NASA Technical Reports Server (NTRS) 20000088651: Structural Similitude and Scaling Laws for Plates and Shells: A Review

12 Pages·0.66 MB·English

by  NASA Technical Reports Server (NTRS)

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AIAA-2000-1383
STRUCTURAL SIMILITUDE AND SCALING LAWS
FOR PLATES AND SHELLS: A REVIEW
G.J. SIMITSES" - University of Cincinnati
Cincinnati, Ohio 45221-0070
J.H. STARNES, JR. + - NASA Langley Research Center
Mail Stop 190, Hampton, Virginia 23665
J. REZAEEPAZHAND _- Ferdossi University of Mashhad
Mashhad, Iran
Abstract Introduction
This paper deals with the development and use Aircraft and spacecraft comprise the class of
of scaled-down models in order to predict the structural aerospace structures that require efficiency and wisdom
behavior of large prototypes. The concept is fully in design, sophistication and accuracy in analysis and
described and examples are presented which numerous and careful experimental evaluations of
demonstrate its appficability to beam-plates, plates and components and prototype, in order to achieve the
cylindrical shells of laminated cons_ction. The necessary system reliability, performance and safety.
concept is based on the use of field equations, which
govern the response behavior of both the small model Preliminary and/or concept design entails the
as well as the large prototype. The conditions under assemblage of system mission requirements, system
which the experimental data of a small model can be expected performance and identification of components
used to predict the behavior of a large prototype are and their connections as well as of manufacturing and
called scaling laws orsimilarity conditions and the term system assembly techniques. This is accomplished
that best describes the process is structural similitude. through experience based on previous similar designs,
Moreover, since the term scaling is used to describe the and through the possible use of models to simulate the
effect of size on strength characteristics of materials, a entire system characteristics.
discussion is included which should clarify the
difference between "scaling law" and "size effect". Detail design is heavily dependent on
Finally, a historical review of all published work in the information and concepts derived from the previous
broad area of stnmtural similitude is presented for step. This information identifies critical design areas
completeness. 'which need sophisticated analyses, and design and
redesign procedures to achieve the expected component
performance. This step may require several
independent analysis models, which, in many instances,
require component testing.
The last step in the design process, before
• Professor of Aerospace Engineering, Fellow going to production, is the verification of the design.
+Fellow This step necessitates the production of large
++Assistant Professor & Head of Mechanical components and prototypes in order to test component
and system analytical predictions and verify strength
Engineering.
and peffom3ance requirements under the worst loading
conditions that the system is expected to encounter in
Copyright© 1999 by GJ. Simitses. Published by AIAA
service.
Inc, with permission.
Clearly then, full-scale testing is in many cases procedures can be used, dimensional analysis and direct
necessary and always very expensive. In the aircraft use of governing equations.
industly, in addition to full-scale tests, certification and
safety necessitate large component static and dynamic Models, as a design aid, have been used for
testing. The C-141A ultimate static tests include eight many years, but the use of scientific models which are
wing tests, 17 fuselage tests and seven empennage based on dimensional analysis was first discussed in a
tests.I Such tests are extremely difficult, time paper by Rayleigh. 3 Similarity conditions based on
consuming and defimtely absolutely necessary. dimensional analysis have been used since Rayleigh's
Clearly, one should not expect that prototype testing time (Macagno4), but the applicability of the theory of
will be totally eliminated in the aircraft industry. It is similitude to structural systems was first discussed by
hoped, though, that we can reduce full-scale testing to a Goodier and Thomson s and later by Goodler. 6 They
minimum presented a systematic procedure for establishing
similarity conditions based on dimensional analysis.
Moreover, crashworthiness aircraft testing
requires full-scale tests and several drop tests of large There exist several books that refer to all
components. The variables and uncertainties in crash elements of structural similitude. Murphy 7, Langha_,
behavior are so many that the information extracted Charlton 9, Pankhurst n° and Guldunan u all dealt with
from each test, although extremely valuable, is similitude and modeling principles, and most of them
nevertheless small by comparison to the expense. dealt with dimensional analysis. Kline 2 gives a
Moreover, each test provides enough new and perspective of the method based on both dimensional
unexpected phenomena, to require new tests, specially analysis and the direct use of the government equations.
designed to explain the new observations. Szucs _2 is particularly thorough on the topic of
similitude theory. He explains the method with
Finally, full-scale large component testing is emphasis on the direct use of the governing equations
necessary in other industries as well. Ship building, of the system. A recent book by Singer, Arbocz and
building construction, automobile and railway car Weller t3 devotes an entire chapter on modeling with
construction all rely heavily on testing. emphasis on dimensional analysis concepts.
