Vax | V-096X | FRY, TA 1 .U4956 - UQ eSpace

Large Deflection and
Post-Buckling Analysis of
Two and Three Dimensional
Elastic Spatial Frames
J. L. MEEK and
FRY,
H. S. TAN
TA
1
.U4956
esearch Report No. CE49
December, 1983
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CIVIL ENGINEERING RESEARCH REPORTS
This report is one of a continuing series of Research Reports published by
the Department of Civil Engineering at the University of Queensland. This
Department also publishes a continuing series of Bulletins. Lists of recently
published titles in both of these series are provided inside the back cover of
this report. Requests for copies of any of these documents should be addressed
to the Departmental Secretary.
The interpretations and opinions expressed herein are solely those of the
author(s) . Considerable care has been taken to ensure the accuracy of the
material presented. Nevertheless, responsibility for the use of this material
rests with the user.
Department of Civil Engineering,
University of Queensland,
St Lucia, Q 4067, Australia,
[Te1:(07) 377-3342, Telex:UNIVQLD AA40315]
LARGE DEFLECTION AND POST -BUCKLING ANALYSIS
OF TWO AND THREE DIMENSIONAL ELASTIC
SPATIAL FRAr·1ES
by
J. L. r1EEK, MS Calif. , BE, BSc, PhD,
Associate Professor of Civil Engineering
and
H. S. TAN, BSc, ACGI,
Post Graduate Student
RESEARCH REPORT NO. CE 49
Department of Civil Engineering
University of Queensland
December, 1983
Synopsis
The safe and economical design of curved spatial
frames necessitates a geometrical nonlinear analysis of
the structure.
In this report,
the nonlinear equations
governing such structural behaviour are derived.
Solution
strategies which are capable of analysing the post­
buckling behaviour of structures are also critically
reviewed.
In addition, an extrapolation stiffness method
that lends itself to a more efficient solution of the
nonlinear equations than conventional Newton-Raphson methods
is presented.
The post-buckling responses of various spatial
frames are analysed and the results found to be in good
agreement with available published work.
•..
"
�-
:: �.·. "\
"
.<-."
;.J
CONTENTS
.a..
l�a.. &)"..�.. ....Q,.
<,.J
<._)
_,.
lC\�4
Fr '6e-r
Page
1
1.
INTRODUCTION
2.
DERIVATION OF GOVERNING NONLINEAR EQUATIONS
3
2.1
GOVERNING EQUATIONS FOR PLANE FRAME
7
2.1.1
7
2.2
Member basic force-displacement relation
GOVERNING EQUATIONS FOR SPACE FRAMES
12
2.2.1
Member basic force-displacement relation
12
2.2.2
Transformation from Member Basic Force/
Displacements to Member Nodal Force/
Displacement (Local Co-ordinates)
16
Transformation from Member Co-ordinates
to Global Co-ordinates
22
Evaluation of Member Rotational Deformations
from Joint Orientation Matrix
23
2.2.3
2.2.4
3.
SOLUTION STRATEGIES FOR TRAVERSING LIMIT POINTS
25
4.
CONSTANT ARC LENGTH STRATEGY OF CRISFIELD
33
5.
EVALUATION OF POST-BIFURCATION EQUILlBRIU�1 PATH
(SECONDARY PATH)
36
6.
EXTRAPOLATED STIFFNESS PROCEDURE
40
7.
NUMERICAL EXAMPLES
42
7.1
CANTILEVER BEAM
42
7.2
WILLIAMS' TOGGLE FRAME
42
7. 3
TWELVE MEMBER HEXAGONAL SPACE FRA�1E
45
7.4
TWENTY-FOUR MEMBER STAR-SHAPED DOME
50
8.
CONCLUSION
50
APPENDIX A
-
NOMENCLATURE
52
APPENDIX B
-
REFERENCES
56
-1-
1.
INTRODUCTION
Curved structural space frames belong to a class of structures
where a large proportion of the applied loading is resisted by the axial
forces in the members.
This together with the trend towards optimum
light weight members, for reasons of economy and aesthetics, has brought
to the fore the stability analysis of this type of structure, as it is
inclined to fail through elastic instability before exhibiting any sig­
nificant nonlinear material response.
The elastic buckling load has in
the main been calculated using linear instability theory, through a
linearised eigenvalue approach.
For structures where the pre-buckling
path is linear or close to being linear, this method will suffice. How­
ever, in the presence of initial imperfections and geometric nonlinearities,
this procedure can grossly overestimate the actual buckling load.
In
addition, the information furnished by a linearised eigenvalue analysis
is limited.
The pre-buckling load-deflection path is not traced out and
more importantly the nature of the bifurcation failure, whether the
failure is catastrophic or there is an increase in post buckling stiff­
ness as the structure deforms further, is not known.
These considerations
lead us to the desirability of considering the geometrical nonlinearities
when formulating the equilibrium equations, the result of which is a
system of nonlinear equations.
The formulation of the nonlinear equations governing the large
deflection behaviour of framed structures has been extensively studied
by various authors and is based on either the finite element method [1-5]
or on the 'beam-column' approach [6-13].
In describing the motion of
the element, a total Lagrangian or an updated Lagrangian description can
be employed.
The finite element formulation is generally used in con­
junction with a total Lagrangian material description.
Analyses based
-2-
on this approach are however 1 imited by the magnitude of the joint
rotations.
To overcome this problem of large joint rotations, an up­
dated Lagrangian approach can be employed which separates the effects
This is
of pure member deformations from the joint displacements.
achieved by introducing a local, convective reference system attached
to the member.
The nonlinear equations can be· solved by utilising the uncon­
strained minimisation algorithms of mathematical programming.
All the
geometrical nonlinearities are included in the potential energy function­
al which is then minimi.sed.
Techniques such as the random search method
[14] and Powell's method of conjugate direction [15] utilise the object
function only;
the method of steepest descent [14] and the conjugate
gradient method [16], in addition makes use of the gradient (first
derivative);
while the quasi-Newton [17-19], secant-Newton [20] and
Newton-Raphson [21] require the evaluation of the 2nd derivates as well.
This approach has been adopted by Berke and Mallet [2] as early as 1969,
and more recently by Papadrakakis [13] and Kamat [22] to trace the
complete load-deflection path of spatial framed structures.
An alternative to solving the nonlinear equations lies in
using an incremental form of the equilibrium equations.
The linearised
incremental-iterative techniques appear to be more efficient than the
minimisation methods for large scale problems.
It however, breaks down
at the first onset of instability due to the singularity of the incre­
mental stiffness matrix.
Earlier work on the large deflection behaviour
of framed structures employing an incremental iterative method has thus
been limited to studying their prebuckling behaviour [8-10,23] and it
is only very recently that robust and efficient techniques that circum­
vent this problem have been developed.
Several good reviews of these
-3-
techniques for traversing the limit points exist in the literature
[24-27].
These solution strategies have invariably been applied to the
analysis of thin shell structures.
To the authors' knowledge the only
published work utilising these nonlinear solution strategies to spatial
frames is that
of Reference
[4,28] on two-dimensional frames and of
Reference [5] on three-dimensional frames;
were presented.
where only limited examples
There is thus scope for further work in applying these
newly developed nonlinear solution strategies to analysing the postbuckling response of spatial frames.
In the linearised incremental-iterative technique, equilibrium
iterations are performed within each increment.
( NR )
In the Newton-Raphson
method, each iteration would entail the assembly and factorisation
of the tangent stiffness matrix.
This is, however, very expensive and
in practise the tangent stiffness is kept constant throughout the incre­
ment or is changed only after a few iterative cycles.
will be needed for the modified Newton-Raphson method
More iterations
( mNR ) ,
but this
may more than outweigh the cost of reforming and refactorising the
incremental stiffness matrix at every cycle.
The authors in Reference
[29] have developed an extrapolated stiffness strategy which was found
to be more efficient than the mNR or NR methods, when tested for plane
frames.
The applicability of the conclusions reached in Reference [29],
to three-dimensional frames is also demonstrated in this report.
2.
DERIVATION OF GOVERNING NONLINEAR EQUATIONS
The 'beam-column' approach has been used here in deriving the
governing nonlinear equations.
Here in the members are assumed to be of
constant cross-section with the external loads applied only at the joints.
