Acta Mech. Sin. (2011) 27(5):720729DOI 10.1007/s10409-011-0455-7

RESEARCH PAPER

An advanced higher-order theory for laminated composite plateswith general lamination anglesZhen Wu Hong Zhu Wan-Ji Chen

Received: 4 May 2010 / Revised: 2 December 2010 / Accepted: 2 December 2010The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011

Abstract This paper proposes a higher-order shear deformation theory to predict the bending response of the laminated composite and sandwich plates with general lamination configurations. The proposed theory a priori satisfiesthe continuity conditions of transverse shear stresses at interfaces. Moreover, the number of unknown variables is independent of the number of layers. The first derivatives oftransverse displacements have been taken out from the inplane displacement fields, so that the C0 shape functions areonly required during its finite element implementation. Dueto C0 continuity requirements, the proposed model can beconveniently extended for implementation in commercial finite element codes. To verify the proposed theory, the fournode C0 quadrilateral element is employed for the interpolation of all the displacement parameters defined at each nodalpoint on the composite plate. Numerical results show thatfollowing the proposed theory, simple C0 finite elementscould accurately predict the interlaminar stresses of laminated composite and sandwich plates directly from a constitutive equation, which has caused diculty for the otherglobal higher order theories.Keywords Laminated and sandwich composites Higherorder theory C0 continuity requirement Transverse shearstressThe project was supported by the National Natural Science Foundation of China (10802052, 11072156), the Program for LiaoningExcellent Talents in University (LR201033), and the Program forScience and Technology of Shenyang (F10-205-1-16).Z. Wu () H. Zhu W.-J. ChenKey Laboratory of Liaoning Province forComposite Structural Analysis of Aerocraft and Simulation,Shenyang Aerospace University, 110136 Shenyang, Chinae-mail: wuzhenhk@yahoo.com.cn

1 IntroductionDue to their specific strength and stiness, laminatedand sandwich composites have been increasingly used inaerospace, naval, automobile and other industries. However,the material mismatch at interfaces as well as the bendingstretching coupling leads to dicult analysis of such laminated structures. In view of this situation, it is desirable topresent advanced laminated plate theories that can predictaccurate global and local responses of laminated compositestructures.Compared to the conventional material, the shear modulus of laminated composites is so low that the transverseshear deformation should be taken into account at formulation level. To consider the eects of transverse shear deformation, a number of higher-order displacement models [15] have been developed for analysis of laminated composite plates. However, the displacement and its slope in thehigher-order models [15] are continuous across the thickness, so that continuity of transverse shear stresses at theinterfaces is unable to be satisfied. To overcome the drawbacks of these models, Di Sciuva [6] proposed a zig-zag theory which is able to a priori satisfy the continuity conditionsof transverse shear stresses at interfaces. Furthermore, thezig-zag theory has been extended to predict the mechanical,thermal and electric behaviors of the smart composite platesand shells [7,8]. A merit of the zig-zag theory is that continuity conditions of transverse shear stresses at interfacescan be a priori satisfied. Nevertheless, the transverse shearstresses can be reasonably predicted from the zig-zag theory only by integrating the three-dimensional equilibriumequations for thin and moderately thick laminated composite plates. Moreover, the transverse shear stresses obtainedfrom the zig-zag theory become gradually inaccurate withdecreasing the length-to-thickness ratio.The global-local higher order theory proposed by Liand Liu [9] can predict accurate transverse shear stresses di-

An advanced higher-order theory for laminated composite plates with general lamination angles