Regardless of the application, a scaled-down A few studies concerning the use of scaled-
(by a large factor) model (scale model) which closely down shell models have been conducted in the past.
represents the structural behavior of the full-scale Ezra_4presented a study based on dimensional analysis,
system (prototype) can prove to be an extremely for buckling behavior subjected to impulse loads. A
beneficial tool. This possible development must be similar investigation was presented by Morgen ns for an
based on the existence of certain structural parameters orthotropic cylindrical shell subjected to a variety of
that control the behavior of a structural system when Static loads. Soede116 investigated similitude for
acted upon by static and/or dynamic loads. If such vibrating thin shells.
structural parameters exist, a scaled-down replica can
be built, which will duplicate the response of the full- Due to special characteristics of advanced
scale system. The two systems are then said to be reinforced composite materials, they have been used
structurally similar. The term, then, that best describes extensively in weight efficient aerospace structures.
this similarity is structural similitude. Since reinforced composite components require
extensive experimental evaluation, there is a growing
Historical Review interest in small scale model testing. Morton .7
discusses the application of scaling laws for impact-
Similarity of systems requires that the relevant loaded cad_on-fiber composite beams. His work is
system parameters be identical and these systems be based on dimensional analysis. Qian el al.ns conducted
governed by a unique set of characteristic equations. experimental studies of impact loaded composite plates,
Thus, if a relation of equation of variables is written for where the similarity conditions were obtained by
asystem, it is valid for all systems which are similar to considering the governing equations of the system.
it.2 Each variable in a model is proportional to the These works and many other experimental
corresponding variable of the prototype. This ratio, investigations have been conducted to characterize the
size effect in material behavior for inelastic analysis
which plays an essential role in predicting the
relationship between the model and its prototype, is (size effects are discussed in alater section).
called the scale factor. In establishing similarity
conditions between the model and prototype, two In recent years, due to large dimensions and
unioue structural design of the vrovosed space station,
small scale model testing and similitude analysis have (bothintension and compression), which isusually
obtainedfrom small specimens affected by scale. In
been considered as the only option in order to
experimental data. Shill et al.19, Letchworth et al. , addition,isthe strength affected by scale? Recognizing
Hsu et al.2_ and McGowan et al.22 discussed the thatbeamsareprimarily designed for strength,the
possibility of scale model testing of space station answer tothe second question isimportant. Onthe
geometries, especially for vibration analysis. Most of other hand, since columns areprimarilydesigned for
these studies have used complete similarity (defined in stiffness (buckling), the answer tothe first question is
alater section) between model andprototype. important
The present authors have j_ublished several Inthiscontext, the use ofthe term size effect
papers (Simitses and_% Remeepazhand issimilartothe termscale effect. On the otherhand,
et al.24, Re2aeepazhand et al.2s, Simitses and one maywish to find theconditions underwhich the
Rezaeepazhand 26,Simitses et al.2_,and Rezaeepazhand behavioral response of asmall size beam andalarge
andSimitses 2s)thatdealwiththe design of scaled-down size beam aresimilar. Inthiscase, the interest istofind
models and the use of test data of these models to the similarity conditions orscaling laws inorderto
predict the behavior of large prototypes. The behavior achieve similarity inresponse. Inthiscontext, the
includes displacements, stresses, buckling loads, and primaryinterest istobe able totesta small scale model,
natural frequencies of laminated beam-plates, plates obtain response characteristics (displacements, buckling
and shells. In these studies, in the absence of model loads,vibration frequencies, etc.) and use the scaling
test data, the authorstheoretically analyzed the models, laws topredictthe behavior of the large prototype. In
and they used the similarity conditions, obtainedby the thissecond case, one can stilluse the termscaling
use of the governing equations, to predict the behavior effects, if heclearly does notreferto size effects on
of the prototype. They then theoretically analyzed the strengthand stiffness.
prototype and they compared these results to the
predictions. In most cases, the compared results were Size Effects
very close to each other and they concluded that the
designed model can accurately predict the behavior of There exist two main sources of recent studies
the prototype. Very recently, Ochoa and her of size effects. First,Ref. 31contains an outline of
collaborators 29'3° applied similitude theory to a paperspresented ata Wod_hop on Scaling Effects in
laminated cylindrical tube under tensile, torsion and Composite Materials and Structures,andsecond, Ref.