Only conservative or displacement independent loading is considered.
-4-
Shear deformations are neglected and the matedal is assumed to be
linearly elastic.
For the three-dimensional frames, the members' cross­
sections are doubly symmetric, thus excluding coupltng of the torsiona 1
stiffness to that of the bending and axial stiffness.
Warping effects
are also neglected.
In developing the nonlinear equations, two types of co-ordinate
systems are employed;
a fixed global set of co-ordinates and a local
convective system which rotates and translates with the member.
The
member deformations are thus separated from the joint displacements.
The basic member force-deformation relations are derived in the local
convective co-ordinates with member deformations assumed to be small
relative to it, through the principle of minimum potential energy.
This
formulation of the equilibrium equation through an updated Lagrangian
approach is thus applicable to analysing structures exhibiting large
rotation small strain behaviour.
The member forces and displacements in
the various co-ordinate systems are illustrated i.n Figures 1 and 2 for
the two and three-dimensional frames respectively.
The rotational degrees of freedom are described by Euler angles
for the case of the two-dimensional frames.
Due to the occurence of
large rotations in post-buckling analysis, this description by Euler
angles is unsatisfactory for the analysis of three-dimensional space
frames.
This stems from the fact that finite rotations in three­
dimensional space are non-vectori.al quantities in that they do no comply
with the rules of vector transformation.
To overcome this difficulty,
the concept of a joint orientation matrix comprising of a triad of
orthogonal unit vectors as presented in References [12 and 30] is employed
to describe the arbitrary large nodal rotations.
-5-
/
Mz
1-P-x1
/
(a) Member basic displacements and forces
M,'
,,
T+\
:__-
L
__
t-
;
�o
x,
i
inal pos tion
sf
(b) Member· intermediate displacements and forces
f
'{6
us
J
XL' L-r�---- "'
'•( i'_,
-
f4
I
u,
x,
(c) Member nodal forces and displacements
(global co·ordmates)
Figure
1.
Member deformations and associated forces
(2-dimensional frame)
-6-
Figure 2.
Member deformations and associated forces
(3-dimensional frame)
-7-
2.1
GOVERNitlG EQUATIONS FOR PLANE FRM1ES
2.1.1
Member basic force-displacement relation
On assuming a cubic lateral deflection curve and that the axial
deformation, e is small compared to the original member length, �
(1)
and the axial lengthening due to bowing is
(2)
=
�
30
( 282
1
- 88
I
2
+
282
2
}
The axial load, P is thus
(3)
where EA
Neglecting the shear strain energy, the
axial rigidity.
=
potential energy due to lateral deflection along the prinicipal axis is
'¥
=
-21
�
I
0
EI
[ l2
�
d2
�
2
dx
dx
1
- t1 8 - �1 8
1 1
2 2
+
I [ 1 ,1
P -21
0
I
d
I�
dx
2
dx
1
(4)
1
Through the principle of minimum potential energy, the member forces
are thus,
M
(5)
1
[ill )
�
_
P� 8
30
1
+
[lli
�
+
)2
4P� 8
30
-
where EI
8
-
flexural rigidity
The basic incremental stiffness matr1x is obtained by partial
differentiating Equations (3) and (5} with respect to the member basic
deformations. Hence
EA
EA
dP
30
T
(48,
k
dM
dr1
J
EA
- 82
30
22
symmetric
(
-
8
+
1
I<
23
k
33
48
)
2
de
de
de
2
where
k
22
k
23
2EI
--�
_
30
300
(
-
282
+
l
68 e
1 2
_
)
282
2
(6)
[kJ {1'- v}
{lis}
2.1.2
EAe + EA�
Transformation from member basic force displacement to
member intermediate force/displacement
Considering gross deformation,
e
=I'(�+
u')2
+
v'2
-
�
8
8
(7)
-9
-
On partial differentiating,
de
{} [ c�+u')2 +v'2 ]� 2(�+u'l }
=
r[ c.�+
+<1
[2
de
de
UI
2
) + VI 2
r�
2V I
1
;_>
)
du'
dv' + (0) de' + (0) de'
2
l
�+u'
v'
__ du' ---- dv' + (1) de'I + (0) de'
2
(�+e)>
c�+e)2
__
=
I
v'
2
(� + e)2
du' - � dv' + (O) de' + (1) de'
I
2
(�+ e)2
Thu s
� +u'
de
de
de
�
v'
I
(�+e)2
2
(�+ e)2
v'
{Liv}
[�]
v'
T+e
0
�+u'
0
(� + e)2
�+u'
(� + e) 2
{Liu'}
0
0
du'
dv'
de'
de'
2
(8)
By the contragredient principle
{ f' }
where
and
(9)
-10 -
{ s}
T
{P, �, ' M }
1
2
(9)
On differentiating Equation
{M'}
[A]T
'
{flu }
[�]
{t.s} +
(10)
where
d
0
0
d
d
0
0
0
0
0
0
0
0
0
0
d
11
21
[�]
12
22
and
1
d
11
d
12
(�
+ e)'
21
2.1.3
(�
- v' 2(�
+ u') T +
[ -v'3
--(�
d
(2(�
+
u')2
(�
+
u')3
u'l (�; u')2
+
2v'2
T-
+ e)"
[(�
+
(
+
v'2
]
)
v•s
+
u')2 v's
s]
+ e)'
d
22
[
)
T-
(�
]
Transformation from member intermediate force/displacement to
nodal global force/displacement
The member intermediate displacements are related to the nodal
displacements by
-
'
u
-cosa
v'
sina
e'
1
e'
2
11
-
sina
0
cosa
sina
0
u
-cosa
0
-sina
cosa
0
0
u
0
1
0
0
0
0
u
0
0
0
0
-
u
u
u
1
2
3
4
5
•
(11)
and
{!.1�'}
=
[!]
(12)
'{!.1�}
By the contragredient principle,
[!]T ([�]T [�] [�]
+
[�]I [!J
{!.I�}
(15)
-12-
2.2
GOVERNING EQUATIONS FOR SPACE FRA"1ES
2.2.1
Nember basic force-displacement relation
In deriving the member basic force-deformation relation, it is
i.
assumed that the lateral deflection curve is cubic in each of the two
principal directions, and that e is small relative to
X
2
X
3
Therefore
and
e
I
(16)
2
with the axial lengthening due to bowing being
,,
=
i
1 [:::r
(
2e
_!
30
2
13
''· •
- e
e
13 23
i
1 [:::r ''.
) (
+
2
28
23
+ _!
30
2
28
12
- e
e
12 22
+ 282
)
(17)
22
The potential energy functional due to lateral deflection along the two
principle axis, neglecting the shear strain, is
'I'=
lJi
2
- �1
EI
e
3
( ]
22 22
d
2
\
dx
1
2
dx
+
.!._
2
1J
0
EI
[ ]
d2x
2
3
dx
I
2
dx
1
- �l
13
e
13
-
�1
8
23 23
- M
e
12 12
-132EI
=
Jl.
3
8
__
2 +
13
2EI
+
+
2 2
-8
Jl.
22
[
P!l.
jiT
2EJ
3
8
Jl.
_
__
8
+
13 23
- M
8
13 13
t·l
-
2El
3
Jl.
__
e
2.3 23
2Et
2EI
2 + � e2 + ____g. e
8
8
12
Jl.
-�
]2 22
-23
_
-
(282 - 8 8
+ 282 l +
12
J2 2.3
23
,. 8
12 J2
{_282
��
8
22 22.
-
- 8
12
2
8
+ 28
J2 23
22
1]
(181
Through the principle of minimum potential energy, this leads us to
M
(
4E I
I3
I2
[
[
22
(
2EI
��
3
_
_
23
��
4P
Jl.
4P!l.
EA
�
Jl.
+
2EI
3
Jl.
)
)
30
8
(
+
+
8
30
4P!l.
)
3
-!1.
-+
3lr
2EI
2
Jl.
8
12
I2
]
P!l.
_
[
+
e13
P!l.
_
where EI and EI
2
respectively.
I3
(
+
4EI
3TIJ
4EI
2EI
2
Jl.
8
PJl.)
-
2
!1.+ �
''
p
)
!l.
- 3
!1.
+ 3lr
[
4EI
)
P!l.
_
30
4P!l.
