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Buckling mode interaction in composite plate assemblies
loannis G Raftoyiannis
Department 0.[ Mechanical and Aerospace Engineering,
W Virginia University, Morgantown WV 265066106 USA
Luis A Godoy
Department 0.[ Civil Engineering, University ofPuerto Rico, Mayaguez,
Puert~ Rico
Ever J Barbero
Department 0.[ Mechanical and Aerospace Engineering,
W Virginia University, Morgantown WV 265066106 USA
The analysis ~fb~ckling mode interaction of fiberreinforced composite columns, modeled as
plat~ .as~embhes, IS presented. The main assumptions are linear elasticity; a linear fundamental
equlhbnum path; the existence of critical states that are coincident or near coincident· and a
coupled path rising from a quadratic combination of modal displacements due to inte:.action. The
f~~ulation a~opted is known as the WfoTmulation, in which the energy is written in terms ofa
shdlng ~et of Incremental c~ordinates, measured with respect to the fundamental path. The
ener~y IS t~~n expres~ed WIth respect to a reduced modal coordinate basis, and the coupled
sol~tlon anslng from Interaction is computed. An example of a pultruded composite Icolumn
subjected to axial compression illustrates the procedure.
INTRODUCTION
Fiber reinforced composite structural shapes are constructed
using thinwalled cross sections to take advantage of the
high strength of the material, and to compensate for its low
modulus. In typical flangeweb assemblies, the flanges may
buckle locally, and the column may buckle globally, at loads
that are close or even coincident for column lengths
commonly used in practice. Experimental evidence (Barbero
and Tomblin 1994) shows buckling loads lower than the
global or local loads predicted from isolated mode analysis.
Since composite materials remain elastic even for large
strains, it was speculated by Barbero and Tomblin (1994)
that the reduction in buckling strength may be caused by
buckling mode interaction.
When two or more modes of buckling correspond to loads
that are close or coincident, interaction between the modes
may lead to postbuckling behavior quite different from the
postbuckling behavior of the participating modes. The
general theory of elastic stability for discrete systems
(Thompson and Hunt 1973)was chosen in this work as the
framework for the study of postbuckling behavior with the
aid of finite element discretization. In this work, perturbation
techniques are used to classify the nature of the bifurcation
points and to follow the initial postbuckling path.
Furthermore, interaction between modes emerging from
bifurcation points, two of which are close or coincident, is
studied to identify tertiary paths emerging from the
secondary paths. The tertiary paths may be quite different
from the secondary paths identified by an isolated mode
analysis, thus revealing a different kind of imperfection
sensitivity of the structure.
Mode interaction in discrete systems was studied by
Chilver (1967), Supple (1967), Chilver and Johns (1971),
Thompson and Supple (1973), Swanson and Croll (1975),
Reis (1977), Reis and Roorda (1979), Maaskant (1989).
Analytical methods were used by Koiter and Pignataro
(1976) for stiffened panels and by Kolakowski (1993) to
study trapezoidal columns. From the literature, it is possible
to identified numerical problems that may arise while using
continuation methods in modecoupling problems.
The finite strip method was used by Mollmann and
Goltermann (1989), while finite elements were used by
Casciaro et al (1992). The finite element method was chosen
in our work because of its superior versatility to model
complex boundary conditions and because it automatically
accounts for the problem of wave modulation (Koiter 1967).
There are several alternatives to carry out modecoupling
postbuckling analysis in composite columns with symmetric
cross section. First, it is possible to combine a local and a
global mode, then compute first order fields. This was done
previously by Barbero et al .( 1993), who showed that this
analysis is not sufficient to detect buckling mode interaction.
Second, it is possible to combine a local and a global mode,
and to compute first and second order fields. This is the
methodology used in this paper. Another possibility is to
part of MECHANICS PANAMERICA 1995, edited by LA Godoy, SR Idelsohn, PAA Laura, and DT Mook
Appl Mech Rev vol 48, no 11 t part 2, November 1995
552
ASME Reprint No AMR177 $96
© 1995 American Society of Mechanical Engineers
Appl Mech Rev vol 48, no 11, part 2, November 1995
Raftoyiannis et at Buckling mode interaction in composite plate
assemb lies
combin e three modes and to comput e first order fields, as
shown by Godoy et al (1994) along with a simplified
~alytical model.
