Coupling effects in bending, buckling and free vibration of generally laminated composite beams

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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 1664–1670 www.elsevier.com/locate/compscitech

Coupling effects in bending, buckling and free vibration of generally laminated composite beams Chai Gin Boay *, Yap Chun Wee School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 10 September 2007; received in revised form 5 February 2008; accepted 10 February 2008 Available online 19 February 2008

Abstract A closed form expression to determine the effective flexural modulus of a laminated composite beam is developed and presented in this contribution. This effective flexural modulus is applied to the bending, buckling and free vibration response of generally laminated composite beams with various boundary supports. The expression was developed using the combination of the Euler–Bernoulli beam and classical lamination theory. In addition the results of an extensive finite element analysis are used to validate the analytical model. The comparison of the analytical results, the finite element results and the experimental results showed good correlation. It is also observed that coupling response is an important variable that must be included in the computation of the effective flexural stiffness of generally laminated beam. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Effective flexural modulus; Composite laminate; Analytical method; Finite element analysis; Coupling effects

1. Introduction The fiber-reinforced composite materials are used widely because of their high strength-to-density and stiffness-todensity ratios as compared with most metals. This is especially true in aerospace industries such as space applications, the making of military and civil aircrafts, where weight saving is very important. Other areas include automotive and sports applications. Furthermore, the cost of fiber-reinforced composite materials has decreased over the years. This is due to the increased manufacturing experience accumulated over the years and more effective manufacturing technologies for mass production. With the high specific stiffness/strength of composites, structural components get more slender and lighter. The structural failure such as buckling and bending becomes dominant in thin and slender structures carrying axial compression and transverse loads. In addition, the current research into the buckling of delaminated composite beams is growing *

Corresponding author. Tel.: +65 6790 5756; fax: +65 6792 4062. E-mail address: [email protected] (G.B. Chai).

0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.02.014

exponentially [1–6] and there is a need to include the effect of composite coupling in the mathematical modeling. The ability to predict how coupling responses in composite structures affect the load carrying capacity thus becomes very urgent and important. Several researchers have dealt into the understanding on how coupling affects the structural behaviour such as Whitney [7], Halpin [8], Iyengar [9], Ashton [10] and Jones [11] but most of them covers the plate and shell structures with a few rare studies on beams and columns. The effect of the complete range of coupling responses on the structural response of composite structures has never been mentioned or analyzed in the literature. Halpin [8] and Iyengar [9] dealt with the effect of coupling responses on buckling behaviour of laminated composite beams by assuming that the variation in the width direction can be neglected in the two-dimensional problem of the beam. The mathematical models presented do not cater for only coupling responses of B11 (a coupling stiffness between in-plane and out-plane response). An excellent review paper by Kapania [12] summarises the available approaches and techniques on analyzing laminated beams and plates at the end of the 80s.

G.B. Chai, C.W. Yap / Composites Science and Technology 68 (2008) 1664–1670

Some recent publications on the effect of coupling on the structural behaviour of laminated beam are by Khdeir and Reddy [13,14] who presented exact solution to the problem of cross-ply laminated beams and, Lifshitz and Gildin [15] who conducted experiments to investigate how the carbon/ epoxy composite beam behaves under cyclic compressive loading. The response of the composite beam from the initial cyclic compressive loading to failure was examined. They developed a closed form solution to the buckling of laminated composite beams with coupling responses, but again not all coupling responses are included. In this contribution, the Euler–Bernoulli beam theory and the classical lamination theory are brought together to develop a closed form solution to the structural problem of a generally laminated beam. This closed form would be very useful in advancing the mathematical modeling of laminated beams or columns with delaminations.

1665

The coupling terms in Eqs. (1) and (2) are calculated using the following formulae: Aij ¼

N X

ðQij Þk ðzk  zk1 Þ

ð3Þ

k¼1

Bij ¼

N 1X ðQ Þ ðz2  z2k1 Þ 2 k¼1 ij k k

ð4Þ

Dij ¼

N 1X ðQ Þ ðz3  z3k1 Þ 3 k¼1 ij k k

ð5Þ

where Qij are the transformed reduced stiffnesses while zk is the distance measured from the mid-plane of the cross-section. Expanding the classical lamination as follows:

