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Large deflection dynamic response of composite laminated plates under lateral loads
Marine Structures 9 (1996) 825-848
© 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0951-8339/96/Sl 5.00 ELSEVIER
0951-8339(95)00048-0
Large Deflection Dynamic Response of Composite Laminated Plates Under Lateral Loads N. G. Tsouvalis & V. J. Papazoglou* ShipbuildingTechnologyLaboratory, Dept. of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou Ave, GR-157 73 Zografos, Greece (Received 23 September 1995; revisedversion received9 November 1995; accepted 18 November 1995) ABSTRACT A very effective analytical tool for the nonlinear large deflection response of simply supported laminated plates under the action of various types of time dependent lateral loads is presented. The solution, based on the Classical Lamination Theory and accounting for moderately large deflections, is obtained by applying the Galerkin procedure. Several numerical results are presented, including a convergence study, comparisons with other published results, and parametric studies. The relationship between duration of load application and lowest plate natural period that leads to a minimum response after load removal is investigated. Copyright © 1996 Elsevier Science Limited. Key words: laminated plates, dynamic response, large deflection. INTRODUCTION Considering the loads acting on a marine structure throughout its operational life, we shall hardly find any exhibiting pure static behaviour. The majority of the loading conditions, including the most crucial ones for the integrity of the structure, are characterised by their dynamic nature, which is usually stochastic. A typical example is the slamming phenomenon at the bow of small, fast power boats, as they run with high speeds in rough or moderately rough seas, as well as high pressures at the hull bottom caused by underwalter explosions or at warship superstructures caused by air-blasts. Analogous conditions are encountered in the aircraft industry, another constructional branch where the use of composite materials is continuously * To whom correspondance should be addressed 825
826
N. G. Tsouvalis, 1I. J. Papazoglou
increasing. For the accomplishment of an effective structural design, these dynamic loads should, in all cases, be taken into account. A dynamic plate bending analysis becomes, therefore, imperative when carrying out a structural design study. The need for considering nonlinear geometric effects in bending stress analysis under dynamic loads became obvious rather early and, thus, numerous studies have been undertaken. An early work on this subject is that by Alwar & Adimurthy, 1 who dealt with simply-supported rectangular sandwich plates excited by lateral pulse or sonic boom loadings. Using Classical Lamination Theory (CLT), Mei & Wentz 2 studied analytically the large deflection response of regular antisymmetric angle-ply laminated rectangular plates, simply supported or clamped at all edges. Considering a one-term expansion of the deflection, w, and the stress function, F, and using the Galerkin method, they arrived at a nonlinear differential equation with respect to time. The solution was obtained by linearizing this equation. Birman & Bert 3 solved the final differential equation with respect to time by means of the Runge-Kutta numerical integration method. Only simply supported boundary conditions were examined. A fair amount of finite element studies have also dealt with the problem under consideration. A nine-node rectangular isoparametric shear-deformable finite element was employed by Reddy 4 to investigate the response of isotropic, orthotropic and layered anisotropic composite plates. The same author modified the above element to include the nonlinear strain-displacement relations of the von K~irm/m plate theory. 5 A Higher Order Shear Deformation theory was employed by Kant e t al. 6 as the basic theory for the development of a shear deformable finite element, to investigate the transient response of laminated composite plates. A detailed review of over 200 references dealing with the geometrically nonlinear behaviour of composite plates has been performed by Chia. 7 The above survey shows the lack of a relatively simple, quick and reasonably accurate analytical solution of the dynamic transient response of laminated plates. The existing analytical solutions either do not account for large deflections or, if they do, they seem not to be very accurate, since they incorporate single term expansions of the displacements. In the present study an analytical solution of the bending behaviour of laminated plates of any stacking sequence is given, under the action of various dynamic lateral loads. The solution is based on the CLT (thus, is valid only for thin plates) and accounts for moderately large deflections, since it employs strain-displacement relations in the von K/trm/m sense. The final set of the nonlinear governing differential equations is solved with the aid of a numerical integration scheme. The laminate is considered to be simply supported at its boundaries with edges free of stresses in the plane of the plate. Initial out-of-plane imperfections, as well as the damping properties of the material, are taken into account.
Dynamic response of compositeplates under lateral loads
827
THEORETICAL ANALYSIS
Governing equations Consider a rectangular plate of length a (x-direction), width b (y-direction) and thickness h (z-direction), subjected to an arbitrarily distributed lateral load q(x,y), which, in addition, varies with time. The plate is assumed to consist of any number of composite material layers with different thicknesses, different material elastic properties and arbitrary fiber orientation angles with respect to the plate geometric axes. Furthermore, let the plate exhibit a steady out-of-plane initial imperfection Wo(X,y),forming any shape and having maximum values in the order of magnitude of the plate thickness. The derivation of the equations governing the dynamic flexural behaviour of the htminate under the action of a time dependent lateral load q, is accomplished with the aid of the principle of minimum total potential energy. 8 Introducing the Airy stress function and through proper nondimensionalisation,9 we obtain the following non-dimensional equilibrium and compatibility equations of the problem, respectively:
o:w
ow
OZ2 q- 2,~(Omn~
04(w - Wo) "l-
O~4
+ d3,~304(W - Wo) •
O~ Or/3
o~: o,/ •N 11/
d4
b-~ o~ ~
O~30,rI
0~20,02
,~404(W - Wo) 04t~ 04(~ 07]4 + bl - ~ + b2/]. O~ 30~
04CI) Jr- b4 ,~3 _ _
+ b3 22 04~
-
l-
,~04(w - Wo) +d22204(W- Wo) "k- dl
.4 04(1)
(1)
2°~ o~w o~O2w) o--U-oo o~o---~~ o¢ ~ ~ - °'~" a3 ~3 0"~(I)
o¢o----~ + a4
o-¢,~ - a, ~ O¢--r~ + a~ ; O~~ O-------7 ~ ( w - Wo) b, 04(m - Wo) -- b6
- b9V
O~4
o'(w- Wo) - ~
7A
-O-~b~
=0
/].40~(I) O-V
;¢ 04(w - Wo)
b8
0¢20rl2 (2)
Wo) O,74
blo 24 04(W -
O~W O2W oe
oe
I O~Wo'X~ 02 Wo ~ o ] + oe
"
828
N. G. Tsouvalis, V. J. Papazoglou
These two equations are coupled partial differential equations with two dependent variables, the non-dimensional total out-of-plane deflection W (including the initial imperfection Wo) and the non-dimensional stress function • . Damping is represented as a linear viscous modal damping with damping ratio ~,10,11, whereas &.,n is the non-dimensional circular natural frequency of the laminate corresponding to mode (m,n), ~ and r/are the nondimensional distances in the x and y directions, respectively, z is the nondimensional time, 2 is the plate aspect ratio, g/is the non-dimensional lateral load and ai, di, bi, Dll, A22 are non-dimensional functions of the plate rigidities. There is no classical closed form solution to the above problem and thus, to proceed further in the solution, we employed the Galerkin technique by assuming double sine Fourier series for the non-dimensional lateral deflection W and initial imperfection Wo, with coefficients W~jand Wo~, respectively, and the following series for the non-dimensional stress function (~:9,12,13 :
E ~mn Xm(¢) Y. (r/). m=l n=l
(3)
Notice that the initial imperfection of the laminate is assumed to be known and thus the coefficients Woij are given. The functions Xi and Y; in the stress function expansion are characteristic beam functions defined as Xi(O = cosh ~xi~-cos 0q~-yi(sinh ~i~-sin ~xi~) Yi(r]) = cosh 0qr/-cos 0qr/-yi(sinh ~07-sin sir/)
(4)
in which ~; and 7, are constants given in Ref. 9 or Ref. 12. We furthermore assume that the non-dimensional lateral load ~/(~,r/,z) can be expanded in a double sine series with coefficients ~mn(Z). For various types of lateral loads, these coefficients become: (a) Uniform load ~7oon the entire plate: {lm,(Z) =
{ 16g/o(Z), m , n = o d d ~2mn O, m, n ---- even
(5)
(b) Sinusoidal load ~(~,O,z)--~o(Z) sin(~O sin(m/): I~o(Z) (~mn(T') = 0
form=n= 1 for m, n/> 2.
(6)
Regarding the time variation of the dimensional applied lateral load q(t), the following commonly encountered forcing functions are taken into account (see Fig. 1), where q(t) = 0 for t < 0 in all cases:
Dynamic response of composite plates under lateral loads
829
q
q,
t
t
(b) Linearly increasing load
(a) Step loading
q°
tl
tl t (d) Blast loading
t
(c) Rectangular pulse
qo
q°
t, t
tl
(e) Half-sine pulse
t
(f) Triangular pulse
Fig. 1. Various time variations of the applied lateral load q.
(a) Step loading: q(t) = qo
(7)
(b) Linearly increasing load: q( t ) = qstt
where qst is the rate of the load increase, measured in pressure/time. (c) Rectangular pulse:
(8)
N. G. Tsouvalis, V. J. Papazoglou
830
q(t)
Sq°'
fort~
I 0,
f o r t > tl
(9)
where tl is the duration of the pulse. (d) Blast loading:
q ( t ) = { q ° ( 1 - - ~) e -~t , , O,
for t ~< tl
(lO)
for t > h
where qo is the peak pressure in excess of the ambient pressure, t~ is the duration of the pulse and ~ is a coefficient. The values of the blast parameters depend on the distance of the plate from the charge location and are usually taken as 0~ = 1.98 and tl = 0.004 sec 14 for air-blast loading, whereas no relevant data exist for underwater explosion loading. (e) Half-sine pulse: .
q(t) =
/tt
qosm t--l' O,
for t ~< t 1
(11)
for t > t~
where tl is the duration of the pulse. (f) Triangular pulse: 2t
q(t) =
qo~,
t~ fort~< ~-
- t) qo 2(tlt------~'
for tl
0,
for t < t~
2 < t <~tl
(12)
where tl is the duration of the pulse. The above defined functions for W and • satisfy simply supported boundary conditions, with the laminate edges being free to deform in the plane of the plate (1-2), since U and V are not constant at the boundaries. However, the functions for W and ~ do not satisfy the zero moment condition at the boundaries (M e = 0 at ¢ = 0, 1 and M,7 = 0 at r/ = 0, 1) for angle-ply laminates. The governing equations are solved by substituting the derivatives of W, Wo and F, multiplying the compatibility equation by Xv( O. Yq(ri) and the equilibrium one by sinpn~.sinqnr/and integrating both equations over the whole plate area. Two boundary integrals included in the equilibrium equation take into account the effects of the unbalanced edge moments.
