak values. On the other hand, the results indicate that in the free motion regime, the rectangular pulse yields is maximum for r = 2 and minimum for r = 1.5. The parametric study continues with the effect of r values on its nonlinear response under the action of a triangular load. Triangular pulse includes just positive impulse (air-blast) for r = 1. The pulse corresponds to a symmetric N-shaped pulse (sonic boom) when r = 2, and for r = 3 the pulse corresponds to a non-symmetric N-pulse where the negative impulse is higher than the positive part. As seen from Fig.13, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the free vibration amplitudes. Thus, maximum deflection amplitude is seen for r = 3, while all cases yield nearly the same response until the tp value is reached.
Fig. 12. Time history of dimensionless (w/h) central deflection under rectangular load.
Fig. 13. Time history of dimensionless (w/h) central deflection under triangular load.
A comparison of deflection time histories for different type loads is demonstrated in Fig.14, from which it is seen that, having smaller deflection for the same peak pressure values, the blast load in general generates less intense oscillating motion. As is well known, the area below the graph demonstrates the impulsive load effect. Needless to say, the sizes of the areas from highest to lowest are rectangular, sine, triangular and blast pulses at the same positive phase duration time tp . Thus, the maximum deflection
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Fig. 14. Comparison of deflection histories for several dynamic loads (Pm = 30 kPa).
peak value is obtained by the rectangular pulse loading, whereas the minimum one is obtained by the blast pulse when the maximum peak pressure value (Pm ) and pulse duration are the same. However, after the strong blast effect, rectangular and triangular pulse oscillations are nearly the same.
VI. CONCLUSIONS In this study, the equations of motion of a laminated hybrid composite plate are derived. In-plane stiffness and inertia effects are considered when formulating the dynamic response of the laminated hybrid composite plate subjected to time-dependent pulses. The geometric nonlinearity effects are taken into account by using the von K´ arm´ an large deflection theory of thin plates. The Galerkin Method is used to obtain a set of the nonlinear differential equations in the time domain. The Newmark Method is used to solve the system of coupled nonlinear equations. An analytic tool was presented for the nonlinear dynamic behavior of a fully-clamped hybrid laminated composite plate under several dynamic loads including in-plane effects. The implications of peak pressure value, aspect ratio, fiber orientation and thickness for the dynamic response of a hybrid laminated composite plate are discussed. It is demonstrated that increasing the peak pressure value increases the amplitude and frequency. Decreasing the aspect ratio results in lower amplitude and higher frequencies while decreasing the plate thickness results in higher deflection and frequencies. The frequencies are significantly affected by the layer thickness. If the larger deflections as a result of thinner plate are considered, it can be deduced that the nonlinear and in-plane stiffness effects will increase in the case of thinner plate and it can result in a higher vibration frequency than that of the thicker plate. However, the fiber orientations have negligible effect on the vibration characteristics. Both the geometrical nonlinearities and the effects of in-plane deformations have been implemented in the structural model. The results obtained can be extended without any difficulty to the determination of the time histories of strain and stress components of the various points of the structure. A parametric study was carried out considering the blast, sine, rectangular and triangular loading. It is seen that, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the free vibration amplitudes for different shock pulse length factor (r) values. However, blast load in general generates less intense oscillating motion compared to other different types of loads. Maximum deflection peak value is obtained by the rectangular pulse loading, whereas the minimum one is generated by the blast pulse when the maximum peak pressure value (Pm ) and pulse duration are the same. These results can be very important in designing structures which are subjected to dynamic loads of known and approximately fixed duration. Different material properties, damping effects and other boundary conditions should be taken into account by using the same method. Future studies may be devoted to these topics.
References [1] Chandrasekharappa,G. and Srirangarajan,H.R., Nonlinear response of elastic plates to pulse excitations. Computers & Structures, 1987, 27: 373-378.
