Higher order models on the eigenfrequency analysis and optimal design of laminated composite structures

1997 0 1998 Elscvicr Science Ltd. All rights reserved Printed in Great Britain 0263-8223/98/$19.00 + 0.00

Composite Structures Vol. 39, Nos 3-4, pp., 231-253,

ELSEVIER

PII:S0263-8223(97)00118-9

Higher order models on the eigenfrequency analysis and optimal design of laminated composite structures V. M. Ranco Correiaa, C. M. Mota Soared’ & C. A. Mota Soared’ “ENIDH-Escola NLiutica Infante D. Henrique, Av. Eng. Bonneville France, 2780 Oeiras, Portugal bIDMEC-Instituto de Engenharia Meclinica-Institute Superior Ttknico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal

This paper deals with the structural optimization of multilaminated composite plate structures of arbitrary geometry and lay up, using single layer higher order shear deformation theory discrete models. The structural and sensitivity analysis formulation is developed for a family of C” Lagrangian elements. The design sensitivities of free vibration response for objective and/or constraint functions with respect to ply angles and ply thickness are presented. The objectives of the design are the maximization of natural frequencies of specified modes and/or the minimization of the structure weight or volume. The accuracy and relative performance of the proposed discrete models are compared and discussed among developed elements and alternative models. Several test designs are optimized to show the applicability of the proposed refined discrete models. 0 1998 Elsevier Science Ltd. All rights reserved.

evaluate these sensitivities efficiently and accurately it is important to have appropriate techniques associated with good structural models. It is well known that the analysis of laminated composite structures using the classical Kirchhoff assumptions can lead to substantial errors for moderately thick plates or shells. This is mainly due to neglecting the transverse shear deformation effects which become very important in composite materials with low ratios of transverse shear modulus to in-plane modulus. This can be attenuated by Mindlin’s first order shear deformation theory, but this theory yields a constant shear strain variation through the thickness and therefore requires the use of shear correction factors in order to approximate the quadratic distribution in the elasticity theory. More accurate numerical models such as three-dimensional finite elements models can be used with refined meshes in order to contemplate reasonable aspect ratios, but these models are computationally expensive. A compromising less expensive model can be achieved by using single layer models, based on higher

INTRODUCTION Laminated composite materials are being widely used in many industries mainly because they allow design engineers to achieve very important weight reductions compared with traditional materials and also because more complex shapes can be easily obtained. The mechanical behavior of a laminate is strongly dependent on the fiber directions and because of this the laminate should be designed to meet the specific requirements of each particular application in order to obtain the maximum advantages of such materials. Accurate and efficient structural analysis, design sensitivity analysis and optimization procedures are very important to accomplish this task. In particular the structural optimization with frequency constraints can be very useful in significantly improving the performance of the structures by manipulating selected natural frequencies. Design sensitivity analysis is important to accurately know the effects of design variable changes on the performance of structures by calculating the search directions to find an optimum design. To 237

238

KM. Franc0 Correia, C. M. Mota Soares, C. A. Mota Soares

order displacement fields involving higher order expansions of the displacement field in powers of the thickness coordinate. These models can accurately account for the effects of transverse shear deformation yielding quadratic variation of out-of-plane strains and therefore do not require the use of artificial shear correction factors and are suitable for the analysis of highly anisotropic plates ranging from high to low length-to-thickness ratios. Pioneering work on the structural analysis formulation for closed form or discrete solutions based in higher order displacement fields can be reviewed in Lo et al. [1,2], Phan & Reddy [3], Reddy [4,5], Mallikarjuna & Kant [6], Bert [7] and Palazotto & Dennis [8] among others. Literature surveys in the field of sensitivity analysis and structural optimization such as those of Adelman & Haftka [9], Haftka & Adelman [lo], and Zyczowski [ 111 reveal some lack of studies carried out on composite structures. Recently, Abrate [12] gave a wide perspective of work carried out by different researchers in the field of optimal design of laminated plates and shells made out of composite materials and Grandhi [13] presented a review covering structural optimization with dynamic frequency constraints. The use of higher order models on design sensitivity analysis and optimal design of laminated composite plates has been carried out on the identification of material properties [14-171 and on shape optimization of axisymmetric shells [18] considering static type situations. The objective of the present work is to present a family of Co nine-node Lagrangian higher order discrete models applied to the eigenfrequency design sensitivity analysis and optimal design of multilaminated composite plates. This work is an extension of a previous one [19] dealing only with static type objectives and constraints. The design variables considered here are the ply orientation angles of the fibers and the ply thicknesses. The design objectives are the maximization of natural frequencies of specified vibration modes and minimization of structure volume or weight. The sensitivities of free vibration response with respect to the design variables can be evaluated either analytically or by using the semi-analytical method. The accuracy and relative computer efficiency of the developed higher order models with respect to first order models is compared and discussed. Several optimization problems are

presented to illustrate the relative performance of the proposed models. HIGHER ORDER DISPLACEMENT FIELDS To approximate the three-dimensional elasticity problem to a two-dimensional laminate problem, the displacement components U, v and w at any point in the laminate space in the X, y and z directions, respectively, are expanded in a Taylor’s series powers of the thickness coordinate z. Each component is a function of X, y, z and t, where t is time. The following higher order displacement fields are considered [5] HSDT 11 u = uo+z~,+z2u~+z3~: v = v,+ze,+z*v;+z3~;

(1)

w = w,+z(p,+z*w; HSDT 9 u = uo+ze,+z2u~+z3~: v = vo+zOY+z%~+z%p; w=w,

(2)

HSDT 7 u = uo+ze,+z%+3: v = v0+z~,+z3ql; w=wg

(3)

of a where uo, vo, wo, are the displacements generic point on the reference surface, t?,, 0,, are the rotations of normal to the reference surface about the y and x axes respectively and cpZ, u&v&w&:,cp~ are the higher order terms in the Taylor’s series expansions, defined at the reference surface. LAMINATE CONSTITUTIVE EQUATIONS The constitutive relations for an arbitrary ply k, written in the laminate (xy,z) coordinate system can be represented by {flXX c yy0 ZZ c .rycrYZ rsXZ)= = Qk 1%Xcyy%&ayZ%,1T The terms of QR matrix of ply k to the laminate axes (x~,z) and can from the Qn matrix referred to the tions (1,2,3) with the transformation Qk = T’QT

(4) are referred be obtained fiber direc[20]

(5) where T is the transformation matrix from the (1,2,3) fiber axes to the (x,Y,z) laminate axes.