23
8
23
a
22
)
2
-!1.+ � 8
22
(19)
are the flexural rigidities about the x and x axis
3
2
The torsional moment and the axial force is given by
_l_
30
2
(28
13
8
+ 282 ) + 1__
8
13 23
23
30
( 282
12
- 8
8
+ 2 82 )
12 22
22
(20)
where EA
=
axial rigidity and GJ
torsional rigidity.
Partial differentiating Equations
(19)
and (2.D) wi.th respect
to the basic member deformations will lead us to the incremental stiffness matrix in the convected co-ordinate system.
-1�.-
Thus
61�
I
K
K
3
.I J
K
6�1
6�1
)3
K·
K
K
3
K
K
K
33
K
K
K
K
12
K
2
22
K
)2
14
1 5
25·
24
K
22
K
..
Symmetry
66
26
3
35
34
68
16.
45
55
23
68
G.
12
K
66
K
68
K
ile
46
K
)3
22
t
56
66
where
K
11
4EI
-T
EA�
4EAe
+ 30 + 3oo
12
EAe EA�
2EI
T- 30- 3oo (28�3
13
900
K
14
900
K
0.0
K
K
15
K
EA� (168 8
)3
EA� (- 48 8
EA
16
12
30
13
12
(48 13 - 8 3 )
2
48 8
(a8�3
68 38
48 8
13 22
+
168 8
+
13 23
13 22
1
23
48 8
+
8 8
23
23
28
)
8
e
2
+
12 23
3 82 )
23
+
12
23 22
)
48 8 )
23 22
-154Ef
4EAe -.EM
8
3oo. c3 3
K
22
T+ � +
K
23
EM (- 48 8
+ e 13 e 22
9TIO
12 �3
K
24
EM (
e
900
13
e
12
48
0.0
K
EA
K
-yL+�+ 900
26
33
( -e
30 .
4EI
+
EA�
J UU
�
2EI
13
4EAe
EA�
2
(8e 12 - 4e 12e 22
EAe
EM
0.0
K
EA (46
- 6 l
3D_ - 12
22 -
K
4EI
EA2
4EAe
-yL + � + 9TIIT (Be�,
K
0.0
44
45
- 4e
e l
23 22
+ 16e 23 822 )
e
12 23
+ 88223 l
+ 3e222 )
K
36
4e
(86 � 3
-yL-
35
+ 16823 e 12
+ 46 23 )
K
34
2
8
+ 88 )
13 23
23
e - 4e e
13 22
23 12
K
25
48
�
30- 9oo c2e�3
e
2
8
+ 26 1
13 23
23
46
j 3
e
2
3
+ 86 2 r
23
-
46
EA (e
30.
12
55
GJ
T
K
K
K
+
46
22
16
-
1
0.0
56
EA
T
K
66
{6$}
2..2..2
(211
{
[�] liV}
Transformation from Mem5er Basic Force/Disolacements to ��ember
Nodal Force/Displacement (Local Co-ordinatesl
From considerations of geometry and equilibrium,
F
F'
2
F3
F
0
0
1
�
�
0
0
Q
Q
0
""1
I
0
M
23
0.
0
11
0
-1
0
M
0
0
Q
M
p
0
-I
�
0
0
0
0
F
1
0
0
0
0
0
F
0
0
0
0
0
1
0
0
Q
Q
1
�
0
Q
�
0
F'
5
1
F
8
�
�
Fs
0
0
F1 o
0
0
0
0
0
0
-
F
]]
-
F
I 2
0
0
3
0
0
0
4
�1
0
0
0
0
Q
0
12
22
t
-
[B l
17
-
(22)
{S}
Using the contragredient principle,
(23}
. {liV}
By differentiating Equation
[BJ
{liS}
(22), its incremental form is obtained,
+
(24}
[ll�]{�}
The matrix[�] is the change in [�] resulting from
{ll�}.
To evaluate
[ll�) let us consider the member moving from position (I} to (II} in
Figure 3, which is equivalent to rotating the translational forces by
p
X
2
=
-
(t.u
-
9
�
(t.u
t.u )
3
p
3
=
-
8
�
t.u )
2
2
>---- x1
Figure 3.
Rotational transformation of member
nodal force (in local co-ordinates)
(25)
-18-
The nodal forces in local co-ordinates are
about the member axis.
thus
F'
1
-P
1
F'
p
0
0
0
0
0
0
0
0
0
F
0
0
Q
0
0
0
0
0
0
0
F
3
2
F'
p
3
-P
a
1
0
0
0
0
0
0
Q
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
Q
0
Q
0
0
0
Q
Q
a
0
F
0
0
0
0
a
0
1
p
Q
0
0
f
0
0
0
0
f
0
l
0
0
0
f
3
F'
4
F'
5
F'
6
F'
-
P
7
F'
3
0
0
0
0
0
0
8
F'
p
3
0
0
0
0
0
0
-P
9
F'
2
0
0
0
0
0
0
0
0
a
0
0
0
F
0
0
0
0
0
0
0
0
Q
Q
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
10
F'
11
F'
12
1 0
11
1 2
(26)
Equation (26) when used to evaluate the matrix [��] will result in an
unsymmetric co-efficient matrix when the term{[��] {�}) is expressed
in terms of the vector {�u}.
To restore symmetry;
in evaluating [��]
we take into account the change in length of the member and that the
moments producing shear are constant rather than the transverse nodal
forces.
If S 0 is the initial transverse force;
and �S0 its variation,
ignoring 2nd order terms
-19-
v1here
t>u
0
- t>u
7
1
{�'>�u} 1�hich is the change in the vector {E} res·ulting frof'l the variation
{t>u} is thus
{C,fu}
t>F
0
t>F
P -o/9.,
1
2
t>F
2
4
t>F
5
t>F
6
t>F
t>F
8
t>F
t>F
t>F
10
11
12
3
p
0
0
0
0
0
0
0
0
Q
F
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
F
2
0 -0/9.,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
-P
p
0
0
0
r:
0
0
0
0
0
0
p -0/9.,
0
0
0
0
F
o - o / 9.,
0
0
0
F
3
0
t>F
-P
3
-P
t>F
{f}
{F'}
0
0
0
0
0
-p
2
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
0
0
0
0
0
0
0
0
0
0
0
0
F
10
11
12
-20-
"p
�P
-P
.,p
..
11
3
T
3
T
2
-�-
2.
T
0
c
:i
.1',2
-6
.1',2
0
0
0
-P
0
0
6
.1',2
6
.1',2
0
0
0
0
0
0
0
"'
0
0
0
0
0
0
•·.1
c
0
0
0
0
0
p
p
3
2
3
2
2
2
2
2
0
0
6
.1',2
6
.1',2
0
0
0
p
3
0
0
6
.1',2
-6
.1',2
0
-p
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
[6B] {S}
p
p
-
p
p
3
2
1 3
r�
23
�
12
22
•t
-21-
0
M
+M
�
1a
M
+M
_ ..ll...-..U.
1a
p
I
p
;:
0
0
0
0
0
M
+M
0 -....!.!....-U
,.
0
0
0
M
+M
�
,.
0
0
0
p
-;:
0
0
0
0
0
0
0
0
0
0
0
0
0
0
...
0
0
0
...
0
0
0
6U,
0
0
0
0
0
0
0
0
0
0
0
p
-;:
� -�
M
+M
M
+M
p
;:
p
;:
S�try
...
'
.
.
... .
...
"
(27)
Substituting Equations
(21), (23) and (27) into (25), the incremental
equilibrium equation in member co-ordinates are
{llF}
[BJ {K} [ll� l
([�) [KJ [BJT
+
+
[NJ {llii}
[�)) {llii}
(28)
-222.2.3
Transformation from Member Co-ordinates to Global Co-Ordinates
The member orientation matrix, which is the direction cosines
of the member axes with respect to the global co-ordinates X1, X2 and X3
is denoted by lrl.
[R]
Let
[r]
0
0
0
0
[r]
0
0
0
0
[r]
0
0
0
0
[r]
We thus have
U.F}
(29)
and by contragredience,
T
[R]
{t>u}
(30)
Substituting Equation {28) into (29)
{t�F}
r�J r�J
{t�u}
(31)
which is the incremental equilibrium eouation in global co-ordinates.