Dynamic response of composite plates under lateral loads
831
Performing all substitutions and integrations in the two governing equations, a final set of equations is obtained in terms of the coefficients W o. and ~,,~. This set of equations contains an infinite number of nonlinear second order differential and algebraic equations to be solved simultaneously. In practice, only a finite number of these equations will be used to obtain the laminate bending behaviour to a specified accuracy. If we ,consider that indices i, j, m and n vary from 1 to M, that is, considering the first M 2 terms of the deflection and the stress function series expansions, the equilibrium eqn (1) results in a set of M 2 nonlinear second order differential equations with respect to time, while the compatibility eqn (2) gives another set of M 2 nonlinear algebraic equations. The general form of these two types of equations is: (a) Equilibrium equations: M
M
d2Wpq+ z, qd~"q+(W,q- Wo,q)A'q+ ~ ~ (W,- Wo~)B~: dz 2
oz
M
i=1 j=l
M
M
M
M
M
q + Z Z +.:c~. + y;= E E E +.:w:=,~ + ~q = 0 m=l n=l
m=ln=l
(13)
i=1 j=l
p,q= 1,2,...,M
(b) Compatibility equations: M
E
M
M
~,E ¢I~mnHPmq-~- E
m=ln=l
M
E
(l/Vii - W,'~l(Pqoijl "~ij
i=1 j=l
M
M
M
M
(14)
+ Z Z E E(w.w~,- Wo.:o~,)L.~,: 0 i=lj=lk=ll=l
p,q= 1,2,...,M
where the various coefficients
Z pq' A pq, nPqij , CPqrrm,OPqmnij, oPq, I-IPqmn, gPqij, LPqijkl are functions of the damping ratio, the dimensions and the rigidities of the laminate and the applied load. 8 S o l u t i o n procedure
Expressing the stress function coefficients t~mnin terms of the deflection ones W,,~, we.,end up with a set of M 2 coupled nonlinear second order differential equations in terms of Wm,, only. s The general form of the ith of these equations is
N. G. Tsouvalis, V. J. Papazoglou
832
d2Wi
dz 2 +
dWi Zi---~z = f ( z , W1, W2,..., WM2)
(15)
subject to the initial conditions (at T = 0) Wi = Wio and d Wg/d'r = 0, signifying that the laminate is at rest before load application. The above equation describes an initial value problem and is solved numerically.
N U M E R I C A L RESULTS The laminates are assumed to be formed by alternate layers of a glass/epoxy composite, having the material and physical properties shown in Table 1. The numerical study was performed using the Runge-Kutta adaptive step integrator, as the most efficient integration tool. This decision was based on a thorough investigation of various alternative numerical schemes, s
Convergencestudy A convergence study was performed to obtain the minimum number of the deflection W and stress function • series terms required for an adequate representation of these two magnitudes, when the lateral load is uniformly distributed over the laminate area. In Fig. 2 the non-dimensional center deflection Wc ( = wdh, where h is the laminate thickness) vs time of a rectangular (a = 50 cm, b = 25 cm, h = 1 mm) antisymmetric cross-ply (ACP) (0°/ 90°)4 laminate has been plotted, for a one, nine and twenty five series terms approximation. A step loading is considered, having a maximum value of qo = 25 Pa, whereas damping effects have been neglected. On the basis of this and numerous other runs, the nine terms approximation seems to give quite accurate results for the deflection of a laminate under a uniform spatial distribution of the lateral load and, thus, all the following numerical results have been obtained with this number of series terms. Nevertheless, this is not the case for the calculated stresses, since, as it can be seen from Fig. 3, convergence has not been attained even for a forty nine series terms (M = 7) TABLE 1 Properties of glass/epoxy composite Young's modulus in direction 1 Young's modulus in direction 2 Shear modulus Poisson ratio Mass density Layer thickness
E l = 38.6 G P a E2 = 8.27 G P a Gi2 = 4.14 G P a vl2 = 0.260 p = 1824 kg/m 3 hi = 0.125 m m
Dynamic response of composite plates under lateral loads
833
~"..0
(0*/90*)4
Step Looding ,
1 Term 9 Terms 25 Terms
- -
1.5
h
1.0
o. l l i_,, ,
°'°,oo
,
,
~
0.02
,
,~
t¢
~
,
,
,
-'v'- o.o4 Time
,
0.06
x ~ ,
,
,
0.08
,
,
,\,. '~o
(see)
Fig. 2. Center deflection time history of a rectangular (0°/90°)4 laminate subjected to a step loading (qo = 25 Pa).
approximation. In this figure, the non-dimensional longitudinal stress a x vs time has been plotted, calculated at the center point of the rectangular ACP laminate previously considered and at the bottom face of the seventh layer from above (0 ° fiber angle). Comparing Figs 2 and 3 for the deflection and stress time histories, respectively, it can be seen that, although stress results follow the global pattern of the deflection time history, significant differences occur an:tong the peak stress values calculated by the various degrees of approxin~tation. This happens despite the fact that, apart from the deflection IV, the stress function @ also converges satisfactorily with the forty nine series terms approximation. The very slow convergence of the stresses is explained by the observation that stresses are expressed in terms of the second derivative of the del]ection and the stress functions, thus not showing the same general behaviour with these two magnitudes. A further increase in the number of series terms leads to unacceptably high computer costs. Figure 3 presents the remarkable feature that results considering 1 and 25 terms are very close to each other, as is the case for those with 9 and 49 terms, although there is a big difference between the two pairs of curves. This happens due to the relative magnitude of the various coefficients of eqn (13) and (14) but the physical interpretation of this phenomenon is still puzzling.
834
N. G. Tsouvalis, V. J. Papazoglou 7.0 6.5
Step
Loading
, (0°/90°),
1 9 25 4-9
--........ - -
6.0 5.5
f!
5.0
Term Terms Terms Terms
:
f"7b
1/ !\
4.5 4-.0 ~x 3.5
l/
3.0
i
qo
'
ti
"',1
,
2.5
/i ,
t
iA'-,
dT,'l i
2.0
f/,' \.;
1.5
',,~
I' /i
{,.
1.0 0.5 0.0 0.00
0.02
0.04
Time (sec)
0.06
0.08
O. 10
Fig. 3. Longitudinal stress ex time history at the center of a rectangular (0°/90°)4 laminate subjected to a step loading (qo = 25 Pa). A closer look at the results of Fig. 2 indicates that, after the instantaneous application of the load, the laminate deflection oscillates about a mean value (the static solution), with a somewhat constant frequency. The non-harmonic nature of this periodic motion is exhibited by the asymmetry of the parts of each oscillation corresponding to the first and the second semiperiod intervals, as well as from the negative values of the deflection W for time values which are selected multiples of the period of oscillation. To better understand and explain these particular characteristics, we have calculated the response of the same laminate in the linear phase, under a small magnitude step load of qo = 1 Pa. The results are shown in Fig. 4, both for the total, and for the constituent deflection terms. Note that, since the laminate is orthotropic and not square ( a / b = 2.0), term W31 dominates over terms Wl3 and W33 which are negligibly small. The Fourier Transform of the deflection time history shown in Fig. 4 (qo = 1 Pa, linear phase) and Fig. 2 (qo = 25 Pa, nonlinear phase) is shown in Fig. 5 and Fig. 6, respectively. The first four natural frequencies of the laminate were calculated to be 29.412, 44.121, 73.695 and 106.804 Hz, corresponding to mode shapes (2,1), (2,3), (2,2) and (3,1), respectively. Comparing the frequencies of the harmonic constituents of the deflection time histories with the natural frequencies
Dynamic response of composite plates under lateral loads
835
0.07
Step Loading , (0°/90°)4 qo= 1 Pa
-
Total W
-
-- -- - - W l l . . . . . . . . W3o
0.06
oo+/l /i /, 1 0.03
/ /
o
t, /
1/.
0.()1 / o.<,o
0.00
0.02
/
~' /
~j/0.04
0.06 Time
....
0.08
O. 10
(see)
Fig. 4. Center deflection time history of a rectangular (0°/90°)4 laminate subjected to a step
loading (qo = 1 Pa). 3.0
~o'/9o'),
29.4.14 I'lZ
2.5
ep
Looding
, q~=,l Po
Total W ii
1.5
2.0 1.0 0.5 0.0
0
I
I
I0
I
|
20
|
I 30
| 40
I
| 50
Frequency
Fig. 5.
I 60
)3.678 HZ
I i, 70
I 80
i
I 90
* 100
(Hz)
Fourier Transform of the deflection time history of a rectangular (0°/90°)4 laminate subjected to a step loading (qo -- 1.0 Pa).
of the laminate, it becomes evident from Fig. 5 that, in the linear phase, the laminate oscillates with a fundamental frequency equal to its first natural frequency and with a superharmonic equal to its third natural frequency. Both these magnitudes are shifted to greater values for the nonlinear case of Fig. 6. Similar results were found in extensive numerical studies for other types of laminates, as well as for isotropic plates. 8
N. G. Tsouvalis, It'. J. Papazoglou
836 0.7
30.271Hz
0.6
(0*/90*h Step Looding Totol W
0.5
, q~=25
Po
0.4
0.3 0.2 0.! 0.0
0
I
I
I0
20
, J ~ ,
30
I
I
40
50
Frequency
I
I
60
I
I
7S.248 Hz e~. I
70
80
I
I
90
t
I00
(Hz)
Fig. 6. Fourier Transform of the deflection time history of a rectangular (00/900)4 laminate subjected to a step loading (qo = 25.0 Pa).