239

Eigenji-equenjl analysis and design of laminates

HIGHER ORDER FINITE ELEMENT FORMULATION The finite element formulation for the displacement field HSDT 11 represented by eqn (4) applied to nine-node Lagrangian quadrilateral elements, will be briefly described. The strain components are given by

au E,,=

-

Ch E,,E

ax

czz=

-

-

ay

t3W

au+ -av

yxv=

-

ay

az

av aw

yvr= -

ax

+-

az

ay

au+ -aw

y.,z= -

aZ

ax

(6)

yielding

I

au, ax

ax

a%

ah,

ay au, av, -+aY ax

4

ay +z aw, e,+-

y.w=

Yxz

ax

:

ay

cm \

2v;+ 7

ae,

a0

ax

au; av; -aY + ax

I

84

3q,;+ -

ay acp 2u;+ 7

+z3

0

ae, --

aq

e,+-aw,

av;

ae

+z

cp, \

au; ax

ae,

+7.*

ax I

ay

aw; 39:+ ax

A vector E;,,, including the bending and membrane terms of the strain components containing the transverse shear terms can be defined in such a way as

$=

(8)

and a vector E:

au, av, au, ---av, au;,av; au; av; sex ae ae -+2 ) 2w0, x ax ' aY 'q7' aY + ax ’ ay ’ ax ’ aY ax ’ ax ’ ay aY

--

i

ae, w:

a$

a4c :

aq::T ax I

(9)

awe awe aw; aw;,T a% aq E;= o,+-,8,+,2v;+ ,2u;+ 2 ,3q;+ ,3&+ ax ax ax I aY aY aY i

(10)

+ ax ’ ax

’

ay ’ ay

240

KM. Franc0 Correia, C. M. Mota Soares, C. A. Mota Soares

The strains can be written as

(11) where Z,, and Z, are matrices containing powers of z coordinate (z” with n = 0,...,3) and zeros defined in accordance with displacement fields and strain relations. Using Co Lagrangian shape functions [21] the displacements and generalized displacements defined in the reference surface are obtained within each element, respectively, by U V

= Z,{

u*v*w*(p*(p* 0 x y j=

u 00v w oxy e 8 cpzoo

(12)

ii W

(U

V 00

W oxy 8 8 ~ zoo U*V*W*~*(p*)== 0 x y

NiQf

in,

where the Z,,, matrix for the displacement

z,=

00 1

0

0

z

0

0

z2

0

0

[0

1

0 1

0 0

z 0

0 z

0 0

z2 0 0 z2

(13) field represented

1

z3

0

0 0

z3 0

by eqn (1) is given by

Ni are the Lagrange shape functions of node i and qf is the displacement is related to the displacement vector of the element, q”, by q”={ ...qy...}T,

i= l,..., 9

(14)

vector of node i which (15)

The strains in eqns (9) and (10) can be represented

as

matrices, respectively, for bending and membrane and where Bbm and B, are the strain-displacements shear, relating the degrees-of-freedom of the element with the strain components. Assuming small displacements and using Hamilton’s variational principle we obtain the following eigenvalue problem for free harmonic vibrations at the element level K’q; - /Z;Me& = 0

(17)

where K’ and Me are the element stiffness and mass matrices, respectively and qz are the eigenvectors associated to the eigenvalues AZ,at the element level. The stiffness and mass matrices of the element are evaluated, respectively, as

dV= [jA N=(s, Z;Z,

dz)N dA=

p,~~~~~~Z,dz)NdetJd5dq

(19)

241

Eigenfequency analysis and design of laminates

Fig. 1. Laminate

geometry and coordinate

axis.

where D is given by

N is the matrix of the Lagrange shape functions, p the material density and pk the material density of kth layer. NL is the number of layers of the laminate, hk is the vector distance from the middle surface of the laminate to the upper side of kth ply (Fig. 1). Finally, det J is the determinant of the Jacobian matrix of the transformation from (5,~) natural coordinates to element (x,y,z) coordinates. FINITE ELEMENT DISCRETE MODELS The finite element model having the displacement field represented by eqn (l), briefly described above, will be referred to as Q9HSDT 11. The remaining elements whose displacement fields are given by eqns (2) and (3), are developed easily from this parent element by deleting the appropriate degrees of freedom leading respectively, to the finite element discrete models referred to as QPHSDT 9 and Q9-HSDT 7. The Co nine-node Lagrangian first order discrete model, referred to as Q9-FSDT 5, can also be obtained by deleting all high order terms and introducing the shear correction factors on the transverse shear terms [20] of matrix D. Two other finite element models are also available in the optimization package. One is the three-node triangular shear flexible plate, based on Mindlin’s theory which was developed by Lakshminarayana et al. [22] usually referred to as TRIPLT, where the nodal degrees of freedom are the displacements and rotations uo, vo, wo, 0,, 0,, and their first derivatives with respect to x and y. The other element is the three-node

triangular discrete Kirchhoff theory multilaminated plate-shell element [23], known in the literature as DKT, first developed for isotropic plates and shells by Batoz et al. [24] and Bathe et al. [25], respectively. The above described finite element models, corresponding degrees of freedom and number of nodes are shown on Table 1. EIGENFREQUENCY ANALYSIS

SENSITIVITY

The problem of free undamped vibrations of a finite element discretized linearly elastic structure is defined by the eigenvalue problem (K-A,M)O,=On=l,...,N

(21) where K and M are respectively the stiffness and mass matrices of the structure and N represents the total number of degrees of freedom. The solution of this eigenvalue problem consists of N eigenvalues A, and corresponding eigenvectors O,, where An = W: and o, is the natural frequency of vibration mode IZ. For single eigenvalues, if we consider a vibration mode Op corresponding to the natural

242

V M. Franc0 Correia, C. M. Mota Soares. C. A. Mota Soares

frequency

~0~ normalized

through the relation of natural frequency with respect to changes in the design variable bi is obtained from OZMO, = 1, the sensitivity

are obtained analytically finite difference.

