Buckling of laminated beam higher order discrete model-main

ELSEVIER

Composite Structures Vol. 38, No. 1-4, pp. 119-131 0 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0263-8223/97/$17.00 + 0.00

PII:SO263-8223(97)00048-2

Buckling behaviour of laminated beam structures using a higher-order discrete model

M. A. Ramos Loja, J. Infante Barbosa ENIDH - Escola N&Utica lnfante D. Henrique, Departamenio de Miquinas Matitimas, Av. Bonneviile France, Pacco de

Arcos, 2780 Oeiras, Portugal

&

C. M. Mota Soares IDMEC - Institute de Engenharia MecBnica, Institute Superior Ttknico, Av. Rovtico Pais, 1096 Lbboa Codex, Portugal

A higher-order shear-deformation theory, assuming a non-linear variation for the displacement field, is used to develop a finite-element model to predict the linear buckling behaviour of anisotropic multilaminated or sandwich thick and thin beams. The model is based on a single-layer Lagrangean four-node straight-beam element. It considers stretching and bending in two orthogonal planes. The most common cross-sections and symmetric and asymmetric lay-ups are studied. The good performance of the present element is evident on the prediction of the buckling of several test cases of thin and thick isotropic or anisotropic beam structures. Comparisons show that the model is accurate and versatile. 0 1997 Elsevier Science Ltd.

INTRODUCTION

Laminated beams are presently used as struc- tural elements in general high-performance mechanical, aerospace, naval and civil applica- tions, where high strength and high stiffness to weight ratios are desired. The beams are made of composite materials which have the ability of being tailored according to specified response constrained requirements to achieve optimum structural objectives. As part of the design process, it is required to predict accurately dis- placements, normal and transverse stresses, delamination, vibrational and buckling behaviour to establish the load and perform- ance capabilities of this type of structural element. Owing to the large elongation to failure allowed by both fibre and resin, buckling is most of the time the governing failure for the most used pultruded structural members. In this paper a refined finite-element model for the linear buckling analysis of composite or sand- wich beam structures is presented. The model is developed for symmetric and asymmetric lay-

ups, and considers the most usual cross-sections used in design.

The present model is based on a higher-order displacement theory using displacement fields proposed by Lo et al. [1,2] for plates, and by Manjunatha & Kant [3], Vinayak et al. [4] and Prathap et al. [5] for rectangular cross-section beam structures in one-plane bending under static loading. The proposed theory enables the non-linear variation of displacements through the composite beam width/depth, thus eliminat- ing the use of shear correction factors. These displacement fields are suitable for the analysis of highly anisotropic beams ranging from high to low length to depth and/or width ratios.

Pioneering work on the buckling analysis of composite beams can be reviewed in Kapania & Raciti [6]. Related work has been carried out by Bhimaraddi & Chandrashekhara [7], Hwu & Hu [8], Barber0 & Tomblin [9], Barber0 & Raf- toyiannis [lo], Wisnom and Haberle [II], Ray & Kar [12], Turvey [13], Rhodes [14] and Bar- bero et al. [15], among others. Recently, Sheinman et al. [16] developed a high-order ele-

119

120 hf. A. Ramos Loja, J. Infante Barbosa. C. M, Mota Soares

ment for pre-buckling and buckling analysis of laminated rectangular cross-section beams and plane frame structures, considering a third- order expansion in the thickness direction for the in-plane displacement and a constant trans- verse displacement throughout the thickness. By deleting degrees of freedom they arrive at various alternative models. A parametric study of the locking phenomenon and the shear- deformation effects was carried out for isotropic and laminated structures. From the surveys one can find very few research publications related to the buckling of multilaminated composite/ sandwich beams using higher-order displace- ment fields and, consequently, comparison with alternative formulations such as Euler-Ber- noulli and Mindlin are also very rare, hence the motivation for the proposed work.