, In this paper, a general formulation is presented for the
finite elemen t analysis of plate assemblies, including
bucklin g and mode interaction. The formulation is written in
tenns of the usual matrices used in finite elemen t analysis,
and it represents an extension of previou s work by the
authors for noninteracting (isolated) modes. Since we are
primari ly interested in applications to fiber reinforced
compos ite columns, a shear deform able finite element is
used to model the structure as an assembly of plate elements.
The finite elemen t implementation is capable of dealing with
crossse ctions of arbitrary shape, as well as symmetric cross
sections. Numerical examples are present ed for the latter
case since it is conside red a critical test case.
where xj are the compon ents of the eigenve ctor.
The solution ofEq (4) is a set ofN eigenva lues, which we
will denote as An (the supersc ript identify ing the mode
number), andeigenvectors xn, for n = I,..., N. These modes
satisfy the following orthogonality conditio ns
n
0
m
Xi • Xi = ,
n:l:m
m i
[V.,,_QF
k] x. X =0
n
mn
A
•
ljl{.
]
(5)
From the critical state, the post bucklin g path will be
followed by perturbation techniques. We select a suitable
general ized coordinate as perturbation parame ter (say q)
and we write the load A and the remain ing general ized
coordinates qi (i I) as
*
.
1
2
A=i\.c+ i\.(l)cq +_A(2)Cq2+
1
FORM ULATI ON
The total potential energy V of a discrete system can be
express ed as a Taylor expansion with respect to a set of
general ized coordin ates Q (usually nodal displacements) as
i
(Raftoy iannis 1994)
A
1
A
A
V = Vo+V.
Q.+V.
I
I
2 l.jO.Q.
I
J
1
1
+6VijkQ;QjQk + 24 Vij,J2;Qj QkQZ
A
A
(I)
~re i, j, k, I = 1,..., N, N being the number of degrees
of
treedo mof the system. The discrete system may result from
a finite elemen t discretization of a continu ous system.
After finding the initial response, the nodal displacements
can be written as
_
(2)
where qi are the incremental displacements, and Qt is the
response in the fundamental path for unit value of the load
parameter. Introduction of Eq (2) into Eq (1) results in the
Wenergy that, after neglecting terms independent of qi and
terms with nonlin ear contribution from Qt, can be written
as
1
1 (2)C 2
ql +qj ql +...
2
...
(1)C
qjqj
(6)
where superscripts (l)C and (2)C indicate first and second
order derivatives with respect to the perturb ation parame ter,
evaluat ed at the critical state.
We next assume that there are two eigenva lues that are
relatively close, or even coincident. We refer to these two
eigenvalues as active, and they play a crucial role in the
interactive bucklin g process. The remain ing N2 eigenva lues
(for n = 3,..., N) are passive, and they play no role in the
analysis.
As a first approximation, we introdu ce a linear
combin ation of the active modes that may be sufficie nt to
reflect mode interaction given as
qj
F
Qi=AQi +q;
S53
I
=X;
k l:
~k
where, i = 1,..., Nand k = 1,2 and ~k are modal amplitudes
(i.e. scalars). The interaction betwee n the X k modes
i
originates new coupled modes, which will be denoted as X kl ,
i
also called higher order fields (arising from the interaction
betwee n modes Xkj and Xli).
If we restrict the analysis to a quadrat ic combin ation, the
initial postcritical path qi may be describ ed as
1
W=~q;+2~F;qj
A
A
1
1
+"6Wijkq;qJqk + 24 Wijkfl;qJqkqZ
A
A
where
~=Vj
A
A
A
F
W;j=Y';j+AVijkQk
A
A
A
F
WiJ1c=Y';]1c +AVij1clQl
WijkZ=Vijkl
.. ne condition of critical stability is
A
A
F
[V ij+i\.VijkQk ] xj=O
(3)
(7)
This combin ation may be found, for exampl e, in the work
of Reis (1977), and Reis and Roorda (1979). For a fixed
value of k, xj k is a vector of dimens ion N (an isolated
eigenvector); while Xi kl is a matrix of dimens ion Nx2, since it
reflects the interaction between mode k and all the remaini ng
modes. If we fix k and I, then X kl is a vector of dimens ion
i
Nxl.
