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Nonstationary random response of laminated composite structures by a hybrid strain-based laminated flat triangular shell finite element
f
FINITE ELEMENTS IN ANALYSIS A N D DESIGN
ELSEVIER
Finite Elements in Analysis and Design 23 (1996) 23-35
Nonstationary random response of laminated composite structures by a hybrid strain-based laminated fiat triangular shell finite element C.W.S. To*, B. W a n g Department of Mechanical Engineering, The University of Western Ontario, London, Ont., Canada N6A 5B9
Abstract The prediction and analysis of response of laminated composite shell structures under nonstationary random excitation is of considerable interest to design engineers in aerospace and automobile engineering fields. However, it seems that there is no known comprehensive published work on such an analysis that employs the versatile finite element method. Thus, the main focus of the investigation reported in this paper is the application of the hybrid strain-based laminated composite fiat triangular shell finite element, that has been developed by the authors, for the analysis of laminated composite shell structures under a relatively wide class of nonstationary random excitations. Representative results of a simply supported cross-ply square plate and a simply supported laminated composite cylindrical panel each subjected to a point nonstationary random excitation, are included.
1. Introduction
The prediction and analysis of response of laminated composite shell structures under nonstationary random excitations is of considerable interest to design engineers in aerospace and automobile engineering fields. The motivation of the present investigation is the analysis of nonstationary random response of laminated composite shell structures on board aerospace systems that are subjected to a wide variety of intensive transient disturbances originating, primarily, from near-missed explosion and impact upon the system. Central to the problem is the nonconservative design and accurate response computation of complex structures. The nonconservative design approach precludes the representation of the intensive disturbances as stationary random process and therefore analysis of nonstationary random processes is required. T h e d e m a n d for a c c u r a t e r e s p o n s e c o m p u t a t i o n of c o m p l e x structures m a k e s analytical * Corresponding author, now at the Department of Mechanical Engineering, University of Nebrasha, Lincoln, NE 68588-0656, U.S.A. 0168-874X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0 1 6 8 - 8 7 4 X ( 9 6 ) 0 0 0 2 0 - 0
24
C.W.S. To, B. Wanq/Finite Elements in Analysis and Design 23 (1996) 23 35
~
D
C
¢9
L
Fig. I. Four-layer cylindrical shell panel with element mesh type considered.
0.6
~-" 0.5 0
2% d a m p i n g
X
A >c
0.4
v
o
0.3
I1) - -
0.2
30 m o d e s * 15 m o d e s a 8 modes
'r, t~
>0.i
0.0
--'
0.0
I
2.0
,
I
I
4.0
6.0
,
8.0
Time (msec) Fig. 2{a). Variance of displacement at centre of nine-layer plate.
10.0
C.W.S. To, B. Wang~Finite Elements in Analysis and Design 23 (1996) 23-35
50.0
,
40.0
F
|
/
tt
]
,
,
25
,
• 25 modes modes
~
11
a 20
o 15 modes
20.0
'~ 10.0
0.0 0.0
2.0
4.0
6.0
8.0
10.0
Time (msee) Fig. 2(b). Variance of velocity at centre of nine-layer plate.
solution impossible and naturally it leads to the application of the versatile numerical technique, the finite element method. Because of its simplicity and various advantages over other existing shell finite elements the hybrid strain-based three-node fiat triangular shell finite elements for isotropic materials proposed by To and Liu [1] are extended by To and Wang [2] for application to laminated composite structures. This paper includes the following sections. Section 2 is concerned with the formulation of the hybrid strain-based laminated composite shell finite element [2]. Section 3 deals with the nonstationary random response of discretized structures. The method adopted is that presented by To and Orisamolu [3, 4]. Section 4 presents computed results of a plate and a typical shell structures, each subjected to a point nonstationary random excitation which is represented as a product of a uniformly modulated function and a zero mean Gaussian white noise process. Finally, concluding remarks are included in Section 5.
2. Laminated composite shell finite element matrices
In the derivation of element stiffness matrix, the Hellinger-Reissner principle is applied: 7~HR(U,e)= ~
[--((re)Tee + 2(ae)T eu -- 2fTu "] dV --
dS
(1)
C.W.S. To, B. Wang/' Finite Elements in Analysis and Design 23 (1996) 23--35
26
2.0 2% damping ×
/~
1.0
X v
o
0.0
o
-l.O
30 modes • 15 modes [] 8 modes
o
-2.0u---------~ 0.0 2.0
4.0
6.0
B.O
10.0
Time (msec) Fig. 2(c). Covariance of displacement and velocity at centre of nine-layer plate.
where {u"} = [4'] {q} is the assumed displacement field, [4'] is the displacement shape function, {e~} = [ P ] { a } is the assumed strain field, [P] is a matrix of strain distributions, {e"} = 5,° {u"} = [B] {q}; 5(' is a linear differential operator, [B] is the strain-displacement relation matrix and
{q} =
I'|
{U 1 Vl W 1 0 r l Osl Otl U 2 • . .,"
(2)
is the generalized nodal displacement vector. The superscripts u and e denote the quantities in the assumed displacement field and strain field, respectively. Other symbols have their usual meanings. Then one can write
v(a~)Te~dV = fa(NTe~ + MT7/ + QIT')da, (3)
fv(a~)Te,"dV = fa(Nre~ + MTzu + Q~7")da, where N, M and Q are, respectively, the membrane, bending and shear stress resultant vectors and em, Z and 7 are the membrane strain, bending curvature and transverse shear strain vectors, respectively.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
27
50.0 I 2% d a m p i n g , 30 m o d e s o 5% d a m p i n g , 30 m o d e s
'o 40.0 X
A V
"-~ 30.0 O
20.0 ID 0
"~t~ 10.0
0.0 0.0
E.0
4.0
6.0
8.0
10.0
Time (msee) Fig. 2(d). Variancesof velocity at centre of nine-layerplate with different dampings.
