A tool for modal analysis of laminated bending plates.

Michnevic, Edvard

Abstract. A new finite element for modelling laminated bending plates was defined based on the effective triangular finite element of the discrete Kirchhoff's theory. The plates can be made of layers arranged in any order and consisting of different but orthotropic materials. The suggested finite element has 6 degrees of freedom in every node, i e 3 linear displacements and 3 rotations about the axis of coordinates. A mathematical model of the element describes stress and strain effects both in the plane of the element or perpendicular to it, except for shear. The suggested element can be used for calculating laminated plates or beams, not subjected to heavy shear stresses. Some numerical case studies are provided, while the results obtained are compared with the well-known analytical and numerical solutions.

Keywords: laminates, composites, composite structures, layered plates, linear analysis, non-linear analysis, finite element.

IRANKIS SLUOKSNIUOTU LENKIAMU PLOKSTELIU MODALINEI ANALIZEI E. Michnevie

Santrauka

Efektyvaus diskretines Kirchhofo teorijos trikampio baigtinio elemento DKT pagrindu suformuluotas naujas baigtinis elementas lenkiamoms daugiasluoksnems plokstelems modeliuoti. Ploksteles gali buti sudarytos is keliu bet kokia tvarka isdestytu sluoksniu, kuriu medziaga gali buti skirtinga bei ortotropine. Naujas trikampis baigtinis elementas turi 6 laisvumo laipsnius kiekviename mazge: 3 linijinius poslinkius ir 3 posukius apie koordinaciu asis. Elemento matematinis modelis apima visus deformaciju ir itempiu efektus tiek elemento plokstumoje, tiek statmena siai plokstumai kryptimi, isskyrus slyti. Elementas gali buti naudojamas sluoksniuotoms lenkiamoms plokstelems arba sijoms, kurioms slyties itaka nezymi, skaiciuoti. Darbe pateikti skaitiniai pavyzdziai, gauti rezultatai palyginti su zinomais analitiniais ir skaitiniais sprendiniais.

Reiksminiai zodziai: laminatai, kompozitai, kompozitines strukturos, sluoksniuotosios ploksteles, tiesine analize, netiesine analize, baigtinis elementas.

1. Introduction

Modern production technologies are used for manufacturing various composite materials. Composites, due to their outstanding mechanical properties, relatively low weight and a possibility to predetermine their characteristics, are widely used not only in high-tech areas, but in civil engineering as well. According to their structural and design characteristics, composite materials can be subdivided into reinforced materials shaped in various metal or non-metal moulds and laminated or layered structures obtained by combining layers of various materials. Therefore, the development of structures made of composite materials largely depends on the ability to model them.

It is hardly possible to review modelling problems associated with all available composite materials, therefore, the present paper addresses only the problems of modelling laminated bending plates. Though laminated structures are widely used, the theories of non-linear deformation and failure of modern layered composites as well as methods of mathematical modelling have not been fully developed and presented in detail yet. Because of anisotropic nature of laminated structures, all tension-compression and bending effects as well as the interaction of bending-membrane and membrane-shear effects should be considered [1-4]. Therefore, only a few assumptions simplifying the stress-strain state for the problems associated with the analysis of thin-wall bending plates can be applied.

A wide variety of finite elements are available for the analysis of structures made of commonly used materials. However, the problems associated with laminated twin-wall bending plates require a special kind of layered finite elements. Precise finite elements [5-9] are often hardly realisable in application programs due to the complexity of a mathematical model. Therefore, the need for sufficiently accurate, efficient and applicable finite elements to be used in laminated bending plates, which do not require any cross-section symmetry, still remains.

In the present paper, new types of matrix expressions allowing for evaluation of all above-mentioned membrane and bending effects are offered for the laminated anisotropic triangular finite element DKT_CST [10]. This finite element can be used for modelling anisotropic laminated bending plates consisting of layers of orthotropic material arranged in any order, as well as the particular zones of similar plates or beams, when shear deformations are insignificant.