The coupled modes should be orthogo nal to the isolated
modes, ie
I
554
MECHANICS PANAMERICA 1995
Appl Mech Rev 1995 Supplement
for m, n, p = 1,2. An additional property of the coupled
modes is that of symmetry,
kl
= Xi
Xi
lk
Notice that the modes considered in the present analysis
need not to be simultaneous, meaning that they do not need'
to correspond to coincident critical loads. Substituting Eq (7)
into the W expression (3), after some manipulation, and
keeping only up to fourth order terms, we get
This is correct, since we have already calculated all the
eigenvectors, and denoted them as x s . Notice that the
coupled modes are afunction of A.
Let us consider the number of systems (Eq 9) to be
solved. If there are M interacting modes, then there will be
Yi(M+M2) systems to obtain the coupled modes.
The above condition (Eq 9) may be written for simplicity
as
K i)"T(A) xj uv=_ (1,iuV_AAuV
Pi
(10)
where,
1 '" A "
F
I I
II II
I II
W=2T(Vij + VijkQk )(qj qj +qj qj +2qj qj )
1 "
A"
F
I I I
II I 1
+~(Vijk + Vij1c/Q/ )(qj qj qk +3qj % qk)
'.
+J...(V
)( I I I 1)
4! i)kl qi qj qkql
We can recognize that some terms vanish due to the
orthogonality condition between modes. Hence, the energy
W can be written as
w=
;! st~s~t(A A~)
0
1"
"
1"
"
(11 )
By substituting Eq (11) into Eq ( 10), the following
conditions are obtained:

F
We write the solution ofEq (10) in the form
+2T(Vlj+AVijkQk )xj Xj ~s~t~u~"
F
stu
+3T(Vijk+AVlj1c/Q/ )xj XjXk ~s~t~u
1"
"
F
st
u
v
+2(Vijk + AVijk/Ql )xj xj x k ~s~t~u~"
+J...(Vlj'kl)X i SXjtXkuXI v~s~t~u~v
4!
(8)
becoming a function of ~s' x j st, and s, t, u, v = 1,2. All the
terms depend on ~s' but the first, third and fifth term are
independent of the higher order fields x j st • The modes arising
from coupling between isolated modes also define paths of
equilibrium in the Aqi space. To obtain the mode shapes x j st
of the coupled modes, we note that the first variation of W
with respect to x ist is zero, for constant values of the
amplitude parameter ;s. Thus,
,I
From the above equation we conclude that
[V,.+AV.
QF] .uv = .!..(V"
+AV'"i)1d'Q1F)Xj UXkv
IJ
11k k Xl
2 ijk
(9)
The member on the right side is clearly nonzero; this
shows that xj uv are not eigenvalues of the tangent matrix
A ' "
[
Vi) +
A
F]
Vljk Qk
A
T
st uv
uv
uv
Kif ( )
Zj
KJ(A)
y/" = P:o'
=(1,;
(12)
Equations (12) are valid for all values of A; however we
are interested in the initial postcritical behavior, so that A
will be restricted to values AcU ~ A ~ AcV.
Next, we consider that the interacting modes are
associated to nearcoin~ident loads, and with negligible error
we may evaluate Eqs (12) at the lowest of the two interacting
loads. If we assume that AcU ~ Acv, then we should set A =
AcU in Eqs. (12). Having chosen A = AcU means that the
matrix KijTl c is singular, and the solution ofEqs (12) requires
some additional conditions, which may be taken as:
F
A
Vi)kQk
Z;
uv
U
Xj =
0
,
Z uV=o
1
,..