Comparison of results Results obtained by the present methodology have been compared with the ones of Bhimaraddi 15 for square plates under the action of a sinusoidally distributed step load. Bhimaraddi proposed a First-order Shear Deformation Theory (FSDT) to study the linear transient response of laminated plates, and he obtained the CLT solution as a limited case of his theory. The nondimensional stress time history #x of a thick (a/h = 10), square (a/b = 1.0), three-layer (00/900/0 °) symmetric laminate is shown in Fig. 7, calculated both by the present method and by Bhimaraddi (FSDT, CLT). The non-dimensionalization considered was tf=trx/qo and T = t(Ex/(a2p)) 1/2. The material properties of the layers w e r e E1/E2 = 25, G12/E2 = 0 . 5 , and vt2 = 0.25, whereas the three layers have not equal thicknesses; the middle one (90 °) is ten times thicker than each of the face ones (0°). It can be concluded from Fig. 7 that the present method correlates well with the CLT solution of Bhimaraddi. Regarding the comparison between the CLT and F S D T results, it is seen that CLT results in a stiffer plate, underpredicting the stress peak values and producing oscillations of higher frequency than the F S D T ones. The finite element computer code A D I N A 16 was also used for comparison purposes. Three-node flat triangular plate/shell elements with six D O F per node were used, with 144 elements in one quarter of the plate. The effect of the magnitude of the time step was investigated by performing the calculations with three time steps, namely At = 4, At = 2 and At = 0.5 ms. The comparison was made for a square orthotropic plate with a -- b = 50 cm, h = 1 mm, and having the material properties given in Table 1. The lateral load was uniformly distributed over the plate surface and applied either as a step or as a blast load with qo = 20 Pa. The blast load duration was taken as tl = 0.1 S. The corresponding results are shown in Figs 8 and 9. The updated Lagrangian formulation used by A D I N A results in a stiffer plate, giving, as
837
Dynamic response of composite plates under lateral loads 120 0"/90"/0* o / . b = 1.0 o / h = 10.0
Laminote
/ /
100
\
_...._ CLT FSDT
,~i \ \\\\
_ _
Present
80 o ¢.o_
== q~
6C,
"~,, \\\
E
lo o z
/ \
¢0,
20
\
\\
/ \
CI, ~ 0
5
10
15
20
'/
/
/
/
/
/ 25
30
Nondimensionol T
Fig. 7. Non-dimensional stress ax time history for a symmetric(0°/90°/0°) laminate subjected to a sinusoidallydistributed step loading. expected, smaller peak deflection values and an oscillating motion of higher frequency than the one arising from the analytical method. The frequency difference has the expected consequence of a phase shift of the A D I N A results with respect to the others. However, the peak differences are not unacceptably high, being about 10-12%, something which, along with the fact that the analytical solution gives conservative results, leads to the conclusion that the present method can be used as a design tool. Regarding time step effects, it can be said that the value of 0.5 ms is adequate for the present case, but great care must be given to the time step selection, if large time intereals are to be integrated.
Parametric study Our attention will first be focused on the effect of the various types of time dependent loads on the nonlinear response of laminates. This response, under three different types of load, is shown in Fig. 10. The lateral load is uniformly distributed and the laminate is a (0°/90°)4 cross-ply one with a = 50 cm, b := 25 cm, h = 1 m m and material properties as given in Table 1, The load m a x i m u m value is qo = 60 Pa and the duration of the rectangular
N. G. Tsouvalis, V..L Papazoglou
838 5.0 4..5 4.0 3.5 3.0 W©
2.5 h 2.0 1.5 1.0 0.5 0.0 0.'
~0
Time (sec)
Fig. 8. Centerdeflectiontime history of a square orthotropic plate subjected to a uniformly distributed step loading. pulse and the blast loading is tl 0.05 S. The solutions corresponding to the step loading and the rectangular pulse go up together to the neighbourhood of 0.05 s (phase I), where the application of the rectangular pulse ends. This is an expected feature, since at this time interval the two loads are identical. The motion is an oscillation, having a mean value near the static deflection solution (wdh = 1.33). After t = 0.05 s (phase II), the step loading response continues its oscillations as previously, while the rectangular pulse response begins a new oscillating motion around the zero deflection value, having a greater amplitude than previously. This amplitude of phase II free vibration motion can be less or more than the phase I one, or even zero, depending on the relation between the pulse duration tl and the first natural period of the plate T. s Differing from the other two, the blast loading response is an oscillating motion around a decreasing mean value, which becomes zero at t = 0.05 s. After that time (phase II), a free vibration motion begins around zero deflection. The rate of increase, as well as the peak values of deflection caused by the blast loading, are smaller than the ones generated by the step loading or the rectangular pulse, since, for the same time interval, the energy passed to the plate by the blast load is less than the corresponding one given by the other two load types. Therefore, in general, blast loadings produce =
Dynamic response of composite plates under lateral loads
839
4.0 Orthotropic Plote , o / b = 1 . 0 Blost Looding q . = 2 0 Po , t 1 = 0 . 1 sec
3.0
W©
,o oo -,.o
,,';"/
:'~ / I
'~
z
&
I
,
"7,kl
o.. I
,,k
i',l
i/I
",
•- ' ~ , /
- 2.0
-3.0 Time (sec)
Present theory _ _ ADINA At=O.O04 ___ ADINA At=O.O02 ........ ADINA At=O.O005
-4.0 Fig. 9. Center deflection time h i s t o r y o f a square o r t h o t r o p i c plate subjected to a u n i f o r m l y distributed blast loading.
smaller mILximum deflections than a step loading or a rectangular pulse with the same maximum pressure. The rectangular pulse response behaviour at phase II (the free vibration motion) depends on the relative magnitudes of the pulse duration tl and the laminate first natural period of free vibrations T, being similar to the corresponding behaviour of a spring-mass single-degree-of-freedom (SDOF) system. It can be readily proven 8 that a zero deflection and velocity condition at t = tl is obtained when h is an integer multiple of the laminate natural period T and, therefore,
XlI=O when t l = n T ,
n = 1,2,3,...
(16)
This period is the one corresponding to the degree of the motion nonlinearity and is not the one calculated by the linear eigenvalue problem, s For example, the (0°/90°)4 laminate of Fig. 10, having the linear natural frequencies mentioned in section 3.1, exhibits the response of Fig. 11 under the action of a rectangular pulse with qo = 60 Pa and tl = 0.03 s. Its linear first natural period is 7"21 = 0.034 s (f21 = 29.412 Hz), and thus, the duration h = 0.03 s
840
N. G. Tsouvalis, V. J. Papazoglou 3.0 (0*/90*),
Lominote
2.5 2.0
,? !io,/oo.i olo
1.5 1.0 0.5 We
-h
0.0 O.q ,o -0.5 -1.0 .
',\
\ /
/:
,
(
-1.5 -2.0 -2.5
....
Step Loodin 9 Rectongulor Pulse Blost Loodin 9 q , = 6 0 Po t , = O . 0 5 see
It
II
t
I
I
I
~n~e (see) -3.0
Fig. 10. Center deflectiontime history for a (0°/90°)4 laminate subjected to various types of time dependentuniform lateral pressure. is near its first natural frequency when w J h ,~ 2.5, resulting in near zero oscillations during phase II, as can be seen in Fig. 11. The nonlinear response of the (0°/90°)4 laminate considered previously, under the action of different pulse types, namely a sine pulse and a triangular pulse with qo = 60 Pa and tl = 0.05 s, is shown in Fig. 12. Comparing this figure with Fig. 10 for the rectangular pulse and blast loadings with equal intensity and duration, it is seen that sine and triangular pulses produce in general less intense oscillating motions, having smaller peak values and a larger period. This fact is due naturally to the gradual increase of pressure for these two pulse types, in contrast to the sudden increase of the step loading and the rectangular pulse. Seeking relations analogous to the ones of eqn (16) between the pulse duration tl and the first natural period T in order for the phase II oscillating motion to vanish, we refer once more to the SDOF system. For the case of a sine pulse of intensity qo and duration tl, in order for the motion at phase II to vanish, the following condition must hold: Xn = 0
T when tl = ( 2 n + l ) ~ -
n = 1,2,3,...
(17)
Dynamic response of composite plates under lateral loads
841
3.0 (0"/90°),
Laminate
2.5
2.0 1.5
1.0 0.5 We
--
h
0.0 O.q:LO
i
o.o~..,/
0.02
x,~
-0.5 -1.0 -1.5 -2.0
Rectangular Pulse q . = 6 0 Po t 1 = 0 . 0 3 sec
-2.5
Time (sec) -3.0
Fig. 11. Ceater deflection time history for a (0°/90°)4 laminate subjected to a rectangular pulse with duration near its first natural period.
As an example, the response of the (00/900)4 laminate under examination is shown in Fig. 13 for two pulse periods, tl = 0.045 s (,~1.5 x T21) and tl = 0.075 s (;~2.5 x T21). It is obvious that the phase II motion is a near zero oscillation. In the case of a triangular pulse of peak value qo and duration tl, the corresponding condition is Xn=0
whentl=2nT
n=1,2,3,...