-232.2.4
Evaluation of Member Rotational Deformations from Joint
Orientation Matrix
The 'joint orientation matrix',
parallel to the global axes.
[�]is initially assumed to be
The incremental rotation matrix for a joint
in global co-ordinates is
-l>O
[l>O]
3
0
(32)
where
and
l>O
l>O
=
I
2
6u , 60
4
L'lu
=
I 0
, l>O
2
=
L'lu , 60
5
=
L'lu
I I
, 60
=
3
=
3
6u
6u
6
I
2
for joint
(1)
for joint
(2)
This results in an increment in the joint orientation matrix which is
(33)
[60] [a]
and thus the updated jo.int orientation matrix, [a]
+
[L'la] can be
evaluated.
The two end sections of the member will in general be not
parallel to each other.
'End section orientation matrices' are thus
defined to describe them with the matrices
section
(1)
and
(2)
respectively.
[P(t)]
and
[PC2l]
for end
They are related to the joint
orientation matrices by
[pl
(34)
-24-
where
[t::(o}]
= member orientation matrix in undeformed configuration.
The first column of the member orientation matrix,
[�] repre­
sents the vector of direction cosines of the member chord (the line
joining the end sections).
This can be obtained from the global trans­
lational displacements of the nodes.
Denoting the first column of [�]
by [r ], the following relation applies for the case of small relative
-·
rotations of the member.
{r }
-·
(35)
and
where
{
p
(i)
j
}
=
j
column of the end orientation matrix of joint i.
To evaluate the second and third columns of [r] , which repre-
1
sents the vector of direction cosines of the 2 member principal directions,
let us define the following incremental rotation matrix for the two end
sections.
-25-
[: (!)]
[
[
[:(zJ]
8
-8
13
8
12
23
8
22
-8
1
2
0
0
8
-8
13
23
-8
22
0
0
l
l
(36)
Due to the member relative rotations, the new end section orientation
2
2
matrices will be [p(1l] [e( 1l] and [p( J] [e( l), The member x2 and
x3 axes could then be defined as the principal direction of an average
cross-section, hence
(37)
[�]
SOLUTION STRATEGIES FOR TRAVERSING LIMIT POINTS
3.
One of the earliest techniques developed was to introduce
spring coefficients into the incremental stiffness matrix [31,32].
This
has the effect of augmenting the stiffness matrix so that it remains
positive definite throughout the entire range of the analysis.
This
augmentation is achieved by adding to the structural stiffness matrix
[K] by the following matrix of rank one;
!._ {Q} {Q} T
82
-
where
k
(38)
-
qiven stiffness of a fictitious elastic spring.
{Q}
reference load vector.
8
the norm of the matrix
[K].
-26The augmented equilibrium equations are thus
{P}
where {P}
=
(39)
total external load of augmented system, and the load
reduction factor, A applied to {P} to obtain the actual external load
is
1 _1.._
62
which can be computed
T
{Q} {r}
-
(40)
-
once Equation
(39)
is solved for {�}.
case of simple load and one spring, as illustrated in Figure
method is easy to apply.
For the
4,
this
However, this technique cannot be easily
justified from a mathematical viewpoint when multiple springs are added.
In addition, for a multi degree of freedom system, the coupling of the
artificial stiffness may destroy the banded nature of the stiffness
matrix.
Also the reference stiffness of the applied springs have to be
obtained by trial and it should not be used for structures with local
buckling or a tendency to bifurcate.
Due to the singularity of the incremental stiffness matrix at
the extremum points, equilibrium iterations easily break down.
To
overcome this problem which has plagued earlier researchers employing
an incremental-iterative solution strategy, Bergan
[24,25,33]
the concept of a current stiffness parameter, S .
With reference to
P
Figure 5, S
p
introduced
is defined as
[�J
i
r�t
(41)
-27-
Augmented structure
Figure 4.
Actual structure
Spring Constant Method
-
28
-
Load,P
Deflection,
Figure 5.
'Current Stiffness Parameter' Method
liP;
with
�
�c
characteristic load, which is chosen to be the reference load
=
,
T
1\r R
- -c
vector, R ref.
Now,
II�;
incremental external load
LIP
s
;
R ref.
llp . llp •
1
1
p
!
llr liR.
-1
-1
llr T IIR
�
llp llp
0
0
r
-29-
[liP;l·2
llp
o
llr
T
-0
K
-0
llr
-0
(42)
t�r "!" K. llr.
_,
_,
-
1
The current stiffness parameter has an initial value of 1.0 and is zero
at a singular point.
This parameter can thus sense in advance the ap­
proach of an instability, and by suppressing equilibrium iterations when
the magnitude of this parameter is less than a threshold value, the
solution is prevented from breaking down.
The solution, by pure load
incrementation, is then made to 'jump' over the limit point and advance
along its path.
On the negative definite portion of the solution path,
the 'current stiffness parameter' is negative and when this occurs the
algorithm reverses the load and displacement increments.
The solution
then proceeds with negative load increments until the next extremum point
is reached.
Due to the low stiffness at the limit points, the load
increment can produce uncontrolled displacements which may cause the
solution to diverge.
Small load increments are thus necessary, and this
also helps to prevent drifting away from the equilibrium path.
with small increments, the displacement can still be too large;
Even
and a
simple method to prevent this is to calculate a norm of the displacement
increment and scale back both load and displacement increments according
to how much a specified maximum value has been exceeded.
A popular method for avoiding the singularities is to increment
the load parameter to the limit point, solving for displacements, then on
beyond this point by incrementing a characteristic displacement to
evaluate the corresponding load parameter.
Argyris
[34],
but unfortunately his approach led to a system of non­
symmetric equations.
Tong
[35],
This was first described by
Yamada
[36]
Alternative approaches were presented by Pian and
and Zienkiewicz
[37]
which circumvented the non-
symmetric matrix problem, but were applicable only for a step-by-step
-�-
procedure without iterating to equilibrium within the displacement
increment.
These also required a modification to the incremental stiff-
ness matrix.
A scheme suggested by Haisler and Stricklin
[38],
which
makes use of an initial value formulation, allows one to iterate to
convergence.
The proposed computational procedure is, however, cumber­
some and a simpler procedure as described by Batoz and Dhatt
[39]
exists.
In the algorithm presented by Batoz and Dhatt, the tangent
stiffness matrix
K
is used to complete 6u which is the incremental
.
-r
displacement due to the residual forces,
�
and also 6�
e
which is that
due to an arbitrary external load increment P i.e.
(43)
The actual external load applied is 6AP and this factor is obtained by
specifying that the displacement increment at degree of freedom, n,
satisfies a displacement, o, where
(44)
The actual incremental displacement vector, 6u is then given by
(45)
6u
The above procedure enables one to obtain the increment at the beginning
of the iteration sequence.
Subsequent iterations are performed in
exactly the same manner except that o is now specified to be zero;
iterating to convergence is achieved through keeping the selected nodal
-31displacement constant.
This particular strategy, in a more generalised
form was also described by Powell and Simons
[40].
From the writers' experience with the above strategy, it has
proved to be both stable and efficient.
However, it has the distinct
disadvantage that the displacement component at the degree of freedom
chosen. to control the incrementation has to be monotically increasing.
If the controlling displacement snaps back from one load level to
another, the strategy will fail to converge.
A proper choice of the
controlling displacement is thus essential and.for some structures this
is not obvious and some experimentation will be necessary.
It has been observed that instead of using a single component
of the displacement vector, it is possible to use some measure of the
complete vector as the controlling parameter.
by Riks
[41)
and Wempner
[42]
This is the path followed
where the load step, �A, is limited by the
following constraint equation.
(46)
In the equation, ��is a prescribed scalar which fixes the length of the
increment.
This constraint equation was originally added to the incre­
mental stiffness expression but this unfortunately destroys the banded­
ness and symmetry of the stiffness matrix.
By adopting a two step
technique used by Batoz and Dhatt, as described above, this problem can
be· overcome.
This modified 'constant arc length' strategy of Riks and
Wempner is presented by Ramm in Reference
[26];
where both iteration
in a 'plane' normal to the tangent and iteration in a 'sphere' are
described.
[43],
This is illustrated in Figure
6.
In the work of Crisfield
instead of applying the constraint equation of
of the following constraint.