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[2] Chandrasekharappa,G. and Srirangarajan,H.R., Parametric study of nonlinear static and dynamic response of composite rectangular plates. Mechanics Research Communications, 1988, 15(3): 177-188. [3] Kazancı,Z., Mecitoˇ glu,Z. and Hacıoˇ glu,A., Effect of in-plane stiffnesses and inertias on dynamic behavior of a laminated composite plate under blast load. In: Proceedings of 9th Biennial International Conference of the Aerospace Division on Engineering, Construction and Operations in Challenging Environments: Earth and Space 2004, 2004, 1: 484-491. [4] T¨ urkmen,H.S. and Mecitoˇ glu,Z., Nonlinear structural response of laminated composite plates subjected to blast loading. AIAA Journal, 1999, 37(12): 1639-1647. [5] T¨ urkmen,H.S. and Mecitoˇ glu,Z., Dynamic response of a stiffened laminated composite plate subjected to blast load. Journal of Sound and Vibration, 1999, 221(3): 371-389. [6] Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load. Journal of Sound and Vibration, 2008, 317(3-5): 883-897. [7] Ba¸s,A., Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear response of a sandwich plate subjected to blast load. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition-2007, Vol 1Advances in Aerospace Technology, 2008, 1: 169-177. [8] Dogan,V. Dynamic multimode response of composite plates to sonic boom and blast loadings. Journal of Aircraft, 2008, 45(2): 498-507. [9] Librescu,L. and Nosier,A., Response of laminated composite flat panels to sonic boom and explosive blast loadings. AIAA Journal, 1990, 28(2): 345-352. [10] Hause,T. and Librescu,L., Dynamic response of anisotropic sandwich flat panels to explosive pressure pulses. International Journal of Impact Engineering, 2005, 31: 607-628. [11] Hause,T. and Librescu,L., Dynamic response of doubly-curved anisotropic sandwich panels impacted by blast loadings. International Journal of Solids and Structures, 2007, 43: 6678-6700. [12] Kazancı,Z., Nonlinear transient response of a laminated composite plate under time-dependent pulses. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies-RAST2009, 2009, 1: 125-130. [13] Dobyns,A.L., Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA Journal, 1981, 19: 642-650. [14] Abrate,S., Transient response of beams, plates, and shells to impulsive loads. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2007, 2008, 9: 107-116. [15] Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear damped vibrations of a laminated composite plate subjected to blast load. AIAA Journal, 2006, 44(9): 2002-2008. [16] Birman,V. and Bert,C.W., Behaviour of laminated plates subjected to conventional blast. International Journal of Impact Engineering 1987, 6: 145-155. [17] Kazancı,Z., Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses. International Journal of Non-Linear Mechanics, 2011, 46(5): 807-817. [18] Tsouvalis,N.G. and Papazoglou,V.J., Large deflection dynamic response of composite laminated plates under lateral loads. Marine Structures, 1996, 9: 825-848. [19] Walker,M. and Smith,R.E., A technique for the multiobjective optimisation of laminated composite structures using genetic algorithms and finite element analysis. Composite Structures, 2003, 62: 123-128. [20] Kuan,H.T., Cantwell,W. and Akil,H.M., The mechanical properties of hybrid composites based on selfreinforced polypropylene. Malaysian Polymer Journal, 2009, 4(2): 71-80. [21] Sayer,M., Bekta¸s,N.B. and Sayman,O., An experimental investigation on the impact behavior of hybrid composite plates. Composite Structures, 2010, 92: 1256-1262. [22] Chen,C.S. and Fung,C.P., Non-linear vibration of initially stressed hybrid composite plates. Journal of Sound and Vibration, 2004, 274: 1013-1029. [23] Ibrahim,H.H., Yoo,H.H. and Lee,K.S., Aero-thermo-mechanical characteristics of imperfect shape memory alloy hybrid composite panels. Journal of Sound and Vibration, 2009, 325: 583-596. [24] Rahul,Sandeep,G., Chakraborty,D. and Dutta,A., Multi-objective optimization of hybrid laminates subjected to transverse impact. Composite Structures, 2006, 73: 360-369. [25] Chen,C.S., Fung,C.P. and Chien,R.D., Analysis of an initially stressed hybrid plate based on a higher-order theory. Composite Structures, 2006, 74: 343-353. [26] Chen,C.S., Fung,C.P. and Yang,J.G., Assessment of plate theories for initially stressed hybrid laminated plates. Composite Structures, 2009, 88: 195-201. [27] Strang,G., Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. [28] Gupta,A.D., Gregory,F.H., Bitting,R.L. and Bhattacharya,S., Dynamic analysis of an explosively loaded hinged rectangular plate. Computers and Structures, 1987, 26: 339-344.
APPENDIX
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Coefficients in the time-dependent nonlinear differential equations (All edges clamped): a0 =
a11 b m, ¯ 18480
a11 b d1 , 18480
a2 =
a5 (A12 + A66 )b5 (−15 + π 2 )2 , 4π 8
a4 =
5a3 [b2 A22 (15 − 4π 2 ) + 3(A12 − A66 )a2 (63 − 4π 2 )] , 64bπ 2
b0 =
ab11 m, ¯ 18480
b2 =
a5 (A12 + A66 )b5 (−15 + π 2 )2 , 4π 8
b4 =
5b3 [a2 A22 (15 − 4π 2 ) + 3(A12 − A66 )b2 (63 − 4π 2 )] , 64aπ 2
c0 =
9ab m, ¯ 4
c2 =
24π 4 (B12 − B66 ) , ab
a =
b =
c =
a1 =
a11 b d2 , 18480
9ab d3 , 4
c3 =
a3 = −
b1 =
c1 =
a9 (33 A11 b2 + a2 A66 π 2 ) 13860b a3 [3b2 B11 + a2 (B12 + 2B66 )](−15 + π 2 ) bπ 2 a5 = −abqx
b9 (33a2 A22 + A66 b2 π 2 ) 13860a
b3 = −
b3 [3a2 B22 + b2 (B12 + 2B66 )](−15 + π 2 ) aπ 2 b5 = −abqy
4π 4 [3b4 D11 + 3a4 D22 + 2a2 b2 (D12 + 2D66 )] a3 b 3
20π 4 [21a4 A22 + 10a2 (3A12 + 4A66 )b2 + 21A11 b4 ] 32a3 b3
a3 [3b2 B11 + a2 (B12 + 2B66 )](−15 + π 2 ) b3 [3a2 B22 + b2 (B12 + 2B66 )](−15 + π 2 ) , c = − 5 bπ 2 aπ 2 5a3 A11 b2 (15 − 4π 2 ) − 3a2 (A12 − A66 )(−63 + 4π 2 ) c6 = 32bπ 2 5b3 A22 a2 (15 − 4π 2 ) − 3b2 (A12 − A66 )(−63 + 4π 2 ) c7 = 32aπ 2
c4 = −
c8 = −abP (t) Coefficients in the Newmark time integration equations: A1 = k0 + a1 , B1 = b2 ,
A2 = a2 ,
B2 = k0 b0 + b1 ,
C1 = c4 + c6 W n ,
A3 = a3 + a4 W n , B3 = b3 + b4 W n ,
C2 = c5 + c7 W n ,
¨n A4 = a5 + a0 k0 U n + a0 k2 U˙ n + a0 k3 U B4 = b5 + b0 k0 V n + b0 k2 V˙ n + b0 k3 V¨ n
C3 = k0 c0 + c1 + c2 W n + c3 (W n )
˙ n + c0 k3 W ¨n C4 = c8 + c0 k0 W n + c0 k2 W
2
ISSN 0894-9166
NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES Mehmet S¸enyer1
Zafer Kazancı2
1
( Turkish Air Force Academy, Aeronautics and Space Technologies Institute, 34149, Ye¸silyurt, Istanbul, Turkey) (2 Turkish Air Force Academy, Aerospace Engineering Department, 34149, Ye¸silyurt, Istanbul, Turkey)
Received 6 December 2010, revision received 26 December 2011
ABSTRACT Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von K´ arm´ an large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.