rectly from constitutive equations without any postprocessing. Moreover, the number of unknown variables involvedin this model is independent of the number of layers. Numerical results show that the global-local higher order theoryhas a good compromise between solution accuracy and eciency. Following this pioneering work, improvements onthe global-local higher order theory [10,11] have been made.Compared to the zig-zag model, the global-local higher ordertheory is able to provide a more accurate description of thestress field without any postprocessing. However, their finiteelement counterparts require C1 interpolation functions, asthe second order derivatives of the transverse displacementare involved in the strain components. To satisfy C1 continuity conditions in the inter-element, the conforming thin plateelement is generally used however such examples availablein literature are very few and quite complex [12]. Therefore,it is desired to present a model which can predict accuratetransverse shear stresses without any postprocessing. Moreover, its finite-element counterparts only require C0 interpolation functions. To this end, this paper aims at proposing anadvanced higher order theory for laminated composite structures with general lamination angle. For the proposed model,the first derivatives of transverse displacements have beentaken out from the in-plane displacement fields, so that theC0 shape functions are only required during its finite elementimplementation.2 Advanced higher order theory for laminate compositeswith general lamination angleAn advanced higher-order theory is proposed to predictthe bending response of laminated composite and sandwichstructures with general lamination configurations. For thek-th layer in-plane displacements u and v are assumed as acombination of the third-order global displacement components and the local displacement components, whereas thetransverse displacement is assumed to be constant in thethickness direction. Thus, the initial displacement fields forthe present model can be given byuk (x, y, z) = uG (x, y, z) + ukL (x, y, z),

ukL (x, y, z) = k uk1 (x, y) + k2 uk2 (x, y) + k3 uk3 (x, y),vkL (x, y, z) = k vk1 (x, y) + k2 vk2 (x, y) + k3 vk3 (x, y),

721

(3)

2zk+1 + zk, bk =. The rezk+1 zkzk+1 zklationship between global coordinates and local coordinatescan be seen in Fig. 1.where k = ak z bk , ak =

Fig. 1 Schematic diagram for the laminated plate segment

Displacement continuity conditionsIn Eqs. (1)(3), it can be found that the number ofunknowns in the initial displacement fields depends on thenumber of layers. To eliminate the layer-dependent displacement variables, the displacement continuity conditions proposed by Li and Liu [9] will be used. Using the continuityconditions of displacements at the interfaces, the followingequations can be given byk1uk2 = uk1 + uk11 + u2 ,k1vk2 = vk1 + vk11 + v2 ,

uk3 = uk13 ,

(4)

vk3 = vk13 ,where k = 2, 3, , n; n is the total number of layers.

(1)

Continuity and boundary conditions of transverse shearstresses

in which, uG and vG are of the global displacement components; ukL and vkL are local displacement components. Theexpressions of the global displacement components can begiven by

After using the continuity conditions of displacements,4(n 1) unknowns have been constrained. For an angle-plycomposite plate, the transverse shear stresses for the k-th plycan be expressed as

vk (x, y, z) = vG (x, y, z) + vkL (x, y, z),wk (x, y, z) = w0 (x, y),

uG (x, y, z) = u0 (x, y) +

3

i=1

vG (x, y, z) = v0 (x, y) +

3

kxz (z) = Q44k kxz (z) + Q45k kyz (z),

zi ui (x, y),(2)zi vi (x, y).

i=1

The local displacement components can be given by

kyz (z) = Q45k kxz (z) + Q55k kyz (z),

(5)

where Qi jk are transformed material constants with respectto the global coordinates for the k-th layer. The transverseshear strains are given by

722

Z. Wu, et al.

kxz (z) =

w0+ u1 + 2zu2 + 3z2 u3 + ak uk1x+2ak k uk2 + 3ak k2 uk3 ,

kk+N11v3 + N12

w0,y

Employing the free conditions of transverse shearstresses at the lower surface, the local displacement parameters u13 and v13 are given by

1 w0+ u1 + 2z1 u2 + 3z21 u3 + a1 u11 2a1 u12 ,u13 = 3a1 x (7)1 w01211v3 = + v1 + 2z1 v2 + 3z1 v3 + a1 v1 2a1 v2 .3a1 y

where coecients Fik , Gki , Hik , Lki , Mik and Nik (i =1, 2, , 12) are given in Appendix.In Eq. (8), it is found that the first derivatives oftransverse displacement, namely w0 /x and w0 /y, appear in the in-plane displacement fields. As the first derivatives of transverse displacement are involved in the in-planedisplacement fields, C1 interpolation functions ought to beused in the finite element implementation. To avoid usingC1 interpolation functions, the derivatives of transverse displacement have to be eliminated by employing the transverseshear free conditions at the upper surface.Using the transverse shear free conditions at the uppersurface, w0 /x and w0 /y can be expressed as