DISPLACEMENT AND STRAIN FIELDS

In the present study, the development of a higher-order discrete model (HSDT) for static and buckling analysis is presented. The model is based on a straight-beam finite element with four nodes and 14 degrees of freedom per node, considering bi-axial bending and stretch- ing. The development takes into consideration non-symmetric lay-ups and the rectangular, I, T, channel and rectangular box beam cross-sec- tions. The present discrete model is part of a package of finite-element programs for the opti- mization of two-dimensional composite or sandwich arbitrary beams. This package also includes Euler-Bernoulli (EBT) and Timo- shenko (FSDT) beam elements. The perform- ance of the model developed is discussed for several buckling applications.

The displacement field considered assumes, for the numerical finite-element model, expansion in the thickness and width co-ordinates for the axial displacement, and a expansion for the transverse displacements. The displacement field can be represented form as

u = ;14; u = [u(x, z, .v) v(x, y) w(x, z)lT

L

1 0 0 0 z --x ? --y2 0 0 0 0 z7 -_v’

L=O1000 0 0 0 _vz 0 _v 0 0 0

00100 0 0 0 020,70 0 I

q = [ZP v” wo 0: 0; 0; uo* lP** vO* wO* p’,’ g) ey* o!*]T

a third-order second-order in a compact

(1) where q is the vector of generalized displacements, representing the appropriated Taylor’s series terms defined along the x-axis and z = 0 and y = 0. The first six terms are related to displacements and rotations as defined in Fig. 1. The remaining parameters are higher-order terms in the Taylor series expansion. They represent higher-order transverse cross-sectional deformation modes which

Fig. 1. Typical laminated beam geometry. Co-ordinate system.

A higher-order discrete model for buckling behaviour 121

are difficult to physical interpret. Considering the kinematic relations for linear elasticity and the HSDT displacement field (eqn (l)), the strain field is obtained as

E = ho; 8 = r&x &?, E, Y,, Yx,lT

i

1 * :

-y* 0 0 z z3 0 -y -y3 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 1 0 0 Flo

0 0 0 0 0 0 0 0

Iz= 0 0 0 0

Y

0 0 0 0 0 0 0 0 000000 0 0 1 z z* 0 0

0 1 0 0 0 0 000000 0 0 0 0 0 1 y y* go = [E; &y q* O O &.v &z ky k.; k, k, k: ky 4, 6, 4:: 4xy 4z.y &;I (2)

where

k a@ = 2w”*k, = - - ax’

k,* = a@’ -; kv =2v”*; 4, =

i3W0 aE ax

- +q; 4;=2u”*-* ax ax’

4: ad*

=3$‘+-; $v= a@ a$

ax -&- -e; 4; = -2l,P**+ ax ; 4; = _3@‘+ $ (3)

CONSTITUTIVE RELATIONS

Considering the orthogonal referential xyz, the constitutive relation for an orthotropic beam layer, which can have an arbitrary fibre orientation, are related to the strains through the relations

I = ‘GM a2 a3 0 GM

312 e22 e23 ’

ii%3 323 e33

226

0 Q3c5

0 0 0 e_T5 0

.e,, 226 G36 0 a6

Neglecting the shear stress z,,=, one obtains for the elastic coefficient & the expression

e;5 = a55 - &@44 (9

where the terms of matrix 3, for the kth layer are explicitly given in Vinson & Sierakowski [1,17]. Integrating the stresses through the depth and width of the laminate one obtains the resultant forces and moments acting on it, as follows

[N] = [N, N: N:* NY NJT= j, ,T yi [cx ox 0.x or gzlT [I Z* Y* 1 11 dY dz * I Y/

[M""] = CM?, M; M,J~ = j, zt i’ [GE 0, c.zlT [z z3 zl dy dz I‘ I Y/

w-1 = [S,, s:, Cl’ = j, zI k I

-1 z,, [ 1 z z*l dy dz

[QX"l = [S, s:y sgl’ = $I J tijyu YY21dYdZ * I

122 M. A. Ramos Loja, J. lnfante Barbosa, C. M. Mota Soares

where N is the number of layers. The constitutive relation then becomes

[;;l_k: ;; E ;.kZ illi] (7)

Matrices A, B”, Di’, s’j, Dtik and Cy with (ij = XZ, .KY; ijk = xyz; n = 1, . . . . 3) are explicitly given in Appendix A for multilaminated rectangular cross-sections. The corresponding matrices for T, I, channel and rectangular box beams, can be found in Loja [18].