2. Mathematical modelling of problems

A finite element may be considered to be completely defined, when it is applied to solve linear and non-linear static problems as well as modal analysis [11] and its accuracy is determined, because all characteristic structural matrices of the element should be generated for these problems. Below mathematical models of the considered problems are provided.

The following equation applies to a linear system of finite elements:

[[K.son.0]][delta] = F, (1)

where [[K.sub.0]] is a linear stiffness matrix; [delta]--a displacement vector; F--a vector of loads applied to the system. Due to membrane stresses, actual plate displacements are much smaller than those determined by the theory of linearity. A discrete problem model may be expressed by a system of non-linear algebraic equations:

[K(delta])][delta] - F = 0, (2)

where the stiffness matrix depends on displacements. The iterative Newton-Raphson method [12, 13] was used for solving a system of non-linear equations. According to this approach, a rough solution [[delta].sub.n], where a connection error of external and internal forces [[psi[.sub.n] [not equal to] 0, was refined by solving a system of linear equations at every iteration:

[DELTA][[delta].sub.n+1] = -[[[K.sub.T]].sup.-1.sub.n][[psi.sub.n], (3)

[[K.sub.T]]=[[K.sub.0]]+[[K.sub.[sigma]]]+[[K.sub.L]], (4)

where [[K.syub.T]] is a tangent stiffness matrix; [[K.sub.[sigma]]]--a matrix of initial stresses; [[K.sub.L]]--a matrix of large displacements. A connection error n o is calculated by the stresses [[sigma].sub.n]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [[bar.B]] is a non-linear matrix relating deformations to displacements. A particular case of a geometrically nonlinear problem is the problem of initial stability [11], when matrix [[K.sub.L]] = 0.

The dynamic problem is expressed by the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [C] and [M] are damping and mass matrices. A particular dynamic problem is an eigenvalue problem [11, 14], when matrix [C] = 0 and F = 0. This problem is expressed by the equation (6) of the form:

[[K.sub.0]][delta] = [lambda][M][delta], (7)

where [lambda] denotes natural frequencies.

3. Definition of the finite element

The stress-strain relation for laminated plates [15, 16] is expressed in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where M N and are membrane and bending stresses; [e.sup.0] denotes midsurface membrane strains; k--curve vector; [[D.sup.pl]],[[D.sup.plb]],[[D.sup.b]]--accumulative constitutive matrices obtained by combining constitutive matrices of layers [17-19]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where k is the layer's number; [[D.sub.k]]--the k-th layer constitutive matrix obtained by transforming the k-th layer elasticity characteristics into a global system of the coordinates; [z.sub.k] and [z.sub.k-1]--the coordinates of the k-th layer.

The finite element is defined as a combination of bending (DKT) and membrane (CST) finite elements [10]. The DKT element [20] has 3 nodes and 3 degrees of freedom, ie a bending flexure and 2 rotations per node, the interpolation functions of the element should meet only C[degrees] continuity requirements, because only the first derivatives of the main variables --slopes to the middle surface--appear in the energy functional. The CST element has three nodes, the 1-st order interpolation function and two degrees of freedom--displacements per node.

Structural element matrices are generated based on the matrices of the membrane (pl) and bending (b) elements by combining them as required by the arrangement of the element degrees of freedom to produce a global element matrix.

The vector of degrees of freedom of any node of the element is expressed as:

[[delta].sub.i] = [{[u.sub.i], [v.sub.i], [[ohm].sub.i], [[theta].sub.xi], [[theta].sub.yi], [[theta].sub.zi}.sup.T], (10)

Where [u.sub.i], [v.sub.i], [[omega].sub.i] are node displacements; [[theta].sub.xi], [[theta].sub.yi], [[theta].sub.zi]--node rotations about the axis of the coordinates.

The stiffness matrix of the element is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [[K.sup.pl.sub.0]], [[K.sup.b.sub.0]] and [[K.sup.plb.sub.0]] are membrane, bending and coupling stiffness matrices.