The constitutive equations for laminated structures are N = [ A ] ~.~ + [ B ] Z,
M = [ B ] e,. + [ D ] Z,
Q = [ E l 7,
(4)
where [AI, [B], [D] and [E] are the membrane, membrane-bending coupling, bending and shear stiffnesses, respectively. The strains from the assumed strain field and the assumed displacement field can be decomposed into membrane, bending and shear parts accordingly as e~ = [P~] {am}, ~, = [ B , ] {q},
z ' = EPb] {ab}, Z~ = [/~2] {q},
7 ' = EP~] {a,} ?~
=
(5)
EB3] {q}.
By substituting Eqs. (2)-(5) into Eq. (1) and taking variation, one has for every element
[kh] {q} -- f [~b]x fdV - f [q~]x fda = 0, 3~
3~
(6)
C.W.S. To, B. Wang l' Finite Elements in Analysis and Design 23 (1996) 23-35
28
8.0 1% dampingl 0
x 6.0
A
V © m
o 4.0 m ©
~ 2.0 t~
0 0.0 0.00
0.03
0.06
20 modes 15 modes -" 8 modes 0 5 modes
0.09
0.12
0.15
Time (see) Fig. 3(a). Variance of displacement at centre of four-layer cylindrical shell panel.
where
[kh]
= [G] T [H]
I[G],
(7)
P~APo P~BPb 0 ] da, 0 [ H ] = f,, P~BPm P~DPb 0 0 P~EP~
J
[G] = fo
PTmAB1 + PTmBB2 PTbBB, + PTbDB2 da. pTEB3
[ B ] = {BtB2B3}
T,
' ~= I
(8)
(9)
(10)
To derive the element matrices, the origin of the local r-s t system is attached to node 1 and at the middle surface of the flat triangular lamina. This arrangement simplifies the derivation of element matrices. The assumed displacement and strain fields are carefully selected. Similar to Allman's triangle [5], the translational displacements are interpolated by quadratic polynomials and the rotations linear ones.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
29
15.0
I
1% d a m p i n g
o
12.0
X A
V
"~
9.0
0
~
6.0 c "
¢
20 modes 15 m o d e s " 8 modes ; 5 modes
z
t~
•r,
3.o
0.0 0.00
0.03
0.06
0.08
0.12
0.15
Time (sec)
Fig. 3(b). Variance of velocity at centre of four-layer cylindrical shell panel.
Thus,
/.-/ ~---Ul~ 1 + U2~ 2 + U3~ 3 "At-fflOtl "[- ff20t2 "~- ff30t3, V = /)1~1 + U2~2 "[- /33~3 -[- (llOtl At- q20t2 -[- q30t3, w = Wl~ 1 --[- w2~ 2 --[- w3~ 3 --[--fflOrl At- ff20r2 "Jr"/530, 3 - -
(lla) ~tlOsl
- - ~ 1 2 0 s 2 - - q30s3,
0 r = Orl~l "~- 0r2~2 -~- 0r3~3 , 0 s = Osl~l "1- 0s2~2 -[- 0s3~3 ,
(llb)
0 t = Oriel -q- 0t2~2 -at- 0t3~3 , w h e r e i f / a n d ~ / ( i = 1, 2, 3) are defined in reference [-1-1, a n d ~1, ~2 a n d ~3 are the area c o o r d i n a t e s .
30
C.W.S. To, B. Wang/' Finite Elements in Analysis and Design 23 (1996) 23-35 2.0
r
~
2
r
1% damping
1.0
S o:
"7-,
0.0
20 modes • 15 modes o 8 modes ,~ 5 modes
-1.0
O
-2.0 0.00
J
.l
0.03
0.06
0.09
0.12
0.15
Time (see) Fig. 3(c). Covariance of displacement and velocity at centre of four-layer cylindrical shell panel.
In the assumed strain field, [Pm] and [eb] are 3 x 3 identity matrices, and [ - s 3 ( l - 2~2) [P=] = --r3(l 242)
s3(1 - 2~1) ( r 3 - - r : ) ( l --2~1)
() ] r2(1--2~3) "
(12)
To include the drilling degrees-of-freedom (DDOF), the following strain energy due to torsional deformation is required [1]:
1.
~t=-~k~l(Grs)khk_
f O tll- - ~ ( ~ r - - u O]2da.
(13)
in which 0, - ½(v~ - u s ) = IT{q},
(14)
where the nodal displacement vector {q} is now understood to be of order 18 x 1. Adding Eq. (13) to Eq. (1) and performing similar procedure outlined above, one has the element stiffness matrix as [k] = [kh] + [kt],
(15)
C.W.S. To, B. Wang~Finite Elements in Analysis and Design 23 (1996) 23-35
31
15.0
Co
12.0
x A
% V
"-" 9.0 i1) m o. o 111
0
6.0 i1) ¢.)
"~, 3.0 t~ IV. damping, 20 m o d e s o 5% damping, 20 m o d e s
0.0 0.00
0.03
0.06
0.09
0.12
0.15
Time (see) Fig. 3(d). Variance of velocity at centre of four-layer cylindricalshell panel with differentdampings.
in which the element stiffness matrix due to the D D O F is
[kt]= ~-'~(G,s)khkf yTy da, k-1
where (Grs)kis the in-plane shear modulus for kth layer and hk is the thickness of the kth layer. Applying other interpolation functions similar to those adopted by To and Liu [1], more element stiffness matrices are derived but not included here. For brevity, the derivation of the element mass matrix is also not presented here. It may be appropriate to mention that the element stiffness and mass matrices have been obtained explicitly by making use of a combination of manual and symbolic computer algebraic manipulation. The symbolic computer algebra package employed is MACSYMA. Thus, no numerical integration is required for the evaluation of element matrices.
