Greschik, "Inflated-Wall Members and Guidelines for Cross Section Design"

7/27/2019 Greschik, "Inflated-Wall Members and Guidelines for Cross Section Design"

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52nd Structures, Structural Dynamics, and Materials Conference, April 47, 2011, Denver, CO

Inflated-Wall Members

and Guidelines for Cross Section Design

Gyula Greschik

TentGuild Engineering Co., Boulder, CO 80303

In order to improve the strength of inflated members, the structuring of the member walls is considered.

Put forth is the optionof a cellular memberwall which consists of compartments separatedby partitions that

comprise a hierarchical load bearing structure. It is the compartments of this wall, rather than the enclosure

within (around the member centerline), that are pressurized when deployed. Straight inflated-wall members

with prismatic compartment geometries of stepwise rotational symmetry are examined. In particular, patterns

of interlocking inner- and outer rib compartments are studied. Via the principles of maximum pressurized

volume and equilibrium, some characteristics of the deployed shapes are studied, as well as the need for, anda procedure to ensure, cross section stability are established. The design procedure presented is developed

semi-empirically: via a derivation that relies on an observation made during numerical studies. It is also

shown that member strength is greatly improved over the capacity of pressurized members with smooth walls.

Technological mass penalties(suchas rigidization overheads) are not considered, nor are fabrication challenges

addressed.

Nomenclature

E Youngs modulus.

h Height above base line.

l Length in the cross section context.

n Rib number (number of inner or outer ribs).R Radius; cross section radius; radial position.

r Local radius of curvature.

t Film thickness; member wall thickness.

Subscripts

0 Reference to the web.

1, 2 Reference to the inner and outer ribs.

cr Reference to limit (critical) state.

e Reference to equivalent monocoque tube.

Symbols

Unit (fundamental) cell central angle, = n. Rib arc central angle.

, Auxiliary variables to simplify expressions. Poissons ratio.

Angle between web and cell border-line. Stress.

Design Engineer, Senior Member AIAA.

Copyright c 2011 by Greschik. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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American Institute of Aeronautics and Astronautics

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference
19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-196

Copyright 2011 by Greschik. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
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I. Introduction

BROAD ideas for, as well as detailed specifics of, inflatable (and inflatable-rigidizable R/I) structures in space

have attracted interest since the dawn of the space age. In the literature accumulated over the decades, low weight

is repeatedly mentioned as a characteristic feature of space inflatable technology. In fact, along with low stowage

volume, low mass may be the most frequently claimed advantage of pressurized structural concepts.

Already in his 1964 review of deployable (expandable) structures and applications, Forbes mentions lightness asthe first advantage when summarizing the characteristics of pressurized solutions a feature he doesnt even list for

mechanically deployable (variable geometry) options [1][Figs. 2025]. The claim that the low weight and packaged

volume of inflatables ... has long been known opens Thomas and Frieses paper [2] on pressurized space antennas in

1980. Freeland, et. al., begin their 1998 inflatable space technology review [3] by highlighting low weight (along with

low cost, packaging efficiency, and deployment reliability) for this technology. Darooka and Jensen in 2001, in the

first paragraph of their structures concept review paper [4], also claim that overall mass can be reduced with inflatable

rigidized solutions. Yet another example is the 2004 inflatable deployment test report [5] by Campbell, et. al., in the

Introduction of which the potential for weight minimization via (unique solutions of) inflated structures is mentioned.

A recent example is the statement by Cobb, et. al., who, when introducing their description of a space experiment [6],

state that inflatable structure concepts are a low mass alternative to conventional hardware.

Comments like those just cited, while not pervasive (e.g., see Ref. [7] for a work where no mass advantage is

mentioned), occur quite often. These statements are trivially true for applications to which alternative technologies are

hardly adaptable. (Examples include bulky but topologically simple enclosures such as habitat modules [8, 9] as wellas large near-perfect spherical objects for calibration [10], passive communication [11], etc.)

However, for applications with meaningful technological tradeoffs (most subsystem-level components: booms,

struts, trusses, antenna- and other device structures, etc.), this narrative can be misleading if specifications, perfor-

mance, andvarious overheadsare not discussed. In particular, the impression of superior performance is givenbecause,

if little attention is paid to mass overheads and stiffness and strength, the last issues appear to be deemed insignificant.