FINITE-ELEMENT MODEL

In the present work a four-node straight-beam element is Lagrangean shape functions are used to interpolate the ment, as follows

developed for static and buckling analysis. generalized displacements within the ele-

N,=- z(s+l) (4-l)

The generalized displacements (eqn (1)) can then be represented as

u0 = ;$, N; u;OY” = ;$, N; 0;;

(9)

(10)

One can then represent the displacement field, by

u, = INq, q; = [$ “0 h’o #‘, 0; #” u()* u()**z “o+ UP* /f$ p; oy* @y (i= 1) 2, 3, 4) ; (11)

where N is the shape function matrix and qe the element nodal displacement vector. By differentiat- ing (eqn (10)) in accordance with the generalized strain field yields

8’ = B, qe; k”’ = B,>; q‘s; k-” = B, q,; I\ c#P = Bs q,; (p-“‘ =Bs qe 1. r> W) where matrices BM, BF,;, BF,,, Bsxz and Bs,, relate the degrees of freedom to the generalized strains, for membrane, flexnre and shear.

The total potential energy for the eth element is

(13)

where the first and the second terms correspond to the first- and second-order strain elastic energy, oyj denotes the stress components associated to the initial state of stress, which are previously calculated by means of a linear static analysis, V is the volume of the element, and the comma within the subscripts denotes the partial derivatives. The second term is expanded as shown in Moita et al.

A higher-order discrete model for buckling behaviour 123

[19] for plate structures, and Q, is the element load vector. Substituting eqns (10) and (12) into eqn (13), one obtains

I-l - + cr:(K+~)s, - s:Q, e- (14)

The application of the minimum potential energy variational principle yields the following equi- librium equations

&q,+K%I, = Q, (15) where & and c are the element stiffness and geometric matrices. These matrices are respectively given as

K, = i’ (BT, AB,+B: _ BXzTBM+B& B”” B, -1 I_ +B; BXyTB,+BT, BXY BF xz x, +Bg D”” BF _ I” IL =.

+B:X>, DXY BFX;fBzX, DxYz BFX,+BzX,, DXyZT BFXz+BT, CTy BsXv+BzXv CTyT BM

+B:_ C;’ Bs_

+B:_ C?’ B,_+B:_ C?’ Bsxv+B:_ C:” B,_+B:_ S”” BsX;+B:“, S”’ Bs_) J d5 (16)

@ = i’, GTzG J d4: (17)

Matrices r and G are shown in Appendix B and Appendix C, respectively. The Jacobian operator, relating the natural co-ordinate derivatives to the local co-ordinate derivative, is J = L/2 for equally spaced nodes. Load vector Q,, when distributed loading is acting within the element, geometric element matrix e and terms relating to stretching, bending and bending-stretching of element stiffness matrix K, are evaluated analytically in the t direction using symbolic manipulator Maple V. [20] The last two sub-matrices of eqn (16), relating to transverse shear elastic strain energy, are evaluated numerically using three Gauss points. The degrees of freedom oXi (i = 1, . . . . 4) are related with angles of twist on a plane normal to the x-axis of the element. Then, assuming that they do not affect displacements other than their own, the stiffness and geometric matrices for a four-node Lagrangean bar element in free torsion are superimposed onto eqns (16) and (17), in the usual assembly way. The equilibrium equations for the whole and discretized beam for static and linear buckling analysis are then

Kq=Q (18) Kq,+ljKGq; = 0 (19)

where Q is the system load vector, K and KG are the system stiffness and geometric matrices, q is the system displacement vector and qi is the eigenvector associated with the & eigenvalue, which is a function of the applied loading. The smallest 5 corresponds to the critical buckling load parameter. Equations (18) and (19) can easily be solved once the boundary conditions are introduced.