bending loads and underexternal and internal pressure. 32isacompilation of papers dealingwith, primarily,
They demonstrated the validity of developing a scale fracturescaling.
model, testing it and use the similarity conditions to
predict the behavior of the prototype. Fromthe conclusions, of virtually all
presenters attheworkshop3!who dealt with size
Scaling Effects in Composites effects, one cansay thatthe size effect on stiffness is
almost nonexistent.33"u'3s Similarly, the size effect on
Considerable renewed interest hasbeen strengthhascreatedsome controversy. Jackson33
exhibited inthe broadfield of scaling inthe recent concludes thatthere isconsiderable size effect on
years, asevidenced bythe multitude ofresearch papers strength. Grimes_ contends thatfor solid laminates,
thathaveappearedinthe technical literature. Before the largest size effect on staticstrength isless than
discussing any and allefforts, we musthave agood 4.5% Furthermore,hestatesthatthe cause ofscale
understanding, forclear discussion of the meaningof effects isnotsize butother factors suchas poorquality
the words thathavebeen used. These words arescaling tooling, differences inenvironmental exposure, etc. A
orscale effects, similarity conditions orscaling laws similarconclusion was reachedby O'Brien 3_who
andsize effects. claims thatthe effect ofscaling is notbecause of size,
butbecause different damage sequence occurs intwo
by
Scaling effects mean the effect of changing the different sizes. Inaprivatecommum_on L.B.
geometric dimensions ofastructureorstructural Greszczuk_ ofMcDonnell Douglas Space Systems Co.,
component onthe response toexternal causes. The he stated,quote "If the small and big parts are made by
external causes include alltypes of forces. Examples of the sameprocess, there isnot size effect neither on
theabove isabeam made outof metallic materialor stiffness noron mength." He further explained thatthe
man-made composite and subjected tobending. The tests performed athis company on specimens with
mainquestions associated withpredicting the response twelve toone ratio inthickness 0ammates), revealthat
of the beam are: Arestiffness and strength affectedby the effect on stiffness isnonexistent, while theeffect on
scaling? This means isthe effective Young's modulus strengthisless than4%. The_ used were
3
carefully manufacturedby the same process andthey response of all similar systems to similar input can be
had the same filament volume fraction and porosity. predicted.
The objective of most papers in Ref. 32 is to The behaviorofaphysicalsystemdependson
study the size effect on fracture of ice, concrete and many parameters,i.e.geometry, materialbehavior,
notched composite beams. Inthese papers, the dynamic response and energy characteristics of the
conclusion is that size does affect fracture and crack system. The nature of any system can be modeled
propagation. mathematically in terms of its variables andparameters.
A prototype and its scale model are two different
One particular paper in Ref. 32, that by Daniel systems with similar but not necessarily identical
and Hsiao 39,dealt with the thickness effect on parameters. The necessary and sufficient conditions of
compressive strength of unnotched laminates. Itis an similitude between prototype and its scale model
experimental study that used various sizes and layups require that the mathematical model of the scale model
and it concluded that the size effect is extremely small. can be transformed to that of the prototype by a bi-
Further evidence that size has negligible effect on unique mapping or vice versa (Szucs_2). It means, ff
stiffness isprovided bythe tests performedby vectors Xp and X, are the characteristic vectors of the
Jackson *)on graphite/epoxy beams atNASA Langley. prototype and model, then we can find a transformation
The scale varied from one-sixth to full and she matrix A such that:
employed unidirectional and quasi-isotropic layups.
Y_ =AX. or X. =A-tY_ (1)
Clearly, thegnone can atthis junction say with
confidence that size effect on stiffness is negligibly The elements of vector X are all the parameters and
small and that more work on strength needs to bedone variables of the system. A diagonal form of the
in order to explain the reasons forthe conflicting transformation mat_ A is the simplest form of
conclusions (if there is an effect, what causes it). transformation. The diagonalelements of the matrix
are the scale factors of the pertinent elements of the
Inview of the above, the authors embarked characteristic vector X
into aresearch program on structural similitude based
on the following premises: (a) both model and Xxl O ... 0
prototype are governed by the same field equations
(equilibrium, kinematic relations and constitutive 0 Xx2 "'" O
equations, subject to boundary conditions), (b) the only A ___ , (2)
: : "'. i
set of equations that may be affected by size arethe
constitutive relations. It has already been concluded 0 0 .. Xxn
though thatstiffness isnot affected by size and
therefore one is safe to use the same constitutive
relations for model and prototype up to but not in the where L_ = xc/xn denotes the scale factor of xi. In
vicimty of strength limits, (c) damage accumulation for general the transformation matrix is not diagonal.