It is implied that inflatable technology can generally match with lower mass the structural performance of alternative

solutions. This message may raise unrealistic expectations for the technology if application in a structurally critical

role is considered. (Concurrently, the spotlight is stolen from a truly pivotal advantage of inflatable structures: the

unparalleled flexibility of stowage design coupled with the simplicity and robustness of deployment.)

Contrary to what is suggested by some broad comments on mass advantages, structural performance (inversely,

the mass needed to achieve specified strength, stiffness, precision) is more often a liability than an asset for inflated-

rigidized structures. This is easily seen for R/I columns (masts, booms, beams), the performance of which directly

suffers from two causes. One, R/I materials are poor compared to those fabricated rigid in controlled conditions:stiffness moduli (on which column strength also depends) of even advanced fabrics, rigidized after deployment, are

worse and less uniform than traditional materials. Two, to be inflatable the column must be tubular: a shape less than

ideal for performance [12]. (More efficient, skeletal, boom structures such as lattices or isogrids could only be used if

stretched within [13] or wrapped on [14, 15] an inflated tube, reducing the latter to non-structural overhead.)

The present work addresses the strength deficiency of inflatable columns via the revision of the prevailing R/I

member configuration. Namely, the traditionally smooth and uniform tube wall, Fig. 1, is replaced with a cellular-

corrugated structure, Fig. 2. During deployment, the set of cavities within the wall, rather than the tube interior, is

pressurized. The resulting set of interconnected wall segments and partitions together resist local buckling better than

a standard wall of the same mass would. Thus member strength, which is the result of the interplay of local and

global stability phenomena [16], can be significantly increased.

Attention in this paper is directed to a simple class of wall corrugation patterns. Namely, prismatic compartment

geometries of interlocking inner- and outer rib compartments with stepwise rotational symmetry are examined. (Such

a structure is exemplified in Fig. 2.) Design guidelines and procedures are established, and the achievable strength

improvement is estimated.

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cross section

pressurized:tube interior

load bearing structure:solid smooth cylindrical wall

R

Figure 1: Customary inflated member.

pressurized:wall chambers

cross section

load bearing structure:"tubelet ribs" of structured wall

Rm

tm

tM

Figure 2: Inflating the member wall.

II. Novelty and Significance

THE idea to deploy a surface structure (shell, plate, wall, etc.) via the pressurization of its structured wall is not

new. Air mattresses and -beds are perhaps the most trivial examples, but there are also several pieces of inflatable

architecture with the walls, rather than the interior space, pressurized. Most common of the last are inflatable sport

installations and tents such as those shown in Fig. 3.

Figure 3: Inflatable tennis court and tent. (Sources: www.surebeatswork.com, www.tradevv.com .)

Space concept examples include tension trusses or drop lines internal to inflated envelopes such as the airmat [1] or

the ISIS idea [13], as well as actual compartmentalization [17]. A recent example of the last is the Inflatable Reentry

Vehicle Experiment (IRVE [18]) launched on August 20, 2009, with a funnel-shapedballute bodyof (at some locations

structurally, elsewhere structurally and pneumatically separated) ring compartments.

Airborne applications where an envelope is pressurized through its internal structure can also be found. Blimp

appendages with interior tension-reinforcement are one common example. Less common but also well known arepressure-deployableairfoils such as inflatable wings [19] with compartments within.

However, simple structural members such as struts or beams with their walls, rather than the interior space, pressur-

ized are unknown to the writer. While there exists a structural concept with a longitudinal-cellular a tubular members

wall, this solution is achieved via poltrusion, not inflation. In fact, this innovation (which is referred to as an artifi-

cial stem, technische Pflanzenhalme in German) by the Institute for Textile Technology and Process Engineering

Denkendorf (ITV Institut fr Textil- und Verfahrenstechnik der Deutschen Institute fr Textil- und Faserforschung

Denkendorf) has been put forth [20, 21] as a biomimetic (biologically inspired) achievement, Fig. 4.