3. Nonstationary random response of discretized composite structures Consider a discretized complex laminated composite structure having n-degree-of-freedom (nDOF). It is excited by nonstationary random forces {F(t)}. The governing equation of motion for
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
32
this system is given by [M]{Jf} + [ C ] { ) ( ] + [K] [X ~, =- {F(t) I ,
{16)
where [ M ] is the assembled mass matrix, [C] is the assembled d a m p i n g matrix, [ K ] is the assembled linear stiffness matrix, and IX], I-£ } and ~X I are the r a n d o m acceleration, velocity and displacement vectors, respectively. Assuming that the r a n d o m excitations are all nonstationary, and they are modelled as products of uniformly modulated functions and zero mean Gaussian white noise processes
Fi
(i = 1,2 . . . . . n),
= eiWi
(17)
where ei is the envelope modulating function for the ith excitation while Wi is the corresponding zero mean Gaussian white noise process. Note that with the above definition a relatively wide class of nonstationary r a n d o m disturbances can be modelled, while any stationary r a n d o m excitations are simply special cases in which the envelope modulating functions are of unity. By following the usual procedure of modal analysis and normalizing the m o d e shapes [6], a complete set of n normalized m o d e shapes [g%] is obtained. The elements of the ith column of the normal m o d e matrix [~bm] is defined by ta~i~ }'¢mJ
1 --
~
I,nu)~ [-r j
(i = 1,2,
....
n),
(18)
X/ Mni
where {g,0~] is the ith column of the modal matrix and M,~ is the diagonal term of the matrix [~b]t[m] [ 4 ] corresponding to the m o d e shape {4)¢i~. Introducing the transformation
[X},,., = [eb,,],×M{q},vt.l
(19)
to Eq. (16), with ~tqi~ being the vector of normal coordinates, and premultiplying the resulting matrix equation t h r o u g h o u t by [q)m] t, it leads to a set of uncoupled equations ~)r + 2~r~o~G + co~2 q~ =J~(t),
r = 1,2 . . . . . M.
(20)
where 2~rO/r = {qb~)}w[c] [qb~l l,
u)ff = [4'~'}t [ K ] '~am, ~ " j
L(t) = ~a't~'*tcrt'*~ 1,'*" m j i - - ~ , % , )
and the integer M is the total n u m b e r of modes considered in the analysis. In general, M is less than n. Now, application of the 'N-modal' analysis [3, 4] is made. By considering any two consecutive modes r and s at a time, the equations of motion in terms of normal coordinates are
"G + 2~,~%G + ~o2q~ =J~(t),
(21a)
~)~ + 2~,(0,4,. + ~o~q.~ =.)'~(t).
(21b)
Eq. (21) is solved by the 'N-modal" approach [3,4] for the following second-order statistical m o m e n t s of the modal responses: (f.,2}\
/I,421
~ 2 --)" "21 I • I 1 ! [ , [ ({qsG}), \t,,,j), ( tqs }), \ ~qs ), (iq,'q,'J ), ( tq,'qs~), (tq~q~),({O~q,}), (~.q,'q~I), '' ''
where the angular brackets denote the ensemble average of the enclosing quantity.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
33
In the digital computer program developed, steps have been taken to ensure that the correct number of pairs of modal responses are evaluated. It may be noted that in Eq. (20) the modal excitation in vector form becomes
{f}M×l = {@~'}M×IF1 + "'" + {@~'}M×,F,,
(22)
where {q~)} (i = 1, 2, ..., n) is the ith column of [ t ~ m ] T x n . That is, the total modal excitation is evaluated by summing up all the excitations due to the excitations, F1 through F,, in Eq. (16). In order to obtain the statistical moments in the original coordinates, Eq. (19) has to be applied. Then, the covariance matrix of displacements in the original coordinate system is given by
({X(t)} {X(t)} T) = ( [~m] ({q} {q}T) [(~rn]T~ = [~m] ({q} {q}T) [t~m]T"
(23)
The covariance matrix for the velocity responses, and the covariance matrix of the displacement and velocity responses may be evaluated similarly.