By imposing the continuity conditions of transverseshear stresses at interfaces, 2(n 1) unknowns can be eliminated. The local displacement variables for the k-th ply canbe written as

w0= A1 u11 + B1 u12 + C1 u1 + D1 u2 + E1 u3x+F1 v11 + G1 v12 + H1 v1 + I1 v2 + J1 v3 ,

uk1 = F1k u11 + F2k u12 + F3k u1 + F4k u2 + F5k u3

w0= A2 u11 + B2 u12 + C2 u1 + D2 u2 + E2 u3y

w0+ v1 + 2zv2 + 3z2 v3 + ak vk1y

kyz (z) =

(6)

+2ak k vk2 + 3ak k2 vk3 .

w0k+ F7k v11 + F8k v12 + F9k v1 + F10v2xkk w0+F11v3 + F12,y

+F2 v11 + G2 v12 + H2 v1 + I2 v2 + J2 v3 ,

+F6k

in which coecients in Eq. (9) can be found in Appendix.After substituting Eqs. (8) and (9) into Eq. (1), the finaldisplacement fields can be given by

uk2 = Gk1 u11 + Gk2 u12 + Gk3 u1 + Gk4 u2 + Gk5 u3

uk = u0 + k1 (z)u11 + k2 (z)u12 + k3 (z)u1 + k4 (z)u2

w0+ Gk7 v11 + Gk8 v12 + Gk9 v1 + Gk10 v2+Gk6xw0,+Gk11 v3 + Gk12yuk3

=

H1k u11

+

H2k u12

+

H3k u1

+

H4k u2

+

+k5 (z)u3 + k6 (z)v11 + k7 (z)v12 + k8 (z)v1+k9 (z)v2 + k10 (z)v3 ,vk = v0 + 1k (z)u11 + 2k (z)u12 + 3k (z)u1 + 4k (z)u2

H5k u3

w0k+ H7k v11 + H8k v12 + H9k v1 + H10v2xkk w0,+H11v3 + H12y

+5k (z)u3

w0+ Lk7 v11 + Lk8 v12 + Lk9 v1 + Lk10 v2xw0,+Lk11 v3 + Lk12y+Lk6

vk2

=

M1k u11

+

M2k u12

+

M3k u1

+

M4k u2

+

M5k u3

w0k+ M7k v11 + M8k v12 + M9k v1 + M10v2xkk w0,+M11v3 + M12y+M6k

+

N7k v11

+

N8k v12

+

N9k v1

+

kN10v2

+

7k (z)v12

+

(10)

8k (z)v1

wk = w0 ,(8)

in which ki and ik are the functions of material constantsand thickness of laminates. Expression of ki and ik is givenin Appendix.3 Finite element formulationIn the present work, the four-node isoparametric element isemployed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate.Thus, the displacement variables in the present model arediscretized as followsu0 =

4

Ni u0i ,

u11 =

i=1

vk3 = N1k u11 + N2k u12 + N3k u1 + N4k u2 + N5k u3w0+N6kx

+

6k (z)v11

k+9k (z)v2 + 10(z)v3 ,

+H6k

vk1 = Lk1 u11 + Lk2 u12 + Lk3 u1 + Lk4 u2 + Lk5 u3

(9)

uj =

4

i=1

4

Ni u11i ,

u12 =

i=1

Ni u ji ,

v0 =

4

i=1

4

Ni u12i ,

i=1

Ni v0i ,

v11 =

4

i=1

Ni v11i ,

(11)

An advanced higher-order theory for laminated composite plates with general lamination angles

v12 =

4

Ni v12i ,

vj =

i=1

4

Ni v ji ,

w0 =

i=1

4

ei = [u0i v0i w0i u11i u12i u1i u2i u3i v11i v12i v1i v2i v3i ],(i = 1, 2, 3, 4).