According to the inflated-wall paradigm examined in the present work, a wall structure somewhat similar to the

artificial stem is achieved via the pressurization of an otherwise collapsible fabric or membrane (composite) shroud.

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(a) (b) (c)

Figure 4: Cross section of scouring rush and artificial stems by the Inst. for Textile Techn. and Process Eng., Germany.

(Images from Ref. [21] and the German Fabric Research Foundation web site www.textilforschung.de.)

Thus a pivotal advantage this concept offers over the artificial stem is deployability: the key to large space applica-

tions. The other structural advantages of wall-inflation over the artificial stem higher stiffness, precision, largerdimensions, and fundamentally different fabrication technologies set the two concepts even further apart. In fact,

the two are entirely different except that both improve member strength by increasing the level of structural hierarchy

downward, by topological refinement. (By better trading various modes of failure against the amount of material

used, advanced structural hierarchies can generally increase performance [12, 22].)

However, the concept of such a hierarchicalenrichment for an inflatable tube wall hasnt been explored before. Ac-

cordingly, the design for such a member calls for the consideration of new trades, and for new means of optimization.

These issues are explored in the remainder of this paper.

III. Basic Considerations for Cross Section Design

Atube wall can be made inflatable with many kinds of internal structuring. Attention here is restricted to prismatic

solutions (where the member cross section is uniform). Some geometries are shown in Fig. 5.

p1p1 >p2

simply-stackedconfig.

simply inter-locked

curvedwebs webscaffolds

(c)

(a)

(d)

(b)

Figure 5: Some cross section options.

The simplest topology is the basic stacked pattern shown inFig. 5 (a) which echoes the air mattress idea called a dual-wall

structure by Bair [17]. (If the chambers are circular, this cross sec-

tion reduces to a ring of circles the member becomes a tube of

tubes.) In the slightly more complex configuration of Fig. 5 (b)

the chambers are wedge shaped and are placed in an alternating-

interlocking pattern. The circular continuity of the zig-zagging par-

titions in this arrangement achieves a healthier structural integration

between the rings of bulging exterior walls: those facing toward the

member centerline and outward. This should increase performance

and robustness (e.g., stiffen the counter-rotational vibration modes

of the inner and outer walls). If, in the latter design, the pressures

in the inward- and outward-facing compartments differ, the partition

walls curve, Fig. 5 (c), increasing their bucklingstrength. The cost of

this performance improvement is the need to maintain (at least dur-

ing rigidization, if applicable) different pressures in the two sets of

compartments. Geometrically, even the hierarchy could be further

increased with more delicate partition patterns, cf. Fig. 5 (d). In ref-

erence to the skeletal microstructures called bone tissue scaffolding

in anatomy, the name web scaffolds could be used for this last option.

This paper focuses on the simple interlocking compartment pattern of Fig. 5 (b) as this configuration is deemed

to best balance the potential to improve tube strength against design complexity. The practical consequences of

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complexity (e.g., technical challenges of fabrication, pressurization, and rigidization if applicable) are beyond the

scope of this work.

A. Simply Interlocking Chambers

web

outer rib

inner rib

l1

l2l0

chambers(rib com-partments)

possible definitions ofcell ofrotationalsymmetry:

unit cell with linelengths:

Figure 6: Nomenclature and cells of symmetry.

The tube wall architecture herein considered con-

sists of interlocking wedge-shaped inner- andouter rib compartments, Fig. 6. The partition

walls separating these chambers are herein called

webs, and the chamber wall sections exterior to

the inflated wall, bulging from the internal pres-

sure, the ribs. The latter are called inner or outer

depending on whether they face the tube interior

or exterior. Equal pressure is assumed in all com-

partments, rendering the webs straight.

The unit cell of general symmetry which char-

acterizes the entire cross section geometry con-

sists of the adjacent halves of an inner and an

outer rib with the web between, delimited by two

cross section radii. Denote the angle betweenthese radii as , as shown in Fig. 6 (upper right

corner). (However, in the strict context of [step-

wise] rotational symmetry the fundamental cell is

twice the above: it includes a full inner and outer rib and its definition is not unique, cf. the lower right part of Fig. 6.)