or by using forward

OPTIMAL DESIGN

This expression level yielding dw,, db;

--

is evaluated

at the element

The eigenfrequency structural optimization problem can be stated in one of the following forms min { W(b)} subject to: (26)

1

- 205, where of

(23)

where 07: is the pth mode vector for element 8 and E is the set of elements for which the design variable bi is defined. In the case of multiple eigenvalues they are not, in general, differentiable with respect to design variables and eqns (22) and (23) become inapplicable. This is because the eigenvectors corresponding to the repeated eigenvalues are not unique. In fact any linear combination of the eigenvectors will satisfy the eigenvalue problem represented by eqn (21). In this case the sensitivities can be obtained by calculating the directional derivatives [26-281 evaluated by considering a simultaneous change of all the design variables. In the present work the design variables are the ply angles and the ply thicknesses of the laminate and the sensitivities C3Ke/3biand 8M”/ abi are evaluated from

(24)

x(~~~ZTZdl)]NdetJd5dq where the derivatives aDlabi and

(25)

or

max 10~1

bf Ibilbi"

i=l,

. . . , n dv

tij (Q, b) 5 0

j=l,

... ) m

the objective

function

W(b)

~j(q,b) are the m inequality constraint equations, bf and by are respectively, the lower and upper limits of the design variables, ndv is the total number of design variables and ok the natural frequency of kth vibration mode. In the second form the most common objective is the maximization of the lowest natural frequency but it can be also considered the case of the maximization of an frequency or the difference higher order between two consecutive frequencies. In this case additional frequency constraints have to be included in order to prevent that the lower frequencies converge to a zero value. A typical frequency constraint can be represented by

$(q,b) = 1 - o,JO,~I 0 (27) where o, is the natural frequency of vibration mode s and W,Vis the specified value of sth natural frequency. The constrained optimization problems are solved by using the method of feasible directions [29] and the unconstrained problems are solved by using the BFGS [29] (BroydonFletcher-Goldfarb-Shanno) method. Ply angles and ply thicknesses can also be designed separately at two levels of optimization [23,30,31]. In this case the global objective is to minimize the weight/volume of the structure. At the first level of optimization, the weight/volume of the structure is kept constant and the optimal fiber directions for maximum natural frequencies of given modes are determined. At the second level the optimal ply thicknesses are found assuming the previously obtained ply angles. In this second level the weight/volume of the structure is minimized subject to natural frequency constraints as well as lower and upper bounds on design variables.

243

Eigenfrequency analysisand design of laminates

NUMERICAL APPLICATIONS Free vibration of simply supported laminated plates To compare the accuracy on the prediction of natural frequencies of higher order models ‘(HSDT) and first order models (FSDT) a few test cases are presented emphasizing the effect of the degree of orthotropy of individual layers (Young’s modulus ratio E,/E,) and the effect of side-to-thickness ratios. The results obtained are compared with alternative solutions when available.

Table 1. Finite element discrete models available for comparison purposes Number of nodes

Degrees-of-freedom per node

Finite element discrete model

9 9 9 9 3

u 07v0,w0,0XJe,“’cp u* v* w* cp:,cp; ~O1~O,~O,~~x,~“~~~~~~o~~rr~D, f’ ,o’ 0,’ 9’ Ul?,vo>wo,~x&C~: &I, vo>wo, cp,, Vy

Q9-HSDT 11 Q9-HSDT 9 Q9-HSDT 7 QPFSDT 5 TRIPLT

~o,vo,~o,Qx,~

aud%

avdk

awda)c, aox& atp, au&y,

DKT “The rotation

oz is used only for shell problems.

Table 2. Nondimensional plates

fundamental

natural frequencies

To analyze the effect of the degree of orthotropy of individual layers we consider the free vibration problem of a simply supported laminated square plate having the following material properties: E,/E, = 3, 10, 20, 30 and 40; E, = E, = 10 GPa; G,-JE, = G,JE2 = 0.6; GZ3/ E, = 05; \I,~= v,~ = vz3 = 0.25. The side-tothickness ratio is a/h = 5. A quarter plate with a 4 x 4 finite element mesh is used. The following symmetric stacking sequences are considered: 3-ply [00/900/07, 5-ply [oO/9~/oO/90°/Oo]and 9-ply [O”/900/oO/900/oO/900/oo/900/Oo]. Table 2 compares the results for the nondimensional fundamental natural frequency 0 obtained with HSDT, FSDT, discrete Kirchhoff model [23] (DKT) and a 3-D elasticity solution [32]. The antisymmetric stacking sequences: 4-ply [O”/900/Oo/907, [o”/90°/o”/90”/oo/900] and lo-ply 6-PlY [oO/90”/oO/90°/oO/900/Oo/900/o”/90”]are also analyzed in Table 3. From Tables 2 and 3 it can be seen that the results obtained with DKT model are not acceptable with errors increasing for higher El/E, ratios. The errors obtained with FSDT models increase when higher E,/E, ratios are used, specially for antisymmetric cases achieving errors of about 5.8%. In Table 4 are presented the nondimensional natural frequencies corresponding to the sym-

(W = w~~h’/E, x 10) of symmetric

Degree of orthotropy

3

5

9

TRIPLT DKT

2.6474 2.6315 2.6239 2.6242 2.6244 3.0863

(-0.60%) (-0.89%) (-0.88%) (-0.87%) (16.6%)

Noor (Ref. 32) Q9-HSDT 911 QPHSDT QPHSDT 7 QPFSDT 5 TRIPLT DKT

2.6587 2-6435 2.6359 2.6359 2.6348 2.6348 3.0875

(-0.57%) (-0.86%) (-0.86%) (-0.90%) (-0.90%) (16.1%)

Noor (Ref. 32) Q9-HSDT 911 Q9-HSDT 7 Q9-FSDT 5 TRIPLT DKT

2.6640 2.6466 2.6389 2.6389 2.6382 2.6380 3.0878

( -0.94%) -0.65%) ( -0.94%) ( -0.97%) ( -0.98%) (15.9%)