F
Vi)kQk
Yi
uv
yt =0
V
uo
Xj  ,
(13)
where there is no summation on u in Eqs (13). The second
and fourth constraints indicate that if the perturbation
parameter chosen to follow the path associated to Acu is qlU,
then ZI uv =0 and YIuv = O. The solution of Eq (9) can now be
obtained in terms ofxj uv •
We now return to Eq (8) for the energy W. Since the
coupled modes xj UV are now known, the complete path could
be traced if we calculate the mode amplitudes ~s. In other
words, we have constructed. a basis of vectors (the modes x·JS
and the coupled modes x/t) to express the equilibrium
condition, and we now seek to find how these amplitudes
change along the evolving path. To obtain the values of ~s
we note that at an equilibrium state W must be stationary
with respect to all possible variations of the generalized
~I Mech Rev vol 48. no 11, part 2. November 1995
Raftoyiannis et a/: Buckling mode interaction in composite plate assemblies
,I"coordinates qj" Using Eq (7), the variation is written in tenns
~,{'[~lof~
{E}
{Eo} + {E 1 }
V..x +W.x
V
1"
"
F
+[ "2(Vijk +AVijk/Q / )xi
"
"
F
stu
Xj Xk
st
+[2(V~;+AVijkQk )X i Xi
"
"
u.,}' +w Jl
ll.y +V .x
..
W.S =o=[(A A~)ost]~t
F
st
]~t~u
8
u
xk
W
1 ~
s t u v]~ ~ ~
+(Vijkl)X; Xi Xk Xl ~t~u~v
w
+2(Vijk+AVij1dQI )X; Xi
0
+
~
.:..
8 x.y 8y.x
v
6
.y
_~
2w ..xw J'
x_~
8y .Y
uv
8
y
8 x
8_
::
2
.y
for s= 1,2 or using Eq (8),
2
2
U..x
0
0
0
0
0
This leads to the following two equilibrium equations in
modal space
~ +A
(AAC)o
s
st~t
stu~ ~
+B
stuv~~ ~
=0
(17)
containing three inplane strains (Ex' E y' Yxy) at the midsurface, three curvatures (Kx' Ky, K xy)' two out of plane shear
strains (Yyz and Yzx)' and the inplane rotation 9z Note that
for s, t, u, v = 1,2. Eq (14) can be written explicitly as
Eq (17) contains not simply the vonKarman equations, but
also the terms av/ax and aulay, which are included as in the
work of Benito and Sridharan (1985). These additional terms
are necessary to represent correctly the nonlinearities at the
junction of flange and web of structural members, mainly for
the modal interaction analysis.
The linear strain matrix [Bo] is formulated, and also the
(15) nonlinear strain matrix [B (q)], which is' a function of the
1
nodal displacements q. We can further write the nonlinear
where
part of the strains as {E 1} = ~ [A] [9] = [B 1(q)] {g}, where
1 ~
~
F
stu
the matrices [A] and [9] are still a function of the nodal
A stu="2(Vijk+ AVijk/Q/ )xi XjX k
displacements. The matrix [9] can be written as [9] =
[G]{q}. Next, we write d{E 1} = [A][G]d{q}, from which it
"
"
F st uv
B sruv =2(Vij+AVijkQk)X; Xi +
follows that [2B 1] = [A][G]. On the other hand, the matrix
"
~
F
st u v 1 V
s r u v
[A(q)] can be written as the product of two matrices, the first
I
+2(Vijk+AVijk/Q/ )Xi Xj X k +'6 ijkfiXjXkX
containing the derivatives of the shape functions and the
(16) second the nodal displacements.
The total potential energy W of the plate, subject to inThus, we have two cubic equations (15) in two unknowns plane and transverse loading, can now be written as
(~1 and ~2). The solutions are a function of A through the
coefficients given in Eq (16).
~s~u
t u v
(14)
o
DISCRETIZED SYSTEM
The structure is discretized using ninenode Lagrangean
plate elements based on first order shear deformation theory.
The element has six degrees of freedom per node and allows
for the modeling of plate assemblies arbitrarily oriented in
space.
