ArticlePDF Available

Abstract

Air-inflated fabric structures fall within the category of tensioned structures and provide unique advantages in their use over traditional structures. These advantages include light weight designs, rapid and self-erecting deployment, enhanced mobility, large deployed-to-packaged volume ratios, fail-safe collapse, and possible rigidification. Most of the research and development pursued in air-inflated structures can be traced to space, military, commercial, marine engineering and recreational applications. Examples include air ships, weather balloons, inflatable antennas and radomes, temporary shelters, pneumatic muscles and actuators, inflatable boats, temporary bridging, and energy absorbers such as automotive air bags. However, the advent of today's high performance fibers combined with continuous textile manufacturing processes has produced an emerging interest in air-inflated structures. Air-inflated structures can be designed as viable alternatives to conventional structures.
NUWC-NPT
Reprint
Report
11,774
5
November
2006
Air-Inflated
Fabric
Structures
Paul
V.
Cavallaro
NUWC
Division
Newport
Ali
M.
Sadegh
The
City
College
of
New
York
NEWPORT
Naval
Undersea
Warfare
Center
Division
Newport,
Rhode
Island
Approved
for
public
release;
distribution
is
unlimited.
Reprint
of
a
chapter
in
Marks'
Standard
Handbook
for
Mechanical
Engineers,
Eleventh
Edition,
McGraw-Hill,
New
York,
2006.
TABLE
OF
CONTENTS
Page
L
IST
O
F
IL
L
U
ST
RA T
IO
N
S
....................................................................................................... 11i
IN
T
R
O
D
U
C
T
IO
N
...........................................................................................................................
I
D
E
S
C
R
IP
T IO N
................................................................................................................................
1
FIBER MATERIALS
AND
YARN
CONSTRUCTIONS
........................................................ 3
EFFECTS
OF
FABRIC
CONSTRUCTION
ON
STRUCTURAL BEHAVIOR
........................
4
O
P
E
RA T
IO
N
...................................................................................................................................
6
INFLATION
AND
PRESSURE
RELIEF VALVES
.................................................................
7
CONTINUOUS
MANUFACTURING
AND
SEAMLESS
FABRICS
......................................
7
IMPROVED DAMAGE
TOLERANCE
METHODS
............................................................... 8
R
IG
ID
IF
IC
A
T
IO
N
..........................................................................................................................
9
A
IR
B E
A
M
S
....................................................................................................................................
9
D
R
O
P-ST
IT
C
H
ED FA
B
RIC S
..................................................................................................
15
EFFECTS
OF AIR
COMPRESSIBILITY
ON
STRUCTURAL STIFFNESS
.........................
16
EXPERIMENTS
ON
PLAIN-WOVEN
FABRICS
.................................................................. 17
A
STRAIN
ENERGY-BASED
DEFLECTION
SOLUTION
FOR
BENDING
OF
AIR
BEAMS
WITH
SHEAR
DEFORMATIONS
...........................................................
20
ANALYTICAL
&
NUMERICAL MODELS
..........................................................................
22
U
nit
C
ell
N
um
erical
M
odels
.................................................................................................
22
Exam
ple
of
a
U
nit
C
ell
M
odel
..........................................................................................
23
Structural
A
ir
B
eam
M
odels
............................................................................................
25
C
onclud
in
g
R
em
ark
s
..................................................................................................................
26
R
E
F
E
R
E
N
C E
S
..............................................................................................................................
2
6
LIST
OF
ILLUSTRATIONS
Figure
Page
I
Various
fabric
architectures
used
in
air-inflated
fabric
structures
..............................
2
2
Yarn
tensile
testing
using
Instron®
textile
grips
........................................................
4
3
Examples
of
Pierce's
geometric
model for
plain-woven
fabrics
w
ith
bi-directional
and
uni-directional
crim
p
........................................................
4
4
R
ip-stop fabric
architectu
re
........................................................................................
8
5
Yarn
tensions
in
a
plain-woven
pressurized
fabric
cylinder
and
definition
of
yam
density
ratio
................................................................................................
. . 10
6
Idealized
distribution
of
warp
yarn
forces due
to
bending
of
a
plain-woven
fab
ric
air
b
eam
...................................................................................................
. . 1 1
7
Combined pressure
and
bending
induced forces
in
warp
yarns
at
various
distances
along
a
plain-woven
air
beam
based
on
a
simple
stress
balance
analysis
...........
12
8
Superposition
of
pressure
and
bending
induced yarn forces
in
plain-woven
air
beam
a
triaxial
braided
air
beam
and
a
dual
axial
strap-reinforced
braided
air beam
......
12
9
4-Point
flexure test
on
a
6
inch
diameter plain-woven
air
beam
constructed
of
3,000-denier,
2:1
Y
D
R
V
ectran fabric
.................................................................
13
10
Experimental
load
vs.
deflection
plot
for
an
uncoated
6-inch
diameter
air
beam
constructed
of
3,000-denier non-twisted
Vectran
yarns
in
a
plain-woven
2:1
Yarn
Density
Ratio
fabric
using
a
37-inch
span
between
load
points
and
an
85-inch span
betw
een
support
points
...................................................................
14
11
Plot
of
total
load
vs
mid-span
deflection
for
a
2-inch
diameter
plain-woven
air
beam
constructed
of
1,500-denier,
2:1
YDR
Vectran
fabric
...............................
14
12
Comparison
of
2-inch
diameter
Vectran
and
PEN
plain-woven
air
beams
subjected
to
4-point
bending
tests at
various
load
point displacement
rates
...... 15
13
Section
view
of
an
example
drop-stitch
construction
for
air-inflated
fabrics
...........
16
14
Stages
of
axial
stiffness
for woven
fabric subjected
to
tension
.................................
17
15
Picture
frame test
fixture for
pure
shear
loading
of
fabrics
........................................
18
16
Stages
of
shear
stiffness
for
pure
shear
loading
of
a
2:1
Yarn
Density
Ratio
w
oven
fabric
................................................................................................
. . . . . 18
17
Combined
biaxial tension
and
in-plane
shear
test
fixture.
(U.S.
Patent
N
o.
6,860,156)
........................................................................................
. . 19
18
4-Point
bending arrangement
for
shear deformable
beam
deflection equation
...... 20
19
Treatment
of
yam
kinematics
in
unit
cell
models
......................................................
23
20
Example
unit
cell
model
and
loading
procedure
......................................................
23
21
Example
of
an
air
beam
global
finite
element
model
subjected
to
4-point
bending
......
25
ii
AIR-INFLATED
FABRIC
STRUCTURES:
A
CHAPTER
FOR MARKS'
STANDARD
HANDBOOK
FOR
MECHANICAL
ENGINEERS
INTRODUCTION
Air-inflated
fabric
structures
fall
within
the
category
of
tensioned structures
and
provide
unique
advantages
in
their
use
over
traditional structures.
These
advantages
include
light
weight
designs,
rapid
and
self-erecting
deployment,
enhanced
mobility,
large
deployed-to-packaged
volume
ratios,
fail-safe
collapse,
and
possible
rigidification.
Most
of
the
research
and
development
pursued
in
air-inflated structures
can be
traced
to
space,
military, commercial, marine
engineering
and
recreational applications.
Examples
include
air
ships,
weather
balloons,
inflatable
antennas
and
radomes,
temporary
shelters,
pneumatic
muscles
and
actuators,
inflatable boats,
temporary bridging,
and
energy
absorbers
such
as
automotive
air bags.
However,
the
advent
of
today's
high
performance
fibers
combined
with
continuous
textile
manufacturing processes
has
produced
an
emerging
interest
in
air-inflated
structures. Air-inflated
structures
can
be
designed
as
viable
alternatives
to
conventional
structures.
Because these
structures
combine
both
the
textile
and
structural
engineering disciplines,
the
structural
designer
should
become
familiar
with the
terminology
used
in
textile
materials
and
their
manufacturing processes.
A
glossary
is
provided
in
reference
[1].
DESCRIPTION
Air-inflated
fabric
structures
are
constructed
of
lightweight
fabric
skins that enclose
a
volume
of
pressurized
air.
The
fabric
is
typically
formed
in
a
variety
of
textile
architectures
including
those shown
in
figure
(1).
Each
architecture
has
its
own
design,
manufacturing,
tooling
and
cost
implications.
Structurally,
these
architectures
will
behave
differently
when
subjected
to
loads.
The
plain-weave
architecture
provides
orthogonal
yarn
placement
resulting
in
extensional
stiffness
along
the two
yarn
axes;
however,
it
lacks
shear
stiffness
for
off-axis
loads.
While
the
braided
architecture
provides
the
fabric with
shear
stiffness
due
to
the
non-orthogonality
of
the
yarns,
it
lacks
extensional stiffness.
The angle
between
the
braid
axis
and
the
yarns,
0,
is
referred
to
as
the
braid
angle
or
bias
angle.
Both
the
triaxial braid
and axial
strap-reinforced
braid
architectures
behave
similarly
in
that
they
afford the
fabric
with
extensional
and
shear
stiffnesses.
The
air
pressure
develops
a
biaxial
pre-tensioning
stress
throughout
the fabric.
