Lattice booms and platforms composed of flexible members or large diameter rings which may be stiffened by cables in order to support membrane-like antennas or reflector surfaces are the main components of some large space structures. The nature of these structures, repetitive geometry with few different members, makes possible relatively simple solutions for buckling and vibration of a certain class of these structures. Each member is represented by a stiffness matrix derived from the exact solution of the beam column equation. This transcendental matrix gives the current member stiffness at any end load or frequency. Using conventional finite element techniques, equilibrium equations can be written involving displacements and rotations of a typical node and its neighbors. The assumptions of a simple trigonometric mode shape is found to satisfy these equations exactly; thus the entire problem is governed by just one 6 x 6 matrix equation involving the amplitude of the displacement and rotation mode shapes. The boundary conditions implied by this solution are simple supported ends for the column type configurations

Anderson, M. S.

English

NASA Technical Reports Server

BUCKLINGANDVIBRATIONOFPERIODICLATTICESTRUCTURES
Melvin S. Anderson
NASALangley ResearchCenter
Hampton,Virginia
Large Space SystemsTechnology-1980
SecondAnnual Technical Review
November18-20,1980
35
https://ntrs.nasa.gov/search.jsp?R=19810010672 2020-03-21T09:27:45+00:00Z
INTRODUCTION 
Proposed concepts  f o r  l a r g e  space  s t r u c t u r e s  are t y p i f i e d  i n  t h e  two photos  
of f i g u r e  1. L a t t i c e  booms and p l a t f o r m s  composed of  f l e x i b l e  members o r  l a r g e  
diameter  r i n g s  which may b e  s t i f f e n e d  by c a b l e s  i n  o r d e r  t o  suppor t  membrane- 
l i k e  antennas o r  r e f l e c t o r  s u r f a c e s  are  t h e  main components of t h e s e  l a r g e  
space  s t r u c t u r e s .  Analysis  of t h e s e  s t r u c t u r e s  w i t h  a complete f i n i t e  element 
model may b e  p r o h i b i t i v e l y  expensive o r  i m p r a c t i c a l .  However, t h e  n a t u r e  of 
t h e s e  s t r u c t u r e s  ( r e p e t i t i v e  geometry w i t h  few d i f f e r e n t  members) makes p o s s i b l e  
r e l a t i v e l y  s imple s o l u t i o n s  f o r  buckl ing  and v i b r a t i o n  of a c e r t a i n  c lass  of 
t h e s e  s t r u c t u r e s .  This  theory  a long  w i t h  t y p i c a l  r e s u l t s  w i l l  be  d i s c u s s e d  i n  
t h i s  paper. 
36 
F i g u r e  1 
PERIOD IC LATTICE CONFIGURATIONS ANALYZED 
Exact buckling and vibration solutions have been obtained for the 
configurations shown in figure 2. 
reference 1 and a simple extension yields vibration results as well. 
configurations are loaded with uniform axial load and are simply supported at 
the ends. If the ring is cable stiffened, the attachment point at the central 
mast must be assumed rigid. 
expense of increasing the problem size. 
The theory for buckling was developed in 
The column 
The flexibility of the mast may be included at the 
Figure 2 
37 
BASISFORTHEORY
The theory in its simplest form is applicable to any configuration where the
relative geometric relationship between nodes is identical for all nodes. This
relation is true for all the configurations of figure 2. (The nodes at the ends
of the mast of the ring configuration are assumedto have zero displacement, so
they do not enter in the analysis.) Each memberis represented by a stiffness
matrix derived from the exact solution of the beamcolumn equation. This
transendental matrix gives the current memberstiffness at any end load or
frequency. It is not necessary to have intermediate nodes to insure accuracy.
Using conventional finite element techniques, equilibrium equations can be
written involving displacements and rotations of a typical node and its
neighbors. The assumptions of a simple trigonometric modeshape is found to
satisfy these equations exactly; thus the entire problem is governed by just one
6 x 6 matrix equation involving the amplitude of the displacement and rotation
modeshapes. The boundary conditions implied by this solution are simple
supported ends for the column type configurations.
O EACH NODEHAVINGIDENTICALGEOMETRICALRELATIONSHIPWITH
ITSNEIGHBORS
