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Advanced buckling analyses of beams with arbitrarycross sectionsCitation for published version (APA):Erp, van, G. M. (1989). Advanced buckling analyses of beams with arbitrary cross sections. Eindhoven:Technische Universiteit Eindhoven. https://doi.org/10.6100/IR307439
DOI:10.6100/IR307439
Document status and date:Published: 01/01/1989
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ADVANCED BUCKLING ANALYSES OF BEAMS
WITH ARBITRARY CROSS SECTIONS
G.M. VAN ERP
CJP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG
Erp, Gerardus Maria van
Advanced buckling analyses of beams with arbitrary cross
sections / Gerardus Maria van Erp. - [S.J.] : [s.n.J. - 111.
Thesis Eindhoven. - With ref. - With summary in Dutch.
ISBN 90-9002808-ü
SISO 694.3 UDC 624.075.2(043.3)
Subject heading: thin-walled beams buckling analysis.
Printed by: Febodruk, Enschede
ADVANCED BUCKLING ANALYSES OF BEAMS
WITH ARBITRARY CROSS SECTIONS
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Eindhoven,
op gezag van de rector magnificus, prof.ir. M. Tels,
voor een commissie aangewezen door het college van decanen
in het openbaar te verdedigen
op dinsdag 23 mei 1989 om 16.00 uur
door
GERARDUS MARIA VAN ERP
geboren te Vught
Dit proefschrift is goedgekeurd door de promotoren
prof.dr.ir. J.D. Janssen
en
prof.ir. J.W.B. Stark
copromotor
dr.ir. C.M. Menken
Het onderzoek, beschreven in dit proefschrift, werd gesteund door de Stichting
Technische Wetenschappen (STW).
The research, reported in this thesis, was supported by the Netherlands
Technology Foundation (STW).
Voor Ans,
Christel
en Michael
V
CONTENTS
Notation viii
1 Introduetion
1.1 General 1.1
1.2 The adopted approach 1.3
1. 3 Research strategy 1.4
2 Local and distortional buckling of flat plate structures
2.1 Introduetion 2.1
2.2 Spline approximation 2.2
2.2.1 General 2.2
2.2.2 Cubic splines 2.3
2.2.3 Basic cubic B3 ~spline 2.3
2.2.4 Adapted B3 -spline representation 2.5
2.3 General procedures in the spline finite strip method 2.6
2.4 Section knot coefficients 2.7
2.5 Displacement functions 2.8
2.6 Buckling theory for flat plates 2.9
2. 6.1 Basic assumptions 2.9
2.6.2 Strain and curvature displacement relations 2.10
2.6.3 Stress strain relations 2.12
2.6.4 The perfect structure 2.12
2.7 Stiffness and stability matrices 2.14
2. 7.1 The reference displacer:~ent field 2.15
2. 7.2 Eigenvalue model 2.17
2.8 Solution process 2.18
2. 8.1 Equilibrium model 2.18
2.8.2 Eigenvalue model 2.19
2.9 Numerical examples 2.19
2.9.1 Symmetrie I-column loaded in compression 2.19
2.9.2 Plate strip loaded in pure shear 2.21
2.9.3. Plate girder in uniform bending 2.22
2.9.4 T-beam loaded by a concentraled force 2.24
2.10 Conclusions and recommendations 2.27
vi
3 Interaction between buckling modes
3.1 Introduetion
3.2 Initial post-buckling theory for simultaneous and nearly
simultaneons buckling modes
3.2.1 Perfect structure
3.2.2 The influence of small geometrie imperfections
3.3 Matrix formulation and computer implementation
3.3.1 General
3.3.2 Determination of the second order displacement
fields
3.4 Determination of the equilibrium paths
3.4 .1 The perfect structure
3.4.2 The imperfect structure
3.5 Numerical examples
3.5.1 Simply supported square plate under uniform
compression
3.5.2 Channel section under uniform compression
3.5.3 Plate girder in uniform bending
3.5.4 T-beam loaded by a concentraled force
3.6 Conclusions
4 The elastic flexura.l-torsiona.l buckling of mono and douhly
symmetrie beams with a. large initia.l bending curvature
4.1 General
4.2 The nonlinear flexural-torsional behaviour of straight
elastic beams
4 .2.1 Kinematics of straight slender elastic beams
4 .2.2 The general potential energy functional
4.2.3 The strain energy
4 .2.4 The potential energy functional
4 .2.5 The rotation matrix and curvature expressions
4.2.6 Alternative warping formulation
4.3 Pormulation of the bifurcation problem
4.3.1 General
4 .3.2 The prebuckling state
4.3.3 The bifurcation criterion
3.1
3.5
3.5
3.10
3.13
3.13
3.13
3.17
3.18
3.20
3.21
3.22
3.24
3.30
3.33
3.35
4.1
4.1
4.2
4.6
4.7
4.8
4.11
4.13
4.14
4.14
4.14
4.15
4.4 Numerical approach
4.4.1 Finite element formulation
4.5 Numerical results
vii
4.5.1 Simply supported, laterally unrestrained beam in
pure bending
4.5.2 Simply supported beam in pure bending with lateral
end restraints
4.5.3 Simply supported, laterally unrestrained beam loaded
by a concentraled force at midlength
4 .5.4 T -beam under moment gradient 4.6 Geometrical constants of beams with arbitrary cross
sec ti ons
4. 6.1 General
4.6.2 Relevant definitions and expressions
4.6.8 Numerical examples
4. 7 Conclusions
5 Summa.ry a.nd conclusions
Appendices
2.1 Matrices used in the derivation of the buckling modes of
chapter two
3.1 Derivation of P[ai,v,.,\]
4.1 Influence of prebuckling deformations on the buckling behaviour
of simply supported beams in uniform bending
4.2 Matrix formulation of the buckling problem of
chapter four
References
Samenvatting
Levensbericht
4.18
4.18
4.20
4.20
4.21
4.22
4.23
4.24
4.24
4.25
4.27
4.29
5.1
A2.1
A3.1
A4.1
A4.3
NOTATION
General
c eeT , eeT
' ~
{,(C
~·d t*d td {·d {·ll
det(C)
I tl v I,II
0
viii
scalar
column, transpose of the column
matrix, transpose of the matrix
vector
second order tensor, conjugate of the secor1d order tensor
inner product of two veetors
cross product of two veetors
dyadic product of two veetors
inner product of a tensor and a vector
inner product of two tensors
determinant of {
magnitude of ~
gradient operator
unit matrix, unit tensor
zero column
ehapter two and three
A
Ati•A2i
Aiik,Aiikl b
B,B1,B2
eiïk'eiikl
D
ê,ei E
fe Ge,G h
hij
sealing factor of buckling mode i
amplitude of imperfection mode i
magnitude of deflection
area of undeformed middle surface
matrices with displacement derivatives
third and fourth order coefficients in the energy function
width
matrices with displacement derivatives
third and fourth order coefficients in the I-th equilibrium
equation
property matrix
unit vector, component of unit vector in direction i
Young's modulus
column with kinematically consistent nodal forces for strip e
strip -, global geometrie stiffness matrix
section length
column with displacement derivatives
K
Mx,My,Mxy
ni N
Nt,N2 pi p
u,v,w
u,v,w,Dx u,v,w
üi,vi;wi,oi
ui
u ij u
x,y,z
x',y',z'
ix
buckling coefficient
strip -, global linear stiffness matrix
length
linear, quadratic and bilinear operator
number of sections
column resulting from orthogonality condition
number of interacting modes
bending and twisting moment per unit length
column with membrane stresses
matrix with transverse shape functions
stress matrices
energy functional of state i of the perfect structure
energy functional of the imperfect structure
load columns associated with displacement field i, ij
rotation matrix
strip thickness
unit vector, component of unit vector in direction i
displacements in x,y ,z direction
section knot coefficients
displacement fields
columns with the displacement parameters of noclal line i
buckling mode i
second order displacement field
geometrie imperfection field
local coordinate axes of the strip
global coordinate axes of the strip
section knot coefficient
column with section knot coefficients
angle
Lagrange multiplier
first variation
column with displacement parameters of the strip
global displacement column
in-plane strains
column with generalized strains
displacement fields
rotation about x-axis
~>x,~>y,~'>xy
À
).i
Àb
Ào
1J
rr
subscripts
0
11
Chapter four
A
Bo
B
c
D,D
êi eij
E
IE
f
IF
g
*
G G(O),G(l)
h
h
H
x
curvatures and twist of the middle surface
loading parameter
critica} value i of the loading parameter
lowest critica! value of the loading parameter
value of the loading parameter used in the computation of the
second order fields
Poisson's ratio
Energy functional associated with uii
column with generalized stresses
B3-spline, amended B3-spline
column with m+3 B3-splines
matrix of B3-spline functions
reference state
pretuekling path
bifurcated path
undeformed cross sectional area
boundary of the cross section
bimoment
centroid
geometrie constants
unit base vector
cartesion components of the Green-Lagrange strain tensor
Young's modulus
Green-Lagrange strain tensor
normal warping displacements
deformation tensor
warping amplitude
shear modulus
undeformed, deformed configuration
element length
height of the beam
higher order torsion constant
.. i i
I2,I3
Is J
L
Mi
i\,ni N
Ni
Pi
Pi p q
~ 10, 1
i'
s
s s ü
u V
Vo
w x
y,z
y,z a:,/3,/
xi
unit vector
second moments of area about the y,z a.xes
polar second moment of area about the shear centre
de Saint Venant torsion constant
length
moment about axis i
unit normal vector, component of i\ in direction i
normal force
shape function
component of surface traction in direction i
component of concentrated force in direction i
force vector
distributed force per unit length
body force per unit mass
beam a.xis in undeformed and deformed state respectively
radial distance of a point on the cross section to the shear
cent re
rotation matrix
components of rotation matrix
rotation tensor
are length
shear centre
global stiffness matrix
average displacement of the cross section
displacement vector of a material point
component of displacement vector in direction i
displacement components of the shear centre
straîn energy
energy functional
volume of undeformed beam
prebuckling displacement
coordinate along the beam a.xis
position vector of a material point in the undeformed and
deformed configuration respectively
coordinates along the principal centroidal a.xes
coordinates along the a.xes through the shear centre
Euler angles
V
Ç(y,z)
n p
p 0}
tP(y,z) n (')
subscripts
s
0
geometrie constauts
warping constant
xii
column with the displacement parameters of the beam
strain
curvature vector
curvature components
loading parameter
Poisson's ratio
warping function with respect to y,z axes
potential energy functional
mass densi ty
axial vector
normal stress
de Saint Venant warping function
the potential of the loads
differentiation with respect to x
shear centre
undeformed state
reference state
1.1
1 INTRODUCTION
1.1 General
Beams with thin-walled cross sections are used in many structures, such as in
buildings, aeronautical structures, glasshouses etc .. Innovations in extrusion and
cold forming techniques have enhanced the industrial utilization of these beams
significantly. The search of engineers for new structural forms, and for
refinements to these forms, to enhance their practical structural efficiency,
constantly increases the complexity of the cross sections (Figure 1.1 ).
Fig. 1.1 Typical cross sections of aluminium bearns.
The need to develop efficient and powerfut techniques to study the behaviour
of these beams is obvious. One of the most important problems faced by the
structural designer concerns their stability.
Tests, performed at the Eindhoven University of Technology [Seeverens,l982;
Winter,1983; Maquine,1983J, showed that effects which are not taken into
account in the classica! buckling analysis of beams, may have a significant
influence on the elastic flexural-torsional buckling behaviour of thin-walled
beams. These effects are.
(i) Distortion of the cross section during buckling.
(ii) Interaction between buckling modes.
(iii) Large in-plane deflections before buckling ( especially with aluminium
beams).
(i) Distartion of the cross section
In the classica! buckling analysis of beams it is assumed that the cross section
does not distort, so that a one dimensional theory (beam theory) can be used
to obtain the buckling loads. However, when the length of the beam decreases
1.2
this assumption often ceases to be valid and the distortion of the cross section
has to be taken into account too. Research into the influence of this effect
[Hancock,1978; Hancock e.a.,1980; Roberts and Jhita,1983; Bradford,1985] has
been restricted mainly to I-shaped beams loaded in pure bending. It has been
shown that this effect may lead to a significantly reduced elastic critica! load
for 1-beams of certain dimensions. Unfortunately, the methods of analysis used
in these investigations are either unsuitable for beams with other cross sections
loaded in bending andfor shear, or require an excessive amount of computer
time.
(ii) Interaction of buckling modes
When cross sectionat deformations are taken into account, the elastic buckling
modes of beams can be classified into either local or distortional [van Erp and
Menken,l987].
Local buckling is characterized by changes in the cross sectionat geometry
without overall lateral displacement or twist, while distortional buckling modes
combine lateral displacement and twist with local changes in the cross
sectionat geometry.
Simultaneons or nearly simultaneons buckling loads may result in a nonlinear
interaction between the buckling modes. The interaction between long-wave
and short-wave buckling modes has been shown to have a destabilizing
influence on the post-buckling behaviour [Koiter,l976]. Consequently,
unavoidable imperfections may significantly rednee the load carrying capacity
of thin-walled beams. Some studies which illustrate this type of interaction
were presented by Van der Neut [1969], Tvergaard [1973] and Koiter and
Pignataro [1976].
The interaction behaviour of thin-walled structural elements has received a
great deal of attention in recent years (see review in [Benito,1983]). That
research, however, was mainly restricted to structural elements loaded in
compression. Interaction between the buckling modes of thin-walled beams
loaded in bending andfor shear has, so far, been given little attention.
(iii) Large in-plane deflections befare buckling
In the classica! analysis of flexural-torsional buckling, it is assumed that the
prebuckling in-plane deformations of an initially straight beam are small
enough to be neglected [Prandtl,l899; Bleich,1952; Timoshenko and Gere,1961].
1.3-
However, due to the low Young's modulus of aluminium (E ~ 70000 N/mm2),
aluminium beams may exhibit large in-plane deflections before buckling. The
effect of in-plane deformations on flexural-torsional buckling has been
investigated by a number of research workers (see Chapter 4), but their
investigations were based on the assumption that finite but small prebuckling
deformations occurred. In the case of long aluminium beams, the in-plane
deflections may become relatively large (in the order of the beam height),
while the material remains elastic. It is questionable whether the results
obtained by other researchers in this field are still valid for these large
in-plane deflections.
In view of the many unanswered questions concerning the elastic flexural
torsional buckling behaviour of thin-walled beams with arbitrary cross sections,
a research project was started at Eindhoven Gniversity of Technology, in order
to study this subject more thoroughly. In this research project, both
experiments and computer simulations play an important role. This thesis
mainly deals with the computer sirnulations.
1.2 The adopted approach
Buckling of perfect thin-walled beams is either of the bifurcation type or of
the limit point type (Figure 1.2). In this research project, only beams which
exhibit bifurcation buckling in the perfect case, have been studied. Due to
unavoidable imperfections, the actual buckling behaviour of a beam differs
frorn that of the (hypothetical) perfect one.
Laad Laad
Dis lacement Dis lacement
Fig. 1.2 Bifurcation point. Limit point.
- 1.4-
Koiter [1945] showed that, in the case of small imperfections, this modified
behaviour is amenable to a simple pertubation-type analysis of the idealized
bifurcation behaviour. The physical insight obtained by this type of analysis is
considerably more than that from a general incremental finite element analysis.
In the latter case, no distinction is made between bifurcation and post
bifurcation regimes; in fact, the structure is modelled with certain
imperfections so that potential bifurcation points are converted into limit
points. With this approach, the buckling problem looses its special qualities
and becomes just another nonlinear analysis.
The buckling behaviour of structures that exhibit mode interaction is, in
genera!, very sensitive to imperfections. The strict study of imperfect structures
can still be made by an incremental analysis, but the great variety of possible
imperfections will require a large number of analyses in order to find the most
critica! ones and this will become prohibitive in cost.
Koiter's method, on the other hand, is particularly suitable for studying the
sensitivity of the buckling load to small imperfections.
Another principal virtue of Koiter's metbod of analysis is that the remairring
nonlinear problem is transformed into a sequence of linear problems, after the
prebuckling path has been determined. The computational effort required for
these linear problems is far less than that for a direct nonlinear analysis.
Given these advantages, Koiter's method was considered to be an attractive
alternative for studying the elastic buckling behaviour of thin-walled beams
with arbitrary cross sections.
1.3 Research strategy
Fortunately, not every type of beam is influenced to the same extent by the
effects mentioned earlier. To show this, the buckling behaviour of a perfect
thin-walled beam is shown in Figure 1.3. The horizontal axis of this figure is
divided into three regions, each repcesenting a certain class of behaviour,
namely, short beams, beams of intermediate length and long beams.
The buckling behaviour of short beams, in many cases, is strongly influenced
by plasticity effects and therefore they will not be considered bere.
Beams of intermediate length may suffer from cross sectional distortion, as
well as, the effects of mode interaction and these effects may occur
simultaneously. The magnitude of the prebuckling deformations will in general
remain small for this class of beams.
1.5 -
It is characteristic of long beams that the effects of mode interaction and
distartion of their cross section are negligible. The buckling load, therefore, can
be predicted to within engineering accuracy by beam theory. This class of
beams, however, may undergo large in-plane deflections before buckling occurs.
Buckling load
\-- Beam theory \
Local btJCkling \\ Oistortional buckling
Len th
I short beams I beams of intermediate length I long beams
Fig. 1.3 The buckling lead versus the length of a perfect beam.
Given the above divisions, the following approach was adopted.
First, a computer program was developed to determine the bifurcation loads
and the associated local and distortional buckling modes of (perfect) beams
with arbitrary cross sections. The influence of the prebuckling deformations
was neglected in this computer program (Chapter 2). Then, these modes were
used to determine the post-buckling and mode interaction behaviour of the
beams, with the aid of Koiter's general stability theory (Chapter 3). In order
to study the effects of large in-plane deflections in the case of long beams, a
general nonlinear beam theory was derived, which is applicable to beams that
undergo arbitrary large deflections and rotations. Based on this theory, a
computer program was developed in order to determine the bifurcation load of
these beams. In addition to this computer program, a second computer
program was developed to determine all the geometrie properties, which play a
role in this nonlinear beam theory (Chapter 4).
-2.1-
2 LOCAL AND DISTORTIONAL BUCKLING OF FLAT PLATE ASSEMBLIES
2.1 Introduetion
For the study of the local and distortional buckling behaviour of beams with
complex cross sections, the sections are considered to be composed of au
assemblage of flat plates. The plates are assumed to be made of an isotropic,
linear elastic, homogeneons material and to be loaded in an arbitrary way.
The finite element metbod provides a general framework for studying
distortional and local buckling of flat plate assemblies under arbitrary loading,
but for slender beams with complex cross sections the computational costs are
often extraordinary high. For beams that have constant cross sectional
properties along one a.xis, these costs can be reduced by using a finite strip
approach. The structural memher instead of being divided into a discrete
number of elements, is divided into a discrete number of longitudinal strips. In
contrast to the standard finite element method, the finite strip metbod as
developed by Cheung [1976], uses simple polynomials in the transverse
direction and continuous Fourier series functions in the longitudinal direction,
with the latter satisfying a priori the boundary conditions of the strip.
This semi-analytica! finite strip metbod has proved to be accurate ancl
efficient for analysing the buckling of prismatic structural members and
stiffened plates under compression [Plank and Wittrick,l974; Graves Smith ancl
Sridharan, 1978}, but the metbod has some disadvantages when analysing the
buckling of beams loaded in bending and/or shear.
(i) Difficulties are experienced when dealing with non-periodic buckling
modes (e.g. due to concentrated loads).
(i i) The buckling analysis of plate assemblies loaded in shear is
probiernatie [Mahendran and Murray,1986].
(iii) Thin-walled beams may have many different buckling modes and since
it is not known in advance which mode is critica!, it becomes
necessary to solve the same eigenvalue problem for different series of
Fourier terrus in order to obtain the minimum buckling load. Especially
in the case of beams with complex cross sections, this repetitive proces
may be time consuming [Mahendran and Murray,1986}.
The spline finite strip metbod recently developed by Fan [1982] replaces the
Fourier series by a linear combination of local splines while still retaining the
transverse interpolation functions. A buckling model based on these
-2.2-
interpolation functions does not suffer from the problems mentioned above [Lau
and Hancock,l986]. The number of degrees of freedom associated with this
spline model is considerably larger than for the classica! finite strip method,
but it is still about 40% smaller than that of a comparable finite element
approach. Consiclering the increased speed of modern computers, the spline
finite strip method seems to be an interesting alternative to study the local
and distortional buckling in plate assemblies which are loaded in bending
andfor shear.
2.2 Spline approximation
!?.!?.1 General
The method of splines was initiated in 1946 by LJ. Schoenberg and has sirree
found various applications. Spline approximation is a piecewise polynomial
approximation. This means that a function f(x) given on an interval a <:: x <:: b
is approximated on that interval by a function g(x) as follows.
The interval is subdivided into m subintervals with common endpoints, called
knots (Figure 2.1).
f(x) l
a I·
I I I
I I ! m •I section knots
I I I I I I
m subintervals
Fig. 2.1 Function f(x) with m subintervals.
x
The function g(x) is given by polynomials, one polynomial per subinterval,
such that at those endpoints g(xd = f(xi) and g(x) is several times
differentiable.