This
pre-
tensioning
stress
enables
the
structure
to
generate
its
intended
shape,
provides stiffness
to
resist
deflections
and
affords
stability
against collapse
from
external loads. Fabric
materials
can
often
be
idealized
as
tension-only
materials
for
design purposes, that
is,
their
in-plane
compressive
moduli
and
bending
moduli
are
considered negligible.
Fy
FxF
Weave
Braid
Triaxial
Braid
Strap-Reinforced
Braid
Figure
1.
Various
fabric
architectures
used
in
air-inflated
fabric
structures.
Stiffness
of
the
structure
is
primarily
a
function
of
the
inflation
pressure.
As
the
inflation
pressure
increases,
the
pre-tensioning
stresses
throughout
the
fabric
increase
and,
in
turn,
stiffen
the
structure.
Once external
loads
are
applied,
stresses
from
these
loads
superimpose
with
the
pre-tensioning
stresses
in
the
fabric.
As
a
result,
a
complete
redistribution
of
stress
occurs.
This
stress
redistribution
balances
the
loads
and
maintains
the
structure
in
a
state
of
static
equilibrium.
Depending
upon
the
type
of
air-inflated
fabric
structure
(i.e.;
beam, arch,
etc.)
and applied
loads
(i.e.;
tension, compression,
shear, bending, torsion,
etc.),
the
redistribution
of
stresses
can
either
increase
or
decrease
the net
tension
stresses
in
the
fabric
skin.
However,
stability
of
the
structure
is
only ensured when
no
regions
of
the
fabric
experience
a
net
loss
in
tensile
stress.
Otherwise,
if
the
stresses
from
applied
loads
begin
to
relax
the
pre-tensioning
stress
(i.e.;
the
tension
approaches
zero),
the
onset
of
wrinkling
is
said
to
have
occurred within
the
structure.
Wrinkling
decreases
the
structure's
load
carrying
capability
and
upon
further loading,
eventual
buckling
or
collapse
will
result.
There
are
two
significant
and
unique
characteristics
that
air-inflated
fabric
structures
provide over conventional structures.
First,
upon
an
overload condition,
a
collapse
of
an
air-
inflated
structure
does
not
necessarily
damage
the
fabric
membrane.
When
the
overload
condition
is
removed,
the
air-inflated
fabric
structure
simply
restores
itself
to
its
design
load
configuration.
Second, since
wrinkling
can
be
visually
detected,
it
can
serve
as
a
warning
indicator
prior
to
collapse.
2
FIBER
MATERIALS
AND
YARN
CONSTRUCTIONS
Proper selection
of
fiber materials
and
yam constructions
are
important
factors
that
must
be
considered
in
the
design
of
air-inflated
fabric
structures.
Both
should
be
optimized
together
to
achieve
the
desired
performance characteristics
at
the
fabric and
structural
levels.
Many
of
today's
air-inflated
fabric
structures
use
yams
constructed
of
high
performance
continuous
fibers
such
as
Vectran®
(thermoplastic
liquid
crystal
polymer)
PEN
(polyethylene
naphthalate),
DSP®
(dimensionally
stable
polyester),
and
others.
These
fibers
provide
improved
structural
performance
(high
strength,
low
elongation, fatigue, flex-fold,
cyclic
loadings,
creep,
etc.) and
enhanced environmental resistance
(ultraviolet
rays,
heat,
humidity,
moisture,
abrasion,
chemicals,
etc).
Other
fibers
used
in
air-inflated
fabric
structures
include
Kevlar®,
Dacron®,
nylon,
Spectra®
and
polyester.
Hearle[21
provides
additional
information
on
fiber
materials
and
their
mechanical properties.
Yams
are
constructed
from
fibers
that
may
be
aligned
unidirectionally
or
arranged
in
a
number
of
twisted bundles.
Twist
is
used
to
improve
the
handling susceptibility
of
the
yarns
by
grouping
the
fibers
together
especially during textile
processing.
Twist,
which
is
measured
in
turns
per
unit
yam
length,
affects
the
yam
tensile
properties.
For
discontinuous
or
staple
fiber
yams,
twist can
increase
the
yarn
breaking
strength because
the
internal forces
at
the
ends
of
a
fiber
can
transfer
to
neighboring
fibers
via
inter-fiber
shear
forces.
However,
twist
in
continuous
fiber yarns
can
reduce
the
yarn
breaking
strength
as
observed
by
Hearle
31
.
Therefore,
a
minimal
twist
is
recommended
for
providing
adequate
handling
protection
to
continuous
fiber
yams.
Hearle[31
experimentally
investigated
the
effects
of
twist
on the
tensile
behavior
of
several
continuous
fiber
yams.
His
results showed
that
yam
tenacity
(defined
as
tensile
strength
measured
in
grams-force
per
denier
or
grams-force
per
tex)
decreased
with
increasing
twist
for
3
prescribed yam
tensions
used during
twist
formation.
In
general,
the
yam
modulus
decreased
with
increasing
twist,
yarn
elongation
at
break increased
with
increasing
twist,
and
yam
elongation
decreased
with
increasing
yam
tension.
A
difference
in
the
load-extension
behavior
of
twisted
and
non-twisted
yams
is
that
a
twisted
yarn
when
subjected
to
tension
will
undergo
compaction
of
its
cross
section
through
migration
of
its
fibers
and
develop greater inter-fiber
frictional
forces
than
a
non-twisted yam.
Like fabrics,
the
structural
performance
of
yams
can
be
tailored
by
changes
in
their architecture
and
processing.
Once
the
yarns
are
processed
into the
fabric
(i.e.;
by
weaving,
braiding,
etc.),
it
is
recommended
that
tensile
tests
be
performed
on
sample
yarns
removed
from
the
fabric.
This
will
allow
the
"as-processed"
yam
properties
to
be
compared
to
the
design
requirements.
For
example,
the
"as-woven"
tensile
properties
of
continuous-fiber,
non-twisted
yarns
removed
from
a
plain-woven
fabric
air
beam were measured
4
]
using
an
Instron
machine
configured
with
textile
grips
as
shown
in
figure
(2).
The
cross
sectional areas
of
the
yams
were
computed
based
on
fiber
diameter
and
quantity.
The tests
revealed
that
the
average
breaking
stress
of
the
weft
yarns
was
nearly
20%
less
than
that
of
the
warp
yams.
The
reduction
in
breaking
stress
of
the
weft
yams
was
due
to
fiber
damage caused
by
the
use
of
higher
tensions
in
these
yarns
during
weaving.
3
Figure
2.
Yarn
tensile
testing
using
Instron®
textile
grips.
EFFECTS
OF FABRIC
CONSTRUCTION
ON
STRUCTURAL
BEHAVIOR
Fabric
materials
are
constructed
from
yams
that
cross
over
and
under
each
other
in
a
repetitive,
undulating
pattern.
The
undulations
shown
in
figure
(3)
are
referred
to
as
crimp,
which
is
based
on
Pierce's
geometric
fabric
modelI
5
.
02-
h12
VVeft
Yarns
d
at
h
D -
PWarp
Yarns
Bi-Directional
Crimp
Uni-Directional
Crimp
Figure
3.
Examples
of
Pierce's
geometric
model
for
plain-woven
fabrics
with
bi-directional
and
uni-directional
crimp.
Pierce's
geometric
model
relates these
parameters
as
they
are
coupled among
yarn
families.
The
crimp height,
h,
is
related
to
the
crimp
angle,
a, and
yam
length,
L,
as
measured
between
yarns,
and the
sum
of
yarn
diameters
at
the
cross
over points
by
the
following
equations
described
by
Grosberg
[3]:
p
=
(L-Da)cosa+Dsina
h
=(L
-
Da)sin
a
+
D(L
-
cos
a)
(1)
D
=
hi
+
h,.
4
These
equations
were
based
on
idealized
geometry
and
assumptions
such
as
restricting
the
yams
to
circular
cross
sections
and
no
consideration
of
force
or
stiffness
effects.
Plain-woven
and
braided
fabrics
behave
differently
under
load
because their
yam
families
are
aligned
at
different
angles.
Plain-woven
fabrics
utilize
a
nearly
orthogonal
yarn
placement
of
warp
and
weft
(or
fill)
yams.
By
general
textile
definitions,
warp
yarns
are
identified
as
those
yams
running
parallel
to
the
selvage
and
are
virtually
unlimited
in
their
length.
Weft
yarns
are
at
right
angles
to
the
selvage
and
are
limited
in
length
by
the
width
of
the
weaving equipment.
On
the
other hand,
braided
fabrics
have
a
+0/-0
yam
placement,
where
0
is
commonly
referred
to
as
the
bias
angle,
with
respect
to
the
braid
axis.
The biaxial
stress-strain
behavior
of
plain-woven
fabrics
is
initially
dominated
by
crimp
interchange
rather
than
yam
elasticity.
Because the
factors
of
safety
used
in
air-inflated
fabric
structures
are
typically within
the
range
of
4-6,
the
operating
stresses
are
designed
to
be
low
with
respect
to
fabric
strength.
Therefore,
the
structural performance
of
fabrics
must
address
the
influences
of
crimp
interchange.
The
relative
yarn
motions
(slip
and
rotation)
affect
the
stiffness
properties.