NUMERICAL APPLICATIONS

The higher-order finite-element model (HSDT) is applied to several illustrative beams subjected to compressive axial loads. Buckling predictions are validated against results obtained by other researchers [23], and also with predictions of two available beam finite-element models based on Euler-Bernoulli formulation (EBT) and first-order shear-deformation theory (FSDT), and a higher-order shear-deformation plate finite-element model (HSDT) [19]. For all cases

but one (see next section where discretization was considered for 10 beam finite elements.

Clamped-free isotropic T-beam

This example shows the influence of the slen- derness ratio (length of column/least radius of gyration of the cross-section) on the critical load of a clamped-free isotropic wide-flange T- beam. The material and geometrical data are:

E = 200.0 GPa (Young’s modulus); v = 0.3 (Poisson’s ratio);

124 M. A. Ramos Loja, J. Infante Barbosa, C. M. Mota Soares

h web = 0.102 m; b = 0.102 m; t 2% x lo-” m (where t is the thickness of

web and flange). In Table 1 one can observe the comparative influence of the slenderness ratio on the critical buckling load, for the different models, and the Euler critical load, given by Pcrit = TC*EZIL*. AS one can observe, for the different slenderness ratios, the EBT model gives very good results when compared with the closed form solution. If one considers low ratios, it is clear from Table 3 that the critical loads become lower than the analytical solutions for the FSDT and HSDT models. This fact is more evident in the HSDT case, which is not surprising because of its greater transverse shear-deformation influ- ence.

are compared with the buckling loads obtained by WennerstrGm & BBcklund [21] and with the mechanics of materials solution, including shear effects, which is evaluated using the expression [22] P,, = P,l( l+P,kl(Gbh)), where P, = x2EZIL2 and k = 5/6. The HSDT model presents a good agreement for the different cases studied when cornDared with the closed form solution, leading to lkwer critical loads because of flexibility.

Simply-supported composite I-beam

A simply-supported composite I-beam is con- sidered. This test case intends to compare the critical loads of different commercially available laminated wide-flange I-beam sections. The fol- lowing mechanical parameters are used:

Simply-supported isotropic beam

In this test case an isotropic rectangular cross- section beam is considered in order to study the shear-deformation effect on the buckling load. The material and geometric properties used are:

E = 1.379 x lo9 Pa; L = 0.0254 m; h = 0.00254 m (thickness); b = 0.003048 m (width).

Table 2 shows predictions for the present (HSDT) model for several discretizations which

E, = 20.632 GPai G,* = 1.985 GPa; L’,~ = 0.318; E, = E,; G23 = G,, = ($2;

“13 = ” 23 = “12.

E, = 4.433 GPa;

Table 3 shows the critical buckling load predic- tions for several I-sections. It can be seen that there is a good agreement between the HSDT model and the experimental values of Barber0 & Tomblin [9] and the corresponding critical buckling loads evaluated by the Southwell asymptote technique of the experimental meas- urements [22,24].

its greater

Table 1. Effect of the slenderness ratio on the critical buckling load. T-column (kN)

Slenderness ratio Euler load EBT FSDT HSDT (k = 516)

30 2863.501 2863.506 2778.604 2243.168 50 1030.860 1030.862 1019.598 924.558

100 257.715 257.715 257.004 247.141 200 64.429 64.429 64.384 62.691 500 10.309 10.309 10.307 10.081

1000 2.577 2.577 2.577 2.525

EBT, Euler-Bernoulli theory; FSDT, first-order shear-deformation theory; HSDT, higher-order shear-deformation theory; k, shear correction factor.

Table 2. Convergence and influence of shear on critical loads (kN)

WennerstrGm & Bscklund [21] - FSDT

Present model - HSDT

Elements A B A

2 86.6758 73.8210 86.0859 4 85.5166 72.9727 85.4426

%alytical [21] 86.1043 85.4477 73.3295 72.9131 85.4321

FSDT, first-order shear-deformation theory; HSDT, higher-order shear-deformation theory. A - G = 0.6895 x lo9 Pa (transverse elasticity modulus); B - G = 0.6895 x lo8 Pa.