The strains are written as the summation of a linear plus a
'llinear part, in the form
where the load vector {f} is assumed to be incremented by a
single load factor A. The stress vector {a} is, in this case,
{crt {Nx,N y , N xy ' Mx' My, MXY' Qy' Qx' Mz}T, where N x'
Ny, and N xy are the inplane stress resultants, M x' My, and
M are the moment resultants; Qy and Qx are the outofx
pla~e shear stress resultants and Mz is the inplane moment.
For the case of a plate made of laminated FRP material,
the constitutive law is given by Raftoyiannis (Eq 5.15,
1994) • The plate stiffness A..,
IJ B..IJ and D··IJ are computed using
•
Classical Lamination Theory, and a very small number C· IS
used to represent the inplane rotational stiffness (9z)·
=
S56
Appl Mach Rev· 1995 Supplement
MECHANICS PANAMERICA 1995
Materials with nonvanishing bendingextension coupling
terms Bjj are modeled here with an approximate linear
fundamental path. The limitations of this approach are
currently being investigated.
The first step in the analysis is to obtain the linear
fundamental path {QF}; this is done by solving the system
[Ko]{QF} = {Pl. The second step is the detection of the
critical states along the fundamental path, which is done by
solving the classical eigenvalue problem, to obtain the
critical load AC and the corresponding eigenvector {x}.
Next, attention is given to the study of the post critical path
passing through the bifurcation point. We must detennine
whether the bifurcation is symmetric or asymmetric. For
that, it is necessary to compute the matrix [D1(x)] contracted
by the eigenvector {x}, which can be written in terms of
finite element matrices as
The coefficients A stu (Eq 16) of the quadratic terms in the
equilibrium equations are related to the contracted matrix
[D.(x)] in the following manner
s[D 1(X. t)] xku
A stu=.!2 x·I
}
Since the contracted matrix [D1(x)] is afunction of the
load parameter A, we can write
where
[E1(x)] =f{[2B 1(O)f[C] [Bo]{x}
+[2B 1(x)]T[C][Bo]+[Bo]T[C][2B 1(x)] }dv
i
[D1(x)] = f{[2B 1(O)f[C][Bo+2AB 1(QF)]{X}
v
+[2B 1(x)]T[C][B o+2AB 1(Q F)]
+[Bo+2AB 1(Q F)]T[C][2B 1(x)]}dv
[Eix )] =f{[2B 1(o)f[C][2B1(qF)]{X}
+[2B 1(x)]T[C][2B 1(q F)]
(18)
where 0i is the Kroneker delta, with values OJ ~ 1, if i = j, and
OJ =0, ifi :;; j. The matrix [28 1i(Oj)] can be computed from [2
Btl =[A][G]. The coefficient C may now be computed as
C
(19)
+[2B l(q F)]T[C][2B 1(x)]}dv
or
A stu=Fstu+AG stu
where
= {x}T [D1(x)]{x}
If C = 0, the critical state is a symmetric bifurcation,
while for C 0 the bifurcation is asymmetric. To follow the
post buckling path, a perturbation analysis is carried out
from the critical state. The variables qi and A are expanded
in terms of a perturbation parameter ql' as indicated in Eq
(6). In symmetric bifurcation, the slope A(1)C of the post
buckling path at the bifurcation point vanishes, and {q(l)C} ==
{x}. As a next step, the {q(2)C} coefficients in Eq (6) are
computed from
*
[KT]
{q(2)C}
= [D 1(x)] {x}
aUV = ~ [E1(x U)]{x V}
I
C
where [KT ] = [Ko ] + A[Ka ]is the tangent stiffness matrix at
the critical point. The value of one of the components in the
vector of the second derivatives of the displacements has to
be chosen (ie, Ql(2)C = 0 if the perturbation parameter is ql)'
Finally, the curvature of the post buckling path at the