The
crimp
ratio,
which
is
denoted
as
C,
is
defined
as
the
waviness
of
the
yarns
and
is
obtained
by
measuring
the
length
of
a
yarn
in
its
fabric
state,
Liobric
,
and the
length
of
that
same
yam
after
it
has
been
extracted
from the
fabric,
L
.
and
straightened
out
according
to
equation
(2).
C
L-
-Llabric
(2)
L
ab,.ic
The
following
equation
described
by
Grosberg relates
the
crimp
height
to
C.
h
24
I
C2.
(3)
Consider
a
plain-woven
fabric
subject
to
a
tensile
load
along
the
direction
of
one
yam
family.
The loaded yarns
will
attempt
to
straighten,
decrease
their
crimp
heights
and
elongate
their effective
lengths.
However,
the
yams
of
the
crossing
family
are
forced
to
increase
their
crimp
heights
resulting
in
contractions
of
their
effective
lengths.
The
effect associated
with
changes
in
crimp
is
referred
to
as
crimp
interchange
and
is
similar
to
the
Poisson's
effect
exhibited
in
metals.
Crimp interchange
is
a
coupling
effect
exhibited
between
yam
families.
When
a
fabric
is
loaded
in
tension,
the
crimp
contents
become
mutually
altered
as
the
yams
attempt
to
straighten.
In
tests
of
plain-woven
fabrics
along
the
direction
of
a
given
yam
family,
crimp
interchange
can be
easily
observed
by
reducing
the
width
of
the
specimen.
Crimp
interchange
is
a
source
of
nonlinear load-extension
behavior
for
fabrics.
Backerr3I
describes
a
limiting
phenomenon
to
crimp
interchange.
As
the
biaxial
tensile
loads
continually
increase
for
a
given
loading
ratio,
a
configuration
results
in
which
yarn
kinematics
(i.e.;
slip
at
the
cross
over
points) cease
and
the
spacing
between
yarns converge
to
5
minimum values.
This
configuration
is
referred
to
as
the
extensional
jamming
point,
which
can
prevent
a
family
of
yams
from
straightening
and
thereby
not
achieve
its
full
strength.
Crimp
interchange depends
upon
the
ratio
of
initial
crimp
between
yam
axes
and the
ratio
of
stress
between
yam
axes
rather
than
the
levels
of
stress.
Crimp interchange
introduces
significant
nonlinear
effects
in
the
mechanical response
of
the
fabric.
Now
consider
the
plain-woven
fabric
subjected
to
shearing
by
applying
a
uniaxial
load
at
±45
degrees
to
either
yarn
family. The
yam
families
will
rotate
at
the
crossover
points
with
respect
to
each
other
and
become
increasingly
skewed
as
the
angle
between
the
yams
changes.
The change
in
angle
is
referred
to
as
the
shear angle.
At
larger
shear
angles,
the
available
space
between
yam
families
decreases
and
rotational
jamming
(locking)
of
the
yam
families occur.
This
phenomenon
is
known
as
shear-jamming
and
the
angle
at
which
the
yam
families
become
jammed
is
referred
to
as
the
shear-jamming
angle.
The
shear-jamming
angle decreases
with
increasing
yarn
density
ratio
and
can
be
estimated
from
Pierce's
geometric
fabric
model
or
obtained
experimentally
with various
trellising
or
biaxial test
fixtures.
Continued
loading
beyond
the
onset
of
shear-jamming
will
produce
shear
wrinkles
leading
to
localized
out-of-plane
deformations.
It is
important
to
determine
the
extension-
and
shear-jamming
points
for both
fabric
manufacturing
and
structural
stiffness
concerns.
In
general,
jamming
is
related
to
the
maximum
number
of
weft yams
that
can
be
woven
into
a
fabric
for
a
given
warp
yarn
size
and
spacing.
OPERATION
Today's
air-inflated
fabric
structures
can
be
operated
at
various pressure
levels
depending
upon
fabric
architecture,
service
loads
and
ambient
temperatures.
Woven
structures
are
generally designed
to
operate
at
pressures
up
to
20 psi
while
triaxial
braided
and
axial strap-
reinforced braided
structures
are
generally
capable
of
higher
inflation
pressures.
Blowers,
which
may
be
used for
inflation
pressures
up
to
3
psi,
provide
a
high
volumetric
air
flow
rate.
However,
air
compressor
systems
will
be
required
for
higher
inflation
pressures.
Air
compressors
are
available
in
single
and
dual
stage
configurations.
The
single
stage
air
compressor
delivers
a
high-pressure
capability
but
at
a
low
volumetric
air
flow
rate.
The
volumetric
flow
rate
is
measured
in
standard
cubic
feet
per
minute
(scfm).
Dual
stage
air
compressors
are
designed
to
provide
an
initial
high
volumetric
air flow
rate
at
low
pressures
in
the
first
stage
and
can
be
followed
by
a
low
volumetric
air
flow
rate
at
high
pressure.
When
a
dual
stage
air
compressor
is
used,
the
first stage
is
most
beneficial
to
erecting
the
fabric
structure.
At
this time,
no
appreciable resistance
to
the
design
loads
is
available. However,
the
second
stage
mode
of
the
compressor
is
used
to
fully
inflate
the
fabric
structure
to
its
proper
operating
pressure
so
that
the
design
loads
can
be
supported.
Because
the
dual stage
compressor
can
achieve
the
desired
inflation
pressure
much
more
rapidly
than
the
single
stage
compressor,
it
is
more
appropriate
for
those
applications
in
which
the
need
for
rapid
deployment
justifies
the
additional cost.
6
INFLATION
AND
PRESSURE
RELIEF
VALVES
Valves
designed
for
use
in
air-inflated
fabric
structures
must
consider
several
criteria
including
locations, orifice
size,
pressure
relief
controls
and
potential strength
degradation
in
the
surrounding
fabric
regions.
Ideally,
valves should
be
positioned
in
structural elements
that
interface
with
the
fabric
and
bladder
such
as
metal
clamps,
discs
or end
plates
at
the end
termination
zones
of
the
structure.
This
avoids
the
need for
cutting
additional
penetration
holes
through
the
fabric
or
repositioning
the
yams
to
accommodate
holes
for
valve
placement,
thus
preventing
stress
concentrations.
When
fibers
must
be
removed
to
accommodate
insertion
of
valves,
fabric
reinforcement
layers
should
be
bonded
and
stitched
to
the
main
fabric
in
the
surrounding
region.
The
valves
should
be
readily
accessible
for
user
access but should
not
jeopardize
the
integrity
of
the
fabric
when
the
structure
is
subjected
to
handling
events
(such
as
drops,
impacts,
etc.).
Valve locations should
be
further optimized
to
mitigate
the
effects
of
interference
with
other objects
when
the
structure
is
deployed
or
stowed.
A
variety
of
pneumatic
valve designs
including threaded valves,
quick
connect-
disconnect
valves,
check valves
(one-way,
two-way),
pressure
relief
valves,
etc.
are
readily
available.
The
use
of
pressure
relief
valves
is
recommended
to
avoid
accidental
overpressures
and
pressure
increases
due
to
changes
in
ambient
temperatures.
Valves
can
also
be
configured
to
manifold
inflation
lines
in
air inflated
structures
containing
multiple
pressure
chambers.
This
also
provides
capabilities
for
self-erecting
and
controlled
deployments
by
sequencing
the
inflation
timing
of
multiple chambers.
Sizing
of
the
valve
orifices
should
be
matched
to
provide
the
optimal
inflation
and
deflation
times
for
the
air
volumes
and
operating pressures required
by
the
structure.
CONTINUOUS MANUFACTURING
AND
SEAMLESS
FABRICS
Prior
to
continuous circular weaving
and
braiding
processes
in
use
today,
air-inflated
fabric
structures
were
constructed
using
adhesively bonded,
piece-cut manufacturing methods.
These
methods
were
limited
to
relatively
low
pressures
because
of
fabric
failures
and
air
leakage
through
the
seams.
Continuous
weaving
and
braiding
processes
can
eliminate
or
minimize
the
number
of
seams
resulting
in
improved
reliability,
increased
pressure capacities
and
greater
structural
load
carrying
capability.
However, when
seams
can not
be
avoided,
the
seam
construction
should
be
designed
in
such
a
way
that
failure
of
the
surrounding
fabric
occurs
rather
than
the
seam.
For
the
safe
and
reliable
use
of
air-inflated
fabric
structures,
sufficient
factors
of
safety
against
burst
and
seam
failures must
be
provided.
To
guard
against
burst,
for
example,
a
minimum
factor
of
safety
of
4-6
is
used
on
yam
strength
for
each
yam
family.
Like
traditional
composite
materials,
fabrics
can
be
tailored
to
meet
specific structural
performance requirements.
Fiber
placement
can be
optimized
for
air-inflated
fabric
structures
by
varying
the
denier
of
the
yams
(defined
as
the
mass
in
grams
of
a
9,000 meter
length
yam)
and
yarn
counts
along
the
each
direction.
For
instance,
consider
a
pressurized
fabric
cylinder.
The
ratio
of
hoop stress per
unit
length
to
axial
stress
per
unit
circumference
is
2:1.
One can
ensure
equal
factors
of
safety against
yam
burst
in
the
weft
(hoop)
and
warp
(longitudinal)
yams
by
weaving
twice
as
many
weft
yams
per
unit
length
of
air
beam
than
the
number
of
warp
yams
per
7
unit
circumference.