Hence instead of approximating f(x) by a single polynomial on the entire
interval a <:: x <:: b , f(x) is approximated by m polynomials. In this way
approximating functions g(x) are obtained which are more suitable in many
problems of approximation and interpolation. Functions thus obtained are
called splines.
-2.3
e.e.e Gubic splines
There are different spline functions available for different applications [De
Boor,l978]. From a practical point of view the cubic splines are probably the
most important ones. By definition, a cubic spline g(x) on a 5 x 5 b is a
continuons function g(x) which has continuons first and second derivatives
everywhere in that interval and, in each subinterval of that partition, is
represented by a polynomial of degree not exceeding three. Hence, g(x) is
composed of cubic polynomials, one in each subintervaL
If f(x) is given on an interval a 5 x 5 b and a partition
a = x0 < x1 < ... < ~ = b (2.1)
is chosen, a cubic spline g(x) which approximates f(x) is obtained by requiring
that
(i = O,l...,m) (2.2)
and
(2.3)
where the prime denotes differentiation with respect to x, and k0 and km are
given numbers. In most cases, however, k0 and km will represent the first
derivative of f(x) at x0 and xm.
1!.2.3 Basic cubic B3 -.<;pline
Several cubic splines with different section lengtbs and values of k0 and km
have been developed over the years [Prenter,l975]. The spline adopted by Fan
[1982] is the basic cubic B3-spline.
In the case of equidistant knots, the B3-spline approximation g(x) of f(x) is
given by
g(x) (2.4)
where the 1/Ji(x) represent the locally hill shaped B3-splines as shown in Figure
2.2 and o:1 are the coefficients at the knots which have to be determined.
A local B:r-flpline function is a piecewise polynomial that is twice continuously
differentiable and has non-.zero values over four consecutive sections, with the
section knot x = xi at the center. A local B3-spline is defined by
- 2.4-
0 X<Xi-2
(X-X i -2 )3 x i _ 2~x~xi-1
1 h3+3h2(x-xH )+3h(x-xi_ 1)2-3(x-xi-t )3 Xi-l~X~Xi (2.5) tPï =-
h3 +3h2( x i. 1-x)+3h(xi+cx)2-3(xi•cx)3 6h3 xi~x~xi•I
(x i+ 2-x) 3
xi+t~x~xi+2
0 xi+2<x.
Xi-2 h Xi-1 h Xi r. · 1· - ,.
Fig. 2.2 Basic cubic B3--spline.
The values of 1/Ji and its first and second derivative at the section knots are
shown in Table 2.1.
x = Xî-2 X = Xi-1 X = Xi x = xi•t x
tPï(x) 0 1 4 1
0 6 6 I 1 - 1 tPï(x) 0 2h 0 2h 0
I! 1 2 1 tPï(x) 0 h2 ~ h2 0
Table 2.1 The values of 1/Ji and its first and second derivative at the section knots.
xi+2
If the interval a ~ x ~ b is subdivided into m subintervals, m+3 Bs-splines
are required for g(x) to be uniquely determined. The m+ 1 knots of the
interval [a,b] are therefore extended by two additional knots outside tbe
interval (Figure 2.3). These extra knots result from the requirement (2.3).
41-1 4lo ' /
\ I
\' t\
I \ / \
2.5-
Fig. 2.3 A linear combination of local Brsplines.
2.2.4 Adapted B3 -spline representation
There are many different methods for modifying the local boundary splines in
order to satisfy the prescribed boundary conditions. In this chapter a form has
been chosen which is considered to be the most suitable in a finite strip
environment.
The knot coefficients O!i of (2.4) are strongly related to the function values
g(xi) at the knots. According to Table 2.1 g(xi) can be written as
(2.6)
The first derivative of g(x) at knot i can be expressed in terms of O!i too (see
Table 2.1)
(2. 7)
Wi th (2.6) and (2. 7) it is possible to replace the variables 0!_1, 0!0 and O!m,
O!m• 1 in (2.4) by g(x0), g'(x0) and g(xm), g'(xm) respectively. Through this
replacement (2.4) transfarms into
(2.8)
where 7/J_1 -2h 11'-1 + ~h 1/10
"i/Jo = ~~Po TPt = lP-1 - ~~Po + lPt
'Vim-1 = 1/Jm-1 ~ ;1/Jm + 1/Jm+l
'Vim = ~1/Jm
'Vim+! = - ;h1/Jm + 2h1/Jm+l'
2.6-
When tbe B3-spline representation given in (2.8) is used, the incorporation of
prescribed kinematic boundary conditions at x0 and xm into the B3-spline
interpolation is straightforward.
Equation (2.8) can be written in concise form as
g(x) = 1/JT a (2.9)
where 1/J is a column with m+3 local B3-splines.
(2.10)
and a is a column witb displacement parameters
(2.11)
2.3 General procedures in the spline finite strip metbod
The general procedures in tbe spline finite strip metbod were described by Fan
[1982], but for the sake of completeness, tbe most important features are
repeated bere.
(i) In a spline finite strip analysis, a structure is divided into a finite
number of strips such that the displacement and stress states of the
whole structure can be described in terms of tbe bebaviour of the strip
di vision lines, called 'nodal lines'. These lines are furtber subdivided into
sections, tbe lengths of whicb are equal.
(ii) The behaviour of each nodal line is described by the displacement
parameters which are tbe section knot coefficients of the B3-spline
representation of that line. Tbe behaviour of a strip is then defined by
simple polynomial interpolation of the behaviour of its two boundary
nodal lines.
(iii) The displacement function for a strip is expressed as a product of the
longitudinal spline representation and the transverse interpolation
polynomials. Consequently, the strains and stresses can be expressed in
terms of the displacement parameters througb tbe kinematica! and
2.7-
constitutive equations respectively.
(iv) Based on the chosen displacement functions, the stiffness matrices and
load columns can be obtained through the standard variational method or
its equivalences such as the virtual work principle or the principle of
minimum total potential energy.
(v) The stiffness matrices and load columns of the individual strips are
then assembied to form the global stiffness matrix and load column.
(vî) Once the displacement parameters of each nodal lîne are solved, the
displacements, strains and stresses at any location in the structure can be
calculated through the kinematica! and constitutive equations.
2.4 Section knot coefficients
For folded plate structures, the membrane action of a flat strip will affect the
bending action of its neighbouring strip and vice versa if they are not situated
in the same plane. As a result both membrane and bending characteristics
should be considered. Based on the buckling theory of flat plates, there is no
interaction between in-plane membrane and out-of-plane bending behaviour of
a single flat strip. Therefore, a flat shell strip can be modelled through a
simple combination of a bending strip and a plane stress strip.
Both lower and higher order flat shell strips may be formulated. In this
chapter a lower order flat shell strip is chosen.
Fig. 2.4 Strip with section knot coefficients.
This strip is obtained by combining a third order bending strip and a linear
(in the transverse direction) plane stress strip. Hence, each section knot has
-2.8-
four degrees of freedom which are related to the two out-of-plane
deformations w and Ox and the two in-plane displacements u and v. The
nodal lines and section knots for a strip are shown in Figure 2.4.
The strip is subdivided longitudinally into m sections using m+l knots. The
two additional knots outside the length of the strip are required to fully define
the B3-spline function over the length of the strip (see (2.2.4)). Consequently,
the total number of degrees of freedom in a spline finite strip analysis of a
folded plate structure is 4n(m+3), where n is the number of nodal lines.
2.5 Displacement functions
The displacement functions of a strip are expressed as products of the
longitudinal equidistant Br-splines and transverse polynomials as follows
N .il- + N .iJu= l'f'ui 2'f'uj
v = Nl't/JTvi + N2't/JTvi
T- N T-0 T- .,,T-0 w = N3 't/J wi + 4 't/J i + N5 't/J wj + N6 'f' j
where N1 = 1-y
N2 = y
N3 = l-3y2+2y3
N4 y(l-2y+y2)
N5 = 3y2-2y3
N6 = y(y2-y)
y t·
(2.12)
't/J represents the column given in (2.10) and üi, ..... ,0j are the displacement
parameter columns for the nodal lines i and j respectively
(2.13)
where u represents the real displacement, while ui represents a section knot
coefficient
(2.14)
The same holds for V Î'w i and oi.
-2.9-
Equation (2.12) can be written in matrix form as
u (2.15)
where N and \}1 are given in appendix 2.1 and the local displacement column 6
of length 8(m+3) is given by
(2.16)
2.6 Buckling thoory for flat plates
2. 6.1 Basic assumptions
The following assumptions are adopted.
(i) The structure is Ioaded by a dead load, in such a way that the
singularity in the load displacement curve of the associated hypothetical
perfect structure is a bifurcation point.
(ii) The prebuckling state of the perfect structure can be described by the
linear theory of elasticity.
(iii) The composing plates are made of an isotropic, linear elastic,
homogenrous materiaL
Consider the plate shown in Figure 2.5.
_Lt T
Fig. 2.5 Thin-walled plate with loca.l coordinate system.
Kirchhoff's hypotheses are assumed to be satisfied, this implies.
(i) The plate is thin.
(ii) Material lines normal to the undeformed middle surface remain normal to
the deformed middle surface.
(iii) Changes in length of these lines may be neglected.
(iv) The state of stress is approximately plane and parallel to the middle
surf ace.
2.10
Furthermore, since the deformations (shears and elongations) are negligible in
comparison to unity, the magnitude of the components of the stress tensor
referred to the undeformed configuration ( 2nd Piola-Kirchhoff stress tensor)
and referred to the deformed configuration (Cauchy stress tensor) are the
same. This group of assumptions implies that the actual three dimensional
problem may be treated as a two dimensional one, where the generalized
strains and stresses are given by
( t x ' t y ' Î xy ' /l, x ' /l, y ' /l, xy J (2.17)
(2.18)
where ~'x• ~'y and Îxy are the in-plane normal and shearing strains respectively,
K,x and K,Y denote the curvature and /l,xy for the twist of the middle surface.
The 'stresses' are defined in terms of the usual resultants, Nx= axt where ax
is the average membrane stress and t is the plate thickness, etc ..
2.6.2 Strain and curvature displacement relations
For a formulation in Lagrangian coordinates (x,y,z denote the coordinates of a
point in the undeformed configuration) the in-plane strains may be written as
~'x = ~ + H(~)2 + (~)2 + (~)2]
{y = ~ + He~? + c~? + (~)2]
[ au (}v au au (}v (}v aw aw J Îxy = TFj + ax + ax TFj + ax TFj + Ox Oy .
(2.19)
Assuming that the values of ~ and ~ are of the order of the strains, they
may be neglected compared to unity. Equation (2.19) then changes to
~'x = t + H (~)2 + (~)2 ]
( = (}v + .!.[ (au)2 + (aw)2 ] y TFj 2 ':]!i!_ Oy (2.20)
- 2.11
These relations differ from those used by von Karman by the underlined
terms. These terms, however, must be taken into account whenever a plate
undergoes in-plane buckling displacements.
The curvature displacement relations a.re taken in the linear form as
K, = x K, = y "xy = iPw -2 11Xóy'
The stra.in displacement relations can be written in short notation as
where 1 1 and 1 2 are a linea.r and a quadratic operator respectively.
a 0 0 o (~)2 (~x-)2 ax 0 a 0 (~;? 0 (~)2 w a 0
[!] 0 0 2a a
[!] 1 1(u) = oy ax 12(u) = ax7JY
()2 0 0 -1JXI 0 0 0
0 0 a2
0 0 0 7JYi 0 0
a2 -2axoy 0 0 0
Besides 11 and 12 a bilinear operator 1 11 is defined by
such that 1u(u,v) = 1 11(v,u) and 1 11(u,u) = 1 2(u)
OVt av2 + Owt Ow2 OxOx OxOx OUt au2 + Owt Ow2 OyOy OyOy aw1 aw2 + Ow! 0w2
1u(ul,~) = OxOy OyOx 0
0
0
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
- 2.12-
2.6.3 Stress strain relations
A linear stress strain relation is adopted
q =De (2.26)
where the matrix D is given by
1 IJ 0 0 0 0
IJ 1 0 0 0 0
Et 0 0 l-IJ 0 0 0 D= 1-IJ2 -r (2.27)
t2 t2 0 0 0 ï2 'TI 0
0 0 0 t2 t2
0 '12 12
0 0 0 0 0 (1-~1t2
E is Young's modulus and IJ is Poisson's ratio.
2.6.-f The perfoet structure
A dead load, controlled by a single loading parameter >., in the form q >.q0
is applied. The load q0 represents a reference load. As stated before, attention
is focussed on bifurcation problems. They are characterized by the existence of
a fundamental state of equilibrium I. An alternative solution of the equations
of equilibrium branches off from this fundamental state at the critica! point,
the so--called adjacent state of equilibrium 11. A load displacement diagram for
a bifurcation point is shown in Figure 2.6.
Dis lacement
Fig. 2.6 Bifurcation point.
A linear response before buckling is assumed. The displacement field u1
of
state I may therefore be writ ten as
- 2.13-
(2.28)
where Uo is a reference displacement field, which is assumed to be known in
the following.
The displacement field uil of a bifurcated path, for a load level À in the
vicinity of the critica! point can be expressed as
The potential energy associated with this displacement field is given by
Pil= Af~ {lf. dA- À Af u~1q0 dA (with u Df.)
where A represents the middle surface of the undeformed structure.
Equilibrium on a bifurcated path requires
where /5,/PII) represents the first variation of P11 with respect to fl.
Using (2.29), P11
can be written as
(2.29)
(2.30)
(2.31)
(2.32)
where P 1
is the poten ti al energy associated with the displacement field ..\u0.
Since P1
does not depend on fl, attention can be confined to P[q]. Using (2.28)
and (2.29), P[ q) can be written as
P[TJ] =AH ~Lt('l)TDLI(TJ) + ~ÀLt(Uo)TDL2('7)
+ ~L1( q) T DL2( q) + ~L2( 77? DL2( TJ) ] dA (2.33)
where terms quadratic in Uo and terms containing L11(1Jo,f7) are neglected,
because they are of a smaller order [van der Heijden, 1979]. Equation (2.33)
may be written in a short form as
(2.34)
where P2[q), P3[q] and P4[q] represent terms which are respectively of degree
2, 3 or 4 in 1f, terms linear in fJ are absent because the fundamental state is
an equilibrium configuration.
2.14-
The load displacement curve is assumed to be continuous so that the
post-buckling displacement 1J on a bifurcated path can be made as small as
wanted. In the neighbourhood of the bifurcation point, the higher order terms
in 1J may be neglected. The equilibrium equation in this neighbourhood may
therefore be written as
(2.35)
This equation defines a linear eigenvalue problem. The associated eigenvalues
and eigenveetors correspond to the bifurcation loa!ls and buckling modes.
Alternatively equation (2.35) can be obtained from the energy criterion for
stability [Koiter, 1945] which states that at the critica! point
(2.36)
lt is assumed that there are M buckling modes ui (i l,M) which satisfy
(2.35) for a load factor Ài. The amplitudes of ui are indeterminate and
therefore the buckling modes are normalized as
~AJL1(uiDL1(ui)dA = 1. (2.37)
Substitution of (2.37) into the first part of (2.36) will give
à..xiAJL1(u0)TDL2(ui)dA = 1. (2.38)
The M modes may be taken to be mutually orthogonal in the sense that
AJL1(uJTDL1(uj)dA = 0 i :/: j
(2.39)
i :/: j.
2. 7 Stiffness a.nd stability matrices
The displacements within a strip are described in terms of the section knot
coefficients of the nodal lines by means of interpolation functions. These
interpolation functions have been chosen so that no singularities exist in the
integrands of the functionals. Therefore, the total potential energy of the
2.15-
structure and the second variation of this energy will be the sum of the energy
contributions of the individual strips.
Therefore, equation (2.35) may be written as
Ó q( e~: ~A) [ L,( q) T DL,( q) + AL,("<>) T D!,.,(q) l dA,} = 0 (2.40)
where ns represents the number of strips.
The salution process of equation (2.40) consists of two parts.
(i) Salution of a linear equilibrium problem in order to evaluate the
reference displacement field u0.
(ii) Salution of a linear eigenvalue problem in order to evaluate the critica!
load level >..
f!. 7.1 The reference displacement .field
The contribution to the potential energy of strip e, associated with the
displacement field Uo is given by
P~ = ! JL1(u0)TDL1(u0)dAe- JlioTCJodAe. Ae Ae
(2.41)
The linear part of the generalized strains can be written in terms of a
displacement vector t5o as
(2.42)
The matrix B is derived by appropriate differentlation of (2.12) using (2.23)
and is given in appendix 2.1.
Substitution of this relation tagether with (2.15) into (2.41) yields
P~ = 45~[~ JBTDBdAebo- J-tTNTq0dAe]. Ae Ae
(2.43)
Equation (2.43) bas been derived in terms of a set of local axes (x,y,z). In
folded plate structures, however, two plates will in general meet at an angle,
and in order to establish the equilibrium of nodal forces at nodal lines
common to non-coplanar strips, a comm~n coordinate system is required.
Therefore a set of global axes is introduced.
A plate strip inclined at an angle {J to the global axes is shown in Figure 2. 7.
- 2.16-
x
z Fig. 2.7 Transformation from global to local coordinates.
The displacements /i0 in the local coordinll~te system are related to those in the
global system lió by
/i0 = R é0 (2.44)
where R= [! ~] I 0 0 0
and ii= 0 cl si 0 c = cos fJ 0 ~si cl 0 s = sin {J
0 0 0 I
I is a (m+3)x(m+3) identity matrix.
In the finite strip method, both the in-plane and out-of-plane displacement
components have the same variation in longitudinal direction and a rotation of
the coordinate axes y and z will consequently not affect the compatibility of
displacements at the nodal lines. This point stands out in favour for the finite
strip method, since many flat shell finite elements use different order
polynomials for the v and w displacement components. Even if bath v and w
are compatible displacements when examined individually, after transformation
they wil! be combined in a certain proportion depending on the direction
cosines of the element considered and compatibility of displacements wil! in
general be lost.
Substitution of (2.44) into (2.43) yields:
(2.45)
For a structure in an equilibrium state, the first variation of the potential
- 2.17-
energy functional equals zero.
The first variation with respect to the global section knot coefficients, for strip
e can be written as
(2.46)
where fe represents the column with kinematically consistent nodal forces for
the strip and Ke is termed the strip stiffness matrix.
The summation of the terms in (2.46) over all the strips, when equated to
zero results in a system of equilibrium equations for the complete structure.
The solution of these equations defines the reierenee displacement field u0•
2. 7.2 Eigenvalue model
Once the displacement field u0 is known, the reierenee stresses can be determined using the following equation
(2.47)
Substitution of this equation, tagether with (2.42) into the integral of (2.40)
gives for strip e
(2.48)
where ó represents the displacement vector and u,v,w are the displacements of
the additional displacement field '11·
The second part of this equation can be written in matrix form as
The columns [~ ~]T and [t *]T are expressed in terms of 6 by
- 2.18-
(2.50)
where B1 and B2 are obtained by appropriate differentiation of (2.12). Both
matrices are presented in Appendix 2.1.
The stress matrices are written in short form as
(2.51)
Substitution of (2.50) and (2.51) into (2.48) yields
P~ = ~bT[ JBTDB + À(B1TN1B1 + B2TN2B2)dAe] b. Ae
(2.52)
In terms of the global section knot coefficients, P~ can be written as
P~ = ~h'T[RT J(BTDB + À(B1TN1B1 + B2TN2B2)]dAe R] b'. Ae
(2.53)
According to (2.36), the first variation of P 2(q] will be zero at the critica!
point. Performance of this variation process for strip e, with respect to the
global section knot coefficients of the strip, results in
!!ft,= [RT J(BTDB + À(B1TN1B1 + B2TN2B2)]dAe R] 8' Ae
= (Ke + ,\Ge) h' (2.54)
where Ke is the linear stiffness matrix of strip e (see 2.46) , À is the toading
parameter and Ge is termed the geometrie stiffness matrix of strip e.
The summation of (2.54) for all the strips, when equated to zero, results in a
linear eigenvalue problem for the complete structure.
2.8 Solution process
!!.8.1 Equilibrium model
The matrix Ke and column fe can be obtained analytically. The integrations
needed for detemining Ke are separated into integrations which only contain
the products of the transverse polynomials and their derivatives, and
integrations with only coupling terms of B3-spline expressions as integrands.
- 2.19
Integrations of the polynomials can be done easily, while the integrations of
the B3-spline expressions may be reduced to summation operations on
standard integration tables which are functions of the section length h only
[Fan,l982]. The resulting system of linear equations is solved using a skyline
algorithm for Gauss decomposition.
~.8.~ Eigenvalue model
Once the displacements due to the reference load are known, the reference
stresses at each point can be determined by using (2.47). Since an arbitrary
prebuckling stress distribution is allowed, numerical integration is used to
determine the geometrie stiffness matrix Ge of a strip. The integration over
the area of the strip is subdivided into m integrations, one for each subregion
with length h. A 3x3 Gauss quadrature is used for each subintervaL
The eigenvalues and eigenveetors of the resulting generalized eigenvalue
problem are obtained using a subspace iteration procedure.
2.9 Numerical examples
The spline finite strip buckling model described in this chapter is capable of
solving many of the bifurcation problems associated with thin-walled plate
assemblies. An extensive treatise on its applications is beyond the scope of thls
thesis, and therefore the method is illustrated by just a few examples.