[[K.sup.pl.sub.0]]= A[[[B.sup.pl.sub.0]].sup.T][[D.sup.pl]][[B.sup.pl.sub.0]], (12)

where A is the area of the element; [[B.sup.pl.sub.0]]--a linear membrane geometric matrix [10]. The elements of matrix [[K.sup.pl.sub.0]] are calculated in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [d.sup.pl.sub.ij] denotes the elements of the accumulative membrane constitutive matrix [[D.sup.pl]]; [b.sub.i] and [c.sub.i] are geometric coefficients. A global matrix is obtained from 9 sections of this type.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where [[B.sup.b.sub.0]]] is a linear bending geometric matrix [10].

After the rearrangement [18], any element of matrix [[K.sup.b.sub.0]] is expressed in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i = 1, 2, ..., 9, j = 1, 2, ..., 9, (15)

where [d.sup.b.sub.ij] denotes the elements of accumulative bending constitutive matrix [[D.sup.b]]; [N.sub.i] denotes the interpolation functions; [X.sub.i,ii] and [Y.sub.ii,i] , are coefficients of the interpolation functions; [b.sub.iii] and [c.sub.iii] stand for geometric coefficients. The values of the integrals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are calculated by the software package Mathematica and stored in the data files. To retrieve them, indices iiz = i-3/3 3 + ii and jjz = j-1/3 3 + jj, calculated according to integer calculation, are used. Then, the elements of the stiffness matrices are numerically synthesised. This method of matrix generation is more advantageous than numerical integration because it is time-saving and allows us to avoid the errors associated with numerical integration.

Flexural strains cause plane deformations, and vice versa (8). This effect is determined by a coupling stiffness matrix [[K.sup.plb.sub.0]]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

When the above rearrangement is made, the elements of matrix [[K.sup.plb.sub.0]] can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where [d.sup.plb.sub.ij] denotes the elements of the accumulative coupling constitutive matrix [[D.sup.plb]].

The matrix of the initial stresses of the element is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where matrix [G] depends only on the coordinates [14]; [T] is a matrix of the membrane stresses.

When the rearrangement is made [18], a single element of matrix [[K.sup.b.sub.sigma]] can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

i = 1, 2, ..., 9, j = 1, 2, ..., 9,

where [T.sub.x], [T.sub.y], [T.sub.xy] are the membrane stresses.

The stiffness matrix of large displacements of the element is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

Where [[K.sup.b.sub.L]] and [[K.sup.plb.sub.L]] are non-linear matrices of bending and coupling stiffness.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

After the rearrangement [18], the expressions for individual elements of matrices [[K.sup.b.sub.LB] and [[K.sup.b.sub.LC] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

i = 1, 2, ..., 9, j = 1, 2, ..., 9.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

i = 1, 2, ..., 9, j = 1, 2, ..., 9.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [[B.sup.b.sub.L] is a non-linear geometric bending matrix [11].

After the rearrangement [18], the following expressions are obtained for the elements of matrix [[K.sup.plb.sub.L]]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

i = 1, 3, 5, l = 1, 2, 3, j = 1, 2, ..., 9,

where [[delta].sup.b.sub.k] is the displacement or turn of the bending element node (10).

A connection error of external and internal forces (5) is calculated in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

where i is the iteration; matrices [[K.sup.plb.sub.L]], [[K.sup.b.sub.LB]] and [[K.sup.b.sub.LC]] are generated according to the calculated displacements [[delta].sup.pt.sub.i] and [[delta].sup.b.ub.i] in iteration (i-1).

Stresses are calculated in this way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Where [[sigma].sub.0] and [[sigma].sub.L] are vectors of linear and non-linear stresses.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[[sigma].sup.pl.sub.L] = 1/2[[D.sup.pl]][[B.sup.b.sub.L]][[delta].sup.b], (34)

[[sigma].sup.b.sub.L] = 1/2[[[D.sup.plb]].sup.T][[B.sup.b.sub.L]][[delta].sup.b]. (35)

4. Numerical examples

The quality of the finite element developed for solving geometrically linear and non-linear static and eigenvalue problems will be demonstrated by standard tests of bending plates. Square symmetrical and nonsymmetrical laminated plates which were fixed or hinged were tested by applying concentrated or distributed loads. The calculation results were compared as dimensionless values.

