1. ELSEVIER
CompositeStructuresVol. 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

2. 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.

3. 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 10
0 0 0 000000 0 0 0 0 0 1 y y*
go=[E;&yq*OO&.v&zkyk.;k,k,k:ky4,6,4::4xy4z.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[cxox 0.x or gzlT[I Z*Y* 1 11dY dz
* I Y/
[M""] = CM?,M; M,J~ = j, zti’[GE0, c.zlT [zz3zl dydz
I‘ I Y/
w-1 = [S,, s:,Cl’ = j, zIk I
-1z,, [1zz*ldydz
[QX"l = [S, s:ysgl’ = $I J tijyuYY21dYdZ
* I

4. 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 qe1. r> W)
where matrices BM, BF,;, BF,,, Bsxzand 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.

5. 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, BXYBFxz x, +Bg D”” BF _I” IL =.
+B:X>,DXYBFX;fBzX,DxYzBFX,+BzX,,DXyZTBFXz+BT,CTyBsXv+BzXvCTyTBM
+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);

6. 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] 86.104385.4477 73.329572.9131 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

7. 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)/124x 100.

8. 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.
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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.
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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.
19. Moita, J.S., Mota Soares, C.M. and Mota Soares,
C.A., Buckling behaviour of laminated composite
structures using a discrete higher-order displacement
model. Composite Struct. 1996, 35, 75-92.
20. Maple V Release 4, Waterloo Maple - Advancing
Mathematics. Waterloo Maple Inc., Waterloo,
Ontario, Canada.
21. Wennerstrom, H. & Backlund, J., Static, free vibra-
tion and buckling analysis of sandwich beams. Report
86-3, Department of Aeronautical Structures and
Materials, Royal Institute of Technology, Stockholm,
Sweden, 1986.
22. Southwell, R.V., On the analysis of experimental
observations in problems of elastic stability. Proc. R.
Sot. Lond. 1932, A135,601-616.
23. Reddy, J. N., Mechanics of Laminated Composite
Plates - Theory and Analysis. CRC Press, Boca
Raton, Florida, 1997.
24. Southwell, R.V., An Introduction to the Theory of Elas-
ticity, 2nd edn. Oxford University Press, New York,
1941.
APPENDIX A
ELASTIC COEFFICIENT MATRICES FOR RECTANGULAR MULTILAYERED CROSS-
SECTIONS
B”” = B” =

10. 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

11. 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
II
T
721 = 212
where the forces and moments resultants are given by
[till d a: dA
[L:, @, L:,, @,I Lzz2tiz,l =j d a: [z z2z3z4z5z61TdA
[L:,@yL&,@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: [YZy2z y3z z2y z2y2 z2y31TdA

13. 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