O STIFFNESSOF EACHMEMBERREPRESENTEDBY "EXACT"FINITEELEMENT
MODELTHATACCOUNTSFOR FREQUENCYANDBEAM COLUMNEFFECT
O PERIODICMODESHAPE
| EIGENVALUESOF 6 x 6 DETERMINANTFOR BUCKLINGOR VIBRATION
Figure 3
38
BUCKLING OF LATTICE ISOGRID CYLINDER
A simple approach to analysis of lattice structures containing a large
number of members is to develop equivalent beam, plate, or shell stiffness so
that they may be analyzed with a continuum theory. The present discrete lattice
theory can be used to evaluate the accuracy of such an approach. In the example
given in figure 4, the buckling load of an isogrid cylinder similar to that
shown in figure 2 is given in terms of the slenderness ratio of an individual
member. The load is normalized by the load obtained from continuum theory; thus,
a value of 1 means agreement between the two approaches. Two orientations of
the isogrid are shown, one with one set of members aligned axially and the other
orientation has one set aligned circumferentially. For each orientation two
curves are given, one for all members straight, the other applicable to
configurations having helical members curved to lie in the surface of the
circular cylinder containing the vertices. The results of the figure show the
following: (I) as member slenderness ratio increases, %he discretness effect
compared to continuum theory is much more significant; (2) the highest load
capability is for straight members having a principal direction in the
circumferential direction. This orientation results in lower individual member
loads for a given end load and thus greater resistance to buckling; (3) buckling
loads are further reduced if helical members are curved. If there is a
compressive force in the curved member which occurs for the + 30 °, 90 °
configurations the reduction in buckling load is the greatest.
P
PEQ
1.0--
.9-
.8 --
.7 --
.6 --
.5
0
0°
_N_NN43 CIRCUMFERENTIAL NODES
"\ \,\ m
\ V_go, +60 °
STRAIGHT MEMBERS \ 50 CIRCUMFERENTIAL NODES
---- CURVED MEMBERS
I I I I I
20 40 60 80 i00
ELEMENT SLENDERNESS RATIO, //p
Figure 4
39
BUCKLING OF TRIELEMENT BEAM
A beam configuration that has been proposed for many applications because of
its structural efficiency for carrying axial load consists of three longerons
with cross-braced diagonals. The buckling characteristics for a simply supported
three element beam loaded in axial compression are shown in figure 5. The
buckling load is plotted as a function of buckle length. Three different modes
are possible. Buckling of an individual member as an Euler column supported at
the joint locations is illustrtated on the left. The load for all buckle
lengths has been normalized with respect to the local buckling load corresponding
to a buckle length equals bay length £. If the column is sufficiently long, in
this case somewhat greater than 17 bays, it will buckle in an overall column
mode. The classical Euler load based on continuum stiffnesses is shown dashed
and indicates some effect of the shear flexibility of the diagonals. A third
mode is possible when the diagonals are small with respect to the longerons. In
this case, a moderate length buckle involving movement of the diagonals can
result in lower loads than assuming the diagonal force nodes. The discrete
lattice buckling theory gives all these results from the same analysis as a
function of the wavelength parameters.
P
3PE
5
2
.2
EAd
.I - EA
/P
.05
I
=_ _____f -
___ "_ \ _'\
n=O
, \\
n=l,_ .
- .05 _
- \ /,
5 I0 20 50 I00
BUCKLE HALF WAVELENGTH _/l.
BAY LENGTH
Figure 5
4O
STRUCTURAL EFFICIENCY OF COLUMN CONFIGURATIONS
A study was made of the column mass required to sustain a given load for the
three element column discussed in the previous slide. Another column
configuration sometimes referred to as a geodesic beam is also shown in figure
2. The configuration is somewhat unusual in that there are no pure axial
members. The load carrying members are inclined to the axis of the column and
provide the transverse shear stiffness as well as contribute to the buckling
stiffness. The battens are loaded in tension when axial compression is applied.
A comparison of the mass of the two different configurations is shown in figure 6.
The mass parameter is plotted against a compressive loading index. The results
were obtained by systematically varying proportions in the Duckling analysis and