Alternatively,
the
same
can
be
accomplished
by
doubling
the
denier
of
the
weft
yams
in
comparison
to
the
denier
of
the warp
yams.
IMPROVED
DAMAGE
TOLERANCE
METHODS
Assorted
methods
are used
to
enhance
the
reliability
of
air-inflated
fabric
structures
against various
damage
mechanisms.
Resistance
to
punctures, impacts,
tears
and
abrasion
can
be
improved
by
using high-density
weaves,
rip-stop
fabrics
and
coatings.
High-density
weaves
are
less
susceptible
to
penetrations
and
provide greater coverage
protection
for
bladders.
Rip-stop
fabrics
have
periodic
inclusions
of
high
tenacity
yams
woven
in
a
cellular
arrangement
used
to
prevent
fractures
of
the
basic
yams
from
propagating
across
cells
as
shown
in
figure
(4). (The
breaking
strength
of
a
yam
is
referred
to
as
tenacity
which
is
defined
in
units
of
grams-force
per
denier.
Denier
is
a
mass
per
unit
length
measure
expressed
as
the
mass
in
grams
of
a
9,000
meter
long
yam.)
The
high
tenacity
yams
contain fractures
of
the
basic
yarns
and
prevent
fractures
from
propagating
across
cells.
High
Tenacity
Yarns
Basic
Yarns
Z
,=,
Fractured
Yarns
Figure
4.
Rip-stop
fabric
architecture.
Additionally,
coatings
protect
the
fabric
against
environmental
exposure
to
ultraviolet
rays,
moisture,
fire,
chemicals,
etc.
Coating
such
as
urethane,
PVC
(polyvinyl chloride),
neoprene,
EPDM
(ethylene
propylene
diene
monomer)
are
commonly
used.
Additives
such
as
Hypalon
further
enhance
a
coating's
resistance
to
ultraviolet
light
and
abrasion. Coatings
can
be
applied
in
two
stages. First,
coating
the
yams
prior
to
forming
the
fabric
by
a
liquid bath
immersion provides
the
best
treatment
to
the
fibers.
Second,
coatings
can be
applied
by
spraying,
painting
or
laminating
directly
to
the
fabric
after forming.
This stage
coats
the
yams
and
bridges
the
gaps
formed
between adjacent
yams.
The
maximum protection
is
achieved
when
both
stages
are
utilized.
While
protective
coatings
have
been
shown
to
increase
the
stiffness
of
the
fabric
by
restricting
relative
yam motions,
they remain
flexible
enough
to
not adversely
impact
the
stowing
and
packaging
operations
of
the
structure.
Additional
information
on
coating
materials
and
processes
is
provided
by
Fung~
6
.
8
RIGIDIFICATION
Air-inflated
fabric
structures
can
be
rigidified
by
coating
the
fabric
with
resins
such
as
thermoplastics,
thermosets,
shape
memory polymers,
etc.
Prior
to
inflation, these
particular
coatings
are
applied
to
the
fabric and
remain
initially
uncured,
thus
acting
as
flexible
coatings.
After
the
structure
is
inflated
and
properly
erected,
a
phase
change
is
triggered
in
the
coating
by
a
controlled
chemical
reaction
(curing
process) activated
by
exposure
to
elevated
temperature,
ultraviolet
light,
pressure, diffusion,
etc.
Once
the
phase
change
is
fully developed,
the
coatings
bind
the
yarns
together,
stiffen
the fabric
in
tension,
compression
and
shear,
and
behave similar
to
a
matrix
material found
in
traditional
fiber-reinforced
composites.
The
air-inflated
fabric
structure
is
now
rigidified
and
no
longer
requires
the
inflation
pressure
to
maintain
its
shape
and
stiffness.
Depending
upon
the
coating
used,
the
transition
process
may
be
permanent
or
reversible
(except
where
thermoset plastics
are
used).
Reversible
rigidification
is
especially
suited
for
those
applications
requiring multiple
long-term
deployments.
The
performance
of
the
rigidified
fabric
structure
can
be
assessed
using
laminated
shell
theory (LST)
as
described
by
Jones
71
.
Unlike
the
inflated
version
of
the
structure,
the
elastic
and
shear
moduli
of
the
rigidified
structure
are
based
on
the
constituent
properties
of
the
fibers
and
cured
coating.
As
such,
the
elastic
and
shear
moduli
can
be
readily
estimated
by
using
LST.
However,
the
rigidified structure
may
have
different
failure
modes
than
its
pressurized
counterpart.
The
designer
must
guard
against
new
failure
modes
including
localized
shell
buckling
and
compression rather
than
wrinkling.
Rigidification
is
of
particular
interest
for space
structures because
of
restrictions
on
payload
weights
and
stowage
volumes.
Cadogan
et
al0
8
'
9
1
have
pursued
the use
of
shape
memory
composites
for
rigidizing deployable
space
frames.
AIR
BEAMS
Fabric
air beams
are
examples
of
air-inflated
structural elements
that
are
capable
of
supporting
a
variety
of
loads
similar
to
conventional
beams.
To
date,
seamless
air
beams
have
been
constructed
using
continuous
manufacturing
methods
that
have
produced diameters
ranging
up
to
42
inches.
They
are
constructed
of
an
outer
fabric
skin that
contains
an
internally
sealed
film
or
bladder
made
of
an
impermeable elastomer-like material. The
bladder contains
the
air,
prevents
leakage
and
transfers
the
pressure
to
the
fabric.
The
air
beam
has
a
cylindrical
cross
section and
its
length
can
be
configured
to
a
straight
or
curved
shape such
as
an
arch.
The
ends
are
closed using
a
variety
of
end
termination
methods consisting
of
bonding,
stitching,
mechanical
clamps,
etc.
depending
upon
the
inflation
pressure
and
loading
requirements.
Clamping methods
are
available
to
permit
disassembly,
repair and
replacement
of
the
bladder
and
fabric
layers.
Once
an
air
beam
is
sufficiently
pressurized,
the
fabric
becomes
pre-tensioned
and
provides
the
air
beam
with
a
plurality
of
stiffnesses including
axial,
bending,
shear,
and
torsion.
Upon
inflation,
the
ratio
of
hoop
(cylindrical)
stress per unit
length
of
beam
to
the
longitudinal
stress per unit
circumference
is
2:1
as
shown
in
figure
(5).
9
weft
Weft
yarn
tension
per
unit
length
of
cylinder
=
pr
Warp
yarn
tension
per
unit
circumference
=
pr/2
#
weft
yarns per
unit
length
of
cylinder
Yarn
density
ratio
(YDR)
=
#
warp
yarns
per
unit
circumference
Figure
5.
Yarn
tensions
in
a
plain-woven
pressurized fabric cylinder
and
definition
of
yarn
density
ratio.
In
the
context
of
plain-woven
air
beams,
the
warp
yams
are
aligned
parallel
to
the
longitudinal
axis
of
the
air beam
and
resist
axial
and
bending
loads.
The weft
yams
spiral
through
the
weave
and
are
located
at
nearly
900
to
the
warp
axis,
thus
lying
along
the
hoop
direction
of
the
air
beam.
Weft
yarns
provide
stability against
collapse
by
maintaining
the
circular
cross
section
of
the
air
beam.
For
braided
air
beams,
the
braid axis
is
aligned
with
the
longitudinal
axis
of
the
air
beam.
If
the
ends
of
a
braided
air
beam
are
unconstrained
from
moving
in
the
longitudinal direction,
the
fibers
will
exhibit
a
scissoring
effect
causing
the
length
of
the
beam
to
expand
or shorten with
pressure
depending
upon
the
selection
of
0.
Eventually,
the
braided
yams
will
achieve
a
maximum
rotation
and
the
fabric
will become
fully
jarmned.
This
phenomenon
can
be
easily
demonstrated
with
the
well-known
Chinese
finger
trap
toy.
Unlike
plain-woven
fabric
structures,
the bias
angle
of
braided
fabric
structures
can
be
controlled
to
allow
expansion
or
contraction
of
the
structure when
inflated.
For
an
unconstrained
braided
air
beam
to
resist
axial
tension,
longitudinal reinforcements
such
as
distributed
axial
yarns
(i.e.;
triaxial
braid)
or
axial
straps,
webbing,
belts,
etc.
must
bonded
on
at
multiple locations
around
the
circumference.
Various design
methods for
constructing
axial-reinforced
braided fabric
air
beams
were
developed
by
Brown['°1
and
Brown
and
Sharpless['
1
,
The
bending stiffness,
El,
for
metal
beams
is
written
as
the
product
of
elastic
modulus,
E,
and area
moment
of
inertia,
I,
and has
units
of
(force
x
distance
2
).
However,
in
air
inflated
fabric
structures,
the
fabric
elastic
modulus,
Ef,
is
commonly denoted
in
units
of
(force
per
unit
length)
and
therefore,
EJ
I
is
denoted
in
units
of
(force
x
distance
3
).
Likewise,
fabric
strengths
are
denoted
in
units
of
(force
per
unit
length).
10
Now,
consider
the
air
beam
subjected
to
transverse
loads.
Upon
loading,
the
pre-tension
stresses
and the
bending
stresses
will
algebraically
add.
