Nonlinear Transient Response of A Laminated Composite Plate Under Time-Dependent Pulses

Zafer Kazancı Turkish Air Force Academy,

Aeronautical Engineering Department, Yeşilyurt, 34149, İstanbul, Turkey

z.kazanci@hho.edu.tr Abstract—This paper includes in-plane stiffness and inertia effects on the motion of a laminated composite plate subjected to explosive pressure pulses. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Transverse shear stresses are ignored. The in-plane stiffness is taken into account as a difference of the analysis made before. All edges simply supported boundary conditions are considered in the analyses. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The influence of loading effects on the nonlinear dynamic response has been predicted for all edges simply-supported plates. The displacement functions are chosen by considering the results of static large deformation analysis of laminated composite by using a commercial finite element code ANSYS 10.0 software. A FORTRAN program was described for automating the application of explosive pressure pulses to a simply supported laminated composite plate. A detailed analysis of the influence of different type of pressure pulses on dynamic response is carried out.

Keywords-nonlinear analysis; large deflection; laminated composite plate; finite difference method; time-dependent pulses

I. INTRODUCTION Advanced laminated composite plates are widely used in the

construction of advanced supersonic/hypersonic flight vehicle and of reusable space transportation systems due to their high stiffness and strength-to-weight ratios, long fatigue life, resistance to electrochemical corrosion, and etc. With the advent of the new composite material structures and their increased use in space, there is a need to reconsider the problem of the nonlinear dynamic behavior of laminated composite plates exposed to time-dependent pulses, such as sonic boom and blast loadings.

Several studies related to the effects of blast loading on the panel structures are investigated in the literature. Hause and Librescu [1] have addressed the problem of the dynamic response in bending of flat sandwich panels exposed to time-dependent external blast pulses. Librescu and Nosier [2] have investigated the response of laminated composite flat panels to sonic boom and explosive blast loadings. Kazancı and

Mecitoğlu [3] have studied on the nonlinear damped vibrations of a fully-clamped laminated composite plate subjected to blast load. Kazancı et al. [4,5] have considered in-plane stiffness and inertias in the analytical solution of the laminated composite plates under the blast load.

Several studies related to the effects of explosive pressure pulses on the panel structures are investigated in the literature. However, only a few studies in the nonlinear response of simply supported laminated composite plates subjected to explosive pressure pulses are investigated. Birman and Bert [6] considered the response of simply supported anti-symmetrically laminated angle-ply plates to explosive blast loading. Chen et al. [7] developed a semi-analytical finite strip method for the analysis of the nonlinear response to dynamic loading of simply supported rectangular laminated composite plates. Kazancı and Mecitoğlu [8] studied on the nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load.

Present work includes in-plane stiffness and inertia effects on the motion of a laminated composite plate subjected to explosive pressure pulses. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. All edges simply supported boundary conditions are considered in the analyses. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The influence of loading effects on the nonlinear dynamic response has been predicted for all edges simply-supported plates. In this study, the displacement functions are chosen by considering the results of static large deformation analysis of laminated composite by using a commercial finite element code ANSYS 10.0 software. A FORTRAN program was described for automating the application of explosive pressure pulses to a simply supported laminated composite plate.

The effects of the blast pressure character on the dynamic behavior are investigated. The results of approximate-numerical analyses are presented.

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II. GOVERNING EQUATIONS

A. Mathematical Model A mathematical model for the laminated composite plate

subjected to blast load is presented in this section. The rectangular plate with the length a, the width b, and the thickness h, is depicted in Figure 1. The Cartesian axes are used in the derivation.

Figure 1. Laminated composite plate.

Displacement functions of a thin plate can be expanded in

the series as mentioned by the Weierstrass [9]. If the first few terms in the series are taken, the displacement functions can be approximated as follows

0

0 wu u zx

∂∂

= − , 0

0 wv v zy

∂∂

= − , 0w w= (1)

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 Kármán plate can be written as

0

x x xzε ε κ= + , 0y y yzε ε κ= + , 0

xy xy xyzε ε κ= + (2) where

20 00 1

2xu wx x

∂ ∂ε∂ ∂

⎛ ⎞= + ⎜ ⎟

⎝ ⎠,

2 0

2xwx

κ ∂= −∂

(3a)

20 00 1

2yv wy y

∂ ∂ε∂ ∂

⎛ ⎞= + ⎜ ⎟

⎝ ⎠,

2 0

2ywy

κ ∂= −∂

(3b)