The accuracy of the solution is governed by the number of strips and sections
used in the analysis; their influence is evaluated, so that some general
guidelines are available for providing results for a desired level of accuracy. To
be able to compare the results with analytica! ones, the examples are
restricted to memhers with simply shaped cross sections.
~.9.1 Symmetrie I-column loaded in compression
The first example to be considered is that of a symmetrie I-column loaded by
a uniform compressive stress. The dimensions, loading conditions and strip
configuration are shown in Figure 2.8. The critica! stress of a compressed
column is given by Allen and Bulson [1980] as
~Et2
IJc = K 12(1-v2)h2 (2.55)
where the buckling coefficient K depends on the boundary and loading
conditions and h represents the distance between the flanges.
-r---1~!!L~ Geometry
---;A; ~-l __ 1500m4
laadirq condition
-x
2.20
1 1
-2 3
2
2 1
strip contiguration
L w(O) = w(L) = Bx(O) = Bx(L) 0
2. v(O) v(L) w(O) w{L) Bx(O) = Bx(L) = 0
3. u(O) = v(O) = v(L) = w(O) =: w(L) = Ox(O) = Ox(L)
Fig. 2.8 Geometry, loading condition and strip conflguration of the l~olumn.
0
The analytically obtained buckling coefficient for the present I-column equals
2.64. The numerical results for different numbers of sections and strips are
presented in Table 2.2 and 2.3. The buckling mode of the column, obtained
with 12 strips and 10 sections is shown in Figure 2.9. For the sake of clarity
both the strip and section subdivision are shown.
nr. of strips nr. of strips nr. of buc k l. coef. buckl. coef. in flange in web sec tions numerical analyt i cal
4 2.66
6 2.65 4 4 2.64
8 2.64
10 2.64
Table 2.2 Buckling coefficients of an uniformly compressed I-column for a different number of sections.
nr. of strips nr. of strips nr. of buc k I. coef. buckl. coef. in flange in web sections numerical analyt i cal
2 2 2.66
4 4 10 2.64 2.64
6 6 2.64
Table 2.3 Buckling coefflcients of a.n uniformly compressed I-column for a different number of strips.
- 2.21-
Fig. 2.9 Buckling mode with a cross sectiona.l view at midlength of an uniformly compressed l~olumn.
!!.9.!! Plate strip loaded in pure shear
In this example, the behaviour of a plate strip in pure shear is evaluated. The
geometry and loading conditions are shown in Figure 2.10.
-t {
- - -i~------------------~ ---1000mm
t=1mm -I
Fig. 2.10 Plate strip loaded in pure shear.
The critica} shear stress of a plate is given by
rEt2
(axy)ç = K 12(1-v2)h2 (2.56)
where b represents the width of the plate. The buckling coefficient K of the
present plate strip equals 5.38 [Allen and Bulson,l980J. The numerical results
are presented in Table 2.4 and a contour plot of the out-of-plane
displacements, obtained with 5 strips and 30 sections, is shown in Figure 2.11.
of numerical buckl. coef. analyt i cal nr.
of se ctions strips nr. buckl. coef. 10 20 30
5 5. 67 5.40 5.39 5.38
10 5. 65 5.39 5.38
Table 2.4 Buckling coefficients of a plate strip in pure shear.
- 2.22
Fig. 2.11 Contour plot of the out-of-plane displacements of a plate strip in pure shear.
The good performance of the present approach in the case of structures loaded
in pure shear is also demonstrated by Raijmakers (1988], who used the
computer program developed by the present author to study the buckling
behaviour of folded plate structures.
In the classica! flexural-torsional buckling analysis of thin-walled beams it is
assumed that the cross section does not distort, and a one dimensional theory
(beam theory) is used to obtain the buckling loads. However, if the length of
a beam decreases this assumption often ceases to be valid and the distartion of
the cross section should be taken into account too. Research into the influence
of this effect has been restricted mainly to 1-shaped beams loaded in uniform
bending (Hancock,1978; Hancock e.a.,l980; Robers and Jhita,l983; Bradford,
1985]. It has been shown that the distartion may lead to a significantly
reduced elastic critica! load for l-beams of certain dimensions. The influence of
this effect on the behaviour of other beams under different loading conditions
has received little attention so far.
The spline finite strip bnckling model described in this chapter is a very
suitable tooi for stndying the influence of cross sectional distartion of beams
with arbitrary shapes and toading conditions. This is demonstrated by the
following two examples. The first example considers a plate girder in uniform
bending and in the second example the behaviour of a T -beam, loaded by a
concentrated force at midlength, is evaluated.
2.9.3 Plate girder in uniform bending
The dimensions, loading conditions and strip configuration of the girder are
shown in Figure 2.12. Girders of 3, 5, 10 and 20 m. are evaluated for a
different number of sections and 20 strips. The numerical results, obtained
with a Young's modulus of 2.1E5 Nfmm2 and a Poisson ratio of 0.3, are
presented in Table 2.5, tagether with some analytica! results. The first
analytica! valnes are obtained using the approximate method of Hancock e.a.
[1980), and the second using bearn theory (which neglects the influence of the
distortion) (Timoshenko and Gere,1961]. The metbod of Hancock e.a. yields
2.23-
very good results, except for beams with a large heigth/length ratio, where too
conservative values are obtained. The cross sections of the buckling modes are
shown in Figure 2.13. The results clearly demonstrate the increasing influence
of the distartion with decreasing length.
+--1-Geometry
(~rr:--. ------..Ä~_! ;1---·---'----t-
loading condition
z 2 2
1 1
1 2 1
Strip contiguration
l. w(O) = w(L) = llx(O) = llx(L) = 0
2. v(O) = v(L) = w(O) := w(L) = llx(O) = llx(L) :: 0
3. u(O) = v(O} = v(L) = w(O) = w(L) = llx(O) = llx(L) = 0
Fig. 2.12 Geometry, loading condition and strip configuration of the plate girder.
gi r der nr. of
numerical analyt i cal beam
Jen gth strips
nr. ofsections Hancock theo r y
fml 4 6 10 e.a.
3 8.92E7 8 .87E7 8.87E7 7.64E7 9. 78E7
5 3.80E7 3 .78E7 3. 78E7 3. 74E7 4.15E7 20
10 1.52E7 1.50E7 1.50E7 1.50E7 1.58E7
20 0.73E7 0 .70E7 0. 70E7 0. 70E7 0. 70E7
Table 2.5 Buckling moments in Nmm for a plate girder in uniform bending for a different number of sections and 20 strips.
3m 5m 10m 20m Fig 2.13 Cross sectional views at midlength of the buckling modes for
plate girders of different lengtbs in uniform bending.
- 2.24
8.9.-f T-beam loaded by a concentrated force
In this example, the buckling behaviour of T-beams of different lengtbs is
evaluated. In addition to the global buckling mode, the local mode with the
lowest critica! load is considered too. The geometry, loading condition and
strip configuration are shown in Figure 2.14. Beams with lengths of 550, 500
and 450 mm are evaluated. The numerical results obtained with a different
number of sections and 15 strips are presented in Table 2.6. The buckling
modes determined with 30 sections are shown in Figures 2.15 - 2.20.
I• 38mm .. 1
1mm
1mm
Ge ometry
E E
IJl
"'
Loading condition
1111121111 I I I I I I I t I I
2
3
2
2
2
2
10 strips in flan ge
5 strips in web
E 70960 N/mm2 11 "' 0.321
L w(O) == w(L) == IJx(O) lix(L) = 0
2. v(O) == v(L) = w(O) = w(L) = lix(O) = lix(L) = 0
3. u(O) v(O) v(L) = w(O) = w(L) = lix(O) = lix(L) = 0
Fig 2.14 Geometry and loading conditions of the T-beams.
nr. of nr. of 1st buckling load 2nd buckli n g load
strips sect i ons x 103 N x 103 N
550 mm 500 mm 450 mm 550 mm 500 mm 450 mm
10 1.88 2.31 2.80 2. 72 2.88 3.13
15 20 1.87 2.30 2.74 2.48 2.68 3.01
30 1.87 2.30 2. 73 2.45 2.65 2.99
Table 2.6 Buckling loads of T-beams with different lengths.
Figure 2.15 and 2.16 show that, for the beam of 550 mm, the first and the
second mode can be classified as a global and a local buckling mode
respectively. When the length of the beam decreases, the cross section of the
'global' mode starts to distort near the concentrated load. This local
2.25-
Fig. 2.15 First buckling mode of a T-beam with a length of 550 mm.
Fig. 2.16 Second buckling mode of a T-beam with a length of 550 mm.
Fig. 2.17 First buckling mode of a T-beam with a length of 500 mm.
2.26
Fig. 2.18 Second buckling mode of a T-beam with a length of 500 mm.
Fig. 2.19 First buckling mode of a T-beam with a length of 450 mm.
Fig. 2.20 Second buckling mode of a T-bearn with a length of 450 mm.
2.27-
distortion, produces sarnething that looks like a local buckle at midlength of
the 'global' mode. The local mode, on the other hand, exhibits increasing
overall displacements with decreasing length. As a result of these two effects,
the difference between the two types of modes varrishes as the length
decreases, so that the usual subdivision of the modes into global and local
buckling modes is no Jonger possible. Now, both modes fall into the category
'distortional mode'. This example indicates that cross sectional deformations
may have a marked influence on the buckling behaviour of thin-walled beams
loaded by concentrated forces.
2.10 Conclusions and recommendations
The previous examples demonstrate the accuracy and efficiency of the present
approach for predicting the local and distortional buckling load of thin-walled
plate assemblies under arbitrary loading. The simplicity of the semi-analytica!
finite strip method is preserved, while the problems of dealing with non
periadie buckling modes, shear and non--simple support are eliminated.
The number of degrees of freedom required for a spline finite strip analysis is
considerably larger than for the semi-analytical finite strip method, but it is
still approximately 40% smaller than that of a comparable finite element
analysis.
A great part of the computer time is required for the determination of the
eigenvalues and eigenveetors of the matrix K+AG. The replacement of the
present lower order flat shell strip by a higher order strip with one internal
line, may result in a reduction of the computer time. This, because the
number of strips needed to accurately describe the linearly varying stress
distribution of a memher in bending is greatly reduced, while the additional
displacement parameters of the internal nodal line do not add to the total
number of degrees of freedom through the process of static condensation
[Cheung,1976]. The extra computer time associated with this condensation
probably wil! be less than that gained by the reduced number of strips.
The combination in longitudinal direction of strips with different section
lengths can be easily accomplished, as demonstrated by Fan [1982]. Extending
the present computer program with this option will result in a reduction of
the computer time for structures with concentrated short wave local buckles.
The present approach uses a 3x3 Gauss quadrature per section length to
2.28-
construct the geometrie stiffness matrix. However, orientating calculations
indicate that a lower order Gauss quadrature may in many cases also lead to
satisfactory results.
The following quidelines can he used to determine the number of sections
needed in a. buckling analysis.
(i) Approxima.tely two sections per half wave are required in order to
describe a local buckling mode to within engineering accuracy.
(ii) The number of sections neerled for modes, with dominating overall
displacements, ranges from 4 to 10 (the minimum number of sections,
possible, equals 3).
-3.1-
3 INTERACTION BETWEEN BUCKLING MODES
3.1 Introduetion
In thin-walled plate assemblies, both local buckling of the plate elements and
global buckling of the whole structure is possible. The occurrence of
simultaneons or nearly simultaneons buckling loads may result in an
interaction between the buckling modes. The interaction between long wave
and short wave huckling modes has been shown to have a destabilizing
influence on the post-buckling behaviour [Koiter,1976]. Consequently,
unavoidable imperfections may reduce the load carrying capacity of thin~walled
structures significantly.
The interaction behaviour of thin-walled structural elements loaded in
compression has received a great deal of attention. The first detailed
investigations of the interaction between global and local buckling of a column,
are due to Van der Neut [1969] and Graves Smith [1969]. Van der Neut
created a simple mechanica! model of a column, whose two plate flanges were
capable of independent local buckling. This model exhibited a rather strong
interaction with overall buckling, resulting in a marked sensitivity to
imperfections. Graves Smith studied a square tube and the interaction
appeared to be of minor importance. Several other authors, notably Koiter,
Tvergaard, Pignataro, Sridharan, and Hancock contributed to the further study
of compressed members.
Research into interaction buckling of memhers loaded in bending andfor shear
has so far received little attention. Cherry [1960] presented a simple analytica!
model and test results for beams in uniform bending, whose compression
flanges had prematurely buckled locally. More recent studies were made by
Reis and Roorda [1977], Wang e.a. [1977] and Bradford and Hancock [1984].
There are basically two strategies for studying interaction buckling.
(i) The stiffness of the locally buckled memher is calculated first, and then
this stiffness is used to evaluate the overall buckling.
(ii) The analysis of the interaction is performed on the basis of the general
Koiter theory [Koiter,l945].
The studies of interaction buckling under bending by Cherry, Reis and Roorda,
Wang e.a. and Bradford and Hancock, belong to the first category. In all these
cases the concept of the effective width was used to account for the post-
3.2-
buckling stiffness of the locally buckled plate component. Koiter, Tvergaard,
Pignataro, Sridharan and Benito used the second approach.
Although the first approach is very popular among engineers, and yields
reliable results for a variety of cases, it is not suited to properly explain the
mechanics of the interaction phenomena. On the other hand, application of
this approach, if possible, in the case of structural elements with complex cross
sections is very difflcult. The second, more fundamental, approach is not only
applicable to every type of structure but is also much more suited to obtain
an insight into the interaction phenomena.
Koiter's pertubation approach was first publisbed in 1945 (in Dutch), but it
remairred relatively unknown until 1960. The theory considers conservative
systems, which exhibit bifurcation buckling in the perfect case. The system is
described by a potential energy expression, that is expanded in integrals of
functions of the displacements and their derivatives. A bifurcation point is
identified with the vanishing of the quadratic term of the potential energy. In
order to obtain this point, a linear eigenvalue problem has to be solved. The
displacement field for the post-buckling equilibrium configuration is
decomposed into the buckling mode, multiplied by a sealing factor, and a
residual displacement, orthogonal to the buckling mode. The residual
displacement is computed by making stationary the increase of the potential
energy for a fixed value of the amplitude of the buckling mode. This field is
quadratic in the amplitude to the Jo west order of approximation. Thus the
potential energy is reduced to an algebraic function of the buckling mode
amplitude. The equilibrium path in the vicinity of the bifurcation point is
obtained by requiring the first derivative of this function to be zero.
Due to unavoidable imperfections, the buckling behaviour of the actual
structure differs from that of the hypothetically perfect one. By including only
the first order effect of smal! initia! deflections, asymptotically exact estimates
of the ultimate load of an imperfect structure can be obtained through a
simple pertubation-type analysis of the ideal bifurcation behaviour. Koiter also
modified his metbod for the case of simultaneons buckling loads.
The pertubation technique of Koiter is exact in an asymptotic sense. lts range
of validity, however, may be quite smal!, in partienlar if the fundamental state
has higher bifurcation points .-\2, .-\3 etc., close to the lowest bifurcation point
.-\ 1, whose associated buckling modes couple with the critica! modes at .-\ 1• A
first step to overcome this difflculty was already suggested by Koiter !1945].
-3.3-
The result of this modification is a replacement of the potential energy
function by another function in the amplitudes, with a more complicated
dependenee on the load factor À. The general approach presented by Byskov
and Hutchinson [1977J is, in essence, a reformulation of this modified method.
Unfortunately, no general criterion seems to be available to access the accuracy
of the refinement. The method, however, has been applied with success to
several buckling problems with nearly simultaneous buckling loads [Koiter,1964;
Byskov and Hutchinson,1977; Byskov,l979; Benito,l983J.
Koiter's approach has been transferred into the framework of the finite element
method by several authors [Lang and Hartz,1970; Haftka e.a.,l971; Carnoy,
1980]. In axially compressed prismatic plate structures, however, it is possible
to describe the buckling modes and the residual displacement field in terms of
harmonie functions of the axial coordinate. This gives rise to a semi-analytica!
approach with the discretization confined to the transverse direction only. The
classica! finite strip technique has been successfully employed in this context.
The combination of Koiter's method and the classica! finite strip technique is
capable of analysing the interaction between simultaneous and nearly
simultaneous buckling modes in the preserree of initia! imperfections for
structural members with arbitrary cross sectionat profiles [Sridharan and
Benito,1984; Pignataro e.a.,1985; Ali,l986; Kolakowski,1987].
Although the publisbed numerical approaches based on the above combination
has proved to be simple and effective, they all suffer from one or more of the
following limitations.
(i) Only structures loaded by an ax:ial stress distribution can be analysed.
(ii) Distartion of the cross section in the global buckling mode is not
accounted for.
(iii) Localized non-periodic buckling modes are very difficult to describe.
(iv) The result is sensitive to the choice of harmonies which are considered
in the analysis.
Due to these limitations, these approaches are less suited to study the
interaction buckling of members loaded in bending andfor shear. However, it is
clear that this type of interaction will be of great importance in the design of
thin walled beams. Therefore, in this chapter, a metbod is presented which
combines the spline finite strip method of Chapter 2 with the stability theory
of Koiter. This combination does not suffer from the limitations mentioned
-3.4
earlier and therefore, it is suitable for studying the interaction buckling of
compressed members, as well as, of memhers in bending and/or shear.
Recently, some interesting papers were published which showed that, for an
accurate evaluation of the interaction between local and global buckling modes,
sometimes, more than one local mode should be taken into account
[Sridharan,l983; Sridharan and Ali,l987; Pignataro and Luongo,l987]. The
spline finite strip model of this chapter, therefore has been designed in such a
way that an arbitrary number of buckling modes can be considered in the
interaction.
-3.5-
3.2 Initial post-buckling theory for simultaneons and nea.rly simultaneons
hDckling modes
3.:U Perfoet structure
Suppose that there are M simultaneons or nearly simultaneons interacting
modes Di (i=1,M) which satisfy (2.35) for a load factor \· The lowest of the
M eigenvalnes is called ..\b.
Following Koiter's theory, the post-buckling displacement field is expanded in
the form
n = a.u. + v (*) 'f 1 I (3.1)
where ai is a measure for the 'amount' of buckling mode ui which is contained
in q, and v is a displacement field orthogonal to the buckling modes Di in the
sense of (2.39).
In the sequel, the load factors ..\i and the buckling modes ui are assumed to
be known and orthonormalized according to (2.37)-(2.39). The displacements 1f
from the fundamental state are assumed to be small, so that aiui and v are
also smal!.
Substitution of (3.1) into the potential energy functional of the bifurcated path
(2.33) and neglecting terms of higher order of smallness yields (appendix 3.1)
M ..\ J[ 1 T E (1-x )a1a1 + 2L1(v) DL1(v) I=l I A
+ ~aiajL1(v)TDL11(ui,uj) + ~aiajakL1(ui)TDL11(uj,uk)
+ ~aiajaka1L11 (ui,ujfDL11(uk,u1 )] dA.
(3.2)
Equilibrium configurations are characterized by stationary valnes of the
potential energy functional (3.2). The equilibrium equations are derived in two
steps. First, the stationary valnes of (3.2) are determined for arbitrary
constant valnes of ai. By this condition the dependenee of the function v on
the parameters ai is determined. By substitution of this relation into equation
(*) Throughout this chapter, except in section 3.3, a repeated lower-case index wil!
denote summation from 1 to M, unless it only appears within the operators L.
A repeated upper-case index is not to be sumrned unless indicated.
-3.6-
(3.2) the energy will he known as a function of ai; P(ai,.X.). The values of ai
for which stationary values of this function are obtained and the corresponding
functions v then yield the displacements for the equilibrium configuration.
The terms in (3.2) which contain the displacement field v are
AH ~L 1(v)TDL 1(v) + ~,X.L1(u0)TDL2(v)
+ aiajL1(ulDL11(uj,v) + ~aiajL 1(v)TDL11(ui,uj)]dA. (3.3)
The first two terms in (3.3) are quadratic in v and their sum is positive
definite (under the orthogonality conditions for v) for 0$-X.<.X.m+t [Koiter,1945]
where Àm•t>Àm (.X.m is the load factor associated with bockling mode M). The
other terms are linear in v. The functional P[ai,v,.X.] may therefore he
minimized with respect to v, at least for sufficiently small fixed values of the
amplitudes ai of the bockling modes. This results in
AH L1(v)TDL1(év) + .X.L1(u0)TDL11(v,óv)
+ aiaj{L1(ulDL11(uj,év) + àL1(év)TDL11(ui,uj)}]dA = 0 (3.4)
with the orthogonality conditions
(i=l,M).
Due to the orthogonality restrietion for óv, (3.4) is not yet equivalent to a
system of differential equations and boundary conditions for the functions v. In
order to obtain this eqcivalence, the functions óv are replaced by
kinematically possible functions óu, which are not subjected to the ortho
gonality condition. These latter functions can he written as
bu = tiui + óv with AJL1(ui)TDL1(év)dA 0 (i=l,M). (3.5)
Substitution of the first part of (3.5) into the second part and using (2.37)
yields
ti= àAJL1(uiDL1(óu)dA (i=l,M). (3.6)
Substitution of (3.5) and (3.6) into (3.4) yields
-3.7-
Af[L1(v?DL1(óu) + H 1(u0?DL11(v,óu)
+ aiadL1(uiDL11(uj,.5u) + ~L1(óu)TDLu(ui,uj)} (3.7)
- aiajtk{L1(ui?DL11(uj,uk) + ~L1(uk)TDL 11(ui,uj) }]dA = 0
(i=l,M).