That
is,
the
compressive
bending
stresses
will
subtract
(relax)
from
the
pre-tension
on
the
compressive
surface
of
the
air
beam
while
the
tensile
bending
stresses
will add
to
the
pre-tension
along
the
tensile
surface
as
shown
in
figure
(6).
The
instant
at
which
any
point
along
the
length
of
the
air
beam
develops
a
net
zero
longitudinal
tensile
stress,
the
structure
is
said
to
have
reached
the
onset
of
wrinkling.
The
corresponding
bending
moment
is
referred
to
as
the
wrinkling moment,
Mu,.
Prior
to
wrinkling,
the
moment-curvature
relationship
is
expectedly linear
in
the
absence
of
both
material
nonlinearities
and
notable
changes
in
air
pressure
and
volume.
A
stress
balance
analysis
based
on
strength
of
materials
equations
can
generally
be
used
to
compute
M,.
Such
a
stress
balance
analysis
was
used
to
show
the
variation
of
warp
yam
tensions
for
a 2"
diameter plain-woven
air
beam
subjected
to
4-point
bending
loads
as
depicted
in
figure
(7).
Once
wrinkling
has
occurred,
the
air
beam
moment-curvature
relation
behaves
nonlinearly
because
with
further
loading,
the
cross
section
loses
bending stiffness,
the
neutral
axis
displaces
away
from
the
centroidal
axis
of
the
cross
section
and
eventual
collapse
occurs.
The spread
of
wrinkling
around
the
circumference
is
similar
to
the
flow
of
plasticity
in
metal beams subjected
to
bending.
Loading
beyond
the
wrinkling
onset
will
result
in
an
increase
of
pressure
due
to
loss
of
volume
and
the
work
done
on
the
air
will
affect
the
post-wrinkled
bending
stiffness.
Thus,
the
post-wrinkled
response
becomes
nonlinear.
The
superposition
of
the
pressure
and
bending-induced
forces
is
shown for
each
architecture
in
figure
(8).
•---.._•....Nth
yarn
.Ib
yarn
. /Fib M
N
M
Fib'Yi
where:
A-
=Fb
YN
Yi
I
and:
YN=2
S~2M
therefore:
Fib
=d
N
S(sin
(a
(0)))2
Figure
6.
Idealized
distribution
of
warp
yarn
forces
due
to
bending
of
a
plain-woven
fabric
air
beam.
11
25
-a
otoJj4 P
Si
O'
2
A
n
AJ4
.2
C
Pressure
Induced
A'4
a.
0
Warn
V'n
Nombenno
rp
yarn
force
= 0)
0
0
10
20
30
40
50
60
70
Warp
yarn
#
(ito
63)
2"
dia,
63
Warp
yarns,
96",
=
44.375",
A
14.625",
Pressure
20
psi,
P.
4.3
Ibs,
Moment
=
31.44
in-lbs
Figure
7.
Combined
pressure
and
bending
induced
forces
in
warp yarns
at
various
distances
along
a
plain-woven
air
beam
based
on
a
simple stress
balance
analysis.
if
Fc.II+F
F
1
'
F
1
.
1
-0
tr-'t01
.
- 0
/
F1F5}
h
II
C'',
0*
-H
+1
* 4F
seam
Pstr
2
P7tr
2
Pxr
2
la'
r.f.rs to
iI
y.orrs j - refer. to braided
yaro.
Fort.
P0 j - refers to braided yams
4
Force trots
booing
- - Fore, hors
prn.ir.
- - Force trots
rr..nro.
Force horn treniog Forte trots bendog
Plain
Weave
Triaxial
Braid
Braided
vVith
Top
&
Bottom Axial
Straps
(U.S.
Patent
No.
5.735.083)
Figure
8.
Superposition
of
pressure
and
bending
induced
yarn
forces
in
plain-woven
air
beam,
a
triaxial braided
air
beam
and
a
dual
axial
strap-reinforced
braided
air
beam.
12
Consider
now
a
plain-woven
air
beam
subjected
to
inflation
pressure
P.
The
force
in
the
warp
yams,
FwA,,
is
shown
in
equation
(4)
as
simply
the
pressure
times
the
cross sectional
area
divided
by
the
number
of
warp
yams,
Nw
4
Rps.
The
force
in
the
weft
yams,
FWEFT,
was
idealized
by
the
assumption
of
orthogonal yam
placement
and
is
shown
in
equation
(5).
The
wrinkling
moment,
Mw,
for
a
woven
air
beam
was
derived
from
a
simple
force
balance
of
the
pressure
and
bending-induced
yam
forces
and
is
shown
in
equation
(6).
=,r
(4)
WARP
PZ
NWARPS
F
PT
rL
(5)
NWEFTS
Pit
r
3
MW
=
(6)
2
Note
that
Mw
is
independent
of
the
fabric
material
properties.
The
global
flexure
response
of
plain-woven
air
beams
was
experimentally
investigated
for
use
in
inflatable
military
shelters'
3
"" 1.
The air beams
were
constructed
of
non-twisted yams
and
were
inflated
to
various operating
pressures.
Operating
pressures
were
considered
to
be
safe
when
the
resulting
yam
tensions
did
not exceed
30%
of
their
breaking strength.
The
experimental
set
up
is
shown
in
figure
(9). A
plot
of
the
experimental
load
versus
deflection
is
shown
in
figure
(10)
which
exhibited
a
significant
influence
on
inflation pressure.
Figure
9.
4-Point
flexure
test
on
a
6
inch
diameter
plain-woven
air
beam
constructed
of
3,000-denier,
2:1
YDR
Vectran fabric.
13
300
40
psi
250
0
Theoretical
wrinkling
onset
*l200
3
s
'a
150
20
psi
(13
"*
100
010
psi
50
0.0
10.
2.0
3.0
4.0
5.0
60
7.0
8.0
Load
point
displacement
(in)
Figure
10.
Experimental
load
vs.
deflection
plot for
an
uncoated
6-inch
diameter
air
beam
constructed
of
3,000-denier non-twisted
Vectran
yarns
in
a
plain-woven
2:1
Yarn
Density
Ratio
fabric
using
a
37-inch
span
between
load
points
and
an
85-inch
span
between
support
points.
A
second
series
of
4-point bending
tests[4
1
was
conducted
on
a
2-inch
diameter,
uncoated
plain-woven
Vectran air beam
to
measure
the
midspan
deflections
as
a
function
of
inflation
pressures ranging
from
10-100 psi.
The
load
versus
mid-span deflection
curves
are
shown
in
figure
(11).
100
Tolal
Span
= 1127 mram (44
375)
Load
Po,ntto-Load
Point
= 314 2
mm
(15
125)
Support
Poin-to-Load
Pont
= 371
5
mm
(14 62-6)
Inflathon
Poessues
= 0 138 - 0
69
MPa
(W-TOO pm,)
E, Onoall Length
-243B4
mm (-%I
0
--
o
0
345
MPa
0
414mMPa
50
<
4
0
0 6
2
1.
M I
N
...
.
. .
...
".
.
-- ---*•
0
689 MPa
.- ....
I-
30
20~
10
Indicates
Theoretical
Onset
of
Wnnkhing
0
0
10
20 30
40
5o
W
Mid-span
deflection,
mm
Figure
11.
Plot of
total
load
vs.
mid-span
deflection
for
a
2-inch
diameter
plain-woven
air
beam
constructed
of 1,500-denier,
2:1
YDR
Vectran
fabric.
14
In
a
third
series
of
tests[
4
],
the
flexure
behavior
of
uncoated
plain-woven
Vectran
and
PEN air
beams
were
compared
for
2
pressures
using
a
constant
load
point
displacement
rate.
Both
air
beams
had
identical
crimp
in
the
warp
and
weft
yams,
yarn
density
ratios
(YDR),
denier,
diameter
and
length. (The
yarn
density
ratio
is
defined
as
the
number
of
weft
(hoop)
yams
per
unit
beam
length
divided
by the
number
of
warp
(longitudinal)
yams
per
unit
circumference.)
The
graph
of
figure
(12)
shows
the
total
applied
load
versus
the
mid-span
deflection responses.
25
Total
Span
=
2206.625
mm
(86.875')
Load-Point
to
Load-Point
=
835.025
mm
(32
875")
Support-Point
to
Load-Point
=
685
8
mm
(27')
Inflation
Pressure
= 0
138.0 207
MPa
(20,30
psi)
20
Overall
Length
=
2438
4
mm
(96")
Z
Displacement
Rate
=
12
7 mm/mirn
'a
Tensile
Modulus
of
'As-Woven'
Yams:
_ .
O
15
E',•
,RC
ymNs
=
62.260
MPa
"EpEmv•yANs
13.762
MPa
< 10
--
Vectran
0.
138
MPa
-0.-
Vectra
0
207
MPa
5
-A-
PEN
0
207
V1Pa
0
25 50
75
100
125
150
175
200
225
Mid-span deflection,
mm
Figure
12.
Comparison
of
2-inch
diameter
Vectran
and
PEN
plain-woven
air
beams
subjected
to
4-point bending
tests
at various
load
point displacement
rates.