KEY WORDS nonlinear vibration, hybrid composite plate, Newmark method, dynamic loads
I. INTRODUCTION In recent years, requirements for high performance materials in aircraft and aerospace structures and development of numerical methods to solve structural problems have made it imperative to rapidly develop the technology of composite materials. With this improvement advanced laminated composites are playing their roles in various engineering applications such as space station structures, aircraft, automobiles and submarines. Conventional applications allow using the same composite materials in the structure of laminated plates. Hybrid composite laminated plates which should be exposed to timedependent pulses, such as blast loads occurring from fuel and nuclear explosions, gust and sonic boom pulses provide more advantages to the minimization of weight, cost and displacement. A fairly large amount of the nonlinear responses of laminated composite plates subjected to the exponentially decaying (blast load) pulses have been reported in several studies[1–12] . In addition, Dobyns[13] investigated the static and dynamic analysis of orthotropic plates with the transient loading conditions taken into consideration. Serge Abrate[14] examined the transient response of beams, plates, and shells to impulsive loads using the modal expansion technique for pulse shapes typically observed during impact and explosion. Kazancı and Mecitoˇglu[15] studied the nonlinear damped vibration of
Corresponding author. E-mail: [email protected]
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a laminated composite plate subjected to blast load. Birman and Bert[16] considered the response of simply supported anti-symmetrically laminated angle-ply plates to explosive blast loading. Kazancı[17] investigated the dynamic response of orthotropic sandwich composite plates impacted by time-dependent pulses. An analytical tool for the nonlinear large deflection response of simply supported laminated plates is presented by Tsouvalis and Papazoglou[18]. While the previous studies mentioned above are mainly on laminated composite plates, several are related to hybrid plates laminated with various materials as reported in the literature. Walker and Smith[19] used a methodology for using genetic algorithms with the finite element method to minimize a weighted sum of the mass and deflection of fiber-reinforced structures subjected to a uniformly distributed load over the entire surface. Kuan et al.[20] observed the properties of composite materials based on two types of self-reinforced polypropylene (SRPP) and a glass fiber-reinforced polypropylene under quasi-static and dynamic loading conditions. Sayer et al.[21] investigated the impact behavior of hybrid composite plates. Chen and Fung[22] derived non-linear partial differential equations based on the Reissner–Mindlin plate theory for large amplitude vibration of a hybrid composite plate in a general non-uniform initial stress state. Ibrahim et al.[23] studied a nonlinear finite element model for the aero-thermal buckling and free vibration behavior of an imperfect SMAHC panel under combined thermal and aerodynamic loads. Rahul et al.[24] used the genetic algorithm (GA) to solve an optimization problem related to the design of symmetric laminated composite plate subjected to transverse impact. Chen et al.[25] investigated the natural frequency and buckling loads of hybrid Al/GFRP/Al plates in a general non-uniform initial stress state. Chen et al.[26] tackled numerical solutions to hybrid laminate plates in a general non-uniform initial stress state based on various plate theories. In this study, the behaviors of hybrid composite laminated plates under the action of different types of time-dependent lateral loads are analyzed. The geometric nonlinearity effects are taken into account with the von K´ arm´ an large deflection theory of thin plates. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate displacement functions are assumed for the space domain by considering the nonlinear static deformation obtained using ANSYS software. They are substituted into the equations of motion and then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The Newmark Method is applied to solve the system of coupled nonlinear equations and a program code in MATLAB software is written for use in calculation. The transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thickness. The results of approximately numerical analysis are obtained and discussions are made about the displacement-time histories.