(18)
The parametric study continues with the effect of the fiber orientation angle of an angle-ply laminate on its nonlinear response under the action of a step loading. This response is shown in Fig. 14 for a (+0)4 glass/epoxy angle-ply laminate with a = 50 cm, b = 25 cm, h = 1 mm and having the material properties of Table 1. Note that the zero direction is the one parallel to the long edge of the laminate. The spatial distribution of the load over the plate area is assumed to be uniform. As expected, Fig. 14 demonstrates that the increase of the fiber angle up to 90 ° leads to stiffer plates, exhibiting smaller deflection peak values and a smaller period of the oscillating motion. The effect of increasing maximum load value on the response of the previous angle-ply laminate with 0 = 30 ° can be seen in Fig. 15. The load
N. G. Tsouvalis, V. J. Papazoglou
842 2.0
~'x'~'~ i
(0°/90°),
Laminate
1.5
1.0
0.5
1// i/
Wc O.N
~\
/// k\
II
0
I
k
/
-0.,'
- 1.( - .... -1.5
Sine Pulse T r i a n g u l a r Pulse q o = 6 0 Pa h = O . 0 5 sec
T/me (sec) -2.0
Fig. 12. Center deflection time history for a (0°/90°)4 laminate subjected to uniform lateral sine and triangular pulse pressure. type is a blast with duration tl = 0.05 s and maximum values qo varying from 10 to 75 Pa. Two general remarks can be made by observing Fig. 15. The oscillating motions corresponding to higher applied loads have a smaller period (higher frequency) of vibration. The form of each oscillation is symmetric and smooth for small loads (small nonlinearities), becoming nonsymmetric as the load increases. This change is due to the fact that the relative magnitude of the deflection series terms other than the dominant one (W13, W31, W33 in this case) with respect to the dominant one (W11) increases with increasing nonlinearity. The effect of damping is investigated in Fig. 16 for a (900/00)4 glass/epoxy cross-ply laminate with a = 50 cm, b = 25 cm, h = 1 mm and having the material properties given in Table 1. The laminate is subjected to a sinusoidally distributed step loading of intensity qo = 60 Pa. Information about damping characteristics and properties of composite materials are rarely presented in the literature. A lot of experimental study has yet to be made in this field for a better understanding of the phenomenon, as well as for the collection of a fair amount of damping properties data. The damping ratio (= c/cc) has been reported to range between 0.005 and 0.05 for the common type of composites used in aircraft structures. 2 Results for four different
Dynamic response of composite plates under lateral loads
843
2.2 1.8
~
(0"/90=)s, L a m i n a t e
0.6
Wc
0.2
,
h
002
,
",,
~
o
0.06
,
,
" ",, , "/~.--""7'~, o.
-0.6 - 1 . 0 ~-
Sine P u l s e
" -1.4 . -~1.8
~ ....
qo=60 Po L1=0.045 h=0"075 Time (sec)
-2.2
Fig.
13. Center deflection dme history for a (0°/90°)4 laminate subjected to two uniform lateral sine pulses with tl = 0.045 s and tl = 0.075 s.
values o1" the damping ratio ( are presented in Fig. 16. All results finally converge to the static solution wc/h = 0.92, as expected. The parametric study is completed with some stress results. Due to the fact that no acceptable stress convergence is obtained for a uniformly distributed lateral load, the study has been performed for a sinusoidaUy loaded laminate, where a one term deflection expansion accurately represents the solution. A rectangular (a = 50 cm, b = 25 cm, h = 1 ram) (90°/ 0°)4 laminate is considered having the material properties given in Table 1 and loaded by a rectangular pulse of intensity qo = 60 Pa and duration tl = 0.04 s. "['he resulting deflection time history response is shown in Fig. 17, where the one term solution produces a smooth and symmetric oscillating motion. Stresses have been calculated at the laminate center, at its lower surface. The time history of the transverse stress gy is shown in Fig. 18, where both the total stress and its membrane and bending components are plotted. It can be clearly seen in this figure that the membrane stress dominates the bending one and is the major component of the total stress. It can also be seen that the bending stress follows an oscillating motion with alternating sign, similar to the one for deflection, whereas the membrane stress, although presenting exactly the same oscillating motion (same
844
N. G. Tsouvalis, IF. J. Papazoglou 2.5
~T
~/-Ct)4 Laminate ep Loading , q o = 2 5
~ 0 = ....... .... ___
Pa
_
2.0
_
0 " 0=30* .~=45 ° 0=60 ° 0=90 °
1.5
we h 1.0
i/,
,f,
111j
0.5
I /
0.0 0.00
",, , , ,',?:!,X,::\", 0.02
" j~.a"
-0.~4
\
.~l's
"
008
010
Time (see)
Fig. 14. Center deflection time history for a (+0)4 angle-ply laminate subjected to a uniform
lateral step loading.
frequency), preserves its sign, being always tensile. Therefore, we see that stress o'y at the bottom of the laminate is always tensile, irrespective of the positive or negative corresponding deflection values. This is also the case for the longitudinal stress ¢rx.s
CONCLUSIONS An analytic tool was presented in this paper for the nonlinear static and dynamic response calculation of simply supported composite laminated plates under the action of lateral loads. The laminate can have any stacking sequence, whereas its edges were assumed to be stress-free, modelling in this way the less stiff possible configuration of a simply supported plate. Taking into account the fact that, in marine structures made of ordinary composite materials, the governing criterion is that of maximum permissible deformations due to the low elastic moduli of these materials, the results of the present method are conservative with respect to any other type of boundary conditions, thus constituting an upper limit for all possible configurations.
Dynamlc response of composite plates under lateral loads
845
2.5
2.C I.~
I.¢
0.~ we -0.¢
h
C
- I ,(
-t.!
-2.(
Pig. 15. Center deflection time history for a (4-300)4 angle-ply laminate subjected to a uniform lateral blast loading of various intensities.
qo
It
t (a) Step loading
q
t (b) Unearly increasing load
q q,
m
tl (c) Rectangular pulse
t
m
tl
t
(d) Blast Ioodin 9
Fig. 16. Center deflection time history for a (90°/0°)4 laminate subjected to a sinusoidal step loading for various damping ratios.
846
N. G. Tsouvalis, IF. J. Papazoglou 2.C
1.C
0..' We h
O.C C -0.~
- 1 .(
-1 .~
-2.(
Fig. 17. Center deflection time history for a (90°/0°)4 laminate subjected to a sinusoidal lateral rectangular pulse.
0.25 (go=/o*)4 Laminate a/b=2.0
rt
Rectangular Pulse q===60 Pa . t1=0.04 sec
0.20 Total Stress --Membrane Stress ........ Bending Stress
I i
0.15
?
o
f
~
-t
0.10
J
II
I
0.05 ,°, 0.00
-0.05 0.00
m
0.02
,.,,/
0.04
0.06
0.06
0.10
0.12
', ,,'
0.14
0.16
'%/,
0.16
0.20
Time (see)
Fig. ]8. Stress ay time history (x = a/2, y = b/2, z = - h / 2 ) for a (90°/0°)4 laminate subjected to a sinusoidal lateral rectangular pulse.
Dynamic response of composite plates under lateral loads
847
An extensive parametric study was carried out, revealing that the undamped response of a laminate under a step loading is an oscillating motion with fundamental frequency near its lowest natural frequency. This fundamental frequency increases slightly with increasing degree of nonlinearity. Referring to the various types of load time variations, the maximum deflection peak values are produced by the step and the rectangular pulse loadings, whereas the minimum ones are produced by the triangular pulse, when the,, load maximum value and pulse duration are the same. For the rectangular, sine and triangular pulses, special relations between the duration of load application and the lowest plate natural period, result in very small deflections after the removal of the load. This characteristic can be very significant in designing structures which are subjected frequently to pulse loads of known and approximately fixed duration, since, with a proper selection of material and geometric configuration, fatigue problems can be minimized. Furthermore, special attention must be paid to the fact that maximum deflections may occur even after the load removal, during the free oscillation phase of the plate, and, therefore, proper consideration must be made when selecting the time interval for our calculations. A furtlaer development of the method should incorporate deflection and stress function series expansions capable of modelling more types of boundary conditions, and resulting in faster convergence for the stress calculations. An extensive experimental verification of the results presented here is considered to be necessary, as it will help in acquiring more confidence for the special characteristics revealed in the present study.
REFERENCES 1. Alwar, R.S. & Adimurthy, N.K., Non-linear dynamic response of sandwich panel,~ under pulse and shock type excitations. J. Sound and Vibration, 39 (1975) 43-54. 2. Mei, C. & Wentz, K.R., Large-amplitude random response of angle-ply laminated composite plates. AIAA J., 20 (1982) 1450-1458. 3. Birman, V. & Bert, C.W., Behaviour of laminated plates subjected to conventional blast. Int. J. Impact Engng, 6 (1987) 145-155. 4. Reddy, J.N., Dynamic (transient) analysis of layered anisotropic compositematerial plates. Int. J. Num. Meth. Engng, 19 (1983) 237-255. 5. Redd'.¢,J.N. Geometrically nonlinear transient analysis of laminated composite plates AIAA J., 21 (1983) 621-629. 6. Kant, T., Ravichandran, R.V., Pandya, B.N., & Mallikarjuna, Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory. Comp. Struct., 9 (1988) 319-342. 7. Chia, C.Y., Geometrically nonlinear behavior of composite plates: A review. Appi. Mech. Rev., 41 (1988) 439-451.
848
N. G. Tsouvalis, V. J. Papazoglou
8. Tsouvalis, N. G., Static and dynamic response of simply supported composite laminated plates under the action of in-plane compressive and/or lateral loads. PhD thesis, Dept. of Naval Architecture and Marine Engineering, National Technical University of Athens, 1993. 9. Chia, C.Y. & Prabhakara, M.K., Postbuckling behavior of unsymmetrically layered anisotropic rectangular plates. J. Appl. Mechl, 41 (1974) 155-162. 10. Nayfeh, A. H. & Mook, D. T., Nonlinear Oscillations, J. Wiley and Sons, New York, 1979. 11. Clough, R. W. & Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1975. 12. Zhang, Y. & Matthews, F.L., Postbuckling behaviour of curved panels of generally layered composite materials. Comp. Struet., 1 (1983) 115-135. 13. Zhang, Y. & Matthews, F.L., Postbuckling behavior of anisotropic laminated plates under pure shear and shear combined with compressive loading. AIAA J., 22 (1984) 281-286. 14. Gupta, A.D., Gregory, F.H., Bitting, R.L. & Bhattacharya, S., Dynamic analysis of an explosively loaded hinged rectangular plate. Computers & Struct., 26 (1987) 339-344. 15. Bhimaraddi, A., Static and transient response of rectangular plates. Thin- Walled Struct., 5 (1987) 125-143. 16. ADINA--A Finite element program for automatic dynamic incremental nonlinear analysis: User's Manual. Report AE 84-1, ADINA Engineering, Massachusetts, 1984.