0 0 0 00xy

u v w wy x x y

∂ ∂ ∂ ∂ε∂ ∂ ∂ ∂

= + + , 2 0

2xyw

x yκ ∂= −

∂ ∂ (3c)

The effective elastic constants are used for defining the constitutive model of the laminated composite. The stress-strain relations of a lamina in the laminate coordinates (x,y,z) are given by

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

Q Q QQ Q QQ Q Q

σ εσ εσ ε

⎧ ⎫ ⎧ ⎫⎡ ⎤⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭ ⎩ ⎭

(4)

where the ijQ are the components of the transformed lamina stiffness matrix. Force and moment components of the plate can be written as

[ ][ ]

[ ] [ ][ ] [ ] [ ]

0N A BM B D

ε

κ

⎧ ⎫⎡ ⎤⎧ ⎫ ⎡ ⎤⎪ ⎪ ⎪ ⎪⎣ ⎦=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭ ⎣ ⎦ ⎪ ⎪⎩ ⎭

(5)

The coefficients in the matrices are given by

11

( ) ( )n

ij ij k k kk

A Q h h −=

= −∑ ,

2 21

1

1 2 ( ) ( )n

ij ij k k kk

B Q h h −=

= −∑ (6)

3 31

1

1 3 ( ) ( )n

ij ij k k kk

D Q h h −=

= −∑

which represents the laminate extensional, coupling and bending stiffness, respectively. Here, hk denotes the kth ply thickness.

B. Equations of Motion 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

0 0 0 0 011 12 13 1( ) 0xL u L v L w N w mu q+ + + + − =

0 0 0 0 0

21 22 23 2( ) 0yL u L v L w N w mv q+ + + + − = (7) 0 0 0 0 0 0 0

31 32 33 3( , , ) 0zL u L v L w N u v w mw q+ + + + − =

where Lij and Ni are linear and nonlinear operators. m is the mass of unit area of the mid-plane, qx, qy and qz are the load vectors in the axes directions, respectively. The explicit expressions of the operators can be found in Kazancı and Mecitoğlu [8].

BLAST

Top surface

126

C. Boundary Conditions The boundary conditions for a simply-supported plate can

be written as

0 0 0 0(0, , ) ( , , ) ( ,0, ) ( , , ) 0u y t u a y t u x t u x b t= = = = , 0 0 0 0(0, , ) ( , , ) ( ,0, ) ( , , ) 0v y t v a y t v x t v x b t= = = = , 0 0 0 0(0, , ) ( , , ) ( ,0, ) ( , , ) 0w y t w a y t w x t w x b t= = = = ,

0 at x = 0, axM = 0 at y = 0, byM =

Initial conditions are given by 0 0 0( , ,0) 0, ( , ,0) 0, ( , ,0) 0u x y v x y w x y= = = , 0 0 0( , ,0) 0, ( , ,0) 0, ( , ,0) 0u x y v x y w x y= = =

III. LOAD CASES In this study, nonlinear transient responses of laminated

composite plates are investigated for various types of explosive pressure pulses. Sonic boom effect could be modeled as an N-shaped pressure pulse and may be described as [2]

( )1 / 0( , , ) ( )

0 0m p p

p

p t t t r tP x y t P t

t ve t r t

⎧ − < <⎪≡ = ⎨< >⎪⎩

(8)

where r denotes the shock pulse length factor. The shape of the pulse is determined by the change of r.

A. Case I For r=1 the sonic boom transformed into a triangular pulse,

and for r=2 the pulse corresponds to a symmetric N-shaped pulse.

B. Case II Rectangular pressure pulse may be defined as

0( , , ) ( )

0m p

p

p t rtP x y t P t

t r t

≤ ≤⎧⎪≡ = ⎨ >⎪⎩ (9)

C. Case III If the blast source is distant enough from the plate, the blast

pressure can be described in terms of the Friedlander exponential decay equation as [10]

/( ) (1 / )m ppt tP t p t t e α−= − (10)

where the negative phase of the blast is included. Here pm is peak pressure, tp is positive phase duration and α is waveform parameter.

IV. SOLUTION METHOD A. Finite Element Solution

The laminated composite plate is analyzed using ANSYS finite element software. The plate is discretized using by 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.

B. Approximate-Numerical Solution The equations of motion given with (7) can be reduced into

time domain by choosing some approximation functions for displacement field and applying the Galerkin method. The coupled-nonlinear equations in time domain are solved by using the finite difference method. The approximation functions are selected so as to satisfy the natural boundary conditions.