Due to the linearity of v in (3.4), the solution can be written in the form
(3.8)
The fuiflilment of (3. 7) for arbitrarily admissible variations óu, requires the
independent varrishing of the coefficients of the parameters ai. Therefore, for
each i and j, (3.7) decouples to
AJ[L1(uij)TDL1(óu) + AL1(Uo)TDL11(uij•8u)
+ HL1(ui)TDL11(uj,óu) + L1(u/DL11(ui,fu)
+ 11( óu)TDL11(ui,uj)} àtk{L1(uiDL11(uj,uk) (3.9)
+ L1(uj)TDL11(ui,uk) + L1(uk)TDL11(ui,uj) }]dA 0
(k=l,M).
Note that (3.9) is written in a form which is symmetrie in ui and ui, resulting
in
(3.10)
Other equivalent expressions for (3.9) are possible, but due to the symmetry
condition mentioned above, the number of displacement fields to compute is
considerably reduced. The fields uij are called second order fields.
In the computer program, which will be discussed later, the displacement fields
uij are obtained through a minimization process of a functional which can be
deduced from (3.9)
-3.8
II[uîi"\] = AJH L1(uîj)TDL1(uij) + .XL1(u<lDL2(uij)
+ {L1(uiDL11(ui,uij) + L1(ui?DL11(ui,uij)
+ L1(uii)TDL11(ui,ui) }]dA
with the condition AJL1(uk)TDL1(uij)dA = 0 (k=l,M).
(3.11)
Inserting the orthogonality condition is carried out using Lagrangian
multipliers, leading to the corrected functional
ft[uii,/Jk,.X] = AJHL1(uii)TDL1(uii) + .XL1(u<lDL2(uii)
+ {L1(uiDL11(uj,uij) + L1(uj)TDLu(ui,uii) (3.12)
+ L1(uij)TDL11(ui,uj)}]dA + /JkAJL1(uk)TDL1(uii)dA
where {Jk represents a Lagrangian multiplier.
Requiring ft[uii,{Jk,À] to be stationary, for constant values of À, results in
AJ[L1(uij)TDL1(8u) + ÀL1(Uo)TDL11(uij•óu)
+ ML1(uiDL11(ui,8u) + L1(ui)TDL11(ui,8u) (3.13)
+ L1(óu)TDL11(ui,ui)}]dA + {JkAJL1(uk)TDL1(óu)dA = 0
together with
AJL1(udTDL1(uij)dA 0 (k=l,M).
The Lagrangian parameters are obtained by putting óu
(3.13), leading to uk (k=l,M) in
-3.9-
/3k = - iAJ[L,(uJDL11(uj,uk) + L1(uj)TDL11(ui,uk)
+ L1(uk)TDL11(ui,uj)] dA.
Substitution of (3.14) into (3.13) again produces (3.9).
(3.14)
Once the displacement fields uij have been obtained, the potential energy is a
function of ai and À only. To write this function in a concise form, bv in (3.4)
is replaced by v, leading to the following equation
Af[L,(v)TDL1(v) + H 1(u0)TDL2(v)]dA =
- aiajAf[L,(uJDL11(ui,v) + ~L 1(v)TDL11(ui,uj)JdA. (3.15)
Substitution of (3.15) tagether with (3.8) into (3.2) converts the potential
energy functional into the form
(3.16)
where Aijk and Aijkl are identified by
Aijkl = i Af[ L1(uJDL11(uj,uk1) + L1(uj)TDL11(ui,uk1) (3.17)
+ L1(uk1)TDL11(ui,uj) + à L11(ui,u/DL11(uk,u1)] dA.
By requiring ?ai= 0, the equilibrium equations to compute the values of ai,
are obtained. This results in
(I = 1,M) (3.18)
where cl and cl are given by jk jkl
-3.10-
cljkl i AH Ll(u/DLu(uj,ukl) + Ll(uj)TDLu(~,ukl)
+ L1(uk1)TDL11(upuj) + ~ L11(ui'ui)TDL11(uk,u1)
+ L1(uj}TDL11(uk,u1) + L1(uk}TDL11(upu11) (3.19)
+ L1(~/DL11 (uj,uk) + ~ L11(uj,uk)TDL11(upu1)]dA.
9.2.2 The injluence of smaU geometrie imperfoctions
The buckling of a structure having a Iinear prebuckling state is always of the
bifurcation type. Unavoidable irregularities in the actual structure, such as
geometrie imperfections, will result in a nonlinear behaviour before buckling
and possibly premature buckling as wel!.
In the case of small imperfections, the imperfect structure wil! have an
equilibrium state, whose displacements will differ slightly from the
displacements >.tto of the perfect structure. The total displacement of the
imperfect structure therefore may be written as
u = >.tto + (. (3.20)
The primary effect of initia! imperfections is that the fundamental state of the
perfect structure, described by the displacement field >.u0, wil! not represent an
equilibrium configuration of the imperfect structure. With equation (3.20), the
potential energy functional associated with the displacement field u can be
written as
P = P1 + E[<J (3.21)
where P 1 does not depend on (.
The displacement field ( for which u is an equilibrium field must satisfy
(3.22)
Confining attention to smal! geometrie imperfections, the difference between
the unloaded perfect and unloaded imperfect structure may be described in
terrus of a displacement field !!: To simplify the comparison between the
behaviour of a perfect and imperfect structure, the unloaded perfect structure
is chosen as the reference state for the imperfect structure too. The part E! (] of the potential energy functional may then be written as
- 3.11 -
f.[ (] = ~ AJ[ ( (2- !_lD( (2- !_) - ((I- !_lD( (1- !_)] dA - À AI (T QodA (3.23)
where A is the undeformed area of the perfect structure, and i' c1 and c2 are
respectively given by
1 i = Ll!!) + 2Lz( !!_)
1 c1 = L1(!!_+Àu0) + 2L2(!!_+.Xu0) (3.24)
1 c2 = LI(!!_+ÀUo+() + 2Lz(!!.+Àuo+().
For small geometrie imperfections !!_, the displacement field ( will also be
small, so that, for a first approximation of the difference between the response
of the perfect and imperfect structure, the energy functional f.[ (] may be
written as [v.d. Heijden, 1979]
(3.25)
where P[(] is equivalent to the functional P[7J] of the associated perfect
structure, given by (2.33).
The further analysis is restricted to geometrie imperfections which are of the
same pattem as the buckling modes ( among these geometrie imperfections are
the most harmful ones when the buckling modes coincide [Koiter,1974]).
u= a.u. - _) 1
where ~i is the amplitude of the 'imperfection mode' i.
(3.26)
In the vicinity of the bifurcation point of the perfect structure, the
displacement field ( may also be expressed in terms of the buckling modes of
the perfect structure
(3.27)
where ui represents a buckling mode, ai is the amplitude of mode i, and w is
a displacement field which is orthogonal to all the buckling modes in the sense
of (2.39).
Substitution of (3.26) and (3.27) into (3.25), neglecting terms of higher order
- 3.12-
of smallness and using the orthogonality conditions, yields
~[(] = P[ai,w,>.] + a~j AJ >.L1(no?DL11 (ui,uj)dA
where P[ai,w,>.] is given by (3.2).
(3.28)
With the equations (2.37)-(2.39) this functional can be simplified further into
(3.29)
Camparing (3.29) with (3.2) shows that the lowest-order influence of geometrie
imperfections can be obtained by adding the following term.
M ). E 2-x~!l:I
1=1 I
to the potential energy functional of the perfect structure.
(3.30)
Since the additional term in (3.29) does not contain the displacement function
w, the dependenee of the function w on ai is the same as the dependenee of
the function v, of the perfect structure, on ai. The potential energy functional
of the imperfect structure therefore may be written as
The field quantities Aijk and Aijkl are the same as those for the perfect
structure and are given by (3.17).
The equilibrium equations of the imperfect structure are obtained by requiring
the first derivative of (3.31) with respect to ai, to be zero.
The equilibrium equations (3.18) are then modified to
(3.32)
The field quantities C1jk and C1jkl are also unchanged and are given by (3.19).
- 3.13-
3.3 Matrix formulation and computer implernentation (*)
9.9.1 General
In order to determine the equilibrium paths for the structure, the coefficients
Aijk• Aijkl• Ch and C1jkl should be known. For determing Aijk and C1.k only
the buckling ihodes u; are needed, while for the determination of A;/kl and
C1jkl the secoud order displacement fields uii have to be known too. The
ingredients needed to calculate these secoud order fields are the buckling
modes and the reference displacement field Uo· Both these ingredients can be
calculated with the spline finite strip model of chapter 2. The same approach
can also be used to determine the secoud order displacement fields uij as wel!
as the coefficients mentioned above.
9.9.2 Detennination of the second order displacement ftelds
In the following it is assumed that the reference displacement field u0 and the
buckling modes u; are known in terms of the section knot coefficients.
Since the interpolation functions are chosen so that no singularities exist in the
integrands of the functionals involved, the secoud order fields can be
determined by requiring the following functionals to be stationary (see (3.12))
fi[u;j,,Bk,>.] = ~s { IH L1(u;i)TDL1(u;i) + >.L1(u0)TDL2(u;i) e=1 Ae
+ L1(u;)TDL11(uj,uij) + L1(uj)TDL11(u;,U;j) (3.33)
Representing the discretized strip displacement of the field U;j in the local
coordinate system by the displacement column 5;j and in the global coordinate
system by 5jj, the first two terms of (3.33) can be written in matrix form as
(see (2.41 )-(2.53))
~ I L1(u;j)TDL1(u;j)dAe = ~5iT[ RT I BTDBdAe R] 5jj Ae Ae
(3.34)
(*) Note, in this section the summation convention is dropped.
~ J >.L1(u0fDL2(uij)dAe = Ae
- 3.14-
~6{1[ RT J >.(BiN1B1 + B1N2B2)dAe R] ó{j· (3.35) Ae
The matrices involved in these two terms are the same as those for the
eigenvalue model in chapter 2.
The third term of (3.33) can be expressed in terms of the displacements as
(3.36)
where u,v,w, and ui,vi,wi are the displacement components in the local
coordinate system of uij and ui respectively, and the membrane stresses NL, N~i and N~yi are given by
with
N!i = C (~i + ~i)
N~i C (~i + {iï) NI . = C 1-v (&ui + /Jvi)
XYI ay 0X
Et C = l-v2·
The righthand side of equation (3.36) may be written in matrix form as
(3.37)
~A)~[~~][ ~i 0 0 l + [~~][~i 0 ~i ]] [ ~~~ ldAe. (3.38)
0 &u. 0 0 iJw. i}w. Yl ;J;;J "!l:":"J "!l:":"J I v,y uy uX Nxyi
With the abbreviations
[
~j 0 0 l 0 ~j 0
(3.39)
- 3.15-
(3.38) may be written in terms of the global coordinate system as
For the fourth term of (3.33) holds in the same way
~ JL1(ui)TDL11(ui,uij)dAe Ae
With the notation of (2.42), the fifth term in (3.33) can be written as
~ JL1(uij)TDL11(ui,uj)dAe = Ae
öjT[~RT A J(BTDhij)dAe] = öj}q~i e
where hij represents the following column
(3.40)
(3.41)
(3.42)
With the notation of (2.42), the terms of (3.33) which contain a Lagrangian
multiplier, change to
M J T E {Jk L1(uk) DL1(uii)dAe k=1 Ae
öj}k~/k[RTA J(BTDB)dAe Rök] e
(3.44)
where ök represents the column with the strip displacement parameters of the
buckling mode k with respect to the global coordinate system.
Using the notations introduced above, the contribution of strip e to fi, can be
written as
(3.45)
3.16
Performance of the variation process for strip e, with respect to the global
displacements ó{j and the Lagrangian multipliers fJk gives
and
M == (Ke + ÀGe)ó{j + q~j + E fJkm~
k==l
(k==l,M).
(3.46)
(3.47)
For equation (3.33) to be stationary, both the summation of (3.46) and the
summation of (3.47) over all the strips must equal zero. These requirements
results in the following system of linear equations
K+ ÀG
(3.48)
............ T··········:· ....... . ml . . T 0 0
mm
where K is the global stiffness matrix, G is the global geometrie stiffness
matrix, À is the loading parameter, i\ ij is the global displacement column of
the field uij and qij is a global 'load' column.
The matrices K and G are the same as those of the eigenvalue model in
chapter 2. The component qii is calculated using numerical integration. In the
same manner as in chapter 2, the integration over the total area of the strip
is carried out with m separate integrations, one over each section with length
h. A 3x3 Gauss quadrature is used for each section. During this integration
process, the coefficients Aijk• C1
_k are calculated, as wel! as those parts of
Aijkl and C1jkl which do not de~nd on uw The columns mk are calculated as
- 3.17-
( 3.49)
where ók represents the column with global displacements of buckling mode k.
The components of the matrix in (3.48) depend on the value of the loading
parameter ..\, which in this case determines the load about which the
asymptotic expansion is performed. The choice of this value will be discussed
in the examples. Once this value is known, the system of equations can be
solved. When ,\ coincides with an eigenvalue \ of the matrix K+..\G, the
upper left part of the matrix in (3.48) will be singular. In that case, pivoting
is necessary in the solution process of (3.48). Consequently, the structure of
the matrix will be changed and the solution procedure, applied in chapter 2
for the determination of u0 can no longer be used. Therefore, a procedure is
applied which uses a sparse variant of Gaussian elimination together with a
pivotal strategy which is designed to campromise between maintaining sparsity
and cantrolling loss of accuracy through round off [NAG Fortran Library
Mk12, routine F04AXF].
When the displacement fields uii are known, the parts of Aijkl and C1ikl
which depend on it, can be determined.
The part of Aijkl which depends on uii is given by
AJH L1(u/DL11(ui,ukl) + L1(u/DL11(ui,uk1)
+ L1(uk1fDL 11(ui,uj)J dA. (3.50)
If these terms are compared with the third, fourth and fifth term of (3.33), it
will be seen that (3.50) can be written in discretized form as ( (3.36)-(3.42))
(3.51)
Similarly, the part of C1jkl which depends on uii can be written as
1 T 1 T 2óklqij + 2óilqjk' (3.52)
3.4 Determination of the equilibrium paths
The final step is to obtain the equilibrium paths for the structure. A perfect
structure with simultaneous or nearly simultaneous buckling loads has several
equilibrium paths that branch off from the trivial prebuckling state. The
- 3.18-
most critical among them being the post-buckling path of steepest descent or
smallest rise.
The introduetion of imperfections not only changes the equilibrium equations
but also the nature of the behaviour of the structure; now the system has a
single solution path and loss of stability is associated with a limit point in the
load displacement curve. Given this different type of behaviour, both cases are
discussed separately.
3.4.1 The perfect structure
The system of nonlinear equilibrium equations is given by
0 (I = 1,M). (3.53)
Following Koiter's approach [Koiter,l974], the amplitudes ai are regarded as
components of a vector lt in Euclidean M-space. The vector lt is written as
lt = aê (3.54)
where ê is a unit vector and a represents the magnitude of the deflection from
the fundamental state.
With these notations, the equilibrium equations of the perfect structure change
to
(1-~ )e1 + aC1. eJ.ek + a2C e-ekel = 0 "r Jk Ijkl J (I = 1,M) (3.55)
with the condition: eiei= l.
This system of equations has more than one solution in generaL Koiter [1974]
showed that, for a structure with simultaneous buckling modes, the directions
ê of the post-buckling paths coincide with the unit veetors t for which the
cubic form Aijktitjtk or the quartic form Aijkttitjtktl (see (3.18)) takes a
stationary value on the unit sphere I tI = 1 (the quartic form is only
considered by Koiter when the cubic form equals zero). The post-buckling
path of steepest descent or smallest rise coincides with the unit vector t for
which the cubic or quartic form takes its absolute minimum on the unit
sphere. With these theorems it is possible to obtain the solution of the
post-buckling paths in closed form [Koiter, 1974].
Unfortunately, no such theorem exists for structures with nearly simultaneons
buckling loads. Benito [1983] showed that when only two interacting modes are
considered some closed form solutions of the post-buckling paths can also be
3.19-
obtained, but in general the equilibrium equations must be solved numerically.
In the computer program, described in this chapter, the nonlinear system of
equations is solved using an iterative solution procedure which chooses its
correction at each iteration as a convex combination of the Newton and scaled
gradient directions [NAG Fortran Library Mk12, routine C05NBF].
As already mentioned, the most critica! equilibrium path is the post-buckling
path of steepest descent or smallest rise. In order to trace this particular path
for a structure with nearly simultaneous buckling loads, the following approach
is adopted.
The numerical solution procedure is started at a large value of a with the
minimizing directions ti of the cubic form Aijktitjtk or the quartic form
Aijkltïtht1 as the initia! estimates for the direction components ei. The initia)
estimate for >. is taken as 0.8 >.b for a deseending path and 1.2 >.b for a rising
path. Once the equilibrium values of ei and >.b for the prescribed value of a
have been obtained, the value of a is decreased by an amount b.a and the
values of >. and ei at the previously obtained equilibrium point are used as the
starting value for the next step. This process is repeated until a becomes zero.
The large starting value of a is chosen such that the post-buckling deflection
is approximately equal to the average plate thickness of the structure.
This s~alled 'backward search' approach is chosen for the following reasons.
(i) The difference between the equilibrium paths of the structure with nearly
simultaneous and simultaneous buckling loads deercases for increasing
values of the post-buckling deflections. Therefore, for large values of this
deflection, the minimizing directions ti of the cubic or quartic term can
be used as a reasonable initia! good estimate for the post-buckling path
of steepest descent or smallest rise of a structure with nearly
simultaneous buckling loads.
(ii) Because the equilibrium paths of a structure diverge for increasing values
of the post-buckling deflections, the initia! estimates have to be less
accurate for a large value of this deflection, to yield a point on the
desired path.
(iii) The salution procedure uses the values of >. and ei at the equilibrium
point of the previous step as the starting value of the present step,
resulting in a tendency to follow the original equilibrium path. If
3.20-
another equilibrium path branches off from this path, additional measures
must be taken in the case of a forward search in order to be sure that
the procedure follows the bifurcated path. If the procedure starts at a
large value of the post-buckling deOeetion and searches backwards, it
will automatically follow the correct path.
To obtain the minimizing directions of the cubic or quartic form, a
minimization (optimization) procedure is used. This procedure minimizes an
arbitrary smooth object function, subjected to constraints, using a sequentia!
quadratic programming algorithm, in which the search direction is the solution
of a quadratic programming problem [NAG Fortran Library Mkl2, routine
E04UCFJ. The user has to supply an initia! estimate of the solution. The
absolute valnes of the initia] estimates used in the present computer program
are given by
1 MJM (i=l,M). (3.56)
All 2M combinations of positive and negative values of ti are used as a
starting value. The minimum value of the object function resulting from these
starting values is considered to be the absolute minimum for this function.
Both the cubic and quartic terros are taken into account in the numerical
solution of the equilibrium equations. Depending on the magnitude of these
terms, either the minimizing directions of the cubic or the quartic form are
used as the initia! estimate for the post-buckling path of steepest descent or
smallest rise. When there is doubt about the dominant form, the equilibrium
equations are solved twice, once with the minimizing directions of the cubic
form as the initia! estimate and once with those of the quartic form.
9.-f.2 The imperfect strocture
The system of equilibrium equations of the imperfect structure is given by
(I = l,M). (3.57)
Substitution of (3.54) yields
À 2 À (1-x
1)ae1 + a C1jkeiek + a
3C1jkleieke1 = Xl-r (I = l,M). (3.58)
- 3.21 -
The introduetion of initial imperfections changes the nature of the system,
which now has a single salution path for given values of the imperfections !!:.i
(i=l,M). The iterative salution procedure used for solveing the equilibrium
equations of the perfect structure can be used also for solving those of the
imperfect structure. For the imperfect structure, however, the procedure is
started at a small prescribed value of a, which is gradually increased by a
known amount ~a during the salution process. The starting value for the
direction components ei coincides with those of the imperfections and the
initia! estimate for .\ is taken as zero. Once an equilibrium point is obtained,
the values of ei and .\ at this point are used as the initia! estimates for the
next step, and so on.
Koiter showed that the most detrimental imperfections for a structure with
simultaneous modes, are those in the direction of the post-buckling path of
steepest descent or smallest rise. Although this theorem no longer holds when
both the cubic and quartic terms are considered and/or when the buckling
loads do not coincide, it still indicates the direction in which the most harmful
imperfections must be sought.
3.5 Numerical examples
The spline finite strip method described in the preceding sections can be used
to solve many of the post-buckling and interaction problems associated with
prismatic plate structures. An exhaustive study of its applications is beyond
the scope of this thesis, and the method is illustrated by just a few examples.
The first example, having been fully investigated by analytica! methods, is
used to demonstrate the accuracy and convergence of the method, while the
second example compares the results of the present method with those of
Benito [1983]. The third and fourth example demonstrate the performance of
the spline finite strip method in the case of structures loaded in uniform and
non-uniform bending.
The accuracy of the numerical salution is governed by a number of factors
(number of strips, number of sections etc.); their influence is also evaluated in
this chapter so that some general guidelines are available for providing results
with a prescribed level of accuracy.