4.1. A linear problem

Test 1. A hinged square three-layer plate (Fig 1a) subjected to uniformly distributed pressure q = 001 , 0 was considered. The plate was calculated for 3 different materials making the intermediate layer. The difference between the material properties of the intermediate and upper and lower layers is 1 (further (1:1:1)), 10 (further (1:10:1)) and 50 (further (1:50:1)) times. The thickness of the layers is [h.sub.1], = 10,0, [h.sub.2] = 80,0 and [h.sub.3] = 10,0, respectively; L=1000,0. The orientation of the orthotropy axes of the layers is 0/45/0. The data on the properties of the material of the layers are given in Table 1.

[FIGURE 1 OMITTED]

One fourth of the plate was modelled by 72, 128 and 200 finite elements DKT_CST. A comparison of the calculation results obtained at the deflection point A (Fig 1) by using the DST element and analytical solutions [21] is provided in Table 2.

As shown in Table 2, the results obtained by using the element DKT_CST are monotonically approaching the accurate analytical and DST-based solutions. The accurate deflection values were obtained by subdividing a quarter of the plate into 128 elements.

Test 2. Two laminated non-symmetrical square fixed and hinged plates were considered. The plates consisted of 4 layers (Fig 1b) of overall thickness 04 , 0 = h and layers thickness [h.sub.1] = 0,011, [h.sub.2] = 0,09, [h.sub.3] = 0,1, [h.sub.4] = 0,1, respectively, and L = 1,0. The data on the properties of the material of the layers are presented in Table 3. The orientation of the orthotropy axes of the layers is 0/90/0/90. The plates were acted upon by a concentrated force P = 100,0 applied at the point A.

A fourth of the plate was modelled by 8, 18, 32, 50, 72, 128 and 200 finite elements DKT_CST. A comparison of the calculation results obtained by using the TRIPLT element at the deflection point A is provided in Fig 2. The solutions based on the use of the element TRIPLT were obtained when the calculations of the fourth part of the plate divided into 18 elements had been made.

[FIGURE 2 OMITTED]

One per cent difference between the solutions based on the use of TRIPLT and DKT_CST elements was obtained for a hinged plate by subdividing one-fourth of the plate into 128 DKT_CST elements. The results of similar accuracy were obtained for a fixed plate by subdividing one-fourth of the plate into 200 DKT_CST elements.

4.2. A geometrically non-linear problem

The calculations were made for a fixed nonsymmetrical laminated square plate (Fig 1b). The plate consisted of 3 layers of the thickness: [h.sub.1] = 0,1, [h.sub.2] = 0,65, [h.sub.3] = 0,25, L = 10,0. The data on the properties of the material of the layers are presented in Table 4. The orientation of the orthotropy axes of the layers was 0/90/0. The deflection of the plate at point A was considered by changing the value of the concentrated load P from 0,01h to 0,7h.

One-fourth of the plate was modelled using 128 DKT_CST finite elements. The deflections of the plate observed at point A were compared with the solutions obtained for the triangular element SHELL91 by calculating plates subdivided into 128 elements, using the software package ANSYS.

The values of deflections are given in Fig 3 in terms of the overall plate thickness h.

[FIGURE 3 OMITTED]

The calculations show that the membrane strains occur when the deflection at point A is more than 0,2 h. When the linear deflection was 0,25 h, the difference between the non-linear solutions obtained by using the elements DKT_CST and SHELL91 made 0,08 %. Then, the linear deflection reached 2,5 h, while the difference between the non-linear solutions was 2 %.