4. Computed results This section considers a cross-ply square plate and a cross-ply cylindrical panel. They are both symmetrically laminated, simply supported at all sides and have a point nonstationary random excitation applied at the centre. The finite element mesh type employed in the present analysis is indicated at the lower right-hand corner of the panel shown in Fig. 1. The shear correction factors are hc4 = / £ 5 = N/~6"Various meshes have been used in the study but for brevity only results for the case of 6 x 6 mesh for the whole plate or shell panel is presented here. With this mesh the finite element model for both plate and shell have 85 nodes and 144 elements. The finite element used is identified as HLCTS qd. The plate has nine layers with fibre orientation (0/90/0/90/0/90/0/90/0). The side length, L, is 1.0 m. The total thickness is h = 0.01 m. The thickness of each 0 ° and 90 ° layer are h/lO and hi8, respectively. The material is the high modulus graphite/epoxy composite with El/E2 = 40, G12/E2 = 0.6 and G13/E2 = G 2 3 / E 2 = 0.5, in which the modulus of elasticity E1 = 2.0685 x 1011 N/m 2, density p = 1605 kg/m 3 and Poisson's ratio v12 = 0.25. As it is a plate bending problem, all in-plane D O F and D D O F are constrained. Consequently, there are 203 unknown equations to be solved. In the free vibration analysis the first three natural frequencies, for example, of the plate obtained by using the present finite element are 344.4, 970.7 and 1053.8 rad/s, respectively. More natural frequencies and mode shapes were calculated but not included here for brevity. It may be appropriate to note that the analytical solution for the fundamental frequency of the same problem given in Ref. I-7] is 338.5 rad/s. The nonstationary random excitation in this case is
Fj(t) = 9.48148(e- 900t - - e- i20ot) W,
(24)
where the subscript j designates the nodal D O F at the centre of the plate, and W is a zero mean Gaussian white noise process with autocorrelation, 2~zf(z), where 6(z) is the Dirac delta function. It may be appropriate to note that the theory presented in Section 3 and the digital computer
34
C.W.S. To, B. Wan,q/Finite Elements in Analysis and Design 23 (1996) 23-35
program developed for the work are general and can be applied to distributed nonstationary random excitations. For brevity, representative results are given in Fig. 2. These results are of 2% and 5% damping for every mode. The size of time step utilized for the numerical integration, employing the fourth-order Runge-Kutta algorithm for solving the first-order differential equations of statistical moments, is 0.02 ms. In the figure the symbols ~XjX~), (Vj Vj) and (X~ V~) represent the variance of displacement, variance of velocity and the covariance of displacement and velocity at the centre of the plate, respectively. Clearly, the results with the first 25 modes considered in the computation are in excellent agreement with those including the first 30 modes. In Fig. 2(a) the maximum variance of diplacement at the centre of the plate is 5.75 x 1 0 - 9 m 2. One can conclude that in this particular case the first 25 modes have to be included in the computation if accurate responses are required. It may be noted that in the foregoing and subsequent nonstationary random response computations, the number of modes included is determined first by performing several computations of variances of velocities with different numbers of mode shapes. As concluded in Ref. [8], if variances of displacements were evaluated first instead of variances of velocities incorrect number of modes to be included in the computations would be identified. The cross-ply cylindrical shell panel shown in Fig. 1 has four layers (0°/90°/90°/0°). For the case of 6 x 6 mesh with simply supported boundary conditions applied the finite element model has 426 unknowns. The geometrical properties of the shell panel are: length L of the sides of the cylindrical shell panel is 10 in, radius R = 50 in, and thickness h = 1.0 in. The angle ~0 = 5.7392 °. The material properties are: El~E2 = 2 5 , Glz/E2 = G13/E2 = 0.5, Gz3/E2 = 0.2, V12 = V13 = 0 . 2 5 , E1 = 7.5 x 106 psi, and density p = 1 lb/in 3. For this case the first natural frequency using the result of Reddy and Liu [9] is 64.8 rad/s while the present finite element solution is 64.3 rad/s. The nonstationary random excitation applied to the centre of the panel is
F~(t) = 9.48148e -4s' - e - 6 ° ' ) W
(25)
in which the autocorrelation of the zero mean Gaussian while noise process is the same as that for the composite plate structure studied above. The nonstationary random responses obtained are presented in Fig. 3. These results are of 1% and 5% damping for every mode. The time step is 1.25 ms. For this case the results with the first 15 modes considered in the variance of velocity computation are identical to those considering the first 20 modes. Therefore, the first 15 modes have to be included in the variance of displacement computations.
5. Concluding remarks In this paper application is made of the hybrid strain-based laminated composite flat triangular shell finite element, that has been developed by the authors, for the analysis of laminated composite plate and shell structures under a relatively wide class of nonstationary random excitations. Representative results of a simply supported laminated composite square plate and cylindrical panel, each subjected to a point nonstationary random excitation, are included. It is believed that this is the first attempt in applying a combination of the hybrid strain-based laminated composite flat triangular shell finite element and the nonstationary random response analysis to laminated composite shell structures.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
35
Acknowledgements The results presented above were obtained in the course of research supported by the Natural Sciences and Engineering Research Council of Canada. A shorter version of this paper has appeared in the Proceedings of the A.S.M.E. Design Technical Conferences and Conference on Mechanical Vibration and Noise, 17-22 September Boston, Massachusetts, 1995.
References [1] C.W.S. To and M.L. Liu, "Hybrid strain based three node flat triangular shell elements", Finite Elements Anal. Des. 17, pp. 169-203, 1994. [2] C.W.S. To and B. Wang, "Analysis of laminated composite shell structures by hybrid strain based laminated flat triangular elements", Proc. Canadian Congress of Applied Mechanics, 28 May-2 June, Victoria, British Columbia, Vol. 1, pp. 226-227, 1995. [3] C.W.S. To and I.R. Orisamolu, "Response of a two-degree-of-freedom system to random disturbances", Int. J. Compu. Struct. 25, pp. 211-320, 1987. [4] C.W.S. To and I.R. Orisamolu, "Response of discretized plates to transversal and in-plane non-stationary random excitations", J. Sound Vib. 114, pp. 481-494, 1987. [5] D.J. Allman, "A compatible triangular element including vertex rotations for plane elasticity analysis", Int. J. Comput. Struct. 19, pp. 1-8, 1984. [6] R.W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hilt, New York, 1975. [7] A.K. Noor and M.D. Mathers, "Shear-flexible finite element models of laminated composite plates and shells", NASA TN D-8044, 1975. [8] C.W.S. To and B. Wang, "Response analysis of discretized plates under in-plane and external nonstationary random excitations", Proc. ASME Design Technical Conferences and Conference on Mechanical Vibration and Noise, DE-Vol. 56 (Dynamics and Vibration of Time-Varying Systems and Structures), 19-22 September, Albuqerque, New Mexico, pp. 51-62, 1993. [9] J.N. Reddy and C.F. Liu, "A higher order shear deformation theory of laminated elastic shells", lnt. J. Eng. Sci. 23, pp. 319-330, 1985.