If the rib number (the number of inner or outer ribs) is n then

= / n (1)

The rib number n (equivalently, the central wedge angle ) along with the lengths of the inner and outer half-ribs andof the web, l1, l2, and l0 as shown in Fig. 6, fully define any symmetric hardware design. (Fabrication procedures can

be developed from the geometric definition these four parameters provide.)

B. The Shape of the Cell of Symmetry

(a)

(b)

(c) 1

1

2

l2

r2

l1l0

e

f

a1,h1'

h2

h1a2

w2

2

2

1

R1

R2

h2'

l0fr1

l0e

Rw1

Rw2

w1

Figure 7: Unit cell position and variables.

Given a particular set of hardware geometry parameters (n, l1,

l2, and l0 as described in Section A), the shape of the unit cell

in the context ofn-step rotational symmetry depends on the lo-

cation of the ribs-web assembly within the principal wedge. As

alluded to in Figs. 7 (a) and (c) with gray arrows, this location

can be interpreted as how(to what extent) the walls slide inward,

toward the member centerline, or outward, away from the latter.

(The geometry of Fig. 7 (b) is shown in Figs. 7 (a) and (c) with

dot lines.)

Geometric variables are also shown in order to avoid clut-

ter, only in one of the three sub-figures even if generally appli-

cable. As indicated in Fig. 7 (a), each rib wall contour length li(with i being 1 for the inner, 2 for the outer rib) is divided into

a free, bulging, part ai and a contact section wi which presses

against the similar section of the adjacent cell.

li = ai + wi i = 1,2 (2)

Obviously, wi=0 means no contact, cf. Figs. 7 (b) and (c).The heights of the web endpoints and of the rib contact line endpoints hi and h

i over the wedge border-lines e and

f are also shown. The distances from the cross section center to the ends of the rib contact regions are denoted with

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Expressionsfor the inflated volume have been subsequently derived, and the configuration was solved via maximiz-

ing this metric with numerical optimization. Primary control for the results was provided via equilibrium conditions.

Moreover, for an additional level of control, the solution has been independently programmed and executed both with

MicroSoft Excel and a custom C program.

2. Rib Shapes

1

a1=l1 a2=l2

a2=l2

a1

(b)

(a)

w1

l0

r2r1

2h

l0

r2

r1

2

1=/2

Figure 8: The degenerate case of=0.

The condition of maximum volume in the planar contextof an

cross section wedge leads to two simple rules for the rib shapes:

The freely bulging section of the rib (the section free of con-

tact with the adjacent cell) has a circular arc contour.

If there is an active contact region w> 0 (part of the rib ispressed against the other in the adjacent cell), then the cir-

cular arc contour of the freely bulging rib section osculates

the line of contact at their shared endpoint. (Cf. Figs. 7 (a)

and 8 (b), the latter showing the special case of a degenerate

geometry.)

While these rules of thumb postulate elementary conditions of

membrane mechanics, their explicit acknowledgment (and, in

case of the second rule, formal proof)in the present work formed

the initial premises on which the geometric derivations have been developed.

3. Special Case: Parallel Wedge Boundaries

If the wedge angle diminishes ( 0) the rib number approaches infinity (n) then the compartmentpattern becomes infinitesimally fine. A cell of balanced proportions for this case degenerates into one with parallel

wedge borders, Fig. 8, with 1=2= and the web endpoint radial positions Ri become immaterial (they increasebeyond all bounds with respect to the cell dimensions). Within this special scenario, the equilibrium of the simple case

with no contact regions shown in Fig. 8 (a) can be formalized with the condition

l0cos 2

cos = l1

cos 1

2+ l2

cos 2

2(30)

which is still implicit as it also involves i in addition to the section lengths li. Casting Eq. 30 in the sole context ofthe li lengths is possible only via complicated transcendent relations, limiting the design utility of this relation.

Moreover, the onset of a contact region, Fig. 8 (b), already destroys the simplicity even of the implicit form Eq. 30,

further reducing utility. Apparently, even for the degenerate case of parallel wedge boundaries, a symbolic solution

has severe limitations. This illustrates the need for numeric approach.

4. Shape Solutions for Given Rib Numbers

In the process of exploring characteristic cell geometries, cell shapes have been determined for various rib numbers n

and wall section contour length li. The results of this exercise are illustrated via examples in Fig. 9.