2. Analytical model The Euler–Bernoulli beam and traditional classical lamination theory are brought together to obtain a closed form solution for the determination of the effective flexural stiffness of a generally laminated composite beam. The laminated composite beam with clamped ends shown in Fig. 1a is subjected to an axial compressive force, P at one end and the x-axis starts from the left end of the beam to any point along the length of the beam. Fig. 1b shows a simply-supported laminated composite beam carrying a concentrated load F at mid-span. Using the classical lamination theory as a starting point: 9 2 8 38 0 9 2 3 A11 A12 A16 > B11 B12 B16 > = = < Nx > < ex > 6 7 6 7 Ny ¼ 4 A12 A22 A26 5 e0y þ 4 B12 B22 B26 5 > > > > ; : : 0 ; N xy A16 A26 A66 B16 B26 B66 cxy 9 8 > = < jx > ð1Þ  jy > > ; : jxy 9 2 8 38 0 9 2 3 B11 B12 B16 > D11 D12 D16 > = = < Mx > < ex > 6 7 6 7 My ¼ 4 B12 B22 B26 5 e0y þ 4 D12 D22 D26 5 > > > ; ; : : 0 > M xy B16 B26 B66 D16 D26 D66 cxy 9 8 > = < jx > ð2Þ  jy > > ; : jxy

N x ¼ A11 e0x þ A12 e0y þ A16 c0xy þ B11 jx þ B12 jy þ B16 jxy

ð6Þ

N y ¼ A12 e0x þ A22 e0y þ A26 c0xy þ B12 jx þ B22 jy þ B26 jxy

ð7Þ

N xy ¼

A16 e0x

Mx ¼

B11 e0x

My ¼

B12 e0x

M xy ¼

þ

A26 e0y

þ

B12 e0y

þ

B22 e0y

B16 e0x

þ

þ

A66 c0xy

þ B16 jx þ B26 jy þ B66 jxy

ð8Þ

þ

B16 c0xy

þ D11 jx þ D12 jy þ D16 jxy

ð9Þ

þ

B26 c0xy

þ D12 jx þ D22 jy þ D26 jxy ð10Þ

B66 c0xy

þ D16 jx þ D26 jy þ D66 jxy ð11Þ

B26 e0y

þ

Since the analysis is confined to the bending behaviour of beams subjected to loads applied only in the x–y plane, Eqs. (7), (8), (10) and (11) can be set equal to zero and solving them as simultaneous equations yields: e0y ¼ C 11 e0x þ C 12 jx c0xy ¼ C 21 e0x þ C 22 jx

ð12Þ

jy ¼ C 31 e0x þ C 32 jx jxy ¼ C 41 e0x þ C 42 jx

where Cij are constants containing the stiffnesses of Eqs. (3)–(5). The expressions for the Cij are compiled in Appendix A. Substituting Eq. (12) into Eq. (6) gives a form of solution for strain e0x as: e0x ¼ C 51 jx þ C 52 N x

ð13Þ

where C51 and C52 are constants containing the constants Cij of Eq. (12). As it will be seen later that the constant C52 can be ignored. The important constant here is constant C51 which is obtained as: C 51 ¼

C 12 A12 þ C 22 A16  B11  C 32 B12  C 42 B16 A11 þ C 11 A12 þ C 21 A16  C 31 B12  C 41 B16 F

y

y

P x L

(a) A clamped beam

x L

(b) A simply-supported beam

Fig. 1. Laminated composite beam: (a) a clamped beam and (b) a simply-supported beam.

ð14Þ

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Finally substituting Eq. (13) into Eq. (9) yields the conventional form of the moment equation:

Table 1 Non-dimensional parameters for use in Eq. (21)

o2 w M x ¼ EIjx þ C 61 N x ¼ EI 2 þ C 61 N x ox

Supports

ð15Þ

where EI is the effective bending modulus of the beam per unit width and C61 is an insignificant constant which will be discarded as mention later. The effective bending modulus is of this form: EI ¼ D11 þ ðC 31 C 51 þ C 32 ÞD12 þ ðC 41 C 51 þ C 42 ÞD16  B11 C 51  ðC 11 C 51 þ C 12 ÞB12  ðC 21 C 51 þ C 22 ÞB16

ð16Þ

Based on the transverse equilibrium of an element in a beam under transverse loads, axial compression and free vibration, the governing equation can be derived as: 2

2

2

o Mx ow ow  2 2 þq¼0 þ P 2  mx 2 ox ox ox

ð17Þ

 is mass per where w is the deflection in the y-direction, m unit length of the beam, x is the natural vibration frequency of the beam and q is the transverse load acting on the beam. Substituting the second derivative of Eq. (15) with respect to the variable x and substituting that into the governing Eq. (17) yields: 4