observing the minimum mass required for a given end load. The minimum mass
conventional three element beam with diagonal bracing is significantly lighter
than the geodesic beam. However, the proportions for minimum weight are much
different. The area of the diagonal stiffeners is less than 1% of the longerons
whereas the geodesic beam has all areas essentially equal. An optimum
configuration with equal areas would be of advantage for small loads that lead
to minimum size members. The upper curve is for the trielement beam with all
members equal area. In this case, it is much heavier than the geodesic beam.
10-6
MASS
INDEX
M
pL3
10-7
_. ALL MEMBERS--N--///
10-13
I I I I I I II I t I I 1 II!
10-12 i0-ii i0-I0
P
LOADINDEX [L2
Figure 6
41
BUCKLINGOFCABLESTIFFENEDRING
The polygonal ring configuration with central mast used to attach cable
stiffening has been proposed to support thin reflector or antenna surfaces
The pretension of the cable puts the ring membersin compression; additional
compression is caused from the inward radial loads due to stretching a membrane
like surface in the interior of the ring. For very large structures, the
prevention of ring buckling can be a significant design requirement. The
buckling problem for this configuration has been solved and typical results
shownin figure 7. The radial buckling load Q which is applied at each vertex
is plotted against a parameter proportional to the pretension in the cables.
Also shownis the internal force in the ring, P. Both P and Q are expressed in
terms of classical ring buckling parameters. The dashed line at the lower part
of the figure indicates the load capability without cable stiffening. As
pretension is increased, the buckling load increases. The single line on the
left represents the point at which the cables go slack. The maximumload is
reached whena general modeappears even while cables are still in tension. For
this case the critical modehas 5 circumferential waves. Note that the n=2 mode,
which is the lowest for an unstiffened ring, is higher than the n = 5 mode. Once
the general buckling modeoccurs, further increases in pretension cause a slow
drop in the buckling load Q but the force in the ring remains essentially constant.
2
P r
r
El r
Qr 3
/-rElr
200
n=2 ._r
2
100 EA n r
c s = 41,100 Q
Elr
50
2O
10
), =80 ° n =5
r - .769
El r
1 2
NED RING
I I I I I
5 I0 20 50 i00 200
PO _(.._.r) 2
nsEA c
I
500
I
I000
Figure 7
42
VIBRATION OF CABLE STIFFENED RING
The vibration characteristics of the same ring studied in figure 7 are shown
in figure 8. Frequency is plotted against pretension for various values of the
radial load Q. No frequency is shown at values of pretension lower than that
required to prevent cables from going slack due to the load Q. The frequency
then drops slowly until the buckling condition is approached. It drops very
rapidly then to zero which represents the buckling solution of the previous
figure. One effect that is missing from these calculations is the mass of the
membane that is exerting the force Q. A method of including this effect is
currently under development.
fl
Hz
D
1 --
0
/--U_!S]IFFE[]ED
-/-I RI_IGI1
2 5
2O
3O
F RIGIDMAST
CABLE_
Q -----_ ...__Q
RINGJ T___r-___
I I
i0 20
I I
50 i00
Po
200
Figure 8
500 I000
43
SUMMARY
An analysis for buckling and vibration of repetitive lattice structures has
been developed. The results are essentially exact for ring configurations and
for column configurations having simply supported ends. A wide variety of
configurations are possible within the framework of one analysis. Results were
given for typical configurations. In many areas, discrete effects not possible
to determine with simple theory were identified. Yet the solution is no more
complicated than the eigenvalues of a 6 x 6 matrix.
RE FERENCE
I. Anderson, Melvin S.: Buckling of Periodic Lattice Structures. Presented at
AIAA/ASME 21st Structures, Structural Dynamics, and Materials Conference
Seattle, Washington. May 12-14, 1980. AIAA Paper No. 80-0681. To be
published in AIAA Journal.
| ESSENTIALLYEXACTBUCKLINGAND VIBRATIONANALYSISDEVELOPED
e APPLICABLETO A NUMBER OF LATTICECONFIGURATIONSWITH REPETITIVE
GEOMETRY
| COMPLEX BEHAVIORACCURATELYMODELED
| SOLUTIONREQUIRESONLY EIGENVALUESOF 6 X 6 MATRIX
Figure 9
44


Buckling and vibration of periodic lattice structures

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