II. EQUATIONS OF MOTION In this section, a mathematical model for the laminated hybrid composite plate subjected to dynamic loads is presented. A rectangular plate with length a, width b, and thickness h is depicted in Fig.1. The Cartesian axes are used in the derivation. The displacement functions of a thin plate can be expanded in the series. If the first few terms in the series are taken, the displacement functions can be approximated as follows: ∂w0 ∂x 0 ∂w v = v0 − z ∂y 0 w=w
u = u0 − z
(1a) (1b) (1c)
where u, v and w are the displacement components in the x, y and z directions. ( )0 indicates the displacement components of reference surface. The strain-displacement relations for the von K´arm´ an plate can be written as εx = ε0x + zκx εy = ε0y + zκy
(2a) (2b)
εxy = ε0xy + zκxy
(2c)
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Fig. 1. Laminated hybrid composite plate.
where ε0x =
1 ∂u0 + ∂x 2
∂w0 ∂x
2
2 1 ∂w0 ∂v 0 + = ∂y 2 ∂y 0 ∂v 0 ∂w0 ∂w0 ∂u + + ε0xy = ∂y ∂x ∂x ∂y ε0y
(3a) (3b) (3c)
and ∂ 2 w0 (4a) ∂x2 ∂ 2 w0 (4b) κy = − ∂y 2 ∂ 2w0 (4c) κxy = −2 ∂x∂y The effective elastic constants are used for defining the constitutive model of the laminated composite. The constitutive equations can then be expressed as ⎧ ⎫ ⎡ ⎫ ⎤⎧ ¯ 11 Q ¯ 12 Q ¯ 16 ⎨ εx ⎬ Q ⎨ σx ⎬ ¯ 12 Q ¯ 22 Q ¯ 26 ⎦ εy σy = ⎣Q (5) ⎩ ⎭ ¯ ¯ ¯ 66 ⎩ εxy ⎭ σxy Q16 Q26 Q ¯ ij ’s are the elastic constants for a laminated composite. where σx , σy and σxy are stress components, Q The force and moment components of the plate can be written as
0 ε [A] [B] {N } = (6) [B] [D] {M } {κ} κx = −
The coefficients in the matrices are Aij = Bij = Dij =
n
¯ ij )k (hk − hk−1 ) (Q
k=1 n
1 2
1 3
k=1 n k=1
(7a)
¯ ij )k (h2 − h2 ) (Q k k−1
(7b)
¯ ij )k (h3 − h3 ) (Q k k−1
(7c)
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where Aij , Bij , and Dij are the extensional, coupling and bending matrices, respectively. Using the constitutive equations and the strain-displacement relations in the virtual work and applying the variational principles, nonlinear dynamic equations of a laminated composite plate can be obtained in terms of mid-plane displacements as follows: L11 u0 + L12 v 0 + L13 w0 + N1 (w0 ) + m¨ ¯ u0 − qx = 0
(8a)
L21 u0 + L22 v 0 + L23 w0 + N2 (w0 ) + m¨ ¯ v 0 − qy = 0
(8b)
L31 u0 + L32 v 0 + L33 w0 + N3 (u0 , v 0 , w0 ) + m ¯w ¨ 0 − qz = 0
(8c)
¯ is the mass of unit area of where Lij and Ni denote linear and nonlinear operators, respectively. m the mid-plane, qx , qy and qz are the load vectors in the axial direction. The explicit expressions of the operators can be found in Kazancı and Mecitoˇ glu[15] . The boundary conditions are in the following form for a clamped plate: u0 (0, y, t) = u0 (a, y, t) = u0 (x, 0, t) = u0 (x, b, t) = 0 ∂u0 ∂u0 ∂u0 ∂u0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y v 0 (0, y, t) = v 0 (a, y, t) = v 0 (x, 0, t) = v 0 (x, b, t) = 0 ∂v 0 ∂v 0 ∂v 0 ∂v 0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y w0 (0, y, t) = w0 (a, y, t) = w0 (x, 0, t) = w0 (x, b, t) = 0 ∂w0 ∂w0 ∂w0 ∂w0 (0, y, t) = (a, y, t) = (x, 0, t) = (x, b, t) = 0 ∂x ∂x ∂y ∂y and initial conditions are given by u0 (x, y, 0) = 0, u˙ 0 (x, y, 0) = 0,
v 0 (x, y, 0) = 0, v˙ 0 (x, y, 0) = 0,
w0 (x, y, 0) = 0 w˙ 0 (x, y, 0) = 0
III. METHODS OF SOLUTION 3.1. Finite Element Solution The laminated composite plate is analyzed using ANSYS finite element software. The plate is discretized by using the eight-node laminated shell elements (SHELL 91) which have geometric nonlinear capability. Four hundred elements are used for the discretization. Large deformation static analyses and transient response analyses are performed for the laminated composite plate under the blast. Transient response analysis is based on the Newmark Method. 3.2. Present Solution (Approximate-Numerical) The equations of motion given as Eqs.(8) can be reduced to a time domain by choosing some approximation functions for the displacement field and applying the Galerkin method. The couplednonlinear equations in the time domain are solved by using the Newmark Method. The approximation functions are selected so as to satisfy the natural boundary conditions. I J 0 u = Uij (t)φij (x, y) (9a) i=1 j=1
v0 =
K L
Vkl (t)ψkl (x, y)
(9b)
k=1 l=1
w0 =
M N m=1 n=1
Wmn (t)χmn (x, y)
(9c)