both model and prototype is minimal. On this basis one
can use similitude theory and obtain the similarity In establishinsgimilaritcyonditionsbetween
conditions. themodel and prototypetwo procedurescan be used,
dimensional analysis and directuse of governing
Theor_ of Similitude equations. The similarity conditionscanbe established
eitherdirectlfyrom thefieldequationsofthesystemor,
if it is anew phenomenon and the mathematical model
Similitude theory is concerned with of the system is not available, through dimensional
establishing necessary and sufficient conditions of analysis. In the second case, all of the variables and
similarity between two phenomena. Establishing parameters, which affect the behavior of the system,
similarity between systems helps to predict the behavior must be known. By using dimensional analysis, an
of a system from the results of investigating other incomplete form of the characteristic equation of the
systems which have already been investigated or can be system can be formulated. This equation is in terms of
investigated more easily than the original system. nondimensioual products of variables and parameters of
Similitude among systems means similarity in behavior the system. The_., similarity conditions can be
in some specific aspects. In other words, knowing how established on the basis of this equation.
a given system responds to a specific input, the
In our studies, we consider only direct use of
d2w
the governing equations procedure. This method is du _ D11 =0
more convenient than dimensional analysis, since the Mxx = Bll dx dx 2
resulting similarity conditions are more specific. When (7)
governing equations of the system are used for
establishing similarity conditions, the relationships
among variables are forced by the governing equations
Equation (1) can be written as:
of the system.
d4w
The field equations of a system with proper
(A11D11 - B I)
boundary and initial conditions characterize the dx--T = qA11
behavior of the system in terms of its variables and
(8)
parameters. If the field equations of the scale model
and its prototype are invariant under transformation A
and Aj, then the two systems are completely similar.
By applying similitude theory, the resulting similarity
This transformation defines the scaling laws (similarity conditions are:
conditions) among all parameters, structural geometry
and cause and response of the two systems. Examples
of the direct use of governing equations is offered _'AI! _'Dll _'w = _211 _w = _'All _,4 _,q
below.
(9)
or
Bending of Laminated Beam-Plates
= _2 (10)
_'AII _'Dll _'w Bll
Consider a laminated beamplate of length a
and width b and simply supported at both ends. We
desire to find the maximum deflection of this
_'w )_Dll = _4_.q (11)
beamplate. The beamplate is subjected to a transverse
line load. By assuming that the displacement functions
are independent of y, or u--u(x), v-=0, w=w(x) Similarly from Eqs. (5), (6) and (7) we have:
(cylindrical bending), from Ashton and Whitney 4_,the
governing differential equations and boundary
_.AII_LU_LX = _w_.BII , (12)
conditions are reduced to:
_.BII_.U_.X =_.w2LDll , (13)
d4w qAll
,(3)
dx 4 AIIDI1 -B_I The condition depicted by Eq. (13) can be obtained by
combining Eqs. (I0) and (12). So, Eqs. (10) through
(12) denote the necessary conditions for complete
d3u Blld4w similarity between the scale model and its prototype, as
(4)
far as deflectional response is eoncemed.
dx 3 AlldX 4
Note that the similarity conditions, Eqs. (10)-
(12) are three, while the number of geometric and
and the B.C.s at x=O, a are: material parameters, cause parameter (load) and
response parameters (u and w) is much larger than
three. This means that there is freedom in designing
w= o (5) models for a given prototype. In addition, if, in
projecting the data of the model to predict the behavior
du d2w of the prototype, all three scaling laws are used, then we
--=0 have complete similarity. If only one (or two) scaling
Nxx = All_-x -B11
dx 2 laws are used, then we have partial similarity.