Ni w0i ,

i=1

=

1where j = 1, 2, 3, Ni = (1 + i )(1 + i ), (i = 1, 2, 3, 4).4After using linear strain-displacement relationships, thestrain for the k-th layer can be given by

k = u k = B 1 B 2 B 3 B 4 e,(12)where e = [ e1B i =

e2

e3

e4 ]T ,

x

0

0

y

yx

0

0

0

0

0

Niy

NiyNix

0

0

0

0

0

0

0

Nix

Niy

k1

Nix

1k

Niy

k1

NiNi+ 1kyx

k1Niz

1kNiz

k2

Nix

2k

Niy

k2

NiNi+ 2kyx

k2Niz

2kNiz

k3

Nix

3k

Niy

k3

NiNi+ 3kyx

k3Niz

3kNiz

k4

Nix

4k

Niy

k4

NiNi+ 4kyx

k4Niz

4kNiz

NiNik5+ 5kyx

k5Niz

5kNiz

Ni5ky

k6

Nix

6k

Niy

k6

NiNi+ 6kyx

k6Niz

6kNiz

k7

Nix

7k

Niy

k7

NiNi+ 7kyx

k7Niz

7kNiz

k8

Nix

8k

Niy

k8

NiNi+ 8kyx

k8Niz

8kNiz

k9

Nix

9k

Niy

k9

NiNi+ 9kyx

k9Niz

9kNiz

Nix

k10

Niy

k10

Nik Ni+ 10yx

k10Niz

k10Niz

k10

After the formation of strain matrix B of the quadrilateral element is established, the element stiness matrix K ecan be given byn i

B T Q i B dxdy dz.(14)Ke=i=1

z0x

T0 .z y

The strain matrix can be written as follows

Nix

Nik5x

723

i1

In terms of following equation, the vector of nodal dis-

T .

placement e can be obtained by

K e e = P ,

(13)

(15)

in which P is a loading vector.After obtaining the vector of nodal displacement, thenodal stresses can be obtained directly from the constitutiveequations without any postprocessing approaches.

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Z. Wu, et al.

4 Numerical examplesIn this section, the performance of the proposed higher ordertheory is assessed by the following numerical examples. Theentire plate is used to analyze the static response of angle-plylaminated composite and sandwich plates. Mesh configuration can be found in Fig. 2, and the material constants aregiven as follows.

Fig. 2 Mesh configuration of 44

Material 1: laminated plates [13]E1 = 172.4 GPa,

E2 = E3 = 6.89 GPa,

G12 = G13 = 3.45 GPa,

G23 = 1.38 GPa,

( x , xy ) = ( x , xy )h2 /q0 a2 ,

( xz , yz ) = ( xz , yz )h/q0 a,

u = 100ET uh2 /q0 a3 ,

( x , xz ) = ( x , xz )/q0 ,

where a is the length of laminated plate strip; h is the thickness of laminates.Firstly, a laminated composite plate with lay-up [30 /30 /30 ] and material 1 is to be studied. The distributions of stresses through the thickness of a laminated plateare plotted in Figs. 36. It is found that the results obtained from the present model agree well with the threedimensional elasticity solution [14], whereas the results obtained from the first order shear deformation theory (FSDT)are less accurate, especially for the transverse shear stresses.Moreover, numerical results show that the proposed modelexactly satisfies the continuity conditions of transverse shearstresses at interfaces and the transverse shear free conditionson the lower and the upper surfaces. Subsequently, the laminated composite plate with lay-up [0.5 /90.5 ]S is taken intoaccount. In Figs. 7 and 8, the present results are comparedwith the results obtained from the enhanced first-order theory (EFSDTM) based on the mixed variational theorem [13].Numerical results show that the present results are in goodagreement with the three-dimensional elasticity solutions.However, the EFSDTM and the FSDT are less accurate incomparison with the exact solutions.

To further assess the range of applicability ofthe present theory, a sandwich plate with lay-up[0.05 /Core(0.05 )/0.05 ] and material 2 is also taken intoaccount. The distributions of in-plane displacement andtransverse shear stresses are shown in Figs. 9 and 10, respectively. Numerical results show that the present theory isable to model the severe zig-zag and kinky variation alongthe thickness in both u and xz . Moreover, the present resultsagree well with the exact solution.

v12 = v13 = v23 = 0.25.Material 2: sandwich plate [13]Face sheetsE1 = 172.4 GPa,

E2 = E3 = 6.89 GPa,

G12 = G13 = 3.45 GPa,

G23 = 1.38 GPa,

v12 = v13 = v23 = 0.25.Core materialE Xc = EYc = EZc = 0.1 GPa,GcXY = GcXZ = GcYZ = 0.04 GPa,vcXY = vcXZ = vcZX = 0.25.Example 1 Cylindrical bending of laminated composite andsandwich plate strip.Stresses for laminated plates are normalized as follows