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As even the simplest multi-term approximation results in hundreds of integral terms during the application of the Galerkin procedure, it is impractical. Therefore, one term approximation functions for the displacement components are used in this study. As mentioned by Strang[27] , choosing the approximation functions is the crucial point that should be most important for the one-term solutions. The approximation function should closely resemble the first mode of the plate. It can be determined by considering the results of static large deformation analysis of laminated composite plate under the uniform pressure load by using ANSYS software. The approximation functions are determined by examining the finite element results obtained from the static large deformation results as follows: 2πy a 0 2 2 u = U11 (t)x (x − a) x − 1 − cos (10a) 2 b b 2πx y 2 (y − b)2 y − (10b) v 0 = V11 (t) 1 − cos a 2 2πy 2πx 0 1 − cos (10c) w = W11 (t) 1 − cos a b Applying the Galerkin method to the equations of motion given in Eqs.(8), the time-dependent nonlinear differential equations can be obtained: ¨ + a 1 U + a2 V + a3 W + a4 W 2 + a 5 = 0 a0 U
(11a)
b0 V¨ + b1 V + b2 U + b3 W + b4 W 2 + b5 = 0
(11b)
¨ + c1 W + c 2 W 2 + c 3 W 3 + c 4 U + c5 V + c 6 U W + c 7 V W + c 8 = 0 c0 W
(11c)
The coefficients of the equations are given in the Appendix. The dot denotes the derivative with respect to time. The initial conditions can be expressed as U (0) = 0,
V (0) = 0,
W (0) = 0
U˙ (0) = 0,
V˙ (0) = 0,
˙ (0) = 0 W
The nonlinearly coupled equations of motion are solved by using the Newmark Method. We may arrange Eqs.(11) in the matrix format: ¨ + [KL ] {Q} + [KN L ] {Q} = {F } [M ] {Q}
(12)
¨ = {U ¨ V¨ W ¨ }T denote the displacement and acceleration where {Q} = {U V W }T and {Q} vectors, respectively. In Eq(12), [M ], [KL ], and [KN L ] matrices are ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ a1 a2 a3 0 0 a4 W a0 0 0 ⎦ 0 b4 W [M ] = ⎣ 0 b0 0 ⎦ , [KL ] = ⎣ b2 b1 b3 ⎦ , [KN L ] = ⎣ 0 (13a) 2 0 0 c0 c6 W c 7 W c 2 W + c 3 W c4 c5 c1 and {F } vector is {F } =
a5 b 5 c8
T
(13b)
The Newmark Method uses forward difference expansions in the time interval Δt, in which it is assumed that ¨ n} + δQ ¨ n+1 ]Δt {Q˙ n+1 } = {Q˙ n } + [(1 − δ){Q
1 ¨ n } + α{Q ¨ n+1 } (Δt)2 − α {Q {Qn+1 } = {Qn } + {Q˙ n }Δt + 2
(14a) (14b)
Here α and δ are Newmark integration parameters. Since the primary aim is the computation of displacement {Qn+1 }, the governing Eq.(12) is evaluated at time tn+1 as ¨ n+1 } + [KL ] {Qn+1 } + [KN L ] {Qn+1 } = {F } [M ] {Q
(15)
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The solution for the displacement at time tn+1 is obtained by rearranging Eq.(14), such that ¨ n+1 } = k0 ({Qn+1 } − {Qn }) − k2 {Q˙ n } − k3 {Q ¨ n} {Q
(16a)
¨ n } + k7 {Q ¨ n+1 } {Q˙ n+1 } = {Q˙ n } + k6 {Q
(16b)
where k0 = 1/[α(Δt)2 ], k2 = 1/[α(Δt)], k3 = 1/(2α) − 1, k6 = Δt(1 − δ), k7 = δΔt. The equations for ¨ n+1 } are then combined with Eq.(15) to form {Q˙ n+1 } and {Q
¨ n} (k0 [M ] + [KL ] + [KN L ]) {Qn+1 } = {F } + [M ] k0 {Qn } + k2 {Q˙ n } + k3 {Q (17) Once a solution is obtained for {Qn+1 }, the velocities and accelerations are updated as described in Eqs.(16). Finally, if the matrices and vectors given in Eqs.(13) are substituted into Eq.(17), the equations of motion are reduced to A1 U n+1 + A2 V n+1 + A3 W n+1 = A4 B1 U n+1 + B2 V n+1 + B3 W n+1 = B4
(18)
C1 U n+1 + C2 V n+1 + C3 W n+1 = C4 The coefficients in the equations are given in the Appendix. Linear equations in Eq.(18) are solved by the Gaussian elimination algorithm to calculate the displacement {Q} = {U V W }T .
IV. DYNAMIC LOADINGS Advanced supersonic/hypersonic flight vehicles are subjected to several dynamic loadings such as gust, sonic boom pulses and nuclear explosion. The analysis and design of structures subjected to blast loads require a detailed understanding of blast phenomena and the dynamic response of structures. In this section, various types of time-dependent external blast pulses are summarized and demonstrated in Figs.2-5.
Fig. 3. Sine load. Fig. 2. Exponentially decaying (blast) load.
4.1. Blast Loading If the blast source is distant enough from the plate, the blast pressure can be described in terms of the Friedl¨ ander exponential decay equation as[28] t e−αt/tp P (t) = Pm 1 − (19) tp where the negative phase of the blast is included. In this equation, Pm is the peak blast pressure, tp is positive phase duration, and α is waveform parameter.
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Fig. 4. Rectangular loading.
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Fig. 5. Triangular loading.