4.3. The eigenvalue problem

The calculations were made for a non-fixed square three-layer plate (Fig 1). The whole plate was analysed to take into account all forms of oscillations, while the conditions of symmetry were neglected. The thickness of the plate layers was [h.sub.1] = 0,1, [h.sub.2] = 0,15, [h.sub.3] = 0,25,, respectively;

L = 4,0; moduli of elasticity were [E.sub.11] = [E.sub.22] = 2,0 x [10.sup.6]; Poisson's ratio was [v.sub.12] = [v.sub.21] = 0,3; shear moduli were [G.sub.12] = 0,77 x [10.sup.6] and density [rho] = 1000 The orientation of the layers orthotropy axes was 0/0/0.

The plate was modelled using 8, 32, 72, 128 and 200 DKT_CST elements. A comparison of the first three calculation results relating to non-zero eigen frequencies with analytical solutions is presented in Fig 4. The first six zero forms of the plate match the movements of a solid body. Non-zero forms are shown in Fig 5.

[FIGURES 4-5 OMITTED]

The convergence curves of the results presented in Fig 4 demonstrate that the solutions obtained using DKT_CST element quickly converge towards an accurate solution. The accurate value of the first eigen frequency was obtained by subdividing the plate into 128 elements.

5. Conclusions

All finite element structural matrices were generated using an effective analytical-numerical method of matrix development. This helped to avoid difficulties and errors involved in currently used numerical integration, to increase the accuracy of calculations and to obtain analytical expressions which can be easily implemented in software programmes for generating the element stiffness, initial stress and large displacement matrices and vectors of stresses.

The suggested finite element was numerically tested by comparing the results obtained in solving geometrically linear and non-linear static as well as eigenvalue problems with the well-known analytical solutions or the data obtained by applying the finite element method. The analysis of the numerical tests shows that the element has good convergence characteristics as well as being sufficiently accurate and saving the time of calculation.

The suggested finite element can be used for modelling laminated anisotropic bending plates, their separate zones or beams when shear strains are insignificant.

The program developed for modelling laminated bending plates can be used in design offices and at industrial enterprises.

Received 20 June 2006; accepted 8 Sept 2006

References

[1.] Damkilde, L.; Gronne, M. An improved triangular element with drilling rotations. In: Proc of the 15th Nordic Seminar on Computational Mechanics, Aalborg, Denmark, 2002, p. 135-138.

[2.] Duan, H. Y.; Liang, G. P. A locking-free Reissner-Mindlin quadrilateral element. Mathematics of Computation, 73, 2004, p. 1655-1671.

[3.] Nikolic, Z.; Mihanovic, A. Thin plate quadrilateral element with independent rotational DOF. International Journal for Engineering Modelling, 16(3), 2003, p. 256-269.

[4.] Piltner, R.; Joseph, D. S. A mixed finite element for plate bending with eight enhanced strain modes. Communications in Numerical Methods in Engineering, 17(7), 2001, p. 443-454.

[5.] Sheikh, A. H.; Haldar, S.; Sengupta, D. A high precision shear deformable element for the analysis of laminated composite plates of different shapes. Composite Structures, 55(3), 2002, p. 329-336.

[6.] Chakrabarti, A. A new plate bending element based on higher order shear deformation theory for the analysis of composite plates. Finite Element in Analysis and Design, 39(9), 2003, p. 883-903.

[7.] Auricchio, F.; Lovadina, C.; Sacco, E. Analysis of mixed finite elements for laminated composite plates. Computer Methods in Applied Mechanics and Engineering, 190(35-36), 2001, p. 4767-4783.

[8.] Alfano, G.; Auricchio, F.; Rosati, L. MITC finite elements for laminated composite plates. International Journal for Numerical Methods in Engineering, 50(3), 2001, p. 707-738.

[9.] Lakshminarayana, H. V.; Sridhara Murthy, S. A. Shearflexible triangular finite element model for laminated composite plates. International Journal for Numerical Methods in Engineering, 20(4), 1984, p. 591-623.