Although
the
ratio
of
the
as-woven elastic
moduli
of
the
Vectran
and
PEN
yarns obtained
through
"as-woven"
yarn
tensile
tests was
4.7-to-1.0,
there were
minimal
differences
in
deflections
of
these
air
beams
at
the
inflation
pressures
used.
It
was
observed
that
the
global
bending
behavior
of
the
air
beams
was:
(1)
invariant
with
Ea,,r,
for
the
range
of
inflation
pressures considered,
(2)
dominated
by
the
kinematics
of
crimp
interchange
and
(3)
subjected
to
considerable
transverse
shearing
deformations.
The
effects
of
crimp
interchange
and
shearing
deformations
precluded
the
use
of
traditional
strength-of-materials
design
methods
for
plain-
woven
fabric
air
beams.
DROP-STITCHED
FABRICS
Drop-stitch
technology,
originally
pursued
by the
aerospace
industry,
extends
the
shapes
that
air-inflated
fabric
structures
can
achieve
to
include
flat
and
curved panels with
moderate
to
large
aspect ratios
and
variable
thickness.
Drop-stitched
fabric
construction consists
of
external
skins
laminated
to
a
pair
of
intermediate
woven
fabric layers
separated
by
a
length
of
perpendicularly
aligned fibers
(Fig.
13).
During
the
weaving
process
of
the
intermediate
layers,
fibers
are
"dropped"
between
these layers.
Upon
inflation,
the
intermediate
layers
separate
forming
a
panel
of
thickness
controlled
by
the
drop-stitched
fiber lengths.
Flatness
of
the
inflated
panel
can
be
achieved
with
a
sufficient distribution
of
drop-stitching.
The
external skins
15
can
be
membranes
or
coated fabrics which
serve
as
impermeable
barriers
to
prevent
air
leakage,
thus
eliminating
the
need for
a
separate bladder.
Air-inflated
structures incorporating
drop-
stitched
fabrics include
floors for
inflatable
boats,
energy
absorbing
walls,
temporary
barriers,
lightweight
foundation
forms, and
a
variety
of
recreational
products.
Laminated
Woven
Skin
Fabric
Drop
Woven
Stitching
Fabric
S'Laminated
Skin
Figure
13.
Section
view
of
an
example
drop-stitch
construction
for
air-inflated
fabrics.
EFFECTS
OF
AIR
COMPRESSIBILITY
ON
STRUCTURAL
STIFFNESS
The
load-deflection
response
of
air-inflated
fabric
structures
may
depend
upon
additional
stiffening
influences other
than
the
initial
inflation
pressure. These
may
include
contributions
from
nonlinearities
in
the
fabric
stress-strain behavior,
the
work
done
on
the
air
by
external
loads,
and
other
phenomena.
If
appreciable
changes
in
pressure
or
volume
occur
during loading,
as
in
the
case
of
energy
absorbers, work
is
performed
on
the
air
through
compressibility,
which
stiffens
the
structure.
Air
compressibility
can
be
modeled
from
thermodynamic principles
in
accordance
with the
Ideal Gas
Law
shown
in
equation
(7).
P
V
=
mRT,
(7)
where:
P
the
absolute pressure
V=
volume
m
=
mass
R
=
gas
constant
for
air
T=
temperature
('K)
For
a
quasi-static,
isothermal process,
the
work
done
on the
air
by
compression
is:
V2
_v'
mRT
V
kjWj
f
PdV
=J-_----V
=
mRT
In-.
(8)
16
The
total
energy
of
the
structure,
E,,,,,,
is
the
work
done
by
all
external forces,
Ve,,,
which
is
related
to
the
total
strain
energy
of
the
fabric,
U_,
and
WV,r
as
shown
in the
energy
balance
of
equation
(8).
E
W
=
U
+
Wi"
(9)
total
ext
-
f
a9r
For
homogeneous
films
and
membranes,
the
elastic
and
shear moduli
are
readily
determined
by
standardized
tests.
However,
for the
case
of
plain-woven
and
braided
fabrics,
the
elastic
and
shear
moduli
can
vary
not
only with
pressure
but also with fabric
architecture
(plain
weave, harness
weave,
braid,
yam
densities,
etc.),
external
loads
(crimp interchange),
and
coatings
if
present.
With
increasing pressure,
slack within
the
yams
is
removed,
the
yarns
begin
to
straighten
(decrimp)
and
contact between
intersecting
yarns
at
the
crossover
points causes
compaction.
EXPERIMENTS
ON
PLAIN-WOVEN
FABRICS
Hearle
3
1
identified
3
distinct
regions
of
stiffness
for
tensile
loading
of
a
plain-woven
fabric
as
shown
in
figure
(14).
For
safe
practice,
air-inflated
fabric
structures
are
designed
to
operate
with
yam
forces
ranging between
regions
I
and
II.
Breaking
load
tftf
III
-
Yarn
e'xlension
reglion
CU
"-• ""P
ange
of
saf
S-
Decrimpingload
fr
l
- Inter-fiber
frictnon
regaon
Axial
displacement
4
,
4
Figure
14.
Stages
of
axial stiffness
for
woven
fabric
subjected
to
tension.
This
ensures
that
the
yarns
are
loaded
well
below their
breaking
strengths
to
provide sufficient
factors
of
safety
against
burst. Once
the
fabric
is
biaxially
stressed
and
subjected
to
an
in-plane
shear stress,
the
yarns
will
shear
(rotate)
with
respect
to
their
original
orientations.
The shear
stiffness
(i.e.;
resistance
to
yam
rotations)
results
from
inter-yam
friction
and
compaction
at
the
17
crossover
points. Hence,
the
shear
modulus
is
actually
a
system property
rather
than
a
constitutive
(i.e.;
material)
property.
As
the
shear rotations
increase,
an
upper
bound
is
reached
when yarns
of
both
directions
become
kinematically
locked
resulting
in
what
is
referred
to
as
shear-jamming.
For
pure
shear
loading,
the
trellising
fixture
shown
in
figure
(15)
can
be
used.
Figure
(16)
shows
the
shear
force versus
shear
angle
plot
of
an
uncoated
Vectran
fabric.
Further
shear loading induces localized
shear
wrinkling
instabilities
that
lead
to
increased
out-of
plane
deformations.
Ji
Figure
15.
Picture
frame
test
fixture
for
pure
shear
loading of
fabrics.
500
Onset
of
Yarn
Shear
Jamming
Wrinkling
400
stage
Z
Shearing
?6
300
stage
p
oInitialasAL
stage
-~200
..LLLLTl
0
0
10
20
30 40
50
Shear
angle,
deg.
Figure
16.
Stages
of
shear
stiffness
for
pure shear
loading of
a
2:1
Yarn
Density
Ratio
woven
fabric.
Farboodmanesh
et
al['
51
conducted shear
tests
on
rubber-coated,
plain-woven
fabrics
and
established
that
the
initial
shear
response
was
dominated
by
the
coating
and
with increased
shearing,
the
behavior
of
a
coated
fabric
transitions
to
that
of
an
uncoated
fabric.
18
Unlike
fabric
structures,
the
homogeneity
of
membranes
excludes
the
kinematic motions
associated
with
crimp
interchange
and strain
energy
is
developed
throughout
all
stages
of
biaxial
tensile loading.
However,
membrane structures composed
of
homogenous
materials
are
also
susceptible
to
both
bending
and
shear
wrinkling
instabilities.
Many
testing
methods
have been
developed
for
evaluating
the
mechanical
properties
of
yams
and
fabrics
including
ASTM
standards,
Kawabata
Evaluation System,
military
specifications (MIL-Spec),
British standards,
etc.
It
is
recommended
that
the
structural engineer
become
familiar
with
the
applicability
of
these
standards
to
the
design
and
testing
of
fabric
materials
as
applicable
to
air-inflated
fabric
structures.
Saville[161
provides
a
description
of
many
standard
tests
used
in
evaluating
fiber,
yarns
and
fabrics.
A
recently
developed
multi-axial tension
and
shear
test fixture
[131
was
developed
to
permit
simultaneous measuring
of
Ewarp,
Ewe/i
and
Gjfor
structural
fabrics.
The
fixture,
as
shown
in
figure
(17),
was
designed
for use
with
conventional
tension/torsion
machines
to
characterize
the
elastic
and
shear
moduli
of
fabrics
as
functions
of
biaxial
loads.
It
utilizes
a
cruciform-
shaped
specimen
and
was
designed
to
evaluate
both
strength
and
stiffness
properties
of
various
fabric
architectures
such
as
weaves, braids
and
knits
subjected
to
biaxial
loads, shear loads
or
combined biaxial
and
shear
loads.
For
fabrics
constructed
of
2
principal fiber
directions,
the
fixture utilizes
2
rhombus-shaped
frames
connected
with
rotary
joints
as
shown
in
figure
(17)
with
the
biaxial
tension
and
shear modes
illustrated
for
a
plain-woven
fabric.
For
triaxial
braided
fabrics,
the
fixture uses
a
third
rhombus-shaped
frame
with
additional
rotary
joints.
Biaxial
tension
mode
In-plane
shear
mode
Figure
17.
Combined
biaxial
tension
and in-plane
shear
test
fixture.
(U.S.
Patent
No.