4.2. Sine Loading The sine pulse can be represented as ⎧ πt ⎨ Pm sin (0 < t < rtp ) P (t) = rtp ⎩ 0 (t < 0 and t > rtp )
(20)
where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. 4.3. Rectangular Loading For the rectangular pulse, the load function is given as Pm (0 < t < rtp ) P (t) = 0 (t < 0 and t > rtp )
(21)
where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. 4.4. Triangular Loading Sonic boom effect could be modeled as an N-shaped pressure pulse and may be described as ⎧ t ⎨ (0 < t < rtp ) Pm 1 − (22) P (t) = tp ⎩ 0 (t < 0 and t > rtp ) where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r. For r = 1 the sonic boom transformed into a triangular pulse (air-blast), for r = 2 the pulse corresponds to a symmetric N-shaped pulse (sonic boom), and for r = 2 the pulse corresponds to a non-symmetric N-pulse.
V. NUMERICAL RESULTS The solution of nonlinear-coupled equations given by Eqs.(11) is obtained by writing a MATLAB computer code based on the Newmark Method. First of all, the results obtained from the present method are validated with ANSYS results. The plate is discretized using the eight-node laminated shell elements (SHELL 91) with the geometric nonlinear capability. A six-layered hybrid composite laminated plate is chosen for comparison. The ply material properties used in the validation are given in Table 1. Unless otherwise stated, every ply has the same thickness which is 0.4×10−3 m. The stacking sequence of lamination is demonstrated in Fig.1. Fiber orientation is taken as [45/ − 45/45/ − 45/45/ − 45]. The dimensions of the plate are a = 0.3 m, b = 0.3 m, and h = 2.4 × 10−3 m. Friedl¨ander exponential decay equation given in Eq.(19) is used as the blast load. The maximum blast pressure Pm is taken to be 30 kPa. Parameters are chosen as α = 0.35 and tp = 0.0018 s. Comparison of the results for the non-dimensional deflection at the center of six-layered square plate with ANSYS is shown in Fig.6. The numerical results presented here have considered different loading conditions. In Fig.7, the dimensionless central deflections of the plate subjected to exponentially decaying (blast load) for different peak values are shown. 15 kPa, 30 kPa and 45 kPa are selected as blast peak pressure Pm for the clamped
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Table 1. Ply material properties
Ply material Woven-glass/epoxy Kevlar/epoxy(Aramid) E-glass/epoxy
E1 (GPa) 29.7 87 39
E2 (GPa) 29.7 5.5 8.6
Fig. 6. Comparison of the dimensionless deflection timehistories of the plate center.
G12 (GPa) 5.3 2.2 3.8
γ12 (Poisson’s ratio) 0.17 0.34 0.28
Density (kg/m3 ) 2200 1380 2100
Fig. 7. Time history of dimensionless (w/h) central deflection under blast load.
plate, respectively. Increasing the peak pressure value increases the amplitude and frequency, as expected. Figure 8 displays the effect of aspect ratio on dimensionless central deflection that corresponds to a blast load for selected aspect ratios. The mid-plane area of the plate is preserved as a constant value for all the aspect ratios. It was demonstrated that decreasing the aspect ratio results in lower amplitude and higher frequencies. In Fig.9, the effects of fiber orientation are depicted. Three lamination schemes are investigated. However, there is no significant discrepancy as seen in the chosen orientation and loading condition.
Fig. 8. Effect of aspect ratio on dimensionless (w/h) central deflection under blast load.
Fig. 9. Effect of fiber-orientation on dimensionless (w/h) central deflection under blast load.
In Fig.10, the dynamic response of the plate by changing the plate thickness is presented. The dimensionless center deflections of the plates for various ply thicknesses (0.3, 0.4 and 0.5 mm) are investigated. Decreasing the plate thickness results in higher deflection and frequencies. The frequency is significantly affected by the layer thickness. Figure 11 displays the effect of the shock pulse length factor (r) on the dynamic response of hybrid plate subjected to a sine load. The shape of the pulse is determined by the change of r as can be seen in Fig.4. As stated above, tp is positive phase duration which defines the extension in time of the forced motion in this case. Maximum deflection amplitudes are nearly the same for the first peak
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Fig. 10. Effect of thickness on dimensionless (w/h) central deflection under blast load.
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Fig. 11. Time history of dimensionless (w/h) central deflection under sine load.
values. Meanwhile, there is a special case for r = 2. The deflection amplitude for r = 2 is significantly lower than the others since the strong blast effect ends when the amplitude is close to the equilibrium condition. Thus, the hybrid plate oscillation is close to the equilibrium condition when the strong blast effect ends. In Fig.12, the dynamic response to a rectangular pulse for different r values is highlighted. Similarly, as previously mentioned for Fig.11, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the deflection amplitude. In the forced motion regime, the deflection amplitudes are almost the same for the first peak values. On the other hand, the results indicate that in the free motion regime, the rectangular pulse yields is maximum for r = 2 and minimum for r = 1.5. The parametric study continues with the effect of r values on its nonlinear response under the action of a triangular load. Triangular pulse includes just positive impulse (air-blast) for r = 1. The pulse corresponds to a symmetric N-shaped pulse (sonic boom) when r = 2, and for r = 3 the pulse corresponds to a non-symmetric N-pulse where the negative impulse is higher than the positive part. As seen from Fig.13, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the free vibration amplitudes. Thus, maximum deflection amplitude is seen for r = 3, while all cases yield nearly the same response until the tp value is reached.
Fig. 12. Time history of dimensionless (w/h) central deflection under rectangular load.
Fig. 13. Time history of dimensionless (w/h) central deflection under triangular load.