( )I J

0ij ij

i 1 j 1

u U (t) x, y= =

= φ∑∑ (11a)

( )K L

0kl kl

k 1 l 1

v V (t) x, y= =

= ψ∑∑ (11b)

( )M N

0mn mn

m 1 n 1

w W (t) x, y= =

= χ∑∑ (11c)

The simplest multi term approximations even results in the hundreds of integral terms during the application of the Galerkin procedure and therefore they are impractical. Therefore, one term approximation functions for the displacement components are used in this study. Choosing the approximation functions is a crucial point. It 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 10.0 [11] software. Fig. 2 – Fig. 4 show the variations of u(x), v(x), and w(x), respectively. While y = b/2 line on the plate is considered for the variations of u(x) and w(x), y = b/4 line on the plate is taken for the variation of v(x).

The approximation functions are determined by examining the finite element results obtained from the static large deformation results as follows,

0 2 211

2 xu U (t)sin y (y b)aπ= − (12a)

0 2 211

2 yv V (t)x (x a) sinbπ= − (12b)

011

x yw W (t)sin sina b

π π= (12c)

Fig. 2 – Fig. 4 also include the variations of the first term of approximation functions given in (12). The approximations look closely agreeing with the finite element results.

Applying the Galerkin method to the equations of motion given in (7), the time dependent nonlinear differential equations can be obtained as

20 1 2 3 4 5 0a U a U a V a W a W a+ + + + + =

20 1 2 3 4 5 0b V bV b U b W b W b+ + + + + = (13)

2 30 1 2 3 4 5 6 7 8 0c W cW c W c W c U c V c UW c VW c+ + + + + + + + =

The dot denotes the derivative with respect to time.

127

Figure 2. Variation of u(x). (y=b/2).

Figure 3. Variation of v(x). (y=b/4).

Figure 4. Variation of w(x). (y=b/2).

Nonlinear-coupled equations of motion (11) are solved by using finite difference method. Solution methodology was also described in Kazancı and Mecitoğlu [8].

V. NUMERICAL RESULTS A seven-layered fiber-glass fabric with (0º/90º) fiber

orientation angle for one layer is used in the numerical analyses. Ply material properties used in the analyses are given as E1 = 24.14 GPa, E2 = 24.14 GPa, G12 = 3.79 GPa, ρ = 1800 kg/m3, and υ 12 = 0.11. The dimensions of the plate are a = 0.22 m, b = 0.22 m, and h = 1.96 mm. The analyses are performed for the uniform blast pressure. The maximum blast pressure is taken to be 28.9 kPa for the plate all edges simply supported. Variation of blast pressure by time at the plate center is shown in Kazancı and Mecitoğlu [3]. The other parameters of the Friendlander’s exponential decay function given in (10) are chosen as α = 0.35 and tp = 0.0018 s.

A. Static Analysis Results Figure 5 shows the strain results under the uniform static

pressure. They are obtained from the finite element model of the laminated plate by using ANSYS software. The variation of the several strain terms along the plate span under the static load would be helpful to understand the dynamic behavior under the blast load.

Figure 5 shows the variation of normal strain xε by x coordinate on the y=b/2 line of the bottom and top surfaces of plate for the linear and nonlinear cases. Nonlinear effects decrease the absolute strain values except near the plate edges, as shown in.

Figure 5. Variation of strain ( xε ) by length x for linear and nonlinear case

(y=b/2).

B. Transient Response Results The solution of nonlinear-coupled equations given by (13) is

obtained by writing a FORTRAN program based on the finite difference schema explained in the previous section. The convergence studies are the goal of a close proximity numerical solution. Time step convergence studies were conducted on

max

m

ax

max

128

plates subjected to 28.9 kPa blast load. It was found that time increment of no more than 0.002 milliseconds were adequate for such a numerical solution.

First, rectangular load as a function of time is highlighted in Figure 6. While in Figure 7 central deflection is presented, in Figure 8 the time-history of the microstrain is presented. The results reveal after the rectangular pulse effect, the oscillation amplitudes remain constant, but the forced and free motion regimes have different values, as expected.