- 3.22-
9.5.1 Simply S?Jpported S(j'/Jare plate under uniform compression
The first example to be considered is a simply supported square plate under
uniform compression. The dimensions and loading conditions are shown in
Figure 3.1.
t::1 mm x b::500mm
y .&>
l9
I· b ·I Fig. 3.1 Simply supported square plate under uniform compression.
The bifurcation stress 0"1 and the bucking mode u1 of the square plate are
given by the well-known formula
i!Et2
3(1-v2)h2
w 1 = fsin~os.q; where f is an undetermined amplitude.
(3.59)
0) (3.60)
The post-buckling equilibrium equation, for the perfect plate reads (see (3.16))
(1-i) + a2Cun = 0. 1
(3.61)
Sirree the bifurcation is symmetrie, C111 is zero. The value of C1111 depends,
besides on the buckling mode, on the displacement field u11 • To evaluate this
secoud order field, the value of À in (3.9) must be fixed; this is equivalent to
prescrihing the load about which the pertubation expansion is made. This
point is discussed to some extend in the following example. For a one-mode
analysis, however, this value is taken equal to the buckling load.
The analytica! solution of u11 and C1111 are given by Koiter and Kuiken [1971]
for two types of in-plane boundary conditions.
(i) All edges remain straight (boundary conditions of Marquerre and Trefftz).
- 3.23-
(ii) The compressed edges remain straight, while the longitudinal edges are
free to wave in-plane (boundary conditions of Hemp ).
Both these boundary conditions have been considered in the numerical analysis.
The influence of the strip and section subdivision is evaluated using different
numbers of strips and sections. The values shown in Table 3.1 are obtained for
10 sections and a varying number of strips, whlle those in Table 3.2 are
determined with 10 strips and a different number of sections. The Young's
modulus and Poisson ratio used in the calculations are respectively 2.1E5
N/mm2 and 0.3. The strips are taken parallel to the loading direction.
nr. of of b oundary conditions
nr. M&T HEMP
sections strips numer i c. analyt. nume r i c. analyt.
10 0. 0930 0. 0652
10 20 0. 0916 0.0911 0. 0635 0.0629
30 0. 0913 0. 0631
Table 3.1 Post-buckling coefficient Cnu of a uniformly compressed square plate obtained with 10 sections and a different number of strips.
of nr. of b oundary conditions
nr. M&T HEMP
strips sect i ons numer1c. analyt. numer 1 c. analyt.
4 0. 0929 0. 0653
10 6 0. 0930 0.0930 0. 0652 0.0652
8 0. 0930 0. 0652
Table 3.2 Post-buckling coefficient Cuu of a uniformly compresseá square plate obtained with 10 strips and a different number of sections.
The resemblance between the results is not only confined to the overall
behaviour, but also at a local level it is significant. The difference between the
analytically and numerically obtained post-buckling displacements, for a plate
with 20 strips and 10 sections, did not exceed 0.5%.
- 3.24-
3.5.2 Channel section under uniform compression
To compare the results of the present method with those of Benito [1983], the
channel having the geometry shown in Figure 3.2 is considered. The structure
is simply supported and subjected to a uniform compressive stress. The loading
condition and the strip configuration are also shown in Figure 3.2. The
interaction buckling of this channel has been stuclied by Benito, using the
classica! finite strip method within the frame of Koiter's theory of stability.
A
B
2 2 2 3 y.v
E = 2.1Eá N/mm2 V = 0.3 iz,w -tf'r-,------"'71- _x.lL - l=900 mm #. I. w(O) = w(L) IJx(O) = lixtLJ = 0
-1-__;::._.:_.::..:....;::::.:.;.. __ _,-r-+- 2. v(O) = v(L) w(O) = w(L) = llx(O)
z,w 3. u(O) = v(O) = v(L) w(O) = w(L)
IJx(L) = 0
lixCOl = II,(L) = 0
Fig. 3.2 Geometry, loading condition and strip contiguration of the channel.
Benito evaluated the behaviour of the channel for a different number of strips,
in order to get two digits accuracy, 10 strips for half the section were found
to be enough (due to the symmetry, it is sufficient to consider only one half
of the cross section). The same number of strips were used in the spline finite
strip analysis. Local bucking in a mode with 11 half waves occurs at a critica!
stress of 141.9 N/mm2 and a maximum amplitude of 0.029 mm at point A.
Global buckling takes place at a stress of 146.3 Nfmm2 with a maximum
amplitude of 0.2 mm at point B. The buckling modes of half the cross section
are shown in Figure 3.3 and 3.4.
Fig. 3.3 Local buckling mode UJ of half the channel section with a cross sectional view at midlength.
/ ______ .
~-
- 3.25-
Fig. 3.4 Global buckling mode u2 of half the channel section with a cross sectional view at midlength.
As mentioned in the previous example, the value of the load factor À in (3.9)
must be fixed in order to evaluate the second order fields. This value will be
denoted by À0
in the following. Benito [1983] showed that the results may
greatly depend on this value, especially in the case of varrishing third order
post-buckling coefficients. Since À0 indicates the load about which the
expansion is made, the most accurate values for the maximum load carrying
capacity will be those, for which À0 coincides with the maximum value of À
(Au) in the load displacement curve of the imperfect structure [Carnoy,1981].
In this example, however, the value is taken equal to one of those used by
Benito, namely 0.6 Àb. The influence of the number of sections is evaluated
using 22, 44 and 66 sections along the length of the channel. The most
important non-zero post-buckling coefficients for the different number of
sections are presented in Table 3.4 and 3.5. The second order fields Uw u12
and u22, obtained with a section subdivision of 44, are respectively shown in
Figure 3.5, 3.6, and 3. 7. To improve the readability of these figures, they are
presented on a different scale.
nr. of nr. of th i rd order post-buckl. coe f.
strips sec ti ons C11 2 C211 C22 2
22 -D. 88E-3 -0.44E-3 0. 57E-3
10 44 -{). 88E-3 -0.44E-3 0. 56E-3
66 -{). 88E-3 -0.44E-3 0. 56E-3
Table 3.3 Third order post-buckling coefficients of a uniformly compressed channel section.
- 3.26-
nr. of nr. of fou rt h order pos t-buckl. coe f.
strips se ct i ons cll 11 Cu22 C2112
22 0. 29E-3 -{). 17E-3 -o .17E-3
10 44 0 .16E-3 -{) .17E-3 -o .17E-3
66 0.16E-3 -{) .16E-3 -o .16E-3
Ta.ble 3.4 Fourth order post-buckling coefficients of a. uniformly compressed channel section.
Fig. 3.5 Second order field uu of half the channel section with a cross sectional view at midlength.
Fig. 3.6 Second order field UJ2 of half the channel section with a cross sectionat view at midlength.
Fig. 3.7 Second order field U22 of half the channel section with a cross sectional view at midlength.
'
-----···-· - ---
- 3.27-
From analytica! analyses [Graves Smith and Sridharan,1978; Koiter and
Kuiken,1971] it is known that the second order field of the local mode of
simple structures has a periodicity which is twice as large as that of the local
buckling mode. Taken this into account, it follows from the values of Table
3.4 that approximately two sections per half wave are required to obtain
results within engineering accuracy.
Benito [1983] presented a load displacement curve for an imperfect column
with an imperfection amplitude in the local mode of 0.05 mm and in the
global mode of 0.4 mm. The same imperfect channel has been evaluated with
the spline finite strip model of this chapter and the results are shown in
Figure 3.8. The maximum value of A obtained with the spline finite strip
approach is approximately 4% less than that obtained by Benito. The
differences between Benito's result and that of present method are mainly due
to the fact that the present approach treats the effect of the amplitude
modulation more accurately. Because the axial strains due to the global
buckling vary along the length of the channel, the amplitude of the local
buckling mode is modified in
Benlto
Present approach
40 20
1. 00 À/ Àb
0.40
0.20
0.0
Benito
Present approach
17 34
Fig. 3.8 Load displacement behaviour of imperfect channel.
the post-buckling range. This effect is known by the name 'amplitude
modulation' [Koiter and Kuiken,1971]. In the spline finite strip model this
effect is fully taken into account, as is illustrated by the out-of-plane
displacements of the web in the second order field u12 (Figure 3.9).
- 3.28-
Fig. 3.9 Vertical web displacement of the second order field u12.
The hehaviour of the imperfect channel as predicted hy the spline model was
also found hy Ali [1986], who used Koiter's theory in conjunction with a
combination of the classica! finite strip metbod and a one-dimensional finite
element model which takes full account of the amplitude modolation too.
As mentioned in the previous section, the computer program of this chapter
can also he used to determine the post-huckling path of the perfect structure.
This path, together with the equilibrium path of the imperfect channel as
ohtained hy Benito and hy the spline method are displayed in Figure 3.10.
Note that the first part of the post-huckling path of the perfect structure
(near >./ >. = 1.0) rises, hut that the path starts to deseend at a value of b
>.j \ ~ 1.01 . The point where the rising path changes to a deseending path
indicates a second hifurction point. At that point, the load carrying capacity
in the post-huckling range of the local mode is undermined by the global
mode. 1.20
Àf.Ab 1.001-----
0.80
0.60
0.40
0
0.0 20.0 40.0 60.0
Fig. 3.10 Load displacement behaviour of perfect and imperfect channel.
- 3.29-
In analytica! analyses and in the classica! finite strip method it is assumed
that the displacements for each plate, as well as the normal stress resultants
perpendicular to the plate junctions, are zero for a local mode. Although these
two assumptions are incompatible, it has been shown by Van Benthem [1959]
that in view of the smallness of the ratio t/b, where t represents the plate
thickness and b the width of the plate, this assumption is reasonable. Due to
this assumption, the bifurcation in the local mode becomes symmetrie and all
the post-buckling coefficients which 'contain' the local amplitude to an odd
degree are zero. In the spline finite strip model, however, this assumption is
not used and the behaviour of the junction lines is described more realistically.
As a result, the displacements perpendicular to the junction lines are, although
very small, not equal to zero, see Figure 3.11.
Fig. 3.11 Global u-<lisplacements at the junction line of the channel near the support.
Due to a disturbance in the displacement pattem of the junction line near the
support, the energy associated with an outward buckle is not the same as that
with an inward buckle. Therefore, the symmetry of the bifurcation will depend
on the number of half waves in the buckling mode. The bifurcation is
symmetrie for an even number of half waves and asymmetrie for an odd
number. This is demonstrated by the third order coefficients Clll of local
modes of the channel with different numbers of half waves, which are
presented in Table 3.5. The influence of this effect on the load displacement
behaviour of the channel is small however.
half waves in critica) stress ct 11 local mode
[N/mm2]
10 142 0 85 0.53E-12
11 141 0 86 0.20E-4
12 142 0 59 0.46E-12
13 144 0 70 0.48E-5
Table 3.5 Third order post-buckling coefficients for buckling modes of the channel with a different number of half waves.
- 3.30
9.5.9 Plate girder in uniform bending
In most papers on interaction, the global buckling mode is determined, using a
one-dimensional theory which assumes that the cross section of the merober
does not distort. In some cases, however, the distartion of the cross section
may play an important role in the global buckling behaviour of thin-walled
members. The interaction model which is presented in this chapter takes the
distartion of the cross section fully into account, by using the spline finite
strip buckling model of chapter 2, to determine the buckling modes.
In this third example, the influence of the distartion of the cross section on
the post-buckling behaviour of the global mode of the plate girder considered
in example 2.3 is evaluated. This girder is loaded in uniform bending. The
geometry, loading condition and strip configuration are shown in Figure 2.12.
Girders of 3, 5, 10 and 20 m. length are evaluated, using a different number
of sections and 20 strips. The post-buckling coefficients are given in Table 3.6.
The coefficient Cm is zero in all cases, because the bifurcation is symmetrie.
gi r der cll •• length [mj 6 sect. 10 se ct. 20 sect.
3 0.112E-5 0. 970E-6 0.980E-6
5 0.564E-5 0.555E-5 0.555E-5
10 0.153E-4 0.147E-4 0.146E-4
20 0.191E-4 0.134E-4 0.130E-4
Table 3.6 Post-buckling coefficients of the plate girder.
The cross sectionat shapes of the global buckling modes are presented in
Figure 3.12 and those of the second order fields are shown in Figure 3.13. A
three dimensional view of the second order field of the girder with a length of
3 meter is shown in Figure 3.14. The displacements of the second order fields
are very small compared to those of the buckling modes, and are therefore
shown on an enlarged scale.
- 3.31
3m 5m 10m 20m Fig. 3.12 Cross sections at midlength of the buckling modes of the plate girder .
. -- T--. T --
3m 5m 10m 20m
Fig 3.13 Cross sections at midlength of the second order fields.
Fig. 3.14 Second order field of the plate girder with a length of 3 m.
Increments of the prebuckling displacements appear in a positive definite form
in the buckling equations, and therefore they can only increase the critica!
load. Consequently, these displacements are zero in the buckling mode. The
junction lines of the plate girder will therefore not undergo a displacement in
vertical direction during buckling and the roller support at x = L does not
- 3.32
undergo a displacement in axial direction during buckling. The second order
displacement field u11 is a first order correction to the global buckling mode in
the post-buckling range. From Figure 3.13 and 3.14 it can be seen that the
two types of displacements mentioned above, which do not enter into the
buckling mode, constitute the major part of the second order field.
The fact that the vertical displacement of the top flange is larger than that of
the bottorn flange can be explained as follows. An increase of the load in the
post-buckling range induces a vertical displacement of the centroid. This
displacement is the same for both flanges. A rigid rotation of the cross section
during buckling lowers the top flange with respect to the bottorn flange over a
distance ~w1 , which equals h(l-cosrp) see Figure 3.15a. When the cross section
distorts, the top flange is lowered with respect to the bottorn flange over a
distance ~w2 which, for the shape of Figure 3.16b, is approximately equal to. h
~w2 ~ ~ j(~)2dz. (3.62) 0
..c:
(a) (b)
Fig 3.15 Lowering of the top flange with respect to the bottorn flange.
Most distortional modes are a combination of the shapes shown in Figure
3.15a and 3.15b and the difference in displacement of the flanges therefore will
be due to a combination of ~w1 and ~w2 too.
The effect of the distortion on the load displacement behaviour of the different
girders is demonstrated by the ratio >../ >..b for a horizontal displacement ratio
b/L equal to 0.02, where ó is the horizontal displacement at midlength of the
upper flange and L is the length of the girder.
girder I en th (rn) 3 5 10 20
>.f>.b 1.013 1.052 1.099 1.068
Table 3.7 Load increase in the post-buckling range of the plate girders (or a horizontal displacement ratio Ó/1 = 0.02.
- 3.33-
3.5.4 T -beam loaded by a concentrated force
In this example, the interaction behaviour of the T-beam of example 2.4 with
a length of 450 mm. is evaluated. The geometry, loading condition and strip
configuration are presented in Figure 2.14. The first and the second buckling
load of the beam are equal to 2.73E3 N and 2.99E3 N respectively. The
associated buckling modes are shown in Figure 3.16 and 3.17. The maximum
deflection (amplitude) of the first mode occurs in the top flange at midlength
and equals 0.178 mm, while that of the second is located at the bottorn of the
web and has a value of 0.188 mm. Both modes exhibit global as well as local
deformations and therefore they must be classified as distortional modes. The
third order post-buckling coefficients are so small that they can be neglected.
The fourth order coefficients obtained with 30 sections and different values of
.À0
, are presented in Table 3.8.
Fig. 3.16 First buckling mode of the T-beam.
Fig 3.17 Second buckling mode of the T-beam.
- 3.34-
f ou r t h orde r _!l_O s t bu c k I i ng co e ff i c i en ts x 10- a )..o/)..1
clltt C1112 C1122 C1222 c2ttt C2112 C2122 C2222
0.95 2.94 ~.57 0.90 0.10 ~.19 0.90 0.30 0.34
1.00 2.93 ~.57 0.91 0.10 ~.19 0.91 0.32 0.32
1.10 2.92 ~.57 0.92 0.10 ~.19 0.92 0.31 0.30
Table 3.8 Fourth order post-buckling coefficients of the T-beam for different values of À. 0 .
The above values demonstrate that the value of À. 0 in this case has little
influence on the fourth order coefficients. The second order fields obtained with
À. 0=À1 are shown in Figure 3.18, 3.19 and 3.20.
Fig. 3.18 Second order field u11 with a cross sectional view at midlength.
Fig. 3.19 Second order field u12 with a cross sectional view at midlength.
- 3.35-
Fig. 3.20 Second order field u22 with a cross sectional view at midlength.
The load displacement curves of the perfect and an imperfect structure with
imperfection amplitudes ~1=-().277 and ~=0.2 are presented in Figure 3.19.
The curves are obtained for À0=À1. The discontinuity in the load displacement
curve, is the result of intervention by the second buckling mode.
1. 50
1. 00
0.50
1 Imperfect I
'
0.0 10.0
a 20.0 30.0 40.0
Fig 3.19 Load displacement curves for a. perfect a.nd an imperfect T-bea.m with imperfection amplitudes i!t=-{).277 and 1!2=0.2.
3.6 Conclusions
The examples of this chapter demonstrate that the combination of the spline
finite strip method and Koiter's stability theory results in a numerical tool
which is capable to analyse the post-buckling and interaction behaviour of
structures under arbitrary toading conditions. Especially the possibility to
study the interaction behaviour of structures loaded in bending and/or shear,
- 3.36-
with cross sectional distortions taken into account, opens a new research area.
Comparison of the results of the present method with analytica! ones and
those of other research workers, shows the accuracy and versatility of the
present combination. However, to check the validity of the approach for
structures loaded in bending and/or shear, comparison with test results is
required. Sirree there is a need for experimental result of (interactive) buckling
experiments of beams under uniform and non-uniform bending, a test program
is presently carried out at the Eindhoven University of Technology. Results of
interactive buckling test with plate assemblies in non-uniform bending are
unfortunately not yet available.
One of the basic assumptions of the interaction theory of this chapter is that
both aiui and v are small. For structures with simultaneous and nearly
simultaneous modes these assumptions are satisfied in the vicinity of the
bifurcation point. The method, however, has been applied to structures with
well separated buckling loads too [Sridharan and Ali,l986]. The validity and
accuracy of the method in these cases is still open to discussion. More research
work and, especially, well controlled experiments will be needed to answer
these questions.
The computer time needed by the present approach depends on the number of
strips and sections used in the analysis. In Chapter 2 it is shown that
approximately two sections per half wave are required to describe a local mode
correctly. The periodicity of the second order field of a local mode is
approximately twice as large as that of the associated mode. This means that
four sections per half wave of the local mode are required to describe the
second order field to within engineering accuracy. Consequently, the
computational effort needed by the present method to analyse structures with
a large number of half waves (> 20) in the local mode will be considerable.
For such structures, preferenee must, at the time being, be given to a
semi-analytica! finite strip approach such as that presented by Benito [1983]
and Ali [1986].
-4.1-
4 THE ELASTIC FLEXURAL-TORSIONAL BUCKLING OF MONQ
AND DOUBLY SYMMETRIC BEAMS WITH A LARGE INITIAL
BENDING CDRVATURE
4.1 General
The effects of distartion of the cross section and mode interaction, discussed in
the preceding sections, are mostly negligible for slender beams. When this is
the case, the flexural-torsional buckling load can be predicted to within
engineering accuracy by a one dimensional theory (beam theory). In that
approach it is normally assumed that the major axis rigidity is very large, so
that the small in-plane prebuckling deformations of an initially straight beam
can be neglected. Flexural-torsional buckling tests on simply supported
aluminium beams, performed at the Eindhoven University of Technology
[Seeverens,1982; Winter,1983; Macquine,1983], showed that these beams may
exhibit relatively large in-plane displacements before buckling. The effect of
in-plane deformations on flexural-torsional buckling has been studied by a
number of research workers [Davidson,1952; Trahair and Woolcock,1973;
Vacharajittiphan e.a.,1974; Vielsack,1974; Roberts and Azizian, 1983]. Their
investigations, however, are based on the assumption of finite but small
prebuckling displacements and are mostly restricted to doubly symmetrie cross
sections. In the case of long aluminium beams the prebuckling displacements
may become relatively large (in the order of the height of the beam), while
the material remains elastic. Because it is questionable whether the results
obtained by the other research workers are still valid for these large in-plane
displacements, the effect of these deformations on the flexural-torsional
buckling behaviour of simply supported beams is investigated in this chapter.
First a nonlinear beam theory is derived which is generally applicable to
situations where the strains are small and the Bernoulli hypotheses are valid.
Then this theory is used to derive the energy functional which governs the
flexural-torsional buckling of mono- and doubly symmetrie beams with a large
initial bending curvature. Finally the buckling problem is solved using a finite
element formulation, and some numerical examples are shown.
4.2 The nonlinear flexural--torsional behaviour of straight slender elastic beams
During the past 15 years several articles on the nonlinear flexural-torsional
behaviour of beams have been published [Ghobarah and Tso,1971; Roik e.a.,
1972; Rosen and Friedmann,1979; Roberts,1981; Attard,1986]. In the majority
-4.2-
of these articles, a potential energy functional in terms of displacements and
rotations is used, which is mostly derived in the following manner. First a
displacement field containing the components of the rotation matrix is
determined, then this field is used to calculate the nonlinear strain components
and finally these strains are used to determine the potential energy functional.