(6)
For this particular application, e._rimental
data was supplied by Professor Sierakowski : for tests
performed on beam plates. The total number of
laminateussedisten.InSimitses
43,some beam plates pertinent values of asmall scale model. The model has
are considered as models and some as prototypes. the same stacking sequence as the prototype but with a
Similitude theory is used and the results are compared smaller number of layers. The prototype and its scale
to the test results of the prototypes (see Ref. 43 for model have the following characteristics:
details). Partial similarity is used in the comparison. Prototype (0(9010/...)96: a = 90 in. b = 100 in.
h = 0.858 in. N =96,
In addition to the above, similitude theo_' is
employed in a case where experimental results do ztot model (0(90/0/...)t6: a = 5.0 m_ b = 6.139 in.
exist. In this case, the theoretical results of the mo_!el h = 0.143 in. N = 16,
are treated as test data, then a scaling law (par_
similarity) is used to predict the behavior of t?le scale factors: L, = 18 Lb= 16.29
prototype and the predictions are compared to the Lh=6 Xr_= 6.
theoretical results of the prototype. If these tv.o
compare well, success has been achieved in desigm_g In designing the model, we assume that it is made of the
the model and in using similitude theory. same material as the prototype and that ).q=3_,. By
employing only the similarity condition of Eq. (11)
Consider a cross-ply laminated E-Glass/El_:, <y (partial similarity), the results are plotted on Fig. 1. For
plate composed of 96 orthtropic layers (0190/0/...)_ as details, see ref. 23.
the prototype. We desire to find the maxim=m
defleclion of the nrototv_ by extrapolating the
12.0
J
..... th.(p) ./__!
9.0
_**°* ..... *_**°***-***e**** .... --°-° .... -**-_-- _o*****-°* ....... ....¢.. ...... °......°........
c • , _
c
o
-6 6.0 ..... °° ......... _.... .i°°°...°..°°° ..... •...............
¢o
t_
0
3.0
_lJ it • s
0.0
0.0 40.0 eo.o ,20.0 Iso.c
Locd (Ib/in)
Fig. 1. Theoretical and predicted maximum deflections of prototype (O°/90°/O°...)s_ when the model is
(O*/90°/O*..-h, [_'s,, = XE_, = _',,,, = 1; _, = 18; _'b = _'q = 16.92; _._ = XN = 6].
6
Buckling and Vibrations of Plates Application to Shell Configurations
Consider a simply supported, rectangular, Complete similarity and partial similarity were
symmetric, cross-ply laminated plate. The governing applied to laminated cylindrical shell
differential equation for buckling and vibration analyses (_oil_gurations. 24"27"28 Detailscan be found in these
isgivenby: references,but some basic equations and steps are
presented, herein, for completeness. The buckling
equation for a symmetric, laminated, cross-ply (Bii =
DI iW,xxxx+2D12w,xxyy +D22w,yyyy Nxw,xx
Dr6 = D26= At, = A26 = 0), cylindrical shell (Ref. 45) is
= PW,tt given by:
(14)
(2T12T13T23 - TI IT223- T22T?3 )
For buckling alone the characteristic equation is: T33 4
(T11T22 - T?2) (20)
x 2 vl2i (2mo)122( )
= __xxrl 2_ _yy_2
05)
( n'_4( a _2 where _=_x/L, _=5/R,
For free vibrations the characteristic equation is TII =AIIrl 2 + A66_, 2
T12 = (A12 + A66)_n
(16)
By applying similitude theory to Eq. (14), we obtain:
T22 =A22 _2 +A66_2
X4m _ Z'2mXn2 Xt
Xp_-2 =_DII X'-_-a-;LDI2 XaXb2---_=xD22 -xb T23 = A22
R -
(17)
which yield the following scaling laws: T33 = Dllrl4 + 2D12_2vl 2 + D22_ 4 +A2--2----_
R2
_Dll _L4 and
DI2 =DI2 +2D66
2 2
_D12 gmkn The lowest eigenvalue corresponds to the buckling
load, and minimization with respect to integer values of
m and nyields the critical load.
_'L- _'D22 Xt (18) Equation (20) represents the buckling response
of both prototype and its models. Applying similitude
XFq2X_
theory to the preceding equalion, Eq. (20) yields the
following scaling laws for symmetric, cross-ply,
['22 = b4°2 P (19) laminated cylinders:
x4 E22 h3
For details and results, see Refs. [25], [2611 (22)
and I441.