A comparison of deflection time histories for different type loads is demonstrated in Fig.14, from which it is seen that, having smaller deflection for the same peak pressure values, the blast load in general generates less intense oscillating motion. As is well known, the area below the graph demonstrates the impulsive load effect. Needless to say, the sizes of the areas from highest to lowest are rectangular, sine, triangular and blast pulses at the same positive phase duration time tp . Thus, the maximum deflection
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Fig. 14. Comparison of deflection histories for several dynamic loads (Pm = 30 kPa).
peak value is obtained by the rectangular pulse loading, whereas the minimum one is obtained by the blast pulse when the maximum peak pressure value (Pm ) and pulse duration are the same. However, after the strong blast effect, rectangular and triangular pulse oscillations are nearly the same.
VI. CONCLUSIONS In this study, the equations of motion of a laminated hybrid composite plate are derived. In-plane stiffness and inertia effects are considered when formulating the dynamic response of the laminated hybrid composite plate subjected to time-dependent pulses. The geometric nonlinearity effects are taken into account by using the von K´ arm´ an large deflection theory of thin plates. The Galerkin Method is used to obtain a set of the nonlinear differential equations in the time domain. The Newmark Method is used to solve the system of coupled nonlinear equations. An analytic tool was presented for the nonlinear dynamic behavior of a fully-clamped hybrid laminated composite plate under several dynamic loads including in-plane effects. The implications of peak pressure value, aspect ratio, fiber orientation and thickness for the dynamic response of a hybrid laminated composite plate are discussed. It is demonstrated that increasing the peak pressure value increases the amplitude and frequency. Decreasing the aspect ratio results in lower amplitude and higher frequencies while decreasing the plate thickness results in higher deflection and frequencies. The frequencies are significantly affected by the layer thickness. If the larger deflections as a result of thinner plate are considered, it can be deduced that the nonlinear and in-plane stiffness effects will increase in the case of thinner plate and it can result in a higher vibration frequency than that of the thicker plate. However, the fiber orientations have negligible effect on the vibration characteristics. Both the geometrical nonlinearities and the effects of in-plane deformations have been implemented in the structural model. The results obtained can be extended without any difficulty to the determination of the time histories of strain and stress components of the various points of the structure. A parametric study was carried out considering the blast, sine, rectangular and triangular loading. It is seen that, after the strong blast effect (forced motion regime), the distance from the equilibrium condition affects the free vibration amplitudes for different shock pulse length factor (r) values. However, blast load in general generates less intense oscillating motion compared to other different types of loads. Maximum deflection peak value is obtained by the rectangular pulse loading, whereas the minimum one is generated by the blast pulse when the maximum peak pressure value (Pm ) and pulse duration are the same. These results can be very important in designing structures which are subjected to dynamic loads of known and approximately fixed duration. Different material properties, damping effects and other boundary conditions should be taken into account by using the same method. Future studies may be devoted to these topics.
References [1] Chandrasekharappa,G. and Srirangarajan,H.R., Nonlinear response of elastic plates to pulse excitations. Computers & Structures, 1987, 27: 373-378.
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[2] Chandrasekharappa,G. and Srirangarajan,H.R., Parametric study of nonlinear static and dynamic response of composite rectangular plates. Mechanics Research Communications, 1988, 15(3): 177-188. [3] Kazancı,Z., Mecitoˇ glu,Z. and Hacıoˇ glu,A., Effect of in-plane stiffnesses and inertias on dynamic behavior of a laminated composite plate under blast load. In: Proceedings of 9th Biennial International Conference of the Aerospace Division on Engineering, Construction and Operations in Challenging Environments: Earth and Space 2004, 2004, 1: 484-491. [4] T¨ urkmen,H.S. and Mecitoˇ glu,Z., Nonlinear structural response of laminated composite plates subjected to blast loading. AIAA Journal, 1999, 37(12): 1639-1647. [5] T¨ urkmen,H.S. and Mecitoˇ glu,Z., Dynamic response of a stiffened laminated composite plate subjected to blast load. Journal of Sound and Vibration, 1999, 221(3): 371-389. [6] Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load. Journal of Sound and Vibration, 2008, 317(3-5): 883-897. [7] Ba¸s,A., Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear response of a sandwich plate subjected to blast load. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition-2007, Vol 1Advances in Aerospace Technology, 2008, 1: 169-177. [8] Dogan,V. Dynamic multimode response of composite plates to sonic boom and blast loadings. Journal of Aircraft, 2008, 45(2): 498-507. [9] Librescu,L. and Nosier,A., Response of laminated composite flat panels to sonic boom and explosive blast loadings. AIAA Journal, 1990, 28(2): 345-352. [10] Hause,T. and Librescu,L., Dynamic response of anisotropic sandwich flat panels to explosive pressure pulses. International Journal of Impact Engineering, 2005, 31: 607-628. [11] Hause,T. and Librescu,L., Dynamic response of doubly-curved anisotropic sandwich panels impacted by blast loadings. International Journal of Solids and Structures, 2007, 43: 6678-6700. [12] Kazancı,Z., Nonlinear transient response of a laminated composite plate under time-dependent pulses. In: Proceedings of the 4th International Conference on Recent Advances in Space Technologies-RAST2009, 2009, 1: 125-130. [13] Dobyns,A.L., Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA Journal, 1981, 19: 642-650. [14] Abrate,S., Transient response of beams, plates, and shells to impulsive loads. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2007, 2008, 9: 107-116. [15] Kazancı,Z. and Mecitoˇ glu,Z., Nonlinear damped vibrations of a laminated composite plate subjected to blast load. AIAA Journal, 2006, 44(9): 2002-2008. [16] Birman,V. and Bert,C.W., Behaviour of laminated plates subjected to conventional blast. International Journal of Impact Engineering 1987, 6: 145-155. [17] Kazancı,Z., Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses. International Journal of Non-Linear Mechanics, 2011, 46(5): 807-817. [18] Tsouvalis,N.G. and Papazoglou,V.J., Large deflection dynamic response of composite laminated plates under lateral loads. Marine Structures, 1996, 9: 825-848. [19] Walker,M. and Smith,R.E., A technique for the multiobjective optimisation of laminated composite structures using genetic algorithms and finite element analysis. Composite Structures, 2003, 62: 123-128. [20] Kuan,H.T., Cantwell,W. and Akil,H.M., The mechanical properties of hybrid composites based on selfreinforced polypropylene. Malaysian Polymer Journal, 2009, 4(2): 71-80. [21] Sayer,M., Bekta¸s,N.B. and Sayman,O., An experimental investigation on the impact behavior of hybrid composite plates. Composite Structures, 2010, 92: 1256-1262. [22] Chen,C.S. and Fung,C.P., Non-linear vibration of initially stressed hybrid composite plates. Journal of Sound and Vibration, 2004, 274: 1013-1029. [23] Ibrahim,H.H., Yoo,H.H. and Lee,K.S., Aero-thermo-mechanical characteristics of imperfect shape memory alloy hybrid composite panels. Journal of Sound and Vibration, 2009, 325: 583-596. [24] Rahul,Sandeep,G., Chakraborty,D. and Dutta,A., Multi-objective optimization of hybrid laminates subjected to transverse impact. Composite Structures, 2006, 73: 360-369. [25] Chen,C.S., Fung,C.P. and Chien,R.D., Analysis of an initially stressed hybrid plate based on a higher-order theory. Composite Structures, 2006, 74: 343-353. [26] Chen,C.S., Fung,C.P. and Yang,J.G., Assessment of plate theories for initially stressed hybrid laminated plates. Composite Structures, 2009, 88: 195-201. [27] Strang,G., Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. [28] Gupta,A.D., Gregory,F.H., Bitting,R.L. and Bhattacharya,S., Dynamic analysis of an explosively loaded hinged rectangular plate. Computers and Structures, 1987, 26: 339-344.
APPENDIX
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Coefficients in the time-dependent nonlinear differential equations (All edges clamped): a0 =
a11 b m, ¯ 18480
a11 b d1 , 18480
a2 =
a5 (A12 + A66 )b5 (−15 + π 2 )2 , 4π 8
a4 =
5a3 [b2 A22 (15 − 4π 2 ) + 3(A12 − A66 )a2 (63 − 4π 2 )] , 64bπ 2
b0 =
ab11 m, ¯ 18480
b2 =
a5 (A12 + A66 )b5 (−15 + π 2 )2 , 4π 8
b4 =
5b3 [a2 A22 (15 − 4π 2 ) + 3(A12 − A66 )b2 (63 − 4π 2 )] , 64aπ 2
c0 =
9ab m, ¯ 4
c2 =
24π 4 (B12 − B66 ) , ab
a =
b =
c =
a1 =
a11 b d2 , 18480
9ab d3 , 4
c3 =
a3 = −
b1 =
c1 =
a9 (33 A11 b2 + a2 A66 π 2 ) 13860b a3 [3b2 B11 + a2 (B12 + 2B66 )](−15 + π 2 ) bπ 2 a5 = −abqx
b9 (33a2 A22 + A66 b2 π 2 ) 13860a
b3 = −
b3 [3a2 B22 + b2 (B12 + 2B66 )](−15 + π 2 ) aπ 2 b5 = −abqy
4π 4 [3b4 D11 + 3a4 D22 + 2a2 b2 (D12 + 2D66 )] a3 b 3
20π 4 [21a4 A22 + 10a2 (3A12 + 4A66 )b2 + 21A11 b4 ] 32a3 b3
a3 [3b2 B11 + a2 (B12 + 2B66 )](−15 + π 2 ) b3 [3a2 B22 + b2 (B12 + 2B66 )](−15 + π 2 ) , c = − 5 bπ 2 aπ 2 5a3 A11 b2 (15 − 4π 2 ) − 3a2 (A12 − A66 )(−63 + 4π 2 ) c6 = 32bπ 2 5b3 A22 a2 (15 − 4π 2 ) − 3b2 (A12 − A66 )(−63 + 4π 2 ) c7 = 32aπ 2
c4 = −
c8 = −abP (t) Coefficients in the Newmark time integration equations: A1 = k0 + a1 , B1 = b2 ,
A2 = a2 ,
B2 = k0 b0 + b1 ,
C1 = c4 + c6 W n ,
A3 = a3 + a4 W n , B3 = b3 + b4 W n ,
C2 = c5 + c7 W n ,
¨n A4 = a5 + a0 k0 U n + a0 k2 U˙ n + a0 k3 U B4 = b5 + b0 k0 V n + b0 k2 V˙ n + b0 k3 V¨ n
C3 = k0 c0 + c1 + c2 W n + c3 (W n )
˙ n + c0 k3 W ¨n C4 = c8 + c0 k0 W n + c0 k2 W
2