(46),
use was made
-32-
,---I-- nitial solution
.----Normal to tangent
·<
r-----r-Final solution
a::
UJ
1UJ
::E
<(
a::
L_
Spherical path
ct.
Cl
<(
0
....J
DISPLACEMENT, u
Figure 6.
Rik's method with iteration about a 'sphere'
or iteration about a 'normal plane'
(47)
which from his experience appears to be numerically better.
The
'constant arc length' method, by generalising the constraint equation
from that for a single component of the displacement vector to that for
the scalar product of the vector, is thus able to overcome the problem
of the choice of a monotonically increasing displacement component
associated with the single displacement component control techniques.
Equilibrium iterations near a singularity do not pose any particular
problems, as is the case with the 'current stiffness parameter' of Bergan.
By iterating about a constant arc length or a constant displacement
component as in the Batoz and Dhatt strategy, we are iterating in the
externa1 1 oad space as we 11 as in the d isp1 acement space, and this is
more efficient than conventional Newton-type strategies at constant load.
This point is illustrated by the work of Crisfield in Reference
[43].
From this brief review, the writers conclude that in terms of generality,
-33-
effectiveness and robustness, the 'constant arc length' method is
superior to the other described strategies.
:I
I
4.
CONSTANT ARC LENGTH STRATEGY OF CRISFIELD
J
At the start of an
arc
mental displacement vector, t>:\ (
o)
length increment, 6u (
o)
is the incre­
the corresponding load multiplier and
6� the prescribed generalised arc length.
Application of the arc length
constraint results in
(48)
with
(49)
K
=
the incremental stiffness matrix, and
vector.
Solving
Crisfield
[43)
(48)
and
and Ramm
(49)
[26),
�
=
an arbitrary external load
0
will then yield t>:\ •
In the paper of
it was suggested that the sign of 6:\
0
should follow that of the previous increment unless the sign of the
determinant of the incremental stiffness matrix has changed.
l
t
From the
examples presented by them, this criterion appears to be.satisfactory.
However, in the case of structures exhibiting multiple negative eigen­
values behaviour, this simple criterion does not always work and it
appears better to base the sign change on that of the sign of the incre­
mental work done, t>W where
6W
(50)
-34If the sign of
sign of n"A (
o
)
W has changed from that of the previous increment, the
is reversed from that of its previous value.
After the initial increment, the displacement is then iterated
to convergence.
nu (
-e
At the i th iteration
i)
K
-�
(5 1 )
p
and
nu (
-r
i
K � R ( i-1
)
)
(52)
(i )
where nu
displacement increment due to the arbitrary externa 1 1 oad,
-e
i-l }
i
nu ( )
displacement increment due to the residual force vector, R (
-r
=
=
at the end of the
matrix
�
( i-1 )
th iteration.
In the Newton-Raphson method the
is reformed after every iteration, while in the modified Newton­
Raphson iterative scheme it remains constant or is reformed after a
number of iterations.
i
If K is constant, then so also is nu ( 1 throughout
-e
the increment and there i s thus no need t o recalculate it at every iteration.
At the i th iteration, the iterative displacement, nu (
i)
, is
(53)
and the incremental displacement up to the
th iteration, ou (
i
)
is
(54)
This incremental displacement is made to satisfy the arc length con­
straint of Equation
(48)
and thus
-35-
(55)
which leads to the following quadratic
where
a
a
a
1
2
(56)
3
i
Two values of �A( ) are possible and to void 'doubling-back' the solution
i
path, the correct choice of �A( ) is that in which the scalar product of
i
the incremental displacement vector, o�( -l) is a positive quantity.
In
i
the event that both choices of �A( ) yield a positive result, the correct
root is the one nearest to the linear solution.
i
6A( )
a /a
3
2
(57)
-36-
The success of this nonlinear solution strategy depends on the
quadratic of Equation
(56)
yielding real roots.
Imaginary roots will
occur if
a2 - 4a a
I
2
3
<
0
and this will indeed be so if a norm measure of
i
than that of llu( ).
-e
(58)
liU(
i
) is much 1 arger
-r
This will happen if the stiffness in the direction
of the degree of freedoms .not acted on by the external force vector
become very small relative to that of the degree of freedoms carrying
the external loads.
In the majority of cases, this is unlikely to occur,
and from the writers' experience this will probably occur only in
structures exhibiting multiple instability directions at a point.
A
structure showing this type of behaviour is described in the numerical
examples.
EVALUATION OF POST-BIFURCATION EQUILIBRIUM PATH (SECONDARY PATH)
5.
To investigate the stability of an equilibrium configuration it
is necessary to formulate the quadratic form
(59)
where
V
V
potential energy of the system.
q
generalised co-ordinates
ij
coefficient matrix of the quadratic form which is
equivalent to the incremental stiffness matrix,
�
defined above.
-37-
According to the theory of stability of cons-ervative systems, the
stalltltty of the equilibrium state ts· ens-ured if the above quadratic
form is positive definite.
E"quation
(59)
is indefinite for an un­
stable configuration while the transition between a stable and an
unstable point of the equilibrium path is denoted by a positive semi­
definite Equation
(59).
The critical equilibrium state is in general
either a limit point or a bifurcati.on point and they are illustrated
in Figure 7.
A necessary but not sufficient condition for the
existence of a critical state is the vanishing of the determinent of
the incremental stiffness matrix.
det l�l
=
(601
0
This can be easily obtained during the factorisation of the tangent
stiffness matrix in the incremental-iterative solution strategy, since
it is the product of all terms of the diagonal matr.ix in the L D L
decomposition.
The precise definition of a critical equilibrium
T
state
is given by the solution of the following eigenvalue problem
(61)
where a(k ) denote the eigenvector and
the critical states.
n(k)
the eigenvalues which define
A limit point is distinguished from a bifurcation
point in that it is characterised by a vanishing load increase.
The incremental-iterative solution strategy described in the
sections above is capable of continuing the computation of the fund­
amental (basic) equilibrium path beyond the critical points.
However,
to follow a secondary path after a bifurcation point, will require a
modification to the described solution strategy at the arc length
increment immediately after the bifurcation point.
The vector
�F
-
38
-
load
Parameter
load
Parameter
Deflection
(a)
limit point
Deflection
(b)
Figure 7.
Bifurcation point
Critical points
tangential to the fundamental path at the bifurcation point, B can be
calculated approximately as
�F
where 6gm
=
=
1
2
(
6g
m
m+1
+ 6g
)
{62)
incremental generalised displacement vector at point m.
The unit vector ! ' colinearwith � is thus
F
F
�F
According to References [41 and 44] the vector
(63)
!s
secondary post bifurcation path at B is given by
tangential to the
-39-
t
a{�
-s
+
�!F}
(64)
where
�=
eigenvector of the incremental stiffness matrix,
�
and the
coefficient � is a function of the third derivative of the potential
energy
[41].
This evaluation of the third derivative is difficult and
thus an approximation to
gonal to
!F
is used.
!s
such that the approximation,
�
! i)
is ortho­
Therefore
(65)
The displacement increment for the first iteration of the arc length
increment after the critical point,
9c
can be assumed to be
(66)
where��= prescribed generalised arc length.
Iteration to equilibrium is then performed as described in Section
4,
with the modification that the incremental stiffness matrix used is
�(9c g8)
+
to prevent the solution from coming back to the fundamental
equilibrium path.
This described technique of superimposing a fraction
of the eigenmode on the displacement field at the bifurcation point can
be called the 'perfect approach'.
A simpler alternative without the need to compute the eigen­
vectors wi11 be to impose either a sma 11 perturbation in 1 oad or geometry
to the structure.
This 'imperfect approach' will yield a load-deflection
path which approaches the perfect post-bifurcation path with increased
-
40
-
deformation of the structure (_see Figure 7).
6.
EXTRAPOLATED STIFFNESS PROCEDURE
Let us consider the efficiency of the linearised incremental.,
iterative arc length technique.
It is apparent from Figure 8 that for
the one degree of freedom cases, the point (a) which is the result after
the first iteration, is a poor guess of the final solution.
If instead
of using the gradient to the solution path at the beginning of the arc
length increment, it is possible to use the gradient of the point half
arc length ahead, it would appear that the result after the first iter­
ation will give a much better estimate to the final solution.
illustrated in Figure 8 where m is midway along the arc a-b.