Noor (Ref. 32) Q9-HSDT 11 Q9-HSDT 79 Q9-FSDT 5

Percentage

20

10

3

Model

NL

of individual

simply supported square

layers E,/E, 40

30

3.2841 3.3155 3.3078 3.3052 3.3068

(0.95%) (0.72%) (0.64%) (0.69%)

3.8241 3.8131 3.8058 3.8036 3.8063 5.7055

(-0.29%) ( - 0.48%) ( -0.48%) (-0.54%) (49.2%) (-0.47%)

4.1089 4.1057 4.0989 ((-0.08%) - 0.24%) 4.1022 ( -0.24%) 4.1057 (-0.16%) (-0.08%) -

4.3006 4.3056 4.2993 4.2993 4.3108 4.3149 7.7243

((0.12%) - 0.03%) ( -0.03%) (0.24%) (0.33%) (79.6%)

3.4089 3.3995 3.3917 3.3917 3.3899 3.3914

(-0.28%) (-0.51%) (-0.51%) (-0.56%) (-0.51%)

3.9792 3.9816 3.9744 3.9744 3.9805 3.9836 5.7176

(0.06%) (-0.12%) (-0.12%) (0.03%) (43.7%) (-0.11%)

4.3140 4.3265 4.3200 (0.29%) 4.3200 (0.14%) 4.3390 (0.14%) (0.58%) 4.3431 (0.68%) -

4.5374 4.5577 4.5519 4.5519 4.5847 4.5896 7.7443

(0.45%) (0.32%) (0.32%) (1.04%) (1.15%) (70.7%)

3.4432 3.4253 3.4176 3.4176 3.4158 3.4173

-052%) ( -0.74%) ( -0.74%) ( -0.80%) ( -0.75%)

4.4210 4.4097 ( -0.26%) 4.4036 ( -0.39%) 4.4035 ( -0.39%) ( -0.40%) 4.4078 ( -0.30%)

4.6679 4.6601 4.6548 4.6548 4.6556 4.6608 7.7490

( -0.17%) ( -0.28%) ( -0.28%) ( -0.26%) ( -0.15%) (66.0%)

errors with respect to Noor solution

[32].

4.0547 4.0328 4.0398 ( - 0.37%) 4.0328 ( --0.54%) 4.0315 ( 0.54%) ( -0.57%) 4.0346 ( -0.49%) 5.7206 (41.1%)

244

c! M. Franc0 Correia, C. M. Mota Soares, C. A. Mota Soares

Table 3. Nondimensional plates

fundamental

natural frequencies

(ti = w\lz

x 10) of symmetric

simply supported

square

Degree of orthotropy of individual layers E,/E, NL

Model

4

Noor (Ref. 32) Q9-HSDT 11 Q9-HSDT 9 QPHSDT 7 Q9-FSDT 5 TRIPLT DKT

2.6182 2.6059 2.5983 2.6004 2.6018 2.6047 3.0174

( - 0.47%) ( - 0.76%) (-0.68%) ( -0.63%) ( - 0.52%) (15.2%)

Noor (Ref. 32) QPHSDT 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT DKT

2.6440 2.6287 2.6211 2.6224 2.6229 2.6240 3.0466

( - 0.58%) ( - 0.87%) ( -0.82%) ( - 0.80%) ( - 0.76%) (15.2%)

Noor (Ref. 32) Q9-HSDT 11 QPHSDT 9 Q9-HSDT 7 QPFSDT 5 TRIPLT DKT

2.6583 2.6409 2.6332 2.6338 2.6336 2.6337 3.0615

( - 0.66%) ( - 0.94%) ( - 0.92%) ( - 0.93%) ( - 0.92%) (15.2%)

6

10

3

10

20

30

3.2578 3.2594 3-2513 3.2780 3.2899 3.2999

(0.05%) (-0.20%) (0.62%) (0.99%) (1.29%)

3.7622 3.7871 3.7793 3.8502 3.8755 3.8873 5.2975

(0.66%) (0.46%) (2.34%) (3.01%) (3.33%) (40.8%)

3.3657 3.3546 3.3467 3.3622 3.3674 3.3726

( - 0.33%) ( - 0.57%) (-0.11%) (0.05%) (0.21%)

3.9359 3.9342 3.9267 3.9673 3.9772 3.9841 5.5364

( - 0.04%) ( - 0.23%) (0.80%) (1.05%) (1.22%) (40.7%)

3.4250 3.4066 3.3988 3.4051 3.4054 3.4082

( - 0.54%) ( - 0.76%) (-0.58%) (-0.57%) ( - 0.49%)

4.0337 4.0176 4.0105 4.0269 4.0256 4.0301 5.6556

( - 0.40%) (0.57%) ( - 0.17%) ( - 0.20%) ( - 0.09%) (40.2%)

40

4.0660 4.1094 4.1022 4.2135 4.2480 4.2598

(1.07%) (0.89%) (3.63%) (4.48%) (4.77%)

4.2719 4.3289 4.3224 4.4683 4.5084 4.5198 7.0879

(1.33%) (1.18%) (4.60%) (5.54%) (5*80%) (65.9%)

4.2783 4.2848 4.2782 4.3420 4.3532 4.3607

(0.15%) ( - 0.002%) (1.49%) (1.75%) (1.93%)

4.5091 4.5223 4.5164 4.6005 4.6106 4.6184 7.4655

(0.29%) (0.16%) (2.03%) (2.25%) (2.42%) (65.6%)

4.4011 4.3880 4.3818 4.4075 4.4024 4.4078

( - 0.30%) (-0.44%) (0.15%) (0.03%) (0.15%)

4.6498 4.6398 4.6344 4.6683 4.6578 4.6640 7.6528

( - 0.22%) (-0.33%) (0.40%) (0.17%) (0.30%) (64.6%)

Percentage errors with respect to Noor solution [32].