[10.] Belevicius, R.; Michnevic, E. A laminated DKT element: modal analysis. Aviation (Aviacija), 4, 2000, p. 94-99 (in Lithuanian).

[11.] Zienkiewicz, O. C.; Taylor, R. L. The finite element method. New York: McGraw-Hill, Vol 2, 1991. 790 p.

[12.] Crisfield, M. A. Non-linear finite element analysis of solids and structures. New York: John Wiley & Sons, Inc, 1991. 345 p.

[13.] Hinton, E. Introduction to nonlinear finite element analysis. Glasgow: Bell and Bain Ltd., 1992. 380 p.

[14.] Bathe, K. J.; Wilson, E. L. Numerical method in finite element analysis. New Jersey: Prentice-Hall Inc, 1976. 524 p.

[15.] Barbero, E. J. Introduction to composite materials design. London: Taylor & Francis, Inc, 1998. 352 p.

[16.] Reddy, J. N. Mechanics of laminated composite plates:: theory and analysis. 2nd ed. London: CRC Press, LLC, 2003. 856 p.

[17.] Jones, R. M. Mechanics of composite materials. 2nd ed. London: Taylor & Francis, Inc, 1998. 519 p.

[18.] Belevicius, R. Computer algebra in finite element method. Vilnius: Technika, 1994. 154 p.

[19.] Belevicius, R.; Rusakevicius, D. The optimization of the composite bending plates. Journal of Civil Engineering and Management, 8(3), 2002, p. 143-152.

[20.] Batoz, J. L.; Bathe, K. J.; Ho, L. W. A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15(12), 1980, p. 1771-1812.

[21.] Birger, I. A.; Panovko, J. G. Strength, stability, vibrations [TEXT NOT REPRODUCIBLE IN ASCII]. Moscow:

Mashinostrojenije, Vol 3, 1968. 567 p. (in Russian).

Edvard MICHNEVIE. Assoc Prof of Dept of Engineering Mechanics, Vilnius Gediminas Technical University, Lithuania. PhD (2001) at VGTU. Research interests: finite element methods, modelling laminated structures.

Edvard Michnevic

Dept of Engineering Mechanics, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania. E-mail: [email protected]

Table 1. A symmetrical composite: properties of the materials of the
layers

 Shear Poisson's
 Modulus of elasticity modulus ratio

 [v.sub.12],
Layer No [E.sub.11] [E.sub.22] [G.sub.12] [v.sub.21]

1, 3 3,4156 1,7931 1,0 0,44
a) laminated orthotropic material (1:1:1)
2 3,4156 1,7931 1,0 0,44
b) properties of the material of the layer (1:10:1)
2 0,34156 0,17931 0,1 0,44
c) properties of the material of the layer (1:50:1)
2 0,06831 0,03586 0,02 0,44

Table 2. Central deflection of symmetric layered plate

 Solutions [omega]
 (according to the number of elements)

 DST DKT_CST
Test 72 Analytical 72 128 200

1:1:1 166,94 168,38 166,93 167,56 167,78
1:10:01 30,96 31,24 30,955 31,071 31,088
1:50:1 6,77 6,76 6,702 6,723 6,729

Table 3. A non-symmetrical composite: properties of the materials of
the layers

 Modulus of elasticity
Layer
No [E.sub.11] [E.sub.22]

1 3,0 x [10.sup.8] 3,0 x [10.sup.8]
2, 3, 4 3,0 x [10.sup.7] 3,0 x [10.sup.7]

 Poisson's
 Shear ratio
 modulus
Layer [v.sub.12],
No [G.sub.12] [v.sub.21]

1 1,2 x [10.sup.8] 0,25
2, 3, 4 1,2 x [10.sup.7] 0,25

Table 4. Properties of the materials of the layers

 Shear Poisson's
 Modulus of elasticity modulus ratio

Layer [v.sub.12],
No [E.sub.11] [E.sub.22] [G.sub.12] [v.sub.21]

1, 3 3,4156 1,7931 1,0 0,44
2 1,7931 3,4156 1,0 0,44