(a) n = 18 (b) n = 18 (c) n = 36 (d) n = 60l1/0/2 = 40/40/40 mm l1/0/2 = 36.45/40/54.31mm l1/0/2 = 30/30/30 mm l1/0/2 = 20/30/25 mm

tM/Rmax = 41 % tM/Rmax = 41 % tM/Rmax = 31 % tM/Rmax = 14 %

Figure 9: Cell shapes (shown at a different scale each) for some rib number and contour section lengths, as indicated.

The cells are arranged according to the rib numbers n. Examples Figs. 9 (a) and (b) correspond to the same n=18(i.e., =10o ) they differ in the contour section lengths only. The latter areuniform in Fig. 9 (a), l1=l0=l2= 40 mm,

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but not so in Fig. 9 (b) where l1=36.45, l0=40, and l2= 54.31 mm. The former case results in an active contact regionon the inner rib but no contact outside. The non-uniform values shown are set to approach the contact limit state on

both sides: the wedge borders osculate the rib arcs with infinitesimal contact lengths.

Uniform lengths are used in Fig. 9 (c) with n=36, twice the number in the preceding examples. Accordingly, themaximum-volume solution features a contact region on the inner rib shorter, than before, with respect to the other

lengths. There is no contact on the outer rib.

The last configuration, Fig. 9 (d), features the highest rib number, n=36, and a web contour length greater than theribs. Consequently, the ribs are not in contact with those of adjacent cells.

Cell designs with concurrently active inner and outer rib contacts can also be achieved with webs sufficiently short

in comparison to the rib contours. Such configurations, however, have little practical relevance.

Also indicated in each figure legend is the ratio of the gross wall thickness, tM=RmaxRmin (cf. Fig. 2), to themaximum radius Rmax. The values range from 41 to 14%, highlighting that the wall depths occupy significant portions

of the cross section radii.

l l

Figure 10: Flattening of collapsed wall if l1= l0= l2.

A seemingly attractive feature of configurations with

uniform section lengths l1 = l0 = l2 is that the ribs andthe web can smoothly collapse, Fig. 10. Such crease-

and fold-free flattening may be convenient during fabri-

cation and storage. However, this paper stops short of

recommending this or any other configuration, for two

reasons. First, attention is herein limited to conceptualexploration. Acknowledging the challenges of fabrica-

tion, stowage, and deployment, care is exercised to im-

ply no judgment on these issues. The second reason why flattenable walls are not presented as desirable is that, in

the framework of the general design recipe discussed in Section IV.A, they dont guarantee cross section stability.

IV. Symmetry and Stability

THE results discussed so far derive from the assumption of symmetry: the kinematics considered were strictly

confined to specific wedge geometries directly defined by the rib number. However, symmetry for a pressurized

configuration shouldnt be assumed a priori, even if the hardware itself is symmetric (cf. asymmetric pumpkin balloon

configurations [23, 24]).

Figure 11: The degenerate case of=0.

If the maximum inflated volume in the global configurationspace is outside the subspace of symmetry, the cross section will

assume that asymmetric shape whenever the opportunity arises.

For example, the cross section may ovalize (collapse) unidirec-

tionally as alluded to in Fig. 11, or it may take some other non-

circular shape, if pressurized volume is gained with the transi-

tion.

Quantitative insight into such phenomena could be gained

only with a model that captures the cross section in its entirety.

Moreover, for an even more faithful representation of practical

reality, the spatial interaction between member walls and end constraints would also be desirable to model.

However, there exists an approach to identify robust cross sections without the analysis of complex interactions:

cross sections immune to pressurization instabilities can be designed relying on the analysis framework of a single

wedge, discussed above, as opposed to more complex models. This design procedure, used in the present work, isdescribed next.

A. Stability and Unconstrained Cell Shape

The procedure herein given for the design of stable cross sections is based on one basic observation. Namely, a cross

section is bound to be stable if all of its cells take the shape that maximizes their own individual volumes. (Clearly, if

this condition is satisfied, then no wall deformation can increase total pressurized volume.)