having the following material properties: E,/ E, = E,IE, = 40, G,,IE, = G,JE2 = O-6, G,,IE, following = O-5, v12 = v,~ = vz3 = O-25. The stacking sequences are considered: non-symmetric 2-ply [W/907 and symmetric 4-ply [O”/90°/9~/Oo]. For the 2-ply non-symmetric situation the results for the nondimensional fundamental natural frequency obtained with HSDT, FSDT and DKT models are compared in Table 5 with alternative values from an elasticity solution [33], a local high order model [34], a higher order theory using a displacement field that satisfies the stress-free boundary con-

metric modes higher than the fundamental one for the 4-ply antisymmetric and 5-ply symmetric lamination sequences assuming an E,/E, ratio equal to 40 and a/h = 5. Discrepancies of FSDT models with respect to higher order model Q9HSDT 11 are presented. It can be observed that FSDT models accuracy on the prediction of natural frequencies decrease when higher vibration are considered achieving modes discrepancies of about 10%. To analyze the effect of side-to-thickness ratio we consider the free vibration problem of a simply supported laminated square plate Table 4. Nondimensional natural frequencies ness ratio a/h = 5 and El/E, = 40

(ti = oL;a

x 10) of a simply supported square plate with side-to-thickVibration mode 5

12.0385 12.0559 12.8706 12.7115 (5.6%) 12.7586 (6.0%)

16.5854 16.5516 17.6423 17.3009 (4.3%) 17.5745 (6.0%)

20.7687 20.9041 22.3941 21.3789 (2.9%) 21.6656 (4.3%)

20.7687 20.9041 22.3941 21.3789 (2.9%) 21.6656 (4.3%)

13.4340 13.4316 13.5117 12.7115 (-54%) 12.8009 (-4.7%)

17.1171 17.1091 17.7173 17.3110 (1.1%) 17.5770 (2.7%)

19.9366 19.9179 21.4123 21.2922 (6.8%) 21.5693 (8.2%)

22.9358 22.9376 23.0846 21.4788 (-6.4%) 21.7609 ( -5.1%)

3

Model

4

Q9-HSDT 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT

4.3289 4.3224 4.4683 4.5084 (4.1%) 4.5198 (4.4%)

12.0385 12.0559 128706 12.6612 (5.2%) 12.7586 (6.0%)

5

QPHSDT 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT

4.5577 4.5519 4.5519 45847 (0.6%) 4.5896 (0.7%)

11.5439 11.5214 12.3709 126486 (9.6%) 12.7357 (10.3%)

Percentage errors with respect to Q9-HSDT

6

4

2

1

NL

11 model.

245

Eigenfequency analyszkand design of laminates ‘lhble 5. Nondimensional fundamental natural frequencies (5 = (o&h)&)

of non-symmetric 2-ply [O”/!lOolsimply

supported square plates (E& = 40) Side-to-thickness ratio a/h 5

10

20

50

100

8.7274 (2.5%) 8.7172 (2.3%) 9.0230 (5.9%) 8.8336 (3.7%) 8.9428 (50%) 11.5249 (35%) 10.584 (24%) 8*518 8.527 9.010

10.4323 (1%) 10.4279 (0.9%) 10.5484 (2.1%) 10.4735 (1.4%) 10.5319 (1.9%) 11.7306 (14%) 11.011 (6.6%) 10.333 10.337 10449

1 l-0668 (0*3%), 11.0653 (0.3%) 11.1003 (0.6%) 1 l-0784 (0.4%) 1 l-0971 (0.6%) 11.7825 (6.8%) 11.125 (0.8%) 11.036 11.037 10.968

11.2693 (0.06%) 11.2689 (0.05%) 11.2747 (0.1%) 11.2711 (007%) 11.2769 (0.12%) 11.7970 (4.7%) 11.158 (-0.9%) 11.263 11.264 11.132

11.2993 (0.02%) 11.2990 (0.02%) 1 l-3005 (0.03%) 11.2996 (0.02%) 11.3085 (0.1%) 11.7991 (4.4%) 11.163 (- 1.2%) 1 l-297 11.297 11.156

Model Q9-HSDTl l Q9-HSDT9 Q9-HSDT7 Q9-FSDTS TRIPLT DKT CPT (see Ref. 34) 3-D elasticity (see Ref. 33) Wu and Chen [34] Reddy and Phan [35]

Percentrrors relative to 3-D elasticity solution [33].

ditions [35] and also a closed form classical plate theory (CPT) [34]. The errors with respect to the elasticity solution are presented. The DKT model exhibits a reasonable behavior for thin plates but for very low length-to-thickness ratios achieves errors of 35%. The errors obtained with FSDT models are about 5% for very low length-to-thickness ratios. It should be noted that the behavior of the higher order model HSDT 7 for this non-symmetric stacking sequence is worse than that of the FSDT models. For the 4-ply symmetric situation the nondimensional fundamental frequencies are compared in Table 6 with alternative HSDT models [34,35]. In this case the DKT model can lead to errors of about 110% for very low sideto-thickness ratios. The CPT solution leads to maximum errors of 95%. The results obtained with FSDT models are good with maximum errors of 1.6%. In this symmetric case the behavior of the higher order model Q9-HSDT 7 is similar to the Q9-HSDT 9 model. Table 6. Nondimensional fundamental supported square plates (E1/E2 = 40)

natural frequencies

Table 7 presents the nondimensional natural frequencies corresponding to the symmetric modes higher than the fundamental one for the 2-ply non-symmetric and 4-ply symmetric lamination sequences. It can be seen that the discrepancies of FSDT models with respect to the higher order model Q9-HSDT 11 can go up to 125% on higher modes and about 25% in the fundamental mode.

Sensitivity analysis of a simply supported rectangular plate The sensitivities of fundamental natural frequency in order to ply angles and ply thicknesses obtained analytically are compared to the sensitivities obtained with the semi-analytical technique for all higher order models and first order model Q9-FSDT 5. A 4-ply simply supported rectangular a x b plate with b/a = 2 and a/h = 25 (side-to-thickness ratio) is (ti = (oa’/h)Jz)

for symmetric 4-ply [O”/90“/!W/Oo] simply

Side-to-thickness ratio a/h Model Q9-HSDTl l Q9-HSDT9 Q9-HSDT7 Q9-FSDTS TRIPLT DKT CFT (see Ref. 34) Wu and Chen [34] Reddy and Phan [35]

4

10

20

50

9.2871 (1%) 9.2711 (0.9%) 9.2711 (0.9%) 9.3311 (1.5%) 9.3416 (1.6%) 19.3108 (110%) 17907 (95%) 9.193 9.497

15.1052 (0.24%) 15.0954 (0.17%) 15.0954 (0.17%) 15.0671 (0.01%) 15.0739 (0.03%) 19.3108 (28%) 18.652 (24%) lN69 15.270

176478 (0.07%) 176441 (0.05%) 17.6441 (0.05%) 17.6262 ( - 0.06%) 17.6287 ( - 0.04%) 19.3108 (9.5%) 18.767 (6.4%) 17636 17.668

18.6728 (0.01%) 18.6721 (0.01%) 18.6721 (0.01%) 186684 ( - 0.01%) 18.6704 (0.002%) 19.3108 (3.4%) 18.799 (0.7%) 18.670 17606

Percentage errors relative to Wu and Chen solution [34].