This is
For
a
multi degree of freedom system, by using the tangent stiffness half an
arc length ahead instead of the value at the beginning of the increment,
the same conclusion should apply.
The pertinent question now is;
how to obtain the tangent
stiffness half an arc length ahead since the nodal displacement associ­
ated with the point are not known at the beginning of the increment.
For the strategy proposed herein, this is achieved by extrapolating
forward the tangent stiffness from previous arc length increments.
3
A
point Lagrange interpolation polynominal [45] will be. used and thus in
addition to the tangent stiffness at the beginning of the arc length
increment, its value at the previous two increments must be retained.
The tangent stiffness matrices are stored in 'skyline' linear arrays.
Let k., k.
J
J-1
and k. 1 represent and element of the tangent stiffness at
J-
the beginni.ng of the j, (j-1) and U-2) increment respectively.
k
j
is
the corresponding element of the extrapolated tangent stiffness half an
arc length ahead at the j th increment and it is given by
-41-
k'
j
(2.5 - 1) (2.5 - 2)
k
j-2
(O - 1} (O - 2)
(2.5 - 0) (2.5 - 2)
+ -------
(1 - 0} (1 - 2)
(2.5 - 0) (2.5 - 1)
+ -------
(2 - 0} ( 2 - 1)
0.375 k.
J-2
1.25 k.
J-1
+
1.875 k.
(67)
J
The above extrapolation is repeated for each element of the tangent
stiffness to fonn the complete extrapolated stiffness matrix.
This
matrix is then factorised and used throughout the increment until
convergence to the final solution.
Since the extrapolation procedure
described above required the tangent stiffness for the previous two
increments, this technique can only be applied after the second arc
length increment.
For the initial two increments either the M.N.R. or
the N.R. procedure will have to be used to iterate to convergence.
The usefulness of this extrapolated stiffness strategy rests
on the assumptions that
(i )
for each element of the tangent stiffness, its value
along the solution path does indeed have a relationship
than can be modelled closely enough by a
3
point Lagrange
interpolation polynomial.
( ii )
the extension from the single d.o.f. case to the multi
d.o.f. system that by using the gradient
( tangent
stiff­
ness ) at half an arc length increment ahead gives a much
closer estimate to the final solution;
and
-
( iii )
42
-
this better initial estimate will iterated more efficient­
ly to the final solution than the current Newton-Raphson
methods.
These assumptions have been validated in Reference [29) for two dimen­
sional frames.
The increased efficiency of these extrapolated stiffness
strategy over the mNR method is demonstrated in the following numerical
example for three dimensional space frames.
7.
NUMERICAL EXAMPLES
7.1
CANTILEVER BEAM
Figure 9 depicts a cantilever beam loaded at its free end with
a concentrated load.
The direction of the load is kept constant in a
vertical direction throughout the deformation.
is used to model the beam.
A single frame element
An analytic solution to this problem, obtain­
ed through the use of elliptic integrals, is given by Frisch-Fary [46].
It is observed that the results of the numerical analysis using the two
dimensional frame program is indistinguishable from the analytical
solution.
7.2
WILLIAMS' TOGGLE FRAME
This problem illustrated in Figure 10, has been solved
analytically as well as experimentally tested by Williams [6].
In his
analytical treatment of the frame, Williams took into consideration the
finite change of geometry as 11ell as the effects of the axial forces on
the flexural stiffness and the flexural shortening of the members.
-43-
Result from
first iteration
Loading
parameter,>..
Final solution
Deflection, u
Figure 8.
Constant arc length procedure combined with
extrapolated stiffness strategy
p
l
I.
12
L=
Area=
M.l.:
E=
400cm
42cm2
6482 cm4
0,2x105kN/cm2
L
-
.....,.- =
present analysis
analytiCal method
\
10
_.8
w
;:;-.
-
a.b
4
0
Figure 9.
0,2
0,4
o.a
·
10.0
Load deflection curve for cantilever beam
-44-
�
1
�k
P,v
25, 87 2
.
-=:c:_=-=-.._
-
EA
=
EI
=
-----�-"'+
6
1,85Sx10 lb
3lb/in2
9,27 x 10
70
Present analysis
•
a..
60
+
-
Cl
<
0
_J
•
II
Analy tical
_
Experimental
}
Williams[6]
Wood & Z ienki ewi c z [4]
Papadrakakis[131
50
40
QL---�----_J----�-----L----�-----�----�
0,2
0,3
0,4
0,6
0,7
0,5
0,1
DEFLECTION, v {in.)
Figure 10.
Load deflection curves for Williams toggle frame
-45-
Wood and Zienkiewicz [4] have also investigated this problem employing
an assumed desplacement finite element approach with five elements per
member.
Papadrakakis [13] used a 'beam-column' approach to derive the
nonlinear equilibrium equations, which were solved through a vector
iteration approach.
The authors results, utilising one plane frame element per
member, are in very close agreement with the analytical solution of
Williams and the finite element solution of Wood and Zienkiewicz.
7.3
TWELVE MEMBER HEXAGONAL SPACE FRAME
This hexagonal frame has been experimentally studied by Griggs
!47].
The prebuckling behaviour has also been solved by Chu and
Rempetsreiter [10] while Papadrakakis [13] traced the solution into the
post buckling range.
From Figure 11, it is observed that the present
analysis is in agreement with that of Chu and Rempetsreiter in the pre­
buckling range.
Papadrakakis' curve appears to be in exact agreement
with that of the experimental result, which yielded a snap through load
4% lower than that of the writers.
The post-buckling region of the writers' solution path does
not correspond exactly with that presented by Papadrakakis.
In his
formulation, he treated the rotations as a vector and it appears to the
writers, therefore, that the validity of his results in the presence of
genuinely large rotations, which do occur in the post-buckling range, is
doubtful.
Iteration to convergence was achieved using the mNR method and
the proposed extrapolated stiffness method.
ations
The average number of iter­
per arc length was 3.5 for the mNR method and 2.3 for the
-46-
250
E
=
439800 lb/in2
159000 lb/in2
0,494 in2
12 :0,02 in 2
G
=
Area
=
13
J
=
=
O.D2 in2
0,0331 in4
200
150
Present analysis
..c
•
c
<
Papadrakakis 191
II
Chu- Rampetsreiter
+
Experimental results [34)
[5)
Present analysis, with the 6
0
�
bo undary nodes res t rained
against translational movement
100
i =no. of iterations for mNR
j
=
no. of iterations for
extrapolated stiff ness
(4,2)
so
Figure 11.
Load-deflection curve for hexagonal frame
-47-
extrapolated stiffness strategy.
This increased efficiency is further
illustrated by the cpu times required for the two runs
( implemented
on
the VAX-11/780 computer ) , 22.5 sees for the mNR and 17.8 sees for the
extrapolated stiffness method.
Next, the structure with the six boundary nodes now restrained
against translational movement in all directions, was reanalysed.
As
shown in Figure 11, this has the effect of yielding a larger snap-through
load.
In addition, it was observed at the snap-through load that two
diagonal elements of the triangularised stiffness matrix changed sign
( from
positive to negative ) at essentially the same load level.
resulted in
a
This
positive determinant and would indicate that the matrix is
still positive definite if the determinant is used as the criterion for
determining it.
The incremental stiffness matrix is, however, negative
definite at this stage, as is indicated by considering the sign of the
incremental work done.
At the lowest point of the post-buckling region, the solution
strategy broke down as no real roots could be found for the quadratic of
Equation
( 56 ) .
On examination, it was found that at that load level, one
of the rotational degree of freedoms exhibited a singularity before it
occurred for the vertical displacement of the loaded node.
This resulted
in a norm mea:;ure of the displacement increment due to the residual forces
becoming very much larger than that due to the external applied load
which yielded the condition of Equation
(58 ) .
To prevent the applied
solution strategy from breaking down, wherever the condition of Equation
(58 )
occurs, iteration is suppressed and only pure incrementation is
applied.
-48-
E
=
Area
=
G
I
+
�
r
25
t
i-
x 3, w
=
5
3,03x 10 N/cm2
5
1, 096 x 10 N/cm2
3,17 cm 2
x2,v
25
1
B,216L�
��
�
k
Figure 12(a).