Deriving the potential energy functional in this way, without introducing
various approximations is almost impossible, because both the components of
the rotation matrix and the strain expressions in terms of displacements are
lengthy and complicated in their exact form. Therefore most of the articles on
the nonlinear flexural-torsional behaviour of beams are restricted to a special
class of deformations as a consequence of the approximations made.
In this paragraph a coordinate free dyadic notation is used to avoid
approximations concerning the magnitude of the deflections and rotations. This
enables the derivation of a potential energy functional and curvature
expressions which are generally applicable to situations where the strains are
small and the Bernoulli hypotheses are valid.
4.2.1 Kinematics of straight slender elastic beams
- The undeformed configuration G(O)
A undeformed slender prismatic beam of length L is shown in Figure 4.1. The
beam is made of a homogeneaus isotropie linear elastic materiaL Each material
point of this beam is described by the coordinates (x,y,z) in a rectangular
Cartesian system. The veetors e1,ih,ïh represent unit veetors along the
coordinate axes. The coordinate x coincides with the elastic axis of the beam,
defined as the line which connects the shear eentres (S) of the cross sections.
If the shear centre is not a material point of the cross section, as is often the
case for thin-walled open sections, it is still considered to follow all the
e1,x
Fig. 4.1 (a) Beam with coordinate system. (b) Cross section.
-4.3-
the deformations of the cross sections as if it was a real material point of that
cross section. It is assumed that before deformation the elastic axis is a
straight line. With x representing length along this axis, it can be represented
by
10(x) = xê1 ; x t [O,L]. ( 4.1)
In the undeformed state the cross section is oriented so that ê2 and ê3 are
parallel to the principal axes. The position of an arbitrary material point
before deformation can be expressed as
Xo(x,y,z) = t 0(x) + tt-0(y,z) (4.2)
where f.t-0 = yê2 + zêa, while y and z denote length along the ê2 and ê3 axis.
In addition to the coordinate axes y,z a second set of coordinate axes is
defined in the cross section, parallel to y,z but with its origin located at the
centroid (see Figure 4.1b). The relation between y,z and y,z is given by
y = y-y6 and z = z-z6• ( 4.3)
The gradient operator with respect to the undeformed configuration can be
written as
(4.4)
- The deformed configuration G(l)
After deformation, the position vector of a material point is given by Jt(x,y,z).
In order to determine 5t the following assumptions are made.
(i) The total deformation of the beam is considered to be the result of two
successive motions of the cross sections; first, a rigid translation and
rotation due to bending and warping free torsion; next, a warping
displacement perpendicular to the displaced cross sections.
(ii) The cross section does not distort in its plane during deformation.
(iii) Shear deformation due to transverse forces can be neglected.
With these assumptions the position vector 5t can be expressed as (Figure 4.2)
Jt(x,y,z) = t(x) + tt-(x,y,z) + f(x,y,z) r ,(x)
where
( 4.5)
tt-(x,y,z) yi2(x) + zia(x); t(x) represents the beam axis in the deformed
- 4.4-
configuration; r 2(x),i 3(x) are unit veetors parallel to the principal axes of the
cross section after the warping free motion; f(x,y,z) r t(x) represents small
normal warping displacements (l 1=l2*l3).
G(O)
Fig. 4.2 Beam after warping free motion.
When the displacement of a point on the elastic axis is described by its
components us,vs,ws in the directions el,e2,e3 respectively, r can be expressed
as
( 4.6)
where the indication of the function variables is omitted. The unit tangent
vector at a point of the deformed elastic axis can be obtained from
dr .. Os= n
where s is the are length along the deformed elastic axis.
Differentiating r with respect to x instead of s yields
where
dr cif ds ( ).. [( ).. .. , .. l <IX = Os <IX = 1 + t 5 n = 1 + u~ e1 + v~e2 + w5e3
fs = I fx I - 1 = [(1 + u~)2 + v~2 + w~2j0·5 - 1.
(4.7)
(4.8)
( 4.9)
When the shear deformation due to transverse forces is neglected, the unit
tangent vector n is perpendicular to the cross section through that point.
-4.5-
(4.10) ..
The triad êk (k=1,3) ca.n be tra.nsformed into the triad ik (k=1,3) by means
of a rigid rotation. This rotation ca.n be described by a.n orthogonal tensor IR
IR = IR(x) ; IRC·IR = IR·IRC = n ; det IR = 1 (4.11)
where n represents the unit tensor a.nd IRc is the conjugate of IR. To analyse
the beam deformation, that is to define the flexural and torsional curvatures of .. the beam axis, the derivatives of ik with respect to s are studied .
.. dik diR >t diR IRC 7 as= as'"'k = as· ·Ik. (4.12)
The orthogonality property of IR implies that (diR/ds)·IRc is a skew tensor, and
therefore (4.12) may also be written as
( 4.13)
where ~ is the axial vector of (diR/ds)·IRc. According to the classical definition
[Love,1944] the torsional and flexural beam curvatures are defined as the "t ..,. t
components of the vector ~ with respect to the local basis 11, 12, 13
.. .. .. ~ = Ktil + ~Î2 + Ksi3 (4.14)
where x:1 is the torsional curvature a.nd K2 a.nd K3 are the flexural curvatures . .. Differentiating ik with respect to x instead of s yields
.. .. dik _ dh ds _ (1 + ~ )?.••1• _ ~.·1• <IX - as nx - LS /J k - ,. k ( 4.15)
where 1. = (1 + t5)~.
Combination of (4.14) and (4.15) yields
.. .. .. ~ (1 + t 5}[x;1i 1 + ~i 2 + ~>3 i 3]. ( 4.16)
The deformation can be described completely in terms of a deformation tensor
f, which is given by
( 4.17)
-4.6-
With the relations derived so far f can be expressed as
f [ 1 + (-:t*"*) (-;t.*1 )f of1J'* of1,. of1,. IR = {811 ,.. u + ,.. 11 + oxl 1::1 + oyl 11::2 + (7Zl1t:3 + (4.18)
where IR = (i 1 ~ 1+ i 2~2+ i 3~3).
-1.2.2 The general potential energy functional
Assuming the strain energy density to be a quadratic functional of the
Lagrangian strain tensor components, and taking into account the condition
( 4.19)
the potential energy functional may be written as [van Erp,1987]
II = ~ J[Eei1 + 4Gei2 + 4Gei3JdV- JPo~·udV- aiP0 ·udS (4.20) Va Va SP
Where eii represent the components of the Lagrangian strain tensor with
respect to the undeformed configuration, u represents the displacement vector
from G(O) to G(1), Vo is the volume of the undeformed body, sg is the part
of the boundary where the loads are prescri bed, p0 is the mass density, ~
represents the body forces per unit mass, P Oi are the prescribed external loads,
E is Young's modulus and G is the shear modulus.
The Green-Lagrange strain tensor can be expressed in terms of the
deformation tensor as
( 4.21)
The components of IE which are relevant in beam theory are given by
( 4.22)
Before proceeding, attention is focussed on the warping function f. In the case
of thin-walled open sections, the normalized warping displacements are mostly
described in terms of the s~alled sectorial area w [Timoshenko and Gere,
1961]. In this chapter, however, preferenee is given to the equivalent, but more
-4.7-
genera!, Saint Venant warping function '1/!{y,z) [Timoshenko and Goodier,l970].
If K1 is chosen as the warping 'amplitude', the function f can be written as
f(x,y,z) = '>1(x)tP(y,z) ( 4.23)
No assumptions concerning the magnitude of the deformations have been
introduced so far. In the following, however, the derivation will be restricted
to cases where the strains are so small that they may be neglected compared
to unity (as is the case for most metals). Applying this type of approximation
to ( 4.22) yields [van Erp, 1987]
e ( + ( Y-., + z- "'-) + l(y-2 + -z2).,2 + .. 1'·'· 11 = s - "3 '•:t 2 "1 "' 'I'
e12 = ~K1(* - z)
e13 = ~"'1(~ + y).
Using relation (4.3), e11 may also be written as
- l-2 2 1.1, eu = c Y'>3 + z~ + 2r '>1 + "1'~'
where
and
..j.2.9 The strain energy
In beam theory the strain energy U is given by (see (4.20))
U ~ J[Eeî1 + 4Gei2 + 4Gei3 ]dV. Vo
(4.24)
(4.25)
( 4.26)
( 4.27)
(4.28)
Sirree y and z are coordinates along the principal axes, the following identities
hold
(4.29)
where A is the area of the cross section. Substituting ( 4.25)--( 4.27) into ( 4.28)
yields
-4.8-
L u - 1 J [EA"f2 + EI K~ + EI K-2 + GJK-2 + EfK-' 2 + - 2" 2·-l 3 3 1 1
+ K-~(EI5 "f + EI2,82~ - EI3,83K-3 + Ef,B.,pK-D] dx (4.30)
where
I2 = Jz2dA ; I3 = Jy2dA ; r = JvdA ; I = Jr2dA A A A
5 A
J =Af[(~- z)2 + (Plz + S'?]dA ; H = i AJr4dA
.l-~-.l The potential energy junctional
The total potential energy is the sum of the strain energy U and the potential
of the loads n
rr =u+ n. ( 4.31)
It is assumed that the beam is subjected to conservative surface tractions P1e1
+ p2e2 + p3e3 at both ends (x = 0 and x = L) and distributed loads q2e2 + q3e3 per unit length (Figure 4.3). Beside these 'external' tractions the beam is
loaded by a distributed load qe3 representing the weight of the beam. The
potential of the loads can be written as
t ê3, i I
èi,x Fig. 4.3 Load configuration.
-4.9
where Ui (i=1,3) repcesent the displacement of the point of application of the
loads in the directions ~i and ua( c) is the displacement of the centraid in the
~3 direction. The displacement vector i1 of a material point is given by
( 4.33)
The components of i1 in the directions ~ 1 ,ê2,ê3 can be expressed as
(4.34)
where u = u6 YsR12 - z6Rt3 and Rij are the components of IR with respect
to é1,ê2,ê3 (Rii=êi ·iJ The terms K1 '1/1~1 and K1'1j1R31 in (4.34) may be
neglected according to the assumption of small strains. Combination of ( 4.30)
( 4.32) and ( 4.34) yields
where
II - 1 J1[EA:E2 + EIK~ + EI K2 + GJK2 + EfK' 2 + EHK4 - 2 2·~L 3 3 1 1 1
0
+ KÎ(EI6 t. + EI2{J2K2 - EI3{13K3 + Ef.B 'I/I"'D J dx
-[P1u + P2v6 + P3w6 + M2R13 - M3R12 + BK1R11 (4.35)
Mt2~3 + Mt3R32 + (R22-1\Jp2ydA + (Raa-1) AJp3zdA ]x=O;L
-(}r[q2vs + q2y(~2-l)- mt2~3 + qaws + mt3R32
+ q3z(R33-l) + q(w6 y6R32 - (R33-l)z6)Jdx
- 4.10-
Equation ( 4.35) includes, beside the linear, also the nonlinear contributions
resulting from the movement of the points of application of the loads. This
contribution, which can have a significant influence on the behaviour of the
beam, must be taken into account [Attard,l986; Moore,l986].
In the case of thin-walled open sections, the average displacement of the cross
section is normally written as
(4.36)
where D represents the sectorial origin [Murray,l984]. In this chapter, however,
the following expression is preferred
( 4.37)
It can readily be seen that both (4.36) and (4.37) represent the displacement
of the centroid as no warping occurs. Si nee equation ( 4.37) results in simpler
expressions, preferenee is given to it.
Constitutive · equations for the normal force N and the bending moments
M2,M3 about the centroidal axes in the deformed state are obtained by
integration over the cross section of the normal stress and its moments. This
results in
(4.38)
The bimoment acting on the cross section in the deformed state is defined as
(4.39)
where u1 represents the normal stress in the deformed state. Integration yields
[Elias,l986]
B = Ef~~:t.x + iEr,O.,p"~· ( 4.40)
Consiclering the integrand of the strain energy U (4.30) as a function of ë, ~~:2 ,
11:3 and Kt,x it is readily verified that the constitutive equations of N1, M2, M3
and B are equal to the part i al derivatives of the integrand with respect to ë,
11:2, 11:3 and ll:t,x respectively.
- 4.11 -
Let M be the partial derivative of the integrand of U with respect to fl:1. h:t
The constitutive equation for the torsional moment may then be written as
[Elias,1986]
( 4.41)
where Mts represents the torsional moment with respect to the shear centre.
Calculating M and B,x and substituting the result in ( 4.41) yields 1\:t
An alternative expression for Mts which is often used [Gregory,1961; Ghobarah
and Tso,1971; Moore,1986] is given in (4.43) and is obtained by solving (4.38)
for (, ~ and fl:3 and substituting into ( 4.42)
- I - -Mts = GJfi:t - Effl:t,xx + fi:1(~N + !12M2 - ;J3M3)
+ àE1i:r(4H - il- I2~ - I3~). (4.43)
Equation ( 4.43) is the same as the general nonlinear differential equation for
torsion obtained by Attard,1986; Moore,1986 and Elias,1986. The differential
equations for bending which are used by Moore [1986] do not contain the
terms with ;32 and ;33 ( 4.38), these terms, however, should be taken into
account when the cross section is asymmetrie .
../.2.5 The rotation matrix and the curvature expressions
To express the strain energy and load potential in terms of displacements, the
rotation tensor is stuclied more closely. The tensor IR represents the rigid
rotation which transfarms the triad ê~,ih,ih at a material point of the
undeformed axis into the triad it,i 2,i 3 of that point on the deformed axis. In
this chapter the tensor IR is decomposed into
IR = IR ·IR 11 ·1R 1 ~ -(l' ( 4.44)
where IR -a' IR ;3' IR7
represent successive rota ti ons a bout the axes ê2,ê3,ê1 of
magnitude -a,;3,"(. The tensor IR can now be expressed in matrix notation as
R=
r
cos Be os a
~o:~:~na -sinjkos acos ')'+sin')1lina sin;1cos asin ')'+s inacos 'l' 1
COS'fCO S p -Si ll'(CO S p .
sinps inacos')'+sin ')'COS a -sinps inas in"t+cos ')'COS a
( 4.45)
- 4.12-
-----.__ëj Fig. 4.4 Memher projection on the x,y ,z axes.
The angles a and /3 can be easily expressed in terros of the displaceroents v 5
and w5 (see Figure 4.4), yielding
-(X)SfV 1 Ü-v' 2-w' 2 - w'sin7 sin"v' J 1-v ' 2-w' 2 -J 1-v ' 2-w'2 s s s s I S S S
s s h-v~2
R v' s COS'j'~ 1-v ~ 2 ( 4.46)
- V~W~COS'j' sin')'V~w~ + cos w' s
J 1-v~ 2
Froro (4.15), (4.16) and (4.45) the following relations can be obtained ..
".1 1 di2 13 ·ax = Î' + a'sinf)
(4.47)
The curvature coroponents in terros of the displaceroents v5
and w5
are given by (see Figure 4.4)
(4.48)
-4.13
,f.2.6 Alternative warping lormulation
In section 4.2.2 the warping displacements are expressed as
( 4.49)
However, by postulating that the 'amplitude' of the warping displacements
equals K1, it is also postulated that, at a section where warping is prevented,
the strains e12 and e13 have to vanish too (see (4.25) and (4.26)). This is
certainly not the case in reality and should in general not be assumed, since
these strains are proportional to the stresses 1112 and 1113. To overcome this
problem, an expression similar to the one proprosed by Reissner [1952] for the
case of non-uniform linear torsion can be used
f(x,y,z) g(x)tf;(y,z) (4.50)
where g(x) is a function yet to be determined.
Reissner [1955] showed that in the case of non-uniform linear torsion of
thin-walled beams with open cross sections, the practical improvement gained
by working with ( 4.50) instead of ( 4.49) will in general be negligible. For
non-uniform linear torsion of thin-walled beams with closed or partly closed
cross sections, however, the more accurate equation (4.50) leads to results
which are quite different from what would follow from a use of ( 4.49).
In the case of beams with arbitrary cross sections which buckle in a flexural
torsional mode after relatively large prebockling deformations, (4.50) must also
be expected to lead to more accurate results than ( 4.49). The influence of this
alternative warping formulation on the flexural-torsional buckling behaviour of
bearns is evaluated in the following sections. When ( 4.50) is used to describe
the warping displacements, the strain energy ( 4.30) changes to
L
U =iJ [EA€2 + EI2~ + EI3K~ + Efg'2 + EHKi
+ Ki(EI5€ + EI2f}2~ EI3f}3K3 + Eft1~') ( 4.51)
where
- 4.14-
* The properties D, D and J are related by (Reissner,1955]
* * J = I5 + D + 2D and D = -D . (4.52)
4.3 Formulation of the bifurcation problem
4.3.1 General
The expressions derived in the previous sections are valid for straight beams
with arbitrary cross sections and boundary conditions. In the following,
however, only simply supported beams which exhibit bifurcation buckling in
the perfect case are considered. This requirement is met by beams with mono
and doubly symmetrie cross sections, which are loaded in the plane of
symmetry. The development of the theory is based on the assumption that the
beams are inextensional. Only beams loaded in bending are considered.
4.3.2 The prebuckling state
The only non-zero displacement component in the prebuckling state is w 5• The
curvature and elastic energy associated with this displacement are given by
- - "(1 - '2)-Q•5 "?. = -ws -ws
IJL -U=~ EI2~ dx.
The potential of the loads can be written as (see (4.35))
L f! = - J (qw6 + qz(R33-l)]dx
- 1:(Pw5 + M2R13 + Pz(R33-1)]
(4.53)
( 4.54)
( 4.55)
where z is the distance between the point of application of the load and the
shear cent re, and P, q represent respectively concentrated and distributed loads
along L. The total potential energy II1
of the prebuckling state is the sum of
the strain energy (4.54) and the potential of the loads (4.55)
II1 = ~j1
EI2~ dx -j1(qw5 + qz(R33-l)]dx
(4.56)
4.15-
Requiring this functional to be stationary renders the equations of equilibrium
and natura! boundary conditions of the prebuckling state.
The only approximations in the derivation of the rotation matrix ( 4.46) and
the curvature expresslons (4.48) applied so far, is the replacement of terros of
the order (1+t) by unity, in accordance wlth tbc assumption of small strains.
In this chapter, however, the magnitude of the prebuckling displacements is
limited and expresslons as accurate as ( 4.46) and ( 4.48) are not required. To
determine the order of magnitude of the terros which should be retained in the
present analysis, two different types of approximation were considered. In the
first approach all terros of the order ('w6')4 and higher were neglected
compared to unity, while in the secoud approach terros of the order ( ws')2 were also neglected, thus only retaining terros linear in w6•
A finite element procedure, using Hermite interpolation polynomials was used
to determine the displaceroents in the first approach. The resulting systero of
nonlinear equations was solved using a Newton-Raphson procedure. Gomparing
the displaceroents obtained by the first approach with those of the secoud
(linear) approach, showed that the difference is negligible for the magnitude of
the prebuckling displacements under consideration [van Wanrooy, 1988]. The
possibility of using a linear theory instead of a nonlinear one in order to
deterroine the prebuckling displacements siroplifies the bifurcation problero
considerably .
../.3.3 The bifurcation criterion
Bifurcation under conservative loads in the elastic region may be defined
entirely in terros of the potential energy [Koiter,1945]. The increment in
potential energy for a structure going froro the prebuckling state I to an
adjacent state II may be written as
V ( 4.57)
where v2, v3 and v4 denote terros which are respectively of degree 2, 3 and 4
in the additional displaceroents. The critica! load and the associated buckling
mode can be obtained by requiring V 2 to be stationary i.e.
(4.58)
The potential energy of an inextensional bearo in the adjacent state II can be
written as (see (4.35))
- 4.16-
L II11 = ~ J [ EI2~ + EI34 + GJ~~:~ + Ef~~:i2 + EH~~:f
+ ~~:~(EI2,82~ - EI3.83~~:a + Er .8 tfi"D] dx (4.59)
-L[Pw5 + M2R13 + Pz(R33-l)J -0t[qw5 + qz(R33-l)Jdx
Substitution of (4.46), (4.48), (4.56) and (4.59) into (4.57) yields V in terms of
the displacements components. Retaining only those terms which are quadratic
in the additional displacements, neglecting terms of the order ( w~)2 and higher
compared to unity, and making use of the fact that the fundamental state is
an equilibrium state, yields
L V = J [-EI w''(-w'v'v" + l"'2w" + ""'") + 1EI (v" + .....W5")2 2 2 S SS S ~I S I"S ~ 3 S I""
- 1EI a w"("'' - v'w")2] dx ~ 2~2 S I S S (4.60)
~[M ( I 1 2-1 l_, '2) -IJ 2 fYs + ~~ ws - 2"wsvs
-J1 [ qz( fV~w~ - ~~2) J dx.
When the alternative warping formulation is employed, V 2 changes to
V = JL [-EI w''(-w'v'v" + l "'2w" + ""'") + 1EI (v" + .....W5")2 2 2 S 6 6 6 ~I 6 I"S ~ 3 S I""
+ ~Ef(g')2 - ~EI2,82w~'( 1' - v~w~')2
+ ~{16(1'- v~w~')2 + 2D.g('Y'- v~w~') + Dg2}Jdx
~[M ( I + 1 2-1 1- I '2) - IJ 2 fYs ~~ ws - ~wsvs
L -J [qz("fV~w~-h2)Jdx. ( 4.61)
All terms containing the additional in-plane displacements w5 are omitted
since they are all positive definite and can therefore only increase the value
- 4.17-
of the buckling load. As a result, no displacements w8 appear in the buckling
mode. Requiring the above functionals to be stationary renders the differential
equations of equilibrium and the associated natural boundary conditions.