B

73.0340 72.9421 72.9407

A higher-order discrete model for buckling behaviour 125

Table 3. Critical buckling load, for wide-flange I-beams (kN)

Section Length Experimental Southwell Present (mm) (m) ]91 method [9] method (HSDT)

102 x 102 x 6.4 4.48 12.08 12.46 11.60 102 x 102 x 6.4 2.98 27.21 28.10 26.09 152 x 152 x 6.4 6.03 23.10 23.66 21.15 152 x 152 x 6.4 3.58 64.15 67.11 59.49 152 x 152 x 9.5 6.03 33.38 34.11 31.42 152 x 152 x 9.5 3.89 78.80 82.22 75.02

HSDT, higher-order shear-deformation theory.

Simply-supported orthotropic beam

A simply-supported orthotropic beam with a rectangular cross-section is analysed, consider- ing the following lay-ups: [OO], [0”/90”], [0”/90”/0”], [O”/900/OV900]. The material proper- ties of the beam are:

EI = 181 GPa; El = E3 = 10.3 GPa; G,3 = G12 = 7.17 GPa; G,, = 6.21 GPa; VI2 = 0.28; v*3 = 0.02; v23 = 0.40.

Table 4 shows the results using the following multiplier 1= pJ[h2/L2 E,hl( 1 - v,~v~~)]. The present results are compared with two alterna- tive beam finite-element models Bhimaraddi & Chandrashekhara displacement field (HSDT) [7]:

proposed by [7] using the

u(x, z) =uO+z 4z2 ( ) 1 - ---$ +z$

w(x, z) = w” (20)

and also the FSDT [7] formulation, respectively. As one can see from Table 4, the present model

shows a good agreement with the two alterna- tive solutions.

Orthotropic beam under different boundary conditions

An orthotropic, multilaminated, rectangular cross-section beam is studied to analyse its behaviour when subjected to different boundary conditions, and for various length to thickness ratios. The beam lay-up sequence is [45’/ - 45’1,. The present model critical buckling load parameters are compared to the closed form solutions shown in Reddy [23] The material properties used are:

E,IE, = 25;

G = G2 = o.%,; G23 = 0.2E2; v12 = 0.25.

Table 5 shows the buckling load parameters, II, which were obtained using the following multi- plier R = PC,L2/E2h3. From Table 5 one can see that there is good agreement between the present HSDT results and Reddy’s solutions [23]. As expected, for lower length to thickness ratios, the present finite-element model gives lower critical buckling load predictions.

Table 4. Critical buckling load parameter, 4 for homogeneous and cross-ply beams (L/h = 10)

Model 0” VI90 o”/90°/00 0”/90”/0”/90”

HSDT [7] 11.5255 2.9172 11.0573 5.7511 FSDT [7] 11.5669 2.9297 11.0967 5.7740 HSDT (present method) 11.4179 2.7574 8.4274 5.5855

FSDT, first-order shear-deformation theory; HSDT, higher-order shear-deformation theory.

Table 5. Influence of the L/h ratio and boundary conditions on the critical buckling load parameter. Lay-up [450/-45”],

Llh Clamped-clamped Clamped-free

Reddy Present Reddy Present

v31 HSDT ]231 HSDT

100 5.737 5.847 (1.9%) 0.359 0.363 (1.1%) 20 5.478 5.515 (0.6%) 0.358 0.343 (-4.2%) 10 4.802 4.767 (-0.7%) 0.355 0.294 (- 17.2%)

HSDT, higher-order shear-deformation theory. Deviations (between brackets) calculated as: (A - 124)/124 x 100.

126 M. A. Ramos Loja, J. Infante Barbosa, C. M. Mota Soares

Table 6. Critical buckling load parameter, 1, for angle-ply beams (L/h = 10)

Model 0 15” 30 4.5” 60” 15” 90

HSDT [7] 11.5255 5.4619 2.4584 1.4050 0.9907 0.8414 0.8056 (19.1%) (0.5%) (6.3%) (6.3%) (6.7%) (7.2%)

FSDT [7] 11.5669 10.4370 (;.;;E) 4.1569 1.8142 0.9345 0.8092 (19.5%) (92.0%) (214:8%) (214.6%) (94.7%) (18.5%) (7.6%)

HSDT [19] 9.6755 5.4352 2.3904 1.3212 0.9317 0.7884 0.7517 HSDT (present) 11.4179 10.2600 4.2349 1.4071 0.8095 0.7241 0.6996

(18.0%) (88.8%) (77.2%) (6.5%) (- 13.1%) (- 22.3%) (-6.9%)

FSDT, first-order shear-deformation theory; HSDT, higher-order shear-deformation theory. Deviations (between brackets) calculated as: (& i”)/l” x 100.