© 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0951-8339/96/Sl 5.00 ELSEVIER
0951-8339(95)00048-0
Large Deflection Dynamic Response of Composite Laminated Plates Under Lateral Loads N. G. Tsouvalis & V. J. Papazoglou* ShipbuildingTechnologyLaboratory, Dept. of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou Ave, GR-157 73 Zografos, Greece (Received 23 September 1995; revisedversion received9 November 1995; accepted 18 November 1995) ABSTRACT A very effective analytical tool for the nonlinear large deflection response of simply supported laminated plates under the action of various types of time dependent lateral loads is presented. The solution, based on the Classical Lamination Theory and accounting for moderately large deflections, is obtained by applying the Galerkin procedure. Several numerical results are presented, including a convergence study, comparisons with other published results, and parametric studies. The relationship between duration of load application and lowest plate natural period that leads to a minimum response after load removal is investigated. Copyright © 1996 Elsevier Science Limited. Key words: laminated plates, dynamic response, large deflection. INTRODUCTION Considering the loads acting on a marine structure throughout its operational life, we shall hardly find any exhibiting pure static behaviour. The majority of the loading conditions, including the most crucial ones for the integrity of the structure, are characterised by their dynamic nature, which is usually stochastic. A typical example is the slamming phenomenon at the bow of small, fast power boats, as they run with high speeds in rough or moderately rough seas, as well as high pressures at the hull bottom caused by underwalter explosions or at warship superstructures caused by air-blasts. Analogous conditions are encountered in the aircraft industry, another constructional branch where the use of composite materials is continuously * To whom correspondance should be addressed 825
826
N. G. Tsouvalis, 1I. J. Papazoglou
increasing. For the accomplishment of an effective structural design, these dynamic loads should, in all cases, be taken into account. A dynamic plate bending analysis becomes, therefore, imperative when carrying out a structural design study. The need for considering nonlinear geometric effects in bending stress analysis under dynamic loads became obvious rather early and, thus, numerous studies have been undertaken. An early work on this subject is that by Alwar & Adimurthy, 1 who dealt with simply-supported rectangular sandwich plates excited by lateral pulse or sonic boom loadings. Using Classical Lamination Theory (CLT), Mei & Wentz 2 studied analytically the large deflection response of regular antisymmetric angle-ply laminated rectangular plates, simply supported or clamped at all edges. Considering a one-term expansion of the deflection, w, and the stress function, F, and using the Galerkin method, they arrived at a nonlinear differential equation with respect to time. The solution was obtained by linearizing this equation. Birman & Bert 3 solved the final differential equation with respect to time by means of the Runge-Kutta numerical integration method. Only simply supported boundary conditions were examined. A fair amount of finite element studies have also dealt with the problem under consideration. A nine-node rectangular isoparametric shear-deformable finite element was employed by Reddy 4 to investigate the response of isotropic, orthotropic and layered anisotropic composite plates. The same author modified the above element to include the nonlinear strain-displacement relations of the von K~irm/m plate theory. 5 A Higher Order Shear Deformation theory was employed by Kant e t al. 6 as the basic theory for the development of a shear deformable finite element, to investigate the transient response of laminated composite plates. A detailed review of over 200 references dealing with the geometrically nonlinear behaviour of composite plates has been performed by Chia. 7 The above survey shows the lack of a relatively simple, quick and reasonably accurate analytical solution of the dynamic transient response of laminated plates. The existing analytical solutions either do not account for large deflections or, if they do, they seem not to be very accurate, since they incorporate single term expansions of the displacements. In the present study an analytical solution of the bending behaviour of laminated plates of any stacking sequence is given, under the action of various dynamic lateral loads. The solution is based on the CLT (thus, is valid only for thin plates) and accounts for moderately large deflections, since it employs strain-displacement relations in the von K/trm/m sense. The final set of the nonlinear governing differential equations is solved with the aid of a numerical integration scheme. The laminate is considered to be simply supported at its boundaries with edges free of stresses in the plane of the plate. Initial out-of-plane imperfections, as well as the damping properties of the material, are taken into account.
Dynamic response of compositeplates under lateral loads
827
THEORETICAL ANALYSIS
Governing equations Consider a rectangular plate of length a (x-direction), width b (y-direction) and thickness h (z-direction), subjected to an arbitrarily distributed lateral load q(x,y), which, in addition, varies with time. The plate is assumed to consist of any number of composite material layers with different thicknesses, different material elastic properties and arbitrary fiber orientation angles with respect to the plate geometric axes. Furthermore, let the plate exhibit a steady out-of-plane initial imperfection Wo(X,y),forming any shape and having maximum values in the order of magnitude of the plate thickness. The derivation of the equations governing the dynamic flexural behaviour of the htminate under the action of a time dependent lateral load q, is accomplished with the aid of the principle of minimum total potential energy. 8 Introducing the Airy stress function and through proper nondimensionalisation,9 we obtain the following non-dimensional equilibrium and compatibility equations of the problem, respectively:
o:w
ow
OZ2 q- 2,~(Omn~
04(w - Wo) "l-
O~4
+ d3,~304(W - Wo) •
O~ Or/3
o~: o,/ •N 11/
d4
b-~ o~ ~
O~30,rI
0~20,02
,~404(W - Wo) 04t~ 04(~ 07]4 + bl - ~ + b2/]. O~ 30~
04CI) Jr- b4 ,~3 _ _
+ b3 22 04~
-
l-
,~04(w - Wo) +d22204(W- Wo) "k- dl
.4 04(1)
(1)
2°~ o~w o~O2w) o--U-oo o~o---~~ o¢ ~ ~ - °'~" a3 ~3 0"~(I)
o¢o----~ + a4
o-¢,~ - a, ~ O¢--r~ + a~ ; O~~ O-------7 ~ ( w - Wo) b, 04(m - Wo) -- b6
- b9V
O~4
o'(w- Wo) - ~
7A
-O-~b~
=0
/].40~(I) O-V
;¢ 04(w - Wo)
b8
0¢20rl2 (2)
Wo) O,74
blo 24 04(W -
O~W O2W oe
oe
I O~Wo'X~ 02 Wo ~ o ] + oe
"
828
N. G. Tsouvalis, V. J. Papazoglou
These two equations are coupled partial differential equations with two dependent variables, the non-dimensional total out-of-plane deflection W (including the initial imperfection Wo) and the non-dimensional stress function • . Damping is represented as a linear viscous modal damping with damping ratio ~,10,11, whereas &.,n is the non-dimensional circular natural frequency of the laminate corresponding to mode (m,n), ~ and r/are the nondimensional distances in the x and y directions, respectively, z is the nondimensional time, 2 is the plate aspect ratio, g/is the non-dimensional lateral load and ai, di, bi, Dll, A22 are non-dimensional functions of the plate rigidities. There is no classical closed form solution to the above problem and thus, to proceed further in the solution, we employed the Galerkin technique by assuming double sine Fourier series for the non-dimensional lateral deflection W and initial imperfection Wo, with coefficients W~jand Wo~, respectively, and the following series for the non-dimensional stress function (~:9,12,13 :
E ~mn Xm(¢) Y. (r/). m=l n=l
(3)
Notice that the initial imperfection of the laminate is assumed to be known and thus the coefficients Woij are given. The functions Xi and Y; in the stress function expansion are characteristic beam functions defined as Xi(O = cosh ~xi~-cos 0q~-yi(sinh ~i~-sin ~xi~) Yi(r]) = cosh 0qr/-cos 0qr/-yi(sinh ~07-sin sir/)
(4)
in which ~; and 7, are constants given in Ref. 9 or Ref. 12. We furthermore assume that the non-dimensional lateral load ~/(~,r/,z) can be expanded in a double sine series with coefficients ~mn(Z). For various types of lateral loads, these coefficients become: (a) Uniform load ~7oon the entire plate: {lm,(Z) =
{ 16g/o(Z), m , n = o d d ~2mn O, m, n ---- even
(5)
(b) Sinusoidal load ~(~,O,z)--~o(Z) sin(~O sin(m/): I~o(Z) (~mn(T') = 0
form=n= 1 for m, n/> 2.
(6)
Regarding the time variation of the dimensional applied lateral load q(t), the following commonly encountered forcing functions are taken into account (see Fig. 1), where q(t) = 0 for t < 0 in all cases:
Dynamic response of composite plates under lateral loads
829
q
q,
t
t
(b) Linearly increasing load
(a) Step loading
q°
tl
tl t (d) Blast loading
t
(c) Rectangular pulse
qo
q°
t, t
tl
(e) Half-sine pulse
t
(f) Triangular pulse
Fig. 1. Various time variations of the applied lateral load q.
(a) Step loading: q(t) = qo
(7)
(b) Linearly increasing load: q( t ) = qstt
where qst is the rate of the load increase, measured in pressure/time. (c) Rectangular pulse:
(8)
N. G. Tsouvalis, V. J. Papazoglou
830
q(t)
Sq°'
fort~
I 0,
f o r t > tl
(9)
where tl is the duration of the pulse. (d) Blast loading:
q ( t ) = { q ° ( 1 - - ~) e -~t , , O,
for t ~< tl
(lO)
for t > h
where qo is the peak pressure in excess of the ambient pressure, t~ is the duration of the pulse and ~ is a coefficient. The values of the blast parameters depend on the distance of the plate from the charge location and are usually taken as 0~ = 1.98 and tl = 0.004 sec 14 for air-blast loading, whereas no relevant data exist for underwater explosion loading. (e) Half-sine pulse: .
q(t) =
/tt
qosm t--l' O,
for t ~< t 1
(11)
for t > t~
where tl is the duration of the pulse. (f) Triangular pulse: 2t
q(t) =
qo~,
t~ fort~< ~-
- t) qo 2(tlt------~'
for tl
0,
for t < t~
2 < t <~tl
(12)
where tl is the duration of the pulse. The above defined functions for W and • satisfy simply supported boundary conditions, with the laminate edges being free to deform in the plane of the plate (1-2), since U and V are not constant at the boundaries. However, the functions for W and ~ do not satisfy the zero moment condition at the boundaries (M e = 0 at ¢ = 0, 1 and M,7 = 0 at r/ = 0, 1) for angle-ply laminates. The governing equations are solved by substituting the derivatives of W, Wo and F, multiplying the compatibility equation by Xv( O. Yq(ri) and the equilibrium one by sinpn~.sinqnr/and integrating both equations over the whole plate area. Two boundary integrals included in the equilibrium equation take into account the effects of the unbalanced edge moments.