Fig. 3 In-plane stresses through thickness of [30 / 30 /30 ] plate(a/h = 4)

An advanced higher-order theory for laminated composite plates with general lamination angles

725

Fig. 4 In-plane stresses through thickness of [30 / 30 /30 ] plate(a/h = 4)

Fig. 7 In-plane stresses through thickness of [0.5 /90.5 ]S plate(a/h = 4)

Fig. 5 Transverse shear stresses through thickness of [30 /30 /30 ] plate (a/h = 4)

Fig. 8 Transverse shear stresses through thickness of [0.5 /90.5 ]Splate (a/h = 4)

Fig. 6 Transverse shear stresses through thickness of [30 /30 /30 ] plate (a/h = 4)

Fig. 9 In-plane displacement through thickness of [0.05 /Core(0.05 )/0.05 ] sandwich plate (a/h = 10)

726

Z. Wu, et al.

verse shear stresses of the angle-ply laminated compositeand sandwich plates directly from the constitutive equations,is lacking in the published literature. Thus, the present workfills this gap.

Fig. 10 Transverse shear stress through thickness of [0.05 /Core(0.05 )/0.05 ] sandwich plate (a/h = 10)

Example 2 Square laminated composite plate simply supported on all edges and subjected to a sinusoidal transverseloading q = q0 sin(x/a) sin(y/b), is analyzed.Stresses for laminated plates are normalized as follows

Fig. 11 Transverse shear stress through thickness of [45 /45 /45 /45 ] plate (a/h = 4)

( xz , yz ) = ( xz , yz )h/q0 a.To further evaluate the range of applicability of thepresent theory, a laminated composite plate simply supportedon all edges with lay-up [45 /45 / 45 /45 ] and material 1 is to be taken into account. The distributions oftransverse shear stresses obtained from the present theory arecompared with the three-dimensional elasticity solutions aswell as other published results in Figs. 11 and 12. It can befound that the present results obtained directly from the constitutive equations agree well with the exact solutions. Theresults computed by Cho and Choi [15], however, are lessaccurate in comparison with the exact solutions.5 ConclusionsThis paper proposes an advanced higher-order theory a priori satisfying the continuity conditions of transverse shearstresses at the interfaces of the general laminated composite plates. The merit of the proposed model is that its finiteelement counterparts only require C0 interpolation functions,as the first derivatives of the transverse displacement areonly involved in the strain expression. Thus, the four-nodeisoparametric element is employed for the interpolation ofall the displacement parameters defined at each nodal pointon the composite plate. Numerical results showed that thepresent C0 finite element model is quite capable of analyzing angle-ply laminated composite and sandwich plates under dierent support and loading conditions. In fact, theC0 finite element model, which can accurately predict trans-

Fig. 12 Transverse shear stress through thickness of [45 /45 /45 /45 ] plate (a/h = 4)

AppendixBy using the continuity conditions of transverse shear stresses atinterfaces, the coecients Fik , Gki , Hik , Lki , Mik and Nik can be determined. For k = 1, the coecients can be written asF11 = 1,

1F21 = F31 = F41 = = F12= 0,

G12 = 1,

G11 = G13 = G14 = = G112 = 0,

L17 = 1,

L11 = L12 = = L16 = L18 = = L112 = 0,

M81 = 1,

1M11 = M21 = = M71 = M91 = = M12= 0,

An advanced higher-order theory for laminated composite plates with general lamination angles1H11 = ,3H41 =

2z1,3a1

H21 =

2,3

H51 =

z21,a1

1= 0,H71 = H81 = = H12

1N71 = ,31,N91 = 3a11N11=

z21,a1

H31 =

1,3a1

H61 =

1,3a1

N11 = N21 = = N61 = 0,N81 =

2,3

1N10=

2z1,3a1

1N12=

1.3a1

The coecients for k > 1 can be calculated from the followingrecursive equations 3(1 + k )Hik1Fik = (2 + k )Fik1 2(1 + k )Gk1i+k (Lk1+ 2Mik1 + 3Nik1 ) + S i ,iLki = (2 + k )Lk1 2(1 + k )Mik1 3(1 + k )Nik1i+k (Fik1 + 2Gk1+ 3Hik1 ) + Si ,iGki = Fik + Fik1 + Gk1i ,