bifurcation point results in
NZ)C
We will now be concerned with the coefficients of the
cubic terms in the equilibrium equations (14). Since they
contain the second order field, we must first solve the system
(12) and define the relation (11). The vectors Ct j UV and Pjuv
can be written in matrix form as
PUV=±[Ez(X ·U)]{x V}
where the contracted matrices [E 1] and [E2] are defined in
Eq (19). Hence, the system (12) can be solved along with the
conditions (13) for the vectors Q,juv and pjuv. The coefficients
Bstuv are written in matrix form as
= {x }T[Dz(x,x)]{x} +3 {x} T[D1(x)]{ q (z)C}
3 {x}T[Ko] {x}
The matrix [D 2(x,x)] contracted by the {x} eigenvector
can be computed from
B
o] + A[Ka ])
st} +1\ {y st} )( {z UV} + A {y UV} )
+2([E}(x s)] +A[E2(x s)])( {z tu}
sruv =2([K
x ( {z
+A{y tU}){x V}
[DzCx,x)]
. +
= f{[2B 1i (O)f[C][2B 1(x)]{x}
v
2[2B}(x)]T[C][2B1(x)]}dv
+'!'[Dz(X s,x ~]{x U} {x V}
6
Expanding the expression (40) and collecting the tenns in
likepowers of the load parameter A, we can write
Appl Mech Rev vol 48, no 11, part 2, November 1995
Raftoyiannis 8t al: Buckling mode interaction in composite plate assembIee}
Table 1. Critical modes and corresponding critical loads for the
pultruded Icolumn
B stuv=Mstuv +i\Nsruv+A2p sruv+A3 Q stuv
MS tuv
= 2{z st}T[Ko]{z UV} +2{z st}T[E1(x ~]{x V}
+2{x S}T[D2(x t~ U)] {x V}
6
~
N stuv = 2{z st}T[Ka] {z UV} +2{z st}T[Kol {y UV}
•
+2{y st}T[Kol{z UV} +2{y st}T[E1(x ~]{x V}
+2{z st}T[E2(x U)]{x V}
p stuv
=
2 {y st}T[Kol {y UV} +2 {z st}T[Ka ] {y UV}
+2{y st}T[Kol {z UV} +2{y st}T[E2(x U)] {x V}
Q SlUV = 2 {y st} T[Kol {y UV}
and s, t, u, v = 1,2. The energy terms are also symmetric in
this case, therefore
A stu=A tus=A ust
B stuv=B tuvs=B uvst=B vstu
Finally, Eqs (14) are completely defined and can be
solved using perturbation techniques or using the NewtonRaphson technique.
The imperfection sensitivity of the 'system can be investigated by using the imperfection parameters ;jimp, that repreSF
the amplitude of an imperfection with the shape. of the
mv",e Xi. The equilibrium equations for the imperfect case of
"two interacting modes become
(AAS)o
~ +A stu
C
st t
~ C
t~u
+B sruv
CC C
~t~u~v
=ACimp .
~t
(20)
where s, t, u, v = 1,2. The NewtonRaphson technique is
used in the next section to solve these equations.
NUMERICAL RESULTS
To illustrate the numerical procedure, we consider a Fiber
Reinforced Plastic (FRP) Icolumn produced by pultrusion.
The geometry of the column is defined by b =6 in; h =6 in,
and L = 100 in. The material properties for the flanges are
All = 893,500 Ib/in; A 22 = 343,000 lb/in; A I2 = 130,800
Ib/in; A66 = 113,600 lb/in; 0Il =4,289 Ib in; D22 = 2,029 lb
in; 012 = 807.3 lb in; D66 = 641.5 1b in, A 44 = Ass = 94,670
Ib/in; and A 4s = A S4 = o. For the web the material properties
are All = 893,500 lb/in; A 22 = 343,000 lb/in; A I2 = 130,800
Ib/in; A66 = 113,600 lb/in; 0Il =4,090 Ib in; D22 = 1,863 lb
in; 012 = 73 1.6 lb in; D 66 = 596.1 Ib in, A 44 = Ass = 94,670
Ib/in; and A4s = A S4 = o.
The column is discretized using 45 elements. The
boundary conditions applied to the FE model are as follows:
t!
~nds of the column are simply supported, with free
rotation about the weak axis of the cross section. Hence, a
global mode (Euler) is expected to occur about the weak
axis.