EI

2

S–S C–C

Mode 1 n1 2

p 22.37

Mode 2 n2 2

4p 61.67

Mode 3 n3 2

9p 120.9

Mode 4 n4 2

16p 199.8

Mode 5 n5 25p2 298.5

2

of the buckling load, P CR ¼ K pL2 EI where K is dependent on the support conditions. Traditionally K = 1 for simply-supported or hinged beams and K = 4 for clamped beams. The governing equation for free vibration to bending inertia of the beam and its form of solution are: 2 o4 w 4o w  b ¼0)w ox4 ox2 ¼ f1 sin bx þ f2 cos bx þ f3 sinh bx þ f4

 cosh bx

ð21Þ

 2 mx

where b4 ¼ EI and fi are the undetermined coefficients of amplitude. The solution to Eq. (21) qffiffiffiffi is the well known form where nm is dependent of natural frequencies, xm ¼ nLm2 EI  m on the support conditions. Table 1 shows the value of for the first five natural frequencies of simply supported (S– S) and clamped (C–C) beam. 3. Finite element analysis

2

ow ow ow  2 2 þq¼0 þ P 2  mx ox4 ox ox

ð18Þ

Assuming small deflection, linear elastic behaviour and also that the in-plane load of Nx is either a zero or a constant which is probably true for the cases of bending, axial compression and free vibration analyze in this contribution. Then the second derivative of Nx (hence the related constants of C52 and C61) with respect to the variable x can be ignored. Solving Eq. (18) for the individual case yields the following results. Solution to bending due to transverse loads yields:    Z Z Z Z o4 w qðxÞ 1 ¼  ) w ¼  qðxÞdx dx dx dx ox4 EI EI f1 x3 f2 x2 þ þ f3 x þ f4 ð19Þ þ 6 2 where fi are constants of integration and they can be determined depending on the support conditions. For the case of simply supported and clamped beam of length L carrying a concentrated load F at mid-span, the deflection at FL3 FL3 mid-span is w ¼ 48EI and w ¼ 192EI respectively. These are well known solutions found in many textbooks on engineering mechanics. The governing equation for linear buckling due to axial compression and its form of solution are: 2 o4 w 2o w þ k ¼ 0 ) w ¼ f1 sin kx þ f2 coskx þ f3 x þ f4 ox4 ox2

ð20Þ

P and fi are the undetermined coefficients of where k2 ¼ EI deflection. The solution to Eq. (20) is the well known form

In order to validate the theory presented earlier, laminated composite beams are modeled using a commercially available finite element method software called ANSYS. The length and width of the beam are arbitrary chosen as 500 mm and 10 mm respectively while the thickness per ply is 0.125 mm. The material is carbon fibre reinforced epoxy composites with the following properties: E11 = 150 GPa, E22 = 9 GPa, G12 = 4.8 GPa, m12 = 0.28 and q = 1230 kg/ m3. Eight-noded layered shell elements designed specifically for laminated composites are used in the modeling of the beam. A convergence test performed on the analysis of the finite element model shows that a finite element model of 360 elements is sufficient to give consistent and converged results. All analyses performed are based on linearly elastic behaviour and small deflection approximation. In total, five finite element models are analyzed in details: (1) a clamped laminated composite beam subjected to axial compression, (2) free vibration of a simply-supported laminated composite beam with support condition applied at a node on the ends as shown in Fig. 2a, (3) free vibration of a simply-supported laminated composite beam with support condition applied at all nodes on the ends as shown in Fig. 2b, (4) a simply-supported laminated composite beam subjected to a concentrated load at mid-span with support condition applied at a node on the ends as shown in Fig. 2a and,

G.B. Chai, C.W. Yap / Composites Science and Technology 68 (2008) 1664–1670

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4. Results and discussion The results obtained using the present theory are compared with those of the finite element analysis for the cases of buckling, free vibration and bending of laminated composite beams. The anti-symmetric angle-ply laminates and the symmetric angle-ply laminates are chosen for their severe coupling responses. The anti-symmetric angle-ply contains the coupling terms of B16 and B26 that link the in-plane extensional response to the out-of-plane twisting response of the laminate. The symmetric angleply contains the coupling terms of A16, A26, D16 and D26 where the A16 and A26 terms link the in-plane extensional response to in-plane shear response of the laminate and D16 and D26 terms couple the out-of-plane bending response to the out-of-plane twisting response of the laminate. The results for the buckling of clamped anti-symmetric and symmetric laminated composite beams are shown in Figs. 3 and 4 respectively. The buckling loads are normalized with respect to the buckling load of a [±h]n laminate for the anti-symmetric angle-ply case and a [h]n laminate for the symmetric angle-ply case. The correlation between the results of the present theory with those of the finite element method is excellent. As expected, the solutions to anti-symmetric laminated angle-ply composite plates converges to the specially orthotropic solution (solution that ignored coupling) as the number of ply increases. In Fig. 4, the results for the symmetric laminated angleply are typical regardless of the number of plies in the lam-