The premise just stated is a sufficient condition: if satisfied, it guarantees stability. However, it is not necessarily a

necessary condition also: it says nothing regarding the existence of stable cross sections for which not all cell volumes

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are individually optimized. This simply sufficient condition, nevertheless, fits the needs of elementary design which

merely aims at achieving stability, without requiring that no other stable conditions exist.

The key to the optimization of a cell unconstrained by the wedge angle is that the latter be included in the control

variables, rather than treated as a fixed constant. Thus the pressurized volume calculated from the geometry defined

by Eqs. 3 through 29 must be maximized in terms of both 2 and . The results of this optimization, performednumerically, for the sets of contour section lengths in Fig. 9 are shown in Fig. 12.

(a) n = (b) n = 18 (c) n = (d) n = 32.96l1/0/2 = 40/40/40 mm l1/0/2 = 36.45/40/54.31mm l1/0/2 = 30/30/30 mm l1/0/2 = 20/30/25 mm

Figure 12: Unconstrained shapes (shown at a different scale each) for the cells in Fig. 9. (An infinite rib number,

n = , means that the maximum-volume configuration has a zero wedge angle, = 0o.)

Note a few characteristics of the new, unconstrained, shapes. First, the configuration in Fig. 12 (b) doesnt differ

from that in Fig. 9 (b) quantitative details, not presented here, reveal identity to numerical precision. Apparently,

the design in which the contour lengths were tuned to achieve osculation between border lines and rib arcs was the

absolute minimum energy configuration, despite that it had been developed for a pre-determined wedge angle.

Second, the unconstrained states for the cells with uniform contour lengths, Figs. 9 (a) and (c), possess the degen-

erate geometryof parallel wedgeborders (cf. Section III.B.3). Furthermore, the two results differ only in scale: their

shapes are identical.

The third noteworthy observation pertains the fourth, generic, design, Fig. 9 (d). The unconstrained shape of this

cell is a wedge, Fig. 12 (d), with a central angle 5.46o different from the initial ini=180o/603o value. The

new rib number is thus n=180o/free=32.96, not an integer, revealing that a cross section (a full circle) couldnt beassembled by repetitively applying this unit.

If a member were fabricated with a cross section of cells in Figs. 9 (a), (c), or (d) despite the nonconform uncon-

strained shapes involved, the result may not be stable. Under pressure, each cell would try to better approach its

unconstrained shape of absolute maximum volume, undermining symmetry. On the other hand, a section design in

which each cell assumes its unconstrained configuration, such as Fig. 9 (b), is unconditionally stable.

B. An Empirical Observation

A last, very significant detail of the freely optimized cell shapes shown in Fig. 12 is that, for each, the inner and outer

rib arcs subtend the same angles with the wedge borders. Quantitatively,

1 = 2 5.31o in Fig. 12 (a) (31)

1 = 2 0.00o in Fig. 12 (b) (32)

1 = 2 5.31o in Fig. 12 (c) (33)

1 = 2 19.96o in Fig. 12 (d) (34)

In fact, the equality

1 = 2 (35)

characterized each and every unconstrained cell shape examined during research, regardless of contour lengths, wedge

angles and cell shapes, and whether the unconstrained shape agreed with the constrained one or not.

This observation, therefore, has been empirically accepted as a necessary condition for cell stability. Accordingly,

the decision to add relation Eq. 35 to the design equations formally proven has been made. The set of equations

so extended enabled the derivation of a practical design procedure without sophisticated cell shape optimization, as

described next.

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C. Semi-Empirical Design

In experimental research and design procedures that rely on it, it is common to tailor formulation to match obser-

vations. Coefficients and functional forms are imported in frameworks established with rigorous theory, if reality is

approximated with acceptable accuracy. While this practice is unusual in conceptual-theoretical research, it has been

nevertheless followed in the present work for convenience. As the observation relied upon, Eq. 35, is numerically

established with computational accuracy, the precision of the developedmodel (and of the consequent results) will not

suffer. Further, the validity of Eq. 35 as a condition for wall cell volume maxima has in fact been verified by the results

developed from it. The need for a theoretical proof for Eq. 35 is herein deemed aesthetic and left for later work.