FINITE ELEMENTS IN ANALYSIS A N D DESIGN
ELSEVIER
Finite Elements in Analysis and Design 23 (1996) 23-35
Nonstationary random response of laminated composite structures by a hybrid strain-based laminated fiat triangular shell finite element C.W.S. To*, B. W a n g Department of Mechanical Engineering, The University of Western Ontario, London, Ont., Canada N6A 5B9
Abstract The prediction and analysis of response of laminated composite shell structures under nonstationary random excitation is of considerable interest to design engineers in aerospace and automobile engineering fields. However, it seems that there is no known comprehensive published work on such an analysis that employs the versatile finite element method. Thus, the main focus of the investigation reported in this paper is the application of the hybrid strain-based laminated composite fiat triangular shell finite element, that has been developed by the authors, for the analysis of laminated composite shell structures under a relatively wide class of nonstationary random excitations. Representative results of a simply supported cross-ply square plate and a simply supported laminated composite cylindrical panel each subjected to a point nonstationary random excitation, are included.
1. Introduction
The prediction and analysis of response of laminated composite shell structures under nonstationary random excitations is of considerable interest to design engineers in aerospace and automobile engineering fields. The motivation of the present investigation is the analysis of nonstationary random response of laminated composite shell structures on board aerospace systems that are subjected to a wide variety of intensive transient disturbances originating, primarily, from near-missed explosion and impact upon the system. Central to the problem is the nonconservative design and accurate response computation of complex structures. The nonconservative design approach precludes the representation of the intensive disturbances as stationary random process and therefore analysis of nonstationary random processes is required. T h e d e m a n d for a c c u r a t e r e s p o n s e c o m p u t a t i o n of c o m p l e x structures m a k e s analytical * Corresponding author, now at the Department of Mechanical Engineering, University of Nebrasha, Lincoln, NE 68588-0656, U.S.A. 0168-874X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0 1 6 8 - 8 7 4 X ( 9 6 ) 0 0 0 2 0 - 0
24
C.W.S. To, B. Wanq/Finite Elements in Analysis and Design 23 (1996) 23 35
~
D
C
¢9
L
Fig. I. Four-layer cylindrical shell panel with element mesh type considered.
0.6
~-" 0.5 0
2% d a m p i n g
X
A >c
0.4
v
o
0.3
I1) - -
0.2
30 m o d e s * 15 m o d e s a 8 modes
'r, t~
>0.i
0.0
--'
0.0
I
2.0
,
I
I
4.0
6.0
,
8.0
Time (msec) Fig. 2{a). Variance of displacement at centre of nine-layer plate.
10.0
C.W.S. To, B. Wang~Finite Elements in Analysis and Design 23 (1996) 23-35
50.0
,
40.0
F
|
/
tt
]
,
,
25
,
• 25 modes modes
~
11
a 20
o 15 modes
20.0
'~ 10.0
0.0 0.0
2.0
4.0
6.0
8.0
10.0
Time (msee) Fig. 2(b). Variance of velocity at centre of nine-layer plate.
solution impossible and naturally it leads to the application of the versatile numerical technique, the finite element method. Because of its simplicity and various advantages over other existing shell finite elements the hybrid strain-based three-node fiat triangular shell finite elements for isotropic materials proposed by To and Liu [1] are extended by To and Wang [2] for application to laminated composite structures. This paper includes the following sections. Section 2 is concerned with the formulation of the hybrid strain-based laminated composite shell finite element [2]. Section 3 deals with the nonstationary random response of discretized structures. The method adopted is that presented by To and Orisamolu [3, 4]. Section 4 presents computed results of a plate and a typical shell structures, each subjected to a point nonstationary random excitation which is represented as a product of a uniformly modulated function and a zero mean Gaussian white noise process. Finally, concluding remarks are included in Section 5.
2. Laminated composite shell finite element matrices
In the derivation of element stiffness matrix, the Hellinger-Reissner principle is applied: 7~HR(U,e)= ~
[--((re)Tee + 2(ae)T eu -- 2fTu "] dV --
dS
(1)
C.W.S. To, B. Wang/' Finite Elements in Analysis and Design 23 (1996) 23--35
26
2.0 2% damping ×
/~
1.0
X v
o
0.0
o
-l.O
30 modes • 15 modes [] 8 modes
o
-2.0u---------~ 0.0 2.0
4.0
6.0
B.O
10.0
Time (msec) Fig. 2(c). Covariance of displacement and velocity at centre of nine-layer plate.
where {u"} = [4'] {q} is the assumed displacement field, [4'] is the displacement shape function, {e~} = [ P ] { a } is the assumed strain field, [P] is a matrix of strain distributions, {e"} = 5,° {u"} = [B] {q}; 5(' is a linear differential operator, [B] is the strain-displacement relation matrix and
{q} =
I'|
{U 1 Vl W 1 0 r l Osl Otl U 2 • . .,"
(2)
is the generalized nodal displacement vector. The superscripts u and e denote the quantities in the assumed displacement field and strain field, respectively. Other symbols have their usual meanings. Then one can write
v(a~)Te~dV = fa(NTe~ + MT7/ + QIT')da, (3)
fv(a~)Te,"dV = fa(Nre~ + MTzu + Q~7")da, where N, M and Q are, respectively, the membrane, bending and shear stress resultant vectors and em, Z and 7 are the membrane strain, bending curvature and transverse shear strain vectors, respectively.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
27
50.0 I 2% d a m p i n g , 30 m o d e s o 5% d a m p i n g , 30 m o d e s
'o 40.0 X
A V
"-~ 30.0 O
20.0 ID 0
"~t~ 10.0
0.0 0.0
E.0
4.0
6.0
8.0
10.0
Time (msee) Fig. 2(d). Variancesof velocity at centre of nine-layerplate with different dampings.