Simply-supported angle-ply beam

An angle-ply laminated beam, with the same properties as those of the previous test case, is studied to analyse the effect of the fibre orien- tation angle on the beam buckling behaviour. Table 6 presents the critical buckling loads on the xz plane, for the different fibre orientation angles considered, obtained with the different models using the multiplier ;1= pcrl[h2/L2 E,h/ (1 -v,~v~,)]. From Table 6 it can be seen that there is a fair agreement between the present HSDT predictions and the beam model of Bhimaraddi & Chandrashekhara [7], whose dis- placement field is given by eqn (20), and the results obtained using the plate model described in Moita et al. [19]. A full mesh discretization of 2 x 10 plate elements was been used. Moita et

d’s [19] plate finite-element model is based on a displacement field using a third-order expan- sion in the thickness co-ordinate for the in-plane displacement and a constant transverse displacement. The present HSDT results agree well with the FSDT [7] predictions for all ply orientations. No apparent reasons have been found for the discrepancies observed between the present model, the HSDT [7] beam model and the Moita et al. [19] plate model. The HSDT [7] and Moita et al. [19] models demon- strate a behaviour closer to that expected.

CONCLUSIONS

A single-layer Lagrangean beam finite-element model, based on a higher-order shear-deforma- tion theory which assumes a non-linear variation for the displacement field, is proposed to study the buckling behaviour of anisotropic multilaminates of thick and thin sandwich beams. Its good performance is shown for most of the illustrative cases presented in this paper. From the extended numerical studies carried

out, and comparisons with experimental and/or numerical alternative solutions available, it can be concluded that the proposed model effi- ciently predicts the buckling loads of beams, underestimating them compared with the EBT and FSDT models. For the simply-supported angle-ply beam (see the section on ‘Simply-sup- ported angle-ply beam’), and with no apparent reason, there are some discrepancies between the present HSDT buckling load predictions and the results obtained from the HSDT [7] beam model (eqn (20)) and the HSDT [19] plate model, which in fact perform better.

ACKNOWLEDGEMENTS

The authors are grateful for the financial support received from H.C.M. Project (CHRTX-CT93-0222), ‘Diagnostic and Relia- bility of Composite Material and Structures for Advanced Transportation Applications’, and Funda@o Calouste Gulbenkian.

REFERENCES

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2.

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5 _.

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A higher-order discrete model for buckling behaviour 127

7. Bhimaraddi, A. and Chandrashekhara, K., Some observations on the modelling of laminated composite beams with general lay-ups. Composite Struct. 1991, 19,371-380.

8. Hwu, C. and Hu, J.S., Buckling and postbuckling of delaminated composite sandwich beams. A&I_4 J. 1992,30 (7), 1901-1909.

9. Barbero, E. and Tomblin, J., Buckling testing of com- posite columns. AOtl J. Techn. Notes 1992, 30 (ll), 2798-2800.

10. Barbero, E.J. and Raftoyiannis, I.G., Euler buckling of pultruded composite beams. Composite Struct. 1993,24, 139-147.

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12. Ray, K. and Kar, R.C., Parametric instability of a sandwich beam under various boundary conditions. Comput. Struct, 1995, 55,857-870.

13. Turvey, G.J., Effects of load position on the lateral buckling response of pultruded GRP cantilevers - comparisons between theory and experiment. Compo- site Struct 1996, 35, 33-47.

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15. Barbero, E.J., Godoy, L.A. and Raftoyiannis, I.G., Finite elements for three-mode interaction in buckling analysis. Int. J. Numer Meth. Engng 1996,39, 469-488.