Scaling L,3ws for Lateral Pressure Load
(23)
For the case of a cylinder subjected to lateral
pressure p, Nyy = pR and Bqs. (26)-(30) assume the
following form:
(24)
3.4
= "_rl 3.2 (31)
3.Kyy 3.1 L
(25)
3.D12 ),.2 3.2 (32)
_.Kyy =3.Dl_---
3.W =3.t _'Dll (26)
2 2
(33)
3.u/=_'Dl2 3..q3._ (27)
_.qj = _.D22 _._ (28)
A12
_'Kyy =
(29)
(34)
3.2
AI2
_'W = (30)
3.All3._
(35)
where where Kyy =-NyyL 2/_2Dll, AI2, and DI2,
have already been defined, and
W= -Nxxrl2, - _yy_2, or- (pR/2)(112 + 2_2), 3.Xi =Xip /Xi m
denotes the scale factor of parameter xL
and A12 = AI2 +A66,
Parenthetical remarks: For the case of lateral
DI2 =D12 +2D66.
pressure qJ=-Nyy_ 2. Therefore,
The nine scaling laws, Eqs. (22)-(30) are the
=_'Nyy3.1" Similarly, from the definition of
necessary scaling laws for cross-ply laminated
cylindrical shells for axial compression, lateral
Kyy (Kyy =NyyL 2/x2Dll), one can write
pressure, and hydrostatic pressure. The conditions that
represent structural geometries and mode shapes, Eqs.
(22)-(25) are the necessary scaling laws for symmetric 3.Kyy =-_._yy_. 2/3.DI1 . Use of these two
cross-ply laminated cylinders regardless of the expressions in Eq. (26) yields Eq. (31). In a similar
destabilizinlogad. manner one can derive F.qs. (32)-(35).
As is apparent, the scaling laws are arranged in Equations (31)-(35) are the necessary scaling
the form of different scale factors for each load case laws for symmetric, cross-ply, laminated cylinders
(_). Itshould be pointed out that the presented form of subjected to uniform lateral pressure.
arranging the scaling laws is not unique. However,
previous experience of establishing scaling laws2s The interested reader is referred to Refs. 24
strongly recomn_nds this type of representation. and 27 for results with primarily partial similarity with
8
distortionin number of plies, stacking sequence and . Goodier, J.N. and Thomson, W.T., "Applicability
cylinder length, radius and thickness. Distortion here of Similarity Principles to Structural Models,"
means that prototype and model have different NACA Tech. Note 993, 1944.
parameters (as mentioned above).
. Goodier, J.N., "Dimensional Analysis," Handbook
Discussion of Experimental Stress Analysis (Edited by M.
Hctenyi), pp. 1035-1045, John Wiley & Sons, NY,
It has been demonstrated through the studies 1950.
reported herein, that structural similitude is a powerful
tool in minimizing the need for full scale and large 7. Murphy, G., Similitude in Engineering, Ronald
component testing of structural systems. Future work Press, New York, 1950.
should include the study of systems that exhibit
imperfection sensitivity, extension to sandwich 8. Langhaar, H.L., Dimensionless Analysis and
configurations and validation of the process through an Theory of Models, John Wiley & Sons, New York,
experimental program for laminated plates and shells as 1951.
well as beam plates, plates and shells of sandwich
construction. 9. Charlton, T.M., Model Analysis of Structures. John
Wiley & Sons, New York, 1954.
Through this review, the authors have
demonstrated a procedure that can be used in designing 10. Pankhust, R.C., Dimensional Analysis and Scale
small scale and easily testable models to predict the Factors, Chapman & Hall, London, Reinhold, New
behavior of large prototypes through the use of scaling York, 1964.
laws. These laws are based on the premise that both
model and prototype are governed by the same field 11. Gukhman, A.A., Introduction to the Theory of
equationsand thatthe systemsbehave in a linearly Similarity, Academic Press, New York, 1965.
elasticmanner and they are frec of damage
(dclamin_ons, fiberbreaks,matrix microcracking, 12.Szues, E., Similitude and Modelling, Elsevier, NY,
etc.). This last premise guaranteesthat no sizeeffects 1980.
are presenL
13. Singer, J., AItK_z, I. And Weller, T., Buckling
Experiments, John Wiley and Sons, Chickester,
Acknowledeement England, 1997.
Work was supported by NASA Langley 14. Ezra, A.A., "Similitude Requirements for Scale
Research Center, under Grant No. NASA NAG-I-2071. Model Determination of Shell Buckling under
The financial support provided by NASA is gratefully Impulsive Pressure," NASA TN D-1510, pp. 661-
acknowledged. 670, 1962.
15. Morgen, G.W., Scaling Techniques for Orthotropic
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' 29. Chouchaoui, C.S. and Ochoa, O.O., "Similitude
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