43,3
.
JI"
43,3
� -----+
�
Geometry of 24 member star-shaped shallow dome
x,,u
-
49 -
1000
500
/
z.a
/
I
/
1 t.o
I
I
_,
._/
//
"'
•1,0
-2.0
19l!!ll
·3,0
•4,0
·5.0
•6,0
DEFlECTION, v • tO lc•l
11110
·•
·•
DEflECTION, u •1011cmJ
·6
·4
·2
12!!!..L
Figure 12(b).
Load-deflection curves
·7.0
•1,0
-50-
TWENTY-FOUR MEMBER STAR-SHAPED DOME
7.4
The structure depicted in Figure 12 ( a) has been analysed by
various authors [13,48,49] as a space truss to trace its load-deflection
behaviour into the post-buckling range.
Here, the structure is analysed
as a space frame and its post-buckling path for the loading condition of
Figure 12(b) presented.
The supports of the dome are assumed to be pin­
ned and restrained against translational motion.
The average number of iterations per arc length increment is
3.5 using the mNR method and 2.3 utilising the extrapolated stiffness
method.
The corresponding cpu times used are 46.3 sees for the mNR and
38.0 sees for the extrapolated stiffness technique.
8.
CONCLUSION
An updated Lagrangian large rotation formulation of the non­
linear equilibrium equations governing the large deflection of plane and
space frames, within the confines of 'beam-column' theory has been
presented.
A comprehensive survey of the most recent solution strategies
available for analysing the post-buckling behaviour of structures was
also undertaken.
From this review, it was concluded that the arc length
strategy due to Crisfield [43] is most robust and efficient, and thus
this incremental-iterative method was used in conjunction with the derived
system of nonlinear equations to study the post-buckling behaviour of
spatial framed structures.
The results of the analyses are in good agree­
ment with previously published experimental and analytical work. Modifi­
cations to the described strategy to enable it to trace the secondary
post-bifurcation path are also discussed.
-51-
The solution of the nonlinear equations is expensive in comput­
ing time and this is especially so for large scale problems.
An extra­
polated stiffness method which would reduce the computing time needed to
arrive at a particular solution was presented.
For the space frames
analysed above, it was found to be more efficient than using the conven­
tional mNR method.
For most problems, what is required is the snap-through load
without the need to trace the· post-buckling path.
However, in some cases,
for example in studying the effects of a concentrated load on a restricted
part of the structure, it is important to obtain information on the
nature of the load shedding after the occurrence of a local instability,
in order to assess the behaviour of the whole structure.
Due to the
lack of robust and efficient techniques for critical points, not much
attention has been focused on the problem of analysing the post-buckling
behaviour of spatial frames.
It is hoped that this report will go some
way in helping to remedy this situation.
-52-
APPENDIX A
-
NOMENCLATURE
A
area of cross-section
(k}
a
eigenvector
[A]
displacement transformation matrix
[B I l
member rotation matrix
[B]
transformation matrix relating local member forces to
member basic forces
[�]
{flu}
[LIB]
change in
[�]
stiffness modification matrix to account for the effect
resulting from
of member forces (two-dimensional frame)
.9E_
generalised gradient of load-displacement curve
dr
E
modulus of elasticity
e
axial displacement of member
{FJ, {fl}
intermediate force vector
{FI l
rotated member local force vector
{LIF}, {llf}
incremented joint force vector
�
variation of the vector
{ll ul
displacement
{LIF},
{llf I }
{F}
resulting from incremental
{llg}
incremental joint force vector
GJ
torsional rigidity
I, I
m
moment of inertia
[�], [k]
member basic incremented stiffness matrix
lKl
member incremental stiffness matrix in local co-ordinates
K, K.
tangent stiffness matrix
[ �Al
augmented stiffness matrix
[�Gl' [�Gl
global incremental stiffness matrix
-
-1
-53incremental stiffne.ss matrix correspondi.ng to
displacement vector {gc
+
g6}
j th e 1 ement of tti.e tangent stiffness matrix stored in
a 'skyline' array
stiffness of fictitious· spring
k
undeformed length parameter
generalised arc length parameter
bending moment
torsional moment
stiffness modification matrix to account for the effect
of member force (three dimensional frame)
p
axial force
P, {P}
external load vector
{.1P}
incremental external load vector
[
i
p( )
]
p
ll i
end section orientation matrix for end section
incremental load multiplier
{g }
reference load vector
q
generalised co-ordinates
t>g
i
[R]
incremental generalised displacement vector at point
transformation matrix for local convective co-ordinates
to global co-ordinates
R
-
�c
residual force vector
characteristic load vector
�ref
reference load vector
[�]
member orientation matrix
llr
incremental displacement vector
{S}, {s}
member force vector
so
sP
transverse force
current stiffness parameter
-54-
{LS}, {Ls}
member incremental force vector
LS
0
incremental transvers-e force
[!l
disp 1 acement transformation matrix
{u}
joint displacement vector
{u'}
intermediate displacement vector
u'
relative displacement of member ends measured parallel
to undeformed member axis
{Lu}
incremental joint displacement vector
LU
incremental displacement vector
{Lu'}, {Lii}
intermediate incremental displacement vector
L
incremental displacement vector due to arbitrary external
�e
load increment
incremental displacement vector due to residual forces
Lu
-r
i
.Su( )
incremental displacement from first iterative cycle to
i th iterative
cycle
potential energy of system
v
coefficient matrix of the quadratic form
v ..
lJ
relative displacement of member ends measured perpendicular
v'
to undeformed member axis
{LV}, {Lv}
member incremental displacement vector
LW
incremental work
X ' X ' X
I
X
I
2
'
X
2
'
X
3
3
global coordinate axes
member coordinate axes
a
initial inclination of member to global axes
[a]
joint orientation matrix
[La]
incremental joint orientation matrix
.s ..
lJ
LA
nk
kronecker delta
incremental load multiplier
eigenvalue
-
55
-
[11�]
incremental joint rotation matrix
e
norm of stiffness matrix
e , 8
m
mn
member relative end rotation
e•
joint rotation of node
e
angle of twist of member
m
t
<P
potential energy
A
load reduction factor
m
-56-
APPENDIX B
1.
-
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4
Brittle Fracture of Steel - Perform­
11
steels: C. O'Connor
5
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Buckling in Steel Structures-1. The
12
column:
O'Connor
C.
13
(1965)
design
the
14
Land use prediction in transportation
C. O'Connor
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16
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Meek
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18
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·(1973)
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Wave Climate at Moffat Beach: M.R.
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Two
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10
(1969)
dimensional seepage
plane
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9
(1969)
Buckling in Steel Structures- 2. The
in
8
pro­
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planning: S. Golding and K.B. David­
use of a characteristic imperfect shape
7
element
Ground Water Hydrology: J.R. Watkins
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6
finite
(1969)
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isolated
by
J.L. Meek and G. Carey
use of a characteristic imperfect shape
an
Computer - Axisy­
by
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20
(1911)
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(1983)
395-405.
"Quadratic Convergence of
Crisfield's Method", Computer and Structures, Vol.
17,
pp
69-72.
CIVIL EtlGitlEERING RESEARCH REPORTS
CE
No.
Title
Flood Frequency Analysis:
Logistic Method
Author(s)
Date
BRADY, D.K.
Februar)
1979
for Incorporating Probable Maximum Flood
2
Adjustment of Ph r eatic Line in Seepage
ISAACS,
L.T.
March,
1979
Analysis by Finite Element Method
3
Creep Buckling of Reinforced
Concrete
Coluinns
4
5
6
BEHAN,
J.E. &
O'CONNOR,
Buckling Properties of Monosymmetric
KITIPORNCHAI,
!-Beams
& TRAHAIR,
Elasto-Plastic Analysis of Cable Net
MEEK,
Structures
BROWN,
A Critical State Soil Model for Cyclic
CARTER, J.P.,
BOOKER, J.R. &
Loading
Resistance to Flow in Irregular Channels
An Appraisal of the Ontario Equivalent
10
11
12
Decembet
1979
C.P.
KAZEMIPOUR,
A.K.
O'CONNOR,
C.
Februar)
1980
A.K.
APELT, C.J.