Hik = Hik1 ,

+ Mik1 ,Mik = Lki + Lk1i

Nik = Nik1 ,

whereS 3 = S 6 = k ,S5 =

3z2k k ,

S 4 = 2zk k ,S 9 = S 12 = k ,

S 10 = 2zk k ,

S 11 = 3z2k k ,

S 1 = S 2 = S 7 = S 8 = 0,

S3 = S6 = k ,

S4 = 2zk k ,

S5 = 3z2k k ,

S9 = S12 = k ,

S10 = 2zk k ,

S11 = 3z2k k ,

S1 = S2 = S7 = S8 = 0,

i = 1, 2, , 12,k = 2, 3, , n.

Q44k1 Q55k Q45k1 Q45k ak1,k =2akQ44k Q55k Q45k

Q45k1 Q45k Q44k1 Q55k 1,k = 1 +akQ44k Q55k Q245k

Q55k1 Q45k Q45k1 Q55k ak1,k =akQ44k Q55k Q245k

Q55k1 Q45k Q45k1 Q55k 1k =,akQ44k Q55k Q245k

Q55k1 Q44k Q45k1 Q45k ak1,k =akQ44k Q55k Q245k

Q45k1 Q45k Q55k1 Q44k 1k = 1 +,akQ44k Q55k Q245k

Q44k1 Q45k Q45k1 Q44k ak1,k =akQ44k Q55k Q245k

Q44k1 Q45k Q45k1 Q44k 1.k =akQ44k Q55k Q245kBy applying the free conditions of transverse shear stresses atthe upper surface, the following coecients can be obtained

727

2 (1)1 (12) 1 (1)(2 (12) + 1),2 (2)1 (12) 1 (2)(2 (12) + 1),B1 =2 (3)1 (12) (1 (3) + 1)(2 (12) + 1)C1 =,2 (4)1 (12) (1 (4) + 2zn+1 )(2 (12) + 1),D1 =A1 =

2 (5)1 (12) (1 (5) + 3z2n+1 )(2 (12) + 1),2 (7)1 (12) 1 (7)(2 (12) + 1),F1 =2 (8)1 (12) 1 (8)(2 (12) + 1),G1 =(2 (9) + 1)1 (12) 1 (9)(2 (12) + 1),H1 =(2 (10) + 2zn+1 )1 (12) 1 (10)(2 (12) + 1)I1 =,E1 =

(2 (11) + 3z2n+1 )1 (12) 1 (11)(2 (12) + 1),1 (1)2 (6) 2 (1)(1 (6) + 1),A2 =1 (2)2 (6) 2 (2)(1 (6) + 1)B2 =,(1 (3) + 1)2 (6) 2 (3)(1 (6) + 1),C2 =(1 (4) + 2zn+1 )2 (6) 2 (4)(1 (6) + 1)D2 =,J1 =

(1 (5) + 3z2n+1 )2 (6) 2 (5)(1 (6) + 1),1 (7)2 (6) 2 (7)(1 (6) + 1),F2 =1 (8)2 (6) 2 (8)(1 (6) + 1)G2 =,1 (9)2 (6) (2 (9) + 1)(1 (6) + 1),H2 =1 (10)2 (6) (2 (10) + 2zn+1 )(1 (6) + 1),I2 =E2 =

1 (11)2 (6) (2 (11) + 3z2n+1 )(1 (6) + 1),1 (i) = an Fin + 2anGni + 3an Hin ,J2 =

2 (i) = an Lni + 2an Min + 3an Nin , = (an F6n + 2anGn6 + 3an H6n + 1)nn(an Ln12 + 2an M12+ 3an N12+ 1)nn(an F12+ 2anGn12 + 3an H12)