Mode
Cr. Load
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
193,994.50
191,354.23
189,244.32
171,387.39
175,405.49
174,636.60
196,773.89
198,689.01
179,645.15
160,458.93
142,923.56
27,323.60
 27,461.80
29,048.20
29,432.43
29,761.56
31,698.91
33,527.09
1,42,770.19
36,572.63
129,134.71
122,922.03
40,905.58
36,342.64
44,589.87
55,340.84
38,960.76
60,111.63
40,119.62
42,739.29
43,760.90
47,486.69
117,812.37
131,584.77
115,168.30
128,594.97
Perfect system: The model is first solved for prebuckling displacements. The' eigenvalue problem is then
formulated to solve for, ~he critical buckling loads and the
associated eigenmodes. A 'selected number of critical loads
that are the eigenvalues of the buckling problem are shown
in Table 1.
A mode number is assigned to each eigenvalue with
reference to fig 1, which. contains schematics of the modal
shapes associated to the eigenvalues presented in Table 1.
Each mode is represented by the crosssectional modal
deformation at the x = L/2, the lateral deformation of the
longitudinal axis passing through the centroid for the entire
length of the column, and the lateral deformation of the right
tip of the top flange for the entire length as well.
The lowest eigenvalue (Mode #1242 in Table 1)
corresponds to the local buckling mode, and is Ale = 27,323
lb. The associated eigenmode Xl can be seen in Fig 1. Both
the flanges rotate and the web bends with eight waves along
the length of the column.
p
558
Appl Mech Rev 1995 Supplement
MECHANICS PANAMERICA 1995
stable symmetric (Fig 2), and the curvature is found to be 0.02651. Hence, the following equilibrium equations are
A 1(2)C = 172,182 Ib/in2 • The global mode x 2 (Euler mode defined, and can be solved for specified values of the load
. about the weak axis) is next identified from Figure I (Mode parameter.
# 1246). The crosssection remains undeformed, but the
whole beam buckles with one wave in the global sense. The
critical load is A2C =29,761 Ib (Table 1). The post buckling.
(A +27323)~1+(324516.24+0.15383A)~i
path is also stable and symmetric, and the curvature is
(18522. 72 +O.55554A)~i~2
computed as A 2(2)C = 211 Ib/in2 that is a low value,
compared to the one of the local post buckling path.
+(215253.48 + 15.45645A)~1~~
Next, the eigenvectors of the two selected modes are nor+(895.02+0.02594A)~~=O
malized according to Eq (5), where the highest componen!s
are 4.48 for the global mode Xl and 1.6755 for the local
mode x 2. The coefficients Fstu, Gstu, Mstuv, and Nstuv (where s,
(A +29761)~2(6174.24+0.18518A)~~
t, u, v = 1,2) of the two equilibrium equations in modal
space are then computed. The numerical values of the inter+(215253.48+ 15.45645A)~~~2
action coefficients that are different from zero are: M 1111 =
+(2685.06 +0. 07782A)~ 1~~
324,516.24; MI1l2 = 6,174.24; MI122 = 71,751.16; MI222 =
895.02; M2222 = '74,743.75; Nlltl = 0.15383; NII12 =
+(74743.75 +0.02651A)~; =0
0.18518; NI122 = 5.15215; NI222 = 0.02594 and N2222 =
1242
Notice that by keeping only the first order field, the
values of the coefficients become MIIII = 2,701,272; MI222
= 2,500; MII22 = 64,993; MI222 = 210.6 and M2222 =
174,216; while all Nstuv become zero. Then, the system is
imperfection insensitive, and the interaction phenomenon
cannot be retrieved. In order to study the interaction between
the two modes, we see that the coefficients which couple the
two ~odes are of significant importance.
Next, in order to determine the tertiary (coupled) path, we
expect the system to experience loss of stability. This means
that the tertiary path will be branching from the lower
secondary path, which corresponds to the local mode. A
NewtonRaphson technique for solving the equilibrium
equations is used here. Hence, the bifurcation point is
detected for load A(O) = 28,725 and amplitude of the
primary local mode ~I (0) = 0.06681. The projection of the
tertiary path on A~I plane is plotted in Fig 5. We see that on
the tertiary path A~I' the load carrying capacity decreases
(loss of stability) for increasing ~I (local mode amplitude).