100 18.8366 (O*OOS%) 18.8364 (0.007%) 18.8364 (0.007%) 18.8354 (0.002%) 18.8415 (0.03%) 19.3108 (2.5%) 18.804 (-0.16%) 18.835 18.755

V M. Franc0 Con-eia, C. M. Mota Soares, C. A. Mota Soares

246 Table 7. Nondimensional NL a/h

of a simply supported square plate with E,IE,= 40

natural frequencies (W = (oxz’/h)ip/EJ

Vibration mode Model

1

2

3

NL = 2 Q9 -HSDT 11 8.7274 a/h = 5 Q9-HSDT 9 8.7172 QPHSDT Q9-FSDT TRIPLT

7 5

NL = 4 Q9-HSDT a/h = 4 Q9-HSDT

11 9 7 5

Q9-HSDT Q9-FSDT TRIPLT

4

5

6

28.3796 28.3850 9.0230 31.1851 8.8336 (1.2%) 29.4015 (3.6%) 8.9428 (2.5%) 29.8134 (5.1%)

28.3796 28.3850 31.1851 29.4015 (3.6%) 29.8242 (5.1%)

39.7852 46.8458 39.6478 46.9617 43.1978 47.5582 40.6330 (2.1%) 49.3446 (5.3%) 41.5051 (4.3%) 52.7198 (12.5%)

47.4878 47.4370 48.1734 49.9509 (5.2%) 52.7347 (11.1%)

9.2871 22.6618 9.2711 22.5682 9.2711 22.5682 9.3311 (0.5%) 25.0942 (10.7%) 9.3416 (0.6%) 25.2618 (11.5%)

27.1165 27.1072 27.1072 25.4654 (-6.1%) 25.6452 (-5.4%)

34.1463 39.2705 34.0878 39.1383 34.0878 39.1383 34.4599 (0.9%) 42.4348 (8.1%) 34.9811 (2.4%) 42.9830 (9.5%)

46.2667 46.2617 46-2617 42.9719 (-7.1%) 43.5268 (-5.9%)

Percentage errors with respect to Q9-HSDT

11 model.

and efficiently obtained first order models.

for all higher order and

assumed. The total thickness of the laminate is 0.04 m, all plies have the same thickness and the lamination scheme is [45”/ - 45”/ - 45”/45”]. The material properties are: E, = 142.5 GPa, E, = E, = 9.79 GPa, G,, = G13 = 4.72 GPa, G,, = 1.192 GPa, v,~ = v13 = vz3 = O-25, p = 2000 kg/ m3. The analytical and semi-analytical sensitivities with respect to ply angles and ply thicknesses are presented, respectively, in Tables 8 and 9. Alternative values obtained from global finite difference (GFD) where the entire analysis is repeated for a perturbed variable are also presented only for comparison purposes. Analytical and semi-analytical sensitivities are accurately

Consider the illustrative case of a cantilever rectangular laminated panel presented in Fig. 2. The length of the panel is 4 m and the width is 1 m. The laminate is made of three plies having equal thickness and the symmetric lamination scheme of [00/9~/00]. The 0” direction is aligned with the longest side of the panel. The material properties are those of a high modulus graphite/ epoxy: E, = 220 GPa, E, = E, = 6.9 GPa, G,2

Table 8. Sensitivities of first natural frequency in order to ply angles ( x 10’ SC’)

Table 9. Sensitivities of first natural frequency in order to ply thicknesses ( x lo5 rad/s.m-‘)

Ply QPHSDT II

1 2 :

QPHSDT 9

1 2 3 4

Q9-HSDT 7 1

32 4

QPFSDT 5 1 2 3 4

Optimal design of a cantilever rectangular panel

Analytical

Semi-analytical

GFD

- 1.8009693 0.50255057 - 0.50255057 1.8009693

- 1.8009696 0.50255004 - 0.50255004 1.8009698

- 1.800912 0.5025527 -0.1800912 0.5025527

- 1.8016872 0.50270647 0.50270647 -1.8016872

- 1.8016875 0.50270595 0.50270595 - 1.8016877

- 1.8016303 0.50268567 0.50268567 - 1.8016303

- 1.8016872

- 1.8016875

- 1.8016303

1

2 3 4

Analytical

Semi-analytical

GFD

- 0.1402666 -0.2016517 -0.2016517 - 0.1402666

- 0.1402666 -0.2016518 -0.2016517 - 0.1402666

-0.1402613 - 0.2016475 -0.2016475 -0.1402613

- 0.1401263 - 0.2015762 - 0.2015762 -0.1401263

-0.1401264 -0.2015763 -0.2015763 -0.1401264

-0.1401210 -0.2015721 -0.2015721 - 0.1401210

-

0.1401263 0.2015762 0.2015762 0.1401263

-0.1401264 -0.2015763 -0.2015763 -0.1401264

-0.1401210 -0.2015721 -0.2015721 -0.1401210

-

0.1368604 0.2029065 0.2029065 0.1368604

- 0.1404887 - 0.2065348 - 0.2065348 -0.1404887

- 0.1404825 -0.2065315 -0.2065315 - 0.1404825

Ply Q9-HSDT 11 1 2 3 4

Q9-HSDT 9 1

2 3 4 QPHSDT 7

0.50270647 - 1.8016872

0.50270595 - 1.8016877

0.50268567 050268567 - 1.8016303

- 1.8093615 0.51378920 0.51378920 - 1.8093615

- 1.8093612 0.51378973 0.51378973 - 1.8093612

- 1.8093549 0.5 1378844 0.5 1378844 - 1.8093549

a6, = 0*0000175 rad z 0.0001”.