Dynamic response of composite plates under lateral loads
831
Performing all substitutions and integrations in the two governing equations, a final set of equations is obtained in terms of the coefficients W o. and ~,,~. This set of equations contains an infinite number of nonlinear second order differential and algebraic equations to be solved simultaneously. In practice, only a finite number of these equations will be used to obtain the laminate bending behaviour to a specified accuracy. If we ,consider that indices i, j, m and n vary from 1 to M, that is, considering the first M 2 terms of the deflection and the stress function series expansions, the equilibrium eqn (1) results in a set of M 2 nonlinear second order differential equations with respect to time, while the compatibility eqn (2) gives another set of M 2 nonlinear algebraic equations. The general form of these two types of equations is: (a) Equilibrium equations: M
M
d2Wpq+ z, qd~"q+(W,q- Wo,q)A'q+ ~ ~ (W,- Wo~)B~: dz 2
oz
M
i=1 j=l
M
M
M
M
M
q + Z Z +.:c~. + y;= E E E +.:w:=,~ + ~q = 0 m=l n=l
m=ln=l
(13)
i=1 j=l
p,q= 1,2,...,M
(b) Compatibility equations: M
E
M
M
~,E ¢I~mnHPmq-~- E
m=ln=l
M
E
(l/Vii - W,'~l(Pqoijl "~ij
i=1 j=l
M
M
M
M
(14)
+ Z Z E E(w.w~,- Wo.:o~,)L.~,: 0 i=lj=lk=ll=l
p,q= 1,2,...,M
where the various coefficients
Z pq' A pq, nPqij , CPqrrm,OPqmnij, oPq, I-IPqmn, gPqij, LPqijkl are functions of the damping ratio, the dimensions and the rigidities of the laminate and the applied load. 8 S o l u t i o n procedure
Expressing the stress function coefficients t~mnin terms of the deflection ones W,,~, we.,end up with a set of M 2 coupled nonlinear second order differential equations in terms of Wm,, only. s The general form of the ith of these equations is
N. G. Tsouvalis, V. J. Papazoglou
832
d2Wi
dz 2 +
dWi Zi---~z = f ( z , W1, W2,..., WM2)
(15)
subject to the initial conditions (at T = 0) Wi = Wio and d Wg/d'r = 0, signifying that the laminate is at rest before load application. The above equation describes an initial value problem and is solved numerically.
N U M E R I C A L RESULTS The laminates are assumed to be formed by alternate layers of a glass/epoxy composite, having the material and physical properties shown in Table 1. The numerical study was performed using the Runge-Kutta adaptive step integrator, as the most efficient integration tool. This decision was based on a thorough investigation of various alternative numerical schemes, s
Convergencestudy A convergence study was performed to obtain the minimum number of the deflection W and stress function • series terms required for an adequate representation of these two magnitudes, when the lateral load is uniformly distributed over the laminate area. In Fig. 2 the non-dimensional center deflection Wc ( = wdh, where h is the laminate thickness) vs time of a rectangular (a = 50 cm, b = 25 cm, h = 1 mm) antisymmetric cross-ply (ACP) (0°/ 90°)4 laminate has been plotted, for a one, nine and twenty five series terms approximation. A step loading is considered, having a maximum value of qo = 25 Pa, whereas damping effects have been neglected. On the basis of this and numerous other runs, the nine terms approximation seems to give quite accurate results for the deflection of a laminate under a uniform spatial distribution of the lateral load and, thus, all the following numerical results have been obtained with this number of series terms. Nevertheless, this is not the case for the calculated stresses, since, as it can be seen from Fig. 3, convergence has not been attained even for a forty nine series terms (M = 7) TABLE 1 Properties of glass/epoxy composite Young's modulus in direction 1 Young's modulus in direction 2 Shear modulus Poisson ratio Mass density Layer thickness
E l = 38.6 G P a E2 = 8.27 G P a Gi2 = 4.14 G P a vl2 = 0.260 p = 1824 kg/m 3 hi = 0.125 m m
Dynamic response of composite plates under lateral loads
833
~"..0
(0*/90*)4
Step Looding ,
1 Term 9 Terms 25 Terms
- -
1.5
h
1.0
o. l l i_,, ,
°'°,oo
,
,
~
0.02
,
,~
t¢
~
,
,
,
-'v'- o.o4 Time
,
0.06
x ~ ,
,
,
0.08
,
,
,\,. '~o
(see)
Fig. 2. Center deflection time history of a rectangular (0°/90°)4 laminate subjected to a step loading (qo = 25 Pa).
approximation. In this figure, the non-dimensional longitudinal stress a x vs time has been plotted, calculated at the center point of the rectangular ACP laminate previously considered and at the bottom face of the seventh layer from above (0 ° fiber angle). Comparing Figs 2 and 3 for the deflection and stress time histories, respectively, it can be seen that, although stress results follow the global pattern of the deflection time history, significant differences occur an:tong the peak stress values calculated by the various degrees of approxin~tation. This happens despite the fact that, apart from the deflection IV, the stress function @ also converges satisfactorily with the forty nine series terms approximation. The very slow convergence of the stresses is explained by the observation that stresses are expressed in terms of the second derivative of the del]ection and the stress functions, thus not showing the same general behaviour with these two magnitudes. A further increase in the number of series terms leads to unacceptably high computer costs. Figure 3 presents the remarkable feature that results considering 1 and 25 terms are very close to each other, as is the case for those with 9 and 49 terms, although there is a big difference between the two pairs of curves. This happens due to the relative magnitude of the various coefficients of eqn (13) and (14) but the physical interpretation of this phenomenon is still puzzling.
834
N. G. Tsouvalis, V. J. Papazoglou 7.0 6.5
Step
Loading
, (0°/90°),
1 9 25 4-9
--........ - -
6.0 5.5
f!
5.0
Term Terms Terms Terms
:
f"7b
1/ !\
4.5 4-.0 ~x 3.5
l/
3.0
i
qo
'
ti
"',1
,
2.5
/i ,
t
iA'-,
dT,'l i
2.0
f/,' \.;
1.5
',,~
I' /i
{,.
1.0 0.5 0.0 0.00
0.02
0.04
Time (sec)
0.06
0.08
O. 10
Fig. 3. Longitudinal stress ex time history at the center of a rectangular (0°/90°)4 laminate subjected to a step loading (qo = 25 Pa). A closer look at the results of Fig. 2 indicates that, after the instantaneous application of the load, the laminate deflection oscillates about a mean value (the static solution), with a somewhat constant frequency. The non-harmonic nature of this periodic motion is exhibited by the asymmetry of the parts of each oscillation corresponding to the first and the second semiperiod intervals, as well as from the negative values of the deflection W for time values which are selected multiples of the period of oscillation. To better understand and explain these particular characteristics, we have calculated the response of the same laminate in the linear phase, under a small magnitude step load of qo = 1 Pa. The results are shown in Fig. 4, both for the total, and for the constituent deflection terms. Note that, since the laminate is orthotropic and not square ( a / b = 2.0), term W31 dominates over terms Wl3 and W33 which are negligibly small. The Fourier Transform of the deflection time history shown in Fig. 4 (qo = 1 Pa, linear phase) and Fig. 2 (qo = 25 Pa, nonlinear phase) is shown in Fig. 5 and Fig. 6, respectively. The first four natural frequencies of the laminate were calculated to be 29.412, 44.121, 73.695 and 106.804 Hz, corresponding to mode shapes (2,1), (2,3), (2,2) and (3,1), respectively. Comparing the frequencies of the harmonic constituents of the deflection time histories with the natural frequencies
Dynamic response of composite plates under lateral loads
835
0.07
Step Loading , (0°/90°)4 qo= 1 Pa
-
Total W
-
-- -- - - W l l . . . . . . . . W3o
0.06
oo+/l /i /, 1 0.03
/ /
o
t, /
1/.
0.()1 / o.<,o
0.00
0.02
/
~' /
~j/0.04
0.06 Time
....
0.08
O. 10
(see)
Fig. 4. Center deflection time history of a rectangular (0°/90°)4 laminate subjected to a step
loading (qo = 1 Pa). 3.0
~o'/9o'),
29.4.14 I'lZ
2.5
ep
Looding
, q~=,l Po
Total W ii
1.5
2.0 1.0 0.5 0.0
0
I
I
I0
I
|
20
|
I 30
| 40
I
| 50
Frequency
Fig. 5.
I 60
)3.678 HZ
I i, 70
I 80
i
I 90
* 100
(Hz)
Fourier Transform of the deflection time history of a rectangular (0°/90°)4 laminate subjected to a step loading (qo -- 1.0 Pa).
of the laminate, it becomes evident from Fig. 5 that, in the linear phase, the laminate oscillates with a fundamental frequency equal to its first natural frequency and with a superharmonic equal to its third natural frequency. Both these magnitudes are shifted to greater values for the nonlinear case of Fig. 6. Similar results were found in extensive numerical studies for other types of laminates, as well as for isotropic plates. 8
N. G. Tsouvalis, It'. J. Papazoglou
836 0.7
30.271Hz
0.6
(0*/90*h Step Looding Totol W
0.5
, q~=25
Po
0.4
0.3 0.2 0.! 0.0
0
I
I
I0
20
, J ~ ,
30
I
I
40
50
Frequency
I
I
60
I
I
7S.248 Hz e~. I
70
80
I
I
90
t
I00
(Hz)
Fig. 6. Fourier Transform of the deflection time history of a rectangular (00/900)4 laminate subjected to a step loading (qo = 25.0 Pa).