1 0.9

8-ply

0.8

4-ply

0.7 0.6 0.5

specially orthotropic

0.4 0.3 2-ply

0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

Ply Angle, degree Fig. 3. Normalized buckling load for clamped anti-symmetric angle-ply laminated beam (symbols are ANSYS results, light solid lines are present theory results).

1

Normalized Buckling Load, P/Po

(5) a simply-supported laminated composite beam subjected to a concentrated load at mid-span with support condition applied at all nodes on the ends as shown in Fig. 2b.

Normalized Buckling Load, P/Po

Fig. 2. Simply-supported symmetric angle-ply laminate: (a) point constraint and (b) edge constraint.

Present Theory FEM Theory (no coupling)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

Ply Angle, degree Fig. 4. Normalized buckling load for clamped symmetric angle-ply laminated beam.

G.B. Chai, C.W. Yap / Composites Science and Technology 68 (2008) 1664–1670

inate. As can be seen, the agreement of the results of the present theory with those of the finite element method is very close. Again the ‘‘specially orthotropic” results are presented as an indication that the existence of the A16, A26, D16 and D26 coupling terms is devastating to the load carrying capacity for this class of laminate. In a similar fashion, the results are presented for the free vibration of simply-supported anti-symmetric and symmetric laminated composite beams in Figs. 5 and 6 respectively. The results of the present theory are compared with the results of the finite element method and they confirmed the accuracy of the theoretical predictions. The fundamental natural frequencies are normalized with respect to the fundamental natural frequencies of a [±h]n laminate for the anti-symmetric angle-ply case and a [h]n laminate for the symmetric angle-ply case. The observations made for the results in the earlier case of axial compression apply equally well here. One important thing to mention here is that the understanding of support

18.2

Normalized Max Deflection

1668

16.2 14.2 12.2 10.2 8.2

4-ply

6.2

2-ply specially orthotropic

4.2 2.2 8-ply

0.2 0

4-ply

0.9 0.8 0.7

specially orthotropic

0.6 0.5 2-ply

0.4

40

50

60

70

80

90

Present Theory Theory (no coupling) FEM point constraint FEM edge constraint

16 14 12 10 8 6 4 2 0 0

0.3

10

20

30

40

50

60

70

80

90

Ply Angle, degree 0.2 0

10

20

30

40

50

60

70

80

90

Ply Angle, degree Fig. 5. Normalized fundamental frequency for simply-supported antisymmetric angle-ply laminated beam (symbols are ANSYS results, light solid lines are present theory results).

1 Present Theory

Normalized Mode 1 Frequency

30

Fig. 7. Normalized maximum deflection for simply-supported anti-symmetric angle-ply laminated beam (symbols are ANSYS results, light solid lines are present theory results).

Normalized Max Deflection

Normalized Mode 1 Frequency

8-ply

20

Ply Angle, degree

18

1

10

0.9

Theory (no coupling) FEM edge constraint

0.8

FEM point constraint

0.7 0.6 0.5 0.4 0.3 0.2 0

10

20

30

40

50

60

70

80

90

Ply Angle, degree Fig. 6. Normalized fundamental frequency for simply-supported symmetric angle-ply laminated beam.

Fig. 8. Normalized maximum deflection for simply-supported symmetric angle-ply laminated beam.

conditions is an important consideration as can be seen in Fig. 6. By allowing the supporting edge to rotate as indicated in Fig. 2a gave much lower fundamental natural frequencies than that when the simple support resist rotation as of Fig. 2b. The difference between the two results is clearly obvious in the region between ply angle of 0° and 50° as observed in Fig. 6. In Figs. 7 and 8, the respective results for the mid-span deflection of a simply-supported anti-symmetric and symmetric laminated composite beam are presented. The mid-span deflections are normalized with respect to the mid-span deflections of a [±h]n laminate for the anti-symmetric angle-ply case and a [h]n laminate for the symmetric angle-ply case. Again the correlation between the results of the present theory with those of the finite element method is excellent. The results are conclusive the coupling in laminate reduces the bending stiffness and hence giving rise to higher deflections than an equivalent laminate having no coupling responses as in the case of the specially orthotropic laminate. The obvious effect of the edge support conditions is again highlighted here.