1246
Fig 1. Modal shapes for the Icolumn
3
,...
.....
~
Perfect
3.0
Imperfect ( ~l = 0.01, ~ = 0 )
2
.....
2.8
~
~
~
Q
''"
'"
~
~
<
Perfect
,...
,.Q
<
1
o
o
0.1
0.2
Fig 2. Equilibrium path, projection on the
0.3
A~1
0.4
plane (local)
2.6
Imperfect
( ~l = 0.01, ~ = 0 )
2.4
o
0.25
0.50
~
Fig 3. Equilibrium path, projection on the A~? plane (global)
Appl Mech Rev vol 48, no 11, part 2, November 1995
Raftoyiannis
The projection on the A~2 plane (~2 being the amplitude of
the global mode) is shown in Fig 3. A comparison between
the tertiary path obtained by a threeparameter Ritz approxiie solution of the same problem (Godoy et ai, 1994) and
the present finite .element solution is shown in Fig 4.
Thedifference in the results are caused by the approximation
to the boundary conditions assumed in the Ritz model.
Imperfect system: We now consider the Icolumn having
in the unloaded state an initial imperfection similar to one of
the buckled mode shapes, global or local. The imperfection
amplitude is defined by the parameter ~iimp. Then, the
equilibrium equations (20) hold for the imperfect system and
are expressed in the modal space. These equations are solved
Simplified lVIodel
3.0
....
,....,
"Q
2.8
<
2.6
Finite Element Model
o
0.25
0.50
Fig 4. Comparison between simple and FE model (equilibrium
path projection on A~." global)
3.0
at at.
Buckling mode interaction in composite plate assemblies
S59
by the NewtonRaphson method.
The imperfection sensitivity curves for two mode interaction is shown in Figure 5. It can be seen that the Icolumn
has low imperfection sensitivity, ie for a significantly imperfect column, the reduction of load carrying capacity is about
30%.
CONCLUSIONS
Fiber reinforced plastic structural shapes are shown to
exhibit buckling mode interaction. Because of the linearity
of the material up to failure, and the linearity of the
fundamental path, the Wformulation was successfully
applied to study the postbuckling behavior interms of
twomode interaction.
Higher order fields were necessary to detect mode
interaction, and these fields were constructed from the
information contained in the Wenergy of the system. A
numerical procedure, based on the finite element method and
perturbation techniques, was developed for postbuckling
analysis of structures modeled as plate assemblies.
The effect of two interacting modes on the postbuckling
behavior was accounted by studying the equilibrium paths in
the reduced modal space of the active coordinates. The
resulting two nonlinear equations describe the complete
behavior of the system in the vicinity of the equilibrium
paths. These .equations were conveniently solved by the
NewtonRaphson technique, in order to track the perfect
primary, secondary, and tertiary paths, the latter arising from
mode interaction. Both the perfect and the imperfect systems
were formulated and solved, the latter having imperfections
in the shape of one or two the interacting modes. By using a
finite element discretization as plate assemblies, all buckling
modes (local and global) as well wave amplitude modulation
are automatical1y taken into account. A fiber reinforced
compositecolumn was analyzed and the results favorably
agree with the results of a simplified model. While all the
isolated mode secondary paths were found to be stable, the
column was found to be imperfection sensitive once mode
interaction was acknowledged.
ACKNOWLEDGEMENTS
This project was partial1y sponsored by NSF Grant 8802265
and West Virginia Department of Highways Grant RPT699.
The financial support is gratefully acknowledged. Our
recognition to Dr Hota VS GangaRao, Director, Constructed
Facilities Center, for his help and comments about this work.
~=o
REFERENCES
1.8
o
0.2
0.4
0.6
0.8
Imperfection Amplitude
Fig S. Imperfection sensitivity curves
1.0
Sl
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