The constitutive equations for laminated structures are N = [ A ] ~.~ + [ B ] Z,
M = [ B ] e,. + [ D ] Z,
Q = [ E l 7,
(4)
where [AI, [B], [D] and [E] are the membrane, membrane-bending coupling, bending and shear stiffnesses, respectively. The strains from the assumed strain field and the assumed displacement field can be decomposed into membrane, bending and shear parts accordingly as e~ = [P~] {am}, ~, = [ B , ] {q},
z ' = EPb] {ab}, Z~ = [/~2] {q},
7 ' = EP~] {a,} ?~
=
(5)
EB3] {q}.
By substituting Eqs. (2)-(5) into Eq. (1) and taking variation, one has for every element
[kh] {q} -- f [~b]x fdV - f [q~]x fda = 0, 3~
3~
(6)
C.W.S. To, B. Wang l' Finite Elements in Analysis and Design 23 (1996) 23-35
28
8.0 1% dampingl 0
x 6.0
A
V © m
o 4.0 m ©
~ 2.0 t~
0 0.0 0.00
0.03
0.06
20 modes 15 modes -" 8 modes 0 5 modes
0.09
0.12
0.15
Time (see) Fig. 3(a). Variance of displacement at centre of four-layer cylindrical shell panel.
where
[kh]
= [G] T [H]
I[G],
(7)
P~APo P~BPb 0 ] da, 0 [ H ] = f,, P~BPm P~DPb 0 0 P~EP~
J
[G] = fo
PTmAB1 + PTmBB2 PTbBB, + PTbDB2 da. pTEB3
[ B ] = {BtB2B3}
T,
' ~= I
(8)
(9)
(10)
To derive the element matrices, the origin of the local r-s t system is attached to node 1 and at the middle surface of the flat triangular lamina. This arrangement simplifies the derivation of element matrices. The assumed displacement and strain fields are carefully selected. Similar to Allman's triangle [5], the translational displacements are interpolated by quadratic polynomials and the rotations linear ones.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
29
15.0
I
1% d a m p i n g
o
12.0
X A
V
"~
9.0
0
~
6.0 c "
¢
20 modes 15 m o d e s " 8 modes ; 5 modes
z
t~
•r,
3.o
0.0 0.00
0.03
0.06
0.08
0.12
0.15
Time (sec)
Fig. 3(b). Variance of velocity at centre of four-layer cylindrical shell panel.
Thus,
/.-/ ~---Ul~ 1 + U2~ 2 + U3~ 3 "At-fflOtl "[- ff20t2 "~- ff30t3, V = /)1~1 + U2~2 "[- /33~3 -[- (llOtl At- q20t2 -[- q30t3, w = Wl~ 1 --[- w2~ 2 --[- w3~ 3 --[--fflOrl At- ff20r2 "Jr"/530, 3 - -
(lla) ~tlOsl
- - ~ 1 2 0 s 2 - - q30s3,
0 r = Orl~l "~- 0r2~2 -~- 0r3~3 , 0 s = Osl~l "1- 0s2~2 -[- 0s3~3 ,
(llb)
0 t = Oriel -q- 0t2~2 -at- 0t3~3 , w h e r e i f / a n d ~ / ( i = 1, 2, 3) are defined in reference [-1-1, a n d ~1, ~2 a n d ~3 are the area c o o r d i n a t e s .
30
C.W.S. To, B. Wang/' Finite Elements in Analysis and Design 23 (1996) 23-35 2.0
r
~
2
r
1% damping
1.0
S o:
"7-,
0.0
20 modes • 15 modes o 8 modes ,~ 5 modes
-1.0
O
-2.0 0.00
J
.l
0.03
0.06
0.09
0.12
0.15
Time (see) Fig. 3(c). Covariance of displacement and velocity at centre of four-layer cylindrical shell panel.
In the assumed strain field, [Pm] and [eb] are 3 x 3 identity matrices, and [ - s 3 ( l - 2~2) [P=] = --r3(l 242)
s3(1 - 2~1) ( r 3 - - r : ) ( l --2~1)
() ] r2(1--2~3) "
(12)
To include the drilling degrees-of-freedom (DDOF), the following strain energy due to torsional deformation is required [1]:
1.
~t=-~k~l(Grs)khk_
f O tll- - ~ ( ~ r - - u O]2da.
(13)
in which 0, - ½(v~ - u s ) = IT{q},
(14)
where the nodal displacement vector {q} is now understood to be of order 18 x 1. Adding Eq. (13) to Eq. (1) and performing similar procedure outlined above, one has the element stiffness matrix as [k] = [kh] + [kt],
(15)
C.W.S. To, B. Wang~Finite Elements in Analysis and Design 23 (1996) 23-35
31
15.0
Co
12.0
x A
% V
"-" 9.0 i1) m o. o 111
0
6.0 i1) ¢.)
"~, 3.0 t~ IV. damping, 20 m o d e s o 5% damping, 20 m o d e s
0.0 0.00
0.03
0.06
0.09
0.12
0.15
Time (see) Fig. 3(d). Variance of velocity at centre of four-layer cylindricalshell panel with differentdampings.
in which the element stiffness matrix due to the D D O F is
[kt]= ~-'~(G,s)khkf yTy da, k-1
where (Grs)kis the in-plane shear modulus for kth layer and hk is the thickness of the kth layer. Applying other interpolation functions similar to those adopted by To and Liu [1], more element stiffness matrices are derived but not included here. For brevity, the derivation of the element mass matrix is also not presented here. It may be appropriate to mention that the element stiffness and mass matrices have been obtained explicitly by making use of a combination of manual and symbolic computer algebraic manipulation. The symbolic computer algebra package employed is MACSYMA. Thus, no numerical integration is required for the evaluation of element matrices.