Q9-FSDT 5 1

2 3 4

ah = 0.00001 m.

247

Eigenfiequency analysis and design of laminates

divided into eight regions.

= G13 = G23 = 4.8 GPa, p = 1640 kg/m3.

~12 =

~23 =

~13 =

0.25,

The rectangular plate is modeled using a 8 x 4 finite element mesh, divided into eight regions as shown in Fig. 2. The panel is designed for minimum volume subjected to frequency constraints and minimum ply thickness. The design variables are the ply thicknesses in each of the eight regions, giving a total of 24 design variables. The initial ply thickness in all regions is 0.02 m. Two test cases are shown. First, a frequency constraint is imposed on the fundamental vibration mode with a minimum frequency of 100 rad/s. The lower bound imposed on ply thickness is 0.004 m. The optimization results are shown in Table 10. In this first case the

side-to-thickness ratio ranges from about 9 to 80. The optimum design obtained with all HSDT models is approximately the same. The optimum volume obtained with the first order model Q9-FSDT 5 is about 9% lower than the values obtained with HSDT models. The natural frequencies for the first four vibration modes are also presented in Table 10 and it can be seen that for the optimum design the first order model shows discrepancies in the natural frequencies up to about 15% for higher vibration modes. The average CPU time ratios with respect to Q9-FSDT 5 model and the number of iterations needed to achieve the optimal design are presented in Table 11. The vibration modes are presented for initial design in Fig. 3 and for optimal design in Fig. 4.

Table 10. Optimal design for minimum volume of a cantilever rectangular panel (case 1)

Initial design

Optimum

design

Model

Total thickness in regions l-8 (m)

Volume (m”)

Natural frequencies (rad/s)

Q9-HSDTll

0.06

0.240

Q9-HSDT9

0.06

0.240

Q9-HSDT7

0.06

0.240

Q9-FSDTS

0.06

0.240

43.2 267.3 43.2 267.3 43.2 267.3 43.2 267.1

104.7 381.2 104.7 381.2 104.7 381.2 104.7 381.1

99.8 344.7 99.9 341.7 99.9 341.7 99.9 291.5

165.0 390.5 164.8 386.9 164.8 386.9 144.3 340.5

Q9-HSDTll

0.24496

QPHSDT9

0.24409

Q9-HSDT7

0.24409

Q9-FSDTS

0.22190

Table 11. Average CPU ratios for the optimal design of a cantilever rectangular panel (case 1) Q9-HSDT Average CPU ratios wrt Q9-FSDTS Number of Iterations

5.3 4

11

Q9-HSDT 3.1 4

9

Q9-HSDT 1.9 4

7

Q9-FSDT

:,

5

248

V M. France Correia, C. M, Mota Soar-es, C. A. Mota Soares

Mode 1

Mode 3

Mode 2

Mode 4

Fig. 3. Vibration

modes of cantilever

In the second test case, three frequency constraints are imposed on the first three vibration The minimum values imposed on modes. natural frequencies are respectively: 100 rad/s, 150 rad/s and 300 rad/s. The lower bound

panel for initial design.

imposed on ply thickness is 0.005 m. The optimization results are shown in Table 12. The side-to-thickness ratio ranges from about 9 to 80. The optimum volume obtained with the FSDT model is about 5% lower than the values

Mode 1

Mode 3

Mode 2

Fig. 4. Vibration

Mode 4

modes of cantilever

panel for optimal design (case 1).

Table 12. Optimal design for minimum volume of a cantilever rectangular panel (case 2) Total thickness in regions l-8 (m)

Model Initial design

Optimum

design

QPHSDTI

1

Volume (m”)

O-06

0.240

Q9-HSDT9

0.06

0.240

Q9-HSDT7

0.06

0.240

Q9-FSDTS

0.06

0.240

Q9-HSDTll

O-24487

Q9-HSDT9

0.24482

Q9-HSDT7

0.24482

Q9-FSDTS

0.23183

Natural frequencies (rad/s) 43.2 267.3 43.2 267.3 43,2 267.3 43.2 267.1

104.7 381.2 104.7 381.2 104.7 381.2 104.7 381.1

105.7 299.6 105.7 299.5 105.7 299.5 100.9 299.7

163.4 348.6 163.2 348.6 163.2 348.6 152-l 350.7

249

Eigenfrequency analysis and design of laminates

obtained with HSDT models. Discrepancies are found in the optimal design thickness distribuetion obtained with the FSDT model with respect to HSDT models and consequently on natural frequencies for optimal design. The lower accuracy on the prediction of natural frequencies denoted by FSDT model in situations of low side-to-thickness ratios leads to lower structural volumes and therefore unsafe designs. The average CPU time ratios and number of iterations for this second case are presented in Table 13. Optimal design of square plate with central circular hole The optimal design for maximum fundamental natural frequency of a simply supported laminated square plate with a central circular hole is considered. The laminate is made up with six plies of equal thickness with h, = 0.01 m. The are: E, = 138 GPa, properties material E, = E3 = 8.96 GPa, G12 = G,3 = GZ3 = 7.1 GPa, v12 = v23 = v13 = 0.3, p = 2000 kg/m3. The side dimension of the plate is a = 2 m and the hole diameter is a/3. A full finite element mesh with 288 nodes and 64 elements as shown in Fig. 5 is used. The objective function is the maximization of the fundamental natural frequency and the design variables are the ply angles. The weight of the structure is kept conthis unconstrained stant. To accomplish optimization problem the plate is divided into eight regions, 16 regions and 32 regions as

shown in Fig. 5. The ply angles for all elements lying in one region are equal. In the initial design the ply angles are all set to 0” (fibers aligned with x axis). The optimization process is developed in three stages. First, considering the plate divided into eight regions an optimization problem with 48 design variables is solved (six design variables in each region). The optimal ply angles obtained from this first problem are introduced as initial design in the second optimization problem where the plate is divided into 16 regions leading to 96 design variables. Finally the optimal ply angles obtained from the second problem are the initial design in the third optimization problem where the plate is divided into 32 regions and 192 design variables are obtained. The first six vibration modes of the laminate and corresponding natural frequencies obtained with Q9-HSDT 9 model are presented for the initial design in Fig. 6 and for optimal design in Fig. 7. The optimal ply angles obtained with higher order models and first order model are presented in Table 14. Due to the symmetry of the optimal stacking sequence about the midplane of the laminate and the symmetry of the optimal ply angles with respect to x and y axes, only the results for lst, 2nd and 3rd plies and for regions located on 1st quadrant are shown. Fig. 8 presents the optimal ply angles and corresponding objective function obtained considering the laminate divided into eight, 16 and 32 regions. A good agreement between all HSDT models is found and a slightly higher