The differential equations of equilibrium resulting from ( 4.60) are
-EI2(w~'1)" + (EI2v~w~w~" + EI2v~w~'w!')' + EI3(v~'+,W~')"
+ [{ GJ( 1'-v~w~') - Er(7'-v~w~')"}w!'l' = o
-[GJ( 1'-v~w~') - Er( 1'-v~w~')"]' - EI2w~'(v~'+,W~')
+ EI3w~'(v~'+,W~') o.
(4.62)
(4.63)
These equations are complicated and must in general be solved numerically.
However, a closed form solution can be obtained for the buckling of a simply
supported beam which is loaded in uniform bending, by assuming that
v s = Asinî:x and 1 = Bsinî:x ( 4.64)
where A and B represent the amplitude of v5 and 1 respectively. These
displacement functions satisfy the kinematic and natural boundary conditions.
The prebuckling state of this beam is characterized by
and 0. (4.65)
Substitution of (4.64) and (4.65) into (4.62) and (4.63) and representing the
result in matrix form yields
l-aM~+ b
-cM 2
11"2 where a = EI:D (1
2
Requiring the determinant of the matrix in ( 4.66) to be zero results in
( 4.66)
- 4.18-
(4.67)
This equation agrees exactly with that obtained by Vacharajittiphan e.a.
[1974}. To obtain the sa.me result with the energy formulations used by other
research workers [Roik,1972; Roberts and Azizian,1983; Attard,l986} some small
incorrect terms must be neglected. These terms appear due to an incorrect
representation of the torsional curvature and the fact that the term
EI2w~'(w~v~v~') in the second variation of the potential energy (V2) is missing
by the other research workers.
Note that when 13 = 12 the denominater of ( 4.67) is infinite which leads to
the condusion that bea.ms having equal flexural rigidities in the principal
planes of bending can not buckle laterally.
4.4 Numerical approach
4.4.1 Finite element formulation
To study the influence of the initia! curvature and that of the alternative
warping formulation, a computer program with the following options was
developed.
(i) A Linear buckling analysis using the classica! warping formulation.
(ii) A Linear buckling analysis using the alternative warping formulation.
(iii) A buckling analysis including the effect of an initia! curvature using the
classica! warping formulation.
(iv) A buckling analysis including the effect of an initia! curvature using the
alternative warping formulation.
The bea.m is subdivided into a set of line elements, each having two end
nodes. Since the in-plane displacements w s do not enter in the buckling mode,
the only displacement variables which play a role in the buckling analysis are
v6 , 'Y and g. For all three variables, cubic interpolation functions are employed.
The resulting displacement components for each element node are shown in
Figure 4.5 for both of the warping formulations.
- 4.19-
1 2 1 2 • •
v1 , v~ v2,v2 v1, v~ v2,v2
1t' 1~ 12' 12 1t,1~ 12,12
gl' g~ g2,g2
classical warping formulation alternative warping formulation
Fig. 4.5 Displacement components for each element node for different warping formulations.
Assembling all the displacement components of the beam in a column li, the
functional V 2, for each approach, may be written in the form
V2 = ~ls(w5)li. ( 4.62)
Requiring this functional to be stationary yields
( 4.63)
The matrix S can be expressed in terms of the prebuckling displacements as
( 4.64)
where S0 is a linear stiffness matrix, and S1, S2, and S3 are matrices which are
respectively linearly, quadratically and cubically dependent on the prebuckling
displacements. For a linear buckling analysis S2 and S3 are zero. The matrices
S0, S1, S2 and S3 are given in Appendix 4.1 for each buckling and warping
formulation.
For a linear prebuckling state, the displacements w5 can be expressed in terms
of a reference displacement field w r and a non--dimensional load factor À
Substitution of (4.65) and (4.64) into (4.63) results in
[S0 + .XS1(wr) + .X 2S2(wr) + .X3S3(wr)Jé = o.
( 4.65)
(4.66)
Equation ( 4.66) represents a cubic eigenvalue problem. Since no stable and
efficient algorithm is presently available for solving this problem, the following
iterative procedure was adopted.
The basic idea behind the procedure is that, within the elastic region, the
buckling load of a beam obtained by a linear buckling analysis constitutes a
reasonable estimate for that of a beam with an initial curvature. Without loss
in generality, the reference load system may be identified with this 'linear'
- 4.20-
buckling load. The critica! load factor of the linear buckling analysis thus
equals unity. The matrix S(Àwr) can be expanded in a Taylor series about this
value of À as
_ - as 1 2 82 s 1 3 8 3 s S(Àwr) = S(wr) + (À-1)(oxh + 2(À-1) (QX"2"h + 6(À-1) (QX3") 1. (4.67)
For values of À close to unity the higher order terms in (À-1) may be
neglected. Substitution of the remaining part of this series into (4.63) results
in the linearized eigenvalue problem
This equation can also be written in the form
[S0 + À{S1(wr)+S2(wr)+S3(wr)} + (À-1){S2(wr)+2S3(wr)}J5 = 0. (4.69)
The last term of ( 4.69) is also negligible in the neighbourhood of À=l. The
final linearized eigenvalue problem thus reads
( 4. 70)
The advantage of this last equation compared to (4.68) is that the matrix
which does not depent on À, is positive definite. This is very useful when
standard eigenvalue routines are used to solve ( 4. 70).
Once the eigenvalue of ( 4. 70) has been determined, this value is used as the
new point for the Taylor series expansion and the process is repeated until
convergence is obtained.
4.5 Numerical resu1ts
The influence of the initia! curvature on the buckling behaviour of slender
beams, has been evaluated using both warping formulations. The difference
between both approaches turned out to be negligible for the problem under
consideration. For beams with a closed or partly closed cross section, both
formulations lead to different stresses near the support, but this local effect
has a negligible influence on the overall buckling behaviour of slender beams.
4.5.1 Simply supported, laterally unrestrained beam in pure bending
In order to demonstrate the accuracy and versatility of the present method,
some numerical examples are presented. The first example to be considered is
that of a simply supported, laterally unrestrained beam, which is loaded by a
uniform bending moment. The loading, boundary conditions and geometrie
- 4.21 -
E = 206850 N/mm2
G = 82740 N/mm2
I2 = 4.57E7 mm4
r3 = 1.54E7 mm4
J = 222300 mm4 ~--___,iJ I~~
1.. L = 2540 mm •I f = 1.42Ell mm6
Fig. 4.6 Simply supported laterally unrestrained beam in pure bending.
properties are presented in Figure 4.6. The numerical result, obtained with 6
elements is presented in Table 4.1 tagether with the analytica! result based on
equation (4.67). In Table 4.1-4.4 h_ represents the height of the beam.
type of Buckl ing load x 1 0~ Nmm
percentage c lassical incl. in-pi ane w8UL)/~ a na lysis analysis deforma t ion s
mc r ease
numerical 0.556 0.685 23 0.3
analytica! 0.556 0. 685 23
Table 4.1 Percentage increase in the buckling load of a simply supported, laterally unrestrained beam in pure bending .
..{.5.2 Simply supported beam in pure bending with lateral end restraints
In this example, the same beam as in the first example is examined, but now
the beam is laterally restrained at the ends. The applied boundary conditions
are
v8(0) = v~(O) = r(O) = 1'(0) = 0
v8(1) = v~(1) = 1(1) = 1'(1) = 0.
The results are presented in Table 4.2. In this case the in-plane deformations
result in a decrease of the critical load. This example demonstrates that the
influence of the in-plane deformations cannot be accounted for by a correction
term which only depends on the cross sectional properties of the beam, as is
sametimes dorre [Allen and Bulson,1980]. The decrease of 5% for the buckling
load has also been found by Vacharajittiphan e.a. [1974].
- 4.22-
type of Buckl ing load x 1 O!! Nmm pereen tage a n alys is c I ass i cal incl. in-pI a ne increase w5 ( tL)/.h
analys is deformat ion s
numerical 1.966 1.868 -5 0.83
an a lytical 1.965 - -
Table 4.2 Percentage increase in the buckling load of a simply supported, beam in pure bending with lateral end restraints.
,/.5.3 Simply supported, laterally unrestrained beam loaded by a concentraled
force at midlength
Critica! loads were also obtained for the simply supported beam of the first
example with a concentrated load at midlength. The loading and boundary
conditions are shown in Figure 4. 7. The effect of the height of the point of
application above the shear centre on the percentage increase with respect to
the classica! buckling load was also studied. The results are presented in Table
4.3. The values demonstrate that the influence of the initia! curvature on the
buckling load depends on the point of application of the load.
L=2540 mm
v8(0) v8(L) = '}'(0) = '}'(L) = 0
Fig. 4.7 Simply supported beam loaded by a concentrated force.
Buck I i ng load x 106 N pereen tage classical incl. i n-plane w5 ( tL)/.h analys is de format ions mcrease
top f 1 ange 0.737 0. 821 11 0.15
centr o i dal 1.193 1. 4 72 23 0.28
bottorn fl. 1.919 2. 622 36 0.49
Table 4.3 Percentage increase in the buckling load of a simply supported, laterally unrestrained beam loaded by concentrated load with different points of application.
~ 4.23-
4.5.4 T-bcam under moment gradicnt
In this example, the influence of the initia! curvature on the buckling
behaviour of a monosymmetrie beam is investigated. When a monosymmetrie
beam twists during flexural~torsional buckling, a resulting torque is formed by
the bending compressive and tensile stresses. This effect is accounted for by
the monosymmetry parameter {32• The reduced attention which the buckling
behaviour of monosymmetrie beams has received, compared to that of doubly
symmetrie beams, may be partly due to the difficult determination of this
parameter. With the computer program described in section 4.6 this problem is
however completely removed.
The monosymmetrie beam considered in this example is a T -beam under
moment gradient. The loading, boundary conditions and geometrie properties
are presented in Figure 4.8. Both a T-beam and an inverted T-beam were
studied. Kitipornchai and Wang [1986] reported on the classica! buckling
behaviour of these beams. The numerical results obtained with 10 elernents are
presented in Table 4.4. The classica! buckling loads, nurnerically obtained,
agree with those obtained by Kitipornchai and Wang.
E = 200000 N/mm2
G = 80000 N/mm2
12 == 3.32E7 mm4
13 = 2.82E6 mm4
J = 0.72E5 mm4
r = o /32 = -274.5 mm
/(L) = 0
Fig. 4.8 Simply supported T-beam under moment gradient.
I . 1 percentage (.lL)/h c · 1 n-p a ne 1ncrease ws 2 -eformat ion s
k
62. 13 0.14 -1 15.61 0.04 112.0 0 0.13 20.83 0.02
0.0 21.49 1 21.49
0.0
Table 4.4 Percentage increase in the buckling load of a simply supported, laterally unrestrained T-beam under moment gradient.
- 4.24-
The val u es of Table 4.4 show that the lowest critica! load for the T -beam
occurs for a value of k equal to 1 (for doubly symmetrie cross sections k=-1
yields the lowest critica! load). The results obtained with the present approach
confirms the condusion reached by Kitipornchai and Wang [1986] that the
formulae for the moment modification factors given in many design
specifications and standard texts, are potentially unsafe for monosymmetrie
beams under moment gradient.
4.6 Geometrica.l constants of beams with arbitrary cross sections
4.6.1 General
The geometrical constants used in the computer program of the previous
paragraph must be supplied by the user. For beams with complex cross
sections (see Figure 4.9) these constants cannot be calculated analytically, nor
can they be obtained from a handbook. Especially the determination of torsion
related properties may cause problems. In seeking solutions to these 'torsional'
problems for bearns with arbitrary cross sections, it is traditional to use the
stress function representation of the problem as developed by Prandtl [1903].
Fig. 4.9 Some complex cross sections of aluminium beams.
This approach results in a Poisson equation, the salution of which is discussed
in many papers and textbooks on finite elements. The main disadvantage of
this stress function formulation is that no information is obtained with regard
to the warping distribution. This information is however vita! for the
determination of several cross sectional properties. It is therefore much more
convenient to deal with the Saint Venant representation of the torsion
problem, that is in terms of the warping function. The finite element
calculation of the torsional constants using the latter approach is already
described by several authors [Herrmann,1965; Brekelmans and Janssen,1972;
Surana,1979]. However, sirree it is believed that neither designers nor codes
- 4.25-
take sufficient advantage of the computing techniques which are nowadays
available to owners of personal computers, it was felt appropriate to pay some
attention to this item.
-f..6.2 Relevant definitions and expressions
The geometrie properties which play a role in the theory of the preceding
sections are:
- The area of the cross section (A).
- The centraid of the cross section (C).
- The orientation of the principal axes (y,z).
- The second moments of area about the principal axes
12 = AJz2dA and 13 = AJy2dA.
- De Saint Venant torsional constant
J =Af[(~- z? + (flz + y)2]dA.
- The alternative torsional constants
( 4. 71)
( 4. 72)
D = Af[(~? + (flz)2]dA and D* = AJ[(yflz- z~)]dA. (4.73)
- The coordinates of the shear cent re (y 5,z5).
- The polar second moment of area about the shear centre
I = J(y2 + z2)dA. s A
- The warping constant
r = AJ 7/fldA.
- The monosymmetry parameters
~2 = f J(y2 + z2)zdA - 2z5 2 A
~3 = f J(y2 + z2)ydA - 2y s· 3 A
- The symmetry parameter associated with warping
~1/J = t AJ(y2 + z2)1/JdA.
( 4. 7 4)
( 4. 75)
( 4. 76)
(4.77)
( 4. 78)
The warping function 1/J(y ,z) which appears in these constants must satisfy the
following Laplace equation
in A. ( 4. 79)
4.26-
With the Neumann type boundary condition
on Bo
where S represents the boundary of the cross section ,Y and z are given in
( 4.3) and ny,nz are the components of the outward unit vector n normal to
the boundary.
The boundary condition ( 4.80) is given in terms of the coordinates y ,z from
which the origin is located in the a priori unknown shear centre. This problem
can be bypassed through the replacement of the function 'ljl(y,z) by the
function e(y,z), which is given by
e(y,z) = '1/J(y,z) + yz6 - ZYs- ( 4.81)
The terms by which e(y,z) differs from 'l/l(y,z) express a rigid body motion, by
which the dependenee on the axes y,z is replaced by a dependenee on the
centroidal axes y ,z.
Substitution of (4.81) into (4.79) and (4.80) yields
(~)2 + (M)2 = o
Q~ + at:n ayY (Jzz
in A
on Bo. ( 4.82)
For a unique salution of the above equations, an additional condition for
e(y,z) has to be supplied. For practical reasons it is chosen for
Af edA = o. ( 4.83)
Once the function e is known, the coordinates of the shear centre can be
calculated with
and z6 = Î JyedA. 3A
( 4.84)
A finite element formulation with eight-node isoparametrie elements was used
to determine the principal axes with their second moments of area and the
function Ç(y,z). The isoparametrie concept is very useful in this context,
because it facilitates an accurate representation of irregular cross sections (e.g.
domains with curved boundaries). A 2x2 Gauss quadrature was employed to
determine the element properties. The storage and computational effort were
- 4.27-
reduced using a skyline or profile storage for the global stiffness matrix. A
modified Gauss-choleski algorithm was used to solve the linear system of
equations. Once the function Ç has been calculated, the determination of the
depending properties is straight forward.
The final computer program which takes full advantage of the symrnetry of
the cross section requires no more than one or two minutes on an IBM XT to
determine all the geometrie properties of beams with arbitrary shapes [van
Erp,1986].
,f.6.9 Numerical examples
- Reetangwar cross section
In order to demonstrate the accuracy and versatility of the present approach,
three different cross sections are analysed. The first section to be considered is
a rectangle with a width and height of 400 and 800 mm respectively.
z .. ~. 11 19 21 r----·
18 20
14 15 16
10 12 11
13
7 8
... y V
Fig. 4.10 One quarter of a rectangular section modelled with 4 elements.
Due to the symmetry of the section, only one quarter needs to be analysed.
This part of the section is modelled using four elements. The section properties
obtained with the present approach, together with the 'exact' values are
presented in Table 4.5. These values illustrate that even with this coarse mesh
excellent correlation is obtained.
c ros s sec t i on al properties
A [mm2] I2 [mm4] I3[mm4] J [mm4] r [mm6] Ys[mm]
num. 3.2E5 1. 707E10 4.26 7E9 1.1725E10 8.32 6El4 0.0
anal. 3.2E5 1.707 ElO 4.26 7E9 1.1712El0 8.324El4 0.0
Table 4.5 Cross sectional properties of rectangular section
- 4.28-
- Channel BEdion
In the second example, a thin-walled channel section is analysed. This type of
section is chosen in orderto demonstrate the accuracy of the determination of
the shear centre. The results are compared with those obtained with a
thin-walled beam theory [Timoshenko and Gere,1961]. The numerically
obtained warping constant differs slightly from the analytically obtained one,
because the present approach also considers the secondary warping. Six
elements are used for half the channel section (see Figure 4.11).
# m -11
I' ,. LL_:-_::-_::-_::-= .::J
I- 49mm • 1
Fig. 4.11 Element mesh for half the channel section.
cross sectional properties
A[mm2] I 2 [mm4] 13 [mm4] J [mm4] r [mm6]
num. 29 4 1.37 4 2E5 785 31 392 0 1 3.384E7
anal. 29 4 1.37 4 2E5 785 31 39 2 3.363E7
Table 4.6 Cross sectional properties of channel sections.
- Circular thin-walled beam
Yc[mm] y 5 [mm]
17 0 33 -37 0 3
17 0 33 -37 0 3
The last example to be considered is that of a thin-walled open beam with a
circular cross section. The advantage of the eight-node isoparametrie elements
is demonstrated by the limited number of elements needed to describe the
geometry of the cross section accurately (see Figure 4.12). The numerical and
analytica! results are presented in Table 4. 7 . Once again excellent agreement
is obtained with just a few elements.
- 4.29-
Fig. 4.12 Element mesh for half the circular thin-walled beam.
c r os s sec t i on al p r ope r t ie s
A[mm2J I 2(mm4] I 3[mm4J J (mm4] r (mm6] y 8 (mm]
num. 659 3.0 3.63 9E7 3.639E7 2.17 9E5 1. 031 E12 -209.7
anal. 659 7.3 3.64 5E7 3.64 5E7 2.199E5 1. 034E12 -210.0
Table 4.7 Cross sectional properties of circular thin-walled beam.
The examples were restricted to simply shaped cross sections in order to be
able to compare the numerical results with analytica! results. The approach,
however, has also been applied with succes to the complex sections of Figure
4.9. The effect of the mesh size on the accuracy of the results is discussed by
Menken and van de Pasch [1986].
The present method has been extended such that to different elements
different material parameters may be assigned. With this extended version,
homogeneaus as well as inhomogeneous cross sections can be evaluated. This
option is also attractive when the reduced stiffness method is used to study
interaction buckling, because different stiffness moduli may be assigned to
different elements.
4. 7 Conclusions
A potential energy functional for the nonlinear flexural-torsional behaviour of
straight elastic beams has been presented. The result is generally applicable to
beams which undergo arbitrary large deflections and rotations.
This functional has been used to derive the energy functional which governs
the flexural-torsional buckling behaviour of simply supported inextensional
-4.30-
beams which exhibit prebuckling displacements of the order of the heîght of
the beam. A closed form salution obtained with this functional, without
neglecting any term, agrees exactly with that of Vacharajittiphan e.a. [1974],
who used an equilibrium approach. To obtain the same result with the energy
formulations of other research workers some small but incorrect terms must be
neglected. These terms appear due to an incorrect representation of the
torsional curvature and the fact that a term which depends on the magnitude
of the prebuckling displacements is missing in the energy functional of the
other research workers.
The present approach is cast in the form of a finite element model, which has
shown to be an accurate and versatile tooi to study the buckling behaviour of
beams with an initia! bending curvature. An exhaustive study of its
applications was beyond the scope of this thesis, but from the examples
presented in this chapter the following can be concluded.
(i) Representing the influence of the prebuckling deformations on the critica!
load by a correction term which only depends on the geometrie properties
of the beam, as is often done, may lead to unsafe results.
(ii) The formulae for the moment modification factors given in many design
specificatîons and standard texts may result in unsafe results for mono
symmetrie bearns under moment gradient and other specific loading
conditîons, even if the influence of the prebuckling deformations is taken
into account.
Two different warping formulations were used in the present theory, the Saint
Venant formulation and an alternative formulation where the amplitude of the
warping distribution depends on an unknown function g(x). The difference
between both approaches turned out to be negligible for the problem under
consideration.
The last section of this chapter has been devoted to the determination of the
geometrie properties of bearns with complex cross sections. The approach is
based on the Saint Venant representation of the torsion problem, that is in
terrns of the warping function. A finite element model using eight-node
isoparametrie elements was developed to determine all the geometrie properties
of bearns with arbitrary cross sections. The resulting computer program, which
only requires a few minutes computer time on an IBM XT personal computer,
has been shown to be accurate and versatile.
-5.1-
5 SUMMARY AND CONCLUSIONS
This thesis describes recent work performed at the Eindhoven University of
Technology in order to develop buckling, post-buckling and interaction
analyses of beams with arbitrary cross sections.