Shape Effects on Resistance to Flow in
KAZEMIPOUR,
April,
Smooth Rectangular Channels
&
1980
April,
The Analysis of Thermal Stress Involving
BEER,
Non-Linear Material Behaviour
MEEK, J.L.
G. &
1980
Buckling Approximations for Laterally
DUX,
April,
Continuous Elastic !-Beams
KITIPORNCHAI, S.
1980
A Second Generation Frontal Solution
BEER,
May,
P.F. &
G.
1980
Combined
Stiffness for Beam and
Column
O'CONNOR,
May,
C.
1980
Braces
14
Beaches:- Profiles, Processes and
GOURLAY,
June,
M.R.
1980
Permeability
15
16
Buckling of Plates and Shells Using
MEEK,
Sub-Space Iteration
TRANBERG,
The Solution of Forced
Vibration Problems
July,
J.L. &
W.F.C.
Numerical Solution of a Special Seepage
1980
ISAACS,
SeptembE
L.T.
1980
Infiltration Problem
18
19
Shape Effects on Resistance to Flow in
KAZEMIPOUR,
Smooth
& APELT,
Semi-circular Channels
The Design of
Single Angle Struts
1980
August,
SWANNELL, P.
by the Finite Integral Method
17
Februar)l
1980
C.J.
Program
13
1979
1979
Base Length
9
May,
Novembet
P.L.D.
& APELT,
8
S.
N.S.
J.L. &
WROTH,
7
April ,
19 79
C.
WOOLCOCK,
A. K.
S.T.
KITIPORNCHAI,
Novembet
1980
C.J.
&
S.
Decembet
1980
CIVIL ENGINEERING RESEARCH REPORTS
CE
No.
20
21
Title
Date
Author(s)
CARTER,
J.P.
Subjected to Non Axi-s�metric Loading
BOOKER,
J.R.
Truck Suspension Models
KUNJAMBOO,
Consolidation of
Axi-symmetric Bodies
23
C.
Elastic Consolidation Around a Deep
CARTER,
J.P. &
Harch,
Circular Tunnel
BOOKER,
J.R.
1981
An Experimental Study of Blockage
WEST,
G.S.
April,
Inelastic Beam Buckling Experiments
1981
DUX,
11ay,
P.F. &
KITIPORNCHAI,
25
February,
1981
Effects on Some Bluff Profiles
24
January,
1981
K.K. &
O'CONNOR,
22
&
Critical Assessment of the ·International
KORETSKY,
Estimates for Relaxation Losses in
PRITCHARD,
s.
A.V. &
R.W.
1981
June,
1981
Prestressing Strands
26
Some Predications of the Non-homogenous
CARTER,
July,
J.P.
1981
Behaviour of Clay in the Triaxial Test
27
The Finite Integral Method in Dynamic
Analysis :
28
August,
SWANNELL, P.
1981
A Reappraisal
Effects of Laminar Boundary Layer on a
ISAACS,
September,
L.T.
1981
Model Broad-Crested Weir
29
30
31
Blockage and Aspect Ratio Effects on
WEST,
Flow Past a Circular Cylinder for
104 < R < 10 5
APELT,
Time Dependent Deformation in Prestressed
SOKAL, Y.J. &
November,
Concrete Girder:
TYRER,
1981
Measurement and Prediction
Non-uniform Alongshore Currents and
Sediment Transport - a One Dimensional
G.S. &
October,
C.J.
1981
P.
GOURLAY,
M.R.
January,
1982
Approach
32
33
A Theoretical Study of Pore Water Pressures
ISAACS,
L.T. &
Developed in Hydraulic fill in Mine Stapes
February,
CARTER,
J.P.
1982
Residential Location Choice Modelling:
GRIGG,
Gaussian Distributed
Stochastic Utility
July,
T.J.
1982
Functions
34
The Dynamic Characteristics of Some Low
WEST,
August,
G.S.
1982
Pressure Transducers
r,
35
Spatial Choice ModellinG with Mutually
Denendent Alternatives:
GRIGG,
September,
1982
T.J.
Legit Distributed
St � chastic Utility Functions
36
Buckling Approximations
Beams
for Inelastic
DUX,
I'.F.
October,
&
KITIPORNCHAI,
S.
1932
CIVIL ENGINEERING RESEARCH REPORTS
CE
No.
Ti t l e
Author(s)
Date
37
Parameters of the Retail Trade Model:
A Utility Based Interpretation
GRIGG, T.J.
October,
1982
38
Seepage Flow across a Discontinuity in
Hydraulic Conductivity
ISAACS, L.T.
December,
1982
Probabilistic Versions of the Short-Run
GRIGG, T.J.
December,
1982
KOE, C.C.L. &
BRADY, D.K.
January,
SWANNELL, P.
March,
39
Herbert-Stevens Model
40
41
Quantification of Sewage Odours
The Behaviour of Cylindrical Guyed Stacks
1983
1983
Subjected to Pseudo-Static Wind Loads
42
Buckling and Bracing of Cantilevers
KITIPORNCHAI, s.
DUX, P.F. &
RICHTER, N.J.
April,
1983
43
Experimentally Determined Distribution
of Stress Around a Horizontally Loaded
Model Pile in Dense Sand
WILLIAMS,,D.J. &
August,
PARRY, R.H.G.
1983
44
Groundwater Model for an Island Aquif er :
Bribie Island Groundwater Study
ISAACS, L.T.
&
September,
WALKER, F.D.
1983
45
Dynamic Salt-Fresh Interface in an
Unconfined Aquifer: Bribie Islartd
Groundwater Study
ISAACS, L.T.
September,
1983
46
An Overview of the Effects of Creep in
Concrete Structures
SOKAL, Y.J.
October,
1983
47
Quasi-Steady Models for Dynamic Salt­
Fresh Interface Analysis
ISAACS, L.T.
November,
1983
48
49
Laboratory
and
Field Strength of Mine
WILLIAMS, D.J. &
Waste Rock
WALKER,
L.K.
Large Deflection and Post-Buckling
Analysis of Two and Three Dimensional
Elastic Spatial Frames
MEEK, J.L.
TAN, H.S.
&
.November,
198 3
December,
1983
CURRENT CIVIL ENGINEERING BULLETINS
4
Brittle Fracture of Steel - Perform­
11
steels: C. O'Connor
5
blems
(1964)
use of a characteristic imperfect shape
12
column:
O'Connor
C.
13
(1965)
design
the
14
Land use prediction in transportation
C. O'Connor
(1965)
Generated
Wave
observations
Currents
(1965)
Brittle Fracture of Steel-2. Theoret­
16
non-uniform,
material: C. O'Connor
17.
frame
Meek
and
grid
18
J.L.
structures:
Methods:
Traffic
Lucas
C.
and
(1974)
and
R.
Owen
Planning
and
of
Evaluation
K.B. Davidson, et a/.
J.L.
19
Meek
of
Evaluation
Quantitative
a
High
Speed Brisbane-Gold Coast Rail Link:
(1974)
Brisbane Airport Development Flood­
way Studies: C.J. Ape/t
Force Analysis of Fixed Support Rigid
(1968)
·(1973)
(1966)
(1967)
Frames:
(1971)
Wave Climate at Moffat Beach: M.R.
K.B. Davidson
polycrystalline
Two
with a free sur­
(1971)
Assignment
Analysis by Computer-Programmes
for
Methods
Transportation Gravity Models: A. T.C.
Gourlay
ical stress distributions in a partially
yielded,
Element
Philbrick
- Some
made in fixed bed hy­
Finite
face: L. T. Isaacs
15
draulic models: M. R. Gourlay
10
(1969)
dimensional seepage
plane
determinate
of
trusses against buckling in their plane:
9
(1969)
Buckling in Steel Structures- 2. The
in
8
pro­
methods:
planning: S. Golding and K.B. David­
use of a characteristic imperfect shape
7
element
Ground Water Hydrology: J.R. Watkins
son
6
finite
(1969)
and its application to the buckling of
isolated
by
J.L. Meek and G. Carey
Buckling in Steel Structures-1. The
an
Computer - Axisy­
by
Analysis
metric solution of elasto-plastic
ance of NO 1B and SAA A1 structural
20
(1911)
Numbers of Engineering Graduates in
Queensland: C. O'Connor
(1977)
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