Figure 6. Rectangular load as a function of time (p(t) = pm , r=3tp)

-3

-2

-1

0

1

2

3

4

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

w /

h

Figure 7. Time history of dimensionless central deflection subjected to

rectangular load at the centroid (Top surface)

-500

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

Mic

rost

rain

Figure 8. Strain-time history results subjected to rectangular load at the

centroid (Top surface)

Figure 9 shows the triangular load as a function of time for r=1. Figures 10 and 11 show the time-histories of the central deflection and microstrain at the center of the plate subjected to triangular load (r=1). After the strong blast effect, it can be seen from the figures that the variations of the amplitudes and the frequencies are very little for this load case.

Figure 9. Triangular load as a function of time (r=1)

-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

w /

h

Figure 10. Time history of dimensionless central deflection subjected to

triangular load (r=1) at the centroid (Top surface)

-500

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

Mic

rost

rain

Figure 11. Strain-time history results subjected to triangular load (r=1) at the

centroid (Top surface)

As mentioned before, for r=2 the pulse corresponds to a symmetric N-shaped pulse, and it is pointed out in Figure 12. In Figures 13 and 14 the dynamic response to a triangular load (r=2) is highlighted.

129

Figure 12. Triangular load as a function of time (r=2)

-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

w /

h

Figure 13. Time history of dimensionless central deflection subjected to

triangular load (r=2) at the centroid (Top surface)

-500

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

Mic

rost

rain

Figure 14. Strain-time history results subjected to triangular load (r=2) at the

centroid (Top surface)

VI. CONCLUSIONS The equations of motion of a simply-supported laminated

composite plate under time-dependent pulses are derived in the

frame of the von Kármán 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 finite difference method is applied to solve the system of coupled nonlinear equations. The nonlinear static behavior is used to explain the nonlinear dynamic behavior under the blast load. The effect of time-dependent pulse character on the dynamic behavior is investigated.

Different material properties can be used for analyses. Other boundary conditions can be analyzed using the same method. Sandwich composite plates can be investigated using the same procedure. Future studies may be devoted on these subjects.

REFERENCES [1] T. Hause and L. Librescu, “Dynamic response of anisotropic sandwich

flat panels to explosive pressure pulses,” International Journal of Impact Engineering, vol. 31, pp. 607-628, 2005.

[2] L. Librescu and A. Nosier, “Response of laminated composite flat panels to sonic boom and explosive blast loadings,” AIAA Journal, vol. 28 (2), pp. 345-352, 1990.

[3] Z. Kazancı and Z. Mecitoğlu, “Nonlinear damped vibrations of a laminated composite plate subjected to blast load,” AIAA Journal, vol. 44, pp. 2002-2008, 2006.

[4] Z. Kazancı and Z. Mecitoğlu, “Approximate-numerical and finite element solutions for nonlinear vibrations of laminated composite plates under blast load,” The 36th International Congress and Exhibition on Noise control Engineering, (Inter-Noise 2007), August 28-31,2007, The İstanbul Convention & Exhibition Centre (ICEC), İstanbul, TURKEY.

[5] Z. Kazancı, Z. Mecitoğlu and A. Hacıoğlu, “Effect of in-plane stiffnesses and inertias on dynamic behavior of a laminated composite plate under blast load, Proceedings of 9th Biennial ASCE Aerospace Division International Conference on Engineering, Construction, and Operations in Challenging Environments, Earth&Space-2004, Houston, Texas, USA, pp. 484-491, March, 7-10, 2004.

[6] V. Birman and C.W. Bert, “Behaviour of laminated plates subjected to conventional blast,” International Journal of Impact Engineering, vol. 6, pp. 145-155, 1987.

[7] J. Chen, D. J. Dawe and S. Wang, “Nonlinear transient analysis of rectangular composite laminated plates,” Composite Structures, vol. 49, pp. 129-139, 2000.

[8] Z. Kazancı and Z. Mecitoğlu, “Nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load,” Journal of Sound and Vibration, vol. 317, pp. 883-897, 2008.

[9] K. Weierstrass,“Über die Analytische Darstellbarkeit Sogenannter Willkürlicher Functionen Einer Reellen Veränderlichen,” Sitzungsberichte der Akademie zu Berlin, 1885, pp. 633–639, 789–805; also appeared in Weierstrass’ “Mathematische Werke,” Mayer and Muller, Berlin, Vol. 3, 1903, pp. 1–37.

[10] A.D. Gupta, F.H. Gregory, R.L. Bitting, S. Bhattacharya, “Dynamic analysis of an explosively loaded hinged rectangular plate,” Computers and Structures vol. 26, pp. 339-344, 1987.

[11] ANSYS 10.0 Software.

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