Comparison of results Results obtained by the present methodology have been compared with the ones of Bhimaraddi 15 for square plates under the action of a sinusoidally distributed step load. Bhimaraddi proposed a First-order Shear Deformation Theory (FSDT) to study the linear transient response of laminated plates, and he obtained the CLT solution as a limited case of his theory. The nondimensional stress time history #x of a thick (a/h = 10), square (a/b = 1.0), three-layer (00/900/0 °) symmetric laminate is shown in Fig. 7, calculated both by the present method and by Bhimaraddi (FSDT, CLT). The non-dimensionalization considered was tf=trx/qo and T = t(Ex/(a2p)) 1/2. The material properties of the layers w e r e E1/E2 = 25, G12/E2 = 0 . 5 , and vt2 = 0.25, whereas the three layers have not equal thicknesses; the middle one (90 °) is ten times thicker than each of the face ones (0°). It can be concluded from Fig. 7 that the present method correlates well with the CLT solution of Bhimaraddi. Regarding the comparison between the CLT and F S D T results, it is seen that CLT results in a stiffer plate, underpredicting the stress peak values and producing oscillations of higher frequency than the F S D T ones. The finite element computer code A D I N A 16 was also used for comparison purposes. Three-node flat triangular plate/shell elements with six D O F per node were used, with 144 elements in one quarter of the plate. The effect of the magnitude of the time step was investigated by performing the calculations with three time steps, namely At = 4, At = 2 and At = 0.5 ms. The comparison was made for a square orthotropic plate with a -- b = 50 cm, h = 1 mm, and having the material properties given in Table 1. The lateral load was uniformly distributed over the plate surface and applied either as a step or as a blast load with qo = 20 Pa. The blast load duration was taken as tl = 0.1 S. The corresponding results are shown in Figs 8 and 9. The updated Lagrangian formulation used by A D I N A results in a stiffer plate, giving, as
837
Dynamic response of composite plates under lateral loads 120 0"/90"/0* o / . b = 1.0 o / h = 10.0
Laminote
/ /
100
\
_...._ CLT FSDT
,~i \ \\\\
_ _
Present
80 o ¢.o_
== q~
6C,
"~,, \\\
E
lo o z
/ \
¢0,
20
\
\\
/ \
CI, ~ 0
5
10
15
20
'/
/
/
/
/
/ 25
30
Nondimensionol T
Fig. 7. Non-dimensional stress ax time history for a symmetric(0°/90°/0°) laminate subjected to a sinusoidallydistributed step loading. expected, smaller peak deflection values and an oscillating motion of higher frequency than the one arising from the analytical method. The frequency difference has the expected consequence of a phase shift of the A D I N A results with respect to the others. However, the peak differences are not unacceptably high, being about 10-12%, something which, along with the fact that the analytical solution gives conservative results, leads to the conclusion that the present method can be used as a design tool. Regarding time step effects, it can be said that the value of 0.5 ms is adequate for the present case, but great care must be given to the time step selection, if large time intereals are to be integrated.
Parametric study Our attention will first be focused on the effect of the various types of time dependent loads on the nonlinear response of laminates. This response, under three different types of load, is shown in Fig. 10. The lateral load is uniformly distributed and the laminate is a (0°/90°)4 cross-ply one with a = 50 cm, b := 25 cm, h = 1 m m and material properties as given in Table 1, The load m a x i m u m value is qo = 60 Pa and the duration of the rectangular
N. G. Tsouvalis, V..L Papazoglou
838 5.0 4..5 4.0 3.5 3.0 W©
2.5 h 2.0 1.5 1.0 0.5 0.0 0.'
~0
Time (sec)
Fig. 8. Centerdeflectiontime history of a square orthotropic plate subjected to a uniformly distributed step loading. pulse and the blast loading is tl 0.05 S. The solutions corresponding to the step loading and the rectangular pulse go up together to the neighbourhood of 0.05 s (phase I), where the application of the rectangular pulse ends. This is an expected feature, since at this time interval the two loads are identical. The motion is an oscillation, having a mean value near the static deflection solution (wdh = 1.33). After t = 0.05 s (phase II), the step loading response continues its oscillations as previously, while the rectangular pulse response begins a new oscillating motion around the zero deflection value, having a greater amplitude than previously. This amplitude of phase II free vibration motion can be less or more than the phase I one, or even zero, depending on the relation between the pulse duration tl and the first natural period of the plate T. s Differing from the other two, the blast loading response is an oscillating motion around a decreasing mean value, which becomes zero at t = 0.05 s. After that time (phase II), a free vibration motion begins around zero deflection. The rate of increase, as well as the peak values of deflection caused by the blast loading, are smaller than the ones generated by the step loading or the rectangular pulse, since, for the same time interval, the energy passed to the plate by the blast load is less than the corresponding one given by the other two load types. Therefore, in general, blast loadings produce =
Dynamic response of composite plates under lateral loads
839
4.0 Orthotropic Plote , o / b = 1 . 0 Blost Looding q . = 2 0 Po , t 1 = 0 . 1 sec
3.0
W©
,o oo -,.o
,,';"/
:'~ / I
'~
z
&
I
,
"7,kl
o.. I
,,k
i',l
i/I
",
•- ' ~ , /
- 2.0
-3.0 Time (sec)
Present theory _ _ ADINA At=O.O04 ___ ADINA At=O.O02 ........ ADINA At=O.O005
-4.0 Fig. 9. Center deflection time h i s t o r y o f a square o r t h o t r o p i c plate subjected to a u n i f o r m l y distributed blast loading.
smaller mILximum deflections than a step loading or a rectangular pulse with the same maximum pressure. The rectangular pulse response behaviour at phase II (the free vibration motion) depends on the relative magnitudes of the pulse duration tl and the laminate first natural period of free vibrations T, being similar to the corresponding behaviour of a spring-mass single-degree-of-freedom (SDOF) system. It can be readily proven 8 that a zero deflection and velocity condition at t = tl is obtained when h is an integer multiple of the laminate natural period T and, therefore,
XlI=O when t l = n T ,
n = 1,2,3,...
(16)
This period is the one corresponding to the degree of the motion nonlinearity and is not the one calculated by the linear eigenvalue problem, s For example, the (0°/90°)4 laminate of Fig. 10, having the linear natural frequencies mentioned in section 3.1, exhibits the response of Fig. 11 under the action of a rectangular pulse with qo = 60 Pa and tl = 0.03 s. Its linear first natural period is 7"21 = 0.034 s (f21 = 29.412 Hz), and thus, the duration h = 0.03 s
840
N. G. Tsouvalis, V. J. Papazoglou 3.0 (0*/90*),
Lominote
2.5 2.0
,? !io,/oo.i olo
1.5 1.0 0.5 We
-h
0.0 O.q ,o -0.5 -1.0 .
',\
\ /
/:
,
(
-1.5 -2.0 -2.5
....
Step Loodin 9 Rectongulor Pulse Blost Loodin 9 q , = 6 0 Po t , = O . 0 5 see
It
II
t
I
I
I
~n~e (see) -3.0
Fig. 10. Center deflectiontime history for a (0°/90°)4 laminate subjected to various types of time dependentuniform lateral pressure. is near its first natural frequency when w J h ,~ 2.5, resulting in near zero oscillations during phase II, as can be seen in Fig. 11. The nonlinear response of the (0°/90°)4 laminate considered previously, under the action of different pulse types, namely a sine pulse and a triangular pulse with qo = 60 Pa and tl = 0.05 s, is shown in Fig. 12. Comparing this figure with Fig. 10 for the rectangular pulse and blast loadings with equal intensity and duration, it is seen that sine and triangular pulses produce in general less intense oscillating motions, having smaller peak values and a larger period. This fact is due naturally to the gradual increase of pressure for these two pulse types, in contrast to the sudden increase of the step loading and the rectangular pulse. Seeking relations analogous to the ones of eqn (16) between the pulse duration tl and the first natural period T in order for the phase II oscillating motion to vanish, we refer once more to the SDOF system. For the case of a sine pulse of intensity qo and duration tl, in order for the motion at phase II to vanish, the following condition must hold: Xn = 0
T when tl = ( 2 n + l ) ~ -
n = 1,2,3,...
(17)
Dynamic response of composite plates under lateral loads
841
3.0 (0"/90°),
Laminate
2.5
2.0 1.5
1.0 0.5 We
--
h
0.0 O.q:LO
i
o.o~..,/
0.02
x,~
-0.5 -1.0 -1.5 -2.0
Rectangular Pulse q . = 6 0 Po t 1 = 0 . 0 3 sec
-2.5
Time (sec) -3.0
Fig. 11. Ceater deflection time history for a (0°/90°)4 laminate subjected to a rectangular pulse with duration near its first natural period.
As an example, the response of the (00/900)4 laminate under examination is shown in Fig. 13 for two pulse periods, tl = 0.045 s (,~1.5 x T21) and tl = 0.075 s (;~2.5 x T21). It is obvious that the phase II motion is a near zero oscillation. In the case of a triangular pulse of peak value qo and duration tl, the corresponding condition is Xn=0
whentl=2nT
n=1,2,3,...