G.B. Chai, C.W. Yap / Composites Science and Technology 68 (2008) 1664–1670

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5. Concluding remarks

a14 ¼ B16 B326 þ B12 B22 B266  B12 B66 B226  B22 B66 B16 B26

In this paper, a closed form solution using the Euler–Bernoulli beam theory in conjunction with the classical laminate theory is developed to predict the structural response of laminated composite beams. The results were compared with those generated using the finite element method and they showed good agreement. The excellent correlations conclude that the coupling responses can be fully accounted for in the prediction of bending, buckling and free vibration behaviour of generally laminated composite beams. Care has to be taken however when the supports are simply-supported as the results can differ significantly if the support rotations are restricted. The main cause of this problem lies in the coupling responses, the more severe the coupling in a laminate, the more effect the coupling has on its structural behaviour. This solution can be readily

b11 ¼ A22 B266 þ A66 B226  2A26 B66 B26

C 21 ¼

C 22 ¼

b12 ¼ A22 B226 þ A66 B222

 2A26 B22 B26 b13 ¼ 2A26 B226 þ 2A26 B22 B66  2A22 B26 B66  2A66 B22 B26 b14 ¼ 2B226 B22 B66  B222 B266  B426 /11 ¼ B16 B26 B66  B12 B266

/12 ¼ B22 B266  B66 B226

/13 ¼ B22 B16 B26  B12 B226

/14 ¼ B326  B22 B66 B26

/15 ¼ B226 B16 þ 2B12 B66 B26  B22 B66 B16 /16 ¼ B66 A26  B26 A66

/17 ¼ B26 A26  B22 A66

/18 ¼ B16 A26 þ B12 A66

a21 D22 þ a22 D66 þ a23 D26 þ ðA22 A16  A26 A12 ÞðD22 D66  D226 Þ þ a24 b21 D22 þ b22 D66 þ b23 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b24  /21 D22 þ /22 D12 þ /23 D66 þ /24 D16 þ /25 D26

ðA3Þ 

þ/26 ðD22 D16  D12 D26 Þ þ /27 ðD12 D66  D16 D26 Þ þ /28 ðD22 D66  D226 Þ b21 D22 þ b22 D66 þ b23 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b24

ðA4Þ

adapted to the analysis of delaminated composite beams. where Acknowledgements

a21 ¼ A12 B26 B66  A22 B66 B16  A16 B226 þ A26 B26 B16

The authors would like to express their gratitude to Nanyang Technological University for providing the financial aid and for permission to use the laboratory facilities throughout this research work.

a22 ¼ A16 B222  A22 B12 B26 þ A12 B22 B26 þ A26 B12 B22

Appendix A. Expressions for the constants Cij in Eq. (12)

a24 ¼ B12 B326  B22 B16 B226 þ B16 B66 B222  B22 B66 B12 B26

The constants were solved simultaneously using Eqs. (7), (8), (10) and (11). The major coupling stiffnesses Bij are kept in the constants aij, bij and /ij so that they automatically become zero when a symmetrically laminated beam/ column is encountered.

/21 ¼ B16 B226 þ B12 B26 B66

/22 ¼ B326  B22 B66 B26

/23 ¼ B16 B222 þ B22 B12 B26

/24 ¼ B66 B222  B22 B226

C 11 ¼

C 12 ¼

 A26 ðB16 B22 þ B12 B26 Þ þ 2A16 B22 B26

/25 ¼ 2B16 B26 B22  B226 B12  B22 B66 B12

a11 D22 þ a12 D66 þ a13 D26 þ ðA66 A12  A26 A16 ÞðD22 D66  D226 Þ þ a14 b11 D22 þ b12 D66 þ b13 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b14  /11 D22 þ /12 D12 þ /13 D66 þ /14 D16 þ /15 D26

ðA1Þ 

þ/16 ðD22 D16  D12 D26 Þ þ /17 ðD12 D66  D16 D26 Þ þ /18 ðD22 D66  D226 Þ b11 D22 þ b12 D66 þ b13 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b14

a11 ¼ A12 B266  A66 B16 B26 þ A16 B26 B66 þ A26 B16 B66 a12 ¼

a23 ¼ A22 ðB66 B12 þ B16 B26 Þ  A12 ðB226 þ B22 B66 Þ

A12 B226

 A66 B12 B22 þ A26 B26 B12 þ A16 B22 B26

a13 ¼ 2A12 B26 B66 þ A66 ðB22 B16 þ B26 B12 Þ  A16 ðB226 þ B22 B66 Þ  A26 ðB16 B26 þ B12 B66 Þ