Tests, performed at the same university, showed that effects which are not
taken into account by the classica! buckling analysis of beams may have a
significant influence on the elastic flexural-torsional buckling behaviour of
thin-walled beams. These effects are.
(i) Distartion of the cross section during buckling.
(ii) Interaction between buckling modes
(iii) Large in-plane deflections before buckling ( especially with stender
aluminium beams).
For studying the first two effects, a computer program has been developed
which is based on the combination of the spline finite strip method and
Koiter's general theory of stability. This combination proved itself very
valuable for accurately studying the local and distortional buckling, including
the interaction between buckling modes, of thin-walled members under
arbitrary loading. With this spline approach, the simplicity of the semi
analytica! finite strip method is preserved, while problems of dealing with
non-periadie buckling modes, shear and non-simple support are eliminated.
The number of degrees of freedom required in a spline finite strip analysis is
considerably larger than that of the semi-analytica! finite strip method, but it
is still approximately 40% less than that of a camparabie finite element
approach.
The computer time needed by the present approach depends on the number of
strips and sections used in the analysis. In Chapter 2 it is demonstrated that
approximately two sections per half wave are required in order to describe a
local mode correctly. The periodicity of the second order field of a local mode
is about twice as large that of the associated mode. This means that four
sections per half wave of the local mode are required to describe the second
order field to within engineering accuracy. Consequently, the computational
effort needed by the present approach to analyse structures with many half
waves (> 20) in the buckling modes is considerably. For this type of
structures, preferenee should be given to the semi-analytica! finite strip
approach for the time being.
5.2-
The present method is based on the assumption that the buckling loads
coincide or nearly coincide. Other research workers have applied the method to
structures with well-separated buckling loads too. The validity and accuracy
for these cases is still open to discussion. More research work and, especially,
well controled experiments will be needed to answer these questions.
For studying the influence of the prebockling displacements on the buckling
behaviour of slender beams, a nonlinear beam theory has been derived which is
generally applicable to beams undergoing arbitrary large deflections and
rotations. This theory was used to derive the energy functional which governs
the flexural-torsional buckling behaviour of mono- and doubly symmetrie
beams which exhibit relatively large prebuckling displacements (in the order of
the height of the beam). Only inextensional beams loaded in bending were
considered. A closed form solution obtained with this functional, without
neglecting any term, agrees exactly with that of Vacharajittiphan e.a. [1974],
who used an equilibrium approach. To obtain the same result with the energy
formulations of other research workers, some .small incorrect terms must be
neglected. These terms appear due to an incorrect representation of the
torsional curvature and the fact that a term which depends on the magnitude
of the prebruckling displacements is missing in the energy functional of the
other research workers.
The general equations were so~ved using a finite element formulation with
Hermite cubic interpolation 'functions for each element. From the few examples
which were analyseçl with the present method the following could already be
conclud~d.
(i) · Representing the influence of prebuckling deformations on the critica!
load by a correction term which only depends on the geometrie properties
of the beam, as is often done, may lead to unsafe results.
(ii) The formulae for. the moment modification factors given in many design
specifications and standard texts may lead to unsafe results for mono
symmetrie beams under moment gradient and other specific loading
conditions, even if the influence of the prebuckling deformations is taken
into account.
Two different warping formulations have been used in the present work, the
Saint Venant's warping formulation and an alternative formulation where the
amplitude of the warping distribution depends on an unknown function g(x).
-5.3-
The difference between the two approaches turned out to be negligible for the
problem under consideration.
The last part of Chapter four is devoted to the determination of the geometrie
properties of beams with arbitrary cross sections. The resulting computer
program, which only requires a few minutes computer time on a standard IBM
XT personal computer, has been shown to be accurate and versatile.
- A2.1 -
APPENDIX 2.1
Matrices used in the denvation of the buckling model of chapter two.
The matrices N and w of equation (2.15) are given by
[ ~· 0 0 0 N2 0 0
~J N= N, 0 0 0 N2 0 (A2.1)
0 N3 N4 0 0 N5
T/JT OT 0 0 0 0 0 0 0 T/J OT 0 0 0 0 0 0 0 T/J OT 0 0 0 0
W= 0 0 0 T/J OT 0 0 0 (A2.2) 0 0 0 0 T/J 0 0 0 0 0 0 0 0 T/JT 0 0 0 0 0 0 0 0 T/JT OT 0 0 0 0 0 0 0 T/J
where 0 represents a row of length m+3.
The matrix B which is used for the first time in (2.42) has the form
The matrices B1 and B2 which appear for the first time in (2.42) are given by
r ;,VT tl 0 0 0 fixT 0 0 N, N2
B, = x (A2.4)
0 0 0 8N2T/JT 0 0 0 Oy Oy
~T ~T fixT fixT 0 0 N3 N4 0 0 N5 N6 B2 = x (A2.5)
0 0 oN3T/JT oN4T/JT 0 0 8N5T/JT oN5T/JT oy oy oy oy
- A3.1 -
APPENDIX 3.1
Derivation of P[ai,v,.,\]
According to eqn. (2.33), P[TJ] may be written as
P[TJ] = AJ[fr11(TJ)TDL1(TJ) + ~,\L 1(~)TDL2(TJ)
+ ~L 1(TJ)TDL2 (TJ) + ~12(TJ)TDL2(TJ)]dA. Substitution of TJ = aiui + v into (A3.1) yields
P[ai,v,.,\] =
AJ[fraiaiL1(uiDL1(ui) + aiL1(uiDL1(v) + ~L 1(v)TDL1(v) + ~aiaj,\L 1 (u0)TDL 11 (ui,uj) + ai,\L1(u0)TDL11(ui,v)
+ fr,\L 1(u0)TDL2(v) + aiajL1(uiDL11(ui,v)
1 T 1 ( )T ( + 2aiaj11(v) DL11(ui,uj) + 2aiajak11 ui DL11 uj,uk)
+ ~aiajakal 1 11 ( ui,uj) T DL11( uk,u1)
+ ~a 1L 1(aiDL2(v) + a1L1('1)TDL11(üpv) + ~L1('1)TDLk;)
+ ~a1aJakL11(ü~'üJ)TDL 11(ük,'1) + ia1aJL11(ü1,üJ)TDL2(v)
(A3.1)
(A3.2)
+ àa1aJL 11 (ü~'v) T DLn(uJ ,v) + ~a1L 11 (upv) T DL2(v) + ~Lk;) T DL2(v)J dA.
Both aiui and v are small in the vicinity of the bifurcation point. The terms
which are striked out are neglected because they are of a higher order of
smallness.
Using the orthonormality relations (2.37)-(2.39), yields
M À J[ 1 T P[ai,v,.,\] = E (1-x )a1a
1 + 2L1(v) DL1(v)
I=1 I A
+ ~,\L 1(u0)TDL2 (v) + aiajL1(uiDL11(ui,v) (A3.3)
- A4.1-
APPENDIX 4.1
lnfluence of prebucking deformations on the buckling behaviour of simply
supported beams in uniform bending
In section 4.3.3 it is demonstrated that the buckling behaviour of a simply
supported beam which exhibits prebuckling displacements of the order of the
height of the beam can be characterized by the following system of equations
r-aM~+ b
-cM 2
-cM21 r A 1 r 0 1 -dM~+ e B 0
where
The determinant of this matrix is given by
adMi - (ae+bd+c2)M~ + be.
Requiring this determinant to be zero yields
M2 _ (ae+bd+c2) ± J(ae+bd+c2)- 4abde1
2 - 2ad ·
(A4.1)
(A4.2)
(A4.3)
For the salution of this equation the following short farms are introduced
~ = p and 2
( GJ + f1r 2 ) _
~ r;;v - q. 2 2
The terms which occur in (A4.3) can be expressed in p and q as
11"4 ae = V (q-q2)
11"4 bd = V (p-p2)
11"4 2 c2 = V (1-p--q)
4 (ae+bd+c2) = V (l-p-q+2pq)
(A4.4)
(A4.5)
(A4.6)
(A4.7)
(A4.8)
- A4.2-
7r8 4abde = 4v ( qp-p2q---q2p+q2p2)
7r8 (ae+bd+c2)2 - 4abde = V (1-p---q)2
7r2 ad = (EI2)2L2 (1-p)(1---q).
Substitution of (A4.10) and (A4.11) into (A4.3) yields
7r4 7r4 L4 (1-p--q+2pq) ± L4 (1-p---q)
M 2 - -------c;-----"---------2 - 2 7r
(EI2) 2[2 (1-p )(1---q)
Consictering the minus sign in (A4.12) results in
7r2 r 7r2l M~ = p EI3 lGJ + Er~
[1- t~][1- ~L [1 + ~}r~JJ Taking the square root of equation (A4.13) yields
(A4.9)
(A4.10)
(A4.11)
(A4.12)
(A4.13)
(A4.14)
- A4.3-
APPENDIX 4.2
Matrix formulation of the buckling problems
Within each element a local coordinate axis Ç is chosen, which has its ongm
at the centre of the element and is scaled such that Ç=-1 at the left end
node and Ç=1 at the right end node.
Ç = ~- 1 and J\x)dx = J1
f(Ç)~Ç (A4.15) 0 -1
where h represents the element length and x is an element coordinate which
ranges from 0 to h. The displacements within the element can be expressed in
terms of Ç as
ws(Ç) = Ni(Ç)wi
vs(Ç) = Ni(Ç)vi
1(Ç) = Ni(Ç)1i
g(Ç) = NM)gi
i=(1,4)
(A4.16)
(A4.17)
(A4.18)
(A4.19)
where wi,vi,'Yi and gi represent the variables at the element nocles which are
shown in Figure 4.4 and Ni( Ç) represent the Hermite cubic interpolation
polynomials which are given by
Nl(Ç) = 1 -3(Ç!1)2 + 2(Ç!1)3
N2(Ç) = h[Ç11 _ 2(ç!1)2 + (Ç!1)3]
N3(Ç) = 3(Ç!1)2 - 2(Ç!1)3
N4(ç) = h[-(Ç!1)2 + (Ç!1)3].
(A4.20)
(A4.21)
(A4.22)
(A4.23)
Substitution of (A4.16)-(A4.19) into the energy functional V2 with the
classica! warping formulation ( 4.60) yields
V2 = ~ àó![S8 + S! + s~ + S~Jóe (A4.24)
where 88 is a linear stiffness matrix, and sr, s~, and s~ are matrices which are
respectively linearly, quadratically and cubically dependent on the prebuckling
displacements. V 2 can be expressed in terms of the interpolation functions as
1
~ó!S8óe = à I [EI3(~) 3Nj'Nj'vivj + Ef(~) 3Nj'Nj''YiÎj -1
+ GJ(~)NjNj'YïÎj + qz(~)NiNj'Yï'Yj]dÇ - ~[M21V~ - %Pzy] (A4.25)
- A4.4-
~o!SiSe = ~ J1
[-2EI2(~)3Nk'\vkNiNj'lïvj + 2EI3 (~)3Nk'wkNiNj'!ivj -1
- EI2,82(~) 3Nk_'wkNjNjlïÎj - 2GJ(~) 3Nk'wkNiNj·hvi
- 2Ef(~)5(N' 'w- N!'N!''Y·V· + N"'''' N!'N!,..,.v.) h k k' 1 J '1 J k " k' 1 J 11 J
(A4.26)
!w'v' 2) + Pzv'w' 1] 2 s s s s
L·Tge• - l J1[ EI (2)5N"N' -, - N'N" :I0 e 2°e - 2 2 2 h k lwkwl i' j vivj
-1
EI (2)3N"N"- - N N EI (2)3N"N"- - N N - 2 h · k 1 wkwl i jÎiÎj + · 3 ïï k 1 wkwl i jli1j
2EI a (2)5N "N"- - N'N' GJ(2)5N "N"- - N'N' + 2/J2 ïï k 1 wkwl i jÎivj + h k 1 wkwl i jvivi
+ Er(~)1(N:k'Ni'wkw1Nj'Nj'vivi + 2N:k'Nl"wkw1NiNj'vïvi
(A4.27)
(A4.28)
For the linear buckling analysis with the classica! warping formulation, S~ and
Sä are zero and Sr is given by
~s!srse = ~ J1
[-2EI2 (~)3Nk'wkNiNj'livi -1
- EI2,82 (~)3Nk.'wkNjNjlïlj]dÇ. (A4.29)
When the alternative warping formulation is employed the matrices are given
by
~o!S88e = ~ f[EI3 (~)3Ni'Nj'vivi + Ef(~)NiN]gigi -1
+ GIS(~)NjNj1iÎj + 2GD*NiN]gilj + GD(~)NiNjgigj]dÇ
- A4.5-
(A4.30)
~b!S1be = ~ J1
(-2EI2 (~) 3Nk'wkNPj''Yivi + 2EI3 (~) 3Nk'wkNiNj''Yivi -1
- EI2,82 (~) 3Nk'wkNjNj'Yi'Yj - 2GI5(~) 3Nk'wkNjNj'Yivj
- 2GD*(~?Nk'wkNiNjgi'Yj - qz(~)NkwkNjNjvi'Yj]dÇ
~(1M A2-t 1-, t2) p- ,_, J - LJ 2 2'1 ws - 2wsvs + zvsWs'f'
+ 2EI2,82(~) 5Nk'Ni'wkw1NjNj'f'iVj + GI5(~) 5N k'Ni 'wk w1 NjNjv ivi]dÇ
(A4.31)
(A4.32)
(A4.33)
For the linear buckling analysis with the alternative warping formulation, S~
and S~ are zero and 81 is given by (A4.29).
Per element 6 Gauss points were used for the buckling analysis which
considers the influence of the prebuckling deformations, while 4 Gauss points
per element were employed in the linear buckling analysis.
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SAMENVATTING
Bij het knik- (kip) gedrag van konstrukties met dunwandige delen in de
dwarsdoorsnede, kunnen een aantal effecten een rol spelen waarmee in de
'klassieke' stabiliteitsberekeningen geen rekening wordt gehouden. Deze effecten
zijn.
(i) Vervorming van de dwarsdoorsnede.
(ii) Interaktie tussen knikvormen.
(iii) Aanzienlijke doorbuiging voor het kippen (vooral bij slanke aluminium
profielen).
In dit proefschrift worden een aantal numerieke modellen beschreven waarmee
de invloed van deze effecten op het knikgedrag van liggers met een wille
keurige dwarsdoorsnede en belasting bestudeerd kan worden. De ontwikkeling
van deze modellen heeft plaats gevonden in het kader van een onderzoeks
project van de faculteit der Werktuigbouwkunde van de Technische Universiteit
Eindhoven.
Na een inleiding en verantwoording van de gemaakte keuzes in hoofdstuk 1,
wordt in hoofdstuk 2 een eindige strippen programma beschreven. Hiermee
kunnen de bifurcatie punten en de bijbehorende locale en vervormde globale
knikvormen van zowel op druk als op buiging en/of afschuiving belaste
konstruktiedelen bepaald worden. Bij dit computer programma wordt gebruikt
gemaakt van het prismatische karakter van de konstruktie middels een strippen
aanpak en zijn splines toegepast als interpolatie funkties. De mogelijkheden van
het programma worden aan de hand van een aantal voorbeelden gedemon
streerd.
Indien een konstruktie meerdere knikvormen bezit kan er sprake zijn van
interactie tussen deze knikvormen, wat vooral bij samenvallende en bijna
samenvallende kritieke belastingen de belastbaarheid nadelig kan beïnvloeden.
Voor de bestudering van dit fenomeen is een computermodel ontwikkeld dat is
gebaseerd op de combinatie van het spline eindige strippen programma van
hoofdstuk 2 en de algemene stabiliteitstheorie van Prof. Koiter. Met deze
kombinatie kan het initiële naknik gedrag van zowel op druk als op buiging
en/of afschuiving belaste konstruktiedelen met dicht bij elkaar liggende kritieke
belastingen bepaald worden. Evenals in hoofdstuk 2 worden de nauwkeurig
heid en de mogelijkheden gedemonstreerd aan de hand van een aantal
voorbeelden.
In hoofdstuk 4 wordt de ontwikkeling van een eindige elementen model
beschreven dat rekening houdt met de invloed van een relatief grote voor
doorbuiging (orde van de balkhoogte) op de kiplast van slanke balken. Nadat
in het eerste deel van het hoofdstuk de geometrisch niet lineaire balktheorie is
afgeleid die als basis heeft gediend voor het programma, wordt ingegaan op de
bepaling van de uiteindelijke energie funktionaal en de oplossing van het
resulterende derde graads eigenwaarde probleem. Omdat de doorsnede
grootheden die in het komputer programma ingevoerd moeten worden voor
complexe dwarsdoorsnede moeilijk te bepalen zijn, is het laatste deel van dit
hoofdstuk besteed aan de beschrijving van een eindige elementen model
waarmee deze grootheden voor een willekeurige dwarsdoorsnede snel bepaald
kunnen worden.
Tot slot worden in hoofdstuk 5 de conclusies en aanbevelingen, die reeds bij de
betreffende hoofdstukken aan de orde zijn geweest, kort samen gevat.
Levensbericht
1~5-1956
1968-1973
1973-1977
1977-1978
1978-1979
1979-1983
1983-1985
1985-1989
Geboren te Vught
HAVO Maurick College te Vught
HTS s'-Hertogenbosch afdeling Weg- en waterbouwkunde
Uitvoerder bij de Surinaamse Constructie Maatschappij te
Paramaribo, Suriname
Tekenaar-constructeur bij ingenieursbureau DHV te
Amersfoort
Studie bouwkunde aan de Technische Universiteit Eindhoven
Projektingenieur bij ingenieursbureau Witteveen+ Bos te
Deventer.
Wetenschappelijk assistent aan de Technische Universiteit
Eindhoven, afdeling Werktuigbouwkunde
Stellingen
behorende bij het proefschrift
ADV ANCED BUCKLING ANALYSES OF BEAMS
WITH ARBITRARY CROSS SECTIONS
1) De 11 spline eindige strippen 11 methode is een uitstekend hulpmiddel bij de
bestudering van knik (kip) van prismatische plaatachtige constructiedelen.
- Dit proefschrift, hoofdstuk 2.
2) Het in rekening brengen van de invloed van de initiële doorbuiging op de
kiplast van liggers door middel van een factor die alleen afhankelijk is
van de geometrie, kan leiden tot een aanzienlijke overschatting van deze
kritieke belasting.
Allen, H.G. and Bulson, P.S. 1980. Background to buckling.
McGraw-Hill Book Company (UK) Limited.
Dit proefschrift, hoofdstuk 4.
3) Als verschillende belastingstoestanden bij een ligger leiden tot hetzelfde
maximale buigend moment, is het bij liggers met een mono--symmetrische
dwarsdoorsnede niet zo dat bij de belastingstoestand met het grootste
oppervlak onder de momentenlijn altijd de laagste kiplast behoort.
Kitipornchai, S. and Wang, C.M. 1986. Lateral buckling of Tee beams
under moment gradient. Computers & Structures 23, No. 1, 69-76.
Dit proefschrift, hoofdstuk 4.
4) Bij op buiging belaste constructiedelen is het van essentieel belang dat,
bij het onderzoek naar interactie van knikvormen, rekening wordt
gehouden met het feit dat ook de globale knikvorm een vervorming van
de dwarsdoorsnede kan bevatten.
- Dit proefschrift, hoofdstuk 3.
5) Het gebruik van de aanduiding 11 lineaire interactie11 voor een knikvorm,
waarin zowel globale als locale vervormingen voorkomen, moet worden
afgeraden omdat deze vorm direct resulteert als de oplossing van een
eigenwaarde probleem en niet het gevolg is van een structurele interactie.
Bradford, M.A. and Hancock, G.J. 1984. Elastic interaction of local
and lateral buckling in beams. Thin-Walled Structures 2, 1-25.
6) De zogenaamde "gereduceerde breedte" methode is bij onderzoek naar de
interactie van knikvormen slechts beperkt toepasbaar en zal bij afnemende
wanddikten en toenemende complexiteit van de dwarsdoorsnede van
constructiedelen vervangen moeten worden door meer fundamentele
werkwijzen.
Wang, S.T., Yost, M.I. and Tien, Y.L. 1977. Lateral buckling of locally
buckled beams using finite element techniques. Computers & Structures,
Vol. 7, No. 7, 469-475.
7) Hoewel de huidige stabiliteitsprogrammatuur in het algemeen van zeer
goede kwaliteit is, bepaalt de deskundigheid van de gebruiker nog steeds
in grote mate de correctheid van de berekende oplossingen.
Bushnell, D. 1985. Computerized buckling analysis of shells. Martinus
Nijhoff Publishers, Dordrecht.
8) Omdat het gedrag van een op buiging belaste metselwerk wand in wezen
niets gemeen heeft met dat van een lineair elastische isotrope of
orthotrope plaat, is het aan te bevelen de huidige berekeningsmethoden te
vervangen door methoden die beter aansluiten bij de werkelijkheid.
- SBR rapport F5 1986. Op buiging belast metselwerk.
9) Genetische manipulatie is een gebied waar voor de mens veel te vinden
is, maar waar hij eigenlijk niets te zoeken heeft.
10) Als er niet snel iets verandert zal het aantal mensen dat uit een goed
milieu komt in de toekomst aanzienlijk afnemen.
11) Bij te weinig middelen leidt hoger onderwijs voor velen tot kwaliteits
verlies voor allen.
Gerard van Erp, mei 1989