Table 13. Average CPU ratios for the optimal design of a cantilever rectangular panel (case 2) Q9-HSDT Average CPU ratios wrt Q9-FSDT Number of Iterations

5

Fimte element mesh

11

2.7 5

9

Q9-HSDT 1.2 7

1.9 7

8 Regions

Fig. 5. Simply supported

Q9-HSDT

16 Regions

32 Regions

square plate with central circular hole.

7

QPFSDT 1 11

5

2.50

Fig. 6. Vibration

Fig. 7. Vibration

V M. Franc0 Correia, C. M. Mota Soar-es, C. A. Mota Soares

Mode 1

(365.4 radk)

Mode 2

(590.1 radk)

Mode 3

(1014.9 radk)

modes of simply supported

(1071.3 radk)

Mode 5

(I493 6 ml/s)

Mode 6

(1612.5 radk)

plate for initial design (frequencies

Mode 1

(560.4 ml/s)

Mode 2

(8’10.9 radk)

Mode 3

( 1076.8 rad/s)

modes of simply supported

Mode 4

obtained

Mode 4

(1585.7 radis)

Mode 5

( 1835.6 radk)

Mode 6

(2273.9 ml/s)

plate for optimal design (frequencies

obtained

with QFHSDT

with Q9-HSDT

9 model).

9 model).

251

Eigenfiequency analysis and design of laminates Table 14. Optimal ply angles for maximum fundamental circular hole Q9-HSDT

Region

: 17 18 19 20 First natural frequency Initial design Final design Objective increase

objective model.

increase

11

-51.2”-13P-1.8” -47.1”/-13.5”/-1.8” -38.9”/-20.2”/-2.6” -34.3”/-17.2”/-2.6” -52.7”/-13.O”/-1-8” -54.2”/-13G’/-1.8” -41.4”/-21.0”/-2.6” -34.3”/-18.4”/-2+Y

:

natural frequency of simply supported square plate with central

QPHSDT

QPHSDT

9

-51.2”/-13W’/-1.8” -47.2”/-13.5”/-1.8 -38.9”/-20.2”/-2.6” -34.3”/-17.2”/-2.6” -52~7°/-13~Oo/-1~8” -54.2”/-13.0”/-1.8” -41.4”/-21V-2.6” -34.3”/-18.4”/-2.6

Q9-FSDT

7

-51.1”/--13.0”/-1.8” -47.2”/-13.5”/-1.8” -38+3=‘-20.2”/-26” -34.3”/-17.2”/-2.6” -52.6”/-13.0”/-1.8” -54.2”/-13.0”/-1.8” -41@/-21.0”/-2.6” -34.4”/-18.4”/-2.6”

5

-51.6”/-13.2”/-1.9” -47.9”/--13.7”/-1.9” -38W-20.1”/-2.7” -33.9”/-175-2.7 -52.7”/-13.1”/-1-9 -54+?/-13.1”/-1.9” -40.6”/-20,8”/-2.7” -33Y-18.5”/-2.7

(rad/s) 365.44 560.64 53.41%

is

obtained

with

FSDT

CONCLUSIONS Kirchhoff models are unable to predict the behavior of highly anisotropic and/or low length-to-thickness ratio laminated plates. Numerical illustrative applications have shown that higher order models are able to accurately predict the behavior of highly anisotropic and/or low length-to-thickness ratio laminated plates with reasonable advantage

8 Regions

365.41 560.32 53.34%

365.41 560.36 53,35%

16 Regions

365.39 562.47 53.94%

over first order models. Errors presented by first order models in the prediction of fundamental natural frequencies are about 3-5% higher than higher order models. These errors increase for vibration modes higher than the fundamental one achieving about 12% in the examples shown. The quadrangular nine-node Lagrangian elements with higher order displacement fields have shown good accuracy but the computational efficiency decreases when more complex displacement fields are used. The higher order model with nine degrees-of-freedom per node Q9-HSDT 9 seems to represent a reasonable

32 Regions

3” and 4’ Plies

5 18.3 radk

554.9 radls

Fig. 8. Optimal ply angles of simply supported

560.4 radk

plate considering

ObJectwe fimctlon

eight, 16 and 32 regions.

252

KM. Franc0 Cor&ia, C. M. Mota Soares, C.’ A. Mota Soares

compromise between accuracy and computational efficiency. The effect of including the higher order degrees-of-freedom cpZand W: in the transverse direction in Q9-HSDT 11 model is not appreciable on eigenfrequency problems. The analytical sensitivities in order to ply angles and ply thicknesses are easily and efficiently obtained for discrete models based on higher order displacement fields. A good agreement was found between analytical and semi-analytical sensitivities in the examples shown. It is believed that the higher order theories are an improvement over the first order theory in order to accurately predict the behavior of laminated plates with low length-to-thickness ratios and/or high degree of anisotropy. The use of higher order discrete models in the optimal design of multilayered composite plates and sandwich plates is very promising as these models can be easily implemented in existing structural optimization packages,

7. 8. 9. 10.

11.

12. 13.

14.

15.

ACKNOWLEDGEMENTS 16.

Sponsorship from the following grants is gratefully acknowledged: Human Capital and Mobility Project ‘Diagnostic and Reliability of Composite Materials and Structures for Transportation Applications’ Advanced (CHRTX-CT93-0222), Fundacao Calouste Gulbenkian and Fundacao Luso Americana para o Desenvolvimento (FLAD).

17.

18.

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