16. Sheinman, I., Eisenberger, M. and Bernstein, Y., High-order element for prebuckling and buckling

analysis of laminated plane frames. Znt. J. Numer Meth. Engng 1996,39,2155-2168.

17. Vinson, J. R. & Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials. Martinus Nijhoff, Dordrecht, The Netherlands, 1986.

18. Loja, M. A. R., Higher-order shear deformation models - development and implementation of stiff- ness coefficients for T, I, channel and rectangular box beams. Report IDMEC/IST, Project STRDA/C/TPR/ 592192, Lisbon, 1995.

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APPENDIX A

ELASTIC COEFFICIENT MATRICES FOR RECTANGULAR MULTILAYERED CROSS- SECTIONS

B”” = B” =

M. A. Ramos Loja, J. lnfante Barbosa, C. M. Mota Soares

where

, (hf :-hi_,) ni

i ; bi

(bh-b;-,) (i= *, ..., 7) i

and k is the number of layers.

APPENDIX B

z MATRIX

z II =

NP 0 0 0 LI), LI),. M:;? My,. 0 0 0 0

N1’ 0 0 0 0 0 0 M;,. 0 L;,, 0

N’; 0 0 0 0 0 0 M(;, O- Lp.; 00 0 0 0 0 0 0 0

MTz 0;: L:;, , O(y)_, 0 0 0 0

My,. P;: L’;,, 0 0 0 0

My,, P$, 0 0 0 0

My;‘,,, 0 0 0 0

My,, 0 L:,, 0

MT,, 0 L;,,

M:,. 0

Sym. MC

A higher-order diwrete model for buckling behaviour 129

I I

L:z, L:,, Q:z Q% %z $y 0 0 0 0 d 0'

ez 0 0 0 0 A?_ 0 Qzy 0 0

0 0 0 0 0 0 0' ez 0 Q,o, 0 0 0 0 0

T;z2 L:y2 GzJ e, cz4 $yl 0 0 0 0 4z5

0 0 0 o- 0 0 syyl 0 Pg 0 0

0 0 0 0 0 0 0 $!,I 0’ Py, 0

0 0 0 0 0 0 PZ! 0 A$, 0 0

0 0 0 0 0 0 o- e, o- syz 0 I I

T 721 = 212

where the forces and moments resultants are given by

[till d a: dA

[L:, @, L:, , @,I Lzz2 tiz,l = j d a: [z z2 z3 z4 z5 z61T dA

[L:, @y L&, @y, L:,, @,,I =I I, 8 [Y y2 y3 y4 y5 y61T dA

@z 0% o;z2 f$z p;z, ez21= j 1 a: [YZ y2z y3z z2y z2y2 z2y31T dA

A higher-order discrete model for buckling behaviour 131

APPENDIX C

G MATRIX

Gl

-iNi -0 0 0 0 0 ax

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

0 0 0 0 0 0 0

8Ni -0 0 0 0

8X

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0 0 0 0 0 0 0

aNi -0 0 0

i3X

0

0

0

0

0

0

0

0

0

0

0

0 0 0 0 0 0 0 0 0 0

aNi -0 0 ax

0 i3Ni -0

0

0

0

0

0

0

0

0

0

0 0 0 0 0 0 0 0 0 0

ax

0 aNi

ax

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0

0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0 0 0 0 0 0 0 0 0 0 Ni 0

0 0 0 0 0 0 0 0 0 Ni

0

0

0

0

0

0

aNj

ax

0

0

0

0

0

0

0

0

0

0

0

0

0

0

aNi

ax aNi

O- ax

aNi 0 o-

ax

0

0

0

0

0

0

0

0

0

0

aNi 0 0 o-

ax

0

0

0

0

0

0

0

0

0

0

0

aNi 0 0 0 o-

ax

0

0

0

0

0

0

0

0

0

0

0

0

aNi 0 0 0 0 o-

ax

-

0

0

0

0

0

0

0

0

0

0

0

0

0

aNj 0 0 0 0 0 ox