3. Nonstationary random response of discretized composite structures Consider a discretized complex laminated composite structure having n-degree-of-freedom (nDOF). It is excited by nonstationary random forces {F(t)}. The governing equation of motion for
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
32
this system is given by [M]{Jf} + [ C ] { ) ( ] + [K] [X ~, =- {F(t) I ,
{16)
where [ M ] is the assembled mass matrix, [C] is the assembled d a m p i n g matrix, [ K ] is the assembled linear stiffness matrix, and IX], I-£ } and ~X I are the r a n d o m acceleration, velocity and displacement vectors, respectively. Assuming that the r a n d o m excitations are all nonstationary, and they are modelled as products of uniformly modulated functions and zero mean Gaussian white noise processes
Fi
(i = 1,2 . . . . . n),
= eiWi
(17)
where ei is the envelope modulating function for the ith excitation while Wi is the corresponding zero mean Gaussian white noise process. Note that with the above definition a relatively wide class of nonstationary r a n d o m disturbances can be modelled, while any stationary r a n d o m excitations are simply special cases in which the envelope modulating functions are of unity. By following the usual procedure of modal analysis and normalizing the m o d e shapes [6], a complete set of n normalized m o d e shapes [g%] is obtained. The elements of the ith column of the normal m o d e matrix [~bm] is defined by ta~i~ }'¢mJ
1 --
~
I,nu)~ [-r j
(i = 1,2,
....
n),
(18)
X/ Mni
where {g,0~] is the ith column of the modal matrix and M,~ is the diagonal term of the matrix [~b]t[m] [ 4 ] corresponding to the m o d e shape {4)¢i~. Introducing the transformation
[X},,., = [eb,,],×M{q},vt.l
(19)
to Eq. (16), with ~tqi~ being the vector of normal coordinates, and premultiplying the resulting matrix equation t h r o u g h o u t by [q)m] t, it leads to a set of uncoupled equations ~)r + 2~r~o~G + co~2 q~ =J~(t),
r = 1,2 . . . . . M.
(20)
where 2~rO/r = {qb~)}w[c] [qb~l l,
u)ff = [4'~'}t [ K ] '~am, ~ " j
L(t) = ~a't~'*tcrt'*~ 1,'*" m j i - - ~ , % , )
and the integer M is the total n u m b e r of modes considered in the analysis. In general, M is less than n. Now, application of the 'N-modal' analysis [3, 4] is made. By considering any two consecutive modes r and s at a time, the equations of motion in terms of normal coordinates are
"G + 2~,~%G + ~o2q~ =J~(t),
(21a)
~)~ + 2~,(0,4,. + ~o~q.~ =.)'~(t).
(21b)
Eq. (21) is solved by the 'N-modal" approach [3,4] for the following second-order statistical m o m e n t s of the modal responses: (f.,2}\
/I,421
~ 2 --)" "21 I • I 1 ! [ , [ ({qsG}), \t,,,j), ( tqs }), \ ~qs ), (iq,'q,'J ), ( tq,'qs~), (tq~q~),({O~q,}), (~.q,'q~I), '' ''
where the angular brackets denote the ensemble average of the enclosing quantity.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
33
In the digital computer program developed, steps have been taken to ensure that the correct number of pairs of modal responses are evaluated. It may be noted that in Eq. (20) the modal excitation in vector form becomes
{f}M×l = {@~'}M×IF1 + "'" + {@~'}M×,F,,
(22)
where {q~)} (i = 1, 2, ..., n) is the ith column of [ t ~ m ] T x n . That is, the total modal excitation is evaluated by summing up all the excitations due to the excitations, F1 through F,, in Eq. (16). In order to obtain the statistical moments in the original coordinates, Eq. (19) has to be applied. Then, the covariance matrix of displacements in the original coordinate system is given by
({X(t)} {X(t)} T) = ( [~m] ({q} {q}T) [(~rn]T~ = [~m] ({q} {q}T) [t~m]T"
(23)
The covariance matrix for the velocity responses, and the covariance matrix of the displacement and velocity responses may be evaluated similarly.