6,860,156)
19
A
STRAIN
ENERGY-BASED
DEFLECTION
SOLUTION
FOR
BENDING
OF
AIR
BEAMS
WITH
SHEAR
DEFORMATIONS
The air
beam
bending experiments
exhibited
that,
unlike metallic
structures,
deflections
are
functions
of
internal
pressure
and
transverse shear
deformations
may
be
significant.
If
shearing
deformations
become
appreciable,
they
will
increase
the total
deflection
of
air-inflated
fabric
structures.
Therefore,
a
shear-deformable
theory
such
as
that
developed
by
Timoshenko[1
7
]
must
be
employed
in
place
of
Euler-Bernoulli[
8
s
theory
to
compute
air
beam
deflections.
This
section
simulates
the
4-point
bending
experiments
and
analytically
estimates
the
fabric
shear
modulus
when
changes
in
pressure
and
volume
are
negligible.
Fabric
shearing
deformations,
which
were
evident
for
each
of
the
4-point
bending
tests
shown
in
figures
(10-12),
were
observed
to
decrease
with
increasing
pressure.
It
will
be
shown
that
the
shear
modulus
of
the
fabric
is
not
a
function
of
the
warp
yam's
elastic
modulus.
This
was
accomplished
by
deriving
a
shear-deformable
air
beam
deflection
equation
for
the
4-point
loading
arrangement
described
by
figure
(18).
From
that
equation,
and
using
experimentally
obtained
air
beam
deflections,
the
fabric
shear
modulus,
Gf
can
be
computed.
This
equation captured
both
the
bending
and
transverse
shear
deformations
that
comprise
the
total
mid-span
deflection.
I I
ProAwl2
,•
•Pkw
12
Figure
18.
4-Point bending
arrangement
for
shear
deformable
beam
deflection
equation.
The nearly-orthogonal
yarn
directions
were
assumed
to
be
truly
orthogonal,
with
the
warp
axis
being
parallel
to
the
longitudinal
axis
of
the
air
beam. This
uncoupled
the
bending
stresses
from
the
weft
(hoop)
yarns.
The
warp
yarns
were
assumed
to
exclusively
support
the
bending
stresses.
In
a
truly orthogonal
fabric
air
beam,
the
weft
yams
can
be
considered
as
parallel
rings
rather
than
continuously
spiraled
yarns
with
small
lead
angles.
Second,
an
equivalent
cylinder
having
a
cross
sectional
area
equal
to
the total
cross sectional
areas
of
all
the
warp yarns,
Aiola,,
was
used
to
represent
the
actual air
beam.
The
inner
radius
of
the
equivalent
cylinder,
ri,
was
taken
as
the
nominal
air beam
radius
and
the
outer
radius
of
the
equivalent
cylinder,
ro,
was
computed
by:
=2
0
Z
(10)
20
Castigliano's
Second
Theorem[
1 81
was used
and
considered
the
strain
energies
from
both
bending
and
shearing
for
the
equivalent
cylinder.
The
total
strain
energy,
Uo,,,,
was
expressed
as
the
sum
of
the
bending
and
shearing
strain
energies:
U,o,
:
L
M(x)
dx
+
LFS-V(x):
dx
•2ErI ~
2A,
G
0 f Y/1
0 Yvi
where:
A,,,/
=
cross
sectional
area
of
equivalent
cylinder
(equal
to
total
area
of
warp
yams,
Atotal)
Ef
=
fabric elastic
modulus
along
warp
(longitudinal)
axis
measured
from
biaxial
tests
FS
=
shear
strain
correction
factor
(for
tube
=
2.0)
Gf =
pressure-dependent
fabric
shear
modulus
],,
=
area
moment
of
inertia
of
the
equivalent
cylinder about bending
axis
L
=
length
of
air
beam
M(x)
=
bending
moment
V(x)=
transverse
shear
force
along
longitudinal
axis
x
=
longitudinal
position
along
air
beam.
For
thin-walled
cylinders
of
outer
radius
ra,
and
thickness,
tc,.,
Io.i
is
computed
as
shown
in
equation
(10)
by
neglecting
terms
containing
higher-ordered
forms
of
t
I
=
rr
3
t/,.,.
(12)
The
total
mid-span deflection,
5,o,a,,
due
to
the
total
applied,
P,o,,
1
,
is
derived
from
minimizing
the total
strain
energy,
Uto,ol.
This
leads
to:
Iota/
= 1 8a
3
pIol
+
12
ca
2
P,ool
3C
2
ap
+
I
FS
2aP
a
P-FS
(13)
48
E
f
I
o1
4
G
f
A
cYi
where:
a
=
distance
between
support point
and
adjacent
load
point
c
=
distance
between
load
points.
The
first
and
second
terms
of
equation
(13)
represent
the
bending
component,
Shend,
and
the
shearing component,
8
shear,
of
the
total
mid-span
deflection, respectively.
The
fabric elastic
modulus,
Ef,
is
measured
from
biaxial
tension tests
performed
on
the
fabric
using
the
2:1
weft-to-
warp
loading
ratio.
The
magnitudes
of
the
warp
yarn
tension used
in
the
biaxial
tests
must
match
the
warp
yarn
forces
of
the
air
beam
at
the
inflation
pressure
of
interest.
Using
the
second
term,
6
.•her,
the
fabric
shear
modulus,
Gf,
may
be
calculated
from:
a
P,o,,I
FS
G
f
=
2
hear
Acy(
21
Equation
(14)
indicates
that
GI
is
not
a
function
of
the
fabric
elastic
modulus,
E,
whereas
P,),,/
and
6shear
are
functions
of
inflation
pressure.
Alternatively,
by
determining
the
shear-jamming
angle
of
the
fabric
and
the
corresponding
-5hea,-,
a
lower
bound
value
of
Gf
can be
determined.
ANALYTICAL
&
NUMERICAL MODELS
A
variety
of
analytical
and
numerical
models
have
been
developed
to
predict
the
load-
deflection
behavior
for
inflatable
structures[4,13,19-26]
and
for the
load-extension
behavior
of
fabrics
themselves[27,28]
UNIT
CELL
NUMERICAL
MODELS
One
could
create
a
finite
element
model
of
an
air
beam with
all
its
discrete
yams,
bladder,
contact surfaces,
friction
etc.,
however,
research
has
shown
that solutions
to
such
a
model
would
exceed
the
computational
limits
of
today's
hardware.
The
need
to
obtain
a
computationally
efficient
model
that would
not
exceed
hardware
limits
or
encounter
convergence problems
led
to
the
development
of
localized
yam
interaction
models
for
plain-woven
fabrics.
These
models
are
commonly
referred
to
as
unit
cell
models.
Unit
cell
models
are
used
to
develop
the
constitutive behavior
on
the
material
scale rather
than the
structural
scale.
All
sources
of
influence
on
constitutive
behavior
are
present
on
the
material scale.
Once
this
behavior
is
known,
it
can
be
applied
to
the
structural
scale
by
using
homogenization
methods.
Homogenization
methods
simply
employ
the
as-established
constitutive
behavior
from
the
unit
cell
models
in
global
structural
models
constructed
of
homogenous membranes.
This
2-step
approach reduces
the
necessary
computations
and
run
time
for
full-structure
modeling
while
preserving
the
effects
of
yam
interactions
on
material
behavior.
A
unit
cell
of
a
plain-woven
fabric
generally
consists
of
several
rows
of
warp
and
weft
yams.
Since
the
yams
exhibit
relative
displacements
and
rotations
with
respect
to
each
other,
researchers
have
investigated
two
options
for
treating
yam
kinematics:
a)
rotation-only
(pinned
centers
at
the
yam crossover
points),
and
b)
rotation
and
translation
at
the
yam
crossover points
as
shown
in
figtre
(19).
In
the
rotation-only
condition,
the
centers
of
the
contact
areas
are
hinged
and
remain
coincident.
Friction
is
due
only
to
the
relative rotation
of
the
yams
and
no
sliding
is
allowed.
This
kinematic
condition
is
appropriate
for
modeling
pneumatic
muscles
29
1
where
the
fabric
is
braided
and the
relative angle
of
rotation
at
the
yam
crossover
points
is
significantly
large.
In
the
rotation
and
translation condition,
no
kinematic
constraints
are
imposed
at
the
yam
crossover
points. This
enables
interacting
yams
to
slide
and
rotate
with
respect
to
each
other. Therefore,
contact-induced
friction
forces
at
the
yarn
crossover
points
provide resistance
to
both
relative
rotations
and
translations.
The
bladder
generally
does
not
contribute
any
structural stiffness
to
the
fabric
nor
is
it
necessary
to
include
the
bladder
as
a
contacting
media.
In
addition,
the
inclusion
of
a
bladder
contact
surface
was
found
to
create
unnecessary
convergence
difficulties
and
added
computational
expenses.
Since
the
bladder
was
eliminated
from these
models,
the
unit
cell
should
be
subjected
to
a
uniform
pressure
representing
the
air-beam
pressure.
There
are
several
22
methods
of
applying
the
internal
pressure
to
the
unit
cell
model,
depending
on
the
individual
preferences.
Loaded
States
Unloaded
State
Option
- 1
Rotations
permissible
only
(pinned
joints
at
cross-over
points)
-<"2
.Option
-
2"
Rotations
and
translations
permissible
7]
Weft
yarn
contact
area
0
Warp
yarn
contact area
Figure
19.