/26 ¼ B26 A26  B66 A22

ðA2Þ

/27 ¼ B22 A26  B26 A22

/28 ¼ B16 A22  B12 A26 b21 ¼ b11

b22 ¼ b12

b23 ¼ b13

b24 ¼ b14

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G.B. Chai, C.W. Yap / Composites Science and Technology 68 (2008) 1664–1670

C 31 ¼

a31 D66 þ a32 D26 þ a33 b31 D22 þ b32 D66 þ b33 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b34

ðA5Þ

C 32 ¼

/31 D12 þ /32 D66 þ /33 D16 þ /34 D26 þ ðA22 A66  A226 ÞðD12 D66  D26 D16 Þ þ /35 b31 D22 þ b32 D66 þ b33 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b34

ðA6Þ

a43 ¼ A22 ðB66 B12 B26  B16 B226 Þ þ A66 ðB22 B12 B26  B16 B222 Þ

where

þ A12 ðB326  B22 B66 B26 Þ  A16 ðB22 B226  B66 B222 Þ

a31 ¼ B22 ðA26 A16  A66 A12 Þ þ B26 ðA26 A12  A22 A16 Þ

þ A26 ð2B22 B26 B16  B22 B12 B66  B12 B226 Þ

þ B12 ðA22 A66  A226 Þ

/41 ¼ A22 B66 B26 þ A66 B22 B26  A26 ðB226 þ B22 B66 Þ

a32 ¼ B66 ðA22 A16  A12 A26 Þ þ B26 ðA12 A66  A26 A16 Þ  B16 ðA22 A66 

/42 ¼ A22 B66 B16  A66 B26 B12 þ A26 ðB16 B26 þ B12 B66 Þ

A226 Þ

/43 ¼ 2A26 B22 B26  A66 B222  A22 B226

a33 ¼ A22 ðB66 B16 B26  B12 B266 Þ þ A66 ðB22 B16 B26

/44 ¼ A22 B26 B16 þ A66 B22 B12  A26 ðB26 B12 þ B22 B16 Þ

 B12 B226 Þ þ A12 ðB22 B266  B66 B226 Þ

/45 ¼ B12 B326 þ B66 B16 B222  B22 B16 B226  B22 B66 B12 B26 b41 ¼ b31 b42 ¼ b32 b43 ¼ b33 b44 ¼ b34

þ A16 ðB326  B22 B66 B26 Þ þ A26 ð2B66 B12 B26  B16 B226  B22 B66 B16 Þ

References

b31 ¼ A66 B226 þ A22 B266  2A26 B26 B66 b32 ¼ A66 B222 þ A22 B226  2A26 B22 B26 b33 ¼ 2A26 ðB226 þ B22 B66 Þ  2A66 B22 B26  2A22 B26 B66 b34 ¼ 2B22 B66 B226  B426  B222 B266 /31 ¼ 2A26 B66 B26  A22 B266  A66 B226 /32 ¼ A22 B26 B16  A66 B22 B12 þ A26 ðB16 B22 þ B12 B26 Þ /33 ¼ A22 B66 B26 þ A66 B22 B26  A26 ðB22 B66 þ B226 Þ /34 ¼ A22 B66 B16 þ A66 B12 B26  A26 ðB26 B16 þ B12 B66 Þ /35 ¼ B16 B326 þ B22 B12 B266  B12 B66 B226  B22 B66 B16 B26 C 41 ¼

a41 D22 þ a42 D26 þ a43 b41 D22 þ b42 D66 þ b43 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b44

C 42 ¼

/41 D12 þ /42 D22 þ /43 D16 þ /44 D26 þ ðA22 A66  A226 ÞðD16 D22  D26 D12 Þ þ /45 b41 D22 þ b42 D66 þ b43 D26 þ ðA226  A22 A66 ÞðD22 D66  D226 Þ þ b44

ðA7Þ

ðA8Þ

where a41 ¼ B66 ðA12 A26  A22 A16 Þ þ B26 ðA26 A16  A12 A66 Þ þ B16 ðA22 A66  A226 Þ a42 ¼ B22 ðA12 A66  A16 A26 Þ þ B26 ðA22 A16  A12 A26 Þ  B12 ðA22 A66  A226 Þ

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