The solutionof wall cell geometry for contour lengths li and a wedge angle, as a function of2, has beenoutlined

in Eqs. 3 through 29. With the following procedure, the input to this solution can be determined to ensure cross section

stability (maximum volume for each cell). Namely, a recipe to obtain l0 and l2 (web- and outer rib contour lengths)

for a given inner rib contour l1 and arc-to-wedge border angle 1 =2 is given, within the context of a wedge angle. The step-wise equations, applicable to configurations with no inter-cell contact (w1=w2=0), are as follows:

1 = 1 (36)

r1 = l1/1 (37)

h1 = r1 sin 1 (38)

= sin(1+2)/sin

1 cos(2) (39)

2 = 1/2 tan1(sin(2)/) (40)

2 = 1+2 (41)

l0 = h1 /sin 2 (42)

h2 = l0 sin 1 (43)

r2

= h2/sin

2(44)

l2 = r22 (45)

These expressions have been derived from simple geometric relations combined with the force-balance equilibrium

conditions of the web endpoints in the associated border-line directions e and f (cf. Fig. 7), and from the equality

Eq. 35. Their output includes l0 and l2 which complete the input to Eqs. 3 through 29, of which only the ones not yet

covered by Eqs. 36 through 45 need to be solved in order to complete the cell design.

Figure 13: Stable cell, n = 45,l1/0/2=

12.70

/15

.09

/14

.88mm.

The cell for an example design is shown in Fig. 13, for wedge angle =180o/n=4o and cross section wedge contour lengths l1= 12.70 mm (half inch),l0 = 15.09 mm, and l2 = 14.88 mm for the inner rib, web, and the outer rib.The rib contour arc-to-wedge border angles are 1=2=10

o. Accordingly, the

rib walls meet with 21=20o groove angles on both the inner and outer wall

surfaces. The inner cross section radius is Rmin = 125.6 mm, the outer one is

Rmax=153.5 mm. Thus the outer diameter is Douter=307 mm, about a foot, andthe total wall structure thickness is R=24.9 mm, a little less than an inch.

A part of the full cross section with several cells is shown as a contour in Fig. 14, and in the member context in 15.

These images are snapshots of a finite element (FE) model of the design.

Figure 14: Cross section geometry detail. Figure 15: Mesh detail perspective view.

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V. Strength Estimate

BY virtue of their higher local curvatures, the ribbed member wall herein considered is generally able to withstand

without local buckling higher compressive stresses than a weight-equivalent tube with smooth wall would. The

assessment of this strength improvement, however, is not a trivial exercise, not even if the wall material is linear elastic

and isotropic (with Youngs modulus Eand Poissons ratio ) as assumed in the following.

In fact, there are two kinds of local buckling for the partitioned wall: one in which the structured wall, as awhole, buckles, the other when individual wall segments (partitions or ribs) loose stability. Ultimate member strength

depends on how local stability phenomena of these two kinds interact with the global context with global buckling

for sufficiently long members, or with constraints by member support otherwise.

The interaction of the global and two local stability effects, in the context of the cellular wall structure, can only

be captured with third order (geometrically correct) full three-dimensional numerical methodology such as capable

nonlinear FE analysis. The delicate structure of the wall, the shell formulation needed, and the fidelity required

render such an effort computationally expensive. This analysis has been attempted multiple times: a model, with

imperfection seeds, has been incrementally compressed to probe the collapse load and mechanism. However, the

analyses were unable to overcome numerical instabilities early in the solution. The problem is deemed to have arisen

from web buckling which occurs in intricate patterns early (web flatness invites buckling much earlier than the ribs on

which wall strength effectively depends).

Inflated-wall member performance, however, can be estimated via well known design relations, even if in an im-

perfect manner. This is carried out here with the following simplifying assumptions:1. Web contribution to member strength is ignored.

2. Web contribution to member mass is fully accounted for.

3. Member strength is the integral over all rib material cross sections of the minimum critical stress the lower one

of the critical stresses assessed for the inner and outer rib walls.

4. Rib wall critical stress can be estimated via the relations presented for compressed circular-cylindrical columns in

the 1968 Peterson report [26].