4. Computed results This section considers a cross-ply square plate and a cross-ply cylindrical panel. They are both symmetrically laminated, simply supported at all sides and have a point nonstationary random excitation applied at the centre. The finite element mesh type employed in the present analysis is indicated at the lower right-hand corner of the panel shown in Fig. 1. The shear correction factors are hc4 = / £ 5 = N/~6"Various meshes have been used in the study but for brevity only results for the case of 6 x 6 mesh for the whole plate or shell panel is presented here. With this mesh the finite element model for both plate and shell have 85 nodes and 144 elements. The finite element used is identified as HLCTS qd. The plate has nine layers with fibre orientation (0/90/0/90/0/90/0/90/0). The side length, L, is 1.0 m. The total thickness is h = 0.01 m. The thickness of each 0 ° and 90 ° layer are h/lO and hi8, respectively. The material is the high modulus graphite/epoxy composite with El/E2 = 40, G12/E2 = 0.6 and G13/E2 = G 2 3 / E 2 = 0.5, in which the modulus of elasticity E1 = 2.0685 x 1011 N/m 2, density p = 1605 kg/m 3 and Poisson's ratio v12 = 0.25. As it is a plate bending problem, all in-plane D O F and D D O F are constrained. Consequently, there are 203 unknown equations to be solved. In the free vibration analysis the first three natural frequencies, for example, of the plate obtained by using the present finite element are 344.4, 970.7 and 1053.8 rad/s, respectively. More natural frequencies and mode shapes were calculated but not included here for brevity. It may be appropriate to note that the analytical solution for the fundamental frequency of the same problem given in Ref. I-7] is 338.5 rad/s. The nonstationary random excitation in this case is
Fj(t) = 9.48148(e- 900t - - e- i20ot) W,
(24)
where the subscript j designates the nodal D O F at the centre of the plate, and W is a zero mean Gaussian white noise process with autocorrelation, 2~zf(z), where 6(z) is the Dirac delta function. It may be appropriate to note that the theory presented in Section 3 and the digital computer
34
C.W.S. To, B. Wan,q/Finite Elements in Analysis and Design 23 (1996) 23-35
program developed for the work are general and can be applied to distributed nonstationary random excitations. For brevity, representative results are given in Fig. 2. These results are of 2% and 5% damping for every mode. The size of time step utilized for the numerical integration, employing the fourth-order Runge-Kutta algorithm for solving the first-order differential equations of statistical moments, is 0.02 ms. In the figure the symbols ~XjX~), (Vj Vj) and (X~ V~) represent the variance of displacement, variance of velocity and the covariance of displacement and velocity at the centre of the plate, respectively. Clearly, the results with the first 25 modes considered in the computation are in excellent agreement with those including the first 30 modes. In Fig. 2(a) the maximum variance of diplacement at the centre of the plate is 5.75 x 1 0 - 9 m 2. One can conclude that in this particular case the first 25 modes have to be included in the computation if accurate responses are required. It may be noted that in the foregoing and subsequent nonstationary random response computations, the number of modes included is determined first by performing several computations of variances of velocities with different numbers of mode shapes. As concluded in Ref. [8], if variances of displacements were evaluated first instead of variances of velocities incorrect number of modes to be included in the computations would be identified. The cross-ply cylindrical shell panel shown in Fig. 1 has four layers (0°/90°/90°/0°). For the case of 6 x 6 mesh with simply supported boundary conditions applied the finite element model has 426 unknowns. The geometrical properties of the shell panel are: length L of the sides of the cylindrical shell panel is 10 in, radius R = 50 in, and thickness h = 1.0 in. The angle ~0 = 5.7392 °. The material properties are: El~E2 = 2 5 , Glz/E2 = G13/E2 = 0.5, Gz3/E2 = 0.2, V12 = V13 = 0 . 2 5 , E1 = 7.5 x 106 psi, and density p = 1 lb/in 3. For this case the first natural frequency using the result of Reddy and Liu [9] is 64.8 rad/s while the present finite element solution is 64.3 rad/s. The nonstationary random excitation applied to the centre of the panel is
F~(t) = 9.48148e -4s' - e - 6 ° ' ) W
(25)
in which the autocorrelation of the zero mean Gaussian while noise process is the same as that for the composite plate structure studied above. The nonstationary random responses obtained are presented in Fig. 3. These results are of 1% and 5% damping for every mode. The time step is 1.25 ms. For this case the results with the first 15 modes considered in the variance of velocity computation are identical to those considering the first 20 modes. Therefore, the first 15 modes have to be included in the variance of displacement computations.
5. Concluding remarks In this paper application is made of the hybrid strain-based laminated composite flat triangular shell finite element, that has been developed by the authors, for the analysis of laminated composite plate and shell structures under a relatively wide class of nonstationary random excitations. Representative results of a simply supported laminated composite square plate and cylindrical panel, each subjected to a point nonstationary random excitation, are included. It is believed that this is the first attempt in applying a combination of the hybrid strain-based laminated composite flat triangular shell finite element and the nonstationary random response analysis to laminated composite shell structures.
C.W.S. To, B. Wang/Finite Elements in Analysis and Design 23 (1996) 23-35
35
Acknowledgements The results presented above were obtained in the course of research supported by the Natural Sciences and Engineering Research Council of Canada. A shorter version of this paper has appeared in the Proceedings of the A.S.M.E. Design Technical Conferences and Conference on Mechanical Vibration and Noise, 17-22 September Boston, Massachusetts, 1995.
References [1] C.W.S. To and M.L. Liu, "Hybrid strain based three node flat triangular shell elements", Finite Elements Anal. Des. 17, pp. 169-203, 1994. [2] C.W.S. To and B. Wang, "Analysis of laminated composite shell structures by hybrid strain based laminated flat triangular elements", Proc. Canadian Congress of Applied Mechanics, 28 May-2 June, Victoria, British Columbia, Vol. 1, pp. 226-227, 1995. [3] C.W.S. To and I.R. Orisamolu, "Response of a two-degree-of-freedom system to random disturbances", Int. J. Compu. Struct. 25, pp. 211-320, 1987. [4] C.W.S. To and I.R. Orisamolu, "Response of discretized plates to transversal and in-plane non-stationary random excitations", J. Sound Vib. 114, pp. 481-494, 1987. [5] D.J. Allman, "A compatible triangular element including vertex rotations for plane elasticity analysis", Int. J. Comput. Struct. 19, pp. 1-8, 1984. [6] R.W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hilt, New York, 1975. [7] A.K. Noor and M.D. Mathers, "Shear-flexible finite element models of laminated composite plates and shells", NASA TN D-8044, 1975. [8] C.W.S. To and B. Wang, "Response analysis of discretized plates under in-plane and external nonstationary random excitations", Proc. ASME Design Technical Conferences and Conference on Mechanical Vibration and Noise, DE-Vol. 56 (Dynamics and Vibration of Time-Varying Systems and Structures), 19-22 September, Albuqerque, New Mexico, pp. 51-62, 1993. [9] J.N. Reddy and C.F. Liu, "A higher order shear deformation theory of laminated elastic shells", lnt. J. Eng. Sci. 23, pp. 319-330, 1985.