(18)
The parametric study continues with the effect of the fiber orientation angle of an angle-ply laminate on its nonlinear response under the action of a step loading. This response is shown in Fig. 14 for a (+0)4 glass/epoxy angle-ply laminate with a = 50 cm, b = 25 cm, h = 1 mm and having the material properties of Table 1. Note that the zero direction is the one parallel to the long edge of the laminate. The spatial distribution of the load over the plate area is assumed to be uniform. As expected, Fig. 14 demonstrates that the increase of the fiber angle up to 90 ° leads to stiffer plates, exhibiting smaller deflection peak values and a smaller period of the oscillating motion. The effect of increasing maximum load value on the response of the previous angle-ply laminate with 0 = 30 ° can be seen in Fig. 15. The load
N. G. Tsouvalis, V. J. Papazoglou
842 2.0
~'x'~'~ i
(0°/90°),
Laminate
1.5
1.0
0.5
1// i/
Wc O.N
~\
/// k\
II
0
I
k
/
-0.,'
- 1.( - .... -1.5
Sine Pulse T r i a n g u l a r Pulse q o = 6 0 Pa h = O . 0 5 sec
T/me (sec) -2.0
Fig. 12. Center deflection time history for a (0°/90°)4 laminate subjected to uniform lateral sine and triangular pulse pressure. type is a blast with duration tl = 0.05 s and maximum values qo varying from 10 to 75 Pa. Two general remarks can be made by observing Fig. 15. The oscillating motions corresponding to higher applied loads have a smaller period (higher frequency) of vibration. The form of each oscillation is symmetric and smooth for small loads (small nonlinearities), becoming nonsymmetric as the load increases. This change is due to the fact that the relative magnitude of the deflection series terms other than the dominant one (W13, W31, W33 in this case) with respect to the dominant one (W11) increases with increasing nonlinearity. The effect of damping is investigated in Fig. 16 for a (900/00)4 glass/epoxy cross-ply laminate with a = 50 cm, b = 25 cm, h = 1 mm and having the material properties given in Table 1. The laminate is subjected to a sinusoidally distributed step loading of intensity qo = 60 Pa. Information about damping characteristics and properties of composite materials are rarely presented in the literature. A lot of experimental study has yet to be made in this field for a better understanding of the phenomenon, as well as for the collection of a fair amount of damping properties data. The damping ratio (= c/cc) has been reported to range between 0.005 and 0.05 for the common type of composites used in aircraft structures. 2 Results for four different
Dynamic response of composite plates under lateral loads
843
2.2 1.8
~
(0"/90=)s, L a m i n a t e
0.6
Wc
0.2
,
h
002
,
",,
~
o
0.06
,
,
" ",, , "/~.--""7'~, o.
-0.6 - 1 . 0 ~-
Sine P u l s e
" -1.4 . -~1.8
~ ....
qo=60 Po L1=0.045 h=0"075 Time (sec)
-2.2
Fig.
13. Center deflection dme history for a (0°/90°)4 laminate subjected to two uniform lateral sine pulses with tl = 0.045 s and tl = 0.075 s.
values o1" the damping ratio ( are presented in Fig. 16. All results finally converge to the static solution wc/h = 0.92, as expected. The parametric study is completed with some stress results. Due to the fact that no acceptable stress convergence is obtained for a uniformly distributed lateral load, the study has been performed for a sinusoidaUy loaded laminate, where a one term deflection expansion accurately represents the solution. A rectangular (a = 50 cm, b = 25 cm, h = 1 ram) (90°/ 0°)4 laminate is considered having the material properties given in Table 1 and loaded by a rectangular pulse of intensity qo = 60 Pa and duration tl = 0.04 s. "['he resulting deflection time history response is shown in Fig. 17, where the one term solution produces a smooth and symmetric oscillating motion. Stresses have been calculated at the laminate center, at its lower surface. The time history of the transverse stress gy is shown in Fig. 18, where both the total stress and its membrane and bending components are plotted. It can be clearly seen in this figure that the membrane stress dominates the bending one and is the major component of the total stress. It can also be seen that the bending stress follows an oscillating motion with alternating sign, similar to the one for deflection, whereas the membrane stress, although presenting exactly the same oscillating motion (same
844
N. G. Tsouvalis, IF. J. Papazoglou 2.5
~T
~/-Ct)4 Laminate ep Loading , q o = 2 5
~ 0 = ....... .... ___
Pa
_
2.0
_
0 " 0=30* .~=45 ° 0=60 ° 0=90 °
1.5
we h 1.0
i/,
,f,
111j
0.5
I /
0.0 0.00
",, , , ,',?:!,X,::\", 0.02
" j~.a"
-0.~4
\
.~l's
"
008
010
Time (see)
Fig. 14. Center deflection time history for a (+0)4 angle-ply laminate subjected to a uniform
lateral step loading.
frequency), preserves its sign, being always tensile. Therefore, we see that stress o'y at the bottom of the laminate is always tensile, irrespective of the positive or negative corresponding deflection values. This is also the case for the longitudinal stress ¢rx.s
CONCLUSIONS An analytic tool was presented in this paper for the nonlinear static and dynamic response calculation of simply supported composite laminated plates under the action of lateral loads. The laminate can have any stacking sequence, whereas its edges were assumed to be stress-free, modelling in this way the less stiff possible configuration of a simply supported plate. Taking into account the fact that, in marine structures made of ordinary composite materials, the governing criterion is that of maximum permissible deformations due to the low elastic moduli of these materials, the results of the present method are conservative with respect to any other type of boundary conditions, thus constituting an upper limit for all possible configurations.
Dynamlc response of composite plates under lateral loads
845
2.5
2.C I.~
I.¢
0.~ we -0.¢
h
C
- I ,(
-t.!
-2.(
Pig. 15. Center deflection time history for a (4-300)4 angle-ply laminate subjected to a uniform lateral blast loading of various intensities.
qo
It
t (a) Step loading
q
t (b) Unearly increasing load
q q,
m
tl (c) Rectangular pulse
t
m
tl
t
(d) Blast Ioodin 9
Fig. 16. Center deflection time history for a (90°/0°)4 laminate subjected to a sinusoidal step loading for various damping ratios.
846
N. G. Tsouvalis, IF. J. Papazoglou 2.C
1.C
0..' We h
O.C C -0.~
- 1 .(
-1 .~
-2.(
Fig. 17. Center deflection time history for a (90°/0°)4 laminate subjected to a sinusoidal lateral rectangular pulse.
0.25 (go=/o*)4 Laminate a/b=2.0
rt
Rectangular Pulse q===60 Pa . t1=0.04 sec
0.20 Total Stress --Membrane Stress ........ Bending Stress
I i
0.15
?
o
f
~
-t
0.10
J
II
I
0.05 ,°, 0.00
-0.05 0.00
m
0.02
,.,,/
0.04
0.06
0.06
0.10
0.12
', ,,'
0.14
0.16
'%/,
0.16
0.20
Time (see)
Fig. ]8. Stress ay time history (x = a/2, y = b/2, z = - h / 2 ) for a (90°/0°)4 laminate subjected to a sinusoidal lateral rectangular pulse.
Dynamic response of composite plates under lateral loads
847
An extensive parametric study was carried out, revealing that the undamped response of a laminate under a step loading is an oscillating motion with fundamental frequency near its lowest natural frequency. This fundamental frequency increases slightly with increasing degree of nonlinearity. Referring to the various types of load time variations, the maximum deflection peak values are produced by the step and the rectangular pulse loadings, whereas the minimum ones are produced by the triangular pulse, when the,, load maximum value and pulse duration are the same. For the rectangular, sine and triangular pulses, special relations between the duration of load application and the lowest plate natural period, result in very small deflections after the removal of the load. This characteristic can be very significant in designing structures which are subjected frequently to pulse loads of known and approximately fixed duration, since, with a proper selection of material and geometric configuration, fatigue problems can be minimized. Furthermore, special attention must be paid to the fact that maximum deflections may occur even after the load removal, during the free oscillation phase of the plate, and, therefore, proper consideration must be made when selecting the time interval for our calculations. A furtlaer development of the method should incorporate deflection and stress function series expansions capable of modelling more types of boundary conditions, and resulting in faster convergence for the stress calculations. An extensive experimental verification of the results presented here is considered to be necessary, as it will help in acquiring more confidence for the special characteristics revealed in the present study.
REFERENCES 1. Alwar, R.S. & Adimurthy, N.K., Non-linear dynamic response of sandwich panel,~ under pulse and shock type excitations. J. Sound and Vibration, 39 (1975) 43-54. 2. Mei, C. & Wentz, K.R., Large-amplitude random response of angle-ply laminated composite plates. AIAA J., 20 (1982) 1450-1458. 3. Birman, V. & Bert, C.W., Behaviour of laminated plates subjected to conventional blast. Int. J. Impact Engng, 6 (1987) 145-155. 4. Reddy, J.N., Dynamic (transient) analysis of layered anisotropic compositematerial plates. Int. J. Num. Meth. Engng, 19 (1983) 237-255. 5. Redd'.¢,J.N. Geometrically nonlinear transient analysis of laminated composite plates AIAA J., 21 (1983) 621-629. 6. Kant, T., Ravichandran, R.V., Pandya, B.N., & Mallikarjuna, Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory. Comp. Struct., 9 (1988) 319-342. 7. Chia, C.Y., Geometrically nonlinear behavior of composite plates: A review. Appi. Mech. Rev., 41 (1988) 439-451.
848
N. G. Tsouvalis, V. J. Papazoglou
8. Tsouvalis, N. G., Static and dynamic response of simply supported composite laminated plates under the action of in-plane compressive and/or lateral loads. PhD thesis, Dept. of Naval Architecture and Marine Engineering, National Technical University of Athens, 1993. 9. Chia, C.Y. & Prabhakara, M.K., Postbuckling behavior of unsymmetrically layered anisotropic rectangular plates. J. Appl. Mechl, 41 (1974) 155-162. 10. Nayfeh, A. H. & Mook, D. T., Nonlinear Oscillations, J. Wiley and Sons, New York, 1979. 11. Clough, R. W. & Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1975. 12. Zhang, Y. & Matthews, F.L., Postbuckling behaviour of curved panels of generally layered composite materials. Comp. Struet., 1 (1983) 115-135. 13. Zhang, Y. & Matthews, F.L., Postbuckling behavior of anisotropic laminated plates under pure shear and shear combined with compressive loading. AIAA J., 22 (1984) 281-286. 14. Gupta, A.D., Gregory, F.H., Bitting, R.L. & Bhattacharya, S., Dynamic analysis of an explosively loaded hinged rectangular plate. Computers & Struct., 26 (1987) 339-344. 15. Bhimaraddi, A., Static and transient response of rectangular plates. Thin- Walled Struct., 5 (1987) 125-143. 16. ADINA--A Finite element program for automatic dynamic incremental nonlinear analysis: User's Manual. Report AE 84-1, ADINA Engineering, Massachusetts, 1984.