The first assumption is, obviously, conservative. The second and third ones are, simply, realistic: they are neither

conservative, nor non-conservative. The fourth, last, one is also conservative because it prescribes expressions devel-

oped for full, unconstrained, circular cross sections (full tubes) to a more constrained problem: to the wall stability of

cylindrical sections with supported edges (by the webs and the adjacent ribs). Therefore, all in all, the assumptions

spelled out are conservative. True performance should be better than the assessment obtained.

Hardware imperfections are implicitly accounted for in the calculations by the statistical data embedded in thePeterson equations. While the imperfection patterns and magnitudes in an inflated-wall member are likely different

from those in simple tubes (the subject of the Peterson report), this discrepancy is accepted herein as inevitable.

A. Performance for the Example Design

To relate the performance of the design shown in Figs. 13 through 15 to that of a comparable monocoque tube, first,

specify some additional design details for the former and define an equivalent design for the latter. Let the rib and web

wall thicknesses in Fig. 13 be

t1 = t2 = 0.305 mm (12 mil) (46)

t0 = 0.152 mm (6 mil) (47)

Next, define an equivalent monocoque tube with a radius that places the tube wall where the inflated-wall unit cellcenter of gravity is, and a wall thickness that results in the same total material volume as for the inflated wall member.

In particular:

Re = 139.7 mm (48)

te = 1.098 mm (49)

Then assume the same, immaterial, density and Youngs modulus E for both tubes, and evaluate the critical load

by simply applying the Peterson equations [26] for the monocoque tube, and using the four assumptions outlined in

Section V for the inflated-wall member.

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The calculations reveal that this particular design achieves a performance about four times that of an equivalent

traditional member:

cr,inflatedwall = 4.01 cr,monocoque (50)

B. Assessment of Performance Trend

With the use of some rule of thumb assumptions, the performance improvement expected from the consideredmember cross sections can be assessed. For this derivation, described here, use the classic [25, 26] tube wall stability

limit

cr =E

3(12)

t

r(51)

with cr the local-critical compressive stress, and apply this to the bulging rib segments as if the latter were (integralparts of) full tubes.

In the monocoque context of a traditional inflatable tube, the rin Eq. 51 corresponds to the tube radius Re

r = Re (52)

in which the subscript e indicates that a tube equivalentto the inflated-wall member is considered. Further, note that

the rin Eq. 51 will be the (local) ribbulge radius. Assuming that the rib contours are semi-circles one can consequently

approximate the value ofrgeometrically with the rule of thumb

r Re/n (53)

in which n is the rib number, and Re is the equivalent tube radius.

Further, assume that the member wall materials are identical and their total masses are similar accordingly,

summarily take the wall thickness in the considered inflated-wall members outer skin to be a third of that in the

traditional tube

t1 t2 te/3 (54)

From the above relations it follows that the strength of the wall-inflated member relates to that of a comparable

traditional one with the equivalent radius Re according to

cr

cr, e

cr, ribs

cr, e

n

3(55)

which is approximately unity ifn=9 and increases with n thereafter. An order of magnitude improvement is expectedfor n=72, where the central angle from the member axis of each wall chamber is =5o.

VI. Concluding Remarks

THE pressurization of a cellular member wall, rather the member interior, as a means of deployment and to improvepressurized member performance has been considered and investigated theoretically and numerically. Geometricequations for the design of stable cross sections composed of interlocking wedge-shaped chambers have been derived,

and the achievable performance improvement has been shown to be substantial. However, as the direct numerical

modeling of thecollapse mechanismhas not been successful, the independent verification of theperformanceestimates

presented is still yet to be completed. Due to the conservative nature of the assumptions used, however, the strength

numbers developed are deemed to be lower bound performance metrics.

A practical investigation of the studied inflated-wall concept is also left for future work. Fabrication, packaging,

and deployment issues have not been investigated herein.

VII. Acknowledgments

THE initial core idea of boosting inflatable member strength via a structured-pressurized member wall was first pro-

posed by the writer in 2005 in a non-public study commissioned by LGarde, Inc. All further work (illustrations,

derivations, computer programs, numerical studies) was subsequently performed with no corporate or government

support. The writer thanks LGarde for their permission to make the initial idea public.

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Documents

Greschik, "Inflated-Wall Members and Guidelines for Cross Section Design"