Treatment
of
yarn
kinematics
in
unit
cell
models.
Example
of
a
Unit
Cell
Model
In
a
specific
example
of
the
unit
cell,
each
yam
was
represented
by
an
isotropic
thin
shell
with material
properties
of
either
Vectran
or
PEN,
depending
on
the
fabric
being
studied.
The
shell
unit
cell
model
consisted
of
4
warp
yams
and
4
weft
yams
as
shown
in
figure
(20).
The
model
was
subjected
to
internal
pressures
ranging
from
0.0138
MPa
(1
psi)
to
0.276
MPa
(20
psi).
Contact
surfaces were
permitted
to
allow
yam
translation
and
rotations
as
described
by
option-2. Two
compliant
membrane
elements
(Emembrane
<<
Earn)
connecting
the
center points
of
the
contact
regions were created.
These
membrane elements were
used
to
determine
the
overall
stress
and
strain
of
the
shell
unit
cell
model
and
did not
contribute
to
the
structural
stiffness
of
the
model.
Represents
2
membrane
elements,
1
attached
to
horizontal
yarns, 1
attached
to
vertical
yarns.
A
STEP-2A:
STEP-21B:
Fshear
RF2
s___-
y1-
STEP-I
Figure
20.
Example
unit
cell
model
and loading
procedure.
23
A
3-step
computational procedure
was
employed
to
determine
the
elastic
and
shear
moduli
of
the
unit
cell.
First,
the
vertical
yarns were
restrained
and
the
horizontal
yarns
(x-
direction)
were
subjected
to
+1%
displacement.
The
model
was
solved
and the
stress
in
the
x-
direction
and
the
strains
in
the
x-
and
y-directions
were
determined.
Then,
the
elastic
modulus,
E1=
o-/le,
and
Poisson's
ratio,
v
12
6-
2/11,
were
determined.
Second,
the
horizontal
yarns
were
restrained
and
the
vertical
yams
(y-direction)
were
subjected
to
+1%
displacement
in
the
y-
direction.
The
model
was
solved
and
the
stress
in
the
y-direction
and
the
strains
in
the
x-
and
y-
directions
were
determined. Then,
the
elastic
modulus,
E
2
=
l2/E
2
,
and
Poisson's
ratio,
v
21
=
-
8,/c2,
were
determined.
Finally,
the
third
step
was
to
calculate
the
shear
modulus.
This
is
difficult
due
to the
fact
that
the
elastic
and
shear
moduli
of
a
plain-woven
fabric
structure
are
highly
dependent
on
the
internal
pressure
and,
furthermore,
the
elastic
and
shear
moduli
are
independent.
Two methods
are
suggested.
The
first
method
is
for
the
case
where
the fabric
is
coated and
there
is
no
relative
motion
between
the
yams,
then
the
following
approach
is
recommended.
Continuum
Approach
In
this
approach
the fabric
is
assumed
to
be
a
continuum
with
orthotropic
material
properties.
The
unit
cell
model
in
this
case
was
subjected
to
a
+I%
nodal
displacement
at
a
450
angle
(not
shown).
The
model
was
solved
and the
stresses
and
strains
of
the
unit
cell
at
the
450
angle
were
determined.
The
shear modulus
was
calculated according
to
Jones[71
by:
GI
4
-t
I
+
(15)
SE
,
El
E2
El
)
where
F,
is
the
elastic
modulus
in
the
direction
of
the
45'
load.
This
expression
may
be
appropriate
for fabrics
with
stiff
coatings.
Non-Continuum
Approach
The
second approach
of
calculating
Gf
assumes
that
the
woven fabric
is
not
a
continuum
and
that
each
yarn
independently
responds
to
the
external
load.
In
this
case,
the
horizontal
yarns
were subjected
to
a
shear
force
Fshear.
The
shear modulus
was then
calculated
based
on
the
equilibrium
of
the
horizontal
yam
and
its
shear
deformation
as:
7-
GI=
-1
(16)
where:
Th,.a,
=
JF,.COclion
/
Z4;
and
y
arctan
(,.h,e,,.
/
L),
and
where
,
is
the
sum
of
reaction
forces
at
the
support,
6
,hear
is
the
nodal
transverse
displacement,
A
is
the
yam
cross
section
and
L
is
the
yarn
length.
It
is
recommended that
a
number
of
tests
on
the
unit
cell
model
be
performed
in
which
the
coefficient
of
friction
is
varied
and
the
changes
in
the
elastic
and
shear
moduli
are
determined. Therefore,
the
results
of
the
shell
unit
cell
model will
be
parametric
in
both
friction
and
pressure.
24
Structural
Air
Beam
Models
Once
the
elastic
and
shear
moduli
of
the
membrane elements
are
determined
from
the
unit
cell
model,
a
global structural
finite
element
model
of
the air
beam,
using
membrane
elements,
can
be
created
such
as
the
one
shown
in
figure
(21).
Here,
the
fabric
skin
would
be
discretized
using membrane
elements.
However,
nine
elastic
material
constants (El,
E
2
,
E3,
v/12,
V1/3,
V2
3
, G
12
,
G1
3
,
and
G
23
)
are
required,
of
which,
only
El,
E
2
, and
G1
2
had been
calculated
from
the
beam
unit
cell
model.
Because
a
membrane
formulation
is
used for
the
fabric
skin,
the
elastic
modulus
E3
would
not
influence
the
global
beam
deflection
and
was
therefore
chosen
to
be
of
the
same
order
of
magnitude
as
E,
and
E2
(in
the
order
of
E
3
=
64,000
MPa
for
the
current
example).
It
was
also
noted
that
any
load
applied
in
the
direction
of
one
yam
is
decoupled
from
all
orthogonal
yams.
That
is,
the
Poisson's
ratio
is
nearly
0,
therefore,
the
condition
that
v12
=
v13
=
v23
=
0.0
may
be
applied.
No
method
was
developed
to
determine
G1
3
and G
23
.
It
is
hypothesized
that
neither
value
had
a
significant
effect
on the
beam
deflection.
One
could
examine
this
hypothesis
by
a
parametric
study
of
values
for
G1
3
and G
23
.
Figure
21.
Example
of
an
air
beam
global
finite
element
model
subjected
to
4-point
bending.
In
the
abovementioned
example, using
the
analytical
strain
energy
solution,
the
equivalent
in-plane
fabric
shear
modulus,
Gf,
for
the
50.8
mm
(2.0
in)
diameter
Vectran
beam
pressurized
to
0.689
MPa
(100
psi)
was
calculated
as
86.06
MPa.
(12,480
psi).
The
computed
bending
and
shear
components
of
the
mid-span deflection
were
12.39
mm (0.488
in)
and
39.54
mm
(1.557
in),
respectively.
The
elastic
and
shear
moduli
in
the
beam unit
cell
model
were
calculated
for
an
internal pressure
of
0.137
MPa
(20
psi)
as
E,
=
64,033
MPa,
E
2
=
64,098
MPa
and
Gf=
G1
2
=
60.96
MPa.
Similarly,
for
an
internal
pressure
of
0.207
MPa
(30
psi),
the
elastic
moduli
were
calculated
as
E,
=
64,045
MPa,
E2
=
64,049
MPa
and
Gf=
G1
2
=
80.64
MPa.
25
CONCLUDING
REMARKS
The unit
cell
modeling
method
was
used
to
determine
the
influence
of
pressure,
weave
architecture (including
yam
density
ratio),
crimp
interchange,
coefficients
of
friction
and
biaxial
loading
ratios
on
the
effective
fabric elastic
and
shear
moduli.
These
moduli were
shown
to
increase
with
increasing
pressure.
The
coefficient
of
friction between
yams
did
not influence
the
elastic
and
shear
moduli
significantly because
the
relative
displacements
of
the
interacting
yams
were small
compared
to
their
cross
sections.
The
shear
modulus
of
plain-woven
fabrics
was
shown
to
be
a
system property;
it
is
a
kinematic
property
of
the
yam assemblage
in
fabric
form
and
is
independent
of
the
yam
elastic
moduli.
The
use
of
yams
with
higher
elastic
modulus
may
not
significantly
influence
the
deflection
of
air
beams
because crimp
interchange
can
prevent
the
as-woven
yarns
from
developing
their
elastic
behavior.
It
has
also
been
shown
that
coatings
can
stiffen
the
fabric
material
resulting
in
smaller
air
beam
deflections.
Homogenization
is
particularly
suitable
for
developing computationally
efficient
global
models
of
air-inflated
fabric
structures
in
conjunction
with unit
cell
methods.
The
unit
cell
method
captures
the
nonlinear constitutive behavior
of
the
plain-woven
fabric
in
response
to
applied
loads.
This
behavior
is
transferred
to
a
homogeneous
material
in
a
global model
of
the
air-inflated
fabric
structure.
The
studies
have
shown
that
that
air-inflated
fabric
structures
differ
fundamentally
from
conventional
metal
structures
and
fiber/matrix composite
structures.
While
the
plain-woven
fabric
may
appear
to be
an
orthotropic material,
its
mechanical
behavior
indicated otherwise.
It
does not
behave
as
a
continuum,
but
rather
as
a
discrete
yam assembly.
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U.S.
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No.
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26