(an Ln6 + 2an M6n + 3an N6n ).lows

The coecients ki and ik can be detailedly expressed as fol-

ki = Rki k + S ik k2 + T ik k3 + Zi ,ik = Oki k + Pki k2 + Qki k3 + Zi ,

728

Z. Wu, et al.kPk3 = M3k + M6k C1 + M12C2 ,

whereZ3 = z,

Z4 = z2 ,

Z5 = z3 ,

Zi = 0,

(i 3, 4, 5),

Z8 = z,

Z9 = z2 ,

Z10 = z3 ,

Zi = 0,

(i 8, 9, 10),

Rk1S 1kT 1k

=

F1k

+

=

Gk1

+ Gk6 A1

+ Gk12 A2 ,

=

H1k

H6k A1

kH12A2 ,

+

F6k A1

+

kF12A2 ,

+

kRk2 = F2k + F6k B1 + F12B2 ,

S 2kT 2kRk3

=

Gk2

+ Gk6 B1

+ Gk12 B2 ,

=

H2k

+

H6k B1

kH12B2 ,

=

F3k

+

F6k C1

+

k+ F12C2 ,

S 3k = Gk3 + Gk6C1 + Gk12C2 ,T 3kRk4

=

H3k

=

F4k

+

H6k C1

kH12C2 ,

+

+

F6k D1

k+ F12D2 ,

S 4k = Gk4 + Gk6 D1 + Gk12 D2 ,kT 4k = H4k + H6k D1 + H12D2 ,kRk5 = F5k + F6k E1 + F12E2 ,

S 5k = Gk5 + Gk6 E1 + Gk12 E2 ,kE2 ,T 5k = H5k + H6k E1 + H12kRk6 = F7k + F6k F1 + F12F2 ,

S 6k = Gk7 + Gk6 F1 + Gk12 F2 ,kT 6k = H7k + H6k F1 + H12F2 ,kG2 ,Rk7 = F8k + F6k G1 + F12

kQk3 = N3k + N6k C1 + N12C2 ,

Ok4 = Lk4 + Lk6 D1 + Lk12 D2 ,kPk4 = M4k + M6k D1 + M12D2 ,kD2 ,Qk4 = N4k + N6k D1 + N12

Ok5 = Lk5 + Lk6 E1 + Lk12 E2 ,kPk5 = M5k + M6k E1 + M12E2 ,kE2 ,Qk5 = N5k + N6k E1 + N12

Ok6 = Lk7 + Lk6 F1 + Lk12 F2 ,kF2 ,Pk6 = M7k + M6k F1 + M12kQk6 = N7k + N6k F1 + N12F2 ,

Ok7 = Lk8 + Lk6G1 + Lk12G2 ,kG2 ,Pk7 = M8k + M6k G1 + M12kQk7 = N8k + N6k G1 + N12G2 ,

Ok8 = Lk9 + Lk6 H1 + Lk12 H2 ,kPk8 = M9k + M6k H1 + M12H2 ,kH2 ,Qk8 = N9k + N6k H1 + N12

Ok9 = Lk10 + Lk6 I1 + Lk12 I2 ,kk+ M6k I1 + M12I2 ,Pk9 = M10kk+ N6k I1 + N12I2 ,Qk9 = N10

Ok10 = Lk11 + Lk6 J1 + Lk12 J2 ,

S 7k = Gk8 + Gk6G1 + Gk12G2 ,

kk+ M6k J1 + M12J2 ,Pk10 = M11

kT 7k = H8k + H6k G1 + H12G2 ,

kkQk10 = N11+ N6k J1 + N12J2 .

kRk8 = F9k + F6k H1 + F12H2 ,

S 8k = Gk9 + Gk6 H1 + Gk12 H2 ,kT 8k = H9k + H6k H1 + H12H2 ,kkRk9 = F10+ F6k I1 + F12I2 ,

S 9k = Gk10 + Gk6 I1 + Gk12 I2 ,kk+ H6k I1 + H12I2 ,T 9k = H10kkRk10 = F11+ F6k J1 + F12J2 ,kS 10= Gk11 + Gk6 J1 + Gk12 J2 ,kkkT 10= H11+ H6k J1 + H12J2 ,

Ok1 = Lk1 + Lk6 A1 + Lk12 A2 ,kPk1 = M1k + M6k A1 + M12A2 ,kQk1 = N1k + N6k A1 + N12A2 ,

Ok2 = Lk2 + Lk6 B1 + Lk12 B2 ,kB2 ,Pk2 = M2k + M6k B1 + M12kQk2 = N2k + N6k B1 + N12B2 ,

Ok3 = Lk3 + Lk6C1 + Lk12C2 ,

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